A new robust model reference control for a class of multivariable unknown plants

TA411

-

9:40

Proceedings Honolulu, Hawall December 1990 on of Decision the 29th and Conference Control A New Robust Model Reference Control for a Class of Multivariable Unknown Plants

Chiang

-

Ju Chien and

Li

-

Chen

Fu

Department of Electrical Engineering National Taiwan University, Taipei, Taiwan, R.O.C.

ABSTRACT

Motivated by the recent works [2] [14], a new robust model reference control (MRC) scheme for a class of multivariable unknown plants is presented in this paper. The controller is devised using the concept of variable structure design which prevails in the robust control context. Such a new scheme solves the model reference adaptive control (MRAC) problem for a multivariable plant of interest subject t o exactly the same conditions but with better performance. It is shown that the global stability of the overall system is achieved and the tracking errors will converges to a residual set whose size can be directly related to the size of unmodeled dynamics and output disturbances explicitly. Furthermore, in the absence of unmodeled dynamics and output disturbances, the tracking error can be driven to zero in finite time.

1. Introduction :

With the advance in designing robust adaptive controllers for single-input single-output (SISO) uncertain dynamical systems, a multi-input multi- output (MIMO) model reference adaptive control

(MRAC) scheme has been established [l] [2]. Since the parametrization issue was solved, the research problems will be to focus on the design of the controller and the adaptation law for robust MRAC of MIMO plants. The object is achieved in the SISO case, although the control performance are different from one to the other, by using the concept of persistency of excitation, e.g. [3] [4] [5] or by modifying the adaptation law, e.g. [6] [7] [8] [9] [lo]. Also, a general framework is proposed in [11] to analyze a wide class of robust adaptive laws. The methods described in [ll] are further transformed into work in dealing with the robust MRAC of MIMO plants.

Recently, some researches are interested in the controller design with variable structure concept for either SISO plants [12] [13] [14] [15] or MIMO plants 1161. In this paper, we propose a new model reference control (MRC) scheme using variable structure design for a class of MIMO plants. This is an outgrowth of [14] and [15] which use the concept of variable structure adaptation and give 'an improvement in transient response and convergence property. A modified version by fixing the control parameters at some constant

values is used to control a linear fast time-varying unknown plant [17]. Motivated by the researches above, a combined technique is developed for the MIMO model reference control. It can be shown that the well behaved transient performance still holds with all closed loop signals remaining uniformly bounded. Also note that the complexity in a standard MIMO MRAC design is reduced since only a simple control law is used.

The paper is organized as follows: in section 2, we give a detailed problem formulation and the control structure for an MIMO MRC scheme. The resulting error model and the robust controller for a class of plants are presented in section 3. Section

4

gives a simulation to demonstrate the effect of the robust controller. Finally, a conclusion is made in section 5.

2. Problem Formulation and Controller Design :

In this paper, we use the concept of modified left interactor (MLI) matrix and modified right interactor (MRI) matrix [2] for the parametrization of a MIMO plant PO( s).

2.1 Sustem Descrivtion :

Consider an MIMO linear time-invariant plant with N inputs and N outputs described by the following transfer matrix

Pu(s)

=

PO(S)[~

+

PAPI(s)]

+

pAp2(~) Po(s) = Z,(s)R,-Ys) (2.1) where Po(s) represents the nominal plant transfer matrix and pAPl(s) pAPz(s) are multiplicative and additive unmodeled dynamics respectively with some p

>_ 0. The control objective is to design a control law up(t) such that the output yp(t) of the plant tracks the output ym(t) of a linear time-invariant reference model, i.e. ym = Wm(s) ref; where Wm(s) is a stable strictly proper rational matrix and ref (t) is a uniformly bounded reference input signal vector. To make the problem more tractable, several assumptions on the plant and the unmodeled dynamics are made in the following:

( A l ) The MLI ( MRI ) matrix $(s) ( ()s(: ) of

Po(s)

is known.

(A2) An upper bound U on the observability index of fl-1(s)(~(s)P0(s) ( resp. f;1(s)Po(s)t:(s) ) is known.

=

re

>

0 ( r a p . Km

K

=

rr

>

0) is positive definite is known.

(A4)

Po(s)

is nonsingular and has stable zeros. (As) The unmodeled dynamics APl(s) and AP2(s) are stable proper and strictly proper transfer matrices respectively. Furthermore, there exists 7

>

0 such that IlAPl(s)ll, and lIP~-~(s)APz(s)ll,

5 Y,

where

IIH(s)ll,

= i?(H(jw)) and

a(.)

denotes the largest singular value of the argument matrix.

-1 121 : The MIMO linear timeinvariant plant yp( s) =

PO(

s) up( s) can be represented

as

rP

r

YP(4 = fe(s)(Ey(s))-lz(s)

4 s )

= P e ( s ) d s ) (2.2) U P ( 4 = fJ~)t:(S)V(s) (2.3)

or YP(4 = Pr(s)V(4

where Pe(S) = fe-l(s)(?(s)Po(s) ( resp. P,(s) = f;'(S) Po(s)(:(s) ) is an N x N transfer matrix whose MLI ( resp. MRI ) matrix is fe(s)Z( resp. f's)Z) ; fe(s) ( resp. f&s) ) is an arbitrary Hurwitz polynomial of degree de ( r a p . d , ) and de ( resp. d , ) 2 the maximum degree of the elements of <?(s) ( resp.

tF(s)

). 000 ,Remark 2.2 : In this paper, the plant

Po(s)'

will in general be assumed to have a diagonal MLI or MRI matrix which can be specified with only the knowledge

Remark 2.3 : The assumptions ( A l )

-

(A4) are equi- valent to the relative degree, upper bound for the order

of

transfer function, the sign of high frequency gain and minimum phase assumptions in model reference control

scheme for SISO plant. OOO

Furthermore, the plant is assumed to be operated subject to bounded output disturbances

CO

E Rnxi (which are usually modeled in a real system

as

measurement noise), i.e.

iP

= yp

+

(0.

$W:

of relative degree of each matrix entry. 000

By the way of parametrization described in (2.2) and (2.3), we can use the standard MRC structure for

MIMO

plants in a way similar to that in an SISO case. In this section, the MRC structure based on MRI matrix formulation is discussed and that based on MLI matrix can be obtained similarly as in as [2]. We now express the system equation in the following form:

i P ( 4 = G,(s)v(s)

+

C O ( 4

U P ( S ) = f;'(s)l:(s)v(s) (2.4) where

Gr(s) = P,(s)[Z

+

/Laf3r(s)l

+

PAPz~(s)

Pr(s) = ~ , - ~ ( S ) P O ( S ) ~ : ( S ) = z p ~ s ) ~ j k ( s ) APir(s) =

(ty(

s))-'APi(s)(:( S)

APzr(s) = i;'(s)AP2(s)<;(s) (2.5) and 2 (s) and R As) are right coprime polynomial matrices

of

dimension N x N and R

(s)

is column proper. The standard MRC structure will be used and the control input up for the plant is given as:

D = C's,Bl)N;(s)v

+

D , ( s , h , & ) N ~ ( s ) i p

+

KO ref+ vp where

Pr

P

p.T %P =

f;'(4t:(s)~

(2.6) CT(s,Ri) = 8 i l ~ ~ - ~

+

...

+

R1,v-i D r ( S , h , f l ~ ) = 8 2 1 ~ ' - ~

+

...

+

&,U-i

+

BjNT(s)

ol

=

[ell

,...,

B ~ , v - , ] ~

h

=

[hi,...

7

h,.-l]

(2.7) T

and N's) = diag{n's)}; n's) is an arbitrary monic stable polynomial of degree U-1 and KO is a cdnstant matrix. The design of the control input is similar to that in [2] but with a difference in the additional term vp to be specified later. Under this structure and with the condition of p = 0, (0 = 0, and vp = 0, it can be easily shown that there exist constant matrices

&, &,

6

and IG such that the closed loop transfer function matrix matches the reference model

Wm(s).

It can be easily observed that a suitable choice for the reference model is simply fr-l(s)I whose strictly positive real (SPR) property can be easily checked from the order of the polynomial f's). For an analysis on error models and Lyapunov design in the next section, a definition of the concept of " generalized relative degree I' for an

MIMO plant is naturally given as follows: Definition 2.1 :

Consider an MIMO linear time-invariant plant Po(s) to be parametrized

as

either (2.2) or (2.3) so that a simple MLI matrix f e ( S ) Z or MRI matrix

f&s)I

is obtained. Then the plant

Po(s)

is defined

as

a system of generalized relative degree n if f&s) or

f's)

is an n-th

3. Robust MRC Design For MIMO Plonts with

Generalized Relative Degree One :

In this section, the MRI matrix f's)Z is considered, where fJs) is a first order Hurwitz polynomial, such that the reference model fr-l(s)Zis SPR.

3.1 Error Model :

It can be ?asily verified that the state-space representation of (2.4) (2.5) can be given with a minimal realization (Ap, B,, Cp) of the transfer matrix

order polynomial. 000

P's)

as

follows:

Xp = A ~ X ~

+

~ , v

+

BppCi ;

X,

E R~

where p ( i represents the effect of unmodeled dynamics satisfying:

W = A

X c = ACXc

+

Bcv

with (Ac, Be, Cc, De) being a minimal realization of the transfer matrix APlr(s)

+

Po-l(s)APzr(s) which is proper stable from assumptions ( A 4 ) and ( A 5 ) so that Ac is Hurwitz. Also define the signal vector w and b with dimension (2n x v) as : ( i = CcXc

+

DLv

(3.2) re - 0 w2 = , " ' = w + i o (3.3) i P (0 d -wlm - B B ~ C ,

A + B B T ~

B B ; ~ wlm -W2m-

.

BCP 0 A .W2m, = S ( S ) N - I ( S ) V = ( S I - A)-'Bv t~ = S(s)N-I(s)ip =

(sZ-

A)-IBip (3.4) with S(s) = [I,

SI,

.... , S ~ - ' I ] ~ , and det(s1- A ) = nr(s) so that the control input up can be rewritten as:

1) =

sT

b

+

vp ; up = f;1(s)<y(s) v

(3.5) with

6

=

[KO,

6lT,

hT,

&IT.

Thus, a state space representation of the plant loop is given by:

+

(eT

w

+

up) BP' B

IC

ref - 0 ~ i: = Ace

+

B c ( $ T ~

+

vp)

+

Bc1pCi

+

Bc2(fl)l0 eo = yp - ym = Cce

where BCl = [B:, 0 , 0IT, Bc2(6) = [@3B:, 63BT, BTIT, and

3.2 Robust Controller Desion :

A new MRC scheme using variable structure design concept is given in this section. This method is an outgrowth from the one proposed by Fu [14], [15] for SISO MRAC scheme with unknown plant of relative degree one or two whose transient response is drastically improved. But, because of the fact that the control parameters converge to zero at a rapid rate by using variable structure adaptation, a modification which fixes the control parameters

6

at a zero vector ( or zero matrix as MIMO plants are considered ) at the very beginning has been used for the control of linear fast timevarying unknown plant successfully [17]. Furthermore, the complexity due to a lot of computation for adaptation is reduced when the control parameters are fixed. Hence we will focus on the choice of vp in (2.6) so that the overall system is guaranteed to be globally stable and the tracking error is ensured to converge to a residual set whose size is a simple function of p and of the bound on (0 when they are small. The following theorem gives the robust property with global stability and bounded tracking performance.

Theorem 3.1 :

assumption ( A l )

-

( A 5 ) with vp being specified as :

(3.9)

= 6 -

e*

is the parameter error vector.

Consider the error systenl (3.9) satisfying

VP = -I',sign

(ZO)(PO

I

W I

+

PI)

(3.10) where 20 = eo

+

CO

,

sign(&) = [sign(201), sign(&), ... , sign(Zon)lT and

1.1

denotes the two morn of a vector or induced two norm of a matrix. Let e(t) = 0 V t

>

0, then there exists p*

>

0 such that for all ,!L E (O,p*] all signals inside the closed loop system are uniformly bounded and the tracking error will converge to a residual set whose size can be written

as

an explicit class Iifunction of

P

and P where p ?

l(01

is the upper bound for output

disturbances. 000

Proof:

If the control parameter 0 is set to be zero, then $ =

-e*

and the error systein will be:

2 = Ace

+

Bc(-8*T~

+

vp)

+

BcIkCi

+

Bc2C0

eo = Cce (3.11)

where Bc2 = [0, 0 , BTIT. Furthermore, the input signal v simply becomes vp. Construct a Lyapunov function:

V(e,xc) =

'2

eTPee

+

f p x ~ P e x c (3.12) where Pe = P:

>

0 satisfies

PeAc

+

A:Pe = -2Qe ; PeBcKr= C: (3.13) due

to

the SPR property of the transfer matrix Cc(sI

-

,4,)-1BcI(T for some &e

>

0 and Pc

>

0 satisfying

PCAE

+

ATPc = -2Q< (3.14) for some Qc

>

0. Then the time derivative of Valong the trajectories

of

(3.2) and (3.11) subject to vp (3.10) can be computed

as

:

V = -eTQee

+

( 2 0 - CO)~K;~(-B*~W

+

vp)

16.11

5

lwml

+ IGl

le1

+

IC01

5

k14

+

k1lsIel +p (3.24) for some constants k14 and kls

>

0. Hence,

~5

-

Pel

-

kisp - k19,U”~)

I

e12 - p [ qci - k17p1/3) 1zCl2 +

I

, U h

+

kJ)

le1

+

&91x<l

+

h o p (3.25) with suitable constants k16 IV

ho,

where we use the fact

that

pab

5

$

(p4i3&

+

p2I3b2)

j

qei

>

(h6p*

+

kiTp**2 3)

(3.26) Hence, there exists ,U*

>

0 such that

T T

/

+

fie PeBclCi

+

e PeB&

-

P:

Q& +

P$PC,B<~P Let qel, qez, qcl, qc2 be strictly positive such that :

(3.15) -!j qC1> kw*1/3 (3.27) and thus v 5 - ~ q e l l e l 2 - 3 p q < l I ~ ~ l 2 + 1 ma+,p)kz1Iel

+

/&Qlz<l

f b o p (3.28)

x

= (eT, p 1 / 2 q ~ ) ~ (3.29)

>

0 for some

h~

>

0. We now define the vector X as

so that we can see that there exist constants a1 N

such that a11X12

5

V 5

*1xp

v<

-@lX(2

+

~ ~ 1 x 1

+

hop (3.30) where 1 1 1 1

01 = min (3 Pel, 3 P<I)

(3.31) (3.32) az = man

(2

p e 2 , ~ pcz)

with and

Pel1

5

Pe

5

P e d and pciI

5

Pc

5

p c d

1 1

0 3 = min

(2

qel,

2

q ~ )

0 4 = ma3 (,U’/~~IQ, ma2: (p,p)k21)

(3.33) Consequently, the ultimate boundedness property [lS] [19] of the overall system can be concluded now for all p

<

p*. Furthermore, the ultimate bound of e and, hence,

eo

can be shown to be a class

K

function of the following form as a result of [ 181

for some 90

>

0. This completes our proof. Corollam 3.2 :

AAA Consider the system

as

described in (2.1) but in the

that the relation between

w

and e is absence of unmodeled dynamics and output dis-

turbances. Then the controller (3.10) will drive the

where output error to zero in finite time with all closed loop

w =

Wm

+

Ge (3.22)

T T T T T signals remaining uniformly bounded. 000

W, = [ref

,

wml

,

wmz

,

Ym]

Proof:

(3.23) In the absence of unmodeled dynamics and output disturbance, i.e. ,p = 0 and

CO

= 0, the error model described in (3.11) will be modified as the following and

i: = Ace

+

B,(--~*~V

+

vp) ; eo = C,e (3.35) Hence, by a simple Lyapunov function V(e) = eTPe, P = PT

>

0, it can be easily shown by a similar procedure of the proof for Theorem 3.1 that

v(e) 5

-mlV(e) for some positive constant ml which concludes that the state error e will converge to zero at least exponentially fast. Furthermore,

e;

r;

k0

= e;f r:(CcAce

+

C,B~(-O*~W

+

vp))

= e;f I';CcA,e

+

e;f r:h'T~ri,[-szgn(eo)(Pol

wI

+

P1)

- K;le*Tw]

5 lQl1[mzler

- ( P ~ I ~ I

+ P I -

I ~ . w * I I ~ I ) ]

(3.36) for some positive constant mz. Since

le1

approaches zero at least exponentially fast and (3.20) is assumed, there exists a finite T

>

0 such that

e;f

r;

i:o _< - m3leoll (3.37) for all t

>

T and for some m3

>

0, which implies that the switching surface eo = 0 will be reached in finite

time [14]. A A A

4. Simulation

:

A 2 x 2 transfer matrix

Po(s)

is given for computer simulation.

3 1

- -

It can be easily observed that the modified right interactor matrix rT(s) can be chosen as (s+Z)I which is diagonal and the high frequency gain matrix

L-

- ,

is nonsingular and positive definite. Hence, we will choose the reference model as (s+2)-1I and the matrix

K,.

= I for our control design such that v, is

Po(s) is 3. In the following simulations, the initial conditions for

Po(s)

is assumed to be 3 and 2.5 for diagonal elements and in the task of tracking, the reference input reA(t) = 2 and reh(t) = Zsin(1) are applied.

4.1 Ideal case :

In the absence of unmodeled dynamics and output disturbance, the control parameters

PO

and p1 are chosen as 20 and 0 respectively. Fig.1 and Fig.2 are the tracking performance for eo1 and whose finite time convergence property is observed!

4.2 Unmodeled dunamic and outvut disturbance :

Consider the multiplicative and modeled dynamics with p = 0.001

s+ 1 APl(s) =

[;i

0 -s:l]

s+4

APz(s) =

[

$

-

st]

and output disturbance

t n _I" =

c

cos(t)

1

0.1 sin(t)

J

The control parameters are now chosen as

PO

= 20 additive un-

(4.4)

(4.5)

(4.6)

and = 10 respectively. From Fig.3 and Fig.4 we can see that the output tracking performance is still acceptable for the existence of unmodeled dynamics and disturbance.

5. Conclusion:

In this paper, a new model reference control (MRC) scheme for a class of MIMO systems was presented. With moderate unmodeled dynamics and bounded output disturbances, the controller, which combines the characteristics of variable structure design [14] [15] and null adaptation process [17], not only stabilizes the overall systems but also drives the tracking error to a residual set whose size can be directly related to an explicit function of p and p. It is noteworthy, however, that such an MRC scheme solves the same problem as the MRAC schemes usually do but the former remarkably simplifies the complex computation conventionally required by the latter.

In the absence of unmodeled dynamics and output disturbances, the output error will be driven to zero in finite time (modulo some chattering afterwards). As indicated in [14] [15], this convergence property is a remedy to the undesirable slow convergence usually appear in traditional MRAC schemes. From the simulation results, the drastic improvement for convergence performance is observed.

-I

-

0 0.5 i

2

1.5 Fig.2

h

2

6. Rekrences:

[l] Elliott, H. and W.A. Wolovich,

-I

I' A Parameter

Adaptive Control Structure for Linear Multivariable Systems II, IEEE, on Automatic Control, AG.27, (2),

[2] Tao, G. and P.A. Ioannou, 'I Robust Model

Reference Adaptive Control for Multivariable Plants

'I, Report 86

-

11

-

01, EE systems, USC (1986).

(31 Chen, Z.J. and P.A. Cook, " Robustness of Model

Reference Adaptive Control Systems with IJnmodeled Dynamics 'I, Int. j. Control, 39, ( l ) ,

Slow Drift Instability in Model Reference Adaptive Systems

-

An Averaging Analysis 'I, Int. j. Control, 45, (2), 503-527 (1987).

[5] Koust, R.L. and B. Friedlander, 'I Robust Adaptive

Control: Conditions for Global Stability It, IEEE, on

Automatic Control, AG30, (7), 610-624 (1985). [6] Peterson, B.B. and K.S. Narendra, 'I Bounded Error

Adaptive Control 'I, IEEE, on Automatic Control,

[7] Kreisselmeier, G. and K.S. Narendra, I' Stable Model

Reference Adaptive Control in the Presence of Bounded Disturbances 'I, IEEE, on Automatic

Control A G 2 7 , (6), 1161-1168 (1982).

[8] Kreisselmeier, G. and B.D.O. Anderson, Robust Model Reference Control 'I, IEEE, on Automatic

Control, A G 3 1 , (2), 127-133 (1986).

[9] Narendra, K.S. and A.M. Annaswamy, 'I A New

Adaptive Law for Robust Adaptation without Persistent Excitation 'I, IEEE, on Automatic

Control, A G 3 2 , (2), pp 134-145 (1987).

[lo] Ioannou. P.A. and K.S. Tsakalis, 'I A Robust

Direct Adaptive Control 'I, IEEE, on Automatic

Control, AC-31, ( l l ) , 1033-1043 (1986).

Theory and Design of Robust Direct and Indirect Adaptive Control Schemes", Int. j. Control, 47, (3), 775-813 (1988). 340-351 (1982).

201-214 (1984).

[4] Fu, L.C. and S.S. Sastry,

AC-27, (6), 1161-1168 (1982).

[ll] Ioannou, P.A. and J. Sun,

-'I

Fig.

4

015

4

115

t

[12] HSU, L. and R. R. Costa, Variable Structure Model Reference Adaptive Control Using Only Input and Output Measurements : Part 1 'I, Int. j. Control,

49, (2), 339-416 (1989).

(131 Narendra, K. S. and J. D. Boskovic', I' A Combined

Direct, Indirect and Variable Structure Method For Robust Adaptive Control II, Proc. of American

Control Conference, 543-548, (1990).

[14] Fu, L.C. 'I A Robust Model Reference Adaptive

Control Using Variable Structure Adaptation for

a

Class of Plants 'I, Proc. of American Control

Conference, 2291-2296 (1989).

[15] Fu, L.C. 'I Robust Model Reference Adaptive

Control Using Variable Structure Adaptation for Plants with Relative Degree Two II, Proc. of

American Control Conference (1990).

[IS] Tao, G. and P.A. Ioannou, Ii A MRAC for Multi-

variable Plants with Zero Residual Tracking Error

",

Proc. of Conference on Decision and Control, 1597- 1600, (1989).

[I71 Fu, L.C. and C.J. Chien, 'I A New Approach for

Model Reference Adaptive Control for Fast Time Varying Unknown Plants with Relative Degree One

'I, Proc of American Control Conference (1990).

[18] Bodson, M. and S.S. Sastry, 'I Small Signal

1/0

Stability of Nonlinear Control Systems: Application to the Robustness of

a

MRAC Scheme 'I,

Memorandum, No. UCB/ERLM84/70, Electronics Research Laboratory, University of California, Berkeley (1984)

[19] Yoshizawa, T. Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solution 'I,