Adaptive Decoupling Control of MIMO System

Copyright © IFAC Identification and System Parameter Estimation. Beijing. PRC 1988

ADAPTIVE DECOUPLING CONTROL OF MIMO SYSTEM M. M. Bayoumi and Li Mo Department of Electrical Engineering, Queen's University, Kingston. Ontario, Canada

Abstract. This paper describes a new adaptive controller for MIMO systems. The resulting c losed loop system will be stable and decoupled. The system to be controlled can be non-minimum phase, and/or unstable linear systems. The interactor matrix of the system is not required. Instead of employing the certainty equivalent principle to design the adaptive controller, a new controller design procedure is proposed. By using this new scheme, the computation time required for controller design will be drastically reduced. Moreover, the boundedness of the controller parameters is automatically ensured. This is a desired property which is useful in stability analysis. The steady state performance of the controller designed by the proposed method will be the same as that of controllers designed using the certainty equivalent principle. Approximate decoupling can be achieved by underestimating the degree of the decoupling matrix without affecting the stability of the system. To achieve decoupling, we need to specify the denominator of the closed loop transfer function matrix of the system which is assumed to be in the form T(q -1 )-t(q -1)1. Global stability of the adaptive system is obtained. A simulation example is presented at the end of this paper. Keywords.

Multivariable systems, adaptive control, decoupling, pole placement.

INTRODUCTION decoupl ing part does not have any effect on the system stability. The system performance during the adaptation process is governed by the pole placement part of this controller. Decoupling is only used to improve the tracking ability after the adaptation period. In the controller deSign phase, the widely used certainty equivalence principle is not employed. A special controller design procedure is presented, which will substantially save the computation time. It also ensures the boundedness of the controller parameters and will give the same steady state performance as that of those controllers which are designed using the certainty equivalent principle. Besides, if the decoupling part of this controller s designed using the proposed procedure, approximate decoupling can be achieved by underestimating the degree of the decoupling matrix. Finally, it should be noticed that the interactor matrix is not required for the controller design and the global stability of the adaptive system is proven.

This paper presents a decoupled adaptive control scheme for MIMO systems. The class of systems under consideration is quite broad and includes unstable and non-minimum phase systems. Several adaptive controllers have been presented in the past few years. Only some of these schemes, however, deal with the problem of decoupled control. For the control of a general MIMO system, decoupling may be needed in order to control each output independently. The control strategy described by Johansson (1987) needs the a priori knowledge of the non-minimum phase zeros of the controlled system. For the controller developed by McDermott and Mellichamp ( 1986 ) , decoupling can not be achieved exactly for general linear systems and hence the stability of the system can not be guaranteed. Recently, a decoupling control scheme was suggested by Kinnaert et al (1987 ) . The controller design necessitates the factorization of the polynomial matrix A(q-1) into its Smith form. Furthermore one needs to compute the adjoint of the polynomial matrix B( q-1) namely, adjB(q-1). Hence this scheme is computationally intensive.

This paper is organized as follows: in section 2, a non-adaptive controller scheme is presented. In section 3, the new adaptive controller design procedure is presented and some convergence aspects are discussed. Section ~ deals with the global stability of the adaptive system and section 5 presents a simulation example. The conclusion is given in section 6.

The global stability problem for the pole placement MIMO controller has rarely been discussed. This is due to the fact that the controller parameters can not always be kept bounded unless some additional a priori knowledge is added concerning the poles and zeros of the controller system.

FIXED CONTROL STRATEGY Consider a system described by

The controller presented in this paper consists of two parts, the pole placement part and the decoupling part. The pole placement part will stabilize the system and the decoupling part should then decouple that system. Hence the

A(q

109

-1

)y(k)· B(q

-1

luCk)

(1)

M. M. Bayoumi and Li Mo

110

where B(q-1) and B(1) nonsingular.

are assumed to be

The polynomial matrices A(q-1) and

B(q-1) are nrxnr matrices and are given by: + ••••

A

+

n q

-n

In the next section, a new controller design procedure will be presented to evaluate p(q-1) and (2)

a

a

B(q

-1

). B q 1

-1

+ ••••

The polynomial matrix M(q - 1) does exist since an -1-1 -1 obvious choice of M1 (q ) is adj{D(q )B(q )}.

D(q

-1

) as well as M1 (q

-1

+ Bn q -n b b

)z(k)· y(k)

(4)

u(k) • _p(q-1 )z(k)

M(q-1 )r(k)

+

).

In this way, the

computational complexity will be reduced and the stability of the system can be ensured under certain conditions.

The controller is designed according to the following equations: D(q

-1

where r(k) is the command signal.

(5)

The polynomial

matrices D(q-1) and p(q-1) are designed to satisfy the following equation (where T(q-1) is a stable polynomial matrix)

(6)

The polynomial matrices p(q-1) and D(q-1) are of the following form

CONTROL DESIGN PROCEDURE In almost all of the parameter adaptive schemes, the controller design is based on the certainty equivalent principle. This implies that the estimated system parameters are used as if they were the true parameters and the controller design equation is solved based on those estimated parameters. For some controllers such as the one presented in this paper, the controller design procedure may be quite involved and may take considerable computati on time if implemented directly. Actually, it is not necessary to solve the controller design problem exactly since it is based on the estimated system parameters rather than their true values. In the following, a new scheme is presented in order to get the controller parameters recursively. In the transient period, the controller design problem is solved approximately and in the steady state, the controller design problem is solved exactly.

+ •••••• +

+

+ ••••••

(8)

For the pole assignment part of this controller, a Diophantine equation to assign the poles of the closed loop system is to be solved. That is, in every sampling period, we need the solution of the following equation

(9)

( 14)

The polynomial t(q-1) is of the form + t

+ ••••••

n

q

-n

t

t -1

A

The polynomial degrees d, p and n

should satisfy

t

the following conditions: p • n

n

- 1

a

t

< na -

+

n

b

assumed to have the same order as those of A(q

- 1

and B(q

The vector z(k) is an internal variable and its initial conditions are arbitrary. The polynomial matrix M(q-1) is used to decouple the system and hence it should be chosen such that D( q-1)B( q-1) M1 (-1) q • diag «b

1

q -1 ) , ... , b

n

r

'-1

Here the polynomial matrices A(k,q ) and B(k,q ) include the estimated parameters whi ch are obtained from the system identifier and are

(q -1 )) (10)

-1

) respectively. A

-1

'-1

the polynomial matrices A(k,q ) and B(k,q ) must be coprime. This is another disadvantage of using the certainty equivalent principle in the pole placement problem since the existence of any '-1

temporary common factors in A(k,q will cause difficulties in solving equation. On the other hand, a simultaneous equations needs to be -1

-1

-1

) and b (q ), i • 1, ..• ,n represent i r the unknown polynomials.

P(k,q ) . This needs substantial presented here, Let a vector x(k) x (k) •

It is easy to show that y(k) • D(q u(k)

•

-1

)T

-P(q

M(q-1)r(k)

-1

(q

-1

-1

)T

)B(q

-1

-1

(q

)M( q

-1

-1

)B(q

)r(k) -1

)M(q

(12)

-1

)r(k)

+

'

(X

-1

) and B(k,q ) the Diophantine set of linear solved in order -1

) and

process is quite involved and computing power. In the scheme three variables are introduced. be defined as follows: 1

(k), ... , xn (k))

T

(15)

r

At this moment, the vector x(k) can be thought of as a general time sequence. Two vectors z(k) and Zm(k) are defined such that:

(13) z(k )

Since T(q-1) is stable, then the resulting closed loop system will be stable regardless of the positions of the original system poles and zeros.

)

To solve this equation,

to get the polynomial matrices D(k,q

where M(q

-1

DT(k,q-1)~T(k,q-1)x(k)

pT(k,q -1 )ST(k,q -1 )x(k )

+

(16) ( 17)

Adaptive Decoupling Control of MIMO System To simplify the notation. the following definition is used. A A_l q-iA(k.q -1 )x(k) _ A(k.q )x(k-i)

(18)

Our design scheme is based on the following observations: if z(k) - Zm(k) for a general time

details are omitted because of space limitations. Several assumptions have been made in order to get the desired result: A

A(~.q

1.

D(k.q-l) and P(k.q-l) will solve the Diophantine

)

and

A_l

B(~.q

)

are coprime.

degp(k.q-l) • degA(~.q-l)-l 3.

x(k) is of the following form

Since the polynomial matrices D(k.q-l)

and P(k.q-l) are assumed to be unknown. a parameter estimator can be constructed based on the time sequences z(k). Zm(k) and x(k). The polynomial matrices D(k.q

-1

) and P(k.q

-1

this estimator are denoted by D(k.q

-1

P(

1

A

(26)

) and

k. q -1 ) . We call this estimator the "auxiliary estimator". If the RLS estimator is used. our objective is to find the polynomial matrices 1

xm{jN-k) - 0 k - 1.2 ..... 2mn-l.2mn+l ..... (n +2)n r • 1 k. 2mn ·C(k) elsewhere for (nr+2)N
) given by

A

A

-1

degD(k.q-l). degB(~.q-l)-l

2.

sequence x(k). then the polynomial matrices equation.

III

D(k.q- ) and P(k.q- ) recursively such that the mean square error between z(k) and zm(k) is

Where: j.m. are positive integers and C(k) is any arbitrary real quantity. The integer n is given by n - max(na.n b ) N is an integer greater than (n +2)n. r

minimized.

Let us introduce the vector ~(k-l) defined by

To get the expression of z(k) which is suitable for the parameter estimator. we present z(k) in the following form:

-

~(k-1)

x x

T T

- [x

T

A

(k-1)A(~.q

A

-1

A

-1

(k-d)A(~.q (k-p)B(~.q

).x )]

T

-1

) •...•

A

(k)B(~.q

-1

) •.•••

T

(27)

(19) It is not difficult to prove that where ~(k-l)

~(k-l)

is the memory vector and is given by

T

A

- [x (k-1)A(k.q

-1

T

A

-1

T

A

-1

x (k-d)A(k.q x (k-p)B(k.q The parameter vector

) ..... T

A

). x (k)B(k.q )]

jN

~

Li

-1

) .....

T

Where (20)

r ~(k)~T(k) > 0

1

J .. m min jN

(28)

k-l

~min(')

means the minimum eigenvalue of the

corresponding matrix. By using Toeplitz lemma. we can also conclude that

0(k) is given by

L jN r

~lm ~min jN

(29)

k-l

(21) This means The parameter vector 9(k) will be updated once every sampling period based on the standard RLS algorithm. according to the following set of equations:

~lm

jN

P(jN)<~

-1

9 (k) - 9 (k-1) + i i

P(k-1)~(k-l

)e (k). i

i - 1 ..... n

r

(22)

(30) -1

Define DO(q ) and PO(q ) as the solution of the following equation: A -1 -1 A -1 -1 -1 A(~.q )DO(q ) + B(~.q )PO(q ) - t(q )1 (31) Let 0 0 contain the parameters of DO(q

E(k) - (e (k) ••..• e (k»T • Zm(k) - z(k) 1 n

(24)

r

To further reduce the computation time. the memory vector can be simplified as follows:

Po(q

-1

) and

-1

) in the same way 0(k) contains the A -1 A_l parameters of D(k.q ) and P(k.q ) respectively. Hence our estimated controller parameters can be expressed as follows:

A

T A -1 T A • [x (k-l)A(k-l.q ) ..... x (k-d)A(k-1 T A -1 T A -1 T d.q ).x (k)B(k.q ) ..... x (k-p)B(k-p.q )]

~(k-1)

(32)

(25)

From the convergence analysis. it can be shown that the steady state performance of the system using the memory vector given by (20) is the same as that obtained using the memory vector given by (25) •

In the following. the convergence aspect of the auxiliary estimator wil be examined. TediOUS

A T AT w(k) - t(k-1)[(t (k-1) - ~ (k-1»9

o

A_1 + (A(~.q ) -

A(k.q -1 »x(k)]

(33)

From the assumption listed above. we have

Hmw(k) • 0

(34)

M. M. Bayoumi and Li Mo

112

'-1 • -1 • -1 A(k.q )y(k) - B(k.q )u(k) + B(k.q )r(k)(t(q-1)I + Q(k.q-1 ))z(k) (42)

Hence: 9(k) - P(k-1 )P- 1 (0)9(0) + P(k-1)

k

r w(k) j-1 k

- P(k-1)P- 1 (0)0(O) + kP(k-1) ~

r w(k) j-1

(35)

A(k.q-1)y(k) -

Since

( 37)

k

L!~ 1 r w(k) - 0 (Toeplitz Lemma) k

Then

B(k.q-1)u(k).

Since the •

(36)

K

In the following. the argument of Goodwin and Sin (1981) of local stability analysis for SISO system pole aSSignment algorithm is used. Note that the system identifier prediction error is Es -

(38)

j-1

k!m0(k)

=

0

-1

coefficient of the polynomial matrices A(k.q ). B(k.q-1).P(k.q-1) and ~(k.q-1) are all bounded and since Q(k.q-1) goes to zero and t(q-1)I is an exponentially stable matrix that is independent of k. then x(k) will grow no faster than linearly with the 11 norm of Es(k). Using lemma 6.2.1 of Goodwin and Sin (1984). we can conclude that the memory vector of the system identifier is bounded. This means that the input u(k) and output y(k) are bounded all the time and that the 11 norm of the Es(k) goes to zero as time increases.

This means that this controller design procedure will give the correct controller parameters in the steady state and that the controller parameters will always be bounded regardless of the existence of any temporary common factors in the estimated •

-1

'-1

system parameters A(k.q ) and B(k.q ). On the other hand. the computational burden is drastically reduced.

In summary. the system will be globally stable if the order of the system is known and if the system is persistently excited. This is to ensure that the controller parameters are bounded and that the pole assignment equation is solved correctly in the steady state. SIMULATION EXAMPLE

The design of M(q-1) will be exactly the same as that obtained by solving Diophantine equation. The only difference is that we need to specify one of the non-zero parameters in each of the bi(q

-1

).

i - 1 ..... n . The order of M(q-1) can be r underestimated. This will give an approximate decoupling controller. But the order should not be overdetermined. In this case. the polynomial -1

matrix M(q controller convergence Mo and M.M.

) can not be designed by this kind of design procedure. For the detailed analysis of this design method. see Li Bayoumi (1987).

STABILITY ANALYSIS In this section. global stability of this controller will be examined. The Diophantine equation can be put in the following form -1' -1 • -1' -1 -1 • A(k.q )D(k.q ) + B(k.q )P(k.q ) - t(q )1 + Q(k.q-1) (39) -1

where the polynomial matrix Q(k.q ) is due to the approximation introduced by the parameter estimator. If the system is persistently excited and the system order is known. then from the last section. we have Lim Q(k.q-1) - 0

(40)

K+
If the RLS or the projection estimator scheme is used as a system identifier. we then have A

k!~(k.q

_1

_1

) - A(q

)

and

•

k!mB(k.q

-1

) - B(q

-1

The system considered for simulation is

) ( 41)

Multiplying both sides of the Diophantine equation by z(k). we have

A(q

-1

)-3.0q

-1

+1.2q

-2

1 .0+2. Oq

-1

+0. 5q

-2

(43)

1.0+q B(q

-1

-1

2.0-2.0q

-1

(44)

)0.1 +q

-1

0.3+0.7q

-1

It is clear that the system poles are located at 1.51+j2.1.-1.51-j2.1.0.27+jO.46.0.27-jO.46. The system zeros are located at 0.4.0+j3.32. 4-j3.32. This is an unstable. non-minimum phase system. If we want to assign the system poles at 0.5. then the polynomial t(q -1) should be specified as 10.5q

-1

-1

.

The degree of M (q ) is 2 for perfect 1 decoupling. The system output is shown in Fig. 1. Before step 25. the magnitude of the output is large (around 100-200) and hence is not shown here in order to get better graphics. We can see that the system is perfectly decoupled in the steady state. For approximate decoupling. we simply specify

the degree of M1 (q

- 1

) to be 1.

The

system is approximately decoupled and is shown in Fig. 2. In addition. all the estimated parameters converge to their true values except the values for M1 (q

-1

) in the approximately decoupling case.

In this case. the parameters of M (q 1 to some fixed values.

-1

) converged

Adaptive Decoupling Control of MIMO System

113

CONCLUSION An adaptive decoupling controller has been presented, which is suitable for a wide class of linear MIMO systems. The controller is computationally simple and globally stable if the system is persistently excited and if the system order is known. A new controller design procedure is proposed to simplify the computational burden for involved controllers such as the one presented in this paper. A simulation example has been presented to demonstrate this controller.

5.-Or 1__

..

~

InCilO

NQnoo

• • - LllZOO I.wJ

nao

3.

" 11

I~

REFERENCES DcDermott, P.E. and D.A. Mellichamp (1986). A decoupling pole placement self-tuning controller for a class of multi variable process. Optimal Control Application and Methods, Vol. 7, 55 79. Fucks, J.J. (1980). Explicit self-tuning methods. lEE Proc. Part D, Vol. 127, No. 6, 259-264.

Goodwin, G.C. and K.S. Sin (1984). filtering, prediction and control. Hall.

Adapti ve Prentice

Johansson, R. (1987). Parameter methods of linear multivariable systems for adaptive control. IEEE, AC-32, No. 4, 303-313. Klnnaert, M., R. Hanus and J.L. Henrotte (1987). A new decoupllng precompensator for indirect adaptive control of multlvarlable linear systems. IEEE,AC-32, No. 5, 455-459. Mo, L. and M.M. Bayouml. Stable regulation and control of a linear, finite order system. Submitted to the lEE Proceeding Part D for publication .

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e.w-.. . . .

: . 25Q:

~-L..........L-'---'-'-.........-'--'-'-.........-'--'~................-'--'-'

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S TEPS (FIG.

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