Robust Control of Nonlinear Systems Using ‘Self-Adaptive’ Methods and Decoupling Theory

Copyright © IFA C Ad aptive Control of Chemical Processes. Copenhagen. Den mark . 1988

DECOUPLING CONTROL

ROBUST CONTROL OF NONLINEAR SYSTEMS USING 'SELF-ADAPTIVE' METHODS AND DECOUPLING THEORY D. P. Paszkiewicz Departement d'En ergfiiqlle, Institut National des Sciences Appliqllees de Lyol!, 69 62 1 Villeurbann f, France

Abstract. This paper reports on two robust control methods for nonl inear systems based on decoupling theory. Gi ven a plant described by a nonlinear state equation x= f(x) + g(x)u we design a linearizing feedback/decoupling law. Subsequently, for the resulting sUb-systems we propose either an incremental fuzzy logic controller or a so . called 'saturation element' about an operating point . The emphasize is on the application and comparison of these methods. Keywords. Nonlinear control systems; decoupling; computer control; self adjusting systems; fuzzy logic controllers.

INTRODUCTION

Methods developed here should prove useful in solving control problems of a large class of nonlinear systems with strong disturbances.

Given a plant described by a nonlinear dynamical model it is interesting to design a control system which will simplify the structure of the. model by feedback transformations i.e. linearization or decoupling by feedback. However, in the case of complex processes, for instance chemical reactors an inevitable mismatch between plant and model dynamics, poor quality of measurments and time varying parameters make the implementation of these control laws with stability and acceptable dynamics difficult.

PLANT DESCRIPTION The process studied was a neutralisation reaction in a pilot scale CSTR ( of the maximum volume 25 litres) of the strong acid (HCI) at the concentration CA (mols/ l) by addition of the strong base (NaOH) at the concentration CB (mols/l).

The control system of a pilot scale neutralisation reactor with two inputs and two outputs has been considered (First Section). Mathematical model of this reactor has strong nonlinearities and time varying parameters. However both: the decoupling control and linearization are possible. Subsequently, using methods of design of Ray and Majumder, (1985) we propose two fuzzy logic controllers in the so called outer loop. However, the analysis of fuzzy rules can be difficult: the outputs of fuzzy logic controllers are not the real inputs of the plant. The second method uses two 'saturation controllers' about operating points : the input of each outer loop controller is the error between the set points and the measured value of the outputs. The gain coefficient can be adapted according to the dynamics of the system.

Fig.l. CSTR neutralisation pilot plant

In the final section we compare the results of the application of the presented methods to the neutralisation reactor.

This reactor can be described (Paszkiewicz and Prevot, 1988) by the following dynamic model:

REACTOR t!

75

D. P. Paszkiewicz

76

d C(t)

And we have:

1

(6)

l i t = V(t) [CA QA(t) - Cs Os (t) - C (t) [QA (t) + Q B (t) JJ d V(t)

(1)

l i t = QA(t)+ Os(t)-Q(t) Q(t).a C(t)

vet)

a(x)=

.rvw

vet) It 0 -I

P(J:)=A(J:)

where: QA(t) ,

QB(t) are flow rates (l/h) of acid and base solutions within admissible range 0 - 60 l/h

C(t)

is a concentration (mols/l) of the neutralised solution

vet)

is a volume (1) neutralised product reactor

Q(t)

is the outflow rate (l / h)

~t)

of in

V(t) [1 =CA+Cs -1

Cs+ C(t)] V( t) CA-C(t) v (t)

and the decoupling control laws:

the the

The new system of the form:

is the output valve variable

(9)

NONINTERACTING CONTROL LAW The state model (1) ca be written under a general form of a control-linear analytic system: d

\

(7)

:~t) = f (J: (t) ) + i1 (J: (t) ) QA(t) + i, (J: ( t) ) Q B(t)

y(t)=[C(t)]=h(J:(t» V(t)

OUTER LOOP CONTROLLERS

(2)

Using standard methods for the design of control laws for nonlinear systems (Isidori and co-workers, 1981) we design a non interacting feedback control law to obtain a model where one input controls one output only:

(t)]

must be stabilised using linear feedback pole assignment.

[ (t)]

QA W1 [ Os (t) = a (C (t). V (t» + P (C (t). V (t» w, (t)

(3)

where W1(t) and W2 (t) are the new inputs. For characteristic numbers P1 = P2 = 0 we get: the decoupling matrix A(x(t)):

The effectiveness of feedback control laws for nonlinear systems can be assured only if the parameters of a nonlinear system are known exactly, there is no 'lost' nonlinear dynamics and the states are measured with no noise. This is only rarely the case . Even if a tradeoff between acceptable dynamics of the system and its stability can be established by means of a stabilizing linear feedback a simple and robust control law must be designed to compensate for possible mismatch between a model and the system itself: the so called outer loop controllers. ~t

can be easily noted that the design of control laws of the general form: ~ncremental

W=W' +dW (4)

and

where W is a new value of the input vector equal the sum of the previous value plus the desired change of inputs allows for a simple algorithm of controllers since: dW",

(5)

(10)

rl

dU

(11)

and the applied vector of inputs W can be calculated taking into account the maximal and minimal values of the real input vectors Umax ' Umin'

Robust Cont ro l of Nonlinea r Systems

77

r-11)

'-

+

-

GJ

-0 ~ ~~ '( ~

-,~

.....c '-

+,<:>.U

fl

~ )I

-

Process

~

'"

0 U

-

0(

I--

Fig. 2. Control system

Fuzzy logic controllers The first controller studied was an incremental fuzzy logic controller with two inputs: error at an instant k defined as:

Changes of errors are described the following fuzzy sets:

by

(12) ErrorpH[k]=

TABLE 4A FUZZY VALUES OF THE CHANGE OF ERROR l2.!:!

= (pH[k]-Setpoint pH) / Setpoint pH (pH units)

t.EPH

ErrorVolume[k]= (Volume[k]-Setpoint Volume)/Setpoint Volume and a change of error: t.EpH[k] =

big positif medium positif small medium positif small positif nul positif nul negatif

(PB) (PM) (PMS) (PS) (PN) (NN)

big negatif

(NB)

0.5 0.3 0.2 0.1 -0.1

.1EPH >.1E pH >.1E pH >.1E pH >.1E pH <.1E pH

> > > > > <

0.5 0.3 0.2 0.1 0 0

abs(ErrOr p H[k-1])-abs(ErrOr pH [k]»* * Setpoint pH

.1EpH < -0 .5

(1 3 ) t.EV[k]= (abs (Errorv[k]) -abs (Errorv [k-l]» * * Setpoint Volume We define fuzzy values of the error for the controllers (Ray and Majumder, 1985) : f.ex. : TABLE 1-

Fuzzy values of the error Q.t; l2.!:!

(litres)

t.Evolume big positif medium positif small medium positif small positif nul positif nul negatif

(PB) (PM) (PMS) (PS) (PN) (NN)

ErrorpH (E pH ) big positif big medium positif medium positif small medium positif small posi tif small nul positif nul positif nul negatif

(PB) (PBM) (PM) (PMS) (PS) (PNS) (PN) (NN)

big negatif

(NB)

EPH >EPH >EPH >EPH >EpH >EpH >EPH -1%
> > > > > > > <

30% 20 % 10 % 5% 2%

1% 0% 0%

EPH < -30%

and analogically for the fuzzy values of the error of volume.

big negatif

(BN)

0.5 0.3 0.2 0.1 -0.1

.t>Ev >.t>E V >.t.E v >.t>E v >.t.E v <.t.E v

> > > > > <

0.5 0.3 0.2 0.1 0 0

t.Ev < -0 . 5

using the defined tables of errors and of the change of errors and the formula (11) we define ' fuzzy look up tables' (Ray and Majumder, 1985) of the change of input W as a function of a desired change of read inflows of the acid QA and base QB. However, the determination of a fuzzy look up table based on the 'experience of expert' can be difficult and needs a good expert !

78

D. P. Pasz kie wicz

E

toE

NO

I PO

PSN

PS

PMS

PM

PBM

PB

0. 1

0.2

0.22

0.25

0.3

0.6

0.6

~

0.05

0.05

0. 1

0 . 1~

0.25

0,5

0·55

I.

0.02

003

O.O~

0. 15

0.22

0.3

0,5

.0.015

0025

0.05

0. 12

015

0.25

0.4

0.025

0.03

O.O~

0. 12

0.2

0.3

I

PB

I

PM

I

PM S

I

I

PS

I

I

PO

_.

NO

+

-

. -~.- ~.-

Fig. 5. Fuzzy Look up Table: where:

The proposed fuzzy logic con~rol~ers ~ave been applied to the neutral~zat~on p~lot plant and have proved to be robust. However the author fully agrees with Ray and Maj~mder, 1985 that 'long simulatio~s' are needed before a fuzzy log~c controller can be efficiently implemented.

pH 2.8

2.4

~.-

_ . - ~.- _ . -

coefficient KA for

Saturation Contoller We propose here the second incremental controller which has the advantages of the fuzzy logic controller: facility of implementation and robustness but can be designed much faster. The controller is modeled on a saturation element about an operating point with an increment of inputs depending on a current value of error and the maximal / minimal admissible real inputs . We have: du = h* (Error/ Setpoint) * (Umax-u)

2.

when the Error > 0 and:

1.6

du

1.2~

____

o.

~

_____

1000.

~

_______

~

__

2000.

Time (secs)

£

-h* (Error/Setpoint) * (Umin-u ) when the Error < 0

The gain coefficient h determines the convergence towards a desired Set point. It can be easily noticed that the proposed controller needs no model of the process and allows no stationary error. We have applied the proposed controller in the outer loop of the control system and the system turned out to be robust. CONCLUSION

20

10

o.

1000.

2000.

Time (secs)

Fig. 6. Control of pH and Volume using the proposed fuzzy logic controllers

New schemes for control of nonlinear systems: feedback linearization and non interacting control take into account nonlinear character of the process and allow for a good stabilization over a wide range of operating points . We show how a simple incremental controllers in an outer loop can be successfully implemented using these 'novel' methods. The proposed 'saturation' controller can be used as a help for design of the 'expert' fuzzy logic controllers.

Robust Control of Nonlinear Systems

79

Ray, K.S. and D. Majumder (1985). Fuzzy logic control of a nonlinear multivariable steam generating unit using decoupling theory. IEEE Trans SMC, ~, 539-558

pH

2.5

1.5

0.5

o.

1000.

2000.

Time (secs)

i 20

10

o.

1000.

2000.

Time (secs)

Fig. 7. Control of pH and Volume using the proposed saturation controllers REFERENCES Claude, D.C. (1986). Decouplage et linearisation des systemes nonlineaires par bouclages statiques. Ph D. Thesis, Paris Sud, Centre d'Orsay Gustafsson, T.K. and K.V. WaIler (1983). Dynamic modelling and reaction invariant control of pH. Chem. Eng. Sci., 38, 389-398. Isidori, A. (1985). Nonlinear Control Systems, Springer Verlag, Berlin. Isidori, A., A.J. Krener, C. Gori-Giorgi and S. Monaco (1981). Nonlinear decoupling via feedback: a differential geometric approach. IEEE Trans AQ, AC-26, 331-345. Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear parameters. iL.. Soc. Indust. ~ Math., 11, 431-441. Paszkiewicz, D. P. (1987). Commande des procedes non lineaires en temps continu et en temps discret. Application a un reacteur chimique de neutralisation. Ph. D. Thesis, Lyon Insa. Paszkiewicz, D.P. and D. Thomasset (1988). Commande non-interactive d'un reacteur chimique. geme Conf. opt. Syst, Cannes, France. Paszkiewicz, D.P. and P. Prevot (1988). Modelling of a neutralization reaction and sel-adaptive control of pH. Identification ~, Pekin