Robustness degree based controller design for a class of nonlinear systems

Compurers them. Engng, Vol. 13, No. I l/12, pp. 1283-1289, 1989 Printed in Great Britain. All rights reserved

ROBUSTNESS FOR

A

T. TAKAMATSU,’

009%1354/89 $3.00 + 0.00 Copyright 0 1989 Pergamon Press plc

DEGREE BASED CONTROLLER DESIGN CLASS OF NONLINEAR SYSTEMS I. HASHIMOTO,’

M.

OHSHIMA,~

J. CHu3t

and J.-C.

WANG~

‘Department of Chemical Engineering, Kansai University, Suita, Japan 2Department of Chemical Engineering, Kyoto University, Kyoto, Japan 3Department of Chemical Engineering, Zhejiang University, Hangzhou, China (Received

for

publication

19 June 1989)

Abstract-This paper is addressed to the topic of control system design for a class of nonlinear or bilinear systems by taking a new concept of robustness degree into consideration. A new design approach for the nonlinear controller with certain specified robustness degree is proposed, which is called a two-step approach. The first step is to design a linear state feedback controller by considering the system robustness degree based on a linear model, then the nonlinear model is considered for the further design of a nonlinear controller in the second step. The control system designed in such a way for nonlinear systems will be robustly stable with certain robustness degree, and the offset of the system state could be restricted within a desirable range. A bilinear system as a typical example of such nonlinear systems is also studied. Finally, a real chemical reactor is studied for the control system design by the computer simulation.

1. INTRODUCTION

One of the main themes in theory on automatic control system design is the problem of how to design a controller which can guide the system to a desired state region under a certain performance index in the presence of uncertainties, disturbances and/or nonlinearities. In the last two decades, many application theories on process control have been proposed for linear systems, but few on nonlinear systems. But quite often the problems we encounter in real processes are the controller design for those with nonlinear properties. So the question for designing a robust nonlinear controller occurred. Even the work done on nonlinear systems is of less practicability. In recent years, increasing attention has been paid to controller designs for nonlinear systems. Molander and Willems (1980) studied the design of cone-bounded feedback nonlinearities which preserve the stability of a linear open-loop system. Noldus (1982) also discussed the dynamical systems the cone-bounded nonlinearities in containing open loop equation, and derived some results on the robustness of systems both of uncertainties in the state matrix and in the control matrix. Vidyasagar and Kimura (1986) derived some results on the design of robust controllers for uncertain linear multivariable systems. The motivation of controller design with the consideration of system robustness property is natural due to the engineering viewpoint. Most of the practical problems we encounter are those working in the variable and perturbed state. A controller designed .tAuthor to whom all correspondence

should be addressed.

only with a deterministic model will not be sufficient for the system stability when the disturbances and nonlinearities are introduced into the system. This paper is addressed to the topic of the control system design for a class of nonlinear or bilinear systems by considering the concept of robustness degree of control systems which were proposed by Takamatsu et al. (1988). The new concept of robustness degree of a control system was proposed by applying the semigroup theory. With this new simple quantitative measure, a new design approach for nonlinear eontrollers is proposed in this paper, called a two-step approach. The first step is to design a linear state feedback controller based on the corresponding linear model by considering the system robustness degree, then the nonlinear model is considered for further design of the nonlinear controller in the second step. The control system designed in such a way for the nonlinear system will be robustly stable with a certain robustness degree and easily implemented in control computers. The system offset could also be bounded on a desirable range. The bilinear system as a typical example is also studied in this paper. Finally, a real chemical reactor is studied by computer simulation. The result of this paper shows the fact that the method proposed is applicable to nonlinear processes. 2. PROBLEM

FORMULATION

Here we will consider a dynamical system described by a special nonlinear differential equation, where we would like to seek an admissible control which leads the system to having a certain robustness property for tolerating and rejecting any uncertainties, while 1283

1284

T.

maintaining system system satisfies: % = AX

stability.

Suppose

the dynamical

+ BU + F(X, U) + g(X, U, d, t), X(0)

(1)

= x,, .

on [0, 7-l. a real finite time interval. where X is an ~1 x 1 state vector, U an I’ x 1 control vector, d a 4 x 1 disturbance vector, A an n x n constant system matrix, B an n x r constant input matrix assumed to be full rank and F(X, U) an n x 1 nonlinear function vector assumed to be continuously differentiable in its arguments. g(X. CJ. d, t) can be construed as the uncertain nonlinearities, disturbances and/or modelling errors, where d is a disturbance vector. Suppose the origin is an equilibrium point of this system, and the control purpose is to keep the system stable at the origin. Usually we cannot know the detailed knowledge about g(X. U, d, t). but its approximate range: I: g(X.

L, d. t) I/G g.

et al.

TAKAMATSU

(2)

where g is some constant value. Obviously, the problem is to design an cft‘ective controller for system equation (I) in the absence of knowledge of an uncertain function n(X, U, d, t ), but the condition equation (2). It is difficult to design a linear controller for a nonlinear system with satisfactory system behavior. We will utilize the special structure of system equation (1) to design an effective nonlinear controller. For further need. we will briefly introduce the concept of a robustness degree of a control system, then discuss the procedures of a nonlinear controller.

Suppose that all of the possible matrices (A - BK) form an operator set which generates an asymptoT, such that: tically stable semigroup

Ii T. I/ < h4 exp(wt),

with o < 0, I 3 0 as shown by Takamatsu rf al. (198X). Also suppose that the nonlinear function F(X. II) in the system forms another nonlinear operator set which is bounded on some range (F). We can expect the solution for equation (7) to have the form:

X(t)

=

T,X(O)

+

Consider

the

ROBUSTNESS

following

Now if we assume another semigroup,

that S, : X(r)

(4) information a feedback

U = -KX, can

stabilize

system ri: = (A -

(4)

about control

(5) as:

BK)X

the

solution

is

given

= S,X(O)

by

(10)

where F(X, - KX) is denoted by F. Obviously, the operator S, also generates an asymptotically stable semigroup if: UI + M I/F iI;!/X Ii -L 0.

(12)

holds for all t 3 0, that is to say the closed-loop system (7) is robustly stable in the sense of Lyapunov. We rewrite the inequality (12) as: 21 :IX /(= p 11 X !j, M ”

(17)

+ F(X. -

(13) is called as robustness with feedback control 3.1.

Theorem

(6)

KX).

(7)

degree (5).

of the linear

system

(4)

1

A nonlinear system in the form of equation (3) with a linear state feedback control law given by equation (5) can be robustly stabilized (in the sense of Lyapunov) with robustness degree p defined by equation (14), if the norm of nonlinear term F(X. U) satisfies a conic sector type of inequality condition with the form of equation (13). Proof--See

BK)X.

It is asymptotically stable. But actually the real system is in the form of equation (3); we will ask now how about the stability and robustness of system (3) with the linear feedback control (5), that is: _% = (A -

(9)

(3

of robustness analysis of a control because. in practical cases, WC would a state feedback controller based only system without the nonlinear term: jc=r\x+su,

which

dT.

where

system:

e.g. in the case of lacking detailed F(X, U). Suppose we have derived law:

~ KX)

we can obtain the following expression through mathematical treatment. the detailed procedures can bc found in Takamatsu it N/. (1988):

DEGHEE

ii = AX + BU + F(X. U). The problem system arises like to design on the linear

1 T, _;F(X, J to

// F i! < 3.

(8)

Takarnatsu

et nl. (1988).

Q.E.D.

We must find M and Q, i.e. robustness degree p, in order to have the practical use. In the general expression, the operator function can be a transition matrix of (A - BK) if(A - BK) IS a constant matrix, such that: 7; 4 exp[(A

- BK)r],

(15)

Robustness degree ba sed controller design

1285

4.1. Step I

i.e.

(16)

At the first step, we consider

It is difficult to find the precise version of robustness degree as shown in Takamatsu et al. (1988), but an approximate and useful measure, called the matrix measure, can be applied to express the system robustness degree.

%=AX+BU+g(X,U,d,t),

(1T 11= I(ewW

Definition l-Matrix Miles, 1980)

- BK)fl\\ G M evW).

measure

p(A)

p(A)=/iy+(l[I--6AII For

the 2-norm

-

(Harris

1)6 -‘.

= &,,&A

p = -p(A Thus, the following derived.

corollary

system: (20)

to (2) for all t >, 0, and

and

IlUll G MM,

(17)

where M, and Mu are the boundednesses of X and U. Since the exact knowledge of g(X, U, d, t) is not available, the system model actually reduces to:

(22)

A=Ax+BU. +

A*)],

(18)

where A,,,,, denotes the maximum eigenvalue and A* means the conjugate of A. We can prove (Takamatsu et al., 1988) that the robustness degree p in one of its estimates has the form:

Corollary

g(X, U, d, t) is subject

concerned:

P(A)

3.2.

where all

the following

- BK). of Theorem

(19)

The problem is to design a proper controller which possesses certain robustness degree p with the relation:

p a 11 g(X, U, 4 t) for

I

4. A TWO-STEP APPROACH FOR NONLINEAR CONTROLLER DESIGN

Since, in any practical design, the property that the system remains stable must be preserved, the allowable perturbations in the design of robust controllers are limited to those which do not make the system unstable. In other words, nonlinear controller design with the consideration of system robustness may have better characteristics to vanish the effects of uncertainties and disturbances when the exact knowledge about uncertainties and disturbances is not available. In this section, a new approach for nonlinear controller design is discussed for a class of nonlinear systems with the form of equation (1). Motivation of the approach is due to the typical structure of the system concerned. Firstly, if there does not exist the nonlinear function F(X, LJ), it is necessary to design a linear feedback control law which has the robust property to undertake the nonlinear uncertain function g(X, U, d, t), restricted by the inequality equation (2). Then, the nonlinear function F(X, U) is taken into account for further nonlinear controller design in order to obtain better performance. So we will complete the design procedures by two steps.

(!/I/X(t) 11)

all t 3 0, i.e. at least (conservatively): P = g/X,,

1 can be

The robustness degree p, in one of its possible precise forms, for the nonlinear system equation (3) with feedback control equation (5) is equal to the minus of the measure of matrix (A - BK) as shown by equation (19).

(23)

where of the of the degree 1988). By in:

(251

X, denotes the desired final steady state offset system state. We can prove that the real offset system is bounded within X, if the robustness is designed based on equation (25) (Chu, Corollary

1, the control

U= should

have the following p = g/X,

gain matrix

K shown

-KX

(26)

relation:

= -p(A

- BK).

(27)

The solution of K in equation (27) is not unique for a given p. By the definition of the matrix measure: p(A

- BK)

= &,,,{;[(A

- BK)

+ (A - BKjT]}.

(28)

i.e. &ax { $[(A + AT> - (BK it is easy to derive I,,,

+ KTBT)]}

= --p,

(29)

= - 2~.

(30)

the form:

[(A + AT) - (BK + KTB’)]

Consequently, this equation (30) shows us a standard formulation of the pole-assignment problem which cannot be so difficultly solved by the well-known methods for pole-assignment. In order to keep the system stable in the presence of uncertainty g(X, U, d. t), the sufficient condition is to assign all of eigenvalues of matrix [(A + AT - (BK + KTB’)] be less than -2~. So the following theorem can be concluded. 4.2.

Theorem 2

A linear system (23) with a feedback (26) is said to have robustness degree p if the feedback gain

T.

1286 matrix equality:

K

is chosen

&[(A hold, 4.3.

where Step

so

+ A’)

-

2, means

as

(BK

to

let

the

-f KT’B’)]

TAKAMATSU

following

in-

< --2p

Substituting have:

(31)

of the

the ith eigenvaluc

et ul.

BU,

equation

(39)

Into

equation

(3&j,

we

+ F(X,

matrix.

2

the system consists of the nonlinear Actually, function F(X. U), the behavior of the system with only the control law (26) will not be satisfactory when the effect of the nonhnear action is serious. Based on the first design step, we consider the system: % = AX

+ BU + F(X,

U).

(32)

U) = BU + F(X,

U),

(33)

Let V(X, then % = AX + Suppose

the control

law

V(X, then

the

closed

V(X,

can

U).

(34)

be designed

as:

U) = -BKX,

(35)

system I% = (A -

BK)X.

(36)

is the same as the linear system with the linear control law designed in Step 1. This is just what we want, but how about the implementability of control afgorithms (33) and (35)? Rewrite equation (33). we have: BU + F-(X, U) + BKX

= 0.

(42)

(37)

This is a nonlinear control loop. It is very difficult to get the explicit solution U from equation (37) in the continuous case. For the practical implementation, if a process computer is applied, the following discrete algorithm can be rcadily derived. Equation (37) can be rewritten in the discrete form: BU,

+ F(X,.

U,)

+ BKX,

= 0,

(38)

where k denotes the time instant. By the proper selection of the sampling time, the stability of the closed-loop system will not be violated when the discretization is carried out. In order to solve this problem, we take the Taylor series for F(X, . Cl, ) and approximate with only the first-order terms (if much more higher accuracy is needed, the higher order terms can be chosen):

_

RX,,

x ([IA -

CT,_,).

dF

ZLl

The following law.

theorem

4.4.

.J

Theorem

There

L’i)=F(X,_,.CI,~!)+~(X,_,. Cl_,) x(x,--x,-,)+1(x,&,.

-

exists

an

can give the nonlinear

unique

solution

control

to equation

(43):

ifi = M + f.

L’A-I )

where

M+

Proof-See (39)

is a Penrose Gopal

(1984)

generalized or Nashed

(46) inverse

of

M

(1976). Q.E.D.

Robustness

degree based controller

So far we have found the nonlinear control law which makes the system possess a certain robustness degree to reject some uncertainties and disturbances. We note one particular case, i.e. when the nonF(X, U) is independent of U, then the linear function controller is straightforward with a solution in the continuous case.

5. NONLINEAR CONTROLLER DESIGN A BILINEAR SYSTEM

= -F(k

-

x x,-,

1287

I)+$$

-

-

1)u,_,

BK+g(k

+g(k

- 1)

1

-

1) X,

(51)

FOR

The

As a typical example, we consider the controller design for a bilinear system. It will be useful in practical application. Consider the following bilinear system: X = AX

design

control

c5, = M+.P= 6. AN

(47)

+ BU + i C,Xu, + g(X, U, d, t), i=l

final nonlinear

APPLICATION

Mathematical

6. I.

strategy

h(X,,X,_,,

will be shown U,_,).

TO A CHEMICAL CONTROL

as: (52)

REACTOR

models

where A, B and C, (i = 1,. . . , t-j are constant matrices, g(X, U, d, t) is an unknown certain function, which denotes the perturbations and disturbances with the boundedness of:

A continuous stirred tank reactor (CSTR) in which an exothermic second-order reaction takes place is taken into account (Kao and Lier, 1981):

11 gG’L U, 4 tj 11G g.

Through the mechanism analysis and the mathematical treatment, in the presence of disturbances, we can obtain the following system. Two states are the concentration of reactant x, and reactor temperature xz in the variation values about their normal operating points. The control variable is the coolant llowrate I( in the variation value too. The reactor system can be described by the following three types of models: nonlinear, bilinear and linear models with a disturbance vector d.

2 A-+B.

(48)

Firstly, we design a linear control law with feedback gain matrix -K with which the closed linear system is stabilized with the robustness degree to reduce the we can consider the effect of g(X, U, d, r). Then second step as the same procedures in the previous section, here we note: F(X, U) = i C,Xu,, I= I

-

jz= +

2. Bilinear

0.4807 0

0 - 0.06846

1. Nonlinear

1x+[

0 -9.887e

-4

1.1576 - 0.2314(x,

+ 16.05)2 exp(

- 2.316 + 0.463(x,

+ 16.05)* exp( -

model:

A=

(49)

-0.1923 0.2886

1 [

0 -9.887e

x +

-

-4

model:

1 [ u +

1422/(x,

0 0

0 1.618e-4

+ 360.78))

1422/(x, + 360.78))

1[ U+

0 0

0 1.618e-4

1 XU

1 1

+ d.

Xu+d.

13’ Linear model: %=[-i:;;;z;

~;:~::;;TC+[_9.8;7e_,]u+d where

then

6.2.

M(S,_,,U,_,j=B+~(k-1) =B+(C,X,_,:...:C,X,_,),

(50)

d denotes

the disturbance

vector.

Simulation

The system is simulated on a digital computer with the fourth-order Runge-Kutta method. The responses (reactor temperatures shown only in figures) by the nonlinear model (denoted by T-NL),

T.

1288

TAKAMATSU

et al.

1

sysm described

Nonlinear

--

by a model -.-._--

nonlinear

Controller

iiesigrxcxi based on bilinear maAe1

a J

Fig. 2. Simulation chart by the nonlinear

0 0

20

40 t

Fig.

1.

100

80

60 cm5.n)

Responses of nonlinear, bilinear and tinear models to a step u = -50.

the bilinear model (T-BL) and the linear model (T-L) due to a sudden reduction of the manipulated variable to the steady-state temperature are shown in Fig. 1. It is the fact that the bilinear model can agree well with the original nonlinear system without apparent differences between the responses vf two models better than the linear model running. Thus, it is suggested that the hihnear model can be applied to design a controller for the practical process. The purpose of this simulation is to show the usefulness of the nonlinear controller design method proposed above and the role of the robustness degree. For the simulation we will consider the nonlinear model as a real system, and use two controllers designed by the bilinear model and the linear model, respectively, as shown in Figs 2 and 3. It is not difficuh to design a linear state feedback controller based on any known approaches (e.g. LQR controller, pole-placement, etc.). Suppose we have a linear controller with certain given robustness degree: K =[lq

K],

then the nonlinear controller model can be derived as: If(k) = (1.618ufk

-

controller.

l)[xt(k

based

on the bilinear

60 time

Linear

Controtler on

a

the nonzero initial states as sr = 0, _Y-,= - 1. The system rcsponscs by two different controllers (C---L: linear controller, and C-NT,: nonlinear controller) have been shown in Fig. 4. in another simulation run, a constant disturbance initial is considered, dT = [0 -- 0.021. the nonzero states are the same as above. The results by applyand linear controllers are shown in ing nonlinear Fig. 5. Furthermore. we consider the system robustness degree in order to reduce the offset where tberr exists only disturbances. For example, if we know ji d jImaxand the maximum ahowable /jX(x) i!. then the least robustness degree of the control system is !Id /j,,,,,//iX(m) Ij_ Figure 6 shows the result of real //X(CZ)ii vs -t- 9.8877X;s,(k)),!ft

.618.r2(k

- I) -9.X871.

robustness degree with respect to the disturbance d. We can see that the offset will decrease with an

0

20

40 time

(ninl

C-L c-NL

a

Fig. 3. Simulation chart by the linear controfier.

100

80

by

based

- I) - .x2(k)] + 9.887K,x,(k)

In the simulation, firstly we consider no disturbances entering the system. i.e. d = 0. and consider having

40

described

60 fmin)

! Linear ccxltroi1er : Nonlinear cantrol.ler

Fig. 4. System responses by C.-L and C-NL

with disturbance

d = 0.

Robustness degree based controller design

1289

-1

time

time (min)

(minf

C-L : Linear oDntro11er C-NL : Nonlinear controller

System responses by C-L

+.

I

I,,

.05

,

,

,

.l robustnessdegree

,

and C-NL

,

0

increase in the system robustness degree for the certain disturbances. AND

the process response controlled by the nonlinear controller behaved better than that by the linear controller. Since the robustness degree of a control system has been taken into account for the linear controller design, the nonlinear controller designed by this two-step approach will have better properties on the stability and robustness, but the controller must be implemented by a computer because only the discrete control algorithm is often available.

-15

Fig. 6. Real offset of state I!X(co) I/ vs p with respectto d.

7. CONCLUSIONS

without disturbance dT = [O - 0.021.

DISCUSSION

This paper discussed the design problem of robust control systems and proposed a two-step design approach of nonlinear controllers for a class of nonlinear systems. The design procedures of this approach are straightforward for control system design. The two-step design approach far a class of nonlinear systems or bilinear systems considers the system robustness degree in its first step to reject the effect of uncertainties or modelling errors for the resulting system and takes the nonlinear term of the system model into consideration for nonlinear controller design in the second step. The computer simulation for a chemical reactor has shown that

REFERENCES

J., A robustness study and synthesis for a ciass of nonlinear control systems. Ph.D. Thesis, Zbejiang University, China (1988). Gopal M., Nodern Control System Theory, pp. 3440. Wiley, New York (1984). Harris C. J. and J. F. Miles, Smbitity of Linear Systems. Academic Press, New York (1980). Kao Y. K. and F. V. Lier, Controllers design based on a two-time-scale model. JACC 11. TP-7A (1981). Molander P. and 3. C. Willems, Synthesis of state feedback control laws with a specified gain and Phase margin. IEEE Zkms. Autom. Control. AC25, 928 (1980). Nashcd M. 2. (Ed.). Generalized inverses and Applications, pp. 245-302. Academic Press, New York (1976). Noldus E., Design of robust state feedback laws. Inr. J. (1982). Control 35, 935-944 Taknmatsu T., I. Hashimoto, M. Ohshima, J. Chu and J. C. Wang, A new measure for robust control--robustness degree. Preprints of 4th IFAC Symp. Comprrter-AidedL)esfgn in Conirof Systems, 23-25 Aug. Beijing, China, PP. 548-552 (19X8). Vidysagar M. and H. Kimura, Robust controller for uncertain linear muItivariable systems. Azztomuzica22,85-94 (1986). Chu