design of p and pi controllers for quasi linear systems

7/25/2019 design of p and pi controllers for quasi linear systems

1/12

Com.~~uters

t hem. Engn g ,

ol. 14, No. 4/S, pp. 415426, 1990

Printed in Great Britain. All rights reserved

0098-I 354/90 $3.00 + 0.00

Copyright (0 1990 Pergamon Press plc

DESIGN OF P AND PI STABILIZING CONTROLLERS

FOR QUASI LINEAR SYSTEMS

J.-P.

CALVET

and Y.

ARKUN~

School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, U.S.A.

(Received 23 Oc tobe r 1989; rece ived

or

pubk a r ion 29 Novembe r 1989 )

Abstract-The systems studied in this paper are nonlinear systems perturbed by disturbances which are

feedback transformable to quasi-linear systems [i.e. i = AZ + Bv + [ ( r ) d with A, B) controllable]. We

consider the problem of designing stabilizing controllers for perturbed nonlinear systems through their

equivalent quasi-linear systems. With the addition of integral action, we can guarantee not only ultimate

d-stabilization but also zero steady state offset for both the output of the quasi-linear system u = Cz)

and for the equivalent output [y = h x)] of the nonlinear system. Moreover, when the so-called

disturbance matching condition is satisfied, it is shown that all the states of the quasi-linear system and

the nonlinear system) will exhibit zero steady state offset. All the results presented here are for single

control input systems.

1. INTRODUCTION

In recent years, the use of differential geometry to

transform nonlinear systems into linear systems has

received much attention

in the control literature.

Several methods compete and differ in their defini-

tions and the issues they address. One approach is to

transform a nonlinear system in an input/output

sense. This was first investigated by Gilbert and Ha

(1984) and Ha and Gilbert (1987). Another approach

is to transform the nonlinear state equations into an

equivalent linear system without specifying any out-

put. This technique is known as the feedback linear-

ization and has been investigated in particular by Su

(1982) and Hunt er al. (1983). The transformations

required are state and input transformations with a

nonlinear state feedback. Finally, feedback lineariz-

ation with decoupling of outputs was addressed by

Isidori et al. (1981). Various control applications

have appeared in aerospace engineering (Meyer et al.,

1984), robotics (Tarn et al., 1984), power systems

(Marina, 1984) and chemical engineering (Hoo and

Kantor, 1987; Kravaris and Chung, 1987; Calvet and

Arkun, 1988a).

It should be noted that there is very limited knowl-

edge about the robustness characteristics of these

techniques with respect to modeling errors and dis-

turbances. For example, the pioneering work of

Kravaris and Palanki (1988a, b) shows that under

matching conditions it is possible t o design robust

controllers for a transformed (in the input/output

sense) nonlinear system with modeling errors. How-

ever, the design of controllers guaranteeing stability

of the transformed nonlinear system influenced by

modeled disturbances has not yet been investigated.

Our recent work (Calvet and Arkun, 1988a, b) has

___

~__

tTo whom all correspondence should be addressed.

shown that under feedback linearization a perturbed

nonlinear system is in general no longer transformed

to a linear system but to a so-called quasi-linear

system, QLS. The QLS is then affected by nonlinear-

ities only due to a state-dependent perturbation

coupled with the modeled disturbance. In this paper

we address the following problem:

Given a bound on the set of disturbances affect-

ing the nonlinear system, we want to design a

proportional (P) controller for the QLS which will

ultimately stabilize the original nonlinear system.

Furthermore, if an output of the non-linear system

is specified, we want to design a stabilizing propor-

tional integral (PI) controller for the QLS which

will guarantee zero steady state offset for the

output.

The paper is organized as follows. The origin of QLSs

is given in Section 2. In Section 3, a theorem gives the

sufficient condition for the design of (P) stabilizing

gains. The extension to PI stabilization is addressed

in Section 4. Because of the inherent conservation due

to the sufficiency of the theorem, a design procedure

to compute the least conservative stabilizing gains

possible is given in Section 5. In Section 6, a steady

state analysis shows that zero steady state offset can

be guaranteed for

a l l

the states of the original non-

linear system under some disturbance matching con-

ditions. Finally, in Section 7 we apply the design

procedure to control an unstable reactor influenced

by various process disturbances.

2. QUASI-LINEAR SYSTEM

Definition I-A quasi-linear system (QLS) is a

system which is linear or affine with respect to

its control input and the modeled disturbances.

415


2/12

416

J.-P.

CALVE?

and Y.

ARKUN

It is described by:

i =Az+Bv+

z d,

1)

where z E

R

are the states, v E

R

is the control input,

d E RP

is a set of modeled disturbances and c(z) is an

n x p matrix with sm ooth scalar entries. The pair

(A , B)

is a controllable matrix pair. It is usually given

in the Brunovsky canonical form (BCF):

R

It is important to realize that a QLS has the nice

property to be truly linear w hen free of disturbances

(d = 0) .

Also, note that the perturbation term

c(z )d

is

s ta te

dependen t and not pa rame te r dependen t [i.e.

I I < D V t > t o } ,

where /I. 11 h

S t e usual Euclidean norm and

D

is the

know n bound. Note that the set of disturbances could

be time varying but should be bounded at all time by

D .

We now giv e the definitions of &-stabilization

(Schmitendorf, 1988) and ultimate bound edness

(Corless and Leitman n, 1981) as adapted for QLS.

efinition

2-A solution z(-):[t,,

co]+R, z(t,) = z,,

(initial condition) of a QLS is said to be ultimately

bound ed with respect to a closed set

B(S) =

{z E R; ) l z [ I


3/12

Design of P and PI

This assumption is the most crucial one but can

always be satisfied in closed balls B(F) for a finite F.

However, if P tends to infinity, sometimes p and/or cc

may not be finite implying that Assumption 2 will not

be satisfied. In other words, this means that Assump-

tion 2 can always be satisfied locally [in B(6)] but not

globaly (when i-00). This choice of F is thus very

important and involves searching for the pair (p, p)

not in the whole space R but in an adequate ball B(F)

which must be specified judiciously. Also, this as-

sumption introduces some inherent conservatism in

the design of 6 -stabilizing controllers. This point will

be discussed and illustrated later.

We now give an important theorem where the

following notation is used: an eigenvalue of a square

matrix W is denoted by 1( W). If

W

is strictly positive

definite, then 1 *(W) and 1 *(W) are its maximum

and minimum

eigenvalues,

respectively. Also

yw = E, (

W)/ , l * ( W

is its condition number.

Theorem-Given 6 such that J 2 6 B 0; if there

exists a P > 0 (strictly positive definite) solving the

algebraic Riccati equation (ARE):

PA + Ar P - PB(2R- - I )B TP + H /E =O,

for some 6 > 0,

N=HT )O

and

R=Rr>O

satisfying

with i , (H ) - I_c~D%~ > 0, then the control law:

v(l)= -R- B=Pz( t ) ,

(7 )

is a a-stabilizing control for the QLS, for all initial

conditions in

B ( r / & )

with r -c F.

Proof-See Calvet and Arkun (1989). The proof of

the theorem is based on the inspection of the sign of

the derivative of a candidate Lyapunov function

V(z) = zTPz; where P is the solution of an algebraic

Riccati equation whose form is different depending

on whether or not the disturbance matching condi-

tion is satisfied.

Note that when the QLS is a single control input

system, the parameter R is a scalar. Assumption 1 is

a disturbance matching condition similar to the one

given in Corless and Leitmann (1981). However, this

assumption can be relaxed. Indeed, if Assumption 1

is not satisfied, one must verify the following assump-

tion instead of Assumption 2:

Assump t i o n ?-The perturbation term itself must

be bounded in a ball B(F) :

3 (p , p ) E R2 (both finite) s.t.

lli(z)lj2

T* .

(8 )

4 .1 . Selec t ion o f the ou tpu t

It is known that in general an a p r i o r i specified

output of the original nonlinear system (2):

Y(l) = h]x(r)l,

(9)

will not depend linearly on the transformed states of

the QLS, z,, , z,.

This has been recognized as a

disadvantage of the feedback linearization transfor-

mation (Kravaris and Chung, 1987). In the proposed

method, however, the output of the QLS must be a

linear combination of the states as:

y(t)=Cz(t)

CER~.

(10)

We also consider that the output C[z(t)] will render

the unforced QLS observable. Hence without loss of

generality, we can set the output of the QLS to be

Z, ,

i.e. C = [I, 0, . . , 01. Indeed, the unforced QLS can

be easily transformed to such a form through linear

transformation and feedback (Kailath, 1980).

The critical requirements for the output of the QLS

to satisfy simultaneously (9) and (10) can be tackled

in different ways:

Given a desired output (9) for the original

nonlinear system, choose a manipulated vari-

able u which will admit feedback transformation

(4, 5) that gives a QLS with a linear output (10).

This is called an a p r i o r i output selection.

Apply the feedback linearization (4, 5) and then

select a linear output of the QLS (IO) which has

a physical meaning in the original nonlinear

system. This is called an a pos te r io r i output

selection.

If these two approaches fail, one should resort to

partial linearization (Krener et a l . , 1984) or input/

output linearization (Kravaris and Chung, 1987)

which will not be addressed here. The option of

selecting an output a p r i o r i (before applying the

feedback transformation) or Q po s t e r i o r i (after) has

been investigated in Kantor (1986) and applied by the

authors in Calvet and Arkun (1988a). In the reactor

application here, the selection of the output will be

done according to the procedure given above.


4/12

418

J. P. CALVET and Y. ARKIJN

4 .2 . Th e fo rm u l a t i on o f the con t r o l l er design p rob lem

The design problem we investigate is then the

following:

Given the bound D of the set of disturbances, and

specifying 8, construct a control law which will:

1. a-stabilize the QLS (i.e. the norm of the states

will be ultimately bounded by 6).

2. Guarantee zero steady state offset of the output

of the QLS [under ultimately time invariant

perturbations (8)].

It is also important to realize that a-stability for a

QLS will automatically guarantee stability for the

original nonlinear system in a closed contour called

a 6-contour. This contour is obtained through the

inverse transformation [X = T-(z)] of the ball B(6).

Therefore, if stability of the original nonlinear system

is addressed through its equivalent QLS, the choice of

6 should be done judiciously so that it corresponds to

a desired region of stability, the 8-contour for the

original states x [see an application in Calvet and

Arkun (1989)].

The approach we suggest is to augment the dimen-

sion of the QLS and apply the P stabilization results

given in the previous section. First, we introduce an

additional state as:

z,+,(t) =

5

z,( t)dz.

(11)

10

Next, we obtain an augmented QLS described by:

t = ;ir + Bv + c(z )d ,

(12)

with i = [zl..

. . .z,, z,,,]~.

and

i

0 1

. . . 0 0

:

: . :

_ij= (j (j

.I. ; 0

ERf+I,Xc+I>

0 0

. . . 0 0

I 0

. . 0 0

I

and

Finally, the new control law (PI controller) can be

constructed:

v(t) = -&Z(t) = -[K,, . . . , K,, K,+, ]g ( t )

,=n

c

= - c Kjz i ( t ) -K ,+ ,

z , ( t )d T,

(13)

r-1

.I@

where (K, , . . . , K,) and K,+,

are, respectively, the

proportional and integral gains.

The theorem given for P stabilization applies to PI

stabilization as well after replacing A,

B, c(z )

by

1, B, c(z). For example, if the disturbance matching

condition is satisfied by the augmented system (12);

then, one will have to solve the following algebraic

Riccati equation ARE:

P; i + ; i=P - PB (2R - - Z)z P + H /e2 = 0.

Otherwise the following Riccati equation ARE must

be solved:

Pa + ;i=P - P(2BR - BT - Z)P + H Jc2 = 0 .

5. DESIGN PROCEDURE

In this section, we give guidelines which will ease

the search of the b-stabilizing gains. This procedure

applies for both P and PI stabilization of a QLS. It

is a graphical approach and is not analytical mainly

because a closed expression for the condition number

yp as a function of R, H and L is not available when

one solves the above algebraic Riccati equations.

5.1.

Sol v i n g ARE and ARE

One can see that the gains of the s-stabilizing

control K = R - BTP do not depend explicitly on the

constants p and ZJ ntroduced in Assumption 2 (As-

sumptions 2 or 2) but rather on the parameters t,

H

and R of the algebraic Riccati equation. Assumption

1 will determine which Riccati equation needs to be

solved. Then all the possible gains can be computed

as a function of E (parametrized by R and H ) and

independently of p and p. In other words, Assump-

tion 2 need not to be checked before computing the

gains.

If the disturbance matching condition (Assumption

1) is satisfied, then, there always exists a

un ique P > 0

solving the ARE for all H = H > 0 , R = RT with

2R -- - Z > 0 and t > 0. Indeed, one can recognize

that in this case the Riccati equation is similar to the

one encountered in the LQR optimal control prob-

lem (Kwakemaak and Sivan, 1972). As mentioned

earlier, the matrix pair (A,

B )

is usually in the BCF.

As an example, for a 2-D BCF and for R = 1 and

H = Z , we can plot the gains R - BT P as a function

of L as shown in Fig. 1. Also, in Fig. 2, we give the

gains obtained when the QLS is augmented with the

integral action. Note that we can obtain the gains for

all values of es, and they tend to infinity as L tends

to zero.

However, if the disturbance matching condition is

not satisfied, we may not be able to solve the ARE

for all the values of the parameters t, H and R .

Indeed, we can show that for a given H and R there

exists a value of e, say clim elow which ARE does not

have a positive solution P . The value of +,,, depends

on the parameters H and R . In Fig. 3, we give the


5/12

Design of P and PI stabilizingcontrollers

419

Fig. 1. Stabilizing gains from the ARE (for Q S with

disturbancematchingcondition).

gains we calculated for a 2-D BCF for n = I and

R = 0.1.

Also, in Fig. 4 are the gains for the aug-

mented QLS. The parameter R = 0.1 was chosen

because it gives the lowest value of c for which ARE

has a positive solution P with N fixed to the identity

matrix (the value of R was obtained by trial and

error). In comparison with the gains obtained from

the ARE, these gains tend to infinity as t tends to cltm.

Below Ellrn

o gains can be computed.

Clearly, it is more attractive to solve the classical

algebraic Riccati equation (ARE) than the ARE.

Therefore the matching condition (Assumption 1) is

a desired property. In the case that Assumption 1 is

not satisfied one may wonder if either a similarity

transformation (change of state coordinate a = Qz)

or another suitable difleomorphism [z = T(x)]

would render Assumption 1 satisfied. However, this

is not true as one can easily show that the matching

condition Assumption 1 is independent of the

200,

Fig. 2. Stabilizing ains from the ARE (for augmented QLS

Fig. 4. Stabilizing gains from the ARE (for augmented

with disturbance matching condition).

QLS without disturbance matching condition).

0;

A 1

1

0 urn 2 4

s 8

EPS

Fig. 3. Stabilizing gains from the ARE (for QLS without

disturbance matching condition).

diffeomorphism and the choice of coordinate system

in the z-domain.

5.2. Ver i f v i n g Assump t i o n 2

Assumption 2 is the most crucial assumption. In

fact, this assumption is the only one dealing with the

nonlinearities of the QLS. It quantifies in a way the

magnitude of the nonlinear state-dependent per-

turbation through the values of the two constants p

and p. Therefore, the challenge in our design method

is not to solve the algebraic Riccati equation (ARE

or ARE) which is usually an easy task; but rather to

find the constants p and p which are obviously not

unique and are problem dependent. In the applica-

tion we will refer to an algorithm which computes

these constants.

5.3. Conse rva t i sm

In general, assuming that the bound of the distur-

bances D is available and the desired region of

of

I

2

4

EL

2


6/12

420

J. P. CALVET and Y. ARKUN

8 values

E

Fig. 5. Graphical procedure to select the least conservativegains(with disturbancematchingcondition).

stabilization (i.e. 6) is specified, the proposed method

will give higher stabilizing gains than it is necessary.

Hence, given two pairs of constants @,, p,) and

&, p2) satisfying Assumption 2, we will say that one

pair gives less conservative results, if for a given set

of parameters t, n and

R ,

it gives smaller S-stabiliz-

ation gains than the other pair. We then have the

following result that will become clearer later on. If:

a,:.,,(~, H, R) < ap:,,,z(~ H, R),

where 6* is given by (6); then, the pair @, , p,) will

give less conservative (smaller) gains than b2, pr). In

other words (for H and R fixed) if the curve SF,,,, vs

.E s below the curve 6,:. p2; then, the pair @, , p, ) will

be preferred. The curve 6 * vs will have the following

shape:

.

.

5.4.

If the disturbance matching condition is satis-

fied, the 6* curve is strictly increasing with 6.

Also as L tends to zero, S* tends to zero as well

(see the schematic plots in Fig. 5).

If the disturbance matching condition is not

satisfied; then, the S* curve has a minimum

(3E s.t. as*/&),_,=

0) and the 6 curve goes to

infinity as c tends to tlim see schematic plots in

Fig. 6).

Procedu re

In light of the above results, we give two different

procedures depending on whether or not the

disturbance matching condition is satisfied. These

procedures help the designer to get the smallest

stabilizing gains once 6 and D are specified. The gains

obtained will guarantee that the QLS will be ulti-

mately stable in B(8) , and with no steady sate offset

for the output if the QLS is augmented.

With matching condition:

This procedure is schematically illustrated in Fig. 5

where each step number is circled.

1.

2.

3.

4.

5.

Plot the gain curve (K vs C) with usually Ei = I

and

R = 1.

Find @&, , past) so that a&(e) is the lowest

possible (this will avoid conservatism).

Locate S and pick 6 * so that S r 6 * (as required

by the theorem).

Get E (from 6* vs e curve).

Get the stabilizing gains K (from K vs L curves).

One can see in Fig. 5 that for 6 given, we get smaller

gains as 6* tends to 6. Also it is now clear that if

another pair @, JJ) gives a 6 * curve above the one

depicted in Fig. 5 it will give higher gains. Hence Step

2 in the procedure is the most important step which

avoids conservatism. It is important to notice that as

6 tends to zero, then 5* must tend to zero and

henceforth L. The stabilizing gains will then all tend

to infinity. As a result, asymptotic stability of all the

states of the QLS can be guaranteed as K tends to

infinity. This important result is not true when the

disturbance matching condition is not satisfied.


7/12

Design of P and PI stabilizing controllers

421

gains

Fig. 6. Graphical procedure to select the least conservative gains (without disturbance matching

condition).

Also for P controllers only, as c increases, K i

decreases and the amplitude of the overshoot (if it

exists) and the steady state offset will increase. This

can be seen from Fig. 1. Since the offset is related to

S, the steady state offset will decrease as e decreases

(the K i s increase). Also, the addition of integral

action in the control law will increase the values of the

gains. A comparison of Figs 1 and 2 shows that for

a given c, higher gains are obtained with the use of

integral action.

Without matching condition:

This procedure is schematically illustrated in Fig. 6.

1.

2.

3.

4.

5.

6.

Plot the gain curves K vs 6 ) with usually N = I

and

R

is tuned so that slim is the smallest

possible.

Find (Pbeat,~,_,) so that 6 ,(s) is the lowest

possible (thus will avoid conservatism).

Obtain 6 zin

as*

[from 3C s.t. X ~_~

= 0 then S&, = S*(C)].

Select 6* > ~5% and 6 > 6* from the theorem).

Get L (from 6* vs c curve).

Get the stabilizing gains K (from K vs 6 curves).

Hence, for all S > 6 * z=- tin such gains are 6 -stabiliz-

able. Once again, when 6* tends to S, the gains will

be smaller. Also the existence of ~5% shows that 6

cannot be arbitrarily chosen as small as we wish. AS

a result asymptotic stability of all the states of the

QLS cannot be guaranteed contrary to results with

matching conditions.

Also, if the disturbance matching condition is

satisfied but is not detected by the designer, stabihz-

ing controller gains can still be obtained through the

procedure without the matching condition, but in

general, the results will be more conservative.

6. STEADY STATE ANALYSIS OF PI STABILIZATION

The reason we introduce integral action in the

control law is to achieve zero steady state offset of the

controlled output under ultimately time invariant

disturbances. In this section, we show that when the

disturbance matching condition is satisfied, the con-

trol law will lead to zero steady state offset, not only

for the output but also for the other states of the

QLS. Equivalently, by virtue of the state transform-

ation [the diffeomorphism (4)], this means that UN the

states of the original nonlinear system will exhibit

zero steady state offset. We illustrate this by perform-

ing a steady state analysis. With the control law:

,=1

s

aa = - c Kiz,(t) - Km+,

21 (rW7,

r-l

0

the augmented closed loop QLS is given by

t=af+r:


8/12

422

J . P.

CALVET and Y. ARKUN

with

7. APPLICATION

As an illustration, we first show that the standard

CSTR model perturbed by fluctuations in the feed

temperature and the feed concentration falls into the

category of a nonlinear system described by (2, 3) and

can be transformed into a QLS by feedback lineariza-

tion. We will then stabilize the CSTR under the

influence of these disturbances using PI control. We

will make use of the plots of the stabilizing gains

obtained in Section 5.1 and then follow the proce-

dures given in Section 5.4. The only knowledge

required about the set of (ultimately time invariant)

disturbance(s) will be the bounds.

7.1.

T r a n s f or m a f i o n o f a

CSTR model i n t o a QLS

The dimensionless model of a first-order exo-

thermic irreversible reaction taking place in a CSTR

is given by Uppal ef a l . (1976) and Ray (1981):

zp+ 2 &_,&-)d,=O k -2,. . ..n.

i=l

i=n

i=p

K.zPq -

K

1 1

n+ ,zz+ +

C

LiWM = 0,

i=,

zp=o.

(14)

Since the disturbances are all ultimately time invari-

ant, all the terms df , i = 1 , . . . ,p are considered

constant in the steady state analysis. Note that the

last equation is a natural consequence of the fact that

the control was designed to guarantee zero steady

state offset of the output

z, ,

The above equations do

not simplify further. In particular z: for i = 2, . . . , n

cannot be obtained explicitly in terms of the

disturbances d is . However, if we assume that c(z )d

satisfies the disturbance matching condition with

respect to B as:

3~ (z) E R xp s.t. g ). (17)

where (

_ .)

denotes the inner product. Note also

that this state space coordinate transformation maps

xP to the origin in the z-state space [i.e. T(xP) = 01.


9/12

Design of P and PI stabilizingcontrollers

423

Then under these transformations, the CSTR model

is transformed to the QLS:

and the algebraic Riccati equation to solve is the

ARE where we picked n = I and R = I ( see gain

plots in Fig. 2).

In order to verify Assumption 2, we inspect the

inequality in the ball B ( f = 0.2). Among all the

with

where the inverse transformation x = T- (z ) is ob-

tained through the one-to-one mapping of the state

transformation:

x, = Z, + x;

[

2, +

z, + x;)p

x2 = v In

Da(1 -z, - ~7)

II

[

n

z2 ,xy

Da(1 -z, -x9)

I>

20)

Remark-We will consider X, (i.e. the dimension-

less composition) to be the output of the nonlinear

system. According to the state transformation (16)

such output corresponds to z,. Hence, the required

conditions (9) and (10) from Section 4.1 are satisfied

simultaneously and PI stabilization can he applied

with z, as output. However, if we would have chosen

x2 as output for the CSTR, then the residence time

should have been the new manipulated variable r en -

dering the nonlinear system transformable to a QLS

with an output z, corresponding to the dimensionless

temperature. Here, the flexibility to choose a new

manipulated variable as mentioned in Section 4.1

indeed exists.

7.2. P I s t a b i l i z a t i o n unde r f eed t em pe r a t u r e p e r t u r b a -

t i o n s d , = 0 )

Consider that the CSTR is subject to feed temper-

ature perturbations only. With a PI stabilizing con-

troller we then have the augmented QLS described

by:

-1

[

II

2

1 -Daexp ~

1 +x2/v

X T- 1(Z)

(19)

possible pairs @, p) satisfying Assumption 2, the pair

giving the lowest 6* curve is:

(PM, C(M) = (0.45, 1.51).

An algorithm to compute such a pair was developed

by the authors and is available in Calvet and Arkun

(1989). a:__,,.,,

curves as a function of L and

parametrized by various bounds D of the disturbance

d , are displayed in Fig. 7. According to the procedure

(with matching condition) we can now compute the

stabilizing gains of the PI controller. Let d = 0.2 and

6* = 0.199; also we consider ultimately time invariant

disturbance bounded by

D =

0.3. The a-stabilizing

gains are then:

[K , , KZ, K,] = [4.93,3.93,2.37].

A simulation in Fig. 8 with these gains in the control

law show that as expected b o t h states of the CSTR

exhibit zero steady offset under a step disturbance of

d, t ) = 0.3. As a performance criterion we can also

obtain the integral below the curve z, (t) = x,(t) - xyp

vs time. Indeed according to the construction of the

augmented QLS (1 I, 12) and the steady state equa-

tions (15) we have:

z? = lim

C

z, IT) d r = x od , = 0.048

I-CC

Jo ' JG

Remark Note that under the condition of distur-

bance matching condition, it is not necessary to

and the control law is U(Z) = -X:1: K i z i t ) where

(K,,

K2 )

and

K 3

are, respectively, the proportional

specify an a p r i o r i output of the nonlinear system (i.e.

and integral gains to be determined. In this case, the

here dimensionless composition or temperature).

state-dependent perturbation term satisfies the distur-

Indeed, integral action on z, {no matter what its

bance matching condition with:

relationship with the original nonlinear system may

l

x) Da exp[*]]_ r_,(Z1

mean) will guarantee zero steady state offset of all the

X(z)= (1 +&Iv>2

states of the QLS and henceforth of the states of the

original nonlinear system (i.e. here the CSTR).


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424

J.-P. CAL~ET and Y.

ARKUN

0.24

-

0.20

-

0.16

-

*

00 0.12

-

0.06

-

EPS

Fig. 7. 6 * curves vs L parametrized by D (with disturbance

matching condition).

7.3. PI st a b i l i z a t i o n unde r f eed composi t i o n per t u r b a -

t i ons (d , = 0)

If we consider feed composition perturbation only,

then, with a PI stabilizing controller, the augmented

QLS is described by:

0.520

0.515

OH0

z

z

0.505

k

i

3.06.10.06.16.14.12 -

c

x

a495 I 1

I 1

0 2 4

6

3D2

a

Time

Fig. 8. PI stabilization (with disturbance matching condi-

tion).

admissible value of 0.006. Then the stabilizing gains

are:

[K,, K,, &] =

[2.44,

1.86, 1.21.

and the control law is again u(t)= -%I: K,z,(t) .

One can easily see that the disturbance matching

condition is not satisfied. Then the algebraic Riccati

equation to solve is ARE where we picked n = I and

R = 0.1 (see gain plots in Fig. 4). With the algorithm

given in Calvet and Arkun (1989) we get the pair

(p. p) satisfying Assumption 2 in B (P = 0.2) that

gives the lowest 6* curve (for n = I and R = 0.1) vs

L. The pair is:

(P

bcs,,p~brrt) (2.4, 13.22).

dFbt._ curves as a function of L and parametrized by

various bounds

D

of the disturbance

d 2

are displayed

in Fig. 9. One can see that, as a result of the absence

of disturbance matching condition, all the 6* curves

parametrized by

D

have a minimum

S& (D ) .

There-

fore, 6 cannot be as small as we may wish. For

example, a-stabilization with 6 = 0.2 cannot be guar-

anteed for disturbance having a bound larger than

0.0065. This can be seen in Fig. 10 where we plotted

S ,& (D ) as a function of D . Indeed, such curve

gives regions where a-stabilization can or cannot be

implemented.

According to the procedure (without matching

condition), we can now compute the stabilizing gains

of the PI controller. Let d = 0.2 and 6 = 0.199. As

a bound D for the disturbance dZ we picked an

A simulation in Fig. 11 with these gains imple-

mented in the control law shows that, as expected,

on ly

the output of the QLS i.e. z, will exhibit zero

steady state offset. However, the other state z2 will

exhibit a steady state offset.

By virtue of the

diffeomorphism T , this corresponds to zero steady

a30

*

00

t

0.25

0.20

0.15

I

0 10

I

I I

I

I I

1 2 3

Ek

5 6 7

Fig. 9. 6 curves vs c parametrized by D (without distur-

bance matching condition).


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Design of P and PI stabilizing controllers

425

0 .30 -

0 .25 -

8

-stabilization

posaibls

0.20 -

0.15 -

8 stabilization

0.10 not possible

0.00 1

I I

I I IILLI

0.001

0 01

D

Fig. 10. Region of d-stability (without matching condition).

state offset for the dimensionless composition x,

and a steady state offset for x2 the dimensionless

temperature.

8. CONCLUSION

The theory and application of the P and PI stabiliz-

ation of quasi-linear systems (QLS) is presented. The

origin and practical importance of QLSs is intro-

duced, and the concept of ultimate boundedness and

&-stabilization is adapted for such systems. A

theorem and a procedure are given to compute the

stabilizing gains in the least conservative sense for the

proposed methodology. The results permit to stabil-

ize the class of nonlinear systems with bounded

disturbances which are feedback transformable to

QLSs. If the so-called disturbance matching con-

dition is satisfied, we show that PI stabilization will

guarantee zero steady state offset, not only for the

output but also for all the other states of the QLS

(and henceforth for the original nonlinear system).

Simulation results on an open-loop unstable (and

perturbed) CSTR mode1 illustrate and agree with the

theory.

NOMENCLATURE

y = h(x) = (Single) output of a system

R = Set of real numbers

d E RP = Disturbance vector

R = Set of real n vectors

I / x I =

1. = Absolute value of elements in R

24= (Single) control input of a system

Euclidean

Xl_ ,

xf

(usually nonlinear)

norm for x E R , I I x I I =

R

m =

All n x m real matrices

I E R x = Identity matrix

x E

R =

States of a system (usually nonlinear)

z E

R =

States of a system (usually linear or

quasi-linear)

zi= (Single) control input of a system

(usually linear or quasi-linear)

K E R = Stabilizing gains

Fig. Il. PI stabilization (without disturbance matching

condition).

(A, B) = A controllable matrix pair (usually the

BCF)

f(x), g(x) = Smooth vector fields in R (infinitely

differentiable i.e. C)

Y(x) E WXp = Perturbation matrix associated with the

disturbances

z = T(x) = A

nonlinear one-to-one mapping

(diffeomorphism)

dT / a x =

Jacobian matrix of T

u = S(.X, U) = A (single) input nonlinear transform-

ation with state feedback

B(6) = Ball of radius 6

r, P = Real positive constants denoting the

radii of balls E(r), B(T)

6,6* = Real positive constants denoting the

radii of balls B(d), B d * )

p . p = Real positive constants satisfying As-

sumption 2

D =

Real positive number, bound of the

disturbance(s)

R, c

RP =

Set of disturbances

c(z) = aT/a x Y x) = Perturbation of the QLS

X(Z)ER

rp = Row vector satisfying Assumption I

L = Real positive number (parameter of the

ARE)

R, W = Matrices of the ARE (parameters)

,I(~) = Eigenvalue of a square matrix

,I*(.), A* (~) = Maximum and minimum eigenvalues

of a positive definite matrix

v(.) = 1 *(-)/A * (.) = Condition number

P z 0 =

Solution of an algebraic Riccati equa-

tion (ARE or ARE)

Abb re v i a t i o n s

ARE = Algebraic Riccati equation

BCF = Brunovski canonical form

QLS = Quasi-linear system

P = Proportional

PI = Proportional integral

REFERENCES

Calvet J.-P. and Y. Arkun, Feedforward and feedback

linearization of nonlinear systems and its implementation

Calvet J.-P. and Y. Arkun, Feedforward and feedback

using IMC. Znd . Engng Chem. Rex . 27, 1822-1831

linearization of nonlinear systems with disturbances. Zn t .

(1988a).

J . Con t r o l 4 8 , 1551~1559 (1988b).


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design of p and pi controllers for quasi linear systems