Feedback Model-Algorithmic Control

Copyright © IFAC Dynamics and Control of Process Systems. Corfu , Greece, 1998

NONLINEAR MULTIRATE FEEDFORWARDIFEEDBACK MODEL-ALGORITHMIC CONTROL Michael Niemiec and Costas Kravaris

Department of Chemical Engineering The University of Michigan Ann Arbor, Michigan 48109, USA

Abstract: This paper develops a systematic methodology for the synthesis of nonlinear multirate multi variable feedforwardlfeedback Model-Algorithmic Controllers (MAC) . The proposed controller incorporates process disturbance measurements that occur between the multi rate output measurements to efficiently reject the effect of these disturbances . The developed feedforwardlfeedback controller is applied to a continuous polymerization reactor, and its performance is evaluated by simulation . Copyright © 19981FAC Keywords : Multirate, Feedforward, Discrete Time, Nonlinear, Multivariable, Control

1. INTRODUCTION

multirate systems using a state space description has been looked at extensively (Araki and Hagiwara, 1986; Hagiwara and Araki, 1988). A linear state space modelpredictive technique that uses a sub-optimal filter for a dual-rate system was developed, which utilizes both the slow sampling rate primary measurements and the fast sampling rate secondary measurements (Lee et aI., 1992). Additionally, a multirate multi variable control scheme control scheme within a linear modelalgorithmic controller (MAC) framework was developed which uses all the multirate measurements to perform input actions at a single rate (Ohshima et aI. , 1994) . Finally . Gopinath and Bequette (1992) proposed an extension of QDMC for multirate systems, while Niemiec and Kravaris (1998) developed a nonlinear multirate model-algorithmic controller for multi variable systems on the basis of output measurements obtained at different sampling rates .

In the chemical process industry, many of the important process measurements are not available at the same rate . One common scenario arises when some of the controlled process outputs are measured with large sampling periods. This is primarily due to hardware limitations, where compositions and product properties are measured with various forms of analytical instruments. With large output sampling periods, a multirate controller may not be able to efficiently reject process disturbances that occur between samples with only output feedback action . Therefore, it is advantageous to synthesize a multirate controller that utilizes available disturbance measurements to reject the effect of process disturbances on the slowly sampled outputs. Re search on multirate systems began in the late 1950' s. but has received considerable attention in the last decade . The state-space description, transfer characteristics, and Nyquist criterion for linear multi variable systems has been studied by Araki and Yamamoto (1986). More recently. the pole assignment problem for linear

The present paper extends the results of Niemiec and Kravaris (1998) to develop a feedforwardlfeedback nonlinear multirate

295

model-algorithmic controller that utilizes multi rate disturbance measurements along with The paper multirate output measurements. begins with the concept of relative order with respect to both the input vector and a disturbance . A nonlinear multirate feedforwardlfeedback controller is then sy nthesized and applied to a polymethyl methacrylate polymerization reactor example.

[~: ][ aC:P(;~U, d)

r-[a(~~u. I

d )]

$

[0 . .. 0]

(2) Equivalently, r, can be viewed as the smallest number of input actuation periods after which a manipulated input move affects the output Y.. Similarly, the relative order of the output.v, with respect to the measurable disturbance d; is the smallest integer P'I for which :

2. NONLINEAR MULTI RATE FEEDFORW ARD/FEEDBACK CONTROL

[

ah, ][ac:p(X,U,d)] P'I -1[a(X,U,d)] $ 0 (3) ad }

ax

ax

Again , P'I can be viewed as the smallest number of input actuation periods after which the disturbance dl affects the output Y,. With these notions, for every i= 1, . . . ,m, the set of measurable disturbances can be partitioned into three classes : ";,, ~ , and

2. 1 Preliminaries Consider a MIMO nonlinear minimum-phase process described by a discrete-time model of the fonn: x(k + 1) = (x(k) ,u(k) ,d(k))

e,:

(1) v,(k) = hi (x(k)), i=l , . .. ,m where the output and disturbance measurements are available at different rates . In this notation , x=[x, .. .xX denotes the vector of state variables, u=[u , . . .uJ denot;s the manipulated input vector, d=[d,. .. d] represents the vector of P T measurable di sturbances, and y=[y , .. .yJ represents the controlled output vector. It is assumed that X E Xc9\", UE Uc9\ m, and dE Dc9\p. C:P(x,u,d) is an analytic vector function on Xx UxD , and h(x)=[h,(x) ... hn,( x)f is an analytic vector function on X. For each time kD.t, where k is an integer, the manipulated inputs of the system are actuated. Furthennore, at each time that is a multiple of NiD.t, where N, is an integer constant. the output y, is sampled. N, is defined as the ratio of the i-th output sampling period to the input actuation period : N = Sampling Period of Output i , Period of Input Actuation Similarly , at each time that is a multiple of

d}

E

14, ~ Pi} > ri

d )

E

'8 i ~

d} E

P i)

= ri

e ~ Pi} < r i

i

In what follows, dA i will denote the vector of the disturbances in class ,,;, , d~ the vector of the disturbances in class ~ , and de, the vector of the The partitioning of the disturbances in class measurable disturbances into classes can be interpreted as follows :

e,.

• If a disturbance dI is in the class 14,' the disturbance dead-time (P'I -1 )D.t will be greater than the minimum plant dead-time (r,-l)D.t. • If a disturbance dI is in the class '8,' the disturbance dead-time (p'I-1 )D.t will be equal to the minimum plant dead-time ( r ,-l )D.r. • If a disturbance dI is in the class e, the ' disturbance dead-time (p 'I-1 )D.t will be smaller than the minimum plant dead-time (r,-1 )D.t.

N ~J D.t, the disturbance dl is sampled, with N t,

As a result of the disturbance classifications and the definition of relative orders, one can define the following notation :

an integer, defined in the same manner: N J = Sampling Period of Disturbance j ) Period of Input Actuation If some outputs or disturbances are measured at the same rate as the input actuation rate, then the corresponding NI is equal to 1.

6

h?(x, d e, ) =hi( x) 6

h,' (x, de) =h:- (C:P(x, u, d ), de, ), I = 1, ... J , - I I

(4) The above notation clearly indicates the dependence on the state x and the disturbances of class e, only, since it is for these disturbances that the disturbance dead-time is less than the minimum plant dead-time. In addition, each

Relative Order Following Soroush ( 1994), the relative order of the output y, with respect to the manipulated input vector U is the smallest integer r, for which :

296

the output and disturbance measurements are generally available at larger sampling periods than the input actuation period, (9) utilizes the best available process information, which are the available output and disturbance measurements, to predict the future of the outputs. Using this structure, a controller can be synthesized that exhibits the best available set point tracking and disturbance rejection based on the available information . Furthermore, the predicted changes in the outputs (9) can be added to the most recent output measurement to obtain the "closed-loop" predictions for each output:

lI,r,-1(Cl>(x.u.d).d e, ) depends on the input u.

therefore the set of equations:

1

rh~':: ((x ,u,d),d e, ) 1=l~'

l

h,;

(Cl>(x , u, d) , d e",)

(5)

Ym

is locally solvable for u via the implicit function theorem if the characteristic matrix :

~hlrl-I(
t.

) I

C=

(6)

~h~",-I(
) '"

is nonsingular. The implicit function defined as the solution of (5): u='¥(x,dt/,de , y)

y;(k +/)

(7)

will depend on disturbances of classes only .

~

and

e.

+ h: (XM (k) ,d e, (k» - h; ( XM

y; (k + r; ) = Y; <[ : ; ]N;) + h;r,-I (<1>( XM (k),u (k) ,d (k » ,de, (k »

Consider a non linear process described by (1) , where the states and outputs are calculated online by simulating the process model :

= <1>(x M (k) , u(k) , d(k»

Y,M(k)=h; (x M(k» ,

- h;( XM <[ : ; ]N ;»

(8)

i=I , . . . , m

(l0)

A reference trajectory can be defined for each of the outputs as follows : y; (k + r;) =(I - a ; ) YISP + a j; (k + r; - I) (11)

driven by the inputs u(k) and the disturbance measurements d(k) . In the above equation (8), the subscript M is used to indicate calculated quantities from the process model.

where y" p is the set point for output )',' Matching the "closed-loop" predictions to the reference trajectories, one can construct a nonlinear multi rate feedforwardlfeedback MAC by solving the following set of algebraic equations :

Us ing the calculated process states x M ' future changes in the process output y, can be predicted on-line according to: Y,M (k

+ I) - Y;M

<[ : ;]N ; )

= h/ (x M (k) ,d e, (k» - h; (XM

rh,"-'

((x.

(k)' U(k)'~(k»' ~e, (k »)

l

<[ :, ]N ; »

h;;,,-I (Cl>(XM (k) ,u(k),d (k»

l=l, .. . .r,-I Y,M (k + r; ) - Y;M

-h,(x.

el

<[ :; ]N;)

= h;r, -I (Cl>(XM (k),u(k) ,d(k» ,d e, (k»

<[ : ,]N , »

l=l, .. .. r,- l

2.2 Synthesis of a Multirate Feedforwardl Feedback Model-Algorithmic Controller

XM (k + I)

= Y;<[ : ; ]N;)

<[

:1 ]N

I )

1

,d e,,, (k»

+ hi ( XM

<[

:1 ]N

I

»

(9)

<[~},»

where [kiN] denotes the integer part of the real number

kiN

and

dj (k)=d{:I]N I) is 'he

most recent disturbance measurement.

where e,=Y"p- Y" o:=diag{ a l·· ·aml , and In, is the identity matrix .

Since

mXm

297

3. APPLICATION: CONTROL OF A POL YMERIZATION REACTOR

Initiator

Monomer

YI 'r=30000 kglkmol

A V n =0 .00037

Y'sp=350.05 K V=1.20x lO J m '

AV,,=0.0664 kJ

£=-0.108

Zp =4 .917x lO; mJlkmols

j=0 .58

Ep=1.820xI0· kJlkmol

e= 1.815 kJ/kg· K

Z,=1.053x I0 15

Cus =0.263 kmoVm C",=8.986 kmoVm

.....

Polymer

Q ..

...

Si

-&1p =57800 kJ/kmol

l l

Si

E,=1 .284x lO l kJlkmol Z,=9 .800xlO'm Jlkmol·s

Cnon,,=3.984 kmoVm '

E,=2.933x lO' kJlkmol

C,n,,=5 .881 kmoVm '

Z/n,=4.66IxI0 m'/kmol ·s

L=293 .2 K

E/",=7.418xI0 kJlkmol

Tnu =293.2 K

Z,,=I .OlOx lO' m 'lkmol·s

AVR ,=0.019

Fig. I. Polymerization Reactor

Si

9

4

Si

E =4 .769x I04 kJ/kmol IJ

AVRz=0.0038

S·I

A Vl/=O .0008

S· I

Z =3 .956x104 ml/kmol ·s " 4 E,,=-1.71IxI0 kJ/kmol

Table 1. System Parameters Consider the free radical homopolymerization of methyl methacrylate (MMA) in a CSTR as shown in Figure I, with azo-bis-isobutronitrile (AIBN) as the initiator and toluene as the solvent (Soroush and Kravaris, 1993). The model was discretized using Euler's method and is composed of seven state equations:

)1o(k + 1) =)1o(k) + (R IlO [x(k)](1

)11 (k + I) =)11 (k) + (Rill [x(k»)(1

Cm(k + 1) = Cm (k) + (Rm[x(k)] + C mmsFm IV (1

C, (I.: + I)

T(k + 1)

+EX p(k»C,(k)FmIV)t:.t

=C, (k) + «FmCsms + Fi (k)C sis) I V (I

+ EX p(k»C, (k)Fm I V)t:.t

In addition, Cnon" and ( nn are the concentrations of the monomer and solvent in the monomer stream, Cu , and C", are the concentrations of the initiator and solvent in the initiator stream, and £ is the volume expansion factor for this reaction . Fn, is the inlet monomer stream flow rate, L is the ambient temperature, and Tn" is the monomer stream temperature. Finally, AVH are the heat transfer coefficients for the reactor and jacket, m is the mass of the reactor contents, and c is the heat capacity of the contents. The reaction rate equations are as follows :

= T(k) + (Rh[X(k)]V I(me) + TmsFm IV(1

+ EX p (k»T(k)Fm I V)t:.t +

AV RI (T, (k) - T(k»t:.t + AV R2 (T~ (k) - T(k»t:.I T, (k + I)

=T, (k) + (AV

JI

+ EX p(k »)11 (k) Fm I V)t:.t

where Cn,' Ci , Csare the molar concentrations of the monomer, initiator, and solvent, respectively, T is the reactor temperature, ~ is the jacket temperature, )1(1 is the molar concentration of the dead polymer chains, )1, is the mass concentration of the dead polymer chains, V is the reactor volume, and t:.t=30s is the input actuation period .

+EX p(k»Cm(k)F", IV)t:.t

C, (I.: + I) = C, (k) + (R;[x(k)] + CiisF, (k) IV (I

+ EX p(k »)10 (k )F", I V )t:.t

(T(k) - T j (k» +

AV]2 (T~ (k) - T j (k» + AV ]2Q(k»t:.I

298

Rm[x( k )] = -C m (k )AO (k)( k Jm (k) + k p (k»

N'= I. The di sturbance has P,=P:= 1 and therefore belongs to classes e, and e,.

=-k , (k)C , (k) Rh[x(k)] =(-MJ p) k p(k)AO(k)C m(k) R, [x(k)]

R!" , [xC k)] = (k ,d (k )AO (k) + k fm (k)C m (k) +

kIJ (k)C s(k»AO(k) + 0.5k tc(k)AO 2 (k) R!, 1 [X (k ) ] = (k Jm (k) Cm (k) + k IJ (k) Cs(k»A, (k) + k, (k)AO(k)A, (k)

where :

A (k)=[2 fk ,(k)C,(k»)1I2 kt( k)

o

~ (k) = M m(2fk , (k)C, (k)

+ kt, (k)Ao (k)C, (k) +

(k p (k) + k fm (k»Ao (k)C m(k» / (k fm (k)C m (k) + k" (k)C , (k) + k, (k )Ao (k» X (k)=

p

!ll(k) !l1(k)+MmCm(k)

k[ (k) = Z [ Exp[ -E[ I RT(k)], k [ (k)

- - = Z[Exp[-E[ / RT(k»), kp(k)

k , (k) = k,c(k)

The closed-loop profiles for the system were simulated for a disturbance step change applying two different controllers. The first case involves the nonlinear multirate MAC algorithm where the disturbance measurement is not utilized . For this case, the model states are simulated using a constant value of 7 Fn>=2.78·10 m l/s . The second case uses the disturbance measurement in the non linear multirate feedforwardlfeedback MAC algorithm. Both cases set y Isp =30000 kg/kmol and Ylsp=350.05 K, with a ,=a1=0.96. At time equal 15 minutes, F., undergoes a step change from 2.78xlO·7 m lls to 3.00xlO·7 m l/s . The closed-loop profiles for the first case without feedforward disturbance measurements are shown in Figures 2 and 3. [--MW--T :

1= t , p, i

t~

30500 ...-.·_·--_·_· _--

30200 30100

b

30000 29900

-I-

30400

~ .. _ 30300

1= fm,ts

!~

~~

+ k'd (k)

u ..

-E -

~

k" (k)

- .- =Ztc Exp[-E,c I RT(k»)

29800

k 'd (k)

.....

-

.. _--;- 35 1.05

t-I- 350.85 350.65 +350.45

t 350.25 ~ ! 350.05 l! T 349.85 8.

~

_ ~

---"----r 349.65 - 349.45

+-i+1+1""""';-

~ -

~

l-

T 349.25 "I -+--!+--+1-+1....,.-,-+-1~-+I--
O~OON~N~WN~~~WNWW ~OONW ro

66~~

~~~~

~~~~

~~~~

Time (hr)

In this notation, \ is the molar concentration of live polymer chains, AI is the mass concentration of live polymer chains, and Xp is • the conversIOn. k,' P k , and k, are the overall rate constants for the initiation, propagation , and termination , respectively . kIn> and kt, are the rate constants for chain transfer to the monomer and solvent, k", and k" are the rate constants for termination by disproportionation and combination, and f is the initiator efficiency .

Fig. 2. Output profiles for the multirate MAC without feedforward measurements

120E·07

T ..

- 0.525

1.00E·07

+;

.:. 0.52

.. 8.00E·08

.,-

The controlled outputs of this system are the number averaged molecular weight (Y1=!l/!l.) and the reactor temperature (Yl=T). For this system, the molecular weight measurement is available every thirty minutes due to the analytical nature of the measurement, and the reactor temperature is available every thirty The seconds. Thus, N I =60 and Nl=l. manipulated inputs of the system are the initiator flow rate (ul=F) and the rate of heat addition or removal (u 2=Q) . It is easy to verify that r ,=r2=2 . The disturbance measurement of the system is the monomer flow rate (d=FJ which is available every thirty seconds, and so

+ I

i ~

..s _

6 .00E·08

"

4.00E·08

t

2 .00E-08

.l-I-L.............A,.~~.-4.-....

O.OOE +00

1

~

rt-iH+t-If-+4-H-+-i-r-r..,........;·

T' '; . J

..--+1 -+1...,.....;-+--+-I+1 ~I--
_

~

0 .515 ~ N

"

0.505

o~roNwN~mNw~~~Nww~m Nw m

66~~

~~~~

~~~~

~~~~

Time (hr)

Fig. 3. Input profiles for the multi rate MAC without feedforward measurements Figure 2 shows that without the feedforward disturbance measurement, the multirate MAC takes a considerable amount of time for the system to return to the molecular weight set point after the step change in the mono mer flow rate. Figures 4 and 5 show the second case with the multirate feedforwardlfeedback MAC.

299

ACKNOWLEDGEMENT Financial support from the National Science Foundation through Grant CTS-9403432 is gratefully acknowledged .

' -MW--T !

E

30500 -

-,- 351 .05

~OO ~

-~~

t+ ~~~

.~ _ 30300 ~ ~ ~ 30200 _ .. x

~ ~ 30100 -E - 30000 ~ 29900 29BOO

350.45 .,. 35025 350.05 T ~ 349.85 ' . _ .•_ -. -- -.-- - --------.. - - - - f 349.65 .;. t 349.45 t 349.25 -' ------+--_-~_.____,~...._._~----"-I 349.05

---------------<,

-;~

~

8. ~

5. REFERENCES

l-

Araki, M ., and T. Hagiwara (1986). Pole Assignment by Multirate Sampled-Data Output Feedback . Int. 1. Control, 44, 1661. Araki, M., and K . Yamamoto (1986) . Multivariable Multirate Sampled-Data Systems: State-Space Description, Transfer Characteristics, and Nyquist Criterion. IEEE Trans. Autom. Contr., AC-31, 145 . Daoutidis, P., M. Soroush, and C. Kravaris (1990) . FeedforwardlFeedback Control of Multivariable Nonlinear Processes . A1ChE 1.,36,1471. Gopinath, R., and B. W. Bequette (1992). Multirate Model Predictive Control of Unconstrained Single Input-Single Output Processes . Proc. ACC, 2042. Hagiwara, T., and M. Araki (1988). Design of a Stable State Feedback Controller Based on the Multirate Sampling of the Plant Output. IEEE Trans. Autom. Contr., 33,812, Lee, J. H., M . Gelormino, and M. Morari (1992). Model Predictive Control of MultiRate Sampled-Data Systems: A State-Space Approach. Int. 1. Control, 55, 153. Niemiec, M., and C. Kravaris (1998) . Nonlinear Model-Algorithmic Control: A Review and New Developments. In : Nonlinear Model Based Process Control (R. Berber and C. Kravaris (Eds» . Kluwer Academic Publishers, Dordecht. Ohshima, M., 1. Hashimoto, H. Ohno, M. Takeda, T. Yoneyama, and F. Gotoh (1994). Multirate Multivariable Model Predictive Control and its Application to a Polymerization Reactor. Int. 1. Control, 59, 731. Soroush, M. (1994). Discrete-Time FeedforwardlFeedback Control of Multivariable Nonlinear Processes . Proc. ACe, 1349. Soroush, M., and C. Kravaris (1993), Multivariable Nonlinear Control of a Continuous Polymerization Reactor: An Experimental Study . AIChE J., 39, 1920. Soroush, M., and C. Kravaris (1994) . Synthesis of Discrete-Time Nonlinear Feedforwardl Feedback Controllers. AIChE 1.,40,473.

O~OON~NvOONW~vOONW~VOONWOO

oo~~

N~~~

~~~~

~~~~

Time (hr)

Fig . 4. Output profiles for the feedforwardlfeedback MAC

multirate

1 -~U1--u2 i

.- -

~~~:~q -

~gg~:~~ t

~6~~ t

.s

l

----- --------- -- --- -1 ° ·525

tI

---:T O.515 I .•.+ 0 .51

5.00E-08 U,.._ _ _ _ _ _ _ _ _ _ :; 400E-OB -;300E-08 200E.OB .!. ••. .,.. __

1 .00E~08

0.52

-

~

~ "

I

...

O.OOE+OO I.

t

t

t

t

t

t

t

t

t

i 0 .505

O~~~~N~~~~V~~~~W~~~~OO OO~~

NNMM

vv~~

~W~~

Time (hr)

Fig. 5. Input profiles for the feedforwardlfeedback MAC

multirate

Comparing Figures 4 and 2, it is clear that the use of feedforward measurements greatly improves the performance of the multirate MAC. The controller is able to maintain the molecular weight set point at a value of 30000 kg/kmol without large excursions. In addition, Figure 5 shows that the input profiles do not exhibit the spiking that appears in Figure 3.

4. CONCLUSION This paper develops a systematic methodology for the synthesis of nonlinear multirate muhivariable feedforwardlfeedback ModelAlgorithmic Controllers (MAC). The proposed controller incorporates available process disturbance measurements along with available output measurements to efficiently reject the effect of the disturbances on the slowly sampled outputs. The developed feedforwardlfeedback controller is applied to a continuous polymerization reactor, where it is seen that the control system performance is greatly improved using disturbance measurements ,

300