Optimal Control of a Class of Multivariable Nonlinear Systems. Application to a Fermentation Process

Copyright© IFAC Control Science and Technology (8t h Triennial World Congress) Kyoto, Japan, 1981

OPTIMAL CONTROL OF A CLASS OF MUL TIVARIABLE NON LINEAR SYSTEMS. APPLICATION TO A FERMENTATION PROCESS

J.

Alvarez G. and

J. Alvarez G.

Departm ent of Electrz'cal Engz'neering, Advanced Studz'es and R esea rch Cent er, Mexz'co

Abstract. Controllability conditions and a control technique for a class of multivariable nonlinear systems are presented . The application to a continuous culture fermentation process is discussed . It is shown that an appropriate selection of a penalization function qives a control law that includes the conventional PI controller . The steady state error for the close0l oop s ystem is zero and it is not sensitive to the process parameter variations in a neiqhbourhood of the controller parameters . Keywords. 'lonlinear systems; controllability; optimal control; PI control ; fermentation process.

INTROoucrIQl\l

(3)

This system is said t o be controllable from (~ , to) t o (Q , t ) if, for some contr ol f ~(t), to ~ t ~ t , the solution of (3), with f x(t )= x , is such that x(t )= 0, where t is 0 --{) -f f a specified fina l time. By definition , the system is comoletely controllable from (~, t o) to (Q, t ) if it is controllable for f all x E Rn.

In this paper we consioer the class of nonlinear systems represented by the equation: x(t)= a(x) + R(x) u(t) y(t)= C

~(t)

(1)

(2)

n m i where x( t) E R, u(t) E R , y( t) E R are the state, input and output vectors , respectively; a(.) and R( . ) are vector and matrix functions of class c;:o and C is a constant matrix of proper dimensions.

The theorem which is aoinq to be used t o find the control lability conditions of the s ys tem (1) i s the followinq (C£rshwin, S . B., and Jacobson, O. H., 1971):

method for sinthesizing an optimal contro l law for this class of systems is presented after havinq studied the conditions for the system to ~ controllable . I t is shown that a convenient selection of the performance index leads to control laws includinq proportional or/and inteqral terms of the error.

A

Theorem. Let us consider the system represented by (3). If a scalar funct i on V(x , t) exists such that: i)

ClV(x,

dX

The knowledge of the process parameter values is necessary t o sinthesize an optimal control law. Any parameter variati on can dearade the controller per formance and may lead to a nonzero steady state error. It is also shown that for a nonlinear "PI " control law, this error is zero and it is not sensitive t o the process parameter values variations arounr. the controller parameter values.

and aV(~, t) exist for all at

x , t, t:;< t ; f ii)

For all continuous vector function ~(t), ~m ~(t) :;< Q impl ies that t->t f iim V(x( t), t) -> oc ; t->t f

and if there exists a control law u * (.) E U (U: set of admissible controls) such that:

The performance of the control structure is evaluated by simulation with a continuous culture fermentation pr ocess.

iii)

Alonq the traiectories of (3), OV(~,

dt

Let us consider the system represented by:

. t

469 eS T 1 - p .

t)

f

;

t) < M <

00

for all t, t

< t < 0

-

Joaquln Alvarez G. and Jaime Alvarez G.

470

iv) The solution to x(t)= fix , u * (x, t) ,t) , x(t )= x , exists and Tt-is Uniqu-e; -

-0

0

then s ystem (3) is controllable from (x , t ) to (Q, t ). -0 0 f A demonstration of this theorem is aiven is given by C£rshwin and Jacobson (1971). Now, let the system be represented by (1): I..erm>a • T Let us cons ider the set 0= {x (t) I B(x) B (x) is of rank n for all t >-0 } and

t - -> f

00 .

-~hen

system (1) is controllable in

(8)

and a necessary condition for obtainina u * (t) will be the existence of (C B(~))-l. Control law 1 ("P"). . 2 2 = 4K o . Then:

Let assume that h( o)=

p

~*(t)= (C B(~)J-1 (y (t) - Ca(x) + o - -

K

~.

(9)

Demonstration. Let Sit) Q Sn eAt be a real symetric definite positive ~trix for all t, >. > 0 and consider the Lyapunov function v(~ , t)= ~T(t) Sit) ~(t). From the previous theorem, it can be seen that: - (i) is satisfied because aV(x , t) and ax aV(x, t) exist for all ~, t, t 1 t ; f at -

V(~(tf))

- >

00

when

tim t->t

x(t) 1 0, then

Bv hioothesis, (B(x) B~(X))-l exists. ~hen V(x, t) = - ! X~ (t) So eAt-x (t) < 0 for all to-< t < tf;-therefore (iTi) of theor~ is also saEisfie<'!. - Finally, using (4) in (1): x(t)= - AX(t) and its solution exists and Tt is unique. CONTROL OF A CIASS 01"

(t

{Kpln + KI lj

~( T )dT ) .

~~(t) (~T(t) ~(t)t7 } 2

(10)

nxn; KD and K are real constants. This law includes a tetm proportional to the intearal of the error with the objective that this term appears in the control law. Then, usina (10) in (8): u*(t)= (C

B(~) ) -l

lYe - Ca(x)

+ Kp

~(t) +

(t

Kr

J ~( T ) ch)

(11)

o

The performance index is proposed to be the following: t ( 2 J= J (h( r ) + {) ) dt (5) o where h( o) is a positive definite function. If yP(t) is the reference vector and the error is defined as ~ ~ Yo - y, 0 is qiven by:

In this case, u * (t) contains G~e system nonlinearity, a term proportion~l to the error and a term prooortional t o the intearal of the error. Considerinq that the error is a continuous func tion of time, the error eauation is ~(t) + Kp ~(t) + KI ~(t)= Q, Which has G~e oriqen as an asymptotically stable equilibrium point if Kp and KI are oositive. -

& (yo (t) - y(t) )T Q(Zc (t) - y(t) ) = 'T'

= ~- (t) 0 ~(t)

(6)

{) is the "velocity error" and it is related to the control u. () is a real symetric oositive definite matrix. Synthesis of the control laws. By application of Fuler-Laarange equation to (5) and considering the system eauations, we can arrive to: Q {Yo - C

0

Consider the matrix

I'ihere In is the identity matrix of dimension

+

Let us consider the system represented by equations (1) and (2).

C~)T

.

'\101\] -

LINFAR SYSTEMS.

/hi p) + k = (Yo -

Control law 2 ("PI"). function: h( o) In= 4

(4)

-4

From (1), (2) and (9), the error equation is given by e(t)= - Kn e(t). Then, this control law reduces the error differential 8.1Uation to a first order linear differential one. The stability of the system is then assured if K is positive.

2

f

Let u * be defined as: * T T -1 ~ (t)= - B (~) ( B(~) B (~)) ( A~(t) + ~(~))

OPTIMAL

It can be seen that the optimal control has a nonlinear term of the same form that the nonlinearity of the system, plus a term proportional to the error .

p

(ii) is verified.

o

e(t))

p-

(~(~) +

+ B(~) ~} Now, let us consider that dim y= dim u. this case, if '1' - > 00 , from (7):

(7)

In

Steady state error. It has been shown that, for the proposed control la~5, the error converqes to zero when the control oarameters values are the same as the pr~ess parameters. However, if a process parameter variation occurs and if this variation is not detected by the controller, the steady state error may not be zero.

471

A Class of Multivariable Nonlinear Systems

In the case of "P" =ntrol, the error equation becares: e(t) + K e(t) + ClCB(x).

(1- ClCB(x) A(x)) A(x) -

-

0

Y-

-

-

ClCB(x) A(x) Ca(x) +

"-0

-

+ ClCa(x)= 0

(12)

where the existence of A(x)= (Co BO (~) + +ClCB(x) ) -i are equivalent implicit variations of cB(x) and Ca(x) . The equilibrium point (e ) of the system represented by (12) is: -0 e = K- 1 (ClCB(X) A(x) Ca(x) 0

-0

--

- ClCa(x) ) - - t ->

(13)

00

which is not necessarily zero. Then, if a parameter variation exists, the converqence of the error to zero cannot be assured.

is the feed substrate concentration; R is the overall yield factor; ].l is the maximum m

specific arowth rate and K is the MichaelisMenten constant. s In this process it is often necessary to reoulate the concentrations (substrate and bioIl\3.ss). This is due to a limited aeration capability of the process, a limited capability of the installed equipment for bi~ass treatment, a limited capability to remove the heat senerated within the fermentator, a need to =nstrain the substrate level which =uld not be recuperated, etc. The reoulation of the substrate and the biamass may be done with the dilution rate (D) and the feed substrate concentration (Sa). Control synthesis. Let us consider x= y= 'T' * - *-T = (x, s)T, u- = (D, D.S) , V = (x , s ) a-o (a constant vector). From (9), (15), (16) and (J7) ~ (18)

In the case of "PI" control, the error equation is: (I - D(~)) ~(t)

ft o

+ Kp ~(t) + KI

~(T)dT + D(~) Ye - D(x)

(14) Ca(x) + ClCa(x)= 0 - - 1 where D(x)= ClCB(x) (CB (x)r has been assumed to exist~ If t£m t->oo

y (t) is a =nstant and if e(t) is "-0

-

a continuous function of time, the equilibrium point of (14) is ~= Q. This equilibrium point is asymptotically stable when the process parameters are equal to the controller parameters (ClCB(x)= ClCa(x)= 0). Then, there exists a neiqhbourhOOd of the controller parameters in which this point is also stable. APPLICATION TO A FERMENTATION PRCCESS. In this section, the application of the proposed =ntrol laws to a simulated continuous fermentation process for producina sinale cell protein is presented. r--tx1el. A ITDdel for this kind of process is obtained from Il\3.terial balances of the Id-cro organisms and the substrate. The ITDdel used here is the followina (Alvarez J., 1978):

S= D(Sa-S) ].l=].l

S

K +S

1 x+ S + 0 R c D1

--*D1

(!R

e

c

*

+ e )

x

(19)

s

where e x =Cl x *

- s.

For the ideal case, ].lmc= ].lm; Rc= Rand Ksc= = K ; but in oractice, it is natural that a s difference between the process and =ntroller parameters values exists. K is a proportional oain; it is an accelergtion factor of the process. From

(15), (16) and (17):

(11),

t KI f

* and Sal

. l~

c

fl

* (D.S a ) 2

1 p

* D2 (Kp ex + KI

( )0

e

x

Jo

e

s

d T) (20)

x

x + S +

c

1 ~

D2 dT) + K

(t + KI

e

o

P

e

s

+ (21)

dT]

(16) (17)

= (2.73, 0.905)

].lX

~

T<

b. (D.S a )1

Experimental results. The plant is the process ITDdelized by Fqs. (15)~ (16) and (17) with].l = 0.2 (h- 1), R = 0.3, K = 0.1 (g/i). m m s* * T It has been considered y = (x, s ) =

(15)

X= (].l-D) x

*

* and S a1

'T'

"-0

(a/i], as reference vector.

m s

where x and S are the biamass and substrate concentrations; D is the dilution rate; Sa

The process was reoulated with the tw:> =ntrol laws (18) - (21). F'igure 1 ShCMS the evolution of x, s, D and S when there is a a

Joaqu i n Al varez G. and Jaime Alvarez G.

472

chanGe in the refer ence vect or. The values of the YJei qhtinq factors are: "P" control, Kp= 0.25 ; "PI" Control, Kp= 0.5 and K = 0 . 1~ I x

S 1.0

S

--P

0 .'

-PI

0.'

X

AccordinG to the carried out experiments , it has been observed that there is a perfect reoulation when process parameters variations oc=s and that is detected by the controller. HOYJever, if these variations are not detected by the "P" controller, there will be a nonzero steady state error, as it can be observed in Fia. 2 , where has been chanqed from 0.2 to 0.16 (h - l ) .

"m

If we use a "PI " controller with its parameters fixed to the initial values and if the process parameters chancre as in Fia . 3, the process response is as in Piq . 4, where it can be observed the very Good reoulation of the biomass and substrate concentration with this structure. CONCLUSIONS

20

T(h)

30

Sa

D .28

Sa 10

--P

In this paper, the controllability conditions of a class of multivariable nonlinear systems and the synthesis of appropriated control laws have been presented as an extension of the results obtained by Alvarez and GalleGos (1980). It has been shown that it is possible to obtain control laws which have terms proportional t o the error and t o the integral of the error if an appropriated penalization function is chosen.

-PI 11

The application of a "P " control law reduces the error differential equation to a first order linear differential one with the ori gin as a stable equilibrium point i f the error coefficient (Kp) is positive and if

•

the process parameters are known. If a difference between controller and pr ocess parameters values exists , then it may have a nonzero steady state error.

D

.1' o

FiG . 1 .

10

20 T (h)

30

Closed- loop system response to a chanGe in the reference vector.

x S 3.or--------------------------------------, S 2.0

This problem is solved with a "PI" control. With this controller, the error equation is a second order linear differential one. The steady state error is zero ,men the reference vector is constant . This error is also zero when a process parameters variations not detected bv the controller occurs.

Pm

Rm

0. 3°r-------------________________________

0 .25

1.5 2 0.

0.35

:

1'----

i

°1 __--J1r-----!1 L

x

0 .15

~m

10

30

20

System response t o non-detected process parameters vari ations ("P" control).

0.25

0 .100!-----------5-0-----------10-0-----------1-5-0J 0 . 20

T (h)

FiG . 2.

'- _____ 2

0 . 30

Rm

1.0

o

~0 . 40

T (h)

FiG . 3.

Parameters variati ons applied to the process.

473

A Class of Multivariable Nonlinear Systems

in Continuous Culture . Ph . D. dissert a tion (in spanish ) . CI EA-IPN . Mexico . Alvarez , J . ; and J . A. r~lleqos (1980) . Opt imal Nonlinear Control of a Permentati on Process (in spanish). Proc . of the ANIAC Conoress. Mexico . pp. 126- 130. r£rshwin, S . B. , and D. H. Jacobson (197 1) . A Controllability Theory for Nonlinear Systems . IEEE TTans . on Aut . Cont rol . AC-16 . 37-46. Rhoten , R. P. , and R. J . Mul holland (1974) . Optimal Reoula t ion of Nonlinear Plants. Int. J. Control . 19 . 707- 718 .

The proposed control laws have been used to regulate a fermentation process . The "PI" controller has shown a oreat robustness and performance when hard process parameter s var i ations have been introduced. In practice, these variations are not so hard, then a better performance may be expected. REFERENCES

Alvarez , J . (1978) . Fermentation Process Identification ann Static Optimization

s

x

3.or-------------------------------------------------------------------------,2.o 11

1\

2.i

s

,/\ \

2.8

I

\

I

\

,

'.

I

"~".-:-:-:-::-:-=--=-=-:-:-:-:-::-:..,=

,..------- --,

I

/'

I I

,,~

,

" ,,~""_...._...._-_---_-_-_......._-_-fl.0

---- ---

I

I

I

I

I

X

,

I

1.~

\

I

\ \

2 .1

I

O.~

I

\... 1

I

I

\/ 2.~

0~----------2-~----------~~~0----------~7~~~---------1~0~0~--------7.12~5~--------~

0.0

T (h)

Sa

D

r-------------------------------------------------------------------------,13 0.24 ~----------,

/ I"

0.22

I

12

I\

--------,1

I

11

\

\

0 . 20

\

Sa

\

, , - _________ J

"'

I

0 . 18

' .... _---

--- ----

10

I I

D

0.18

8 ~O

Fiq. 4 .

Svste~

7~

100

12~

T(h)

res pon se to parame t ers variations of Fig.

Discussion to Paper 18 . 2 S. Kahne (USA): I would have expected that the more important con s traint would be maxi mum acce l eration of the load to prevent break age due to large forces. Yet the paper seems to emphasize velocity l i mits. Y. Sakawa (Japan ) : Each motor has its own maximum torque and maxi mum speed , which we cons i dered as the torque constraint and the speed cons t raint in ou r formula t ion . Torque constrai n t on the moto r effective l y l i mi ts the acceleration , so we have not specifically cons i dered load acceleration .

3.

We think that the most important factor to prevent breakage is to keep the load swing angle as sma l l as possible.