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Elucidating and handling effects of valve-induced nonlinearities in industrial feedback control loops
Accepted Manuscript Title: Elucidating and Handling Effects of Valve-Induced Nonlinearities in Industrial Feedback Control Loops Author: Helen Durand Robert Parker Anas Alanqar Panagiotis D. Christofides PII: DOI: Reference:
S0098-1354(17)30300-9 http://dx.doi.org/doi:10.1016/j.compchemeng.2017.08.008 CACE 5875
To appear in:
Computers and Chemical Engineering
Received date: Revised date: Accepted date:
23-3-2017 30-6-2017 18-8-2017
Please cite this article as: Helen Durand, Robert Parker, Anas Alanqar, Panagiotis D. Christofides, Elucidating and Handling Effects of Valve-Induced Nonlinearities in Industrial Feedback Control Loops, (2017), http://dx.doi.org/10.1016/j.compchemeng.2017.08.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights • A closed-loop system view of valve stiction-induced nonlinear dynamic behavior. • Impact of stiction on closed-loop dynamics for both classical and model predictive control systems is elucidated. • Stiction compensation strategies are presented for both classical and model predictive control systems.
Ac ce
pt
ed
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an
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• Chemical process examples are used to illustrate the impact of stiction and how it can be handled.
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Elu idating and Handling Ee ts of Valve-Indu ed Nonlinearities in Industrial Feedba k Control Loops ∗,a,b a a a Helen Durand , Robert Parker , Anas Alanqar , Panagiotis D. Christo des
Department of Chemi al and Biomole ular Engineering, University of California, Los Angeles, CA, 90095-1592, USA. b Department of Ele tri al Engineering, University of California, Los Angeles, CA 90095-1592, USA.
ip t
a
cr
Abstra t
us
In this work, we investigate the ee ts of various types of valve behavior (e.g., linear valve dynami s and sti tion) on the ee tiveness of pro ess ontrol in a uni ed framework based on systems of
an
nonlinear ordinary dierential equations that hara terize the dynami s of losed-loop systems in luding the pro ess, valve, and ontroller dynami s. By analyzing the resulting dynami models,
M
we demonstrate that the responses of the valve output and pro ess states when valve behavior
annot be negle ted (e.g., sti tion-indu ed os illations in measured pro ess outputs) are losed-
a priori
due to the oupled and typi ally nonlinear
d
loop ee ts that an be di ult to predi t
te
dynami s of the pro ess-valve model. Subsequently, we dis uss the impli ations of this losed-loop perspe tive on the ee ts of valve dynami s in losed-loop systems for understanding valve behavior
Ac ce p
ompensation te hniques and developing new ones. We on lude that model-based feedba k ontrol designs that an a
ount for pro ess and valve onstraints and dynami s provide a systemati method for handling the multivariable intera tions in a pro ess-valve system, where the models in su h ontrol designs an ome either from rst-prin iples or empiri al modeling te hniques. The analysis also demonstrates the ne essity of a
ounting for valve behavior when designing a ontrol system due to the potentially dierent onsequen es under various ontrol methodologies of having dierent types of valve behavior in the loop. Throughout the work, a level ontrol example and a
ontinuous stirred tank rea tor example are used to illustrate the developments.
Key words:
Model predi tive ontrol, valve dynami s, hemi al pro esses, pro ess ontrol,
sti tion, empiri al modeling
∗
Corresponding author: Tel: +1 (310) 794-1015; Fax: +1 (310) 206-4107; E-mail: pd seas.u la.edu.
Preprint submitted to Computers & Chemi al Engineering
August 18,Page 20172 of 75
1. Introdu tion Valves do not respond instantaneously to ontrol signal hanges (i.e., their response has dynami s) be ause they have me hani al parts that move when the ontrol signal hanges. Typi ally,
ip t
the hemi al pro ess ontrol literature assumes that the valve dynami s are so fast that they an be negle ted, meaning that the valve output is assumed to immediately equal the value requested
cr
by the ontroller. However, in industry, various valve behaviors are observed that impa t ontrol
us
loop performan e, though an engineer entering industry after ompleting his or her undergraduate edu ation typi ally has had limited exposure to valve behavior.
Undergraduate pro ess ontrol
an
ourses often handle valve dynami s with linear transfer fun tion models; su h dynami s an be related to, for example, resistan e of the gas used to apply pressure in a pneumati a tuator to Valve hara teristi s (e.g., linear,
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ow at the top of a valve [Coughanowr and LeBlan (2009)℄.
equal per entage, and square root) may also be reviewed in undergraduate oursework to provide undergraduates with fundamentals regarding valve sizing and the ee ts of installing a valve on the
d
valve's ow hara teristi s (the manner in whi h the ow through the valve is related to the valve
te
opening) [Coughanowr and LeBlan (2009); Ogunnaike and Ray (1994); Riggs (1999)℄.
Though
Ac ce p
there may be some dis ussion of other types of valve behavior des ribed by nonlinear models (e.g., saturation of the valve output at its maximum value, failure of a valve to respond to hanges in the
ontrol signal to the valve for some time after a valve movement dire tion hange due to me hani al parts in a valve (deadband due to ba klash) [Choudhury et al. (2005)℄, or sti tion [Brásio et al. (2014); Armstrong-Hélouvry et al. (1994)℄, whi h refers to valve behavior due to fri tion that an be des ribed by nonlinear dynami equations), time onstraints in a semester/quarter and also a general fo us in undergraduate pro ess ontrol on linear dynami systems do not typi ally permit an in-depth treatment of nonlinear valve behavior and its impa t on pro ess ontrol from a rst-prin iples perspe tive. However, at hemi al plants throughout the world, valve issues su h as sti tion, deadband, saturation, hysteresis, and deadzone prevent adequate set-point tra king [Jelali and Huang (2010); Choudhury et al. (2008)℄. Therefore, it would be bene ial if the various types
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Page 3 of 75
of valve behavior were presented within a single framework so that undergraduates ould learn the mathemati al stru ture needed to allow them to understand and ompensate for any valve behavior that they en ounter in their ourses and their areers. The traditional manner of handling valve behavior, in whi h various types of valve behavior For
ip t
are treated separately rather than within a single framework, is prevalent in the literature.
example, Durand et al. (2014) demonstrates that even linear valve dynami s an be problemati
cr
for a pro ess operated under an optimization-based ontrol design alled e onomi model predi tive
ontrol (EMPC) [Ellis et al. (2014a); Heidarinejad et al. (2012); Amrit et al. (2011); Rawlings et al.
us
(2012); Huang et al. (2012)℄ when the valve dynami s are negle ted in the model utilized by the EMPC for making state predi tions; therefore, that work suggests in orporating the valve dynami s
an
in the dynami pro ess model for the ontroller. Choudhury et al. (2008) analyzes the range of valve travel over whi h linear ontrol design theory would be expe ted to be adequate when a pro ess
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that an be ee tively modeled with a linear model re eives a ow rate from a valve with a square root or an equal per entage inherent valve hara teristi .
Zabiri and Samyudia (2006) develops
d
an MPC-based method for linear pro esses where the valve is subje t to ba klash. The literature
te
analyzing and ompensating for the sti tion nonlinearity is parti ularly extensive, with reviews su h as Brásio et al. (2014) ategorizing the methods, though sti tion ompensation remains an
Ac ce p
important resear h topi with newer works su h as those in Durand and Christo des (2016); Ba
i di Capa i et al. (2017); Munaro et al. (2016) expanding the ompensation literature. Another feature of the valve behavior literature is that the types of pro esses and ontrol laws handled are often limited. For example, despite the re ognition in [Thornhill and Hor h (2007); Jelali and Huang (2010); Choudhury et al. (2005)℄ that the ontroller, pro ess, and valve dynami s all play a role in determining the traje tories of the measured outputs of a losed-loop system with a sti ky valve, a summary table in Thornhill and Hor h (2007) fo uses on ombinations of linear pro esses (integrating and non-integrating) and linear ontrollers (proportional (P) and proportional-integral (PI)) with dierent sti tion hara teristi s for a valve in the ontrol loop to larify whi h ombinations are expe ted to result in limit y ling of the valve output. This is onsistent with mu h of the sti tion ompensation literature, whi h is developed assuming linear pro ess models and li-
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near ontrollers. Furthermore, the sti tion literature generally fo uses on single-input/single-output
ontrol loops. It is widely appre iated, however, that hemi al pro esses are typi ally des ribed by nonlinear dynami models with multiple inputs. Motivated by these onsiderations, the primary ontribution of this work is the development of
ip t
a single, omprehensive systems engineering framework for understanding the impa ts of various types of valve behavior on losed-loop nonlinear pro ess systems with multiple inputs, various
cr
ontrol ar hite tures, and various ontrol designs. This framework not only permits a fundamental understanding of the auses of the negative ee ts observed in ontrol loops with valves for whi h
us
the valve behavior annot be negle ted, but also enables a better understanding regarding how the tools previously developed in the literature for handling these dynami s work. This is riti al for
an
sti tion ompensation be ause a number of sti tion ompensation te hniques are not model-based (and therefore the fundamental bene ts and limitations of su h sti tion ompensation strategies
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must be analyzed within a model-based framework) or are model-based but designed for spe i linear pro ess models, ontrol ar hite tures, and valve nonlinearity models. Furthermore, mu h of
d
the sti tion literature assumes that the system under study exhibits losed-loop os illations due to
te
sti tion; notwithstanding some dis ussion regarding the auses of these os illations in the literature, the onditions under whi h they do and do not o
ur for a general nonlinear hemi al pro ess under
Ac ce p
various ontrol designs must be lari ed.
The developments of this work des ribe not only the
auses of sti tion-indu ed os illations, but also the auses of improvements observed in the response of a losed-loop system with a sti ky valve when a sti tion ompensation method is used.
The
developments are shown to permit the proposal of new sti tion ompensation s hemes and to allow a number of perspe tives on ontrol design in light of valve behavior to be stated.
2. Preliminaries 2.1. Notation
The transpose of a ve tor
u
x
is denoted by
xT .
The notation
u ∈ S(∆)
signi es that the ve tor
is a member of the set of pie ewise- ontinuous (from the right) fun tions with period
notation
tk = k∆, k = 0, 1, 2, . . .,
and the notation
4
t˜j = j∆e , j = 0, 1, 2, . . .,
∆.
The
refer to elements of a
Page 5 of 75
time sequen e separated by sampling time periods of lengths
|·|
∆
and
∆e ,
respe tively. The notation
signi es the Eu lidean norm of a ve tor.
2.2. Class of Systems
ip t
We onsider a nonlinear hemi al pro ess system with the following form:
x˙ = f (x, ua , w)
pro ess inputs,
w ∈ Rl
f : Rn ×Rm ×Rl
omponent
X ), ua ∈ Rm
is a ve tor of bounded pro ess disturban es (i.e.,
is a lo ally Lips hitz ve tor fun tion of its arguments with
ua,i , i = 1, . . . , m,
is the ve tor of
w ∈ W := {w | |w| ≤ θ}), f (0, 0, 0) = 0.
Ea h
of the pro ess input ve tor is an output of a valve that is adjusted
an
and
is the pro ess state ve tor (bounded in the set
cr
x ∈ X ⊆ Rn
us
where
(1)
utilizing a feedba k ontroller for the nonlinear pro ess that outputs a set-point
um,i , i = 1, . . . , m,
for ea h valve output. Be ause the valve output ow rates are bounded by physi al valve onstraints, is bounded between a minimum (ua,i,min ) and a maximum (ua,i,max ) ow rate, with
the resulting input onstraint on
ua,i ≤ ua,i,max, i = 1, . . . , m}). um ∈ Um ,
where
where
um,i
and ea h
ua,i − um,i
ua,i
where
U := {ua ∈ Rm | ua,i,min ≤
depends on the valve behavior. We will onsider
relationship is either stati or dynami . In the ase that the
relationships are stati , the following equation holds:
fstatic,i
ua,i = fstatic,i (um,i )
is a nonlinear ve tor fun tion (fstatic (um )
natively, a dynami model may hara terize the
um,i
ua ∈ U ,
Um := {um ∈ Rm | um,i,min ≤ um,i ≤ um,i,max, i = 1, . . . , m}).
Ac ce p
ua,i − um,i
(i.e.,
Sin e the ow rates out of the valve are bounded, the set-points are
The relationship between ea h valve behavior for whi h the
U
denoted by
te
bounded also (i.e.,
ua
M
ua,i
d
ea h input
and the dynami state ve tor
xdyn,i ∈ Rpi ,
(2)
= [fstatic,1 (um,1 ) · · · fstatic,m (um,m )]T ).
ua,i − um,i
relationship, where
for whi h the omponents
ua,i
Alter-
is related to both
xdyn,i,j , j = 1, . . . , pi ,
states of the valve model. In this ase, the following equations des ribe the dynami s of the
are
i − th
valve:
x˙ dyn,i = xˆdyn,i (xdyn,i , um,i )
5
(3)
Page 6 of 75
ua,i = fdynamic,i (xdyn,i ) where
xˆdyn,i
(4)
is a nonlinear ve tor fun tion hara terizing the dynami s of the internal states of
the model for the
i − th
valve, and
fdynamic,i
the internal dynami states of the valve.
is a nonlinear ve tor fun tion relating
We de ne
ua,i
and
xdyn = [xdyn,1 · · · xdyn,m ]T , xˆdyn (xdyn , um ) =
We assume in this work that the value of ea h
um,i
ip t
[ˆ xdyn,1 (xdyn,1 , um,1) · · · xˆdyn,m (xdyn,m , um,m )]T , and fdynamic (xdyn ) = [fdynamic,1 (xdyn,1 ) · · · fdynamic,m (xdyn,m )]T . is determined utilizing a feedba k ontroller
cr
(e.g., proportional-integral (PI) ontrol or model predi tive ontrol (MPC), whi h will be pla ed
us
in the notation of this se tion in the following two se tions) that utilizes knowledge of at least one pro ess state to ompute ontrol a tions. This means that the value of ea h
an
subset of the state ve tor as follows:
um,i
is ae ted by some
um,i = fcontroller,i (x, ζˆi ) ζˆi ∈ Rri , i = 1, . . . , m,
is a ve tor of internal states of the ontroller al ulating
internal states may be dynami as follows:
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where
d
˙ ζˆi = finternal,i (x, xdyn , ζˆi )
um,i .
These
(6)
te
De ning
(5)
ˆ = [fcontroller,1(x, ζˆ1 ) · · · fcontroller,m(x, ζˆm )]T , ζˆ = [ζˆ1 · · · ζˆm ]T , and finternal (x, xdyn , ζ) ˆ = fcontroller (x, ζ)
Ac ce p
[finternal,1 (x, xdyn , ζˆ1 ) · · · finternal,m (x, xdyn , ζˆm )]T , we an write the pro ess-valve system as follows for ua and um (i.e., Eq. 2 holds): " # ˆ w(t)) x˙ f (x(t), fstatic (fcontroller (x, ζ)), ˙ = ˆ finternal (x, xdyn , ζ) ζˆ
the ase of a stati relationship between
where
xdyn = 0
(7)
when Eq. 2 holds.
For the ase of a dynami relationship between
ua
and
um
(i.e., Eqs. 3-4 hold), the following
pro ess-valve system results:
x˙
f (x(t), fdynamic (xdyn ), w(t)) ˆ x˙ dyn = xˆdyn (xdyn , fcontroller (x, ζ)) ˙ˆ ˆ finternal (x, xdyn , ζ) ζ
(8)
With slight abuse of notation, the right-hand sides of both Eqs. 7 and 8 will be denoted by
6
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fq (q(t), um (t), w(t)) (fq
relationship, and it signi es the right-hand side of Eq. 8 when Eqs. 3-4 hara terize the relationship), where
ˆT q(t) represents the ve tor of pro ess-valve states (i.e., q = [x ζ]
ribes the pro ess-valve dynami s, and
fq
ˆT q = [x xdyn ζ]
ua,i − um,i
when Eq. 7 des-
when Eq. 8 des ribes the pro ess-valve dyna-
is a lo ally Lips hitz ve tor fun tion of its arguments with
fq (0, 0, 0) = 0.
ip t
mi s). We assume that
ua,i − um,i
signi es the right-hand side of Eq. 7 when Eq. 2 hara terizes the
The general prin iples of the ontrol ar hite tures des ribed are illustrated in Fig. 1.
besides
x,
Disturban es ould also be onsidered in other dynami states of the pro ess-valve model su h as in
xdyn ,
cr
Remark 1.
and the analysis presented throughout this work would ontinue to hold.
us
2.3. Classi al Linear Control with Integral A tion
an
Linear ontrol designs with an integral term are designed to drive a sele ted pro ess output (here taken to be a pro ess state, whi h is onsistent with standard industrial pra ti e in the hemi al
it omprises the ve tor
xˆsp ∈ Rnˆ .
xˆ ∈ Rnˆ , n ˆ ≤ n,
M
pro ess industries) to its set-point. Thus, we assume that the pro ess state ve tor or a subset of of measured outputs being driven to the set-point ve tor
When a PI ontroller is used, ea h omponent of
xˆ
is regulated to its set-point by an
The dynami s of the
i − th
te
n ˆ = m.
Ac ce p
and thus
d
individual linear ontroller that outputs a valve output ow rate set-point for an individual valve, PI ontroller are represented by:
um,i = gA,i (ˆ xi , ζi )
(9)
xˆ ζ˙i = Acon,i i + Bcon,i xˆi,sp ζi
The form of these equations follows that in Eqs. 5-6, with
finternal,i
given by the right-hand side of Eq. 10.
Acon,i
and
(10)
ζˆi = ζi , fcontroller,i
Bcon,i
given by
gA,i ,
and
are a matrix and s alar.
2.4. Model Predi tive Control
Model predi tive ontrol [Qin and Badgwell (2003); Ellis et al. (2014a)℄ is an optimization-based
ontrol strategy based on the following optimization problem:
min
um (t)∈S(∆) s.t.
Z
tk+N
Le (˜ x(τ ), um (τ )) dτ
(11a)
tk
x˜˙ (t) = f (˜ x(t), um (t), 0)
7
(11b)
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x˜(t) ∈ X, ∀ t ∈ [tk , tk+N )
(11d)
um (t) ∈ Um , ∀ t ∈ [tk , tk+N )
(11e)
gM P C,1(˜ x(t), um (t)) = 0, ∀ t ∈ [tk , tk+N )
(11f )
gM P C,2(˜ x(t), um (t)) ≤ 0, ∀ t ∈ [tk , tk+N )
(11g)
ip t
Le (x, um )
(11 )
is minimized subje t to bounds on the states (Eq. 11d), bounds on the
cr
The stage ost
x˜(tk ) = x(tk )
inputs (Eq. 11e), equality and inequality onstraints (Eqs. 11f-11g), and the restri tion that the
from a measurement of the state (Eq. 11 ).
≡ 0)
dynami model in Eq. 11b when initialized
x˜(t) represents
us
states must evolve a
ording to the nominal (w(t)
the pro ess state a
ording to the model
an
of Eq. 11b (i.e., in the absen e of disturban es or plant/model mismat h) at time
t; x(t),
on the
other hand, represents the a tual value of the state (whi h is ae ted by disturban es and thus is
x˜(t)).
By setting
MPC be omes a feedba k ontroller.
um (t) is used
to a measurement of the pro ess state at tk , the
in pla e of
ua (t) be ause
the standard formulation of MPC in industry
d
In Eq. 11b,
x˜(tk )
M
not ne essarily the same as
no referen e is made to
ua
in Eq. 11).
sampling periods of length
∆ (N
ua = um ;
therefore,
MPC an, however, handle valve saturation through the
ua = um .
Ac ce p
onstraint of Eq. 11e, assuming
N
te
and the literature negle ts valve behavior in general (i.e., it assumes that
A ve tor of ontrol a tions
um
is omputed for ea h of the
is the predi tion horizon), and only the rst of these ve tors is
applied to the pro ess in a sample-and-hold fashion a
ording to a re eding horizon strategy. The notation
u∗m (t|tk ), t ∈ [tk , tk+N ),
problem initiated at time
signi es the optimal value of
tk (u∗m,i (t|tk )
represents the
i − th
um
at time
t
for the optimization
omponent of this ve tor).
A form of MPC that is ommonly used in the hemi al pro ess industries is tra king MPC, whi h drives
xˆ
to
xˆsp
(though it is not ne essary in this ase that
stage ost with its minimum at the set-point ve tor
um,sp
n ˆ = m)
by utilizing a quadrati
xˆsp with orresponding steady-state input ve tor
as follows:
Le (˜ x(τ ), um (τ )) = (ˆ xsp − x˜ˆ)T QT (xsp − x˜ˆ) + (um,sp − um )T RT (um,sp − um )
8
(12)
Page 9 of 75
where
QT > 0
and
RT > 0
The value of ea h
um,i
um,i
are tuning matri es, and
x˜ˆ
denotes the predi ted value of the ve tor
al ulated by the MPC has the form of Eq. 5 with
as a fun tion of the pro ess state), though the fun tion
fcontroller,i
ζˆi = 0
xˆ.
(whi h leaves
is not expli itly de ned in
this ase but is de ned impli itly by the optimization problem in Eq. 11. We assume that full state feedba k is available for all MPC designs presented in this
ip t
Remark 2.
work. When it is not, state estimation and output feedba k MPC (e.g., Ellis et al. (2014b)) ould
cr
be investigated to determine if they provide a suitable ontrol design for a given system, but this will not be pursued here.
us
3. Pro ess-Valve Model Analysis: Insights on Impa ts of Valve Behavior on ClosedLoop Systems
an
Eqs. 7-8 omprise the uni ed framework that we develop in this work for analyzing the impa ts of valve behavior and valve behavior ompensation on ontrol loop performan e. They reveal that valve
M
behavior hanges the dynami s of a oupled losed-loop system. To see this, onsider the system of Eq. 7, whi h represents a pro ess-valve system subje t to only a stati valve nonlinearity under
d
feedba k ontrol. The stati nonlinearity impa ts the dynami s of
ˆ w(t)), = um ), the dynami s of x˙ would be des ribed by f (x(t), fcontroller (x, ζ), ˙ ˙ and ζˆ are fun tions of x, so than the dynami s in Eq. 7). In addition, both x
whi h is dierent
Ac ce p
fstatic
te
valve nonlinearity (ua
modifying
x (i.e., in the absen e of the stati
ae ts the dynami s of both
x and ζˆ.
the ee t on the losed-loop response of hanging
Due to the fa t that the system is nonlinear,
fstatic
is di ult to determine without performing
losed-loop simulations. This is parti ularly signi ant when there are multiple pro ess inputs related to
um,i
through dierent stati nonlinear fun tions, espe ially assuming that the dynami s
of the omponents of
x
are oupled. Then, ea h
ua,i
ae ts all omponents of
through oupling of those omponents in the ve tor fun tion by all omponents of
x
ontrol (Eq. 5).
to de ne
ua,i
either dire tly or
x
is ae ted
are oupled and at least one of
(Eq. 2) due to the use of state feedba k
Using a similar analysis, it an be dedu ed that hanging the ontrol law (i.e.,
fcontroller
to determine
um,i
x
f , and the value of ea h ua,i
due to the fa t that the omponents of
those omponents is used to al ulate
hanging
ua,i
and
a priori
finternal )
also impa ts the losed-loop response in a manner that is di ult
(without simulations).
9
Page 10 of 75
When the valve dynami s an be des ribed by dynami systems of equations as in Eq. 8, the dynami s of the valve, ontroller, and pro ess are again oupled. In this ase, however, there is an additional omplexity in that the valve dynami s add additional states with nonlinear dynami s (or linear in the spe i ase of linear valve dynami s) that are not present in the ase that
x
is a fun tion of at least one of the omponents of
x,
it is ae ted by the
ip t
other omponents of
um,i
as well, assuming oupling between these omponents. This auses
and
ua,i
ua,i
in Eq. 1 and thus by all valve states
to also be related to
x
(Eqs. 3-4), and the omponents of
xdyn,i , i = 1, . . . , m,
x are
xdyn,i
ae ted by the values of all
cr
Furthermore, be ause
ua = um .
from Eq. 4.
us
The above analysis shows that from a fundamental mathemati al analysis of general equations for a pro ess-valve system, the dynami s of all valves an be seen to be oupled with the dynami s of
an
the other valves and also with the dynami s of the pro ess and the ontroller due to state feedba k (this is not limited to PI ontrol or MPC). Be ause the ontroller dynami s ae t the evolution of
M
the states and thus the pro ess outputs, dierent types of ontrollers would be expe ted to result in dierent responses of the pro ess outputs. Furthermore, the ontrol loop ar hite ture will also
d
ae t the response be ause it will impa t the equations that des ribe the ontroller dynami s. This
te
analysis reveals that the negative ee ts of valve dynami s on ontrol loop ee tiveness are related to the ontroller, pro ess, and valve dynami s, in addition to the ontrol ar hite ture. The next
Ac ce p
two se tions make the notation of Eqs. 7-8 more on rete by showing how linear and sti ky valve dynami s t within that framework.
Remark 3.
In this work, we onsider a general ase in whi h all states of the pro ess-valve model
are oupled. For ases when this does not hold, it may be possible to analyze the dynami s of the spe i pro ess to see if any simpli ations result ompared to the analysis in this work.
3.1. Linear Valve Dynami s
Linear valve dynami s an be denoted as follows:
x˙ ua,i = Ai xua,i + Bi um,i
(13)
ua,i = Ci xua,i
(14)
10
Page 11 of 75
where
xua,i ∈ Rpi
is the ve tor of internal states of the linear valve dynami model for the
Ai , Bi ,
valve, and
and
Ci
i − th
are a matrix and two ve tors, respe tively, of appropriate dimensions.
Combining the pro ess and valve layer models gives the following pro ess-valve model for this ase (omitting the dynami s of the ontroller):
f (x(t), CL xua (t), w(t)) x˙ = AL xua + BL um x˙ ua
Ai , Bi ,
and
and
CL
are matri es and ve tors of appropriate dimensions ontaining the elements
Ci , i = 1, . . . , m,
notation in Eqs. 3-4,
in an appropriate order, and
xdyn = xua ,
and
xˆdyn,i (xdyn,i , um,i )
and
sides of Eqs. 13 and 14, respe tively.
xua = [xua,1 . . . xua,m ]T .
fdynamic,i (xdyn,i )
Using the
equal the right-hand
an
3.2. Sti ky Valve Dynami s
cr
of
AL , BL ,
us
where
(15)
ip t
For the ase that all valves are sti ky (i.e., ae ted by fri tion/sti tion, whi h prevents the
M
valve position from appre iably hanging until the for e applied to the valve moving parts be omes su iently large) and move in a straight line (rather than rotating), the valve position for the
i − th
valve evolve in time a
ording to the following for e balan e:
mv,i
x˙ v,i = vv,i
(16)
1 (aT FO,i + cTi FA,i − Ff ric,i ) mv,i i
Ac ce p
v˙ v,i =
where
and the
d
vv,i
te
valve velo ity
xv,i
is the mass of the moving parts of the
i − th
valve,
FO,i
(17)
is a ve tor of non-fri tion
for es on the valve that are not related to the ontroller output or fri tion for e and whi h have
oe ient ve tor
ai , FA,i
is a ve tor of non-fri tion for es on the valve that are adjusted based on
the ontroller output and have oe ient ve tor
ci ,
and
is the fri tion for e on the
i − th
zf,i
(whi h is a dynami internal state
Ff ric,i = Fˆf ric,i (xv,i , vv,i , zf,i )
(18)
z˙f,i = zˆf,i (xv,i , vv,i , zf,i )
(19)
valve. The fri tion for e is a stati fun tion of
xv,i , vv,i ,
Ff ric,i
and
of the fri tion model), as follows:
11
Page 12 of 75
where
zˆf,i
is a nonlinear ve tor fun tion des ribing the dynami s of the internal states of the fri tion
model. Assuming that
FA,i
is a stati fun tion of
um,i
as follows:
FA,i = fSO,i (um,i ) is a nonlinear ve tor fun tion des ribing the relationship between
um,i
and
FA,i ,
and
FO,i is also a fun tion of the valve model states, the right-hand side of Eq. 17 an be denoted by
vˆv,i (um,i , xv,i , vv,i , zf,i ).
cr
that
fSO,i
ip t
where
(20)
Finally, assuming that the relationship between
ua,i and xv,i an be expressed
(21)
an
ua,i = ff low,i (xv,i )
us
through the following stati nonlinear equation des ribing the valve hara teristi :
we obtain the following pro ess-valve model (omitting the dynami s of the ontroller for the pro ess):
xv = [xv,1 · · · xv,m ]T , vv = [vv,1 · · · vv,m ]T , zf = [zf,1 · · · zf,m ]T , ff low (xv (t)) = [ff low,1 (xv,1 )
te
where
(22)
d
M
f (x(t), ff low (xv (t)), w(t)) x˙ x˙ v vv = v˙ v vˆv (um , xv , vv , zf ) zˆf (xv , vv , zf ) z˙f
· · · ff low,m(xv,m )]T , vˆv (um , xv , vv , zf ) = [ˆ vv,1 (um,1 , xv,1 , vv,1 , zf,1 ) · · · vˆv,m (um,m , xv,m , vv,m , zf,m )]T , In the notation of Eq. 8,
[xv vv zf ]T , fdynamic (xdyn ) = ff low (xv (t)),
is given by the right-hand sides of
Ac ce p
zˆf (xv , vv , zf ) = [ˆ zf,1 (xv,1 , vv,1 , zf,1 ) · · · zˆf,m (xv,m , vv,m , zf,m )]T . and
xˆdyn,i (xdyn,i , um,i )
xdyn =
Eqs. 16-17 and 19.
4. Illustrating a Closed-Loop, Systems Perspe tive on the Impa ts of Valve Behavior on Control Loop Performan e The losed-loop perspe tive on the impa ts of valve behavior on ontrol loop performan e developed above provides a novel platform from whi h to fundamentally understand the onditions under whi h various negative ee ts attributed to valve behavior in ontrol loops o
ur. In the next se tion, we show through an illustrative level ontrol example that a prevalent phenomenon in the literature (sti tion-indu ed os illations) an only be explained through this losed-loop perspe tive. The simpli ity of the example allows us to fo us on the ee ts of the valve dynami s without the
12
Page 13 of 75
added omplexity of a large-s ale nonlinear pro ess model; however, we subsequently elu idate the ne essity of a losed-loop analysis for predi ting the behavior of the measured outputs of a more
omplex pro ess-valve system through a ontinuous stirred tank rea tor (CSTR) example.
4.1. A Closed-Loop Perspe tive on Sti tion-Indu ed Os illations
ip t
In the level ontrol problem (shown in Fig. 2) onsidered, the tank inlet ow rate
ontrolled variable. The dynami s of the tank level are:
A = 0.25 m2
denotes the ross-se tional area of the tank, and
(23)
c1 = 0.008333 m5/2 /s
us
where
is the
cr
√ dh 1 = (ua − c1 h) dt A
ua
is the
an
outlet resistan e oe ient. On an order of magnitude onsistent with an example from Ogunnaike and Ray (1994) that uses these parameter values, the minimum tank height is
0.5184 m,
maximum value of
ua
is
the minimum value of
ua,max = 0.006 m3 /s
ua
is
ua,min = 0 m3 /s
the maximum
(fully losed valve), and the
(fully open valve).
M
tank height is
0 m,
4.1.1. Level Control with Negligible Valve Dynami s
ua = um
d
When the valve dynami s are so fast that
at all times is a reasonable approximation, a
te
well-tuned PI ontroller and an MPC an be designed that ee tively drive the level to its set-point.
where
um
value of
Ac ce p
The PI ontroller for the tank level has the following form:
(24)
um = uas + Kc (hsp − h) + Kc ζ/τI
(25)
is the ontroller output,
ua
τI = 43.2
ζ˙ = hsp − h, ζ(0) = 0
ζ
is the dynami state of the PI ontroller,
before the set-point hange, and
was sele ted that prevents
set-points simulated. Eqs. 24-25 when
um
hsp
is the steady-state
is the level set-point. A tuning
from dipping below
ua,min
Kc = 0.006
or shooting above
ua,max
and
for the
The response of the level of the tank of Eq. 23 under the PI ontroller of
ua = um
is shown in the top plot of Fig. 3, plotted every 100 integration steps, for
the tank level initiated from its maximum (ua and then in reased to
uas
0.20 m
= 0.006 m3 /s, h = 0.5184 m),
(the set-point hange from
13
0.15 m
to
0.20 m
de reased to
0.15 m,
will be the fo us in the
Page 14 of 75
remainder of this se tion to avoid the ee ts of possible initial transients during the rst set-point
hange). Ea h set-point was held for 1040
s.
The dynami system was integrated with the expli it
Euler numeri al integration method and an integration step size of the value of
ζ
was re-set to 0 and the value of
uas
10−3 s.
At the set-point hanges,
was set to the last applied value of
um .
0.15 m
um (t)∈S(∆)
tk+N tk
ua = um
as follows:
˜ 2 + RT (ua,sp − um )2 dτ QT (hsp − h)
p ˜˙ = 1 (um − c1 h) ˜ h A ˜ k ) = h(tk ) h(t
s.t.
when
cr
min
Z
0.20 m
to
us
tra king MPC to drive the level from
ip t
Fo using on the se ond set-point hange from the example above, we also design a well-tuned
integration step size of
RT = 1.
10−3 s
simulation of
2500 s,
Using this MPC to ontrol the pro ess of Eq. 23 with an
within the MPC, an integration step size of
N = 50,
a sampling period of length
d
level, a predi tion horizon of
(26d)
an
and
M
QT = 0.00001
(26b) (26 )
0 ≤ um (t) ≤ 0.006, ∀ t ∈ [tk , tk+N ) where
(26a)
and a set-point
hsp = 0.20 m
10−5 s
∆ = 1 s,
to simulate the
a nal time of the
with its orresponding steady-state ow rate
te
ua,sp = 0.00373 m3 /s, the state pro le in the bottom plot of Fig. 3 is obtained (the results are plotted
Ac ce p
every 10000 integration steps). The nonlinear interior point optimization solver Ipopt [Wä hter and Biegler (2006)℄ was used for the simulations with a toleran e of
10−8
on a 2.40 GHz Intel Core 2
Quad CPU Q6600 on a 64-bit Windows 7 Professional operating system with 4.00 GB of RAM.
Remark 4.
The losed-loop responses of the level under the PI and MPC designs are not meant
to be ompared with one another. The purpose of Fig. 3 is to show that under both PI ontrol and under MPC, when
ua = um ,
the tunings that are used for the level ontrol example result in
very good set-point tra king of the level. The results for the PI ontroller and MPC in Fig. 3 will be ompared with the results utilizing the same tunings when
ua 6= um
for ea h ontroller in the
following se tion.
14
Page 15 of 75
4.1.2. Level Control with a Sti ky Valve When only the valve dynami s hange ompared to the ase in the prior se tion (i.e., the valve be omes sti ky), sustained os illations are set up in the originally well-tuned ontrol loops. For this
ua
in Eq. 23 is the ow rate out of a pressure-to- lose spring-diaphragm sliding-
stem globe valve a tuated by a pressure
P.
If the valve is initiated from its fully open position, no
ip t
sti ky valve ase,
pressure is initially applied and the valve stem is at its equilibrium position
xv = xv,max = 0.1016 m.
Its fully
The dierential equations for the valve
cr
losed position orresponds to
xv = 0 m.
dynami s are [Gar ia (2008)℄:
(27)
us
dxv = vv dt
dvv 1 [Av P − ks xv − Ff ] = dt mv and
ks
are the diaphragm area and spring onstant, respe tively, and the fri tion for e
is determined from the LuGre fri tion model [Canudas de Wit et al. (1995)℄:
M
Ff
Av
an
where
(28)
Ff = σ0 zf + σ1
dzf + σ2 vv dt
(29)
d
dzf |vv |σ0 = vv − zf dt FC + (FS − FC )e−(vv /vs )2
te
(30)
The parameters of the valve dynami model in Eqs. 27-30 are those for the nominal valve in Gar ia
Ac ce p
(2008) and are displayed in Table 1. In addition, we assume that the valve has a linear installed
hara teristi [Coughanowr and LeBlan (2009)℄:
ua =
xv,max − xv xv,max
ua,max
The pressure to be applied to the valve for a given set-point
um − P
um
(31)
is determined from the following
relationship that was developed for a low-sti tion valve in Durand and Christo des (2016)
(though the tunings developed in the no-sti tion ase perform well for the low-sti tion valve):
P =
(um /ua,max ) − 0.70391/0.7042 0.05864 − 6894.76∗0.7042
This relationship has the form of Eq. 20, where
P
is
FA,i
(32)
and the right-hand side of Eq. 32 is
fSO,i (um ).
15
Page 16 of 75
h, um ,
Fig. 4 shows
ua
and
when the valve with the dynami s in Eqs. 27-32 is used to adjust the
ow rate to the pro ess of Eq. 23 under the PI ontroller of Eqs. 24-25. The valve was initiated from its fully open position (i.e.,
zf = 0 m),
h = 0.5184 m, ua = 0.006 m3 /s, ζ = 0 m · s, xv = 0 m, vv = 0 m/s,
and the set-point was hanged to 0.15
m
for 15600
s,
then to 0.20
m
for 15600
s.
t = 15600 s
hange from
qI
from this simulation as
0.15 m
to
0.20 m).
hanged, and the value of
uas
(the initial pro ess-valve state for the level set-point
The value of
ζ
was re-set to zero when the level set-point was
cr
state at
ip t
Be ause the se ond set-point hange is the fo us in this work, we will refer to the pro ess-valve
in Eq. 25 was re-set to the last applied value of
ua
when the set-
method with an integration step size of
10−5 s.
onsiderations are taken into a
ount: if
um > ua,max
ua,min
respe tively.
P < 0, P
is set to 0; if
or
um < ua,min, um
an
respe tively; if
In the simulations of the valve, several physi al
ua > ua,max
or
ua < ua,min, ua
is saturated at
ua,max
or
is saturated at
ua,max
or
M
ua,min
us
point was hanged. The traje tories were obtained using the expli it Euler numeri al integration
Choudhury et al. (2005) notes that the impetus for the os illations is related to the fri tion
FS ,
de reases rapidly when the valve begins to move (a ontributor to this is that the parameter
te
Ff
d
for e dynami s and PI ontroller dynami s. Spe i ally, the fri tion for e dynami s are su h that
whi h represents the stati fri tion oe ient, is larger than
Ac ce p
fri tion oe ient) whi h reates a large and rapid hange in overshoot its value asso iated with
hsp .
FC ,
whi h represents the Coulomb
dvv , leading the valve position to dt
In the following, we dis uss this in detail in terms of
oupling of the pro ess states. The fa t that the dynami s of the various states play a role in the response observed will be exempli ed by analyzing numeri al data from the simulation performed (the exa t numbers reported are related to the integration step size utilized, but the general ee ts would be expe ted to extend qualitatively to other integration step sizes) between the times
17608.72441 s
and
t¯2 = 17608.72597 s
¯1 and t¯2 , Between t
hanges only by pro ess, and from
xv
P
in Fig. 4.
only in reases from 55942.138 to 55942.181
0.0027 N )
at
t¯1
to
P a (the for e due to the pressure
due to the dynami s of the PI ontroller within the oupled nonlinear
hanges only slightly (from
1700.022 N
t¯1 =
1501.874 N
at
t¯2
0.036236 m to 0.036295 m) as well. (i.e., it de reases by lose to
16
However,
200 N ),
Ff
hanges
ausing the right-
Page 17 of 75
hand side of Eq. 28 to in rease from rst two terms related to
P
and
xv
4.1304 N/kg
at
t¯1
to
147.4380 N/kg
at
t¯2
be ause though the
do not hange mu h, the term for the fri tion for e de reases
signi antly. When the right-hand side of Eq. 28 in reases, the valve velo ity in reases, whi h an
ause the valve to move. This shows that though the drop in the fri tion for e played an important
played a role by not allowing
P
xv
and
to adjust rapidly when
P
de reased.
Furthermore, the
as well sin e the PI ontroller depends on
h
in
cr
dynami s of the level play a role in adjusting
Ff
ip t
role in adjusting the valve velo ity, the dynami s of the PI ontroller and the stem position also
relation to its set-point. Additionally, the magnitude of the integral term of the PI ontrol law is
us
signi antly larger than magnitude of the proportional term when the level hanges and overshoots its set-point, whi h may also impa t the length of time that the valve is stu k.
an
The fa t that the negative ee t of sti tion in a ontrol loop is a losed-loop property of a full pro ess-valve system is further emphasized by utilizing the MPC of Eq. 26 (i.e., an MPC that
0.15 m
to
0.20 m
M
a
ounts only for the level dynami s and not the a tuator dynami s) for the set-point hange from for the pro ess-valve system of Eqs. 23 and 27-32, initiated from
10−6 s using the
The pro ess
expli it Euler numeri al integration
d
was integrated with an integration step size of
qI .
te
method. The resulting traje tories of the level under the MPC are shown in Fig. 5, plotted every 1000 integration steps (the values of
h, ua ,
and
um
for this ase are denoted in the legend by U,
Ac ce p
signifying that sti tion is un ompensated in these results be ause the MPC does not a
ount for the a tuator dynami s). No os illations are observed for this level set-point hange as in Fig. 4, demonstrating that the sti tion-indu ed os illations observed under the PI ontroller depend on the
ontroller utilized. Instead of os illations, a persistent oset from the set-point o
urs under the MPC. The reason for this is that be ause the MPC is unaware of the a tuator dynami s, it al ulates values of
um
that orrespond to pressures (through Eq. 32) that do not allow the valve to move
a
ording to the for e balan e (i.e., the MPC expe ts that the ontrol a tions that it al ulates will drive the level toward the set-point be ause it is anti ipating that there is no fri tion in the valve, but due to fri tion the valve annot move with the pressure applied to it). The MPC ontinues to ompute approximately the same ontrol a tion for the rst sampling period of the predi tion horizon at ea h sampling time (whi h is reasonable onsidering that the state measurement that
17
Page 18 of 75
it re eives is approximately the same ea h time sin e the valve is stu k and thus the ow rate out of the valve is not appre iably hanging to adjust the level). This ontrol a tion ontinues to be unable to ae t the level appre iably, resulting in persistent o-set of the level from
hsp
be ause
the MPC has no me hanism for dete ting that the set-points it has al ulated are failing to make
ip t
an impa t on the system. Returning again to the ontrol loop under PI ontrol, we now show that a hange in the ontrol
cr
loop ar hite ture (in this ase, adding ow ontrol to the valve of Eqs. 27-31) alters the response of the pro ess outputs. The ontrol loop ar hite ture in this ase is depi ted in Fig. 6, where the ow
um
is omputed by the PI ontroller of Eqs. 24-25 for the tank level, and be omes
the set-point for a minor PI ontrol loop used to regulate
P
ua
This minor loop al ulates the
um − ua
to be applied to the valve stem based on the error
as follows:
um − ua Kc,p + ζP ua,max τI,p
(33)
M
P = Ps + Kc,p
um .
to
an
pressure
us
rate set-point
um − ua , ζP (0) = 0 ζ˙P = ua,max Ps
is the steady-state value of the pressure, and
d
where
(34)
Kc,p = −82737.09, τI,p = 0.01,
and
ζP
are
te
the proportional gain, integral time, and internal state, respe tively, of the minor loop ontroller.
Ac ce p
The tuning (from Durand and Christo des (2016)) performed su
essfully when
um
was onstant
for some time.
The valve was initially operated without ow ontrol (i.e., Eq. 32 was used to related
P)
for
15600 s
for a level set-point hange from
0.5184 m
was operated under the ow ontroller of Eqs. 33-34 for
0.15 m to 0.20 m. to
Fig. 7 shows the responses of
h, um ,
to
0.15 m
15600 s
and
ua
to rea h
qI .
um
Subsequently, it
for the level set-point hange from
for the set-point hange from
0.20 m (the time axis is short to display the fast response of the valve under ow ontrol).
results were obtained using an integration step size of integration steps. The integral term
ζP
10−5 s,
and
0.15 m These
with the data plotted every 100000
of the ontroller for the valve was re-set to zero when the
level set-point was hanged, and at that point the value of
Ps
was also re-set to the last applied
pressure.
18
Page 19 of 75
The sustained os illations apparent in Fig. 4 do not appear in Fig. 7, though the same pro ess, sti ky valve, and outer loop PI ontroller are used as in Fig. 4. This is parti ularly notable given that the fri tion dynami s and the right-hand side of Eq. 17 have not hanged (i.e., the hange is essentially in the manner in whi h
P
is al ulated; the oupling among the states of the pro ess-
ip t
valve model auses the hange in the system dynami s resulting from the addition of the dynami PI ow ontroller to alter the level response ompared to the ase without ow ontrol). Instead
ua
tra ks
um
well after it initially overshoots
um
in the
cr
of os illations, when ow ontrol is used,
time period immediately after the set-point hange. During these initial overshoots, the pressure
in the initial signi ant overshoots of
ua ,
around the hanging
um
set-point. However, despite these
the ow ontroller is su
essful at stabilizing the level at its set-point (it
an
initial overshoots of
ua
us
applied to the valve hanged rapidly a
ording to Eqs. 33-34, and the for es on the valve resulted
auses the for es on the valve to balan e in su h a manner that
ua
is able to tra k
um
and thus to
M
drive the level to its set-point). The hange of the ontrol loop ar hite ture hanged the number of
oupled dynami states in the system of nonlinear dierential equations des ribing the pro ess-valve
and
zf ,
this means that
it also in ludes
xdyn
ζP
when ow ontrol is used). Returning to the notation of Eq. 8,
te
ζ , xv , vv ,
h,
d
system (i.e., whereas the state ve tor of the pro ess-valve system without ow ontrol in luded
in orporates an extra state and its dynami s when ow ontrol is used, whi h
Ac ce p
overall hanges the response of the measured output (h) of the pro ess-valve system ompared to the ase without ow ontrol.
Remark 5.
The analysis performed demonstrates that the standard valve output- ontroller input
response for a sti ky valve exhibited in Fig. 8 (and displayed in multiple sour es in the literature su h as Choudhury et al. (2005) and Brásio et al. (2014)) an be understood as the response of the valve output when the for e applied to the valve is ramped up and down by a ontroller (i.e., the losed-loop analysis above indi ates that the ontroller output on the standard plots is linked to the for e applied to the valve). Furthermore, Fig. 8 re e ts the transient behavior of the valve after it begins moving (i.e., it shows the slip-jump). This is in ontrast to Fig. 6 in Durand and Christo des (2016), whi h is a plot of the relationship between
ua
and the pressure applied to two
valves, one with low and one with more signi ant sti tion, as the pressure is ramped up and down,
19
Page 20 of 75
but plotting only the nal values of
ua
and
P
at the end of ea h time period for whi h
onstant (rather than throughout the transient as
ua
adjusts to a
P
hange).
P
is held
Furthermore, in
Fig. 8 in Durand and Christo des (2016), the pressure is ramped up and down independently of a ontroller or pro ess (i.e., the valve is analyzed alone); however, in plots like those in Fig. 8 of
ip t
the present work, whi h expli itly assume the use of a ontroller and thereby imply that the plot
omes from measurements of a pro ess-valve system, the oupling of the ontroller, pro ess, and
ua
versus the for e applied to the
cr
valve dynami s for that pro ess-valve system may ause a plot of
valve to be dierent when the same valve is operated under a dierent ontroller or for a dierent
us
pro ess. For example, for the level ontrol problem without ow ontrol, the plot of
ua
versus
P
does not have a signi ant moving phase be ause soon after the valve slips, the ontroller begins
position
xv
(rather than
ua )
an
to drive the valve in the opposite dire tion. Be ause the plot an also be understood as the valve versus the ontroller signal as in Ivan and Lakshminarayanan (2009),
M
a linear valve hara teristi is assumed when the same plot is obtained for signal.
ua
versus the ontroller
d
4.2. A Closed-Loop Perspe tive for a CSTR Example
te
We now demonstrate the ne essity of a losed-loop perspe tive for analyzing a pro ess-valve dynami system when the omplexity of the pro ess model in reases (i.e., it has more states and
Ac ce p
inputs than the level ontrol problem).
For illustration purposes, we examine the produ tion of
ethylene oxide from ethylene in a CSTR with a heating/ ooling ja ket.
Using the rea tion rate
equations for this pro ess from Alfani and Carberry (1970), the following nonlinear pro ess equations are obtained (in dimensionless form) from mass and energy balan es on the rea tor [Özgül³en et al. (1992)℄:
x˙ 1 = ua,1 (1 − x1 x4 )
(35a)
x˙ 2 = ua,1 (ua,2 − x2 x4 ) − A1 exp(γ1 /x4 )(x2 x4 )0.5 − A2 exp(γ2 /x4 )(x2 x4 )0.25
(35b)
x˙ 3 = −ua,1 x3 x4 + A1 exp(γ1 /x4 )(x2 x4 )0.5 − A3 exp(γ3 /x4 )(x3 x4 )0.5
(35 )
x˙ 4 =
B1 B2 ua,1 (1 − x4 ) + exp(γ1 /x4 )(x2 x4 )0.5 + exp(γ2 /x4 )(x2 x4 )0.25 + x1 x1 x1 B4 B3 exp(γ3 /x4 )(x3 x4 )0.5 − (x4 − ua,3 ) x1 x1
20
(35d)
Page 21 of 75
where
x1 , x2 , x3 ,
and
x4
are dimensionless quantities orresponding to the gas density, ethylene
on entration, ethylene oxide on entration, and temperature in the rea tor, and the pro ess inputs
ua,1 , ua,2 ,
and
ua,3
are dimensionless quantities orresponding to the feed volumetri ow rate, feed
ethylene on entration, and oolant temperature, whi h are assumed to be adjusted by individual
ua,1 )
or indire tly (e.g.,
ua,2
may be adjusted by opening or losing
ip t
valves either dire tly (e.g.,
ua,3
valves that allow a more on entrated ethylene stream to mix with a solvent stream, and
may
cr
be adjusted by heating or ooling the oolant using a higher or lower ow rate of another uid past the oolant in a heat ex hanger). Due to the oupling between the states in Eq. 35, and the
regardless of the type of ontroller used to al ulate and
ua,3 = um,3 .
um,1 , um,2 ,
and
um,3 ,
x1 , x2 , x3 ,
and even if
and
x4 ,
ua,1 = um,1 ,
Therefore, if the valves also have dynami s, linear or nonlinear, stati
an
ua,2 = um,2 ,
us
highly nonlinear dynami equations, it is di ult to predi t the evolution of
or dynami , or potentially dierent dynami s for ea h valve, and dierent ontrollers or ontrol loop
M
ar hite tures for ea h valve, the number of oupled states in this system of nonlinear dierential equations in reases and performing simulations will be the best way to understand how ea h pro ess
d
output will respond.
Ac ce p
te
5. Impli ations of a Uni ed Pro ess-Valve System Framework for Valve Behavior Compensation and Modeling The losed-loop perspe tive regarding valve behavior lari es how and when valve behavior
ompensation strategies in the literature work (to demonstrate this, we will review several sti tion
ompensation methods) and enables valve behavior ompensation designs to be proposed that target the oupling of the states that leads to negative ee ts (whi h we will demonstrate by des ribing how two of our valve behavior ompensation strategies, an integral term modi ation method for pro esses with sti ky valves under PI ontrol and an MPC-based general valve behavior
ompensation te hnique, an be understood in a losed-loop ontext). Be ause a full pro ess-valve system model is required to predi t the response of the pro ess and valve outputs under various valve behavior ompensation methods and potentially is needed by those ompensation s hemes to allow them to ompute ompensating signals to adjust the losed-loop response, we also examine the pra ti al question of how an empiri al model that aptures the dominant pro ess-valve dynami s
21
Page 22 of 75
may be obtained when a rst-prin iples model is unavailable or undesirable to use. Finally, we will exemplify the dis ussion through several pro ess examples.
5.1. A Closed-Loop Dynami Analysis of the Sti tion Compensation Literature We analyze three ategories of methods from the sti tion ompensation literature (more thorough
ip t
reviews of sti tion ompensation methods have been provided elsewhere; see, for example, Brásio et al. (2014); Cuadros et al. (2012); Mohammad and Huang (2012)). The novelty of the analysis
cr
of this se tion lies in presenting several of these methods within the losed-loop systems framework
us
des ribed above to provide a fundamental understanding of how they ameliorate the negative ee ts observed in ontrol loops with sti ky valves, and their bene ts and limitations.
an
5.1.1. Sti tion Compensation Methods: Controller Tuning Adjustments
Re-tuning of ontrollers has been advo ated as a method for redu ing losed-loop os illations
M
developed in a ontrol loop ontaining a sti ky valve under lassi al proportional-integral-derivative (PID)-type ontrol (e.g., Mohammad and Huang (2012)). The re-tuning may result in an improved
losed-loop response be ause it hanges the dynami s of the PID-type ontroller, whi h alters the
d
response of the pro ess outputs due to the oupling of the ontroller, pro ess, and valve dynami s.
te
Thornhill and Hor h (2007) highlights the di ulty of determining an appropriate tuning for obtai-
a priori,
the response
Ac ce p
ning a desired response, whi h is onsistent with the di ulty of predi ting, of a nonlinear dynami system.
5.1.2. Sti tion Compensation Methods: Augmented Controller Signal The kno ker [Hägglund (2002); Srinivasan and Rengaswamy (2005)℄ and onstant reinfor ement [Ivan and Lakshminarayanan (2009)℄ adjust the ontrol signal re eived by the valve by adding either a onstant or time-varying signal to the input al ulated by the ontroller. This hanges the manner in whi h the for e applied to the valve is al ulated. For example, onsider the kno ker applied to the level ontrol example without ow ontrol. The PI ontroller has its own dynami s that, in the absen e of the kno ker, di tate the pressure applied to the valve. However, with the kno ker, there are times when the pressure applied to the valve is in reased by an amount determined by the kno ker parameters above the amount output by the PI ontroller, but then after a ertain
22
Page 23 of 75
time period, the kno ker takes away that extra amount of pressure. This allows the PI ontroller to retain its dynami s but permits the pressure to be adjusted using a sour e other than the PI
ontroller as well, whi h hanges the balan e of for es on the valve/the pro ess-valve dynami s and
an result in a dierent losed-loop response than would be obtained without the kno ker. Changes
response as observed in Srinivasan and Rengaswamy (2005).
ip t
to the kno ker parameters hange the losed-loop system dynami s and as a result the losed-loop
cr
Constant reinfor ement similarly augments the output of the PI ontroller, adding a onstant positive signal when the PI ontroller output is in reasing and a onstant negative signal when the
us
PI ontroller output is de reasing, whi h again hanges the right-hand side of the equation for the valve velo ity ompared to not adding su h a signal. One aspe t of the ee t of this on the level
an
ontrol problem might be, for example, that the integral term of the PI ontroller may not need to be ome as large for the pressure from the pneumati a tuation to over ome the stati fri tion
M
for e and ause the valve to move. A method proposed by Cuadros et al. (2012) for turning o the PID-type ontroller and the kno ker is an extension of the kno ker method, but as noted by Wang
d
(2013) and lari ed through the losed-loop analysis in this work, the kno ker hanges the for e
te
balan e/ losed-loop dynami s but that does not guarantee that there will be no oset between the pro ess output and its set-point if the PID ontroller is removed so thus removing the ontroller
Ac ce p
and ompensating pulses may not be appropriate.
5.1.3. Sti tion Compensation Methods: Two Moves Method The two moves method [Srinivasan and Rengaswamy (2008)℄ spe i es ompensating signals to apply to the signal oming from a linear ontroller that will drive the stem position to its set-point. This method is model-based, whi h means that it has a
ounted for the oupling of the pro essvalve dynami s, and a number of assumptions are required to guarantee that the method an drive the valve position to its set-point, in luding that a parti ular data-driven sti tion model is an exa t representation of the sti tion dynami s and that there are no disturban es or plant-model mismat h. A method similar in on ept to the two moves method (it determines how to hange the set-points for a losed-loop system in a manner that brings the valve position to a desired value) is developed in Wang (2013) and is inspired by a pro ess-valve model for pro esses that an be des ribed with
23
Page 24 of 75
a linear model and are under linear ontrol.
5.2. Motivating Valve Behavior Compensation Design with Closed-Loop Dynami Analyses In this se tion, we seek to en ourage further resear h on valve behavior ompensation based on the losed-loop perspe tive on valve behavior developed in this work. To do this, we des ribe how
5.2.1. Sti tion Compensation Methods: Integral Term Modi ation
ip t
two of our re ent sti tion ompensation methods an be motivated by this framework.
cr
The ontroller tuning adjustment methods dis ussed above require a ontroller's tuning to be
us
hanged even if the tuning being utilized is known to work well for the valve when it is not sti ky and thus would be the preferred tuning after valve maintenan e is performed. Furthermore, the
an
nonlinearity of the sti tion phenomenon, the oupling between the pro ess states, and the omplexity of the manner in whi h the for es on the valve balan e and ome out of balan e makes it di ult to dis ern
a priori
what the best tuning to use when the valve is sti ky should be. Therefore, it
M
is reasonable to onsider whether there is another method of adjusting a PI ontroller's dynami s instead of disrupting the desired tuning in an
ad ho
fashion.
d
One su h method is to add a term to the integral a tion of the linear ontroller that an easily be
te
removed or adjusted for any set-point hange to attempt to alleviate sti tion-indu ed os illations.
Ac ce p
Given that a hara teristi of the sti tion-indu ed os illations is that the valve output tra k its set-point
um ,
this added term an be based on the s aled dieren e between
Spe i ally, for a PI ontroller for whi h
ζ
ua
does not
ua
and
um .
signi es the integral of the error between the pro ess
output set-point (assuming a single output denoted by
xˆsp )
and the pro ess output (x ˆ, assumed to
be a omponent of the pro ess state ve tor), the following ontrol law de nes the ontrol a tion with an integral term modi ation:
um = uas + Kc (ˆ xsp − xˆ) + Kc ζ/τI ζ˙ = ζ(0) = 0 where
L, β ,
and
tAW
(ˆ xsp − xˆ) + L(ua − um ), t < tAW (ˆ xsp − xˆ) + Le(−β(t−tAW )) (ua − um ), t ≥ tAW
(36)
(37)
are tuning parameters that an be adjusted by a ontrol engineer to attempt
to mitigate sti tion-indu ed os illations. When
L = 0,
24
Eq. 37 redu es to the standard integral term
Page 25 of 75
in a PI ontrol law, and thus has no ee t. The parameters are best determined using losed-loop simulations and/or on-line adjustments of
L, β , and tAW ; however, the general goals of adjusting the
parameters provide a potential methodology for looking for an appropriate tuning. In parti ular, the goal of this method is to determine a tuning that an de rease
ζ˙
in su h a way that the for es
of the term
(ˆ xsp − xˆ),
L
that auses the term
it is possible to ause
ζ˙
L(ua − um )
in Eq. 37 to have a sign opposite to that
to de rease even before
xˆ = xˆsp
(this would not be
cr
By hoosing a value of
ip t
on the valve equilibrate at a value that auses the ontrolled pro ess output to meet its set-point.
possible with the standard PI ontrol law, for whi h the integral term an only begin to de rease
ζ˙
β
to equal zero and drive
and tAW to zero, and then sear hing for a value of
L
in
L that is able to ause
dvv to zero by providing a onstant for e from the valve a tuation. This dt
may o
ur, however, before
xˆsp
an
Eq. 37 is by rst setting
us
after the set-point is ex eeded). Therefore, a possible strategy for tuning the term ontaining
is rea hed, resulting in oset. Therefore, the value of
tAW
may be
M
set to a time at whi h the for es on the valve appear to no longer be hanging and that allows
xˆ
to begin to approa h its set-point as soon as possible after this has o
urred. Then, various values
β
may be tried to attempt to de rease the term ontaining
L
in
ζ˙ ,
whi h an ause this integral
d
of
um
te
term to hange and thus hanges the for e applied to the valve as a result of the ontrol a tion re eived by the valve. If a value of
β
an be found that hanges the for e applied to the valve
Ac ce p
in a manner that auses the for es to on e again be ome onstant, but this time at a value of the valve position that auses
xˆsp
to be rea hed, then this ontrol strategy is su
essful for the set-point
hange examined. However, some values of up on e again even if the value of
L
β
may even ause sti tion-indu ed os illations to be set
examined was able to attenuate them before the time
tAW ;
this shows that the tuning problem is omplex and that an appropriate tuning annot be de ided
priori.
a
In addition, due to the nonlinearities in the valve and pro ess dynami s, and the potential
for intera tions between states and inputs, there is no guarantee that any appropriate tuning will be found for a given set-point hange, or that the same tuning will work for a variety of set-point
hanges or disturban es; however, losed-loop simulations or on-line adjustment an be attempted to see if there are values of
L, β , and tAW
that are generally appropriate for a given pro ess. Notably,
when the integral term modi ation method is utilized in the ase of bounded, time-varying pro ess
25
Page 26 of 75
disturban es, whi h is the ase onsidered in this work a
ording to the lass of pro ess systems in Se tion 2.2, the notion of driving the output to the set-point does not exist (instead, it would be hoped that the ontroller ould drive the state to an a
eptably small neighborhood of the set-point).
Various tunings ould be evaluated for the integral term modi ation in this ase to
ip t
determine if they are able to do so.
5.2.2. Valve Behavior Compensation Methods: MPC for Valve Behavior Compensation
cr
The losed-loop perspe tive on valve behavior indi ates that a ompensation method that a
ounts for the pro ess-valve dynami s when omputing ontrol a tions would be able to systemati-
us
ally address the root ause of the negative ee ts observed in ontrol loops with valves that have behavior that annot be negle ted. An MPC design that uses the pro ess-valve model to make state
an
predi tions is able to do this. An MPC design with a general obje tive fun tion that in orporates a pro ess-valve model in luding sti ky valve dynami s was developed in Durand and Christo des
M
(2016). It an a
ount for multiple sti ky valves, even with diering levels of sti kiness, and has the ability to in orporate onstraints that guarantee feasibility and losed-loop stability of a nonlinear
d
pro ess operated under the ontroller. In general, any pro ess-valve model an be used within this
te
MPC framework, so it ould be onsidered for valve behavior ompensation in general (i.e., it is not spe i to sti tion). This allows the MPC to handle pro esses where a single valve exhibits
Ac ce p
multiple nonlinearities (e.g., sti tion and also an equal per entage valve hara teristi ) or where the various inputs exhibit dierent dynami s (e.g., one valve is sti ky but another for the same pro ess has linear dynami s) if the dynami s an be modeled and then in luded within the pro ess-valve dynami model.
The formulation of an MPC that in ludes the valve and pro ess dynami s (Eqs. 7-8) is as follows:
min
um (t)∈S(∆) s.t.
Z
tk+N
Le (˜ q (τ ), um (τ )) dτ
(38a)
tk
q˜˙(t) = fq (˜ q(t), um (t), 0)
(38b)
q˜(tk ) = q(tk )
(38 )
q˜(t) ∈ Qv , ∀ t ∈ [tk , tk+N )
(38d)
26
Page 27 of 75
um (t) ∈ Um , ∀ t ∈ [tk , tk+N )
(38e)
gM P C,1 (˜ q(t), um (t)) = 0, ∀ t ∈ [tk , tk+N )
(38f )
gM P C,2 (˜ q(t), um (t)) ≤ 0, ∀ t ∈ [tk , tk+N )
(38g)
of Eqs. 38b-38 and is bounded within the set
Qv
(Eq. 38d; the set
Qv
q˜ follows
the model
ip t
where the notation follows that in Eq. 11. The predi ted pro ess-valve state
is de ned to be the bounds
ua ∈ U
and the state onstraints orresponding to
Eq. 2 (sin e ea h
um,i
sin e
ua
is a fun tion of the states from
is al ulated by a state feedba k ontroller) and Eq. 4).
us
x∈X
cr
on the pro ess-valve model states, whi h may in lude, for example, the state onstraints restri ting
The onstraints
in Eqs. 38f-38g may in lude onstraints related to spe i behavior of the valve.
For example,
an
in Durand and Christo des (2016), it is noted that a bene ial onstraint for a sti ky valve may be one whi h ensures that only physi al values of the for es applied to the valve are predi ted
P˜
of
P
to satisfy
M
(for example, it may require the predi tions
P˜ ≥ 0
for the tank level example).
Theoreti al results regarding disturban e reje tion for an MPC with the form in Eq. 38 but with
d
ertain stability-related onstraints added to the formulation have been developed in Durand and
te
Christo des (2016). Guaranteed robustness to disturban es in that ase requires several onditions to hold, in luding that the disturban e magnitude be su iently small.
In general, there is no
Ac ce p
guarantee that the losed-loop response of a pro ess under an MPC in the presen e of disturban es will be as desirable as that in the ase without disturban es. Two potential limitations of an MPC for valve behavior ompensation are: 1) The sampling period may, for omputation time reasons, be signi antly longer than the time that it takes for the for es on the valve to rea h equilibrium after the deadband and sti kband are over ome. This means that the MPC may not have the exibility to hange the for e from the a tuation rapidly in response to a rapid hange of the fri tion for e (this is dis ussed further in Se tion 5.4.2). 2) It may be di ult to determine all parameters of an appropriate rst-prin iples model of the valve layer for a given valve (the valve layer for valve the relationship between
um,i
and
i
is de ned in this work to refer to all dynami s des ribing
ua,i , i = 1, . . . , m).
Empiri al models for the valve layer dynami s
(and potentially the pro ess dynami s as well) may aid in handling both of these limitations be ause
27
Page 28 of 75
they may not require detailed information on the valve layer dynami s like the details of the fri tion for e model or valve hara teristi , and they may also be less sti than a rst-prin iples model (an example of this is demonstrated in Se tion 5.4.3), resulting in a lower omputation time for the MPC for valve behavior ompensation. If the model utilized within the MPC an apture the dominant
ip t
pro ess-valve dynami s to provide su iently a
urate state predi tions, it would be expe ted to be bene ial in ompensating for sti tion, even if it is not an exa t model, due to the in orporation
cr
of state feedba k. The following equation denotes an empiri al model for use in the MPC-based
y(t) ˙ = fy (y, um) y(t)
ua
is the predi ted value of
hara terized by the ve tor fun tion
fy .
from the empiri al model at time
an
where
us
valve behavior ompensation method:
(39)
t
and has dynami s
This empiri al model an be used to predi t the values of
M
the pro ess inputs in the obje tive fun tion and onstraints of Eq. 38 when a rst-prin iples model is not used.
d
5.3. Valve Behavior Empiri al Modeling Considerations Following Closed-Loop Dynami Analyses
te
Two questions that arise when onsidering how to empiri ally model a full pro ess-valve system
Ac ce p
are: 1) what part of the pro ess-valve dynami s should be aptured in a single empiri al model and 2) is it possible to develop empiri al modeling strategies for valve behavior that are general rather than fo used on a spe i type of valve behavior. To explain the impli ations of the rst question, we rst examine sti tion empiri al modeling te hniques in the literature (e.g., the Choudhury model [Choudhury et al. (2005)℄, the Kano model [Kano et al. (2004)℄, and the He model [He et al. (2007)℄), whi h generally assume that the relationship between
ua
and the for e applied to the valve
is similar to that in Fig. 8, so they use an if-then (bran hed) type stru ture to mimi this (i.e., if the ontrol signal has not hanged enough to un-sti k the valve, then the valve position does not
hange with a hange in the ontrol signal; if the ontrol signal has hanged enough to un-sti k the valve, the valve takes a new position de ned by the empiri al model). These models relate the valve position and the for e on the valve determined by a ontroller. However, this may not eliminate the need to develop rst-prin iples models for some of the dynami s relating
28
um,i
and
ua,i
for the
Page 29 of 75
i − th
valve. For example, if ow ontrol is utilized on the valve as in the example of Fig. 7, some
knowledge of the relationship between the valve output ow rate set-point from the major loop
ontroller and the for e applied to the valve by the minor loop ontroller is required to model the valve layer. If an equal per entage valve hara teristi hara terizes the
ua,i − xv,i
relationship, the
ip t
form of this model must be known. The prin iple demonstrated through this analysis is not spe i to sti tion: to avoid the need to obtain a rst-prin iples model for any aspe t of the valve layer
um,i
to
ua,i .
We show in
cr
dynami s, it may be bene ial to develop an empiri al model that relates
Se tion 5.4.3 that it may be possible to develop an a
eptable relationship even when the
um,i − ua,i
us
dynami s are somewhat omplex (e.g., they ontain the dynami s not only of the valve but also of a ow ontroller for the valve).
to obtain a
relationship an be obtained, it should be determined whether it would be possible
an
um,i −ua,i
um −x relationship as well (i.e., to develop a single empiri al model for the pro ess-valve
system that relates the ontroller output obtain data on the pro ess input ve tor
um
ua
to the pro ess states su h that it is not ne essary to
M
If a
when developing the empiri al model). However, there
d
may arise ases in whi h it may not be possible to easily apture the pro ess and valve dynami s in
te
the same empiri al model. To see why di ulty may arise, onsider a pro ess with multiple inputs, all of whi h are adjusted by sti ky valves. Be ause the valves are sti ky, ea h hange in
ua,i
will not ause in
um,i
to hange appre iably (whi h will ause
Ac ce p
either ause
as if
ua,i
um,i
to hange appre iably (whi h will ause
to respond to the hange in
x
um,i ),
will or it
to ontinue behaving after the hange
had not hanged). Due to the oupling of the dynami s of the states of a pro ess-
valve system, the ombination of des ribing the
x
um,i
um − x
ua,i 's
ae ts the pro ess dynami s, so a bran hed empiri al model
relationship may need to in lude dierent bran hes for every ombination
of sti king and slipping for all valves, whi h may lead to a di ult identi ation task. A solution if this is found to be an issue would be to empiri ally model ea h valve as well as the
ua − x
ua,i − um,i
relationship for the pro ess (this may also be bene ial from a valve
maintenan e perspe tive be ause it permits monitoring of how losely valve (assuming that the
relationship for ea h
ua,i − um,i
ua,i
mat hes
um,i
for ea h
relationship an be assessed frequently or potentially that an
empiri al modeling pro edure for those dynami s ould be automated), allowing valve maintenan e
29
Page 30 of 75
to be performed rst on those valves for whi h
ua,i
tra ks
um,i
the least).
To examine the impli ations of the se ond question above (whether a general empiri al modeling strategy an be used to obtain the
um,i − ua,i
relationship) we return again to the sti tion empiri al
models of the literature. Many of these assume that a spe i trigger auses the valve to sti k or slip
ip t
and they assume a spe i model for the valve response after slipping o
urs that is related to the
hara teristi s of the typi al response of a sti ky valve to ontrol signal hanges. For a general valve
ne essary to onsider whether a
cr
layer (i.e., the type of dynami s and ontrol ar hite ture within the valve layer are not spe i ed), it is
ua,i −um,i relationship an be developed without ne essarily knowing
us
the underlying valve behavior/loop ar hite ture represented through the relationship.
Bran hed
models may be required to do this be ause valves are not ne essarily operated within a range of
an
states throughout whi h linearization of the valve dynami s is adequate for representing the valve response. For example, onsider the model of a valve's dynami s in Eqs. 27-32. The valve velo ity
M
must be for ed through zero every time the dire tion of the valve movement hanges, whi h is a point where Eq. 30 is non-dierentiable due to the absolute value in the right-hand side. Therefore, the
d
model of Eqs. 27-32 annot be linearized at zero velo ity every time the valve movement hanges Though Eq. 30 represents the dynami s for
te
dire tion (whi h has the potential to be frequent).
only one of many fri tion models, its form indi ates that the fri tion nonlinearity may need to
Ac ce p
be represented by more omplex nonlinearities su h as absolute values than are usually present in standard empiri al model stru tures su h as linear or polynomial state-spa e models, and as a result when the latter models are used, bran hed models may be required to apture the ee ts of the nonlinearities of the valves.
When a bran hed model is used, some understanding of the physi s of the valve layer an be exploited by an engineer in developing the triggers for the bran hes, or attempts an be made to automate the determination of triggers from data. If the valve layer has a number of states, this may result in an empiri al model with a signi ant number of bran hes and even with parameters that depend on the inputs to the valve layer. For example, onsider that we want to develop an empiri al model for the relationship between
ua,i
and
um,i , i = 1, . . . , m,
when ea h
um,i
is a valve
output ow rate set-point from an MPC whi h is transmitted to a valve layer ontaining a sti ky
30
Page 31 of 75
valve with a linear valve hara teristi under a PI ow ontroller based on the error between and
ua,i .
We assume that
um,i , i = 1, . . . , m,
is held onstant for a time period
∆
during whi h the
PI ontroller repeatedly omputes new values of the pressure applied to the valve stem to drive to
um,i .
um,i
If the valve output ow rate set-point hange dire tion reverses (e.g., the set-points
ua,i um,i
throughout some or all of the time period
∆
during whi h
um,i
is onstant, depending on whether
∆
to over ome the for e required to
cr
the pressure applied to the valve hanges enough throughout
ip t
were previously in reasing but the next set-point is lower than the previous one), the valve may sti k
move the valve. Also, the PI ontroller will al ulate larger ontrol a tions when the valve output
us
ow rate set-point hanges signi antly (be ause this reates a large error between
um,i
and
ua,i ),
and su h larger ontrol a tions (pressures) are more likely to over ome the deadband/sti kband
∆.
Therefore, it is more likely that the valve will move if the ontroller is
an
within the time period
aggressively tuned or if the set-point hange is large and the error between
um,i
and
ua,i
ae ts the
M
ontroller output. Saturation of the valve (e.g., the pressure from the pneumati a tuation drops to zero so that the valve an no longer move in the dire tion of de reasing pressure) an also o
ur.
d
Therefore, the various bran hes of the response may need to be triggered by onsiderations related
te
to the sti tion dynami s and saturation of the pressure of the pneumati a tuation, with the time that the valve is stu k depending on the set-point hange magnitude due to the ow ontroller.
Ac ce p
The pro edure proposed for empiri al modeling of the feedba k loop for the sti ky valve under
onsideration is as follows:
1. Colle t valve layer input-output data (i.e., data relating
ua,i
and
um,i , i = 1, . . . , m),
ensuring
that data gathered represents all aspe ts of the valve layer response that should have separate equations (bran hes) within the pie ewise model stru ture (e.g., sti king and slipping). 2. For ea h aspe t of the response that requires its own model stru ture, sele t an appropriate stru ture based on the valve layer input-output data trends and determine what a tivates the dierent bran hes of the response (e.g., set-point hange dire tion reversals). 3. Identify the parameters for the dierent bran hes of the model. 4. Develop models for any parameters that an be seen from the valve layer input-output data to be dependent on the valve layer inputs (e.g., parameters that depend on the magnitude of
31
Page 32 of 75
the valve output ow rate set-point hange). 5. Validate the nal pie ewise-de ned model.
It will be shown in Se tion 5.4.3 of this work that Step 2 in the pro edure above may be able to be performed with standard empiri al model stru tures in the hemi al pro ess industries, su h
ip t
as se ond-order or rst-order-plus-dead time models (i.e., they do not take a form dependent on the valve behavior being modeled). Furthermore, the ve steps above are general and therefore are
cr
not ne essarily restri ted to the valve layer with a sti ky valve under ow ontrol above but an be
us
examined for extension to other ontrol loop ar hite tures and valve behavior that hara terize the valve layer.
an
A nal onsideration related to empiri al modeling of a pro ess-valve system is that there may be states that it is desirable to onstrain in an MPC-based valve behavior ompensation strategy that would be known from a rst-prin iples model but are not expli itly modeled within an empiri al
M
model. As a result, onstraints of the MPC-based valve behavior ompensation strategy may need to be adjusted to be in terms of the states of the empiri al model and the inputs. For example,
d
Eqs. 38f-38g may in lude a onstraint on the pressure from the pneumati a tuation to prevent it
um,i
to
ua,i
for a valve layer with a ow ontroller may not have any state related
Ac ce p
model relating
te
from saturating in the ase that a sti ky valve is in the loop. However, an empiri al valve layer
dire tly to this pressure. As a result, the following input rate of hange onstraints an be added to the MPC of Eq. 38 in pla e of Eqs. 38f-38g:
|um,i (tk ) − u∗m,i (tk−1 |tk−1 )| ≤ ǫ, i = 1, ..., m
(40a)
|um,i (tj ) − um,i (tj−1 )| ≤ ǫ, i = 1, ..., m, j = k + 1, ..., k + N − 1
(40b)
These onstraints require the dieren es between the values of periods of the predi tion horizon to be no greater than in the set-points
um,i
ǫ
um,i
at any two subsequent sampling
and therefore an prevent large hanges
between two sampling periods that may saturate the valve a tuation (or wear
the a tuators [Durand et al. (2016)℄). However, applying input rate of hange onstraints in an MPC for valve behavior ompensation
32
Page 33 of 75
is not always bene ial. For example, onsider the sti ky valve without ow ontrol of Eqs. 27-32. As exempli ed in Fig. 5, the valve does not move when the valve is stu k if the hange in
um does not
hange the pressure applied to the valve enough to over ome the stati fri tion for e. This indi ates that even if the valve dynami s were in luded in the MPC for valve behavior ompensation for this
um
that may satisfy
ip t
valve but input rate of hange onstraints were also in luded, the values of
these onstraints may not dier enough from the values at the prior sampling time in some sampling
the pressure applied to the valve is not an expli it fun tion of
cr
periods to allow the valve to move. If the valve is under the ow ontroller of Eqs. 33-34, however,
um , but instead is hanged throughout um
and
us
a sampling period by the ow ontroller. If the dieren e between
ua
in Eqs. 33-34 is large,
it auses the pressure applied to the valve stem to hange signi antly, and may potentially ause
allow the set-point
um
to be rea hed by
ua .
an
the pressure applied by the a tuation to saturate (e.g., hit its minimum bound) and therefore not In Eq. 38, this may be prevented through the a tuation
M
magnitude onstraint; in the ase when the empiri al sti tion model is used and details on the a tuation magnitude are not aptured by the empiri al model so that they an be onstrained,
d
utilizing the input rate of hange onstraints may be bene ial for preventing large hanges in
ua
from rea hing
um .
This emphasizes
te
that ould saturate the a tuation and therefore prevent
um
that be ause sti tion is a losed-loop ee t, onstraints added to an MPC to ompensate for valve
Ac ce p
behavior should be hosen with an understanding of the impa t of the ontroller and ontrol loop ar hite ture on the valve behavior.
Remark 6.
Prior works that have looked at MPC with a general obje tive fun tion with empiri al
models [Alanqar et al. (2017, 2015a,b)℄ have fo used on empiri al models of the nonlinear pro ess (rather than valves) and have indi ated that signi ant omputation time redu tions may result from using empiri al as opposed to rst-prin iples models in MPC. We will demonstrate in Se tion 5.4.3 that empiri ally modeling the valve layer an result in omputation time redu tion even if the pro ess is modeled with a rst-prin iples model be ause it an make the pro ess-valve ombination model less sti.
Remark 7.
One ould examine whether the MPC-based valve behavior ompensation method, par-
ti ularly with an empiri al valve layer model, ould be utilized for ompensating for dynami ee ts
33
Page 34 of 75
(e.g., slower movement of a valve or la k of movement) related to physi al valve issues like oversizing, undersizing, orrosion, leaks through the valve pa king, or diaphragm faults that an ae t valve performan e [Choudhury et al. (2008)℄.
Remark 8.
An appropriate empiri al model stru ture for the valve layer input-output data must be
ip t
sele ted. Many model identi ation te hniques for obtaining linear and nonlinear empiri al models exist that an be evaluated for their suitability for modeling a given valve layer input-output trend,
cr
whi h fall in the ategories of state-spa e and input-output models (see, for example, Ljung (1999); Billings (2013); Van Overs hee and De Moor (1996); Paduart et al. (2010)).
us
5.4. Valve Behavior Compensation Methods: Pro ess Examples
an
We now demonstrate the development and use of several of the valve behavior ompensation methods des ribed above for ontrol loops ontaining sti ky valves.
5.4.1. Level Control Example with a Sti ky Valve: Integral Term Modi ation
under the PI ontroller of Eqs. 24-25 for to rea h
qI .
0.15 m
0.20 m.
to
xˆ = h)
The expli it Euler method with a numeri al
L = 7, β = 0.007,
to
0.20 m).
and
tAW = 22880 s
(i.e.,
7280 s
after the
From omparison with Fig. 4, the addition of the term
L to the integral a tion was able to redu e ontrol loop os illations (though there is some
oset from the set-point for the hosen
ζ˙ = 0
0.5184 m
was used, and the results are shown in Fig. 9 (plotted every 100000
integration steps) for the ase that set-point hange from
to
te
10−5 s
0.15 m
Ac ce p
integration step size of
have
for a level set-point hange from
Subsequently, it was ontrolled using the ontroller of Eqs. 36-37 (with
for a level set-point hange from
ontaining
15600 s
d
0.15 m
M
The pro ess of Eq. 23 with the open-loop valve dynami s in Eqs. 27-32 was initially operated
when
h 6= hsp ).
After
tAW ,
L
before
tAW
the value of
um
be ause the integrator state in Eq. 37 an is able to hange again be ause
ζ˙
be omes
nonzero, and eventually the valve moves and the for es due to this strategy balan e in su h a way that the set-point is a hieved.
5.4.2. Level Control Example with a Sti ky Valve: MPC with a First-Prin iples Valve Layer Model For the level ontrol problem, we develop an MPC for sti tion ompensation as follows:
min
um (t)∈S(∆)
Z
tk+N
tk
˜ 2 + RT (ua,sp − u˜a )2 dτ QT (hsp − h)
34
(41a)
Page 35 of 75
q˜(tk ) = q(tk )
(41 )
0 ≤ u˜a (t) ≤ 0.006, ∀ t ∈ [tk , tk+N )
(41d)
0 ≤ um (t) ≤ 0.006, ∀ t ∈ [tk , tk+N )
(41e)
P˜ ≥ 0, ∀ t ∈ [tk , tk+N )
(41f )
ua,sp = 0.00373 m3 /s, hsp = 0.20 m, QT = 0.00001,
predi tions of
ua
ip t
(41b)
RT = 1. u˜a
and
and
˜ h
denote the
and of the level a
ording to Eq. 41b, respe tively. The notation of Eq. 41 follows
that in Eq. 38. The pro ess-valve state ve tor
q T = [h xv vv zf ]
is modeled for the open-loop valve
us
where
q˜˙(t) = fq (˜ q (t), um (t), 0)
cr
s.t.
using Eqs. 23 and 27-32. The pro ess-valve system was initiated at
150 s
with
∆=1s
and
N = 50.
The level was ontrolled by
An integration step of
an
the MPC of Eq. 41 for
qI .
within the MPC to integrate Eq. 41b, with an integration step of
10−6 s
10−5 s
was used
used outside of the MPC
M
to simulate the pro ess. The onstraints in Eqs. 41d and 41f were enfor ed on e every sampling period. The obje tive fun tion derivatives required by the optimization solver Ipopt were al ulated
d
using a entered nite dieren e, and the Ipopt limited-memory Hessian approximation option was
te
used, so that the non-dierentiability in Eq. 30 did not prevent a solution to the optimization problem from being obtained. The results in Fig. 5 (designated by C be ause the valve dynami s
Ac ce p
are ompensated) are plotted every 1000 integration steps and indi ate that the MPC in luding the valve dynami s drove the level toward its set-point, in ontrast to the MPC that did not in lude the a tuator dynami s (designated by U in this gure). The reason for this is that the MPC of Eq. 41 in orporates the full pro ess-valve model, and thus omputes an input traje tory that a
ounts for the manner in whi h the for es on the valve will balan e under the ontrol a tions al ulated by the MPC.
Though the MPC was able to drive the level toward its set-point (a signi ant improvement with regard to set-point tra king of the level ompared to the ase that the valve dynami s were not a
ounted for within the MPC), the fa t that the valve is operated without ow ontrol redu es the exibility of the MPC to be able to maintain the level at the set-point for all times after it rst approa hes the set-point in Fig. 5. Spe i ally, ea h time that the MPC sets
35
um ,
the pressure
Page 36 of 75
applied to the valve hanges a
ording to the relationship of Eq. 32 sin e the valve is operated without ow ontrol.
However, be ause the MPC implements pie ewise- onstant ontrol a tions
that are held for a sampling period, the pressure that is applied to the valve (a fun tion of
um
al ulated by the MPC) is held onstant throughout a sampling period. The length of the sampling
ip t
period in this example is long ompared to the dynami s of the valve, su h that the dynami s of the valve under a onstant applied pressure di tate the nal position of the valve at the end of a
um
that will drive the
cr
sampling period. The result is that the MPC is not able to nd a value of
valve, subje t to its dynami s during the sampling period that the pressure is held onstant and
us
the MPC annot intervene, exa tly to the valve position orresponding to the steady-state ow rate through the valve when the level set-point is a hieved. Instead, the MPC must ontinuously
um
that allow the valve to sti k and slip in ways that the MPC nds will
an
al ulate new values of
minimize the tra king obje tive fun tion and therefore keep the value of
h
in a region around the
M
set-point over time. Based on this analysis, potential ways of improving the set-point tra king of the level in lude de reasing the MPC sampling period until it is on a time s ale omparable to
d
the time s ale of the valve dynami s, in reasing the predi tion horizon to give the MPC greater
ua
te
foresight to potentially allow it to determine a sequen e of values of
um
that an drive the value of
to its set-point, or adding ow ontrol to the valve and then in luding the dynami s of both the
Ac ce p
valve and the ow ontroller in the MPC as it al ulates set-points
Remark 9.
um
for the ow ontroller.
Fig. 5 demonstrates the ee ts of not a
ounting for the behavior of the valve of Eqs. 27-
32 within an MPC (no signi ant hange of the level for ertain hanges in the valve output ow rate set-point
um )
and the improvement that an be obtained when the dynami s are a
ounted
for. However, other types of valve behavior that are not exhibited by the valve of Eqs. 27-32 an result in dierent types of negative ee ts when the valve behavior is not in luded within the MPC model for making state predi tions. For example, onsider a valve without sti tion but with the following equal per entage valve hara teristi [Coughanowr and LeBlan (2009)℄:
ua = ua,max eln(0.03)xv /xv,max
(42)
developed for the ase that the valve stem is fully retra ted when the valve is fully open and is fully
36
Page 37 of 75
extended when the valve is fully losed. Assume that the valve an be manipulated in su h a manner that the valve position is an expli it fun tion of
um
given by Eq. 31 (i.e.,
though this linear relationship does not re e t the nonlinear
um xv,max ) xv = xv,max − ua,max
xv −ua relationship of Eq. 42.
If an MPC
is used to ontrol the pro ess but is not aware of the mismat h between the valve behavior of Eq. 42
um − xv
relationship that sets the valve position (a similar on ept to the mismat h
between Eq. 32 and the a tual
ua − P
ip t
and the linear
relationship for the sti ky valve of Eqs. 27-31), permanent
ow rate of
0.00323 m3 /s,
The steady-state ow rate for a level of
0.15 m
orresponding to a steady-state
ua,sp = 0.00373 m3 /s.
orresponding to
orresponds to a fra tion
FI,A = 0.5379
of the
through the valve as shown in Fig. 10. For the equal per entage
an
0.006 m3 /s
maximum ow rate of
hsp = 0.20 m
to the set-point
0.15 m,
us
onsider again the set-point hange from an initial level of
For example,
cr
oset of the level from its set-point an result due to the plant-model mismat h.
valve of Eq. 42, the valve position asso iated with the steady-state ow rate for the initial level is a
XI,A = 0.1768
of its maximum. When the set-point of the level is hanged to
M
fra tion
0.20 m,
the
ow rate out of the valve should in rease to a hieve this (ideally it should rea h the fra tion of the
FD = 0.62113 shown Fig. 10).
For the equal per entage valve of Eq. 42, this ow
d
maximum ow rate
te
rate is a hieved at a fra tion of 0.1358 of the maximum stem position. The used to set the stem position based on
fra tion
the linear
um − xv
XD,C = 0.37887
relationship
however, is linear, so when the MPC requests that
relationship moves the valve stem to a position orresponding to a
Ac ce p
um = ua,sp ,
xv ,
um − xv
of the maximum stem position. However, when the fra tion of the stem
position for an equal per entage valve is 0.37887, the fra tion of the maximum ow through the valve is
FD,A = 0.2649
( orresponding to a ow rate lower than the initial value instead of above
it as desired). This example highlights that in luding valve behavior in MPC an be bene ial for many types of valve behavior. Also, the example valve in this se tion has a the form in Eq. 2 sin e the
ua − xv
xv
in the linear
um − xv
um − ua
relationship of
relationship an be substituted in terms of
um
in
relationship of Eq. 42.
5.4.3. Ethylene Oxidation Example with a Sti ky Valve: MPC with an Empiri al Valve Layer Model In this se tion, we return to the ethylene oxidation example of Se tion 4.2 for the ase that
ua,2 = 0.5
and
ua,3 = 1.0
(the values of the other parameters are listed in Table 2). We onsider
37
Page 38 of 75
that the value of
ua,1
is adjusted by a spring-diaphragm sliding-stem globe valve (be ause it is the
only manipulated input onsidered for this pro ess in the following example, we will drop the 1 in the subs ript for this se tion and refer to the pro ess input as ua ). This valve is under ow
ontrol and has the same design as the sti ky valve des ribed in Se tion 4.1 (i.e., it is pneumati ally
open position, it has
xv = 0 m
u a = 0,
xv = xv,max = 0.1016 m
when the
and the valve layer dynami s are des ribed by Eqs. 27-31 and
cr
valve is fully losed with
when the valve is fully open and
ip t
a tuated and pressure-to- lose, with no pressure applied to the valve initially when it is in its fully
33-34) ex ept that the time unit for all parameters and variables is denoted by a dimensionless unit instead of
s for onsisten y with the dimensionless units in Eq. 35 (i.e., all valve model parameter
us
td
values in Table 1 apply for this valve ex ept that ea h instan e of the unit
ζP
and of the steady-state value of the pressure
(ζP is re-set to zero, and
Ps
Ps
ua = ua,max = 0.7042.
are re-set ea h time that
is set to the last applied pressure).
M
values of
an
with td in this example) and the fully open valve position orresponds to
s in that table is repla ed
The value of
um
um
The
hanges
is hanged by
an EMPC (an MPC with an e onomi s-based obje tive fun tion) every sampling period of length
d
∆ = 0.2 td .
te
The ontrol obje tive is to maximize the yield of ethylene oxide utilizing an MPC that a
ounts for the valve dynami s, where the yield is given by the following ratio of the amount of ethylene
Ac ce p
oxide produ ed from the rea tor in a time period of length tf the rea tor in that time:
R tf t0
Y (tf ) =
− t0
to the amount of ethylene fed to
ua (τ )x3 (τ )x4 (τ )dτ R tf 0.5ua(τ )dτ t0
(43)
We also onsider that the valve output ow rate is onstrained between the minimum ow through the valve (0) and the maximum ow (0.7042), and is also required to satisfy the following restri tion on the amount of ethylene that an be fed in a time period of length
1 tf − t0
Z
tf − t0 :
tf
0.5ua (τ )dτ = 0.175
(44)
t0
This onstraint requires that the amount of ethylene fed to the pro ess in a time period of length
tf − t0
must equal the amount that would be fed in that time period under steady-state operation.
38
Page 39 of 75
We seek to avoid fully losing the valve by requiring the MPC to keep the value of
um
between
0.0704 and 0.7042. To a hieve the ontrol obje tives, we will utilize an MPC with an empiri al model of the valve dynami s, and we will ompare the omputation time of that ontroller with the omputation time
ip t
of an MPC that in ludes a rst-prin iples model of the valve dynami s. We develop the empiri al model for the valve layer des ribed in Eqs. 27-31 and 33-34 a
ording to the steps outlined in data, and noti e that when the set-points
cr
um − ua
Se tion 5.3. A
ording to Step 1, we rst gather
repeatedly hange in the same dire tion, the valve responds rapidly to the set-point hange, but
us
when the set-point hange dire tion reverses, there is a delay before the valve responds. Also, there is a greater delay for small set-point hanges than for large set-point hanges when the deadband is
an
en ountered due to the use of the PI ontroller in the valve layer. In addition, the valve layer inputoutput data indi ates that when the valve output set-point is kept onstant for multiple sampling
M
periods, the valve output will not exhibit deadband if the next hange in the set-point is in the same dire tion as the hanges prior to the valve set-point remaining onstant, but will exhibit deadband if
d
the next hange in the set-point is in the opposite dire tion to the last hanges. The valve layer data
te
also suggests that valve output ow rates above about 0.5164 are not a hievable with the pressure available from the pneumati a tuation after the valve rst begins to lose be ause sti tion alters
ua − P
relationship su h that these ow rates would require negative pressures to be rea hed
Ac ce p
the
(i.e., sin e the valve is initialized with open more) until
ua ∼ 0.5164,
ua = 0.7042,
it an only lose (it annot reverse dire tion to
and subsequently annot rea h ow rates above that value).
The above observations are used in Step 2 of the model identi ation pro edure to postulate that the dynami s between
ua
and
um
an be aptured in a pie ewise-de ned model with two bran hes,
one orresponding to the response of
ua when the set-point hanges in um are repeatedly in the same
dire tion (no deadband), and another orresponding to the response when the set-point hanges swit h dire tion (deadband), with a spe ial onsideration for the ase that the set-point does not
hange between two sampling periods. The part of the model orresponding to the ase when there is deadband before the valve moves should have dierent speeds of response of the valve for dierent set-point hange magnitudes. The valve layer input-output data should be gathered while avoiding
39
Page 40 of 75
in reasing the set-point
um
above 0.5164 to avoid gathering data for ow rates where the pressure
is saturated (the de ision was made not to add bran hes to the empiri al model to a
ount for saturation of the a tuation pressure due to the omplexity that this adds to the empiri al model, but to instead seek to avoid saturating the pressure during pro ess operation by utilizing the input
ip t
rate of hange onstraints of Eq. 40 in the MPC used to ontrol the pro ess). Step 3 of the model identi ation pro edure will now be arried out to identify the equations for
cr
the two bran hes of the proposed model. We rst verify that su h a pie ewise-de ned valve layer model is ne essary by showing the results of attempting to identify a single model for the valve
by initializing the valve at its fully open position (ua
= 0.7042, Ps = 0 kg/m · t2d , ζP = 0, zf = 0 m,
and integrating the rst-prin iples valve layer model in Eqs. 27-31 and
an
xv = 0 m, vv = 0 m/td )
us
layer based on the valve layer input-output data. The valve layer input-output data was gathered
33-34 with the expli it Euler numeri al integration method and an integration step of
M
for 19 step hanges in the set-point (the set-point was rst de reased from
hI = 10−6 td
um = 0.7042
to 0.7, and
was subsequently de reased to 0.15 in in rements of 0.05, and then in reased to 0.5 in in rements
d
of 0.05, with ea h set-point held for a sampling period). A subset of the
um − ua
data generated
te
is shown in Fig. 11. Based on the data generated, the valve output response to a set-point hange was postulated to be able to be des ribed by a se ond-order linear dynami model. The values of and
ua
were measured every
10−4
Ac ce p
um
time units (every 100 integration steps; i.e.,
∆e = 10−4 td ),
and the following ARX model was t to the data by using a least-squares regression:
y(t˜j ) = 1.99212y(t˜j−1) − 0.99219y(t˜j−2) + 0.00035um(t˜j−1 ) − 0.00027um(t˜j−2 ) where
y(t˜j )
refers to the predi ted value of
data (i.e., at time
t˜j ).
ua
When the predi tions
they overshoot the values of
for the
y
j − th
(45)
measurement of the valve layer output
are generated from this model and the input data,
ua , and there is poor agreement with ua when the valve velo ity hanges
sign (deadband is rea hed), as shown in Fig. 11. Though it was not possible to identify an adequate se ond-order model using the input-output data for the entire set of 19 set-point hanges, it is possible to su
essfully identify a se ond-order model if only the data orresponding to the set-point de reases between 0.6 and 0.15, for whi h no
40
Page 41 of 75
deadband o
urs, is used to identify the model. In this ase, the following model is obtained:
y(t˜j ) = 1.96209y(t˜j−1) − 0.96249y(t˜j−2) + 0.00038um(t˜j−1) + 0.00002um(t˜j−2 )
(46)
When the de reasing set-points between 0.6 and 0.15 are used as inputs in Eq. 46, the predi tions of the valve output losely mat h the a tual values, as shown in Fig. 12.
ip t
y
To omplete Step 3 of the empiri al modeling pro edure, it ne essary to omplement Eq. 46
Fig. 11 orresponding to the deadband when
um
cr
with a model for the ase that deadband is observed. Based on the valve layer input-output data in
hanges from 0.15 to 0.2, it is postulated that the
us
response of the valve output to set-point hange dire tion reversals an be modeled as a rst-order pro ess with time delay. However, the values of the time onstant
τ
and of the delay
α
in su h a
of the valve layer to a set-point hange in
um
an
model are dependent on the magnitude of the set-point hanges be ause the speed of the response depends on the magnitude of the set-point hange.
M
Closed-loop simulations indi ate that for a set-point hange dire tion reversal, set-point hanges less than approximately 0.02 are unable to ause the PI ontroller to over ome the deadband within
d
a sampling period. To determine the dependen e of the delay on the magnitude of the set-point
um
was de reased from
te
hange, in a
ordan e with Step 4 of the model identi ation pro edure, 0.7042 to 0.15, and subsequently
um
was in reased by set-point hanges of dierent magnitudes.
Ac ce p
The regression method in Ogunnaike and Ray (1994) for the determination of the parameters of a rst-order-plus-dead-time model was applied to the data generated for ea h set-point hange. A plot of the resulting delays against the set-point hanges with whi h they were asso iated was t to the fun tion
a/x
using the MATLAB fun tion lsq urve t, with
τ
asso iated with ea h delay were averaged to give
t. The values of
a = 0.0037 τ = 0.0123
providing the best for the rst-order-
plus-dead-time model. Thus, the rst-order-plus-dead-time model is written in dis rete-time form as:
y(t˜j ) =
y(t˜j−1), t˜j − tk < α y(tk ) + exp(−∆e /τ )(y(t˜j−1 ) − y(tk )) + K(1 − exp(−∆e /τ )) × (um (tk ) − um (tk−1 )), t˜j − tk ≥ α (47)
41
Page 42 of 75
where
α= In Eqs. 47-48,
k
∆, |um (tk ) − um (tk−1 )| < 0.02 a/(|um(tk ) − um (tk−1 )|), |um (tk ) − um (tk−1 )| ≥ 0.02
is the value of
k
that brings
tk
losest to
t˜j (tk ≤ t˜j ),
and
(48)
K = 1.
In orporating the above onsiderations, the following dis rete-time empiri al valve layer model
ip t
was devised and validated to perform well for a number of valve layer input-output data points,
ompleting Step 5 of the model identi ation pro edure:
tk−2 ,
set
y(t) = y(tk )
and
tk−1
t ∈ [tk , tk+1).
for all
and also did not hange between
cr
and
tk
tk−1
us
1. If the set-point has not hanged between
2. If two set-point hanges are hanging in the same dire tion, or if the set-point has been
an
onstant for some time but has now hanged in the same dire tion that it was hanging prior to be oming onstant, use the model of Eq. 46.
M
3. If two set-point hanges are in opposite dire tions, or if the set-point has been onstant for some time but has now hanged in the opposite dire tion to that in whi h it was hanging
d
prior to be oming onstant, use the model of Eqs. 47-48.
Z
tk+Nk
Ac ce p
above is as follows:
te
The MPC-based sti tion ompensation strategy in orporating the empiri al model des ribed
min
um (t)∈S(∆)
s.t.
tk
−y(τ )˜ x3 (τ )˜ x4 (τ ) dτ
(49a)
x˜˙ (t) = f (˜ x(t), y(t), 0)
(49b)
y(t) ˙ = fy (y, um)
(49 )
x˜(tk ) = x(tk )
(49d)
y(t˜0 ) = ua (t0 )
(49e)
0.0704 ≤ um (t) ≤ 0.7042, ∀ t ∈ [tk , tk+Nk )
(49f )
0 ≤ y(t) ≤ 0.7042, ∀ t ∈ [tk , tk+Nk )
(49g)
|um (tk ) − u∗m (tk−1 |tk−1 )| ≤ 0.1
(49h)
|um (tj ) − um (tj−1 )| ≤ 0.1, j = k + 1, ..., k + Nk − 1
42
(49i)
Page 43 of 75
Z
tk+Nk
y(τ )dτ +
tk
Z
tk (p−1)tp
u∗a (τ )dτ = 0.175tp /0.5
where the notation follows that in Eqs. 11 and 38. The notation
signi es the value of
that was determined to be optimal at the prior sampling time and was applied to the pro ess
for the sampling period between
tk−1
and
tk .
Minimization of the obje tive fun tion in Eq. 49a
ip t
um
u∗m (tk−1 |tk−1)
(49j)
maximizes the yield of ethylene oxide when the amount of rea tant fed to the pro ess over the operating period of length
signi es a value of
ua
tp = 1 td
meets the onstraint in Eq. 49j (the notation
u∗a (t)
cr
p − th
that was applied to the pro ess at a past time t). Enfor ing the onstraint of
us
Eq. 49j ensures that Eq. 44 is satis ed by the nal time tf of operation. Two operating periods were simulated under this EMPC; though a longer simulation may redu e the ee ts from the transient
an
on the results, the two operating periods simulated are su ient for demonstrating that an empiri al model of the valve dynami s an readily be used in pla e of a rst-prin iples model in the MPC
Nk
was used in ea h operating
M
for valve behavior ompensation. A shrinking predi tion horizon
period with an initial length of 5 at the beginning of ea h operating period. At ea h subsequent
d
sampling time, the predi tion horizon was de reased by 1. The pro ess model of Eq. 49b (whi h is
ua,1
as mentioned above) is integrated using the expli it
te
the model of Eq. 35 with the single input
Euler numeri al integration method with an integration step size of
Ac ce p
predi tions.
Eq. 49 signi es that the predi tions developed in this se tion.
y
of
ua
hemp = 10−4 td
for making state
in the EMPC ome from the empiri al model
Eq. 49 is written in ontinuous-time form for onsisten y with the
remainder of the manus ript in whi h a ontinuous-time model of the pro ess-valve dynami s is utilized (in the sense that dierential equations are onsidered to onstitute the pro ess model, though their solution must be obtained through numeri al dis retization be ause no analyti solution is available), though the results presented for this example ome from utilizing the dis rete-time empiri al valve layer model developed in this se tion.
Be ause both the numeri ally dis retized
(with expli it Euler) ontinuous-time pro ess dynami model and the empiri al valve layer model evolve every
10−4 td
when state predi tions are made within the EMPC (i.e.,
∆e = hemp ),
the
dis rete-time nature of the empiri al model poses no issues for ombining it with the ontinuous-
43
Page 44 of 75
time pro ess model for making state predi tions. In the simulations, the value of with a state measurement of
ua
y
was not updated
at ea h sampling time but instead evolved in an open-loop fashion
(the notation in Eq. 49e signi es that the initial data required for simulating the valve layer based on the empiri al model (i.e.,
y(t˜0 ) and y(t˜−1)) ua ).
The state onstraint in Eq. 49g was enfor ed every integration
ip t
for all times without feedba k of
are known and used to integrate the empiri al model
step. The input rate of hange onstraints in Eqs. 49h-49i are added to redu e the likelihood that
a tuation to be ome saturated at zero.
cr
the EMPC will request unrea hable ow rates that would ause the pressure from the pneumati The optimization problems were solved using the open-
10−10 . and
y
initiated from
an
ua , um ,
Fig. 13 shows the traje tories of
us
sour e interior-point optimization solver Ipopt [Wä hter and Biegler (2006)℄ with a toleran e of
[x1 x2 x3 x4 xv vv zf ζP ] =
[0.997 1.264 0.209 1.004 0.051 m 2.000 × 10−6 m/td 1.426 × 10−5 m 0]
resulting from the use of the
M
empiri al EMPC. The empiri al model was su
essfully able to apture the behavior of
ua ,
and the
EMPC al ulated set-points that the valve layer ould tra k. Fig. 14 shows the pressure applied to
d
the valve throughout this losed-loop simulation, whi h never saturated at zero with the help of the
EMPC.
te
input rate of hange onstraints. Fig. 15 shows the losed-loop pro ess states under the empiri al
Ac ce p
In addition to al ulating rea hable set-points and preventing pressure saturation in the two operating periods simulated, the MPC-based valve behavior ompensation strategy with an empiri al model was also able to ensure that the integral material onstraint was not signi antly violated. In the rst operating period, the empiri al EMPC used only 0.02% less material than required by the material onstraint, and in the se ond operating period only 0.05% less. A omparison of a simulation of the form of Eq. 49 but with the rst-prin iples valve layer model of Eqs. 27-31 and 33-34 in pla e of the empiri al model of Eqs. 46-48 was formulated, in whi h the rst-prin iples model was simulated with an integration step of
10−5 td
within the MPC,
and state feedba k of the pro ess-valve states was obtained at ea h sampling time. This integration step size is smaller than for the simulation with the empiri al EMPC be ause the rst-prin iples model of Eqs. 27-31 and 33-34 annot be integrated with a step size of
44
10−4 td
using expli it Euler
Page 45 of 75
due to numeri al stability issues (i.e., the simulation does not produ e numeri al results within a reasonable range, whi h o
urred for some smaller integration step sizes as well). The onstraint of Eq. 49g was enfor ed on
ua
every 10 integration steps so that it was enfor ed every
10−4 td
as
for the empiri al EMPC. The nite dieren e approximation used for the gradients of the obje tive
ip t
fun tion and onstraints used a perturbation one order of magnitude smaller than in the empiri al EMPC. The resulting simulation of two operating periods took approximately three times longer to
10−4 td .
Though the
cr
solve than the MPC of Eq. 49 where the integration step within the MPC was
dieren e in omputation time depends on a large number of fa tors su h as the ode used and the
us
integration step size, it is signi ant that the empiri al model is less sti than the rst-prin iples model. 10
.
The omputation time omparison is not meant to be an exa t numeri al omparison
an
Remark
be ause it is di ult to determine the exa t minimum step size that an be utilized within the rst-
M
prin iples EMPC for omparison with the omputation time of the empiri al EMPC. Therefore, we do not fo us on providing exa t omputation time results, but instead highlight the dieren e
d
in stiness of the models, whi h is demonstrated in that an integration step size smaller than that
te
of the empiri al model must be utilized to integrate the rst-prin iples model. In general, a less
Ac ce p
sti model is onsidered to have the potential to be more omputationally e ient, and through the example, we show that empiri al modeling of a valve layer has the potential to develop a more
omputationally tra table model of the valve layer than would be obtained from rst-prin iples. We do not onsider impli it numeri al integration methods instead of expli it methods be ause they are harder to implement. MPC's need to be solved on-line within a sampling period, and the numeri al integration is performed many times to simulate the pro ess throughout the predi tion horizon at every iteration of the numeri al optimization method. Therefore, a simple implementation of numeri al integration is desirable.
6. Perspe tives on Valve Nonlinearity Compensation A on lusion of the results in this work is that an MPC design that utilizes models of both the pro ess and valve behavior for making state predi tions provides a systemati method for driving
45
Page 46 of 75
an output to its set-point that an a
ount for multivariable intera tions in a pro ess-valve dynami system and onstraints su h as valve output and a tuation magnitude saturation that an lead to undesirable losed-loop behavior. This method is not restri ted to linear plant dynami s, and it does not require tuning of ompensator-spe i parameters that are not learly tied to the pro ess output
ip t
responses as do some of the sti tion ompensation methods dis ussed above su h as ow ontrol, the integral term modi ation method, and kno ker-type methods. The bene ts of an MPC in luding
cr
valve dynami s for improving the issues ommonly observed due to valve behavior indi ate that it may be bene ial for industry to onsider wider use of MPC due to analysis not only of whether the
us
hemi al pro ess itself would bene t from being ontrolled by an MPC (whi h is the typi al analysis performed), but also of whether it might provide better valve behavior ompensation in the long
an
run (undesirable behavior like valve sti tion an develop over time) that may redu e e ien y and pro t in the long-term and therefore make MPC a more attra tive option than lassi al regulatory Thus, more analysis of the impa t of a tuator dynami s at the initial design
M
ontrol designs.
phase may allow for better ontroller designs to be hosen that an handle hanges in the a tuator
d
dynami s that often plague pro esses at a later phase and are less straightforward to handle when
te
non-model-based ontrol strategies are attempted to be used to handle nonlinear valve behavior. Though the MPC-based valve behavior ompensation method was shown in the pro ess examples
Ac ce p
in this work to be bene ial at ompensating for valve behavior, it was also shown that it has limitations in handling valve nonlinearities. For example, in Se tion 5.4.2, it was noted that a valve without ow ontrol under MPC a
ounting for valve sti tion may not be able to keep a pro ess output at its set-point for all times when the MPC sampling period is long ompared to the time s ale of the valve dynami s su h that the MPC is not able to regularly adjust the for e applied by the valve a tuation throughout a sampling period. An alternative to this is to use a ow ontroller for the valve or a small sampling period for the MPC to allow the for e from the valve a tuation to be adjusted frequently as the valve position hanges a
ording to its dynami s to try to drive it to the position orresponding to the valve output set-point. However, this may ause signi ant variations in the valve a tuation during the time that the valve position is being adjusted, whi h may in rease a tuator wear [Ivan and Lakshminarayanan (2009)℄. This indi ates that an MPC-based
46
Page 47 of 75
valve behavior ompensation method must seek to balan e a tuator wear and set-point oset for
ertain ontrol ar hite tures and valve nonlinearities through appropriate onstraints and design of parameters su h as the sampling period. Another on lusion of this work is that be ause the ee ts observed in sti ky ontrol loops are losed-loop ee ts, hanging the ontrol design of a system may
ip t
result in dierent pro ess output responses in a ontrol loop in whi h valve dynami s annot be negle ted. This is important to onsider as ontrollers at a plant are re-tuned or as upgrades are
cr
made to the ontrol design.
Finally, the losed-loop perspe tive on valve behavior developed in this work an impa t the
us
sti tion dete tion and quanti ation literature. It gives greater insight into the bene ts and limitations of the dete tion/quanti ation methods for sti tion in the literature, whi h are reviewed
an
in Brásio et al. (2014) and in lude shape-based methods and model identi ation-based methods. Many of the shape-based methods (e.g., Hor h (1999); Choudhury et al. (2006); Singhal and Sals-
M
bury (2005); Srinivasan et al. (2005a)) assume that a spe i pattern exists in the data from the measured outputs of the system (pro ess outputs or valve outputs), often in relation to the onIt has been highlighted that the pro ess and ontroller dynami s will ae t the
d
troller outputs.
te
patterns and thus may redu e the ee tiveness of shape-based methods (e.g., He et al. (2007) notes that the sti tion dete tion method in Hor h (1999) may give dierent results depending on the
Ac ce p
ontroller tuning, and Choudhury et al. (2006) and Jelali and Huang (2010) also note that the pattern-based methods are not always ee tive be ause patterns depend on the ontroller, pro ess, and valve dynami s). The present manus ript gives a general mathemati al framework for analyzing the di ulties noted with pattern-based methods through a pro ess-valve dynami model. It also gives greater insight into the onditions under whi h the assumption that os illations are o
urring in a pro ess output due to sti tion may not hold (e.g., when the ontroller, pro ess, and valve dynami s produ e the un ompensated ase in Fig. 5). Multiple model identi ation-based sti tion dete tion/quanti ation methods (see, for example, Srinivasan et al. (2005b); Jelali and Huang (2010); Jelali (2008)) assume that the pro ess an be des ribed by a linear model, whi h may be a limiting assumption espe ially as the requirement of steady-state operation is being hallenged by the re ent developments in EMPC [Ellis et al. (2014a)℄.
47
Page 48 of 75
The primary goal of sti tion dete tion methods is to identify problemati valve behavior so that maintenan e an be performed on a valve, and quanti ation methods are intended to be used to prioritize valve maintenan e based on whi h valves are most sti ky. The empiri al modeling strategy in this work ould be onsidered as a valve behavior dete tion/quanti ation strategy that is not
i = 1, . . . , m,
um,i − ua,i
relationship ould be developed for every valve if
data is available. The dieren e between
um,i
and
ua,i
ould then be tra ked over
ua,i
um,i
deviates most signi antly from
The valves
cr
time, and when it be omes signi ant, the valve ould be agged for maintenan e. for whi h
um,i − ua,i ,
ip t
limited to sti tion. The
ould be given priority in the maintenan e
us
s hedule. Though measurements of ow through a valve (ua,i ) are not always available in industrial appli ations when the ontrol loop is not a ow ontrol loop [Thornhill and Hor h (2007)℄, this
an
analysis indi ates that new instrumentation to provide measurements of pro ess variables su h as ow (when it is not already measured) may be bene ial long-term for dete ting and ompensating
M
for valve behavior by allowing empiri al models to be developed for a pro ess-valve system when it may be di ult to obtain a pro ess-valve model without the ow measurement (Se tion 5.3).
d
A nal observation is that many ontributions to the sti tion literature have fo used on sti tion
te
as the nonlinearity in the pro ess-valve system (i.e., many works examine linear pro esses and linear
ontrollers); the results of this work indi ate that nonlinear pro esses, espe ially with multiple inputs
Ac ce p
all ae ted by nonlinear valve behavior, may be parti ularly interesting to onsider in future works on sti tion dete tion, quanti ation, and ompensation, due to the multivariable intera tions of the pro ess-valve states, whi h may, as noted above, best be handled with multiple-input/multipleoutput nonlinear ontrol designs.
7. Con lusions
In this work, we analyzed the roles of the pro ess, valve, and ontroller dynami s, and also the ontrol loop ar hite ture, in the losed-loop response of a pro ess-valve dynami system. The
losed-loop perspe tive dis ussed allows a variety of valve behaviors (e.g., linear valve dynami s and sti tion) to be analyzed within a single framework. It was demonstrated to be useful for explaining how a number of sti tion ompensation methods from the literature seek to ompensate for sti tion
48
Page 49 of 75
at a fundamental mathemati al level, and also was shown to be bene ial for developing new valve behavior ompensation te hniques su h as an integral term modi ation sti tion ompensation method and an MPC design in orporating a dynami model ( rst-prin iples or empiri al) of the full pro ess-valve system. A level ontrol example and a ontinuous stirred tank rea tor were used
ip t
to demonstrate the on epts dis ussed throughout the manus ript.
cr
8. A knowledgements
Finan ial support from the National S ien e Foundation and the Department of Energy is gra-
Ac ce p
te
d
M
an
us
tefully a knowledged.
49
Page 50 of 75
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Jelali, M., Huang, B. (Eds.), 2010. Dete tion and Diagnosis of Sti tion in Control Loops: State of the Art and Advan ed Methods. Springer-Verlag, London, England.
Kano, M., Maruta, H., Kugemoto, H., Shimizu, K., 2004. Pra ti al model and dete tion algorithm for valve sti tion, in: Pro eedings of the IFAC Symposium on Dynami s and Control of Pro ess Systems, Cambridge, Massa husetts. pp. 859864.
Ljung, L., 1999.
System Identi ation: Theory for the User.
Prenti e Hall PTR, Upper Saddle
River, New Jersey.
Mohammad, M.A., Huang, B., 2012.
Compensation of ontrol valve sti tion through ontroller
tuning. Journal of Pro ess Control 22, 18001819.
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Munaro, C.J., de Castro, G.B., da Silva, F.A., Angarita, O.F.B., Cypriano, M.V.G., 2016. Redu ing wear of sti ky pneumati ontrol valves using ompensation pulses with variable amplitude, in: Pro eedings of the 11th IFAC Symposium on Dynami s and Control of Pro ess Systems, in luding Biosystems, Trondheim, Norway. pp. 389393.
ip t
Ogunnaike, B.A., Ray, W.H., 1994. Pro ess Dynami s, Modeling, and Control. Oxford University
cr
Press, New York, New York.
Özgül³en, F., Adomaitis, R.A., Çinar, A., 1992.
A numeri al method for determining optimal
us
parameter values in for ed periodi operation. Chemi al Engineering S ien e 47, 605613.
an
Paduart, J., Lauwers, L., Swevers, J., Smolders, K., S houkens, J., Pintelon, R., 2010. Identi ation of nonlinear systems using polynomial nonlinear state spa e models. Automati a 46, 647656.
M
Qin, S.J., Badgwell, T.A., 2003. A survey of industrial model predi tive ontrol te hnology. Control Engineering Pra ti e 11, 733764.
d
Rawlings, J.B., Angeli, D., Bates, C.N., 2012. Fundamentals of e onomi model predi tive ontrol,
te
in: Pro eedings of the 51st IEEE Conferen e on De ision and Control, Maui, Hawaii. pp. 3851
Ac ce p
3861.
Riggs, J.B., 1999. Chemi al Pro ess Control. Ferret Publishing, Lubbo k, Texas.
Singhal, A., Salsbury, T.I., 2005. A simple method for dete ting valve sti tion in os illating ontrol loops. Journal of Pro ess Control 15, 371382.
Srinivasan, R., Rengaswamy, R., 2005. Sti tion ompensation in pro ess ontrol loops: A framework for integrating sti tion measure and ompensation. Industrial & Engineering Chemistry Resear h 44, 91649174.
Srinivasan, R., Rengaswamy, R., 2008. Approa hes for e ient sti tion ompensation in pro ess
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Srinivasan, R., Rengaswamy, R., Miller, R., 2005a. qualitative approa h for sti tion diagnosis.
Control loop performan e assessment. 1. A
Industrial & Engineering Chemistry Resear h 44,
67086718.
Chemistry Resear h 44, 67196728.
Industrial & Engineering
cr
assessment. 2. Hammerstein model approa h for sti tion diagnosis.
Control loop performan e
ip t
Srinivasan, R., Rengaswamy, R., Narasimhan, S., Miller, R., 2005b.
Thornhill, N.F., Hor h, A., 2007. Advan es and new dire tions in plant-wide disturban e dete tion
us
and diagnosis. Control Engineering Pra ti e 15, 11961206.
Subspa e identi ation for linear systems:
Theory-
an
Van Overs hee, P., De Moor, B.L., 1996.
Implementation-Appli ations. Kluwer A ademi Publishers, Norwell, Massa husetts.
On the implementation of an interior-point lter line-sear h
M
Wä hter, A., Biegler, L.T., 2006.
algorithm for large-s ale nonlinear programming. Mathemati al Programming 106, 2557.
d
Wang, J., 2013. Closed-loop ompensation method for os illations aused by ontrol valve sti tion.
te
Industrial & Engineering Chemistry Resear h 52, 1300613019.
Ac ce p
Zabiri, H., Samyudia, Y., 2006. A hybrid formulation and design of model predi tive ontrol for systems under a tuator saturation and ba klash. Journal of Pro ess Control 16, 693709.
54
Page 55 of 75
List of Figures S hemati showing the general ontrol ar hite ture under onsideration in this work. A ontroller re eives measurements of the pro ess state mation as appropriate, so that it an al ulate set-points for
m
m
x,
as well as set-point infor-
individual valve output ow rate
individual valves. The valves themselves have dynami s asso iated
with developing a value of
ua,i , i = 1, . . . , m,
um,i , i = 1, . . . , m. x. . . . . . . . . . .
for a given
outputs of the valves a t on the pro ess to alter the state 2
S hemati depi ting the tank onsidered in the level ontrol example.
3
Closed-loop traje tory of level
h
with referen e to its set-point
The . . .
ip t
1
hsp
. . . . . . . .
56 57
for the pro ess of
Eq. 23 under the PI ontroller of Eqs. 24-25 (top plot) and under the MPC of Eq. 26 Closed-loop traje tories of
h, ua ,
. . . . . . . . . . . . . . . . . . . . . . .
um
and
cr
(bottom plot) with no a tuator dynami s. 4
58
for the pro ess of Eq. 23 under the PI
ontroller of Eqs. 24-25 with the valve dynami s in Eqs. 27-32. This data is plotted
ua,min and ua,max are plotted to indi ate
us
every 100000 integration steps. The values of
the range of allowable ow rates through the valve. 5
Plot of traje tories of
h, ua , um ,
. . . . . . . . . . . . . . . . . .
59
and their set-points for the MPC of Eq. 26 (U,
an
signifying un ompensated) and the MPC of Eq. 41 (C, signifying ompensated) applied to the nonlinear pro ess of Eqs. 23 and 27-32 for a level set-point hange from
0.15 m
to
0.20 m.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ontrol.
In the inner loop, the ow ontroller omputes the pressure
pneumati a tuation, whi h then alters the position
xv
ua
from the
The valve position
xv
is
through the valve a
ording to the linear valve hara teristi
d
related to the ow
P
of the sti ky valve a
ording
to the valve dynami s (labeled as Valve in the gure). of Eq. 31 (labeled as Flow in the gure). Closed-loop traje tories of
h, ua ,
and
te
7
60
S hemati depi ting the ontrol ar hite ture for the level ontrol example with ow
M
6
. . . . . . . . . . . . . . . . . . . . . . .
um
61
for the pro ess of Eq. 23 under the PI
ontroller of Eqs. 24-25 with the valve in Eqs. 27-31 and the PI ontroller of Eqs. 33-
Ac ce p
34 used to ontrol the valve ow rate to its set-point value for the set-point hange from 8
0.15 m
to
0.20 m.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Standard ontroller input-valve output relationship reported for a sti ky valve. The
ontrol signal to the valve hanges but the valve output does not hange appre iably in the regions of deadband and sti kband. The valve output hanges qui kly in the region of slip-jump, and the valve output and ontrol signal are linearly related in the moving phase region of the response.
9
of Eqs. 36-37 with
hsp = 0.20 m. 10
11
. . . . . . . . . . . . . . . . . . . . . . . .
63
h, ua , and um for the pro ess of Eq. 23 under the ontroller L = 7 and β = 0.007, with the open-loop valve (Eqs. 27-32), and
Closed-loop traje tories of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Figure depi ting linear (L) and equal per entage (EP) valve hara teristi s of Eqs. 32 and 42, respe tively, along with several fra tions of the maximum ow rate (FI,A = 0.5379, FD = 0.62113, and FD,A = 0.2649) and of the maximum stem position (XI,A = 0.1768 and XD,C = 0.37887) for the level ontrol example. . . . . . . . . . . Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) from Eq. 45 when 19 set-point hanges are applied (a
65
subset of the data is shown). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
55
Page 56 of 75
12
Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) using the set-points de reasing between 0.6 and 0.15 and Eq. 46 (y overlays
13
ua ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) under the EMPC using an empiri al valve layer model (y almost overlays
ua ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Pressure applied to the valve when the EMPC using an empiri al valve layer model is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
15
Closed-loop pro ess states under the EMPC using an empiri al valve layer model.
70
.
Ac ce p
te
d
M
an
us
cr
ip t
14
56
Page 57 of 75
ip t Valve 1
Controller
ua,2
an
um,2
ua,1
us
um,1
cr
Set-points
Valve 2 . . .
Valve m
. . .
Process
ua,m
Ac ce p
te
d
um,m
M
. . .
x
Figure 1: S hemati showing the general ontrol ar hite ture under onsideration in this work. A ontroller re eives measurements of the pro ess state
x,
as well as set-point information as appropriate, so that it an al ulate
individual valve output ow rate set-points for
ua,i , i = 1, . . . , m, x.
with developing a value of pro ess to alter the state
m
m
individual valves. The valves themselves have dynami s asso iated
for a given
um,i , i = 1, . . . , m.
57
The outputs of the valves a t on the
Page 58 of 75
ip t M
an
us
cr
ua
te
d
h
√
Ac ce p
c1 h
A
Figure 2: S hemati depi ting the tank onsidered in the level ontrol example.
58
Page 59 of 75
ip t cr us
0.5 0.4 0.3
an
h (m), PI
0.6
0.2 0.1 500
1000
1500
2000
M
0
h hsp
d
0.2
te
0.19 0.18 0.17 0.16
Ac ce p
h (m), MPC
Time (s)
h hsp
0.15
0
500
Figure 3: Closed-loop traje tory of level
h
1000
1500
2000
2500
Time (s)
with referen e to its set-point
hsp
for the pro ess of Eq. 23 under the PI
ontroller of Eqs. 24-25 (top plot) and under the MPC of Eq. 26 (bottom plot) with no a tuator dynami s.
59
Page 60 of 75
ip t cr us
hsp 0.2 0.15
an
h (m)
h
0.25
0.1 0.5
x 10
ua,max
2
2.5
3 4
x 10
te
4 2
Ac ce p
u (m /s)
6
1.5
M
−3
3
1 ua um ua,min
d
0
0
0
0.5
Figure 4: Closed-loop traje tories of
1
h, u a ,
and
1.5
2
Time (s) um
2.5
3 4
x 10
for the pro ess of Eq. 23 under the PI ontroller of Eqs. 24-25
with the valve dynami s in Eqs. 27-32. This data is plotted every 100000 integration steps. The values of
ua,max
ua,min
and
are plotted to indi ate the range of allowable ow rates through the valve.
60
Page 61 of 75
ip t cr
0.1 h (C) h (U) hsp
0.05 0 50 −3
x 10
150
d
6
4
te
3
u (m /s)
100
M
0
us
0.15
an
h (m)
0.2
Ac ce p
2 ua (C)
ua (U)
um (C)
um (U)
ua,sp
0
0
50
Figure 5: Plot of traje tories of
100
150
Time (s)
h, ua , um , and their set-points for the MPC of Eq. 26 (U, signifying un ompensated)
and the MPC of Eq. 41 (C, signifying ompensated) applied to the nonlinear pro ess of Eqs. 23 and 27-32 for a level set-point hange from
0.15 m
to
0.20 m.
61
Page 62 of 75
ip t cr us Flow Control
ua
xv
P
Valve
Flow
h Tank
−
d
−
an
PI Control
um +
M
hsp +
Figure 6: S hemati depi ting the ontrol ar hite ture for the level ontrol example with ow ontrol. In the inner
P
from the pneumati a tuation, whi h then alters the position
te
loop, the ow ontroller omputes the pressure
the sti ky valve a
ording to the valve dynami s (labeled as Valve in the gure). The valve position
ua
xv
of
is related to
through the valve a
ording to the linear valve hara teristi of Eq. 31 (labeled as Flow in the gure).
Ac ce p
the ow
xv
62
Page 63 of 75
ip t cr h hsp
0.1 1.56
1.57
1.58
1.59
−3
x 10
1.6
1.61
1.62
1.63 4
x 10
d te
3.8 3.6 3.4
Ac ce p
u (m3 /s)
4
3.2 1.56
an
0.15
us
0.2
M
h (m)
0.25
1.57
1.58
Figure 7: Closed-loop traje tories of
h, u a ,
ua um
1.59
1.6
1.61
Time (s)
and
um
1.62
1.63 4
x 10
for the pro ess of Eq. 23 under the PI ontroller of Eqs. 24-25
with the valve in Eqs. 27-31 and the PI ontroller of Eqs. 33-34 used to ontrol the valve ow rate to its set-point value for the set-point hange from
0.15 m
to
0.20 m.
63
Page 64 of 75
ip t cr
us
Stickband Deadband Slip-Jump
Ac ce p
te
d
M
an
Valve Moving Output Phase
Control Signal
Figure 8: Standard ontroller input-valve output relationship reported for a sti ky valve. The ontrol signal to the valve hanges but the valve output does not hange appre iably in the regions of deadband and sti kband. The valve output hanges qui kly in the region of slip-jump, and the valve output and ontrol signal are linearly related in the moving phase region of the response.
64
Page 65 of 75
ip t cr
0.2
0.15
an
h hsp
0.1 1.8
2
2.2
−3
x 10
2.6
2.8
4
ua um
3
te
5
Ac ce p
4
3 x 10
d
6
u (m /s)
2.4
M
1.6
us
h (m)
0.25
3
1.6
1.8
Figure 9: Closed-loop traje tories of
L=7
and
β = 0.007,
2
h, u a ,
and
2.2
2.4
2.6
Time (s) um
2.8
3 4
x 10
for the pro ess of Eq. 23 under the ontroller of Eqs. 36-37 with
with the open-loop valve (Eqs. 27-32), and
65
hsp = 0.20 m.
Page 66 of 75
ip t
1
cr
L EP
us
0.9 0.8
0.6
an
FD FI,A
M
0.5 0.4
FD,A
0.3
d
ua,max
ua
0.7
te
0.2
Ac ce p
0.1
XI,A
0
0
0.2
XD,C 0.4
0.6
0.8
1
xv xv,max
Figure 10: Figure depi ting linear (L) and equal per entage (EP) valve hara teristi s of Eqs. 32 and 42, respe tively, along with several fra tions of the maximum ow rate (FI,A maximum stem position (XI,A
= 0.1768
and
= 0.5379, FD = 0.62113, and FD,A = 0.2649) and XD,C = 0.37887) for the level ontrol example.
66
of the
Page 67 of 75
ip t cr us
0.28
an
0.26 0.24
u
M
0.22
d
0.2
te
0.18
Ac ce p
0.16
ua um y
0.14
1.8
2
2.2
2.4
2.6
2.8
3
Dimensionless Time
Figure 11: Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) from Eq. 45 when 19 set-point hanges are applied (a subset of the data is shown).
67
Page 68 of 75
ip t us
cr
0.7
an
0.6
u
M
0.5
d
0.4
te
0.3
Ac ce p
0.2
0.1 0.4
ua um y
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Dimensionless Time
Figure 12: Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) using the set-points de reasing between 0.6 and 0.15 and Eq. 46 (y overlays
68
ua ).
Page 69 of 75
ip t cr
0.5
an
0.45
us
um ua y
u
M
0.4
te
d
0.35
Ac ce p
0.3
0.25
0
0.5
1
1.5
2
Dimensionless Time
Figure 13: Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) under the EMPC using an empiri al valve layer model (y almost overlays
69
ua ).
Page 70 of 75
ip t cr us
Pressure/6894.76 (kg/m · t2d )
12
an
10
M
8
d
6
te
4
Ac ce p
2
0
0
0.5
1
1.5
2
Dimensionless Time
Figure 14: Pressure applied to the valve when the EMPC using an empiri al valve layer model is used.
70
Page 71 of 75
ip t cr
0.9968
1.5
2
1
1.5
2
te
0.9966 0
0.5
0
0.5
0
1
1
1.5
2
0.5
1
1.5
2
an
x2
1.5 1.0
M
0.5
0.2
d
x3
0.4
0.0
0.5
Ac ce p
1.0040
x4
us
x1
0.9970
1.0035 1.0030
0
Dimensionless Time
Figure 15: Closed-loop pro ess states under the EMPC using an empiri al valve layer model.
71
Page 72 of 75
List of Tables Valve Model Parameters [Gar ia (2008)℄
2
Ethylene Oxidation Model Parameters
. . . . . . . . . . . . . . . . . . . . . . . .
72
. . . . . . . . . . . . . . . . . . . . . . . . .
73
Ac ce p
te
d
M
an
us
cr
ip t
1
72
Page 73 of 75
ip t cr us an
Table 1: Valve Model Parameters [Gar ia (2008)℄
Parameter
M
1.361 kg 0.06452 m2 52538 kg/s2 0.000254 m/s 108 kg/s2 9000 kg/s 612.9 kg/s 1423 kg · m/s2 1707.7 kg · m/s2
Ac ce p
te
d
mv Av ks vs σ0 σ1 σ2 FC FS
Value
73
Page 74 of 75
ip t cr us
Parameter
Value -8.13 -7.12
M
γ1 γ2 γ3 A1 A2 A3 B1 B2 B3 B4
an
Table 2: Ethylene Oxidation Model Parameters
-11.07 92.80
Ac ce p
te
d
12.66
2412.71 7.32 10.39 2170.57 7.02
74
Page 75 of 75
S0098-1354(17)30300-9 http://dx.doi.org/doi:10.1016/j.compchemeng.2017.08.008 CACE 5875
To appear in:
Computers and Chemical Engineering
Received date: Revised date: Accepted date:
23-3-2017 30-6-2017 18-8-2017
Please cite this article as: Helen Durand, Robert Parker, Anas Alanqar, Panagiotis D. Christofides, Elucidating and Handling Effects of Valve-Induced Nonlinearities in Industrial Feedback Control Loops, (2017), http://dx.doi.org/10.1016/j.compchemeng.2017.08.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights • A closed-loop system view of valve stiction-induced nonlinear dynamic behavior. • Impact of stiction on closed-loop dynamics for both classical and model predictive control systems is elucidated. • Stiction compensation strategies are presented for both classical and model predictive control systems.
Ac ce
pt
ed
M
an
us
cr
ip t
• Chemical process examples are used to illustrate the impact of stiction and how it can be handled.
1
Page 1 of 75
Elu idating and Handling Ee ts of Valve-Indu ed Nonlinearities in Industrial Feedba k Control Loops ∗,a,b a a a Helen Durand , Robert Parker , Anas Alanqar , Panagiotis D. Christo des
Department of Chemi al and Biomole ular Engineering, University of California, Los Angeles, CA, 90095-1592, USA. b Department of Ele tri al Engineering, University of California, Los Angeles, CA 90095-1592, USA.
ip t
a
cr
Abstra t
us
In this work, we investigate the ee ts of various types of valve behavior (e.g., linear valve dynami s and sti tion) on the ee tiveness of pro ess ontrol in a uni ed framework based on systems of
an
nonlinear ordinary dierential equations that hara terize the dynami s of losed-loop systems in luding the pro ess, valve, and ontroller dynami s. By analyzing the resulting dynami models,
M
we demonstrate that the responses of the valve output and pro ess states when valve behavior
annot be negle ted (e.g., sti tion-indu ed os illations in measured pro ess outputs) are losed-
a priori
due to the oupled and typi ally nonlinear
d
loop ee ts that an be di ult to predi t
te
dynami s of the pro ess-valve model. Subsequently, we dis uss the impli ations of this losed-loop perspe tive on the ee ts of valve dynami s in losed-loop systems for understanding valve behavior
Ac ce p
ompensation te hniques and developing new ones. We on lude that model-based feedba k ontrol designs that an a
ount for pro ess and valve onstraints and dynami s provide a systemati method for handling the multivariable intera tions in a pro ess-valve system, where the models in su h ontrol designs an ome either from rst-prin iples or empiri al modeling te hniques. The analysis also demonstrates the ne essity of a
ounting for valve behavior when designing a ontrol system due to the potentially dierent onsequen es under various ontrol methodologies of having dierent types of valve behavior in the loop. Throughout the work, a level ontrol example and a
ontinuous stirred tank rea tor example are used to illustrate the developments.
Key words:
Model predi tive ontrol, valve dynami s, hemi al pro esses, pro ess ontrol,
sti tion, empiri al modeling
∗
Corresponding author: Tel: +1 (310) 794-1015; Fax: +1 (310) 206-4107; E-mail: pd seas.u la.edu.
Preprint submitted to Computers & Chemi al Engineering
August 18,Page 20172 of 75
1. Introdu tion Valves do not respond instantaneously to ontrol signal hanges (i.e., their response has dynami s) be ause they have me hani al parts that move when the ontrol signal hanges. Typi ally,
ip t
the hemi al pro ess ontrol literature assumes that the valve dynami s are so fast that they an be negle ted, meaning that the valve output is assumed to immediately equal the value requested
cr
by the ontroller. However, in industry, various valve behaviors are observed that impa t ontrol
us
loop performan e, though an engineer entering industry after ompleting his or her undergraduate edu ation typi ally has had limited exposure to valve behavior.
Undergraduate pro ess ontrol
an
ourses often handle valve dynami s with linear transfer fun tion models; su h dynami s an be related to, for example, resistan e of the gas used to apply pressure in a pneumati a tuator to Valve hara teristi s (e.g., linear,
M
ow at the top of a valve [Coughanowr and LeBlan (2009)℄.
equal per entage, and square root) may also be reviewed in undergraduate oursework to provide undergraduates with fundamentals regarding valve sizing and the ee ts of installing a valve on the
d
valve's ow hara teristi s (the manner in whi h the ow through the valve is related to the valve
te
opening) [Coughanowr and LeBlan (2009); Ogunnaike and Ray (1994); Riggs (1999)℄.
Though
Ac ce p
there may be some dis ussion of other types of valve behavior des ribed by nonlinear models (e.g., saturation of the valve output at its maximum value, failure of a valve to respond to hanges in the
ontrol signal to the valve for some time after a valve movement dire tion hange due to me hani al parts in a valve (deadband due to ba klash) [Choudhury et al. (2005)℄, or sti tion [Brásio et al. (2014); Armstrong-Hélouvry et al. (1994)℄, whi h refers to valve behavior due to fri tion that an be des ribed by nonlinear dynami equations), time onstraints in a semester/quarter and also a general fo us in undergraduate pro ess ontrol on linear dynami systems do not typi ally permit an in-depth treatment of nonlinear valve behavior and its impa t on pro ess ontrol from a rst-prin iples perspe tive. However, at hemi al plants throughout the world, valve issues su h as sti tion, deadband, saturation, hysteresis, and deadzone prevent adequate set-point tra king [Jelali and Huang (2010); Choudhury et al. (2008)℄. Therefore, it would be bene ial if the various types
2
Page 3 of 75
of valve behavior were presented within a single framework so that undergraduates ould learn the mathemati al stru ture needed to allow them to understand and ompensate for any valve behavior that they en ounter in their ourses and their areers. The traditional manner of handling valve behavior, in whi h various types of valve behavior For
ip t
are treated separately rather than within a single framework, is prevalent in the literature.
example, Durand et al. (2014) demonstrates that even linear valve dynami s an be problemati
cr
for a pro ess operated under an optimization-based ontrol design alled e onomi model predi tive
ontrol (EMPC) [Ellis et al. (2014a); Heidarinejad et al. (2012); Amrit et al. (2011); Rawlings et al.
us
(2012); Huang et al. (2012)℄ when the valve dynami s are negle ted in the model utilized by the EMPC for making state predi tions; therefore, that work suggests in orporating the valve dynami s
an
in the dynami pro ess model for the ontroller. Choudhury et al. (2008) analyzes the range of valve travel over whi h linear ontrol design theory would be expe ted to be adequate when a pro ess
M
that an be ee tively modeled with a linear model re eives a ow rate from a valve with a square root or an equal per entage inherent valve hara teristi .
Zabiri and Samyudia (2006) develops
d
an MPC-based method for linear pro esses where the valve is subje t to ba klash. The literature
te
analyzing and ompensating for the sti tion nonlinearity is parti ularly extensive, with reviews su h as Brásio et al. (2014) ategorizing the methods, though sti tion ompensation remains an
Ac ce p
important resear h topi with newer works su h as those in Durand and Christo des (2016); Ba
i di Capa i et al. (2017); Munaro et al. (2016) expanding the ompensation literature. Another feature of the valve behavior literature is that the types of pro esses and ontrol laws handled are often limited. For example, despite the re ognition in [Thornhill and Hor h (2007); Jelali and Huang (2010); Choudhury et al. (2005)℄ that the ontroller, pro ess, and valve dynami s all play a role in determining the traje tories of the measured outputs of a losed-loop system with a sti ky valve, a summary table in Thornhill and Hor h (2007) fo uses on ombinations of linear pro esses (integrating and non-integrating) and linear ontrollers (proportional (P) and proportional-integral (PI)) with dierent sti tion hara teristi s for a valve in the ontrol loop to larify whi h ombinations are expe ted to result in limit y ling of the valve output. This is onsistent with mu h of the sti tion ompensation literature, whi h is developed assuming linear pro ess models and li-
3
Page 4 of 75
near ontrollers. Furthermore, the sti tion literature generally fo uses on single-input/single-output
ontrol loops. It is widely appre iated, however, that hemi al pro esses are typi ally des ribed by nonlinear dynami models with multiple inputs. Motivated by these onsiderations, the primary ontribution of this work is the development of
ip t
a single, omprehensive systems engineering framework for understanding the impa ts of various types of valve behavior on losed-loop nonlinear pro ess systems with multiple inputs, various
cr
ontrol ar hite tures, and various ontrol designs. This framework not only permits a fundamental understanding of the auses of the negative ee ts observed in ontrol loops with valves for whi h
us
the valve behavior annot be negle ted, but also enables a better understanding regarding how the tools previously developed in the literature for handling these dynami s work. This is riti al for
an
sti tion ompensation be ause a number of sti tion ompensation te hniques are not model-based (and therefore the fundamental bene ts and limitations of su h sti tion ompensation strategies
M
must be analyzed within a model-based framework) or are model-based but designed for spe i linear pro ess models, ontrol ar hite tures, and valve nonlinearity models. Furthermore, mu h of
d
the sti tion literature assumes that the system under study exhibits losed-loop os illations due to
te
sti tion; notwithstanding some dis ussion regarding the auses of these os illations in the literature, the onditions under whi h they do and do not o
ur for a general nonlinear hemi al pro ess under
Ac ce p
various ontrol designs must be lari ed.
The developments of this work des ribe not only the
auses of sti tion-indu ed os illations, but also the auses of improvements observed in the response of a losed-loop system with a sti ky valve when a sti tion ompensation method is used.
The
developments are shown to permit the proposal of new sti tion ompensation s hemes and to allow a number of perspe tives on ontrol design in light of valve behavior to be stated.
2. Preliminaries 2.1. Notation
The transpose of a ve tor
u
x
is denoted by
xT .
The notation
u ∈ S(∆)
signi es that the ve tor
is a member of the set of pie ewise- ontinuous (from the right) fun tions with period
notation
tk = k∆, k = 0, 1, 2, . . .,
and the notation
4
t˜j = j∆e , j = 0, 1, 2, . . .,
∆.
The
refer to elements of a
Page 5 of 75
time sequen e separated by sampling time periods of lengths
|·|
∆
and
∆e ,
respe tively. The notation
signi es the Eu lidean norm of a ve tor.
2.2. Class of Systems
ip t
We onsider a nonlinear hemi al pro ess system with the following form:
x˙ = f (x, ua , w)
pro ess inputs,
w ∈ Rl
f : Rn ×Rm ×Rl
omponent
X ), ua ∈ Rm
is a ve tor of bounded pro ess disturban es (i.e.,
is a lo ally Lips hitz ve tor fun tion of its arguments with
ua,i , i = 1, . . . , m,
is the ve tor of
w ∈ W := {w | |w| ≤ θ}), f (0, 0, 0) = 0.
Ea h
of the pro ess input ve tor is an output of a valve that is adjusted
an
and
is the pro ess state ve tor (bounded in the set
cr
x ∈ X ⊆ Rn
us
where
(1)
utilizing a feedba k ontroller for the nonlinear pro ess that outputs a set-point
um,i , i = 1, . . . , m,
for ea h valve output. Be ause the valve output ow rates are bounded by physi al valve onstraints, is bounded between a minimum (ua,i,min ) and a maximum (ua,i,max ) ow rate, with
the resulting input onstraint on
ua,i ≤ ua,i,max, i = 1, . . . , m}). um ∈ Um ,
where
where
um,i
and ea h
ua,i − um,i
ua,i
where
U := {ua ∈ Rm | ua,i,min ≤
depends on the valve behavior. We will onsider
relationship is either stati or dynami . In the ase that the
relationships are stati , the following equation holds:
fstatic,i
ua,i = fstatic,i (um,i )
is a nonlinear ve tor fun tion (fstatic (um )
natively, a dynami model may hara terize the
um,i
ua ∈ U ,
Um := {um ∈ Rm | um,i,min ≤ um,i ≤ um,i,max, i = 1, . . . , m}).
Ac ce p
ua,i − um,i
(i.e.,
Sin e the ow rates out of the valve are bounded, the set-points are
The relationship between ea h valve behavior for whi h the
U
denoted by
te
bounded also (i.e.,
ua
M
ua,i
d
ea h input
and the dynami state ve tor
xdyn,i ∈ Rpi ,
(2)
= [fstatic,1 (um,1 ) · · · fstatic,m (um,m )]T ).
ua,i − um,i
relationship, where
for whi h the omponents
ua,i
Alter-
is related to both
xdyn,i,j , j = 1, . . . , pi ,
states of the valve model. In this ase, the following equations des ribe the dynami s of the
are
i − th
valve:
x˙ dyn,i = xˆdyn,i (xdyn,i , um,i )
5
(3)
Page 6 of 75
ua,i = fdynamic,i (xdyn,i ) where
xˆdyn,i
(4)
is a nonlinear ve tor fun tion hara terizing the dynami s of the internal states of
the model for the
i − th
valve, and
fdynamic,i
the internal dynami states of the valve.
is a nonlinear ve tor fun tion relating
We de ne
ua,i
and
xdyn = [xdyn,1 · · · xdyn,m ]T , xˆdyn (xdyn , um ) =
We assume in this work that the value of ea h
um,i
ip t
[ˆ xdyn,1 (xdyn,1 , um,1) · · · xˆdyn,m (xdyn,m , um,m )]T , and fdynamic (xdyn ) = [fdynamic,1 (xdyn,1 ) · · · fdynamic,m (xdyn,m )]T . is determined utilizing a feedba k ontroller
cr
(e.g., proportional-integral (PI) ontrol or model predi tive ontrol (MPC), whi h will be pla ed
us
in the notation of this se tion in the following two se tions) that utilizes knowledge of at least one pro ess state to ompute ontrol a tions. This means that the value of ea h
an
subset of the state ve tor as follows:
um,i
is ae ted by some
um,i = fcontroller,i (x, ζˆi ) ζˆi ∈ Rri , i = 1, . . . , m,
is a ve tor of internal states of the ontroller al ulating
internal states may be dynami as follows:
M
where
d
˙ ζˆi = finternal,i (x, xdyn , ζˆi )
um,i .
These
(6)
te
De ning
(5)
ˆ = [fcontroller,1(x, ζˆ1 ) · · · fcontroller,m(x, ζˆm )]T , ζˆ = [ζˆ1 · · · ζˆm ]T , and finternal (x, xdyn , ζ) ˆ = fcontroller (x, ζ)
Ac ce p
[finternal,1 (x, xdyn , ζˆ1 ) · · · finternal,m (x, xdyn , ζˆm )]T , we an write the pro ess-valve system as follows for ua and um (i.e., Eq. 2 holds): " # ˆ w(t)) x˙ f (x(t), fstatic (fcontroller (x, ζ)), ˙ = ˆ finternal (x, xdyn , ζ) ζˆ
the ase of a stati relationship between
where
xdyn = 0
(7)
when Eq. 2 holds.
For the ase of a dynami relationship between
ua
and
um
(i.e., Eqs. 3-4 hold), the following
pro ess-valve system results:
x˙
f (x(t), fdynamic (xdyn ), w(t)) ˆ x˙ dyn = xˆdyn (xdyn , fcontroller (x, ζ)) ˙ˆ ˆ finternal (x, xdyn , ζ) ζ
(8)
With slight abuse of notation, the right-hand sides of both Eqs. 7 and 8 will be denoted by
6
Page 7 of 75
fq (q(t), um (t), w(t)) (fq
relationship, and it signi es the right-hand side of Eq. 8 when Eqs. 3-4 hara terize the relationship), where
ˆT q(t) represents the ve tor of pro ess-valve states (i.e., q = [x ζ]
ribes the pro ess-valve dynami s, and
fq
ˆT q = [x xdyn ζ]
ua,i − um,i
when Eq. 7 des-
when Eq. 8 des ribes the pro ess-valve dyna-
is a lo ally Lips hitz ve tor fun tion of its arguments with
fq (0, 0, 0) = 0.
ip t
mi s). We assume that
ua,i − um,i
signi es the right-hand side of Eq. 7 when Eq. 2 hara terizes the
The general prin iples of the ontrol ar hite tures des ribed are illustrated in Fig. 1.
besides
x,
Disturban es ould also be onsidered in other dynami states of the pro ess-valve model su h as in
xdyn ,
cr
Remark 1.
and the analysis presented throughout this work would ontinue to hold.
us
2.3. Classi al Linear Control with Integral A tion
an
Linear ontrol designs with an integral term are designed to drive a sele ted pro ess output (here taken to be a pro ess state, whi h is onsistent with standard industrial pra ti e in the hemi al
it omprises the ve tor
xˆsp ∈ Rnˆ .
xˆ ∈ Rnˆ , n ˆ ≤ n,
M
pro ess industries) to its set-point. Thus, we assume that the pro ess state ve tor or a subset of of measured outputs being driven to the set-point ve tor
When a PI ontroller is used, ea h omponent of
xˆ
is regulated to its set-point by an
The dynami s of the
i − th
te
n ˆ = m.
Ac ce p
and thus
d
individual linear ontroller that outputs a valve output ow rate set-point for an individual valve, PI ontroller are represented by:
um,i = gA,i (ˆ xi , ζi )
(9)
xˆ ζ˙i = Acon,i i + Bcon,i xˆi,sp ζi
The form of these equations follows that in Eqs. 5-6, with
finternal,i
given by the right-hand side of Eq. 10.
Acon,i
and
(10)
ζˆi = ζi , fcontroller,i
Bcon,i
given by
gA,i ,
and
are a matrix and s alar.
2.4. Model Predi tive Control
Model predi tive ontrol [Qin and Badgwell (2003); Ellis et al. (2014a)℄ is an optimization-based
ontrol strategy based on the following optimization problem:
min
um (t)∈S(∆) s.t.
Z
tk+N
Le (˜ x(τ ), um (τ )) dτ
(11a)
tk
x˜˙ (t) = f (˜ x(t), um (t), 0)
7
(11b)
Page 8 of 75
x˜(t) ∈ X, ∀ t ∈ [tk , tk+N )
(11d)
um (t) ∈ Um , ∀ t ∈ [tk , tk+N )
(11e)
gM P C,1(˜ x(t), um (t)) = 0, ∀ t ∈ [tk , tk+N )
(11f )
gM P C,2(˜ x(t), um (t)) ≤ 0, ∀ t ∈ [tk , tk+N )
(11g)
ip t
Le (x, um )
(11 )
is minimized subje t to bounds on the states (Eq. 11d), bounds on the
cr
The stage ost
x˜(tk ) = x(tk )
inputs (Eq. 11e), equality and inequality onstraints (Eqs. 11f-11g), and the restri tion that the
from a measurement of the state (Eq. 11 ).
≡ 0)
dynami model in Eq. 11b when initialized
x˜(t) represents
us
states must evolve a
ording to the nominal (w(t)
the pro ess state a
ording to the model
an
of Eq. 11b (i.e., in the absen e of disturban es or plant/model mismat h) at time
t; x(t),
on the
other hand, represents the a tual value of the state (whi h is ae ted by disturban es and thus is
x˜(t)).
By setting
MPC be omes a feedba k ontroller.
um (t) is used
to a measurement of the pro ess state at tk , the
in pla e of
ua (t) be ause
the standard formulation of MPC in industry
d
In Eq. 11b,
x˜(tk )
M
not ne essarily the same as
no referen e is made to
ua
in Eq. 11).
sampling periods of length
∆ (N
ua = um ;
therefore,
MPC an, however, handle valve saturation through the
ua = um .
Ac ce p
onstraint of Eq. 11e, assuming
N
te
and the literature negle ts valve behavior in general (i.e., it assumes that
A ve tor of ontrol a tions
um
is omputed for ea h of the
is the predi tion horizon), and only the rst of these ve tors is
applied to the pro ess in a sample-and-hold fashion a
ording to a re eding horizon strategy. The notation
u∗m (t|tk ), t ∈ [tk , tk+N ),
problem initiated at time
signi es the optimal value of
tk (u∗m,i (t|tk )
represents the
i − th
um
at time
t
for the optimization
omponent of this ve tor).
A form of MPC that is ommonly used in the hemi al pro ess industries is tra king MPC, whi h drives
xˆ
to
xˆsp
(though it is not ne essary in this ase that
stage ost with its minimum at the set-point ve tor
um,sp
n ˆ = m)
by utilizing a quadrati
xˆsp with orresponding steady-state input ve tor
as follows:
Le (˜ x(τ ), um (τ )) = (ˆ xsp − x˜ˆ)T QT (xsp − x˜ˆ) + (um,sp − um )T RT (um,sp − um )
8
(12)
Page 9 of 75
where
QT > 0
and
RT > 0
The value of ea h
um,i
um,i
are tuning matri es, and
x˜ˆ
denotes the predi ted value of the ve tor
al ulated by the MPC has the form of Eq. 5 with
as a fun tion of the pro ess state), though the fun tion
fcontroller,i
ζˆi = 0
xˆ.
(whi h leaves
is not expli itly de ned in
this ase but is de ned impli itly by the optimization problem in Eq. 11. We assume that full state feedba k is available for all MPC designs presented in this
ip t
Remark 2.
work. When it is not, state estimation and output feedba k MPC (e.g., Ellis et al. (2014b)) ould
cr
be investigated to determine if they provide a suitable ontrol design for a given system, but this will not be pursued here.
us
3. Pro ess-Valve Model Analysis: Insights on Impa ts of Valve Behavior on ClosedLoop Systems
an
Eqs. 7-8 omprise the uni ed framework that we develop in this work for analyzing the impa ts of valve behavior and valve behavior ompensation on ontrol loop performan e. They reveal that valve
M
behavior hanges the dynami s of a oupled losed-loop system. To see this, onsider the system of Eq. 7, whi h represents a pro ess-valve system subje t to only a stati valve nonlinearity under
d
feedba k ontrol. The stati nonlinearity impa ts the dynami s of
ˆ w(t)), = um ), the dynami s of x˙ would be des ribed by f (x(t), fcontroller (x, ζ), ˙ ˙ and ζˆ are fun tions of x, so than the dynami s in Eq. 7). In addition, both x
whi h is dierent
Ac ce p
fstatic
te
valve nonlinearity (ua
modifying
x (i.e., in the absen e of the stati
ae ts the dynami s of both
x and ζˆ.
the ee t on the losed-loop response of hanging
Due to the fa t that the system is nonlinear,
fstatic
is di ult to determine without performing
losed-loop simulations. This is parti ularly signi ant when there are multiple pro ess inputs related to
um,i
through dierent stati nonlinear fun tions, espe ially assuming that the dynami s
of the omponents of
x
are oupled. Then, ea h
ua,i
ae ts all omponents of
through oupling of those omponents in the ve tor fun tion by all omponents of
x
ontrol (Eq. 5).
to de ne
ua,i
either dire tly or
x
is ae ted
are oupled and at least one of
(Eq. 2) due to the use of state feedba k
Using a similar analysis, it an be dedu ed that hanging the ontrol law (i.e.,
fcontroller
to determine
um,i
x
f , and the value of ea h ua,i
due to the fa t that the omponents of
those omponents is used to al ulate
hanging
ua,i
and
a priori
finternal )
also impa ts the losed-loop response in a manner that is di ult
(without simulations).
9
Page 10 of 75
When the valve dynami s an be des ribed by dynami systems of equations as in Eq. 8, the dynami s of the valve, ontroller, and pro ess are again oupled. In this ase, however, there is an additional omplexity in that the valve dynami s add additional states with nonlinear dynami s (or linear in the spe i ase of linear valve dynami s) that are not present in the ase that
x
is a fun tion of at least one of the omponents of
x,
it is ae ted by the
ip t
other omponents of
um,i
as well, assuming oupling between these omponents. This auses
and
ua,i
ua,i
in Eq. 1 and thus by all valve states
to also be related to
x
(Eqs. 3-4), and the omponents of
xdyn,i , i = 1, . . . , m,
x are
xdyn,i
ae ted by the values of all
cr
Furthermore, be ause
ua = um .
from Eq. 4.
us
The above analysis shows that from a fundamental mathemati al analysis of general equations for a pro ess-valve system, the dynami s of all valves an be seen to be oupled with the dynami s of
an
the other valves and also with the dynami s of the pro ess and the ontroller due to state feedba k (this is not limited to PI ontrol or MPC). Be ause the ontroller dynami s ae t the evolution of
M
the states and thus the pro ess outputs, dierent types of ontrollers would be expe ted to result in dierent responses of the pro ess outputs. Furthermore, the ontrol loop ar hite ture will also
d
ae t the response be ause it will impa t the equations that des ribe the ontroller dynami s. This
te
analysis reveals that the negative ee ts of valve dynami s on ontrol loop ee tiveness are related to the ontroller, pro ess, and valve dynami s, in addition to the ontrol ar hite ture. The next
Ac ce p
two se tions make the notation of Eqs. 7-8 more on rete by showing how linear and sti ky valve dynami s t within that framework.
Remark 3.
In this work, we onsider a general ase in whi h all states of the pro ess-valve model
are oupled. For ases when this does not hold, it may be possible to analyze the dynami s of the spe i pro ess to see if any simpli ations result ompared to the analysis in this work.
3.1. Linear Valve Dynami s
Linear valve dynami s an be denoted as follows:
x˙ ua,i = Ai xua,i + Bi um,i
(13)
ua,i = Ci xua,i
(14)
10
Page 11 of 75
where
xua,i ∈ Rpi
is the ve tor of internal states of the linear valve dynami model for the
Ai , Bi ,
valve, and
and
Ci
i − th
are a matrix and two ve tors, respe tively, of appropriate dimensions.
Combining the pro ess and valve layer models gives the following pro ess-valve model for this ase (omitting the dynami s of the ontroller):
f (x(t), CL xua (t), w(t)) x˙ = AL xua + BL um x˙ ua
Ai , Bi ,
and
and
CL
are matri es and ve tors of appropriate dimensions ontaining the elements
Ci , i = 1, . . . , m,
notation in Eqs. 3-4,
in an appropriate order, and
xdyn = xua ,
and
xˆdyn,i (xdyn,i , um,i )
and
sides of Eqs. 13 and 14, respe tively.
xua = [xua,1 . . . xua,m ]T .
fdynamic,i (xdyn,i )
Using the
equal the right-hand
an
3.2. Sti ky Valve Dynami s
cr
of
AL , BL ,
us
where
(15)
ip t
For the ase that all valves are sti ky (i.e., ae ted by fri tion/sti tion, whi h prevents the
M
valve position from appre iably hanging until the for e applied to the valve moving parts be omes su iently large) and move in a straight line (rather than rotating), the valve position for the
i − th
valve evolve in time a
ording to the following for e balan e:
mv,i
x˙ v,i = vv,i
(16)
1 (aT FO,i + cTi FA,i − Ff ric,i ) mv,i i
Ac ce p
v˙ v,i =
where
and the
d
vv,i
te
valve velo ity
xv,i
is the mass of the moving parts of the
i − th
valve,
FO,i
(17)
is a ve tor of non-fri tion
for es on the valve that are not related to the ontroller output or fri tion for e and whi h have
oe ient ve tor
ai , FA,i
is a ve tor of non-fri tion for es on the valve that are adjusted based on
the ontroller output and have oe ient ve tor
ci ,
and
is the fri tion for e on the
i − th
zf,i
(whi h is a dynami internal state
Ff ric,i = Fˆf ric,i (xv,i , vv,i , zf,i )
(18)
z˙f,i = zˆf,i (xv,i , vv,i , zf,i )
(19)
valve. The fri tion for e is a stati fun tion of
xv,i , vv,i ,
Ff ric,i
and
of the fri tion model), as follows:
11
Page 12 of 75
where
zˆf,i
is a nonlinear ve tor fun tion des ribing the dynami s of the internal states of the fri tion
model. Assuming that
FA,i
is a stati fun tion of
um,i
as follows:
FA,i = fSO,i (um,i ) is a nonlinear ve tor fun tion des ribing the relationship between
um,i
and
FA,i ,
and
FO,i is also a fun tion of the valve model states, the right-hand side of Eq. 17 an be denoted by
vˆv,i (um,i , xv,i , vv,i , zf,i ).
cr
that
fSO,i
ip t
where
(20)
Finally, assuming that the relationship between
ua,i and xv,i an be expressed
(21)
an
ua,i = ff low,i (xv,i )
us
through the following stati nonlinear equation des ribing the valve hara teristi :
we obtain the following pro ess-valve model (omitting the dynami s of the ontroller for the pro ess):
xv = [xv,1 · · · xv,m ]T , vv = [vv,1 · · · vv,m ]T , zf = [zf,1 · · · zf,m ]T , ff low (xv (t)) = [ff low,1 (xv,1 )
te
where
(22)
d
M
f (x(t), ff low (xv (t)), w(t)) x˙ x˙ v vv = v˙ v vˆv (um , xv , vv , zf ) zˆf (xv , vv , zf ) z˙f
· · · ff low,m(xv,m )]T , vˆv (um , xv , vv , zf ) = [ˆ vv,1 (um,1 , xv,1 , vv,1 , zf,1 ) · · · vˆv,m (um,m , xv,m , vv,m , zf,m )]T , In the notation of Eq. 8,
[xv vv zf ]T , fdynamic (xdyn ) = ff low (xv (t)),
is given by the right-hand sides of
Ac ce p
zˆf (xv , vv , zf ) = [ˆ zf,1 (xv,1 , vv,1 , zf,1 ) · · · zˆf,m (xv,m , vv,m , zf,m )]T . and
xˆdyn,i (xdyn,i , um,i )
xdyn =
Eqs. 16-17 and 19.
4. Illustrating a Closed-Loop, Systems Perspe tive on the Impa ts of Valve Behavior on Control Loop Performan e The losed-loop perspe tive on the impa ts of valve behavior on ontrol loop performan e developed above provides a novel platform from whi h to fundamentally understand the onditions under whi h various negative ee ts attributed to valve behavior in ontrol loops o
ur. In the next se tion, we show through an illustrative level ontrol example that a prevalent phenomenon in the literature (sti tion-indu ed os illations) an only be explained through this losed-loop perspe tive. The simpli ity of the example allows us to fo us on the ee ts of the valve dynami s without the
12
Page 13 of 75
added omplexity of a large-s ale nonlinear pro ess model; however, we subsequently elu idate the ne essity of a losed-loop analysis for predi ting the behavior of the measured outputs of a more
omplex pro ess-valve system through a ontinuous stirred tank rea tor (CSTR) example.
4.1. A Closed-Loop Perspe tive on Sti tion-Indu ed Os illations
ip t
In the level ontrol problem (shown in Fig. 2) onsidered, the tank inlet ow rate
ontrolled variable. The dynami s of the tank level are:
A = 0.25 m2
denotes the ross-se tional area of the tank, and
(23)
c1 = 0.008333 m5/2 /s
us
where
is the
cr
√ dh 1 = (ua − c1 h) dt A
ua
is the
an
outlet resistan e oe ient. On an order of magnitude onsistent with an example from Ogunnaike and Ray (1994) that uses these parameter values, the minimum tank height is
0.5184 m,
maximum value of
ua
is
the minimum value of
ua,max = 0.006 m3 /s
ua
is
ua,min = 0 m3 /s
the maximum
(fully losed valve), and the
(fully open valve).
M
tank height is
0 m,
4.1.1. Level Control with Negligible Valve Dynami s
ua = um
d
When the valve dynami s are so fast that
at all times is a reasonable approximation, a
te
well-tuned PI ontroller and an MPC an be designed that ee tively drive the level to its set-point.
where
um
value of
Ac ce p
The PI ontroller for the tank level has the following form:
(24)
um = uas + Kc (hsp − h) + Kc ζ/τI
(25)
is the ontroller output,
ua
τI = 43.2
ζ˙ = hsp − h, ζ(0) = 0
ζ
is the dynami state of the PI ontroller,
before the set-point hange, and
was sele ted that prevents
set-points simulated. Eqs. 24-25 when
um
hsp
is the steady-state
is the level set-point. A tuning
from dipping below
ua,min
Kc = 0.006
or shooting above
ua,max
and
for the
The response of the level of the tank of Eq. 23 under the PI ontroller of
ua = um
is shown in the top plot of Fig. 3, plotted every 100 integration steps, for
the tank level initiated from its maximum (ua and then in reased to
uas
0.20 m
= 0.006 m3 /s, h = 0.5184 m),
(the set-point hange from
13
0.15 m
to
0.20 m
de reased to
0.15 m,
will be the fo us in the
Page 14 of 75
remainder of this se tion to avoid the ee ts of possible initial transients during the rst set-point
hange). Ea h set-point was held for 1040
s.
The dynami system was integrated with the expli it
Euler numeri al integration method and an integration step size of the value of
ζ
was re-set to 0 and the value of
uas
10−3 s.
At the set-point hanges,
was set to the last applied value of
um .
0.15 m
um (t)∈S(∆)
tk+N tk
ua = um
as follows:
˜ 2 + RT (ua,sp − um )2 dτ QT (hsp − h)
p ˜˙ = 1 (um − c1 h) ˜ h A ˜ k ) = h(tk ) h(t
s.t.
when
cr
min
Z
0.20 m
to
us
tra king MPC to drive the level from
ip t
Fo using on the se ond set-point hange from the example above, we also design a well-tuned
integration step size of
RT = 1.
10−3 s
simulation of
2500 s,
Using this MPC to ontrol the pro ess of Eq. 23 with an
within the MPC, an integration step size of
N = 50,
a sampling period of length
d
level, a predi tion horizon of
(26d)
an
and
M
QT = 0.00001
(26b) (26 )
0 ≤ um (t) ≤ 0.006, ∀ t ∈ [tk , tk+N ) where
(26a)
and a set-point
hsp = 0.20 m
10−5 s
∆ = 1 s,
to simulate the
a nal time of the
with its orresponding steady-state ow rate
te
ua,sp = 0.00373 m3 /s, the state pro le in the bottom plot of Fig. 3 is obtained (the results are plotted
Ac ce p
every 10000 integration steps). The nonlinear interior point optimization solver Ipopt [Wä hter and Biegler (2006)℄ was used for the simulations with a toleran e of
10−8
on a 2.40 GHz Intel Core 2
Quad CPU Q6600 on a 64-bit Windows 7 Professional operating system with 4.00 GB of RAM.
Remark 4.
The losed-loop responses of the level under the PI and MPC designs are not meant
to be ompared with one another. The purpose of Fig. 3 is to show that under both PI ontrol and under MPC, when
ua = um ,
the tunings that are used for the level ontrol example result in
very good set-point tra king of the level. The results for the PI ontroller and MPC in Fig. 3 will be ompared with the results utilizing the same tunings when
ua 6= um
for ea h ontroller in the
following se tion.
14
Page 15 of 75
4.1.2. Level Control with a Sti ky Valve When only the valve dynami s hange ompared to the ase in the prior se tion (i.e., the valve be omes sti ky), sustained os illations are set up in the originally well-tuned ontrol loops. For this
ua
in Eq. 23 is the ow rate out of a pressure-to- lose spring-diaphragm sliding-
stem globe valve a tuated by a pressure
P.
If the valve is initiated from its fully open position, no
ip t
sti ky valve ase,
pressure is initially applied and the valve stem is at its equilibrium position
xv = xv,max = 0.1016 m.
Its fully
The dierential equations for the valve
cr
losed position orresponds to
xv = 0 m.
dynami s are [Gar ia (2008)℄:
(27)
us
dxv = vv dt
dvv 1 [Av P − ks xv − Ff ] = dt mv and
ks
are the diaphragm area and spring onstant, respe tively, and the fri tion for e
is determined from the LuGre fri tion model [Canudas de Wit et al. (1995)℄:
M
Ff
Av
an
where
(28)
Ff = σ0 zf + σ1
dzf + σ2 vv dt
(29)
d
dzf |vv |σ0 = vv − zf dt FC + (FS − FC )e−(vv /vs )2
te
(30)
The parameters of the valve dynami model in Eqs. 27-30 are those for the nominal valve in Gar ia
Ac ce p
(2008) and are displayed in Table 1. In addition, we assume that the valve has a linear installed
hara teristi [Coughanowr and LeBlan (2009)℄:
ua =
xv,max − xv xv,max
ua,max
The pressure to be applied to the valve for a given set-point
um − P
um
(31)
is determined from the following
relationship that was developed for a low-sti tion valve in Durand and Christo des (2016)
(though the tunings developed in the no-sti tion ase perform well for the low-sti tion valve):
P =
(um /ua,max ) − 0.70391/0.7042 0.05864 − 6894.76∗0.7042
This relationship has the form of Eq. 20, where
P
is
FA,i
(32)
and the right-hand side of Eq. 32 is
fSO,i (um ).
15
Page 16 of 75
h, um ,
Fig. 4 shows
ua
and
when the valve with the dynami s in Eqs. 27-32 is used to adjust the
ow rate to the pro ess of Eq. 23 under the PI ontroller of Eqs. 24-25. The valve was initiated from its fully open position (i.e.,
zf = 0 m),
h = 0.5184 m, ua = 0.006 m3 /s, ζ = 0 m · s, xv = 0 m, vv = 0 m/s,
and the set-point was hanged to 0.15
m
for 15600
s,
then to 0.20
m
for 15600
s.
t = 15600 s
hange from
qI
from this simulation as
0.15 m
to
0.20 m).
hanged, and the value of
uas
(the initial pro ess-valve state for the level set-point
The value of
ζ
was re-set to zero when the level set-point was
cr
state at
ip t
Be ause the se ond set-point hange is the fo us in this work, we will refer to the pro ess-valve
in Eq. 25 was re-set to the last applied value of
ua
when the set-
method with an integration step size of
10−5 s.
onsiderations are taken into a
ount: if
um > ua,max
ua,min
respe tively.
P < 0, P
is set to 0; if
or
um < ua,min, um
an
respe tively; if
In the simulations of the valve, several physi al
ua > ua,max
or
ua < ua,min, ua
is saturated at
ua,max
or
is saturated at
ua,max
or
M
ua,min
us
point was hanged. The traje tories were obtained using the expli it Euler numeri al integration
Choudhury et al. (2005) notes that the impetus for the os illations is related to the fri tion
FS ,
de reases rapidly when the valve begins to move (a ontributor to this is that the parameter
te
Ff
d
for e dynami s and PI ontroller dynami s. Spe i ally, the fri tion for e dynami s are su h that
whi h represents the stati fri tion oe ient, is larger than
Ac ce p
fri tion oe ient) whi h reates a large and rapid hange in overshoot its value asso iated with
hsp .
FC ,
whi h represents the Coulomb
dvv , leading the valve position to dt
In the following, we dis uss this in detail in terms of
oupling of the pro ess states. The fa t that the dynami s of the various states play a role in the response observed will be exempli ed by analyzing numeri al data from the simulation performed (the exa t numbers reported are related to the integration step size utilized, but the general ee ts would be expe ted to extend qualitatively to other integration step sizes) between the times
17608.72441 s
and
t¯2 = 17608.72597 s
¯1 and t¯2 , Between t
hanges only by pro ess, and from
xv
P
in Fig. 4.
only in reases from 55942.138 to 55942.181
0.0027 N )
at
t¯1
to
P a (the for e due to the pressure
due to the dynami s of the PI ontroller within the oupled nonlinear
hanges only slightly (from
1700.022 N
t¯1 =
1501.874 N
at
t¯2
0.036236 m to 0.036295 m) as well. (i.e., it de reases by lose to
16
However,
200 N ),
Ff
hanges
ausing the right-
Page 17 of 75
hand side of Eq. 28 to in rease from rst two terms related to
P
and
xv
4.1304 N/kg
at
t¯1
to
147.4380 N/kg
at
t¯2
be ause though the
do not hange mu h, the term for the fri tion for e de reases
signi antly. When the right-hand side of Eq. 28 in reases, the valve velo ity in reases, whi h an
ause the valve to move. This shows that though the drop in the fri tion for e played an important
played a role by not allowing
P
xv
and
to adjust rapidly when
P
de reased.
Furthermore, the
as well sin e the PI ontroller depends on
h
in
cr
dynami s of the level play a role in adjusting
Ff
ip t
role in adjusting the valve velo ity, the dynami s of the PI ontroller and the stem position also
relation to its set-point. Additionally, the magnitude of the integral term of the PI ontrol law is
us
signi antly larger than magnitude of the proportional term when the level hanges and overshoots its set-point, whi h may also impa t the length of time that the valve is stu k.
an
The fa t that the negative ee t of sti tion in a ontrol loop is a losed-loop property of a full pro ess-valve system is further emphasized by utilizing the MPC of Eq. 26 (i.e., an MPC that
0.15 m
to
0.20 m
M
a
ounts only for the level dynami s and not the a tuator dynami s) for the set-point hange from for the pro ess-valve system of Eqs. 23 and 27-32, initiated from
10−6 s using the
The pro ess
expli it Euler numeri al integration
d
was integrated with an integration step size of
qI .
te
method. The resulting traje tories of the level under the MPC are shown in Fig. 5, plotted every 1000 integration steps (the values of
h, ua ,
and
um
for this ase are denoted in the legend by U,
Ac ce p
signifying that sti tion is un ompensated in these results be ause the MPC does not a
ount for the a tuator dynami s). No os illations are observed for this level set-point hange as in Fig. 4, demonstrating that the sti tion-indu ed os illations observed under the PI ontroller depend on the
ontroller utilized. Instead of os illations, a persistent oset from the set-point o
urs under the MPC. The reason for this is that be ause the MPC is unaware of the a tuator dynami s, it al ulates values of
um
that orrespond to pressures (through Eq. 32) that do not allow the valve to move
a
ording to the for e balan e (i.e., the MPC expe ts that the ontrol a tions that it al ulates will drive the level toward the set-point be ause it is anti ipating that there is no fri tion in the valve, but due to fri tion the valve annot move with the pressure applied to it). The MPC ontinues to ompute approximately the same ontrol a tion for the rst sampling period of the predi tion horizon at ea h sampling time (whi h is reasonable onsidering that the state measurement that
17
Page 18 of 75
it re eives is approximately the same ea h time sin e the valve is stu k and thus the ow rate out of the valve is not appre iably hanging to adjust the level). This ontrol a tion ontinues to be unable to ae t the level appre iably, resulting in persistent o-set of the level from
hsp
be ause
the MPC has no me hanism for dete ting that the set-points it has al ulated are failing to make
ip t
an impa t on the system. Returning again to the ontrol loop under PI ontrol, we now show that a hange in the ontrol
cr
loop ar hite ture (in this ase, adding ow ontrol to the valve of Eqs. 27-31) alters the response of the pro ess outputs. The ontrol loop ar hite ture in this ase is depi ted in Fig. 6, where the ow
um
is omputed by the PI ontroller of Eqs. 24-25 for the tank level, and be omes
the set-point for a minor PI ontrol loop used to regulate
P
ua
This minor loop al ulates the
um − ua
to be applied to the valve stem based on the error
as follows:
um − ua Kc,p + ζP ua,max τI,p
(33)
M
P = Ps + Kc,p
um .
to
an
pressure
us
rate set-point
um − ua , ζP (0) = 0 ζ˙P = ua,max Ps
is the steady-state value of the pressure, and
d
where
(34)
Kc,p = −82737.09, τI,p = 0.01,
and
ζP
are
te
the proportional gain, integral time, and internal state, respe tively, of the minor loop ontroller.
Ac ce p
The tuning (from Durand and Christo des (2016)) performed su
essfully when
um
was onstant
for some time.
The valve was initially operated without ow ontrol (i.e., Eq. 32 was used to related
P)
for
15600 s
for a level set-point hange from
0.5184 m
was operated under the ow ontroller of Eqs. 33-34 for
0.15 m to 0.20 m. to
Fig. 7 shows the responses of
h, um ,
to
0.15 m
15600 s
and
ua
to rea h
qI .
um
Subsequently, it
for the level set-point hange from
for the set-point hange from
0.20 m (the time axis is short to display the fast response of the valve under ow ontrol).
results were obtained using an integration step size of integration steps. The integral term
ζP
10−5 s,
and
0.15 m These
with the data plotted every 100000
of the ontroller for the valve was re-set to zero when the
level set-point was hanged, and at that point the value of
Ps
was also re-set to the last applied
pressure.
18
Page 19 of 75
The sustained os illations apparent in Fig. 4 do not appear in Fig. 7, though the same pro ess, sti ky valve, and outer loop PI ontroller are used as in Fig. 4. This is parti ularly notable given that the fri tion dynami s and the right-hand side of Eq. 17 have not hanged (i.e., the hange is essentially in the manner in whi h
P
is al ulated; the oupling among the states of the pro ess-
ip t
valve model auses the hange in the system dynami s resulting from the addition of the dynami PI ow ontroller to alter the level response ompared to the ase without ow ontrol). Instead
ua
tra ks
um
well after it initially overshoots
um
in the
cr
of os illations, when ow ontrol is used,
time period immediately after the set-point hange. During these initial overshoots, the pressure
in the initial signi ant overshoots of
ua ,
around the hanging
um
set-point. However, despite these
the ow ontroller is su
essful at stabilizing the level at its set-point (it
an
initial overshoots of
ua
us
applied to the valve hanged rapidly a
ording to Eqs. 33-34, and the for es on the valve resulted
auses the for es on the valve to balan e in su h a manner that
ua
is able to tra k
um
and thus to
M
drive the level to its set-point). The hange of the ontrol loop ar hite ture hanged the number of
oupled dynami states in the system of nonlinear dierential equations des ribing the pro ess-valve
and
zf ,
this means that
it also in ludes
xdyn
ζP
when ow ontrol is used). Returning to the notation of Eq. 8,
te
ζ , xv , vv ,
h,
d
system (i.e., whereas the state ve tor of the pro ess-valve system without ow ontrol in luded
in orporates an extra state and its dynami s when ow ontrol is used, whi h
Ac ce p
overall hanges the response of the measured output (h) of the pro ess-valve system ompared to the ase without ow ontrol.
Remark 5.
The analysis performed demonstrates that the standard valve output- ontroller input
response for a sti ky valve exhibited in Fig. 8 (and displayed in multiple sour es in the literature su h as Choudhury et al. (2005) and Brásio et al. (2014)) an be understood as the response of the valve output when the for e applied to the valve is ramped up and down by a ontroller (i.e., the losed-loop analysis above indi ates that the ontroller output on the standard plots is linked to the for e applied to the valve). Furthermore, Fig. 8 re e ts the transient behavior of the valve after it begins moving (i.e., it shows the slip-jump). This is in ontrast to Fig. 6 in Durand and Christo des (2016), whi h is a plot of the relationship between
ua
and the pressure applied to two
valves, one with low and one with more signi ant sti tion, as the pressure is ramped up and down,
19
Page 20 of 75
but plotting only the nal values of
ua
and
P
at the end of ea h time period for whi h
onstant (rather than throughout the transient as
ua
adjusts to a
P
hange).
P
is held
Furthermore, in
Fig. 8 in Durand and Christo des (2016), the pressure is ramped up and down independently of a ontroller or pro ess (i.e., the valve is analyzed alone); however, in plots like those in Fig. 8 of
ip t
the present work, whi h expli itly assume the use of a ontroller and thereby imply that the plot
omes from measurements of a pro ess-valve system, the oupling of the ontroller, pro ess, and
ua
versus the for e applied to the
cr
valve dynami s for that pro ess-valve system may ause a plot of
valve to be dierent when the same valve is operated under a dierent ontroller or for a dierent
us
pro ess. For example, for the level ontrol problem without ow ontrol, the plot of
ua
versus
P
does not have a signi ant moving phase be ause soon after the valve slips, the ontroller begins
position
xv
(rather than
ua )
an
to drive the valve in the opposite dire tion. Be ause the plot an also be understood as the valve versus the ontroller signal as in Ivan and Lakshminarayanan (2009),
M
a linear valve hara teristi is assumed when the same plot is obtained for signal.
ua
versus the ontroller
d
4.2. A Closed-Loop Perspe tive for a CSTR Example
te
We now demonstrate the ne essity of a losed-loop perspe tive for analyzing a pro ess-valve dynami system when the omplexity of the pro ess model in reases (i.e., it has more states and
Ac ce p
inputs than the level ontrol problem).
For illustration purposes, we examine the produ tion of
ethylene oxide from ethylene in a CSTR with a heating/ ooling ja ket.
Using the rea tion rate
equations for this pro ess from Alfani and Carberry (1970), the following nonlinear pro ess equations are obtained (in dimensionless form) from mass and energy balan es on the rea tor [Özgül³en et al. (1992)℄:
x˙ 1 = ua,1 (1 − x1 x4 )
(35a)
x˙ 2 = ua,1 (ua,2 − x2 x4 ) − A1 exp(γ1 /x4 )(x2 x4 )0.5 − A2 exp(γ2 /x4 )(x2 x4 )0.25
(35b)
x˙ 3 = −ua,1 x3 x4 + A1 exp(γ1 /x4 )(x2 x4 )0.5 − A3 exp(γ3 /x4 )(x3 x4 )0.5
(35 )
x˙ 4 =
B1 B2 ua,1 (1 − x4 ) + exp(γ1 /x4 )(x2 x4 )0.5 + exp(γ2 /x4 )(x2 x4 )0.25 + x1 x1 x1 B4 B3 exp(γ3 /x4 )(x3 x4 )0.5 − (x4 − ua,3 ) x1 x1
20
(35d)
Page 21 of 75
where
x1 , x2 , x3 ,
and
x4
are dimensionless quantities orresponding to the gas density, ethylene
on entration, ethylene oxide on entration, and temperature in the rea tor, and the pro ess inputs
ua,1 , ua,2 ,
and
ua,3
are dimensionless quantities orresponding to the feed volumetri ow rate, feed
ethylene on entration, and oolant temperature, whi h are assumed to be adjusted by individual
ua,1 )
or indire tly (e.g.,
ua,2
may be adjusted by opening or losing
ip t
valves either dire tly (e.g.,
ua,3
valves that allow a more on entrated ethylene stream to mix with a solvent stream, and
may
cr
be adjusted by heating or ooling the oolant using a higher or lower ow rate of another uid past the oolant in a heat ex hanger). Due to the oupling between the states in Eq. 35, and the
regardless of the type of ontroller used to al ulate and
ua,3 = um,3 .
um,1 , um,2 ,
and
um,3 ,
x1 , x2 , x3 ,
and even if
and
x4 ,
ua,1 = um,1 ,
Therefore, if the valves also have dynami s, linear or nonlinear, stati
an
ua,2 = um,2 ,
us
highly nonlinear dynami equations, it is di ult to predi t the evolution of
or dynami , or potentially dierent dynami s for ea h valve, and dierent ontrollers or ontrol loop
M
ar hite tures for ea h valve, the number of oupled states in this system of nonlinear dierential equations in reases and performing simulations will be the best way to understand how ea h pro ess
d
output will respond.
Ac ce p
te
5. Impli ations of a Uni ed Pro ess-Valve System Framework for Valve Behavior Compensation and Modeling The losed-loop perspe tive regarding valve behavior lari es how and when valve behavior
ompensation strategies in the literature work (to demonstrate this, we will review several sti tion
ompensation methods) and enables valve behavior ompensation designs to be proposed that target the oupling of the states that leads to negative ee ts (whi h we will demonstrate by des ribing how two of our valve behavior ompensation strategies, an integral term modi ation method for pro esses with sti ky valves under PI ontrol and an MPC-based general valve behavior
ompensation te hnique, an be understood in a losed-loop ontext). Be ause a full pro ess-valve system model is required to predi t the response of the pro ess and valve outputs under various valve behavior ompensation methods and potentially is needed by those ompensation s hemes to allow them to ompute ompensating signals to adjust the losed-loop response, we also examine the pra ti al question of how an empiri al model that aptures the dominant pro ess-valve dynami s
21
Page 22 of 75
may be obtained when a rst-prin iples model is unavailable or undesirable to use. Finally, we will exemplify the dis ussion through several pro ess examples.
5.1. A Closed-Loop Dynami Analysis of the Sti tion Compensation Literature We analyze three ategories of methods from the sti tion ompensation literature (more thorough
ip t
reviews of sti tion ompensation methods have been provided elsewhere; see, for example, Brásio et al. (2014); Cuadros et al. (2012); Mohammad and Huang (2012)). The novelty of the analysis
cr
of this se tion lies in presenting several of these methods within the losed-loop systems framework
us
des ribed above to provide a fundamental understanding of how they ameliorate the negative ee ts observed in ontrol loops with sti ky valves, and their bene ts and limitations.
an
5.1.1. Sti tion Compensation Methods: Controller Tuning Adjustments
Re-tuning of ontrollers has been advo ated as a method for redu ing losed-loop os illations
M
developed in a ontrol loop ontaining a sti ky valve under lassi al proportional-integral-derivative (PID)-type ontrol (e.g., Mohammad and Huang (2012)). The re-tuning may result in an improved
losed-loop response be ause it hanges the dynami s of the PID-type ontroller, whi h alters the
d
response of the pro ess outputs due to the oupling of the ontroller, pro ess, and valve dynami s.
te
Thornhill and Hor h (2007) highlights the di ulty of determining an appropriate tuning for obtai-
a priori,
the response
Ac ce p
ning a desired response, whi h is onsistent with the di ulty of predi ting, of a nonlinear dynami system.
5.1.2. Sti tion Compensation Methods: Augmented Controller Signal The kno ker [Hägglund (2002); Srinivasan and Rengaswamy (2005)℄ and onstant reinfor ement [Ivan and Lakshminarayanan (2009)℄ adjust the ontrol signal re eived by the valve by adding either a onstant or time-varying signal to the input al ulated by the ontroller. This hanges the manner in whi h the for e applied to the valve is al ulated. For example, onsider the kno ker applied to the level ontrol example without ow ontrol. The PI ontroller has its own dynami s that, in the absen e of the kno ker, di tate the pressure applied to the valve. However, with the kno ker, there are times when the pressure applied to the valve is in reased by an amount determined by the kno ker parameters above the amount output by the PI ontroller, but then after a ertain
22
Page 23 of 75
time period, the kno ker takes away that extra amount of pressure. This allows the PI ontroller to retain its dynami s but permits the pressure to be adjusted using a sour e other than the PI
ontroller as well, whi h hanges the balan e of for es on the valve/the pro ess-valve dynami s and
an result in a dierent losed-loop response than would be obtained without the kno ker. Changes
response as observed in Srinivasan and Rengaswamy (2005).
ip t
to the kno ker parameters hange the losed-loop system dynami s and as a result the losed-loop
cr
Constant reinfor ement similarly augments the output of the PI ontroller, adding a onstant positive signal when the PI ontroller output is in reasing and a onstant negative signal when the
us
PI ontroller output is de reasing, whi h again hanges the right-hand side of the equation for the valve velo ity ompared to not adding su h a signal. One aspe t of the ee t of this on the level
an
ontrol problem might be, for example, that the integral term of the PI ontroller may not need to be ome as large for the pressure from the pneumati a tuation to over ome the stati fri tion
M
for e and ause the valve to move. A method proposed by Cuadros et al. (2012) for turning o the PID-type ontroller and the kno ker is an extension of the kno ker method, but as noted by Wang
d
(2013) and lari ed through the losed-loop analysis in this work, the kno ker hanges the for e
te
balan e/ losed-loop dynami s but that does not guarantee that there will be no oset between the pro ess output and its set-point if the PID ontroller is removed so thus removing the ontroller
Ac ce p
and ompensating pulses may not be appropriate.
5.1.3. Sti tion Compensation Methods: Two Moves Method The two moves method [Srinivasan and Rengaswamy (2008)℄ spe i es ompensating signals to apply to the signal oming from a linear ontroller that will drive the stem position to its set-point. This method is model-based, whi h means that it has a
ounted for the oupling of the pro essvalve dynami s, and a number of assumptions are required to guarantee that the method an drive the valve position to its set-point, in luding that a parti ular data-driven sti tion model is an exa t representation of the sti tion dynami s and that there are no disturban es or plant-model mismat h. A method similar in on ept to the two moves method (it determines how to hange the set-points for a losed-loop system in a manner that brings the valve position to a desired value) is developed in Wang (2013) and is inspired by a pro ess-valve model for pro esses that an be des ribed with
23
Page 24 of 75
a linear model and are under linear ontrol.
5.2. Motivating Valve Behavior Compensation Design with Closed-Loop Dynami Analyses In this se tion, we seek to en ourage further resear h on valve behavior ompensation based on the losed-loop perspe tive on valve behavior developed in this work. To do this, we des ribe how
5.2.1. Sti tion Compensation Methods: Integral Term Modi ation
ip t
two of our re ent sti tion ompensation methods an be motivated by this framework.
cr
The ontroller tuning adjustment methods dis ussed above require a ontroller's tuning to be
us
hanged even if the tuning being utilized is known to work well for the valve when it is not sti ky and thus would be the preferred tuning after valve maintenan e is performed. Furthermore, the
an
nonlinearity of the sti tion phenomenon, the oupling between the pro ess states, and the omplexity of the manner in whi h the for es on the valve balan e and ome out of balan e makes it di ult to dis ern
a priori
what the best tuning to use when the valve is sti ky should be. Therefore, it
M
is reasonable to onsider whether there is another method of adjusting a PI ontroller's dynami s instead of disrupting the desired tuning in an
ad ho
fashion.
d
One su h method is to add a term to the integral a tion of the linear ontroller that an easily be
te
removed or adjusted for any set-point hange to attempt to alleviate sti tion-indu ed os illations.
Ac ce p
Given that a hara teristi of the sti tion-indu ed os illations is that the valve output tra k its set-point
um ,
this added term an be based on the s aled dieren e between
Spe i ally, for a PI ontroller for whi h
ζ
ua
does not
ua
and
um .
signi es the integral of the error between the pro ess
output set-point (assuming a single output denoted by
xˆsp )
and the pro ess output (x ˆ, assumed to
be a omponent of the pro ess state ve tor), the following ontrol law de nes the ontrol a tion with an integral term modi ation:
um = uas + Kc (ˆ xsp − xˆ) + Kc ζ/τI ζ˙ = ζ(0) = 0 where
L, β ,
and
tAW
(ˆ xsp − xˆ) + L(ua − um ), t < tAW (ˆ xsp − xˆ) + Le(−β(t−tAW )) (ua − um ), t ≥ tAW
(36)
(37)
are tuning parameters that an be adjusted by a ontrol engineer to attempt
to mitigate sti tion-indu ed os illations. When
L = 0,
24
Eq. 37 redu es to the standard integral term
Page 25 of 75
in a PI ontrol law, and thus has no ee t. The parameters are best determined using losed-loop simulations and/or on-line adjustments of
L, β , and tAW ; however, the general goals of adjusting the
parameters provide a potential methodology for looking for an appropriate tuning. In parti ular, the goal of this method is to determine a tuning that an de rease
ζ˙
in su h a way that the for es
of the term
(ˆ xsp − xˆ),
L
that auses the term
it is possible to ause
ζ˙
L(ua − um )
in Eq. 37 to have a sign opposite to that
to de rease even before
xˆ = xˆsp
(this would not be
cr
By hoosing a value of
ip t
on the valve equilibrate at a value that auses the ontrolled pro ess output to meet its set-point.
possible with the standard PI ontrol law, for whi h the integral term an only begin to de rease
ζ˙
β
to equal zero and drive
and tAW to zero, and then sear hing for a value of
L
in
L that is able to ause
dvv to zero by providing a onstant for e from the valve a tuation. This dt
may o
ur, however, before
xˆsp
an
Eq. 37 is by rst setting
us
after the set-point is ex eeded). Therefore, a possible strategy for tuning the term ontaining
is rea hed, resulting in oset. Therefore, the value of
tAW
may be
M
set to a time at whi h the for es on the valve appear to no longer be hanging and that allows
xˆ
to begin to approa h its set-point as soon as possible after this has o
urred. Then, various values
β
may be tried to attempt to de rease the term ontaining
L
in
ζ˙ ,
whi h an ause this integral
d
of
um
te
term to hange and thus hanges the for e applied to the valve as a result of the ontrol a tion re eived by the valve. If a value of
β
an be found that hanges the for e applied to the valve
Ac ce p
in a manner that auses the for es to on e again be ome onstant, but this time at a value of the valve position that auses
xˆsp
to be rea hed, then this ontrol strategy is su
essful for the set-point
hange examined. However, some values of up on e again even if the value of
L
β
may even ause sti tion-indu ed os illations to be set
examined was able to attenuate them before the time
tAW ;
this shows that the tuning problem is omplex and that an appropriate tuning annot be de ided
priori.
a
In addition, due to the nonlinearities in the valve and pro ess dynami s, and the potential
for intera tions between states and inputs, there is no guarantee that any appropriate tuning will be found for a given set-point hange, or that the same tuning will work for a variety of set-point
hanges or disturban es; however, losed-loop simulations or on-line adjustment an be attempted to see if there are values of
L, β , and tAW
that are generally appropriate for a given pro ess. Notably,
when the integral term modi ation method is utilized in the ase of bounded, time-varying pro ess
25
Page 26 of 75
disturban es, whi h is the ase onsidered in this work a
ording to the lass of pro ess systems in Se tion 2.2, the notion of driving the output to the set-point does not exist (instead, it would be hoped that the ontroller ould drive the state to an a
eptably small neighborhood of the set-point).
Various tunings ould be evaluated for the integral term modi ation in this ase to
ip t
determine if they are able to do so.
5.2.2. Valve Behavior Compensation Methods: MPC for Valve Behavior Compensation
cr
The losed-loop perspe tive on valve behavior indi ates that a ompensation method that a
ounts for the pro ess-valve dynami s when omputing ontrol a tions would be able to systemati-
us
ally address the root ause of the negative ee ts observed in ontrol loops with valves that have behavior that annot be negle ted. An MPC design that uses the pro ess-valve model to make state
an
predi tions is able to do this. An MPC design with a general obje tive fun tion that in orporates a pro ess-valve model in luding sti ky valve dynami s was developed in Durand and Christo des
M
(2016). It an a
ount for multiple sti ky valves, even with diering levels of sti kiness, and has the ability to in orporate onstraints that guarantee feasibility and losed-loop stability of a nonlinear
d
pro ess operated under the ontroller. In general, any pro ess-valve model an be used within this
te
MPC framework, so it ould be onsidered for valve behavior ompensation in general (i.e., it is not spe i to sti tion). This allows the MPC to handle pro esses where a single valve exhibits
Ac ce p
multiple nonlinearities (e.g., sti tion and also an equal per entage valve hara teristi ) or where the various inputs exhibit dierent dynami s (e.g., one valve is sti ky but another for the same pro ess has linear dynami s) if the dynami s an be modeled and then in luded within the pro ess-valve dynami model.
The formulation of an MPC that in ludes the valve and pro ess dynami s (Eqs. 7-8) is as follows:
min
um (t)∈S(∆) s.t.
Z
tk+N
Le (˜ q (τ ), um (τ )) dτ
(38a)
tk
q˜˙(t) = fq (˜ q(t), um (t), 0)
(38b)
q˜(tk ) = q(tk )
(38 )
q˜(t) ∈ Qv , ∀ t ∈ [tk , tk+N )
(38d)
26
Page 27 of 75
um (t) ∈ Um , ∀ t ∈ [tk , tk+N )
(38e)
gM P C,1 (˜ q(t), um (t)) = 0, ∀ t ∈ [tk , tk+N )
(38f )
gM P C,2 (˜ q(t), um (t)) ≤ 0, ∀ t ∈ [tk , tk+N )
(38g)
of Eqs. 38b-38 and is bounded within the set
Qv
(Eq. 38d; the set
Qv
q˜ follows
the model
ip t
where the notation follows that in Eq. 11. The predi ted pro ess-valve state
is de ned to be the bounds
ua ∈ U
and the state onstraints orresponding to
Eq. 2 (sin e ea h
um,i
sin e
ua
is a fun tion of the states from
is al ulated by a state feedba k ontroller) and Eq. 4).
us
x∈X
cr
on the pro ess-valve model states, whi h may in lude, for example, the state onstraints restri ting
The onstraints
in Eqs. 38f-38g may in lude onstraints related to spe i behavior of the valve.
For example,
an
in Durand and Christo des (2016), it is noted that a bene ial onstraint for a sti ky valve may be one whi h ensures that only physi al values of the for es applied to the valve are predi ted
P˜
of
P
to satisfy
M
(for example, it may require the predi tions
P˜ ≥ 0
for the tank level example).
Theoreti al results regarding disturban e reje tion for an MPC with the form in Eq. 38 but with
d
ertain stability-related onstraints added to the formulation have been developed in Durand and
te
Christo des (2016). Guaranteed robustness to disturban es in that ase requires several onditions to hold, in luding that the disturban e magnitude be su iently small.
In general, there is no
Ac ce p
guarantee that the losed-loop response of a pro ess under an MPC in the presen e of disturban es will be as desirable as that in the ase without disturban es. Two potential limitations of an MPC for valve behavior ompensation are: 1) The sampling period may, for omputation time reasons, be signi antly longer than the time that it takes for the for es on the valve to rea h equilibrium after the deadband and sti kband are over ome. This means that the MPC may not have the exibility to hange the for e from the a tuation rapidly in response to a rapid hange of the fri tion for e (this is dis ussed further in Se tion 5.4.2). 2) It may be di ult to determine all parameters of an appropriate rst-prin iples model of the valve layer for a given valve (the valve layer for valve the relationship between
um,i
and
i
is de ned in this work to refer to all dynami s des ribing
ua,i , i = 1, . . . , m).
Empiri al models for the valve layer dynami s
(and potentially the pro ess dynami s as well) may aid in handling both of these limitations be ause
27
Page 28 of 75
they may not require detailed information on the valve layer dynami s like the details of the fri tion for e model or valve hara teristi , and they may also be less sti than a rst-prin iples model (an example of this is demonstrated in Se tion 5.4.3), resulting in a lower omputation time for the MPC for valve behavior ompensation. If the model utilized within the MPC an apture the dominant
ip t
pro ess-valve dynami s to provide su iently a
urate state predi tions, it would be expe ted to be bene ial in ompensating for sti tion, even if it is not an exa t model, due to the in orporation
cr
of state feedba k. The following equation denotes an empiri al model for use in the MPC-based
y(t) ˙ = fy (y, um) y(t)
ua
is the predi ted value of
hara terized by the ve tor fun tion
fy .
from the empiri al model at time
an
where
us
valve behavior ompensation method:
(39)
t
and has dynami s
This empiri al model an be used to predi t the values of
M
the pro ess inputs in the obje tive fun tion and onstraints of Eq. 38 when a rst-prin iples model is not used.
d
5.3. Valve Behavior Empiri al Modeling Considerations Following Closed-Loop Dynami Analyses
te
Two questions that arise when onsidering how to empiri ally model a full pro ess-valve system
Ac ce p
are: 1) what part of the pro ess-valve dynami s should be aptured in a single empiri al model and 2) is it possible to develop empiri al modeling strategies for valve behavior that are general rather than fo used on a spe i type of valve behavior. To explain the impli ations of the rst question, we rst examine sti tion empiri al modeling te hniques in the literature (e.g., the Choudhury model [Choudhury et al. (2005)℄, the Kano model [Kano et al. (2004)℄, and the He model [He et al. (2007)℄), whi h generally assume that the relationship between
ua
and the for e applied to the valve
is similar to that in Fig. 8, so they use an if-then (bran hed) type stru ture to mimi this (i.e., if the ontrol signal has not hanged enough to un-sti k the valve, then the valve position does not
hange with a hange in the ontrol signal; if the ontrol signal has hanged enough to un-sti k the valve, the valve takes a new position de ned by the empiri al model). These models relate the valve position and the for e on the valve determined by a ontroller. However, this may not eliminate the need to develop rst-prin iples models for some of the dynami s relating
28
um,i
and
ua,i
for the
Page 29 of 75
i − th
valve. For example, if ow ontrol is utilized on the valve as in the example of Fig. 7, some
knowledge of the relationship between the valve output ow rate set-point from the major loop
ontroller and the for e applied to the valve by the minor loop ontroller is required to model the valve layer. If an equal per entage valve hara teristi hara terizes the
ua,i − xv,i
relationship, the
ip t
form of this model must be known. The prin iple demonstrated through this analysis is not spe i to sti tion: to avoid the need to obtain a rst-prin iples model for any aspe t of the valve layer
um,i
to
ua,i .
We show in
cr
dynami s, it may be bene ial to develop an empiri al model that relates
Se tion 5.4.3 that it may be possible to develop an a
eptable relationship even when the
um,i − ua,i
us
dynami s are somewhat omplex (e.g., they ontain the dynami s not only of the valve but also of a ow ontroller for the valve).
to obtain a
relationship an be obtained, it should be determined whether it would be possible
an
um,i −ua,i
um −x relationship as well (i.e., to develop a single empiri al model for the pro ess-valve
system that relates the ontroller output obtain data on the pro ess input ve tor
um
ua
to the pro ess states su h that it is not ne essary to
M
If a
when developing the empiri al model). However, there
d
may arise ases in whi h it may not be possible to easily apture the pro ess and valve dynami s in
te
the same empiri al model. To see why di ulty may arise, onsider a pro ess with multiple inputs, all of whi h are adjusted by sti ky valves. Be ause the valves are sti ky, ea h hange in
ua,i
will not ause in
um,i
to hange appre iably (whi h will ause
Ac ce p
either ause
as if
ua,i
um,i
to hange appre iably (whi h will ause
to respond to the hange in
x
um,i ),
will or it
to ontinue behaving after the hange
had not hanged). Due to the oupling of the dynami s of the states of a pro ess-
valve system, the ombination of des ribing the
x
um,i
um − x
ua,i 's
ae ts the pro ess dynami s, so a bran hed empiri al model
relationship may need to in lude dierent bran hes for every ombination
of sti king and slipping for all valves, whi h may lead to a di ult identi ation task. A solution if this is found to be an issue would be to empiri ally model ea h valve as well as the
ua − x
ua,i − um,i
relationship for the pro ess (this may also be bene ial from a valve
maintenan e perspe tive be ause it permits monitoring of how losely valve (assuming that the
relationship for ea h
ua,i − um,i
ua,i
mat hes
um,i
for ea h
relationship an be assessed frequently or potentially that an
empiri al modeling pro edure for those dynami s ould be automated), allowing valve maintenan e
29
Page 30 of 75
to be performed rst on those valves for whi h
ua,i
tra ks
um,i
the least).
To examine the impli ations of the se ond question above (whether a general empiri al modeling strategy an be used to obtain the
um,i − ua,i
relationship) we return again to the sti tion empiri al
models of the literature. Many of these assume that a spe i trigger auses the valve to sti k or slip
ip t
and they assume a spe i model for the valve response after slipping o
urs that is related to the
hara teristi s of the typi al response of a sti ky valve to ontrol signal hanges. For a general valve
ne essary to onsider whether a
cr
layer (i.e., the type of dynami s and ontrol ar hite ture within the valve layer are not spe i ed), it is
ua,i −um,i relationship an be developed without ne essarily knowing
us
the underlying valve behavior/loop ar hite ture represented through the relationship.
Bran hed
models may be required to do this be ause valves are not ne essarily operated within a range of
an
states throughout whi h linearization of the valve dynami s is adequate for representing the valve response. For example, onsider the model of a valve's dynami s in Eqs. 27-32. The valve velo ity
M
must be for ed through zero every time the dire tion of the valve movement hanges, whi h is a point where Eq. 30 is non-dierentiable due to the absolute value in the right-hand side. Therefore, the
d
model of Eqs. 27-32 annot be linearized at zero velo ity every time the valve movement hanges Though Eq. 30 represents the dynami s for
te
dire tion (whi h has the potential to be frequent).
only one of many fri tion models, its form indi ates that the fri tion nonlinearity may need to
Ac ce p
be represented by more omplex nonlinearities su h as absolute values than are usually present in standard empiri al model stru tures su h as linear or polynomial state-spa e models, and as a result when the latter models are used, bran hed models may be required to apture the ee ts of the nonlinearities of the valves.
When a bran hed model is used, some understanding of the physi s of the valve layer an be exploited by an engineer in developing the triggers for the bran hes, or attempts an be made to automate the determination of triggers from data. If the valve layer has a number of states, this may result in an empiri al model with a signi ant number of bran hes and even with parameters that depend on the inputs to the valve layer. For example, onsider that we want to develop an empiri al model for the relationship between
ua,i
and
um,i , i = 1, . . . , m,
when ea h
um,i
is a valve
output ow rate set-point from an MPC whi h is transmitted to a valve layer ontaining a sti ky
30
Page 31 of 75
valve with a linear valve hara teristi under a PI ow ontroller based on the error between and
ua,i .
We assume that
um,i , i = 1, . . . , m,
is held onstant for a time period
∆
during whi h the
PI ontroller repeatedly omputes new values of the pressure applied to the valve stem to drive to
um,i .
um,i
If the valve output ow rate set-point hange dire tion reverses (e.g., the set-points
ua,i um,i
throughout some or all of the time period
∆
during whi h
um,i
is onstant, depending on whether
∆
to over ome the for e required to
cr
the pressure applied to the valve hanges enough throughout
ip t
were previously in reasing but the next set-point is lower than the previous one), the valve may sti k
move the valve. Also, the PI ontroller will al ulate larger ontrol a tions when the valve output
us
ow rate set-point hanges signi antly (be ause this reates a large error between
um,i
and
ua,i ),
and su h larger ontrol a tions (pressures) are more likely to over ome the deadband/sti kband
∆.
Therefore, it is more likely that the valve will move if the ontroller is
an
within the time period
aggressively tuned or if the set-point hange is large and the error between
um,i
and
ua,i
ae ts the
M
ontroller output. Saturation of the valve (e.g., the pressure from the pneumati a tuation drops to zero so that the valve an no longer move in the dire tion of de reasing pressure) an also o
ur.
d
Therefore, the various bran hes of the response may need to be triggered by onsiderations related
te
to the sti tion dynami s and saturation of the pressure of the pneumati a tuation, with the time that the valve is stu k depending on the set-point hange magnitude due to the ow ontroller.
Ac ce p
The pro edure proposed for empiri al modeling of the feedba k loop for the sti ky valve under
onsideration is as follows:
1. Colle t valve layer input-output data (i.e., data relating
ua,i
and
um,i , i = 1, . . . , m),
ensuring
that data gathered represents all aspe ts of the valve layer response that should have separate equations (bran hes) within the pie ewise model stru ture (e.g., sti king and slipping). 2. For ea h aspe t of the response that requires its own model stru ture, sele t an appropriate stru ture based on the valve layer input-output data trends and determine what a tivates the dierent bran hes of the response (e.g., set-point hange dire tion reversals). 3. Identify the parameters for the dierent bran hes of the model. 4. Develop models for any parameters that an be seen from the valve layer input-output data to be dependent on the valve layer inputs (e.g., parameters that depend on the magnitude of
31
Page 32 of 75
the valve output ow rate set-point hange). 5. Validate the nal pie ewise-de ned model.
It will be shown in Se tion 5.4.3 of this work that Step 2 in the pro edure above may be able to be performed with standard empiri al model stru tures in the hemi al pro ess industries, su h
ip t
as se ond-order or rst-order-plus-dead time models (i.e., they do not take a form dependent on the valve behavior being modeled). Furthermore, the ve steps above are general and therefore are
cr
not ne essarily restri ted to the valve layer with a sti ky valve under ow ontrol above but an be
us
examined for extension to other ontrol loop ar hite tures and valve behavior that hara terize the valve layer.
an
A nal onsideration related to empiri al modeling of a pro ess-valve system is that there may be states that it is desirable to onstrain in an MPC-based valve behavior ompensation strategy that would be known from a rst-prin iples model but are not expli itly modeled within an empiri al
M
model. As a result, onstraints of the MPC-based valve behavior ompensation strategy may need to be adjusted to be in terms of the states of the empiri al model and the inputs. For example,
d
Eqs. 38f-38g may in lude a onstraint on the pressure from the pneumati a tuation to prevent it
um,i
to
ua,i
for a valve layer with a ow ontroller may not have any state related
Ac ce p
model relating
te
from saturating in the ase that a sti ky valve is in the loop. However, an empiri al valve layer
dire tly to this pressure. As a result, the following input rate of hange onstraints an be added to the MPC of Eq. 38 in pla e of Eqs. 38f-38g:
|um,i (tk ) − u∗m,i (tk−1 |tk−1 )| ≤ ǫ, i = 1, ..., m
(40a)
|um,i (tj ) − um,i (tj−1 )| ≤ ǫ, i = 1, ..., m, j = k + 1, ..., k + N − 1
(40b)
These onstraints require the dieren es between the values of periods of the predi tion horizon to be no greater than in the set-points
um,i
ǫ
um,i
at any two subsequent sampling
and therefore an prevent large hanges
between two sampling periods that may saturate the valve a tuation (or wear
the a tuators [Durand et al. (2016)℄). However, applying input rate of hange onstraints in an MPC for valve behavior ompensation
32
Page 33 of 75
is not always bene ial. For example, onsider the sti ky valve without ow ontrol of Eqs. 27-32. As exempli ed in Fig. 5, the valve does not move when the valve is stu k if the hange in
um does not
hange the pressure applied to the valve enough to over ome the stati fri tion for e. This indi ates that even if the valve dynami s were in luded in the MPC for valve behavior ompensation for this
um
that may satisfy
ip t
valve but input rate of hange onstraints were also in luded, the values of
these onstraints may not dier enough from the values at the prior sampling time in some sampling
the pressure applied to the valve is not an expli it fun tion of
cr
periods to allow the valve to move. If the valve is under the ow ontroller of Eqs. 33-34, however,
um , but instead is hanged throughout um
and
us
a sampling period by the ow ontroller. If the dieren e between
ua
in Eqs. 33-34 is large,
it auses the pressure applied to the valve stem to hange signi antly, and may potentially ause
allow the set-point
um
to be rea hed by
ua .
an
the pressure applied by the a tuation to saturate (e.g., hit its minimum bound) and therefore not In Eq. 38, this may be prevented through the a tuation
M
magnitude onstraint; in the ase when the empiri al sti tion model is used and details on the a tuation magnitude are not aptured by the empiri al model so that they an be onstrained,
d
utilizing the input rate of hange onstraints may be bene ial for preventing large hanges in
ua
from rea hing
um .
This emphasizes
te
that ould saturate the a tuation and therefore prevent
um
that be ause sti tion is a losed-loop ee t, onstraints added to an MPC to ompensate for valve
Ac ce p
behavior should be hosen with an understanding of the impa t of the ontroller and ontrol loop ar hite ture on the valve behavior.
Remark 6.
Prior works that have looked at MPC with a general obje tive fun tion with empiri al
models [Alanqar et al. (2017, 2015a,b)℄ have fo used on empiri al models of the nonlinear pro ess (rather than valves) and have indi ated that signi ant omputation time redu tions may result from using empiri al as opposed to rst-prin iples models in MPC. We will demonstrate in Se tion 5.4.3 that empiri ally modeling the valve layer an result in omputation time redu tion even if the pro ess is modeled with a rst-prin iples model be ause it an make the pro ess-valve ombination model less sti.
Remark 7.
One ould examine whether the MPC-based valve behavior ompensation method, par-
ti ularly with an empiri al valve layer model, ould be utilized for ompensating for dynami ee ts
33
Page 34 of 75
(e.g., slower movement of a valve or la k of movement) related to physi al valve issues like oversizing, undersizing, orrosion, leaks through the valve pa king, or diaphragm faults that an ae t valve performan e [Choudhury et al. (2008)℄.
Remark 8.
An appropriate empiri al model stru ture for the valve layer input-output data must be
ip t
sele ted. Many model identi ation te hniques for obtaining linear and nonlinear empiri al models exist that an be evaluated for their suitability for modeling a given valve layer input-output trend,
cr
whi h fall in the ategories of state-spa e and input-output models (see, for example, Ljung (1999); Billings (2013); Van Overs hee and De Moor (1996); Paduart et al. (2010)).
us
5.4. Valve Behavior Compensation Methods: Pro ess Examples
an
We now demonstrate the development and use of several of the valve behavior ompensation methods des ribed above for ontrol loops ontaining sti ky valves.
5.4.1. Level Control Example with a Sti ky Valve: Integral Term Modi ation
under the PI ontroller of Eqs. 24-25 for to rea h
qI .
0.15 m
0.20 m.
to
xˆ = h)
The expli it Euler method with a numeri al
L = 7, β = 0.007,
to
0.20 m).
and
tAW = 22880 s
(i.e.,
7280 s
after the
From omparison with Fig. 4, the addition of the term
L to the integral a tion was able to redu e ontrol loop os illations (though there is some
oset from the set-point for the hosen
ζ˙ = 0
0.5184 m
was used, and the results are shown in Fig. 9 (plotted every 100000
integration steps) for the ase that set-point hange from
to
te
10−5 s
0.15 m
Ac ce p
integration step size of
have
for a level set-point hange from
Subsequently, it was ontrolled using the ontroller of Eqs. 36-37 (with
for a level set-point hange from
ontaining
15600 s
d
0.15 m
M
The pro ess of Eq. 23 with the open-loop valve dynami s in Eqs. 27-32 was initially operated
when
h 6= hsp ).
After
tAW ,
L
before
tAW
the value of
um
be ause the integrator state in Eq. 37 an is able to hange again be ause
ζ˙
be omes
nonzero, and eventually the valve moves and the for es due to this strategy balan e in su h a way that the set-point is a hieved.
5.4.2. Level Control Example with a Sti ky Valve: MPC with a First-Prin iples Valve Layer Model For the level ontrol problem, we develop an MPC for sti tion ompensation as follows:
min
um (t)∈S(∆)
Z
tk+N
tk
˜ 2 + RT (ua,sp − u˜a )2 dτ QT (hsp − h)
34
(41a)
Page 35 of 75
q˜(tk ) = q(tk )
(41 )
0 ≤ u˜a (t) ≤ 0.006, ∀ t ∈ [tk , tk+N )
(41d)
0 ≤ um (t) ≤ 0.006, ∀ t ∈ [tk , tk+N )
(41e)
P˜ ≥ 0, ∀ t ∈ [tk , tk+N )
(41f )
ua,sp = 0.00373 m3 /s, hsp = 0.20 m, QT = 0.00001,
predi tions of
ua
ip t
(41b)
RT = 1. u˜a
and
and
˜ h
denote the
and of the level a
ording to Eq. 41b, respe tively. The notation of Eq. 41 follows
that in Eq. 38. The pro ess-valve state ve tor
q T = [h xv vv zf ]
is modeled for the open-loop valve
us
where
q˜˙(t) = fq (˜ q (t), um (t), 0)
cr
s.t.
using Eqs. 23 and 27-32. The pro ess-valve system was initiated at
150 s
with
∆=1s
and
N = 50.
The level was ontrolled by
An integration step of
an
the MPC of Eq. 41 for
qI .
within the MPC to integrate Eq. 41b, with an integration step of
10−6 s
10−5 s
was used
used outside of the MPC
M
to simulate the pro ess. The onstraints in Eqs. 41d and 41f were enfor ed on e every sampling period. The obje tive fun tion derivatives required by the optimization solver Ipopt were al ulated
d
using a entered nite dieren e, and the Ipopt limited-memory Hessian approximation option was
te
used, so that the non-dierentiability in Eq. 30 did not prevent a solution to the optimization problem from being obtained. The results in Fig. 5 (designated by C be ause the valve dynami s
Ac ce p
are ompensated) are plotted every 1000 integration steps and indi ate that the MPC in luding the valve dynami s drove the level toward its set-point, in ontrast to the MPC that did not in lude the a tuator dynami s (designated by U in this gure). The reason for this is that the MPC of Eq. 41 in orporates the full pro ess-valve model, and thus omputes an input traje tory that a
ounts for the manner in whi h the for es on the valve will balan e under the ontrol a tions al ulated by the MPC.
Though the MPC was able to drive the level toward its set-point (a signi ant improvement with regard to set-point tra king of the level ompared to the ase that the valve dynami s were not a
ounted for within the MPC), the fa t that the valve is operated without ow ontrol redu es the exibility of the MPC to be able to maintain the level at the set-point for all times after it rst approa hes the set-point in Fig. 5. Spe i ally, ea h time that the MPC sets
35
um ,
the pressure
Page 36 of 75
applied to the valve hanges a
ording to the relationship of Eq. 32 sin e the valve is operated without ow ontrol.
However, be ause the MPC implements pie ewise- onstant ontrol a tions
that are held for a sampling period, the pressure that is applied to the valve (a fun tion of
um
al ulated by the MPC) is held onstant throughout a sampling period. The length of the sampling
ip t
period in this example is long ompared to the dynami s of the valve, su h that the dynami s of the valve under a onstant applied pressure di tate the nal position of the valve at the end of a
um
that will drive the
cr
sampling period. The result is that the MPC is not able to nd a value of
valve, subje t to its dynami s during the sampling period that the pressure is held onstant and
us
the MPC annot intervene, exa tly to the valve position orresponding to the steady-state ow rate through the valve when the level set-point is a hieved. Instead, the MPC must ontinuously
um
that allow the valve to sti k and slip in ways that the MPC nds will
an
al ulate new values of
minimize the tra king obje tive fun tion and therefore keep the value of
h
in a region around the
M
set-point over time. Based on this analysis, potential ways of improving the set-point tra king of the level in lude de reasing the MPC sampling period until it is on a time s ale omparable to
d
the time s ale of the valve dynami s, in reasing the predi tion horizon to give the MPC greater
ua
te
foresight to potentially allow it to determine a sequen e of values of
um
that an drive the value of
to its set-point, or adding ow ontrol to the valve and then in luding the dynami s of both the
Ac ce p
valve and the ow ontroller in the MPC as it al ulates set-points
Remark 9.
um
for the ow ontroller.
Fig. 5 demonstrates the ee ts of not a
ounting for the behavior of the valve of Eqs. 27-
32 within an MPC (no signi ant hange of the level for ertain hanges in the valve output ow rate set-point
um )
and the improvement that an be obtained when the dynami s are a
ounted
for. However, other types of valve behavior that are not exhibited by the valve of Eqs. 27-32 an result in dierent types of negative ee ts when the valve behavior is not in luded within the MPC model for making state predi tions. For example, onsider a valve without sti tion but with the following equal per entage valve hara teristi [Coughanowr and LeBlan (2009)℄:
ua = ua,max eln(0.03)xv /xv,max
(42)
developed for the ase that the valve stem is fully retra ted when the valve is fully open and is fully
36
Page 37 of 75
extended when the valve is fully losed. Assume that the valve an be manipulated in su h a manner that the valve position is an expli it fun tion of
um
given by Eq. 31 (i.e.,
though this linear relationship does not re e t the nonlinear
um xv,max ) xv = xv,max − ua,max
xv −ua relationship of Eq. 42.
If an MPC
is used to ontrol the pro ess but is not aware of the mismat h between the valve behavior of Eq. 42
um − xv
relationship that sets the valve position (a similar on ept to the mismat h
between Eq. 32 and the a tual
ua − P
ip t
and the linear
relationship for the sti ky valve of Eqs. 27-31), permanent
ow rate of
0.00323 m3 /s,
The steady-state ow rate for a level of
0.15 m
orresponding to a steady-state
ua,sp = 0.00373 m3 /s.
orresponding to
orresponds to a fra tion
FI,A = 0.5379
of the
through the valve as shown in Fig. 10. For the equal per entage
an
0.006 m3 /s
maximum ow rate of
hsp = 0.20 m
to the set-point
0.15 m,
us
onsider again the set-point hange from an initial level of
For example,
cr
oset of the level from its set-point an result due to the plant-model mismat h.
valve of Eq. 42, the valve position asso iated with the steady-state ow rate for the initial level is a
XI,A = 0.1768
of its maximum. When the set-point of the level is hanged to
M
fra tion
0.20 m,
the
ow rate out of the valve should in rease to a hieve this (ideally it should rea h the fra tion of the
FD = 0.62113 shown Fig. 10).
For the equal per entage valve of Eq. 42, this ow
d
maximum ow rate
te
rate is a hieved at a fra tion of 0.1358 of the maximum stem position. The used to set the stem position based on
fra tion
the linear
um − xv
XD,C = 0.37887
relationship
however, is linear, so when the MPC requests that
relationship moves the valve stem to a position orresponding to a
Ac ce p
um = ua,sp ,
xv ,
um − xv
of the maximum stem position. However, when the fra tion of the stem
position for an equal per entage valve is 0.37887, the fra tion of the maximum ow through the valve is
FD,A = 0.2649
( orresponding to a ow rate lower than the initial value instead of above
it as desired). This example highlights that in luding valve behavior in MPC an be bene ial for many types of valve behavior. Also, the example valve in this se tion has a the form in Eq. 2 sin e the
ua − xv
xv
in the linear
um − xv
um − ua
relationship of
relationship an be substituted in terms of
um
in
relationship of Eq. 42.
5.4.3. Ethylene Oxidation Example with a Sti ky Valve: MPC with an Empiri al Valve Layer Model In this se tion, we return to the ethylene oxidation example of Se tion 4.2 for the ase that
ua,2 = 0.5
and
ua,3 = 1.0
(the values of the other parameters are listed in Table 2). We onsider
37
Page 38 of 75
that the value of
ua,1
is adjusted by a spring-diaphragm sliding-stem globe valve (be ause it is the
only manipulated input onsidered for this pro ess in the following example, we will drop the 1 in the subs ript for this se tion and refer to the pro ess input as ua ). This valve is under ow
ontrol and has the same design as the sti ky valve des ribed in Se tion 4.1 (i.e., it is pneumati ally
open position, it has
xv = 0 m
u a = 0,
xv = xv,max = 0.1016 m
when the
and the valve layer dynami s are des ribed by Eqs. 27-31 and
cr
valve is fully losed with
when the valve is fully open and
ip t
a tuated and pressure-to- lose, with no pressure applied to the valve initially when it is in its fully
33-34) ex ept that the time unit for all parameters and variables is denoted by a dimensionless unit instead of
s for onsisten y with the dimensionless units in Eq. 35 (i.e., all valve model parameter
us
td
values in Table 1 apply for this valve ex ept that ea h instan e of the unit
ζP
and of the steady-state value of the pressure
(ζP is re-set to zero, and
Ps
Ps
ua = ua,max = 0.7042.
are re-set ea h time that
is set to the last applied pressure).
M
values of
an
with td in this example) and the fully open valve position orresponds to
s in that table is repla ed
The value of
um
um
The
hanges
is hanged by
an EMPC (an MPC with an e onomi s-based obje tive fun tion) every sampling period of length
d
∆ = 0.2 td .
te
The ontrol obje tive is to maximize the yield of ethylene oxide utilizing an MPC that a
ounts for the valve dynami s, where the yield is given by the following ratio of the amount of ethylene
Ac ce p
oxide produ ed from the rea tor in a time period of length tf the rea tor in that time:
R tf t0
Y (tf ) =
− t0
to the amount of ethylene fed to
ua (τ )x3 (τ )x4 (τ )dτ R tf 0.5ua(τ )dτ t0
(43)
We also onsider that the valve output ow rate is onstrained between the minimum ow through the valve (0) and the maximum ow (0.7042), and is also required to satisfy the following restri tion on the amount of ethylene that an be fed in a time period of length
1 tf − t0
Z
tf − t0 :
tf
0.5ua (τ )dτ = 0.175
(44)
t0
This onstraint requires that the amount of ethylene fed to the pro ess in a time period of length
tf − t0
must equal the amount that would be fed in that time period under steady-state operation.
38
Page 39 of 75
We seek to avoid fully losing the valve by requiring the MPC to keep the value of
um
between
0.0704 and 0.7042. To a hieve the ontrol obje tives, we will utilize an MPC with an empiri al model of the valve dynami s, and we will ompare the omputation time of that ontroller with the omputation time
ip t
of an MPC that in ludes a rst-prin iples model of the valve dynami s. We develop the empiri al model for the valve layer des ribed in Eqs. 27-31 and 33-34 a
ording to the steps outlined in data, and noti e that when the set-points
cr
um − ua
Se tion 5.3. A
ording to Step 1, we rst gather
repeatedly hange in the same dire tion, the valve responds rapidly to the set-point hange, but
us
when the set-point hange dire tion reverses, there is a delay before the valve responds. Also, there is a greater delay for small set-point hanges than for large set-point hanges when the deadband is
an
en ountered due to the use of the PI ontroller in the valve layer. In addition, the valve layer inputoutput data indi ates that when the valve output set-point is kept onstant for multiple sampling
M
periods, the valve output will not exhibit deadband if the next hange in the set-point is in the same dire tion as the hanges prior to the valve set-point remaining onstant, but will exhibit deadband if
d
the next hange in the set-point is in the opposite dire tion to the last hanges. The valve layer data
te
also suggests that valve output ow rates above about 0.5164 are not a hievable with the pressure available from the pneumati a tuation after the valve rst begins to lose be ause sti tion alters
ua − P
relationship su h that these ow rates would require negative pressures to be rea hed
Ac ce p
the
(i.e., sin e the valve is initialized with open more) until
ua ∼ 0.5164,
ua = 0.7042,
it an only lose (it annot reverse dire tion to
and subsequently annot rea h ow rates above that value).
The above observations are used in Step 2 of the model identi ation pro edure to postulate that the dynami s between
ua
and
um
an be aptured in a pie ewise-de ned model with two bran hes,
one orresponding to the response of
ua when the set-point hanges in um are repeatedly in the same
dire tion (no deadband), and another orresponding to the response when the set-point hanges swit h dire tion (deadband), with a spe ial onsideration for the ase that the set-point does not
hange between two sampling periods. The part of the model orresponding to the ase when there is deadband before the valve moves should have dierent speeds of response of the valve for dierent set-point hange magnitudes. The valve layer input-output data should be gathered while avoiding
39
Page 40 of 75
in reasing the set-point
um
above 0.5164 to avoid gathering data for ow rates where the pressure
is saturated (the de ision was made not to add bran hes to the empiri al model to a
ount for saturation of the a tuation pressure due to the omplexity that this adds to the empiri al model, but to instead seek to avoid saturating the pressure during pro ess operation by utilizing the input
ip t
rate of hange onstraints of Eq. 40 in the MPC used to ontrol the pro ess). Step 3 of the model identi ation pro edure will now be arried out to identify the equations for
cr
the two bran hes of the proposed model. We rst verify that su h a pie ewise-de ned valve layer model is ne essary by showing the results of attempting to identify a single model for the valve
by initializing the valve at its fully open position (ua
= 0.7042, Ps = 0 kg/m · t2d , ζP = 0, zf = 0 m,
and integrating the rst-prin iples valve layer model in Eqs. 27-31 and
an
xv = 0 m, vv = 0 m/td )
us
layer based on the valve layer input-output data. The valve layer input-output data was gathered
33-34 with the expli it Euler numeri al integration method and an integration step of
M
for 19 step hanges in the set-point (the set-point was rst de reased from
hI = 10−6 td
um = 0.7042
to 0.7, and
was subsequently de reased to 0.15 in in rements of 0.05, and then in reased to 0.5 in in rements
d
of 0.05, with ea h set-point held for a sampling period). A subset of the
um − ua
data generated
te
is shown in Fig. 11. Based on the data generated, the valve output response to a set-point hange was postulated to be able to be des ribed by a se ond-order linear dynami model. The values of and
ua
were measured every
10−4
Ac ce p
um
time units (every 100 integration steps; i.e.,
∆e = 10−4 td ),
and the following ARX model was t to the data by using a least-squares regression:
y(t˜j ) = 1.99212y(t˜j−1) − 0.99219y(t˜j−2) + 0.00035um(t˜j−1 ) − 0.00027um(t˜j−2 ) where
y(t˜j )
refers to the predi ted value of
data (i.e., at time
t˜j ).
ua
When the predi tions
they overshoot the values of
for the
y
j − th
(45)
measurement of the valve layer output
are generated from this model and the input data,
ua , and there is poor agreement with ua when the valve velo ity hanges
sign (deadband is rea hed), as shown in Fig. 11. Though it was not possible to identify an adequate se ond-order model using the input-output data for the entire set of 19 set-point hanges, it is possible to su
essfully identify a se ond-order model if only the data orresponding to the set-point de reases between 0.6 and 0.15, for whi h no
40
Page 41 of 75
deadband o
urs, is used to identify the model. In this ase, the following model is obtained:
y(t˜j ) = 1.96209y(t˜j−1) − 0.96249y(t˜j−2) + 0.00038um(t˜j−1) + 0.00002um(t˜j−2 )
(46)
When the de reasing set-points between 0.6 and 0.15 are used as inputs in Eq. 46, the predi tions of the valve output losely mat h the a tual values, as shown in Fig. 12.
ip t
y
To omplete Step 3 of the empiri al modeling pro edure, it ne essary to omplement Eq. 46
Fig. 11 orresponding to the deadband when
um
cr
with a model for the ase that deadband is observed. Based on the valve layer input-output data in
hanges from 0.15 to 0.2, it is postulated that the
us
response of the valve output to set-point hange dire tion reversals an be modeled as a rst-order pro ess with time delay. However, the values of the time onstant
τ
and of the delay
α
in su h a
of the valve layer to a set-point hange in
um
an
model are dependent on the magnitude of the set-point hanges be ause the speed of the response depends on the magnitude of the set-point hange.
M
Closed-loop simulations indi ate that for a set-point hange dire tion reversal, set-point hanges less than approximately 0.02 are unable to ause the PI ontroller to over ome the deadband within
d
a sampling period. To determine the dependen e of the delay on the magnitude of the set-point
um
was de reased from
te
hange, in a
ordan e with Step 4 of the model identi ation pro edure, 0.7042 to 0.15, and subsequently
um
was in reased by set-point hanges of dierent magnitudes.
Ac ce p
The regression method in Ogunnaike and Ray (1994) for the determination of the parameters of a rst-order-plus-dead-time model was applied to the data generated for ea h set-point hange. A plot of the resulting delays against the set-point hanges with whi h they were asso iated was t to the fun tion
a/x
using the MATLAB fun tion lsq urve t, with
τ
asso iated with ea h delay were averaged to give
t. The values of
a = 0.0037 τ = 0.0123
providing the best for the rst-order-
plus-dead-time model. Thus, the rst-order-plus-dead-time model is written in dis rete-time form as:
y(t˜j ) =
y(t˜j−1), t˜j − tk < α y(tk ) + exp(−∆e /τ )(y(t˜j−1 ) − y(tk )) + K(1 − exp(−∆e /τ )) × (um (tk ) − um (tk−1 )), t˜j − tk ≥ α (47)
41
Page 42 of 75
where
α= In Eqs. 47-48,
k
∆, |um (tk ) − um (tk−1 )| < 0.02 a/(|um(tk ) − um (tk−1 )|), |um (tk ) − um (tk−1 )| ≥ 0.02
is the value of
k
that brings
tk
losest to
t˜j (tk ≤ t˜j ),
and
(48)
K = 1.
In orporating the above onsiderations, the following dis rete-time empiri al valve layer model
ip t
was devised and validated to perform well for a number of valve layer input-output data points,
ompleting Step 5 of the model identi ation pro edure:
tk−2 ,
set
y(t) = y(tk )
and
tk−1
t ∈ [tk , tk+1).
for all
and also did not hange between
cr
and
tk
tk−1
us
1. If the set-point has not hanged between
2. If two set-point hanges are hanging in the same dire tion, or if the set-point has been
an
onstant for some time but has now hanged in the same dire tion that it was hanging prior to be oming onstant, use the model of Eq. 46.
M
3. If two set-point hanges are in opposite dire tions, or if the set-point has been onstant for some time but has now hanged in the opposite dire tion to that in whi h it was hanging
d
prior to be oming onstant, use the model of Eqs. 47-48.
Z
tk+Nk
Ac ce p
above is as follows:
te
The MPC-based sti tion ompensation strategy in orporating the empiri al model des ribed
min
um (t)∈S(∆)
s.t.
tk
−y(τ )˜ x3 (τ )˜ x4 (τ ) dτ
(49a)
x˜˙ (t) = f (˜ x(t), y(t), 0)
(49b)
y(t) ˙ = fy (y, um)
(49 )
x˜(tk ) = x(tk )
(49d)
y(t˜0 ) = ua (t0 )
(49e)
0.0704 ≤ um (t) ≤ 0.7042, ∀ t ∈ [tk , tk+Nk )
(49f )
0 ≤ y(t) ≤ 0.7042, ∀ t ∈ [tk , tk+Nk )
(49g)
|um (tk ) − u∗m (tk−1 |tk−1 )| ≤ 0.1
(49h)
|um (tj ) − um (tj−1 )| ≤ 0.1, j = k + 1, ..., k + Nk − 1
42
(49i)
Page 43 of 75
Z
tk+Nk
y(τ )dτ +
tk
Z
tk (p−1)tp
u∗a (τ )dτ = 0.175tp /0.5
where the notation follows that in Eqs. 11 and 38. The notation
signi es the value of
that was determined to be optimal at the prior sampling time and was applied to the pro ess
for the sampling period between
tk−1
and
tk .
Minimization of the obje tive fun tion in Eq. 49a
ip t
um
u∗m (tk−1 |tk−1)
(49j)
maximizes the yield of ethylene oxide when the amount of rea tant fed to the pro ess over the operating period of length
signi es a value of
ua
tp = 1 td
meets the onstraint in Eq. 49j (the notation
u∗a (t)
cr
p − th
that was applied to the pro ess at a past time t). Enfor ing the onstraint of
us
Eq. 49j ensures that Eq. 44 is satis ed by the nal time tf of operation. Two operating periods were simulated under this EMPC; though a longer simulation may redu e the ee ts from the transient
an
on the results, the two operating periods simulated are su ient for demonstrating that an empiri al model of the valve dynami s an readily be used in pla e of a rst-prin iples model in the MPC
Nk
was used in ea h operating
M
for valve behavior ompensation. A shrinking predi tion horizon
period with an initial length of 5 at the beginning of ea h operating period. At ea h subsequent
d
sampling time, the predi tion horizon was de reased by 1. The pro ess model of Eq. 49b (whi h is
ua,1
as mentioned above) is integrated using the expli it
te
the model of Eq. 35 with the single input
Euler numeri al integration method with an integration step size of
Ac ce p
predi tions.
Eq. 49 signi es that the predi tions developed in this se tion.
y
of
ua
hemp = 10−4 td
for making state
in the EMPC ome from the empiri al model
Eq. 49 is written in ontinuous-time form for onsisten y with the
remainder of the manus ript in whi h a ontinuous-time model of the pro ess-valve dynami s is utilized (in the sense that dierential equations are onsidered to onstitute the pro ess model, though their solution must be obtained through numeri al dis retization be ause no analyti solution is available), though the results presented for this example ome from utilizing the dis rete-time empiri al valve layer model developed in this se tion.
Be ause both the numeri ally dis retized
(with expli it Euler) ontinuous-time pro ess dynami model and the empiri al valve layer model evolve every
10−4 td
when state predi tions are made within the EMPC (i.e.,
∆e = hemp ),
the
dis rete-time nature of the empiri al model poses no issues for ombining it with the ontinuous-
43
Page 44 of 75
time pro ess model for making state predi tions. In the simulations, the value of with a state measurement of
ua
y
was not updated
at ea h sampling time but instead evolved in an open-loop fashion
(the notation in Eq. 49e signi es that the initial data required for simulating the valve layer based on the empiri al model (i.e.,
y(t˜0 ) and y(t˜−1)) ua ).
The state onstraint in Eq. 49g was enfor ed every integration
ip t
for all times without feedba k of
are known and used to integrate the empiri al model
step. The input rate of hange onstraints in Eqs. 49h-49i are added to redu e the likelihood that
a tuation to be ome saturated at zero.
cr
the EMPC will request unrea hable ow rates that would ause the pressure from the pneumati The optimization problems were solved using the open-
10−10 . and
y
initiated from
an
ua , um ,
Fig. 13 shows the traje tories of
us
sour e interior-point optimization solver Ipopt [Wä hter and Biegler (2006)℄ with a toleran e of
[x1 x2 x3 x4 xv vv zf ζP ] =
[0.997 1.264 0.209 1.004 0.051 m 2.000 × 10−6 m/td 1.426 × 10−5 m 0]
resulting from the use of the
M
empiri al EMPC. The empiri al model was su
essfully able to apture the behavior of
ua ,
and the
EMPC al ulated set-points that the valve layer ould tra k. Fig. 14 shows the pressure applied to
d
the valve throughout this losed-loop simulation, whi h never saturated at zero with the help of the
EMPC.
te
input rate of hange onstraints. Fig. 15 shows the losed-loop pro ess states under the empiri al
Ac ce p
In addition to al ulating rea hable set-points and preventing pressure saturation in the two operating periods simulated, the MPC-based valve behavior ompensation strategy with an empiri al model was also able to ensure that the integral material onstraint was not signi antly violated. In the rst operating period, the empiri al EMPC used only 0.02% less material than required by the material onstraint, and in the se ond operating period only 0.05% less. A omparison of a simulation of the form of Eq. 49 but with the rst-prin iples valve layer model of Eqs. 27-31 and 33-34 in pla e of the empiri al model of Eqs. 46-48 was formulated, in whi h the rst-prin iples model was simulated with an integration step of
10−5 td
within the MPC,
and state feedba k of the pro ess-valve states was obtained at ea h sampling time. This integration step size is smaller than for the simulation with the empiri al EMPC be ause the rst-prin iples model of Eqs. 27-31 and 33-34 annot be integrated with a step size of
44
10−4 td
using expli it Euler
Page 45 of 75
due to numeri al stability issues (i.e., the simulation does not produ e numeri al results within a reasonable range, whi h o
urred for some smaller integration step sizes as well). The onstraint of Eq. 49g was enfor ed on
ua
every 10 integration steps so that it was enfor ed every
10−4 td
as
for the empiri al EMPC. The nite dieren e approximation used for the gradients of the obje tive
ip t
fun tion and onstraints used a perturbation one order of magnitude smaller than in the empiri al EMPC. The resulting simulation of two operating periods took approximately three times longer to
10−4 td .
Though the
cr
solve than the MPC of Eq. 49 where the integration step within the MPC was
dieren e in omputation time depends on a large number of fa tors su h as the ode used and the
us
integration step size, it is signi ant that the empiri al model is less sti than the rst-prin iples model. 10
.
The omputation time omparison is not meant to be an exa t numeri al omparison
an
Remark
be ause it is di ult to determine the exa t minimum step size that an be utilized within the rst-
M
prin iples EMPC for omparison with the omputation time of the empiri al EMPC. Therefore, we do not fo us on providing exa t omputation time results, but instead highlight the dieren e
d
in stiness of the models, whi h is demonstrated in that an integration step size smaller than that
te
of the empiri al model must be utilized to integrate the rst-prin iples model. In general, a less
Ac ce p
sti model is onsidered to have the potential to be more omputationally e ient, and through the example, we show that empiri al modeling of a valve layer has the potential to develop a more
omputationally tra table model of the valve layer than would be obtained from rst-prin iples. We do not onsider impli it numeri al integration methods instead of expli it methods be ause they are harder to implement. MPC's need to be solved on-line within a sampling period, and the numeri al integration is performed many times to simulate the pro ess throughout the predi tion horizon at every iteration of the numeri al optimization method. Therefore, a simple implementation of numeri al integration is desirable.
6. Perspe tives on Valve Nonlinearity Compensation A on lusion of the results in this work is that an MPC design that utilizes models of both the pro ess and valve behavior for making state predi tions provides a systemati method for driving
45
Page 46 of 75
an output to its set-point that an a
ount for multivariable intera tions in a pro ess-valve dynami system and onstraints su h as valve output and a tuation magnitude saturation that an lead to undesirable losed-loop behavior. This method is not restri ted to linear plant dynami s, and it does not require tuning of ompensator-spe i parameters that are not learly tied to the pro ess output
ip t
responses as do some of the sti tion ompensation methods dis ussed above su h as ow ontrol, the integral term modi ation method, and kno ker-type methods. The bene ts of an MPC in luding
cr
valve dynami s for improving the issues ommonly observed due to valve behavior indi ate that it may be bene ial for industry to onsider wider use of MPC due to analysis not only of whether the
us
hemi al pro ess itself would bene t from being ontrolled by an MPC (whi h is the typi al analysis performed), but also of whether it might provide better valve behavior ompensation in the long
an
run (undesirable behavior like valve sti tion an develop over time) that may redu e e ien y and pro t in the long-term and therefore make MPC a more attra tive option than lassi al regulatory Thus, more analysis of the impa t of a tuator dynami s at the initial design
M
ontrol designs.
phase may allow for better ontroller designs to be hosen that an handle hanges in the a tuator
d
dynami s that often plague pro esses at a later phase and are less straightforward to handle when
te
non-model-based ontrol strategies are attempted to be used to handle nonlinear valve behavior. Though the MPC-based valve behavior ompensation method was shown in the pro ess examples
Ac ce p
in this work to be bene ial at ompensating for valve behavior, it was also shown that it has limitations in handling valve nonlinearities. For example, in Se tion 5.4.2, it was noted that a valve without ow ontrol under MPC a
ounting for valve sti tion may not be able to keep a pro ess output at its set-point for all times when the MPC sampling period is long ompared to the time s ale of the valve dynami s su h that the MPC is not able to regularly adjust the for e applied by the valve a tuation throughout a sampling period. An alternative to this is to use a ow ontroller for the valve or a small sampling period for the MPC to allow the for e from the valve a tuation to be adjusted frequently as the valve position hanges a
ording to its dynami s to try to drive it to the position orresponding to the valve output set-point. However, this may ause signi ant variations in the valve a tuation during the time that the valve position is being adjusted, whi h may in rease a tuator wear [Ivan and Lakshminarayanan (2009)℄. This indi ates that an MPC-based
46
Page 47 of 75
valve behavior ompensation method must seek to balan e a tuator wear and set-point oset for
ertain ontrol ar hite tures and valve nonlinearities through appropriate onstraints and design of parameters su h as the sampling period. Another on lusion of this work is that be ause the ee ts observed in sti ky ontrol loops are losed-loop ee ts, hanging the ontrol design of a system may
ip t
result in dierent pro ess output responses in a ontrol loop in whi h valve dynami s annot be negle ted. This is important to onsider as ontrollers at a plant are re-tuned or as upgrades are
cr
made to the ontrol design.
Finally, the losed-loop perspe tive on valve behavior developed in this work an impa t the
us
sti tion dete tion and quanti ation literature. It gives greater insight into the bene ts and limitations of the dete tion/quanti ation methods for sti tion in the literature, whi h are reviewed
an
in Brásio et al. (2014) and in lude shape-based methods and model identi ation-based methods. Many of the shape-based methods (e.g., Hor h (1999); Choudhury et al. (2006); Singhal and Sals-
M
bury (2005); Srinivasan et al. (2005a)) assume that a spe i pattern exists in the data from the measured outputs of the system (pro ess outputs or valve outputs), often in relation to the onIt has been highlighted that the pro ess and ontroller dynami s will ae t the
d
troller outputs.
te
patterns and thus may redu e the ee tiveness of shape-based methods (e.g., He et al. (2007) notes that the sti tion dete tion method in Hor h (1999) may give dierent results depending on the
Ac ce p
ontroller tuning, and Choudhury et al. (2006) and Jelali and Huang (2010) also note that the pattern-based methods are not always ee tive be ause patterns depend on the ontroller, pro ess, and valve dynami s). The present manus ript gives a general mathemati al framework for analyzing the di ulties noted with pattern-based methods through a pro ess-valve dynami model. It also gives greater insight into the onditions under whi h the assumption that os illations are o
urring in a pro ess output due to sti tion may not hold (e.g., when the ontroller, pro ess, and valve dynami s produ e the un ompensated ase in Fig. 5). Multiple model identi ation-based sti tion dete tion/quanti ation methods (see, for example, Srinivasan et al. (2005b); Jelali and Huang (2010); Jelali (2008)) assume that the pro ess an be des ribed by a linear model, whi h may be a limiting assumption espe ially as the requirement of steady-state operation is being hallenged by the re ent developments in EMPC [Ellis et al. (2014a)℄.
47
Page 48 of 75
The primary goal of sti tion dete tion methods is to identify problemati valve behavior so that maintenan e an be performed on a valve, and quanti ation methods are intended to be used to prioritize valve maintenan e based on whi h valves are most sti ky. The empiri al modeling strategy in this work ould be onsidered as a valve behavior dete tion/quanti ation strategy that is not
i = 1, . . . , m,
um,i − ua,i
relationship ould be developed for every valve if
data is available. The dieren e between
um,i
and
ua,i
ould then be tra ked over
ua,i
um,i
deviates most signi antly from
The valves
cr
time, and when it be omes signi ant, the valve ould be agged for maintenan e. for whi h
um,i − ua,i ,
ip t
limited to sti tion. The
ould be given priority in the maintenan e
us
s hedule. Though measurements of ow through a valve (ua,i ) are not always available in industrial appli ations when the ontrol loop is not a ow ontrol loop [Thornhill and Hor h (2007)℄, this
an
analysis indi ates that new instrumentation to provide measurements of pro ess variables su h as ow (when it is not already measured) may be bene ial long-term for dete ting and ompensating
M
for valve behavior by allowing empiri al models to be developed for a pro ess-valve system when it may be di ult to obtain a pro ess-valve model without the ow measurement (Se tion 5.3).
d
A nal observation is that many ontributions to the sti tion literature have fo used on sti tion
te
as the nonlinearity in the pro ess-valve system (i.e., many works examine linear pro esses and linear
ontrollers); the results of this work indi ate that nonlinear pro esses, espe ially with multiple inputs
Ac ce p
all ae ted by nonlinear valve behavior, may be parti ularly interesting to onsider in future works on sti tion dete tion, quanti ation, and ompensation, due to the multivariable intera tions of the pro ess-valve states, whi h may, as noted above, best be handled with multiple-input/multipleoutput nonlinear ontrol designs.
7. Con lusions
In this work, we analyzed the roles of the pro ess, valve, and ontroller dynami s, and also the ontrol loop ar hite ture, in the losed-loop response of a pro ess-valve dynami system. The
losed-loop perspe tive dis ussed allows a variety of valve behaviors (e.g., linear valve dynami s and sti tion) to be analyzed within a single framework. It was demonstrated to be useful for explaining how a number of sti tion ompensation methods from the literature seek to ompensate for sti tion
48
Page 49 of 75
at a fundamental mathemati al level, and also was shown to be bene ial for developing new valve behavior ompensation te hniques su h as an integral term modi ation sti tion ompensation method and an MPC design in orporating a dynami model ( rst-prin iples or empiri al) of the full pro ess-valve system. A level ontrol example and a ontinuous stirred tank rea tor were used
ip t
to demonstrate the on epts dis ussed throughout the manus ript.
cr
8. A knowledgements
Finan ial support from the National S ien e Foundation and the Department of Energy is gra-
Ac ce p
te
d
M
an
us
tefully a knowledged.
49
Page 50 of 75
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List of Figures S hemati showing the general ontrol ar hite ture under onsideration in this work. A ontroller re eives measurements of the pro ess state mation as appropriate, so that it an al ulate set-points for
m
m
x,
as well as set-point infor-
individual valve output ow rate
individual valves. The valves themselves have dynami s asso iated
with developing a value of
ua,i , i = 1, . . . , m,
um,i , i = 1, . . . , m. x. . . . . . . . . . .
for a given
outputs of the valves a t on the pro ess to alter the state 2
S hemati depi ting the tank onsidered in the level ontrol example.
3
Closed-loop traje tory of level
h
with referen e to its set-point
The . . .
ip t
1
hsp
. . . . . . . .
56 57
for the pro ess of
Eq. 23 under the PI ontroller of Eqs. 24-25 (top plot) and under the MPC of Eq. 26 Closed-loop traje tories of
h, ua ,
. . . . . . . . . . . . . . . . . . . . . . .
um
and
cr
(bottom plot) with no a tuator dynami s. 4
58
for the pro ess of Eq. 23 under the PI
ontroller of Eqs. 24-25 with the valve dynami s in Eqs. 27-32. This data is plotted
ua,min and ua,max are plotted to indi ate
us
every 100000 integration steps. The values of
the range of allowable ow rates through the valve. 5
Plot of traje tories of
h, ua , um ,
. . . . . . . . . . . . . . . . . .
59
and their set-points for the MPC of Eq. 26 (U,
an
signifying un ompensated) and the MPC of Eq. 41 (C, signifying ompensated) applied to the nonlinear pro ess of Eqs. 23 and 27-32 for a level set-point hange from
0.15 m
to
0.20 m.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ontrol.
In the inner loop, the ow ontroller omputes the pressure
pneumati a tuation, whi h then alters the position
xv
ua
from the
The valve position
xv
is
through the valve a
ording to the linear valve hara teristi
d
related to the ow
P
of the sti ky valve a
ording
to the valve dynami s (labeled as Valve in the gure). of Eq. 31 (labeled as Flow in the gure). Closed-loop traje tories of
h, ua ,
and
te
7
60
S hemati depi ting the ontrol ar hite ture for the level ontrol example with ow
M
6
. . . . . . . . . . . . . . . . . . . . . . .
um
61
for the pro ess of Eq. 23 under the PI
ontroller of Eqs. 24-25 with the valve in Eqs. 27-31 and the PI ontroller of Eqs. 33-
Ac ce p
34 used to ontrol the valve ow rate to its set-point value for the set-point hange from 8
0.15 m
to
0.20 m.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Standard ontroller input-valve output relationship reported for a sti ky valve. The
ontrol signal to the valve hanges but the valve output does not hange appre iably in the regions of deadband and sti kband. The valve output hanges qui kly in the region of slip-jump, and the valve output and ontrol signal are linearly related in the moving phase region of the response.
9
of Eqs. 36-37 with
hsp = 0.20 m. 10
11
. . . . . . . . . . . . . . . . . . . . . . . .
63
h, ua , and um for the pro ess of Eq. 23 under the ontroller L = 7 and β = 0.007, with the open-loop valve (Eqs. 27-32), and
Closed-loop traje tories of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Figure depi ting linear (L) and equal per entage (EP) valve hara teristi s of Eqs. 32 and 42, respe tively, along with several fra tions of the maximum ow rate (FI,A = 0.5379, FD = 0.62113, and FD,A = 0.2649) and of the maximum stem position (XI,A = 0.1768 and XD,C = 0.37887) for the level ontrol example. . . . . . . . . . . Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) from Eq. 45 when 19 set-point hanges are applied (a
65
subset of the data is shown). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
55
Page 56 of 75
12
Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) using the set-points de reasing between 0.6 and 0.15 and Eq. 46 (y overlays
13
ua ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) under the EMPC using an empiri al valve layer model (y almost overlays
ua ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Pressure applied to the valve when the EMPC using an empiri al valve layer model is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
15
Closed-loop pro ess states under the EMPC using an empiri al valve layer model.
70
.
Ac ce p
te
d
M
an
us
cr
ip t
14
56
Page 57 of 75
ip t Valve 1
Controller
ua,2
an
um,2
ua,1
us
um,1
cr
Set-points
Valve 2 . . .
Valve m
. . .
Process
ua,m
Ac ce p
te
d
um,m
M
. . .
x
Figure 1: S hemati showing the general ontrol ar hite ture under onsideration in this work. A ontroller re eives measurements of the pro ess state
x,
as well as set-point information as appropriate, so that it an al ulate
individual valve output ow rate set-points for
ua,i , i = 1, . . . , m, x.
with developing a value of pro ess to alter the state
m
m
individual valves. The valves themselves have dynami s asso iated
for a given
um,i , i = 1, . . . , m.
57
The outputs of the valves a t on the
Page 58 of 75
ip t M
an
us
cr
ua
te
d
h
√
Ac ce p
c1 h
A
Figure 2: S hemati depi ting the tank onsidered in the level ontrol example.
58
Page 59 of 75
ip t cr us
0.5 0.4 0.3
an
h (m), PI
0.6
0.2 0.1 500
1000
1500
2000
M
0
h hsp
d
0.2
te
0.19 0.18 0.17 0.16
Ac ce p
h (m), MPC
Time (s)
h hsp
0.15
0
500
Figure 3: Closed-loop traje tory of level
h
1000
1500
2000
2500
Time (s)
with referen e to its set-point
hsp
for the pro ess of Eq. 23 under the PI
ontroller of Eqs. 24-25 (top plot) and under the MPC of Eq. 26 (bottom plot) with no a tuator dynami s.
59
Page 60 of 75
ip t cr us
hsp 0.2 0.15
an
h (m)
h
0.25
0.1 0.5
x 10
ua,max
2
2.5
3 4
x 10
te
4 2
Ac ce p
u (m /s)
6
1.5
M
−3
3
1 ua um ua,min
d
0
0
0
0.5
Figure 4: Closed-loop traje tories of
1
h, u a ,
and
1.5
2
Time (s) um
2.5
3 4
x 10
for the pro ess of Eq. 23 under the PI ontroller of Eqs. 24-25
with the valve dynami s in Eqs. 27-32. This data is plotted every 100000 integration steps. The values of
ua,max
ua,min
and
are plotted to indi ate the range of allowable ow rates through the valve.
60
Page 61 of 75
ip t cr
0.1 h (C) h (U) hsp
0.05 0 50 −3
x 10
150
d
6
4
te
3
u (m /s)
100
M
0
us
0.15
an
h (m)
0.2
Ac ce p
2 ua (C)
ua (U)
um (C)
um (U)
ua,sp
0
0
50
Figure 5: Plot of traje tories of
100
150
Time (s)
h, ua , um , and their set-points for the MPC of Eq. 26 (U, signifying un ompensated)
and the MPC of Eq. 41 (C, signifying ompensated) applied to the nonlinear pro ess of Eqs. 23 and 27-32 for a level set-point hange from
0.15 m
to
0.20 m.
61
Page 62 of 75
ip t cr us Flow Control
ua
xv
P
Valve
Flow
h Tank
−
d
−
an
PI Control
um +
M
hsp +
Figure 6: S hemati depi ting the ontrol ar hite ture for the level ontrol example with ow ontrol. In the inner
P
from the pneumati a tuation, whi h then alters the position
te
loop, the ow ontroller omputes the pressure
the sti ky valve a
ording to the valve dynami s (labeled as Valve in the gure). The valve position
ua
xv
of
is related to
through the valve a
ording to the linear valve hara teristi of Eq. 31 (labeled as Flow in the gure).
Ac ce p
the ow
xv
62
Page 63 of 75
ip t cr h hsp
0.1 1.56
1.57
1.58
1.59
−3
x 10
1.6
1.61
1.62
1.63 4
x 10
d te
3.8 3.6 3.4
Ac ce p
u (m3 /s)
4
3.2 1.56
an
0.15
us
0.2
M
h (m)
0.25
1.57
1.58
Figure 7: Closed-loop traje tories of
h, u a ,
ua um
1.59
1.6
1.61
Time (s)
and
um
1.62
1.63 4
x 10
for the pro ess of Eq. 23 under the PI ontroller of Eqs. 24-25
with the valve in Eqs. 27-31 and the PI ontroller of Eqs. 33-34 used to ontrol the valve ow rate to its set-point value for the set-point hange from
0.15 m
to
0.20 m.
63
Page 64 of 75
ip t cr
us
Stickband Deadband Slip-Jump
Ac ce p
te
d
M
an
Valve Moving Output Phase
Control Signal
Figure 8: Standard ontroller input-valve output relationship reported for a sti ky valve. The ontrol signal to the valve hanges but the valve output does not hange appre iably in the regions of deadband and sti kband. The valve output hanges qui kly in the region of slip-jump, and the valve output and ontrol signal are linearly related in the moving phase region of the response.
64
Page 65 of 75
ip t cr
0.2
0.15
an
h hsp
0.1 1.8
2
2.2
−3
x 10
2.6
2.8
4
ua um
3
te
5
Ac ce p
4
3 x 10
d
6
u (m /s)
2.4
M
1.6
us
h (m)
0.25
3
1.6
1.8
Figure 9: Closed-loop traje tories of
L=7
and
β = 0.007,
2
h, u a ,
and
2.2
2.4
2.6
Time (s) um
2.8
3 4
x 10
for the pro ess of Eq. 23 under the ontroller of Eqs. 36-37 with
with the open-loop valve (Eqs. 27-32), and
65
hsp = 0.20 m.
Page 66 of 75
ip t
1
cr
L EP
us
0.9 0.8
0.6
an
FD FI,A
M
0.5 0.4
FD,A
0.3
d
ua,max
ua
0.7
te
0.2
Ac ce p
0.1
XI,A
0
0
0.2
XD,C 0.4
0.6
0.8
1
xv xv,max
Figure 10: Figure depi ting linear (L) and equal per entage (EP) valve hara teristi s of Eqs. 32 and 42, respe tively, along with several fra tions of the maximum ow rate (FI,A maximum stem position (XI,A
= 0.1768
and
= 0.5379, FD = 0.62113, and FD,A = 0.2649) and XD,C = 0.37887) for the level ontrol example.
66
of the
Page 67 of 75
ip t cr us
0.28
an
0.26 0.24
u
M
0.22
d
0.2
te
0.18
Ac ce p
0.16
ua um y
0.14
1.8
2
2.2
2.4
2.6
2.8
3
Dimensionless Time
Figure 11: Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) from Eq. 45 when 19 set-point hanges are applied (a subset of the data is shown).
67
Page 68 of 75
ip t us
cr
0.7
an
0.6
u
M
0.5
d
0.4
te
0.3
Ac ce p
0.2
0.1 0.4
ua um y
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Dimensionless Time
Figure 12: Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) using the set-points de reasing between 0.6 and 0.15 and Eq. 46 (y overlays
68
ua ).
Page 69 of 75
ip t cr
0.5
an
0.45
us
um ua y
u
M
0.4
te
d
0.35
Ac ce p
0.3
0.25
0
0.5
1
1.5
2
Dimensionless Time
Figure 13: Comparison of valve layer set-point (um ), valve layer output (ua ), and predi tion of the valve layer output (y ) under the EMPC using an empiri al valve layer model (y almost overlays
69
ua ).
Page 70 of 75
ip t cr us
Pressure/6894.76 (kg/m · t2d )
12
an
10
M
8
d
6
te
4
Ac ce p
2
0
0
0.5
1
1.5
2
Dimensionless Time
Figure 14: Pressure applied to the valve when the EMPC using an empiri al valve layer model is used.
70
Page 71 of 75
ip t cr
0.9968
1.5
2
1
1.5
2
te
0.9966 0
0.5
0
0.5
0
1
1
1.5
2
0.5
1
1.5
2
an
x2
1.5 1.0
M
0.5
0.2
d
x3
0.4
0.0
0.5
Ac ce p
1.0040
x4
us
x1
0.9970
1.0035 1.0030
0
Dimensionless Time
Figure 15: Closed-loop pro ess states under the EMPC using an empiri al valve layer model.
71
Page 72 of 75
List of Tables Valve Model Parameters [Gar ia (2008)℄
2
Ethylene Oxidation Model Parameters
. . . . . . . . . . . . . . . . . . . . . . . .
72
. . . . . . . . . . . . . . . . . . . . . . . . .
73
Ac ce p
te
d
M
an
us
cr
ip t
1
72
Page 73 of 75
ip t cr us an
Table 1: Valve Model Parameters [Gar ia (2008)℄
Parameter
M
1.361 kg 0.06452 m2 52538 kg/s2 0.000254 m/s 108 kg/s2 9000 kg/s 612.9 kg/s 1423 kg · m/s2 1707.7 kg · m/s2
Ac ce p
te
d
mv Av ks vs σ0 σ1 σ2 FC FS
Value
73
Page 74 of 75
ip t cr us
Parameter
Value -8.13 -7.12
M
γ1 γ2 γ3 A1 A2 A3 B1 B2 B3 B4
an
Table 2: Ethylene Oxidation Model Parameters
-11.07 92.80
Ac ce p
te
d
12.66
2412.71 7.32 10.39 2170.57 7.02
74
Page 75 of 75