- Email: [email protected]
Control system design based on a nonlinear first-order plus time delay model
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J. Pt'oc. Cemt Vol 7, No. 1, pp. 65 73, 1997 Copyright C 1996 Elsevier Science Ltd Printed in Great Britain All rights reserved 095% 1>24/97 S17O0 + 0.00
ELSEVIER
PII: S0959-1524196)00014-5
Control system design based on a nonlinear first-order plus time delay model Jietae Lee,* Wonhui Cho t and Thomas F. Edgar t* *Department of Chemical Engineering, Kyungpook National University, Taegu 702701, Korea tDepartment of Chemical Engineering, University of Texas, Austin, TX 78712, USA Received 11 December 1995; revised 14 April 1996 Most chemical processes are nonlinear in nature. When large set point changes or load disturbances occur frequently, nonlinear control systems are required. Instead of using the differential geometric method or nonlinear model predictive method, simple gain scheduling may be sufficient for many nonlinear single input-single output (SISO) processes. To design such simple control systems systematically, a nonlinear first-order plus time delay model is proposed for model-based control. A logarithmic transformation which is very useful for control of high purity distillation columns is shown to be effective in general. Several chemical process examples are also given. Copyright © 1996 Elsevier Science Ltd Keywords: nonlinear FOPTD model; output transformation
Almost all chemical processes are inherently nonlinear, Conventional linear control systems are not sufficient for such nonlinear processes when large set point changes and load disturbances occur frequently, and there is growing interest in nonlinear control systems 1. The nonlinear model predictive control method 2 formulates the control problem as a general constrained optimization problem and solves it by nonlinear programming. Since nonlinear programming allows very general objective function and various constraints, the method is quite general and can handle hard constraints explicitly. One critical disadvantage is its computational load. While recent improvements in computer speed and optimization code reliability make the method attractive, the computational load may be viewed as a disadvantage when a simpler method which will provide similar closed-loop performance exists, The nonlinear geometric method ~ 5 globally linearizes the nonlinear process by coordinate transformation and feedback. The state variables of the process are first transformed so that the input can cancel nonlinear terms. The method requires measuring or observing all the states. Such a transformation can be very complex even for approximate linearization 6.7. Furthermore, since it is based on the nonlinear version of pole/zero cancellation s, its load performances are not as good as those expected from the set point responses. The theo*+Author t o
whom
correspondence
should
be
addressed.
retical and computational complexity of the geometric method has limited its application in the process industries. Another approach for the nonlinear process control is the extended linearization metho& .~°. A set of linear controllers is obtained by local linearization of the process model for various operating points. Then a nonlinear controller is designed whose local linearizations around the operating points become the above linear controllers. That is, this nonlinear control method ensures that the locally linearized closed-loop systems along the operating points can satisfy a given performance criterion. In this way, difficult transformation and pole/zero cancellation nature of the nonlinear geometric method can be avoided. Rugh JI applied this method to obtain a nonlinear P I D controller for liquid level processes. Global linearization is usually based on rigorous physico-chemical models, which may be hard to obtain in practical situations. Instead of the rigorous model, an alternative is to use empirical input-output models which will approximate the nonlinear dynamic behaviour well. Block-oriented models which consist of a linear dynamic block and a nonlinear static block are often usedt2.~3; see Figure 1. A model in which the nonlinear static block is followed by the linear dynamic block is referred to as the Hammerstein model; see Lee and Lee ~4 and Eskinat et al.~5 for process control applications. The Wiener model uses the reverse order in that the nonlinear static block follows the linear dynamic
Email:
[email protected] 65
Control system design: J. Lee et al.
66
It
a) Wiener Model
_...E_ u It
linear control laws corresponding to the variation of the operating point to cover the entire dynamic range of the process. Bauman and Rugh 9 presented an analytic formulation of the gain scheduling approach called the extended linearization method. The procedure for extended linearization can be found elsewhere ~0. A nonlinear PID controller can be designed with this method 11. First, a process is linearized with the steady state value of input us as a linearization parameter and the linearized system is assumed to have the ultimate gain K u (u0 and the ultimate period Pu (u~). Then, using the Ziegler-Nichols tuning rule 19, we have linear PID controllers as a function of the steady state value of input us:
PID(us) = Kl(uO + K2(uO/s + K3(u~)
(1)
b) Hammersteln Model Figure 1 Two-blockoriented models and their approximations
where KI(Us) = 0.6K,(us)
block 16. The logarithmic transformation for a high purity distillation column 17 typifies this model. While these block-oriented models can describe nonlinear gain variations, they cannot treat the time constant variations. For chemical processes, nonlinearity can be expressed by the variations of the process gain and the time constant. A nonlinear first-order plus time delay (FOPTD) model can describe the two variations in a simplest way. Ouarti and Edgar TM have compared some nonlinear firstorder models which can describe the process gain variation. Here we investigate the general nonlinear FOPTD model and resulting nonlinear control system. When the process is overdamped, the extended linearization method can be approximated by the set of linear FOPTD models to yield a nonlinear FOPTD model, Hence the extended linearization method supports the nonlinear FOPTD model. In addition, the block-oriented models can also be approximated by this nonlinear FOPTD model structure by replacing the linear blocks with their approximate FOPTD models (Figure 1). First we review briefly the extended linearization method because it serves as the background for our nonlinear FOPTD model and because a simple nonlinear PID controller similar to what we want here can be designed with this method. Then we describe our main results followed by several examples,
E x t e n d e d linearization and nonlinear P I D controller When the operating point is fixed, a nonlinear process can be controlled with a linear controller designed from the model linearized at that operating point. When the operating point is varying, the linear controller should be adjusted accordingly to maintain satisfactory closedloop performance. Gain scheduling utilizes several
Kz(u0 = 1.2Ku(uYP~(uO K3(u0--O.075Ku(uOP~(uO Other model-based PID controllers can be employed as wellJL This parameterized controller can be realized in a nonlinear PID form as shown in Figure 2: ~;(t) = Kz(z(t))e(t ) (2)
u(t) = Kl(z(t))e(t) + z(t) + K3(z(t))b(t ) At steady state, e = 0, u = z. Hence linearization of the nonlinear controller (2)becomes the above parameterized linear PID controller (1). Rugh ~ showed that the nonlinear controller maintains the Ziegler-Nichols response characteristics for a wide range of set points for a liquid level process. However he did not include anti-reset windup, which is necessary for real processes with input bounds. Since z(t) represents u(t), reset windup can be eliminated simply by freezing the integration in Equation (2) when z(t) exceeds the lower and upper bounds of the input. General analysis of the extended linearization method can be found elsewhere 10. One disadvantage is that it is also a local controller and hence the stability region can be smaller than the one from the nonlinear geometric method. Stability of the control system can be guaranteed if the set point change or the load disturbances are sl°w2°.
_ ~ r
Nonlinear PID ~- = K2(z) i
~
u = K, lz)
+ z +
~ Process
a(z)
Figure 2 Rugh'snonlinear PID controller
Control system design: J. Lee et al.
Process u6
~a = A(a)x6 + b(a)u6 = c(a)x6
./.(-c~ )
%
67
_ )]<(.v~ . ~ r( .x-~) K(_v~)
> g(xs ) -
(6)
r( .< )
It
Process u6
k(a) exp(-~ts) r(=) s + I
Y6
>
Y
>
It
Process u
~ = f(x) + g(x) u y = x(t-~l)
Figure 3 Approximation by extended linearization (co linearization parameter)
T h a t is, the nonlinear F O P T D model can be identified from the steady state u,, the gain K and time constant r for various operating points, which can be obtained using open or closed-loop testing. On-line identification methods m a y be applied toi f i x ) and g(xb with a p p r o p r i a t e parameterizations as in the block-oriented models~t However, because Equation (3) is not very accurate, reasonable values o f / ( v ) and g(x) max not be obtained for some processes. Unless the process is evident to be describable well bv the first-order plus time deh/v model, on-line identitication is not recommended. Output trans/brmation and logarithmic traH,ginmatio/~
When the process is overdamped, instead of designing controllers directly from the linear models, we can a p p r o x i m a t e the linear models with the F O P T D models (Figure 3) and then design a controller. The set of linear F O P T D models leads to a nonlinear F O P T D model. Nonlinear F O P T D m o d e l As in linear cases, o v e r d a m p e d nonlinear processes can be a p p r o x i m a t e d by nonlinear F O P T D models, resulting in a very simple control system which will c o m p e n sate for the nonlinearities of the process. Consider the nonlinear F O P T D model: t-(t)
-
For tion have gain
some nonlinear processes, the output transformashown in F(gure 4 is very useful Simulation tests shown that the usual t r a n s t o r m a t i o n inverting the wtriations may not be satisfactor;', tlere another
output transformation is investigated, which inchtdes a logarithmic transformation as a special case. The output transformation :(el - 0(x) - /(v)/,,,(x) results in:
/;(:(t))-(t) =
:(t)
+
,(t)
(7)
It can c o m p e n s a t e for the process gain variations exactly. The corresponding transfer function is:
J{.v(t)) + g(_v(t))u(t) (3)
e o~
./l'(--s
y(t) - x(t - O) where u(t), x(t) and y(t) are the scalar input, state and output variables, respectively, The nonlinear F O P T D model (3) can describe the gain variations and time constant variations according to the operating point. Linearizing the model (3) at the operating point (<, x~), we have i'a(t) = [/"(x,) + g'(x,)u,}xa(t) + g(x,)Uo4t)
(4)
where u~ : /(.vO/g(x O, x~(t) = x(t) - x~ and u~{t) = u(t) u~. Hence the gain K and time constant r are as follows:
)s + I
(S)
A n o t h e r output transformation: : ( t ) = ~p( v(t)) = f g( 1x ) d.v
(9}
results in: -(t) - / i ( : ( t ) ) + u(t)
(10)
The corresponding transfer function is: e o~ (ll)
K(<)
:--
g(-h)
.s+ 1"~(- )
.l"( .< ) + ,,~'(.< )u~
1 r(.v~ )
:
r-'---i
(5)
E.J~(.)~
r-----q
pl t_~
- .1 '(.v~ ) + g (x~)u~
F r o m Equations rewritten as:
(4) and (5), J ( x 0 and g(xO can be Figure 4 Output transformation
J
Proc
Control system design: J. Lee et al.
68
30
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1 / / /
25
/ / /
20
/ //; time ~ l a y = 0.1
m
~15 E
/
.---g-
/-
/ /
/
/
/
/
10 / /
/ / /
f / "
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,
j.~
,
,
,
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,
,
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/
,
/
,
,
,
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101
10 0 a
Figure 5
Ultimate gains for the processes: exp(-Os)/(as + 1) (dashed line) and exp(-Os)/(s + a) (solid line)
To determine which is better, we apply a proportional (P) controller to the linearized systems (8) and (11) and obtain the stability regions. For P controllers the stability boundary lines are just the ultimate gains. In Figure 5, the stability of the system (8) is more sensitive than that of the system (11). For example, when the P controller gain of half of the ultimate gain for the nominal valuesJi(z~) =f'2(zs) = 1 is used, the system (1 1) remains stable for all values off'2(zs). On the other hand, the system (8) can be unstable for smaller values of Ji(z~). For the P control systems designed at the nominal values Ji(z~) = f'2(zs) -- 1, the normalized ISE values when impulse load disturbance occurs are shown in Figure 6, which yields the same conclusion as above. The smaller the time delay 0 the better Equation (11) performs, Since output transformation does not change the time constant, the range of Ji(zs) is the reciprocal off'2(z~), Hence, from Figures 5 and 6, we can conclude that the output transformation (9) is better. When the time constant is not varying, the two output transform-ations are the same and they can linearize the nonlinear process exactly. Other output transformations may be possible but are not considered further because there is no analytical way to evaluate them and a method linearizing the whole nonlinear F O P T D model is available as shown later, Using a linear approximation of g(x):
g(x(t) ~ bo + b~x(t)
(12)
the output transformation (9) becomes:
z( t) = ~( x( t)) = l ln(l + b' x( t)l
hi(
b0
)
The logarithmic transformation will improve the control performance in general over the linear control system designed with the constant g(x). The effectiveness of this logarithmic transformation has been demonstrated for a high purity distillation column 17 and for hot spot temperature control in a tubular reactor 21.
Nonlinear control system design If the change in the process time constant is large, it should be compensated for. A dynamic compensator or integral action with varying integral time can be used for this purpose. The integral action with varying integral time can be realized as shown in Figure 7, which is an extension of a linear scheme implementing integral action 22. It is simple and free from integral windup. Input/output linearizing control with an open-loop observer can also be applied to the nonlinear F O P T D model as shown in Figure 8. For some processes, this input/output linearizing control system suffers from instability due to large elements in the open-loop observer loop. Rugh's nonlinear PID control system has similar problems. The output transformation in the proposed control system reduces such problems.
Examples Nonlinear FOPTD model. First we consider just the nonlinear F O P T D process of Equation (3). When:
(13)
J(x) = x/(lOO + x2)
g ( x ) = X2/(l O0 + X 2)
(14)
Control system design: J. Lee et al.
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time delay =l.k0
~I I1Iii
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[ i m e o e l a y ~ ^~ "
10-1
_
100 a
101
Figure 6 Normalized ISE values For the impulse load responses for the processes: exp(-es)/(as + |) dashed line) and ¢xp( 0s)/(,~ + a) (solid line)
Process
~
-,~*
'~_1 ~] ~ -
0.5--
=f2(z)+U
Figure7
Proposed control system
Y
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~
0
I/0
,
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Linear
,
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,
7-1 21111,
Time
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][1 Process
Figure 8 Input/output linearizing control system (nonlinear firstorder system and open-loop observer are used)
the process has the gain of x~ and the time constant of 100 + x~. For the time delay 0 = 40 and sampling time = 2, set point changes are shown in Figure 9. Controller gains are 0.015, 0.015 and 25 for the PI controller, the input/output linearizing controller and the proposed controller, respectively. Integral time for the PI controller is set to 100. An oscillation in the input/output linearizing controller is seen in Figure 9. This is due to the large elements in the open-loop observer loop and imperfect integration of the Euler method. The oscillation depends strongly on the integration step size, limits
e,ample
response of the pure PI controller is very slow at the set point near zero due to a very small process gain. We can speed up the response with a higher controller gain but the responses at other set points become more oscillatory. When:
f(x) -
g(.r) I
c
1 dx
(15)
g(x)
the process time constant is constant as c > 0 while the process gain varies as cg(&). In this case, the output transformation exactly linearizes the process. The input/output linearizing control system and Rugh's nonlinear PID control system would suffer from similar problems due to large elements as indicated above for some g(x).
Control system design: J. Lee et al.
70
When the process gain and time constant are varying but have the same ratio, g(x) is constant. Then the proposed method does not use the output transformation and is equivalent to the input/output linearizing control method. The output transformation which compensates the steady-state gain can be shown to have very poor results. In this case, it is better not to use the output transformation, C S T R process. A continuous stirred tank reactor where an irreversible first-order reaction occurs is one of the key processes to which various nonlinear methods are applied. It is simple and shows some typical characteristics of nonlinear processes such as multiple steady states 23. A CSTR process studied by Sistu and Bequette 2 is as follows:
x~ .~ 21 = - 0 . 0 7 2 x lexp l + x E / 2 0 ) + xu 22 = 0.576x~ exp [ 1+ xXl l/20 ]
Xl
1.3x~+0.3u
(16)
y = .r~
a u o -a 5
]" ] r-
F
I
4 y a 2 /
0
]~ I \ ,
,
,
,
[
~ ~ J Time
~ I
,
~ ~ ,
I
Figure 10 Set point and load responses for the CSTR example
in the continuous stirred tank reactor. Recently Wright and Kravaris 25 showed that the reaction invariant models of pH processes 26 can be reduced to the following nonlinear first-order models: V2(t) = - F x ( t ) + (1 - x(t))u(t)
(18) where u is the temperature of cooling water, and x~ and x~ are normalized concentration and temperature in the tank, respectively. Xlf is the normalized feed concentration (nominal value is 1) and is used as a load disturbance. At high temperature operation, the concentration dynamics is usually much faster than the temperature dynamics. As in the singular perturbation theory, we may ignore the concentration dynamics (Xlf = 1)as: I (x~ 11 (17) x I = 1 / 1 + 0.072 exp 1 + x 2 / 20
Figure 10 shows control performances of the proposed control system with the reduced nonlinear F O P T D model. At time 20, Xlf of the process is changed from 1 to 1.2. The controller gain is 12 and the sampling time is set to 0.02. By increasing the controller gain, the overshoot can be decreased. In comparison with the results of Sistu and Bequette 2 the new method is promising. We can also see that degradation of the control performances for model mismatch as in Sistu and Bequette 2 is minor, Continuous stirred tank bio-reactors such as fermenters can be modelled by three ordinary differential equations about effluent cell concentration, substrate concentration and product concentration. Henson and Seborg 24 have shown that it is still satisfactory when the nonlinear control system is designed only with a single nonlinear first-order equation about the effluent cell concentration (output variable) ignoring the other two
dynamics. Another well-known nonlinear process is the pH process where acid-base neutralization reactions occur
y(t) = pH = ~ ~(x(t))
where V -- reactor volume, F -- influent flow rate, u -titrant flow rate (input variable), x = scalar reduced state variable and pH is the output variable. The pH equation ~ l(x) is given in Wright and Kravaris 25. The titrating stream is usually much more concentrated than the influent stream. That is, xs = us/(F + us) is much less than 1 and hence x(t) << I. The process time constant becomes nearly constant and so the process can be approximately linearized by the output transformation z(t) = ~ p H ) . It is well-known that the gain variations of pH processes are large, and control methods which do not use the output transformation would be problematic. For example, the ratio of gain changes is as much as 105 for a strong acid-strong base reaction. The conventional PID, the Rugh's nonlinear PID and the input/output linearizing controllers do not work well. Perfect control is possible with the output transformation 25 whenever the pH equation is exact.
Heat exchanger. Counter-current heat exchangers are distributed parameter systems and hence a large number of state equations may be required to describe their dynamics well. Alsop and Edgar 7 proposed a two-state model and applied the approximate linearization method with the reduced two-state model. As shown in Alsop and Edgar 7, the two-state model shows a similar dynamic response and tracks most of nonlinear characteristics of the higher order state models. We used the following two-state model:
2~ = -0.021792q + (32.507 - 0.11019x0u 2,= 0.069871q + 2 0 . 4 9 2 - 0.055882x2
(19)
Control system design: J. Lee et al.
661.7-v t -.v, q = ln[(295- x~)_ ,.'(xI - 366.7)]
a~ u0
where u is the flow rate of the cold stream and .v~ and _,-~ are the outlet temperatures of the cold and hot streams, respectively. Steady state values, the process gain and the process time constant are calculated as shown in Tabh, 1. The process gain and the process time constant are obtained by applying the characteristic area method 22 to the process model. The time delay term is also varying but is set to 5 because its variation is not large compared to tile process time constant. As shown in Tabh, 1, the process gain and the time constant grow as the output wtriable x: grows. The nonlinear functions J(x) and ,g(.v) are fitted as: j(x) = 0.1750 + 0.9189 >( 10:.v 0.1709 X 10 2.v2 + 0.3067 >( 10 ~.V~ g(.v) -
0.455
(20)
0.119x
where x :: w -- 320. Three control strategies were applied and responses are shown in Figure 11. Controller gains are 0.5.2 and 2 for the PI controller, the PI controller with logarithmic transformation and the proposed method, respectively. Integral time for both PI controllers without and with logarithmic transformation is 4v ~ and the sampling time is set to 3. A sluggish response is seen for the PI controller and a set point x> = 317. The logarithmic transformation reduces this problem with its wtriable gain at that set point. More symmetric responses are obtained with the proposed method,
Hi~,/~ purity distillation column. A high purity distillation column is one of the processes where linear controllers usually do not work well. A simple remedy is the use of logarithmic transformation. Here we show that our output transformation leads to this logarithmic transformation. However, it is also shown that the logarithmic transformation is not sufficient if changes to the mid purity ranges are needed. Using the model in Eskinat et al.~L the process gain and time constant were obtained
71
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~
t--=-
/
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a20
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t
t
t
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L
~
.
~1~ L [ t i a~
i
A ]
Time Figure I 1
Set y.omt responses t,~[ tile heat cxcl,~mgc~ e \ a m p lc
from the closed-loop simulations via relay feedback with h vsteresis::. Steady state values are shown in 7~zt~le, 2. The time delay terms are negligible compared to the time constant. Variations of the process ~ain and the time constant as shown in Table 2 are rather complex: the nonlinear functions of.f{ v) and ~v(v) are simple and can be well approximated as: fix)-
0.9993(1 --0.9993.\) (21)
,~(x)
0.7_0_ ( 1
1.0003.v)
where x is the top concentration. Hence the logarithmic transformation ln(1 v) will reduce the nonlinearity of ,~(x) and provide better control performance. Three control systems are applied to the full nonlinear model. Step responses are shown in F(,,,ure 12. Controller gains are 200, ().2 and 0.2 for the PI controller, the Pl controller with logarithmic transformation and the proposed method, respectively. Integral time for both PI controllers without and with logarithmic transfornmtion is 20 and the sampling time is set to 0.2. Slow response and offset at the set point 0.999~ of the Pl controller ale avoided by the logarithmic transformation. Howexer, both the PI controller with and without the logarithmic transformation show very sluggish responses at the set point of 0.995. due to a very large process time constant. This sluggishness can be avoided b\' the proposed method as sho\~n in ["i~4mc /2.
Table I Steady state values, the process gains and the process time constants for the heat exchanger example u 0 I 02 03 04 0.5 06 07 0.8 09 10
x
Gain
332.97 324.52 32151 ~2 00 219.09 318.49 31R07 317 75 317.50 317.30
155.4 44.80 20.15 11.32 7.218 4995 3.659 2.794 .2.203 1.781
Time constant 54.40 39.86 30.10 24.46 20.92 18.53 16.82 15.56 14,60 13.84
Table 2 Stead\ state values, process gains ~md process tm~e constants for the distillation column example it
v
Gain
l'imc c o a s t , a t
1527 1. 507 1.477 1.457 1.427 1,407 1.377
,1.'?g89 , L9986 ,).~)950 ~)9598 O.9079 0.8763 )8~27
tl.t}15 I) 02 I).q~ l 82 164 1.53 1 38
1~73 1~ 56 2823 643(I 24.90 17.20 11.43
Control system design: J. Lee et al.
72
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.991
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lime Figure 12 Set point responses lbr the distillation column example
flow rate to the pH reactor nonlinear function K,K(.) process gain Kl(u~), K2(us), K 3 ( u s ) p a r a m e t e r s in Rugh's nonlinear PID controllers as a function of the steady state of input u~ Ku(us) ultimate gain of a nonlinear process as a function of the steady state of input u s P.(u~) ultimate period of a nonlinear process as a function of the steady state of input us s Laplace variable t time u input variable V volume of the pH reactor x scalar reduced state variable y output variable z transformed output variable
Conclusions Just as a F O P T D model can be used to design modelbased PID controllers, a nonlinear F O P T D model can be employed to design nonlinear control systems for nonlinear overdamped processes. The nonlinear F O P T D model can describe the process nonlinearity via the variations of both the process gain and the time constant. The linear dynamic block of the two simplest block oriented models, the Wiener model and Hammerstein model, can be approximated by the F O P T D model and they become a nonlinear F O P T D model. In addition, locally linearized models in the extended linearization method can also be approximated by F O P T D models and reduced to a nonlinear F O P T D model. An output transformation is a traditional way to compensate for variations of the process gain. It is shown that an output transformation inverting the process gain is not satisfactory. The nonlinear F O P T D model provides a systematic way to obtain an output transformation better than that obtained by inverting the process gain. However, the output t r a n s formation cannot compensate for the variation of the process time constants. For a large variation of time constants, a simple nonlinear control system is also investigated which accounts for the process time constant variations.
Acknowledgements The first author acknowledges the financial support from the Automation Research Center in POSTECH designated by the KOSEF.
Nomenclature c e
a positive c o n s t a n t e r r o r signal in the feedback loop
J(.), Jl (.), f 2 ( . )
nonlinear functions
Subscripts s 6
steady state deviation variable
Superscript '
differentiation
Greekletters T,r(.) 0(.), P(.) 0
process time constant output transformation time delay
References 1. Bequette, B. W. Ind.Eng.
2. 3. 4.
Chem.Res.,1991, 30, 1931.
Sistu, P. B. and Bequette, B. W. AIChE J., 1991, 37, 1711. Kravaris, C. and Chung, C. B. A1ChEJ., 1987, 33, 592. Kravaris, C. and Kantor, J. C. Ind. Eng. Chem. Res. 1990, 29,
2295. 5. Kravaris, C. and Arkun, Y. CPC IV, 1991, 477. 6. Krener, A. J., System & Control Letters, 1984, 5, 181. 7. Alsop, A. W. and Edgar, T. F. Chem. Eng. Comm., 1989. 75, 155. 8. 9.
Kravaris, C. A1ChEJ., 1988, 34, 1803. Baumann, W. T. and Rugh, W. J. IEEE Trans. Automatic Control, 1986, AC-13, 40. 10. Lin. C.-F. Advanced Control System Design, Prentice-Hall, NJ. 1994. l ~ Rugh, W. J. AIChE J., 1987, 33, 1738. 12. Billings, S. A. lEE Proc.. 1980, 127, 272. 13. Haber, R. and Unbehauen, H. Automatica. 1990, 26, 651. 14. Lee, K. S. and Lee, W. K. AIChEJ., 1985, 31, 667. 15. Eskinat, E., Johnson, S. H. and Luyben, W. L. AIChEJ., 1991,
37, 255 16. Pajunen, G. A. Automation, 1992, 28, 781. 17. Georgiou, A., Georgakis, C. and Luyben, W. L. A IChE J., 1988. 34, 1287. 18. Ouarti, H. and Edgar, T. F. ACC, San Francisco, 1993, 2268. 19. Seborg, D. E., Edgar, T. F. and Mellichamp, D. A. Process Dynamics and Control, Wiley, New York, 1989. 20. Kelemen, M. IEEE Trans. Automatic Control, 1986, AC-31, 766. 21. Kozub, D. J., MacGregor, J. F. and Wright, J. D. AIChE J., 1987, 33, 1496.
Control system design: J. Lee et al. 22.
23.
A,strOm, K. J. and Hagglund, T. Automatic Tuning oJPID Controllers, Instrument Society of America, Research Triangle Park, NC, 1988. Uppal, A., Ray W. H. and Poore, A. B. Chem, Eng. Sci,. 1974, 29, 967,
24. 25. 26.
73
Henson, M. A. and Seborg, D. E. AIChE J. 1991, 37, 1065. Wright, R. A. and Kravaris, C. bid. Ene ('hem. Re~., 1991, 30. 1561. Gustafsson, T. K. and Waller, K V (7~em End,, S~i, 1983, 38, 389.
J. Pt'oc. Cemt Vol 7, No. 1, pp. 65 73, 1997 Copyright C 1996 Elsevier Science Ltd Printed in Great Britain All rights reserved 095% 1>24/97 S17O0 + 0.00
ELSEVIER
PII: S0959-1524196)00014-5
Control system design based on a nonlinear first-order plus time delay model Jietae Lee,* Wonhui Cho t and Thomas F. Edgar t* *Department of Chemical Engineering, Kyungpook National University, Taegu 702701, Korea tDepartment of Chemical Engineering, University of Texas, Austin, TX 78712, USA Received 11 December 1995; revised 14 April 1996 Most chemical processes are nonlinear in nature. When large set point changes or load disturbances occur frequently, nonlinear control systems are required. Instead of using the differential geometric method or nonlinear model predictive method, simple gain scheduling may be sufficient for many nonlinear single input-single output (SISO) processes. To design such simple control systems systematically, a nonlinear first-order plus time delay model is proposed for model-based control. A logarithmic transformation which is very useful for control of high purity distillation columns is shown to be effective in general. Several chemical process examples are also given. Copyright © 1996 Elsevier Science Ltd Keywords: nonlinear FOPTD model; output transformation
Almost all chemical processes are inherently nonlinear, Conventional linear control systems are not sufficient for such nonlinear processes when large set point changes and load disturbances occur frequently, and there is growing interest in nonlinear control systems 1. The nonlinear model predictive control method 2 formulates the control problem as a general constrained optimization problem and solves it by nonlinear programming. Since nonlinear programming allows very general objective function and various constraints, the method is quite general and can handle hard constraints explicitly. One critical disadvantage is its computational load. While recent improvements in computer speed and optimization code reliability make the method attractive, the computational load may be viewed as a disadvantage when a simpler method which will provide similar closed-loop performance exists, The nonlinear geometric method ~ 5 globally linearizes the nonlinear process by coordinate transformation and feedback. The state variables of the process are first transformed so that the input can cancel nonlinear terms. The method requires measuring or observing all the states. Such a transformation can be very complex even for approximate linearization 6.7. Furthermore, since it is based on the nonlinear version of pole/zero cancellation s, its load performances are not as good as those expected from the set point responses. The theo*+Author t o
whom
correspondence
should
be
addressed.
retical and computational complexity of the geometric method has limited its application in the process industries. Another approach for the nonlinear process control is the extended linearization metho& .~°. A set of linear controllers is obtained by local linearization of the process model for various operating points. Then a nonlinear controller is designed whose local linearizations around the operating points become the above linear controllers. That is, this nonlinear control method ensures that the locally linearized closed-loop systems along the operating points can satisfy a given performance criterion. In this way, difficult transformation and pole/zero cancellation nature of the nonlinear geometric method can be avoided. Rugh JI applied this method to obtain a nonlinear P I D controller for liquid level processes. Global linearization is usually based on rigorous physico-chemical models, which may be hard to obtain in practical situations. Instead of the rigorous model, an alternative is to use empirical input-output models which will approximate the nonlinear dynamic behaviour well. Block-oriented models which consist of a linear dynamic block and a nonlinear static block are often usedt2.~3; see Figure 1. A model in which the nonlinear static block is followed by the linear dynamic block is referred to as the Hammerstein model; see Lee and Lee ~4 and Eskinat et al.~5 for process control applications. The Wiener model uses the reverse order in that the nonlinear static block follows the linear dynamic
Email:
[email protected] 65
Control system design: J. Lee et al.
66
It
a) Wiener Model
_...E_ u It
linear control laws corresponding to the variation of the operating point to cover the entire dynamic range of the process. Bauman and Rugh 9 presented an analytic formulation of the gain scheduling approach called the extended linearization method. The procedure for extended linearization can be found elsewhere ~0. A nonlinear PID controller can be designed with this method 11. First, a process is linearized with the steady state value of input us as a linearization parameter and the linearized system is assumed to have the ultimate gain K u (u0 and the ultimate period Pu (u~). Then, using the Ziegler-Nichols tuning rule 19, we have linear PID controllers as a function of the steady state value of input us:
PID(us) = Kl(uO + K2(uO/s + K3(u~)
(1)
b) Hammersteln Model Figure 1 Two-blockoriented models and their approximations
where KI(Us) = 0.6K,(us)
block 16. The logarithmic transformation for a high purity distillation column 17 typifies this model. While these block-oriented models can describe nonlinear gain variations, they cannot treat the time constant variations. For chemical processes, nonlinearity can be expressed by the variations of the process gain and the time constant. A nonlinear first-order plus time delay (FOPTD) model can describe the two variations in a simplest way. Ouarti and Edgar TM have compared some nonlinear firstorder models which can describe the process gain variation. Here we investigate the general nonlinear FOPTD model and resulting nonlinear control system. When the process is overdamped, the extended linearization method can be approximated by the set of linear FOPTD models to yield a nonlinear FOPTD model, Hence the extended linearization method supports the nonlinear FOPTD model. In addition, the block-oriented models can also be approximated by this nonlinear FOPTD model structure by replacing the linear blocks with their approximate FOPTD models (Figure 1). First we review briefly the extended linearization method because it serves as the background for our nonlinear FOPTD model and because a simple nonlinear PID controller similar to what we want here can be designed with this method. Then we describe our main results followed by several examples,
E x t e n d e d linearization and nonlinear P I D controller When the operating point is fixed, a nonlinear process can be controlled with a linear controller designed from the model linearized at that operating point. When the operating point is varying, the linear controller should be adjusted accordingly to maintain satisfactory closedloop performance. Gain scheduling utilizes several
Kz(u0 = 1.2Ku(uYP~(uO K3(u0--O.075Ku(uOP~(uO Other model-based PID controllers can be employed as wellJL This parameterized controller can be realized in a nonlinear PID form as shown in Figure 2: ~;(t) = Kz(z(t))e(t ) (2)
u(t) = Kl(z(t))e(t) + z(t) + K3(z(t))b(t ) At steady state, e = 0, u = z. Hence linearization of the nonlinear controller (2)becomes the above parameterized linear PID controller (1). Rugh ~ showed that the nonlinear controller maintains the Ziegler-Nichols response characteristics for a wide range of set points for a liquid level process. However he did not include anti-reset windup, which is necessary for real processes with input bounds. Since z(t) represents u(t), reset windup can be eliminated simply by freezing the integration in Equation (2) when z(t) exceeds the lower and upper bounds of the input. General analysis of the extended linearization method can be found elsewhere 10. One disadvantage is that it is also a local controller and hence the stability region can be smaller than the one from the nonlinear geometric method. Stability of the control system can be guaranteed if the set point change or the load disturbances are sl°w2°.
_ ~ r
Nonlinear PID ~- = K2(z) i
~
u = K, lz)
+ z +
~ Process
a(z)
Figure 2 Rugh'snonlinear PID controller
Control system design: J. Lee et al.
Process u6
~a = A(a)x6 + b(a)u6 = c(a)x6
./.(-c~ )
%
67
_ )]<(.v~ . ~ r( .x-~) K(_v~)
> g(xs ) -
(6)
r( .< )
It
Process u6
k(a) exp(-~ts) r(=) s + I
Y6
>
Y
>
It
Process u
~ = f(x) + g(x) u y = x(t-~l)
Figure 3 Approximation by extended linearization (co linearization parameter)
T h a t is, the nonlinear F O P T D model can be identified from the steady state u,, the gain K and time constant r for various operating points, which can be obtained using open or closed-loop testing. On-line identification methods m a y be applied toi f i x ) and g(xb with a p p r o p r i a t e parameterizations as in the block-oriented models~t However, because Equation (3) is not very accurate, reasonable values o f / ( v ) and g(x) max not be obtained for some processes. Unless the process is evident to be describable well bv the first-order plus time deh/v model, on-line identitication is not recommended. Output trans/brmation and logarithmic traH,ginmatio/~
When the process is overdamped, instead of designing controllers directly from the linear models, we can a p p r o x i m a t e the linear models with the F O P T D models (Figure 3) and then design a controller. The set of linear F O P T D models leads to a nonlinear F O P T D model. Nonlinear F O P T D m o d e l As in linear cases, o v e r d a m p e d nonlinear processes can be a p p r o x i m a t e d by nonlinear F O P T D models, resulting in a very simple control system which will c o m p e n sate for the nonlinearities of the process. Consider the nonlinear F O P T D model: t-(t)
-
For tion have gain
some nonlinear processes, the output transformashown in F(gure 4 is very useful Simulation tests shown that the usual t r a n s t o r m a t i o n inverting the wtriations may not be satisfactor;', tlere another
output transformation is investigated, which inchtdes a logarithmic transformation as a special case. The output transformation :(el - 0(x) - /(v)/,,,(x) results in:
/;(:(t))-(t) =
:(t)
+
,(t)
(7)
It can c o m p e n s a t e for the process gain variations exactly. The corresponding transfer function is:
J{.v(t)) + g(_v(t))u(t) (3)
e o~
./l'(--s
y(t) - x(t - O) where u(t), x(t) and y(t) are the scalar input, state and output variables, respectively, The nonlinear F O P T D model (3) can describe the gain variations and time constant variations according to the operating point. Linearizing the model (3) at the operating point (<, x~), we have i'a(t) = [/"(x,) + g'(x,)u,}xa(t) + g(x,)Uo4t)
(4)
where u~ : /(.vO/g(x O, x~(t) = x(t) - x~ and u~{t) = u(t) u~. Hence the gain K and time constant r are as follows:
)s + I
(S)
A n o t h e r output transformation: : ( t ) = ~p( v(t)) = f g( 1x ) d.v
(9}
results in: -(t) - / i ( : ( t ) ) + u(t)
(10)
The corresponding transfer function is: e o~ (ll)
K(<)
:--
g(-h)
.s+ 1"~(- )
.l"( .< ) + ,,~'(.< )u~
1 r(.v~ )
:
r-'---i
(5)
E.J~(.)~
r-----q
pl t_~
- .1 '(.v~ ) + g (x~)u~
F r o m Equations rewritten as:
(4) and (5), J ( x 0 and g(xO can be Figure 4 Output transformation
J
Proc
Control system design: J. Lee et al.
68
30
.
.
.
.
.
.
i
t
.
.
.
.
.
.
.
1 / / /
25
/ / /
20
/ //; time ~ l a y = 0.1
m
~15 E
/
.---g-
/-
/ /
/
/
/
/
10 / /
/ / /
f / "
0
,
j.~
,
,
,
10 -1
,
,
I
/
,
/
,
,
,
,
,
101
10 0 a
Figure 5
Ultimate gains for the processes: exp(-Os)/(as + 1) (dashed line) and exp(-Os)/(s + a) (solid line)
To determine which is better, we apply a proportional (P) controller to the linearized systems (8) and (11) and obtain the stability regions. For P controllers the stability boundary lines are just the ultimate gains. In Figure 5, the stability of the system (8) is more sensitive than that of the system (11). For example, when the P controller gain of half of the ultimate gain for the nominal valuesJi(z~) =f'2(zs) = 1 is used, the system (1 1) remains stable for all values off'2(zs). On the other hand, the system (8) can be unstable for smaller values of Ji(z~). For the P control systems designed at the nominal values Ji(z~) = f'2(zs) -- 1, the normalized ISE values when impulse load disturbance occurs are shown in Figure 6, which yields the same conclusion as above. The smaller the time delay 0 the better Equation (11) performs, Since output transformation does not change the time constant, the range of Ji(zs) is the reciprocal off'2(z~), Hence, from Figures 5 and 6, we can conclude that the output transformation (9) is better. When the time constant is not varying, the two output transform-ations are the same and they can linearize the nonlinear process exactly. Other output transformations may be possible but are not considered further because there is no analytical way to evaluate them and a method linearizing the whole nonlinear F O P T D model is available as shown later, Using a linear approximation of g(x):
g(x(t) ~ bo + b~x(t)
(12)
the output transformation (9) becomes:
z( t) = ~( x( t)) = l ln(l + b' x( t)l
hi(
b0
)
The logarithmic transformation will improve the control performance in general over the linear control system designed with the constant g(x). The effectiveness of this logarithmic transformation has been demonstrated for a high purity distillation column 17 and for hot spot temperature control in a tubular reactor 21.
Nonlinear control system design If the change in the process time constant is large, it should be compensated for. A dynamic compensator or integral action with varying integral time can be used for this purpose. The integral action with varying integral time can be realized as shown in Figure 7, which is an extension of a linear scheme implementing integral action 22. It is simple and free from integral windup. Input/output linearizing control with an open-loop observer can also be applied to the nonlinear F O P T D model as shown in Figure 8. For some processes, this input/output linearizing control system suffers from instability due to large elements in the open-loop observer loop. Rugh's nonlinear PID control system has similar problems. The output transformation in the proposed control system reduces such problems.
Examples Nonlinear FOPTD model. First we consider just the nonlinear F O P T D process of Equation (3). When:
(13)
J(x) = x/(lOO + x2)
g ( x ) = X2/(l O0 + X 2)
(14)
Control system design: J. Lee et al.
2
•
.
~
,
.
~
,
.
\
.
.
.
.
.
.
69
.
.
.
.
.
.
,
time delay =l.k0
~I I1Iii
NE ljl
[ i m e o e l a y ~ ^~ "
10-1
_
100 a
101
Figure 6 Normalized ISE values For the impulse load responses for the processes: exp(-es)/(as + |) dashed line) and ¢xp( 0s)/(,~ + a) (solid line)
Process
~
-,~*
'~_1 ~] ~ -
0.5--
=f2(z)+U
Figure7
Proposed control system
Y
0.o
,
A
~
,
1
,
,
~
IkJ\
~
0
I/0
,
I
,
t ~
Linear
,
,
,
7-1 21111,
Time
'¢*>
II
][1 Process
Figure 8 Input/output linearizing control system (nonlinear firstorder system and open-loop observer are used)
the process has the gain of x~ and the time constant of 100 + x~. For the time delay 0 = 40 and sampling time = 2, set point changes are shown in Figure 9. Controller gains are 0.015, 0.015 and 25 for the PI controller, the input/output linearizing controller and the proposed controller, respectively. Integral time for the PI controller is set to 100. An oscillation in the input/output linearizing controller is seen in Figure 9. This is due to the large elements in the open-loop observer loop and imperfect integration of the Euler method. The oscillation depends strongly on the integration step size, limits
e,ample
response of the pure PI controller is very slow at the set point near zero due to a very small process gain. We can speed up the response with a higher controller gain but the responses at other set points become more oscillatory. When:
f(x) -
g(.r) I
c
1 dx
(15)
g(x)
the process time constant is constant as c > 0 while the process gain varies as cg(&). In this case, the output transformation exactly linearizes the process. The input/output linearizing control system and Rugh's nonlinear PID control system would suffer from similar problems due to large elements as indicated above for some g(x).
Control system design: J. Lee et al.
70
When the process gain and time constant are varying but have the same ratio, g(x) is constant. Then the proposed method does not use the output transformation and is equivalent to the input/output linearizing control method. The output transformation which compensates the steady-state gain can be shown to have very poor results. In this case, it is better not to use the output transformation, C S T R process. A continuous stirred tank reactor where an irreversible first-order reaction occurs is one of the key processes to which various nonlinear methods are applied. It is simple and shows some typical characteristics of nonlinear processes such as multiple steady states 23. A CSTR process studied by Sistu and Bequette 2 is as follows:
x~ .~ 21 = - 0 . 0 7 2 x lexp l + x E / 2 0 ) + xu 22 = 0.576x~ exp [ 1+ xXl l/20 ]
Xl
1.3x~+0.3u
(16)
y = .r~
a u o -a 5
]" ] r-
F
I
4 y a 2 /
0
]~ I \ ,
,
,
,
[
~ ~ J Time
~ I
,
~ ~ ,
I
Figure 10 Set point and load responses for the CSTR example
in the continuous stirred tank reactor. Recently Wright and Kravaris 25 showed that the reaction invariant models of pH processes 26 can be reduced to the following nonlinear first-order models: V2(t) = - F x ( t ) + (1 - x(t))u(t)
(18) where u is the temperature of cooling water, and x~ and x~ are normalized concentration and temperature in the tank, respectively. Xlf is the normalized feed concentration (nominal value is 1) and is used as a load disturbance. At high temperature operation, the concentration dynamics is usually much faster than the temperature dynamics. As in the singular perturbation theory, we may ignore the concentration dynamics (Xlf = 1)as: I (x~ 11 (17) x I = 1 / 1 + 0.072 exp 1 + x 2 / 20
Figure 10 shows control performances of the proposed control system with the reduced nonlinear F O P T D model. At time 20, Xlf of the process is changed from 1 to 1.2. The controller gain is 12 and the sampling time is set to 0.02. By increasing the controller gain, the overshoot can be decreased. In comparison with the results of Sistu and Bequette 2 the new method is promising. We can also see that degradation of the control performances for model mismatch as in Sistu and Bequette 2 is minor, Continuous stirred tank bio-reactors such as fermenters can be modelled by three ordinary differential equations about effluent cell concentration, substrate concentration and product concentration. Henson and Seborg 24 have shown that it is still satisfactory when the nonlinear control system is designed only with a single nonlinear first-order equation about the effluent cell concentration (output variable) ignoring the other two
dynamics. Another well-known nonlinear process is the pH process where acid-base neutralization reactions occur
y(t) = pH = ~ ~(x(t))
where V -- reactor volume, F -- influent flow rate, u -titrant flow rate (input variable), x = scalar reduced state variable and pH is the output variable. The pH equation ~ l(x) is given in Wright and Kravaris 25. The titrating stream is usually much more concentrated than the influent stream. That is, xs = us/(F + us) is much less than 1 and hence x(t) << I. The process time constant becomes nearly constant and so the process can be approximately linearized by the output transformation z(t) = ~ p H ) . It is well-known that the gain variations of pH processes are large, and control methods which do not use the output transformation would be problematic. For example, the ratio of gain changes is as much as 105 for a strong acid-strong base reaction. The conventional PID, the Rugh's nonlinear PID and the input/output linearizing controllers do not work well. Perfect control is possible with the output transformation 25 whenever the pH equation is exact.
Heat exchanger. Counter-current heat exchangers are distributed parameter systems and hence a large number of state equations may be required to describe their dynamics well. Alsop and Edgar 7 proposed a two-state model and applied the approximate linearization method with the reduced two-state model. As shown in Alsop and Edgar 7, the two-state model shows a similar dynamic response and tracks most of nonlinear characteristics of the higher order state models. We used the following two-state model:
2~ = -0.021792q + (32.507 - 0.11019x0u 2,= 0.069871q + 2 0 . 4 9 2 - 0.055882x2
(19)
Control system design: J. Lee et al.
661.7-v t -.v, q = ln[(295- x~)_ ,.'(xI - 366.7)]
a~ u0
where u is the flow rate of the cold stream and .v~ and _,-~ are the outlet temperatures of the cold and hot streams, respectively. Steady state values, the process gain and the process time constant are calculated as shown in Tabh, 1. The process gain and the process time constant are obtained by applying the characteristic area method 22 to the process model. The time delay term is also varying but is set to 5 because its variation is not large compared to tile process time constant. As shown in Tabh, 1, the process gain and the time constant grow as the output wtriable x: grows. The nonlinear functions J(x) and ,g(.v) are fitted as: j(x) = 0.1750 + 0.9189 >( 10:.v 0.1709 X 10 2.v2 + 0.3067 >( 10 ~.V~ g(.v) -
0.455
(20)
0.119x
where x :: w -- 320. Three control strategies were applied and responses are shown in Figure 11. Controller gains are 0.5.2 and 2 for the PI controller, the PI controller with logarithmic transformation and the proposed method, respectively. Integral time for both PI controllers without and with logarithmic transformation is 4v ~ and the sampling time is set to 3. A sluggish response is seen for the PI controller and a set point x> = 317. The logarithmic transformation reduces this problem with its wtriable gain at that set point. More symmetric responses are obtained with the proposed method,
Hi~,/~ purity distillation column. A high purity distillation column is one of the processes where linear controllers usually do not work well. A simple remedy is the use of logarithmic transformation. Here we show that our output transformation leads to this logarithmic transformation. However, it is also shown that the logarithmic transformation is not sufficient if changes to the mid purity ranges are needed. Using the model in Eskinat et al.~L the process gain and time constant were obtained
71
/ Y
k
[~'"~
[/
& PI O Log. Transformation
~
~
t--=-
/
/£/
A
a20
315
0
I
t
t
t
[
L
~
.
~1~ L [ t i a~
i
A ]
Time Figure I 1
Set y.omt responses t,~[ tile heat cxcl,~mgc~ e \ a m p lc
from the closed-loop simulations via relay feedback with h vsteresis::. Steady state values are shown in 7~zt~le, 2. The time delay terms are negligible compared to the time constant. Variations of the process ~ain and the time constant as shown in Table 2 are rather complex: the nonlinear functions of.f{ v) and ~v(v) are simple and can be well approximated as: fix)-
0.9993(1 --0.9993.\) (21)
,~(x)
0.7_0_ ( 1
1.0003.v)
where x is the top concentration. Hence the logarithmic transformation ln(1 v) will reduce the nonlinearity of ,~(x) and provide better control performance. Three control systems are applied to the full nonlinear model. Step responses are shown in F(,,,ure 12. Controller gains are 200, ().2 and 0.2 for the PI controller, the Pl controller with logarithmic transformation and the proposed method, respectively. Integral time for both PI controllers without and with logarithmic transfornmtion is 20 and the sampling time is set to 0.2. Slow response and offset at the set point 0.999~ of the Pl controller ale avoided by the logarithmic transformation. Howexer, both the PI controller with and without the logarithmic transformation show very sluggish responses at the set point of 0.995. due to a very large process time constant. This sluggishness can be avoided b\' the proposed method as sho\~n in ["i~4mc /2.
Table I Steady state values, the process gains and the process time constants for the heat exchanger example u 0 I 02 03 04 0.5 06 07 0.8 09 10
x
Gain
332.97 324.52 32151 ~2 00 219.09 318.49 31R07 317 75 317.50 317.30
155.4 44.80 20.15 11.32 7.218 4995 3.659 2.794 .2.203 1.781
Time constant 54.40 39.86 30.10 24.46 20.92 18.53 16.82 15.56 14,60 13.84
Table 2 Stead\ state values, process gains ~md process tm~e constants for the distillation column example it
v
Gain
l'imc c o a s t , a t
1527 1. 507 1.477 1.457 1.427 1,407 1.377
,1.'?g89 , L9986 ,).~)950 ~)9598 O.9079 0.8763 )8~27
tl.t}15 I) 02 I).q~ l 82 164 1.53 1 38
1~73 1~ 56 2823 643(I 24.90 17.20 11.43
Control system design: J. Lee et al.
72
4
u
]
~
,
F g(.)
~
0~ 1" p l ~
~
Y
~ o Log. Transformation
.991
0
• Proposed I
I
I
2'0
I
I
I
4'0
lime Figure 12 Set point responses lbr the distillation column example
flow rate to the pH reactor nonlinear function K,K(.) process gain Kl(u~), K2(us), K 3 ( u s ) p a r a m e t e r s in Rugh's nonlinear PID controllers as a function of the steady state of input u~ Ku(us) ultimate gain of a nonlinear process as a function of the steady state of input u s P.(u~) ultimate period of a nonlinear process as a function of the steady state of input us s Laplace variable t time u input variable V volume of the pH reactor x scalar reduced state variable y output variable z transformed output variable
Conclusions Just as a F O P T D model can be used to design modelbased PID controllers, a nonlinear F O P T D model can be employed to design nonlinear control systems for nonlinear overdamped processes. The nonlinear F O P T D model can describe the process nonlinearity via the variations of both the process gain and the time constant. The linear dynamic block of the two simplest block oriented models, the Wiener model and Hammerstein model, can be approximated by the F O P T D model and they become a nonlinear F O P T D model. In addition, locally linearized models in the extended linearization method can also be approximated by F O P T D models and reduced to a nonlinear F O P T D model. An output transformation is a traditional way to compensate for variations of the process gain. It is shown that an output transformation inverting the process gain is not satisfactory. The nonlinear F O P T D model provides a systematic way to obtain an output transformation better than that obtained by inverting the process gain. However, the output t r a n s formation cannot compensate for the variation of the process time constants. For a large variation of time constants, a simple nonlinear control system is also investigated which accounts for the process time constant variations.
Acknowledgements The first author acknowledges the financial support from the Automation Research Center in POSTECH designated by the KOSEF.
Nomenclature c e
a positive c o n s t a n t e r r o r signal in the feedback loop
J(.), Jl (.), f 2 ( . )
nonlinear functions
Subscripts s 6
steady state deviation variable
Superscript '
differentiation
Greekletters T,r(.) 0(.), P(.) 0
process time constant output transformation time delay
References 1. Bequette, B. W. Ind.Eng.
2. 3. 4.
Chem.Res.,1991, 30, 1931.
Sistu, P. B. and Bequette, B. W. AIChE J., 1991, 37, 1711. Kravaris, C. and Chung, C. B. A1ChEJ., 1987, 33, 592. Kravaris, C. and Kantor, J. C. Ind. Eng. Chem. Res. 1990, 29,
2295. 5. Kravaris, C. and Arkun, Y. CPC IV, 1991, 477. 6. Krener, A. J., System & Control Letters, 1984, 5, 181. 7. Alsop, A. W. and Edgar, T. F. Chem. Eng. Comm., 1989. 75, 155. 8. 9.
Kravaris, C. A1ChEJ., 1988, 34, 1803. Baumann, W. T. and Rugh, W. J. IEEE Trans. Automatic Control, 1986, AC-13, 40. 10. Lin. C.-F. Advanced Control System Design, Prentice-Hall, NJ. 1994. l ~ Rugh, W. J. AIChE J., 1987, 33, 1738. 12. Billings, S. A. lEE Proc.. 1980, 127, 272. 13. Haber, R. and Unbehauen, H. Automatica. 1990, 26, 651. 14. Lee, K. S. and Lee, W. K. AIChEJ., 1985, 31, 667. 15. Eskinat, E., Johnson, S. H. and Luyben, W. L. AIChEJ., 1991,
37, 255 16. Pajunen, G. A. Automation, 1992, 28, 781. 17. Georgiou, A., Georgakis, C. and Luyben, W. L. A IChE J., 1988. 34, 1287. 18. Ouarti, H. and Edgar, T. F. ACC, San Francisco, 1993, 2268. 19. Seborg, D. E., Edgar, T. F. and Mellichamp, D. A. Process Dynamics and Control, Wiley, New York, 1989. 20. Kelemen, M. IEEE Trans. Automatic Control, 1986, AC-31, 766. 21. Kozub, D. J., MacGregor, J. F. and Wright, J. D. AIChE J., 1987, 33, 1496.
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