Linearizing Control of a Class of Non-Linear Continuous Processes

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LINEARIZING CONTROL OF A CLASS OF NON· LINEAR CONTINUOUS PROCESSES G. Gilles and N. Laggoune LIl/wm/()irl' (rAII/()II/(//iqlll'.

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Abstract. Nost of the continuous industrial processes working inside 3 large operating range can be modelled by non - linear state equations that are linea r in control, A way for control l ing those processes can be a closed loop linearization. It is shown that, when the solution exists, we get an explicit non - linear state feedback control law. The linearizability condition is demonstrated and it is shown that the control law may in volve some singularities. Two practical applications on chemical and hydraulic pilot plants are presented. Keywords. Non l inear systems control ; hydraulic systems

nonlinear control systems pH control.

NON-LINEAR MODELLING OF CONTINUOUS INDUSTRIAL PROCESSES

optimal control

bilinear

LINEARIZINC CONTROL LAI"

Introduction Most of the continuous processes (electrical, mechanical, hydraulic, thermal, chemical, biochemical , etc ... ) present in general a non - linear globa l be haviour. Only in some special working conditions, it is possible to approach their behaviour by linear models, mathematical abstraction then allowing to get simple properties and analytical developme n ts . Since the variation magnitude of the descriptive variables is l arge, the use of linear models beco mes total l y unsu i table. In genera l, a knowledge mathematical model of a continuous process is got by writing infinitesimal balances of quantities like mass , energy, etc . . . :

storage inside ] [ the volume of the system _ [wha t goes out] the surface

Many recent works have been devoted to systems linear in contro l , also called " linear-analytic systems". Among the works related to their control, most of them are based on differential geometry and Lie algebra. In this fie l d, let us specially men tion the immersion theory of a non - linea r system into another system by Claude, Fliess and Isidori ( 1983), and the model matching of non - linear systems by Di Benedetto and Isidori ( 1984). Recently, the idea of pseudo - linearization, l ocal concept ex tended to a global area, has been developed by Mouyon, Champetier and Reboullet ( 1984). Extremely rare the papers dealing with concrete applications. Apar t from this l ast work applied to an asynch r o nous motor control, let us mention the work of Alvarez Callegos brothers to digi t al computer optimal control of fermentation processes (Alvarez Gallegos, 1982). Much more works are devoted to a special subclass of this type of systems: bilinear systems.

what enters ] [ through the surface +

what is generated] [ inside the volume what is consumed] [ i n side the volume

that is to say dx = fi(~,~)dt - !o(~)dt + ~i(~)dt - ~(~)dt,

Finally, we can recall the work of Cilles ( 1984) which is extended here and which concerns the idea of linearization by considering only state equa tions. In the case where f(x) = Ax, Cilles ( 1983) showed that the linearizi~g-control law minimizes a quadratic criterion. For the special subclass of bilinear processes, the authors (CitIes and Laggoune, 1985) derived a digital linearization by using a discrete model. After having recalled the linearization principles, the authors will empha size on some p r atical aspects of the use of the linearization control law. Applications to 2 pilot plants will be presented.

x denoting the state vector of the system (dimension n) and u be i ng the control vector (dimension

m). The input vector modulates some terms, of the vector i al differential equation, mainly the input flux f .. Mostly, this modulation is expressed as a produ-;:-~ of a "flow- rate" by the r atio of the element which is carried, that can be written under the fol l owing type of state equation: m g. (x). u . , or x f(x) + 1: -J J j =1

x

f(x) + C(x).u

Principles

(1) The control problem of systems linear with respect to the control vector (Eq. (1» can be solved in order to get, by means of an appropriate feedback, a linear system whose dynamic properties are more well - kno\"''TI. ~1oreover, if such a control can be managed, the fact that the closed loop system belongs to the linear class allows to set dynamic performances which remain constant inside the whole ope rating range.

C(x) being a n xm matrix non - linearly depending upon the state - vector x as well as the vector f. Such a non-linear state equation has the property to be linear with respect to the control vector. This quasi - general property is very interesting for designing control systems. In particular, it will be shown that, under certain conditions, an expli cit non-linear control law can linearize those nonlinear dynamic processes.

The closed loop system is linear if its error ~ (t)

l -l')

G. Gilles and \1. Laggoune

150 satisfies a linear dynamic equation

set)

=

guarantee the system stability and prescribe an arbitrary dynamics for the system. s et) is an error vector between the state ~ and a reference on the state --<:

- x(t)

(3)

or can be an error vector between the output = ~

-

~(t)

(4)

m

f'(x) - .1: u £' (~) i Fr+1 (9)

Some e lements of OH are, in general, set in order

to cancel ,,"(x). The remaining elements of Dare chosen in order to set the eigenvalues, which is close to modal control.

The control law involving a matrix inversion, the control vector norm may become infinite for the va lues of ~ such that

det {G'(x)} = 0

-~)

( 10)

So, a system which is linear in control and which satisfies the linearizability condition is linearizable almost everywhere except in a finite number

E = - ~ = - f(x) - G(x).u = D(~ - x) = D(x

x )

--<:

(8)

1)

of singular points.

the input ~ is constant, so :

G(~).~

D' (x

(x)

x

Existence of singularities

In the first case, the control system is said to be a strong linearization (between input and state). In~econd one, it is called a weak linearization (between input and output). In order to simplify the presentation, this paragraph will be only devoted to strong linearization. In the case of regulation problems or step changing in the reference,

or

!

with

(r

~(px1)

and a reference input :

£(t)

(rxr)

(r x 1)

(2)

D is a desired closed loop matrix whose eigenvalues

E(t) = x

[G' (~)r1.'f' (~)

u'

D [ et)

- f(x)

In order to get a robust control, it is necessary

to detect tendencies towards a singularity and apply a temporary auxi liar y control allowing to avoid the singularity.

(5)

The algebraic set of equations (5), non-linearly depending on the state x, is structurally linear with respect to the unknown u. Then, it can be sol ved explicitly in order to get the expression of the control law. Nevertheless, as the solution does

Nevertheless, in fact, we must say that for some

not always exist, its existence must be discussed.

industrial applications, the singularities are not accessible. Moreover, the non-linear processes such that G(~) = G = Cte present no singularities.

Linearizability condition

Optimality properties

Let us call r =(rank G(~) } . m, the Cramer system leads to the unique If r = n solution

Let us recall that, for the subclass of systems such that f(x) = Ax, it has been shown that the

u

=

G-1

(~)

[D (~

-

~)

-

linearizatlon control law minimizes a quadratic criterion :

i. (~) ]

If r = n < m, the system is undetermined of order m - r = m - n. Then, (m - n) control variables can be arbitrarily chosen, the n other control varia bles expressing themselves by means of an inversion of a submatrix (n x n) of G(x). If r < n, the system is undetermined of order (m - r) iff the (n - r) characteristic determinants are equal to zero. If this condition is not satisfied, the solution is impossible.

J =

ui£i

Examples

=

[!'(~)] f"(~)

If x

--<:

x =

)n-r

(6)

The system being solvable iff ,!,"(x) = 0, we deduce: Theorem: A system linear in cont~ol defined by Eq. (1) is closed loop linearizable in the sense of Eq. (5) iff, after the application of a linear transformation T leading to Eq. (6), the second member vector ~( x) is such that

'1''' (x)

= 0

= 0

[~ :] ~ + [~+CX2]u

ax2 + (b + cx )u 2 x

[ )r

(7)

This linearizability condition leads to the following remark. If q>" (x) does not involve non-linear elements, ,!," (x) can-be cancelled by setting some elements of the D matrix. But a sufficient nonlinearizability condition is that, after applying the linear transformation T, f(~) involves more than r non-lin ear elements . When the linearizability condition is satisfied, we get the explicit control law :

(bilinear system followed

Compartmental syst em by an ,ntegrator).

formation T on the rows, we can cancel the matrix u' [G'(~)] o . -

dt (11)

represent a precision criterion and a control ener-

(~)

and get the system:

~ + [G(~) .~JT R[G(~).~]}

gy. This optimal contro l law is especially very useful for bilinear systems (Gilles, 1983) .

G'(x) being the square (r x r) submatrix of G, G"(x) being another (n - r) x r submatrix and u' being a(r x 1) subvector of u. By applying a l~near transG"(~)

p

which is an extension of the well-known result for linear systems and where the 2 terms respectively

This leads to the system under the following form

.~'= D(~-~) - i.(~) -.1.=r+1 ~ [ ~:,i~;] -

r"{~T

2

d11 d 12] = [ d21 d22 ~

leads to

d21 x1 + d 22 x2 d 11 x

1

Obviously, d11 = 0, d 12 = + (d22 - a)x2 u = b + cX 2 d 21 and d 22 are adjusted in order to set eigenva lues which are solutions of the equation A2 - d 22 A - d 21 = O. Compartmental system (linear system followed by a b,l,near system). If ~ = 0, x2 being the output of the first system controlling the second one, the state equations x x

1

a x + b x + c x x 1 2 1 1 2 1 1 a x 2 2

2

+ b u 2

d

x + d x 12 2 11 1

d

x + d x 22 2 21 1

show that

'f" (x) = (d 11 - a 1)x 1 + (d 12 - b 1 )x2 - c 1x 1x 2 cl

"

~,

and this system is not linearizable.

0

:\oll-Iinear ComillllOllS Processes

~EAK

milar r esu lts a r e got . Nevert heless, we cann o t go over d ; -1 /20 sec - l because then the ini ti a l co ntro l value rea ches the maximum flow- rate delivered by the wide open valve.

CLOSED LOOP LIKEARIZATION

In the most important practical cases , a weak linearization fr om i nput t o out pu t is su fficient. In this case , we must co nside r Eq . ( 1) followed by the outp ut eq uati on th at we will con s id e r to be linear

x.

;

C

(1 2 )

x

Th en Eq . (2) and (4) l ead t o :

- X.

E

CX

C· .U~)

Disturbances effec t . Let us assume th e existence o f a l eak act in g as a disturbance qp f O. The s t a t e equation now becomes : 1 "

S tU -

1

q(x) - qpJ

( 18)

D(Yc o r C.G(x)u

( 13)

As in the previ o us parag raph devo t e d to strong linearizat i on , we have t o solve a se t of linear eq ua-

ti on s of th e same st ruct ur e in ord er t o find th e contro l l a w. So, the same results can be applied if the second (n x r) part of the second membe r of Eq. ( 13) tran sfo rmed by T is zero, the lin ear i zation co ntrol law exists and exp li c itl y expr esses as

:

~; -[(C G(~)) ,] - 1 [ (C !(~)) '

+

D'(Yc - 1.)]

( 14)

whi ch corresponds to th e contr ol st ru ct ur e s hown on Fig. 1 which po int s ou t the 3 imp orta nt part s : th e non-linear pre-compensator [ (C G(~)) ,] -1, the non -linear state feedback and th e c l osed loop ga in matri x D' wh i ch se t s the sy s t em performances . Let us notice th at th e cases where r ; m ; p are very common. Then, D' ; D and (C.G) ' ; C.G and a decoup ling control can be achieved by tak i ng a dia gonal D ma tri x . Moreover, thi s kind of l i nearizati on also pre se nt s s ingul a riti es a nd opt imal y properties. APPLICATION TO A HYDRAUL IC PILOT PLANT Pr esent a tion

dx dt q(x)

~(u - q(x)) 5 k Vx

( 15 )

By applying th e linearization technique, we get th e co n t r o l law ( 16) representing a combi na ti on o f state (leve l) and flux (ou tput flow - rate) feedbacks as shown in Fig. 3.

An ana l og reali zatio n of th e co nt ro l law be comes very s impl e because it on ly needs th e use of gain factors and adders - substrac t e r s . This r ealizat i on takes int o account the instrumentation techno l ogy invo l v ing sensors and ac tuat o r s in the 4- 20 mA indu st ri al standard. Exp e ri mental r esults Lar ge ope r ating range linea ri zation. Fig. 4 shows the closed l oop step r esponses of the l evel r elated t o in pu t magnitudes of 30 % and 55 i, of th e tank height, the desired dynamics being d ; -1 /40 sec - l (Eq. 2) . Those r efe r ence magni tud es make th e plant wo rking outsi de its linea rity domain . From the tangen t to the o ri g in of th e step response, we get the time constant ~ ; l / d f o r which x(~) is roughly 64 % of th e fin al va l ue in both cases ; so we can cons id e r th e system to be well linearized . Stat i c e rro r s can be observed ; th ev can be exp l ained by modelli ng e rrors (inst rum ent ation conve r sion f ac to r s) . Othe r arbitrary dynamics can be set and s i-

CSTO - F

So , Xoo < Xc in the case of a disturbance . Fig. 5 shows this e ff ect wh en d ; - 1/20 sec -I. In order to r edu ce the influence of mode llin g e rrors and to cance l static er ror s , the use of a reference

model and an int egrator mus t be presc rib ed . APPLICATION TO A CHEMICAL PILOT PLANT Presenta t ion The pilot plant (Gilles and Laggoune, 1985 ; Neyran and co wo rk e r s , 1984) (Fi g . 6), is a continuou s neutralization process of a s tr ong base (N aOH, co ncentration CB) by a s trong ac i d (HCl, conce n tra tion CA)' The reactor vo lume (V ; 6£) i s maint ai ned constant by local regulation, as we ll as for t he a cid fl ow-rate (QA ; 35 £/ h). The description varia bl es are C (output concentrati on) a nd Qb (base flowrat e) . The mo novariable s tat e model i s of bi l inear type : x

= a . x

+ b u + nu. x

( 19)

where u ; Qb(t) - Qb(O) i s th e input variable devia ti on, x ; - c(t)/(C B + Co) is the s tat e represen ting the no rmali zed concentration in t he react o r (C is th e initi al concen tration ) and c(t) i s the

ou~put concentrati on deviation.

Fi gur e (2) shows a part of the hydr aul i c pilot pl a nt used for level and t emperatur e con tr ol . A tank (area S) is wat e r s uppli ed by the input flow-ratE u ; the output f l ow-rate , den o t ed q(x) , i s a square root function of the level x in th e tank. Thu s , a mass balan c e lea ds to the following mode l without any dis turb ance : and

( 17)

Replac in g u by its exp r ess i on (Eq. 16 ) , we get the new sta ti c le ve l

C.G(x)u

- 1.) - C fIx) - D(Yc - 1.)

15 1

By app l yi ng Eq. (8), the l in ea ri zing control law i s: d(x - x ) - ax u

c b + n x

(20)

This control l aw involves the singularity x ; -bin but we can easi l y show on the knowledge mode l th a t this co rr es ponds to c( t ) ; CB (or the symmetri ca l C(t) ; CA) which is unaccess ibl e . As shown in Fig. 6 , the st at e (norma l ized co nce ntr a t ion) mus t be deduced by convertin g th e Ph value int o concent rati on what can be done by usin g an antilog analog c h ip. Th e desi r ed non - linear contr o l law (Eq. (20)) can be generated by adders/substracte r s , ga in elements and a divide r. We pr efe r ed usi ng a digita l compute r, the contr o l variable being hol den dur i ng ea ch small samplin g period (2 se c .) . Experimental results In th e f ollow ing, th e constant parameters in equa ti ons ( 19 ) and (20) are: a ; - 9 . 27 h- l , b ; 0 .1 7 " -1 and n ; - 0.185 £ - 1. All resul t s are given for an i nitial base fl ow -rate Qb(O) ; 20 X, /h a nd the Ph excursion is limited to th e acid fi e ld. The ave ra ge open loop time constant is greater th an 300 sec . Refe r ence changing. Figures 7 and 8 r e spec tively show th e evolution of the s t ate x i n case of an input r efere nce x ; 0 .1 and Xc ; 0. 15 . Th e dyna mi cs is d ; - 25h - ~ whi ch r e pr esents a c l osed -l oop system approx i mately 2 times faster than the ope nl oop one. The results denote that the linear i zed system is s li ght ly delayed (20 t o 30 sec .) and the f inal value is a l so slight l y different from the reference. The static erro r i s due to t he fact that the model is not exact (Gilles and Laggoune , 1985) , and the use of integrators must improve the result.

c . (; ill es and :\. Laggoul1e

152

The delay is simply due to the Ph measurement location (fig. 6). The desired time constant is 144 sec. (d = - 25 h- 1)and the ex per i menta l dynami cs r ep re-

small enough, thar is to say if the real - time computations are rapidly processed in a parallel way by means of rnultimicroprocessor structures .

sents approximately a time co nstant of 128 sec . in

fig. 7 and 176 sec. in fig. 8. Thus, with r egard t o the modelling errors and the Ph meter pre cision ,

~e

can say that the results are

very satisfactory: the r elat iv e errors are 13 Z and 9 % for the final value , 10 Z and 22 % for th e dynamics resp ectively for Xc = 0.1 (Fig. 7) and Xc = 0.15 (Fig. 8). Fig. 9 shows th e linearizing control variable evolution in the both former cases .

Arbitrary dynamics. Let us assume that the desired time co nstant f o r ttIe closed-loop linearized system

is 80 sec . (4 times faster than the open-loop). When choosing d = - 45 h- 1 , Fi g . 10 shows the state evolution. The ex perimental time cons tant is appro-

ximately 11 8 sec. which cannot be cons id e red as satisfactory (48 Z relativ e er r o r between th e desi red and the obtained values). This illustrates the limits of the method in the pr ese nc e of an important noise and of non negligea-

ble modelling errors. I-Ihen a time delay is not small enough with respe c t to th e desired time constant, it has t o be taken into account in the model in order to build a new control l aw. CONCLUSION AND PROSPECTS It has been shown that, under ce rtain conditions, continuous processes that ar e linea r with re spec t to the control vector can be lin ea rized, inside

its whole operating range , by means of a non - linear state feedback control law. Two applications on different monovariable pilot plants have shown th e interesting possibilities pr ese nted by the linearizing method. Assigning a linear behaviour to the closed loop system simplifies further studies on the system among a more complex plant, guarantees

its stability and leads to a same dynamic behaviour inside the whole operating range. Simple for low order proc esses , this type of control avoids designing an adaptive control with respect to the operating point variation. Noreover, as the linea-

rizing control law i s still valid for non - linear processes inVOlving measurable variable parameters, it would be v e r y ir.teresting to set up experiments on this point of view; for example, on the neu -

tralization pilot plant, it would be possible to take into account the me asurab le variations of the

volume. The differences observed between the theo r e tical and pra c ti cal results are mostly due to modelling errors. In order to r ed uc e their influence it would be useful to introdu ce a reference modei into the linea rizing control structur-;:-FOr"

higher order sys tems, the non - linear s tate f eed back control law ne eds an observation of the whole state vector : th e n it is nec essary t o ex tend the

results got on observers for bilinear sys tems (Lag go un e , 1984) ove r the class of systems lin ea r in control. On multivariable processes, th e most il lustrative application would be the possibility of deco upli.n g control; this co uld be sholm o n the hydraulic pilot plant by using two tanks. By appiying this typ e of co ntr ol on different multiva riable indu st rial processes, a deep analysis of the singularitie s could be managed in order to known

the ir degree of concrete importan ce . Finally, on the realization point of vie~, we sho~ed that ~e can use anal og o r digit~l te Ch nolo gies in th e case of simple pr ocesses. For 10\..' order rnono\'ariable pr ocesses , the design of analog non-linear co ntrol -

lers may be preferable. In th e case o f high order multivariable processes, th e pseudo - analog non-

linear controllers have to be approached with a digital technology if the sampling period can be

ACK..'lQl{LEDGNEKTS This research is supported by grant from Centre National de la Recherche Scientifique for a cooperation project between three teams on non - linear

process control. The authors are grateful to this institution and to their colleagues J. Biston, B. Neyran and D. Thomasset for their fruitful he l p on the use of the pilot plants. REFERENCES Alvarez Gallegos, J. ( 1982). Optimal Control of a class of discrete multivariable non-linear

systems. Application to a fermentation process, Jou r nal of Dynamic Systems, Heasurement

and Control, vol 104, pp. 212-217 . Claude , D., ~1. Fliess and A. Isidori, (1983). Immer sion directe et par bouclage d'un systerne noo-

lineaire dans un lineaire. Compte-r e ndu a 1 'Academie des Sciences, serie 1 , t. 296, pp. 237-240. Di Benedetto, M. and A. Isidori, ( 1984) . The matching of non- linear models via dynamic state feedback,

r ese arch report, universita di Roma

Gilles, G., (1983). Une lo i de commande optimale d ' une classe de systemes bilineaires multiva -

riables, Optimization days, Montreal, Canada. Gilles, G., ( 1984) . Commande optimale linearisante d ' une classe de systemes non-lineaires conti -

nus, workshop on non-linear system theory, Gr enoble, France. Gilles, G. and N. Laggoune , ( 1985). Digital control of bilinear continuous processes . Application

to a chemical pilot plant. 7th IFAC Symp. on Digital computer App l icatio~~c~­ trol, Sept. 1985, Vienna. Laggoune:-N . , (1984). Sur la discretisation de l ' observateur de Hara et Furuta . Internal note,

Nov. 1984, Laboratoire d'Automatique de Lyon 1 Mouyon, P., C. Champetier, and C. Reboullet, (1984) . Application d'u ne nouvelle methode de commande des systemes non- lineaires - la pseudo linearisation -

a un

exemple industriel,

Six~

INRIA international conference on Control and Optimization, Nice, France.

Neyran, B., J. Saade - Castro, D. Thomasset and R. Reynaud, ( 1984). Modelisation dynamique nonlineaire d'un procede chirnique de neutrali sation, Congres S.E.E., Nice , France.

:\ o n -lin ea r Co ntinu o u s Processes

t-

Ye

U

+

1

t

I I

L

Fig .

1 Weak closed- loop linearization block-diagram

x=l (U-q(X))

x

'-------' q(X)

Fig. 3 Linearizing leve l control structure

x

l OO

80

Fig. 2 Descriptive scheme of the tank 60 tank 1 evE' 1

reference step

40

~--------r--------r--------r--------r--------~--~~ t(se c)

Fig . 4 Closed loop level step responses

\ .

~ )

80

t

60

t

end of perturbati0n

start 0f ?erturbaticn

40 t(sec)

o

120

180

2:'0

lOO

Fig. 5 Lev e l re,;u la tion-pcrturbation effect

)60

154

tWN -L l1;[AR AliA LOG CONT ROL LE R

Ph -!:ETER

v c _ _-.J

v--~

LR : Local Regu l at ion F i g . 6 Continuous neut r a li zation p ilo t plant an d con tr o l struc tur e

··:···

176 Sec

.. ... .....

O.1S

O' S

~e c'r-

_______

/

0.10

....

0'0

.. ... ~

(

O.OS

,,

.'

D:-2Sh-"

I

DOS

REfEltENCE ST EP

f

... .

REFERENCE STEP

I

sysr[}1 OUT PUT

SYST H I OU TPUT

./

O.OO ........·-··· ·.:.··-··-··-I·f---~--.,--~----r---~---r----~--T~c:..... 310 480 '60 640

..

Fig . 8 Closed loop s t ep respo n se ( r eference : x = 0 .1 5 )

F i g . 7 Closed l oop step r esponse ( r efe r e n ce: x = 0 . 1) c

c

..........-... .......... . ....... ..-.........-~~

I/ h

O.'S

Ob

- - Xc:O.1S 40

- -- - Xc : 0,10

0.11)

!.

30

0,·4\ h" O.OS

RH I::R.i::: j';CE STEP

10

'60

310

480

640 . 0

Fig . 9 I nput va r iab l e in case of two ref e rence steps (D ~ - 25h - l )

Fig.

'60

HO

480

ID Closed loop step response \"ith a faster dynam i cs

640