Set-membership identification for continuous-time systems with nonparametric uncertainties and disturbances

I

WA3

-

I t 3 0

P d l n g r on of Doclrlon thr 32nd Conference and Control

San Antonlo, 10x88 Docember 1BW

Set -Membership Identification for Continuous- Time Systems

with Nonparametric Uncertainties and Disturbances

Fu-Ming Lee',

I-Kong Fong',

and

Li-Chen

Fu'>~

Room 205, Department

of

Electrical Engineering'

Department

of

Computer Science and Information Engineering2

National Taiwan University, Taipei, Taiwan,

10617, Republic

of

China

Abstract

An on-line set-membership identification problem is formal- ly formulated for linear time-invariant continuous-time systems which have bounded disturbances as well as nonparametric un- certainties. Based on the formulation, an efficient ellipsoidal- bounding al orithm is proposed to estimate the parameter set of the system t o m the available input-output timed data.

1

Introduction

"Set-membership identification" is referred to as a class of techniques for estimating parameters of a system under a prior- i information that constrains the solution to certain set. Many theoretical and applied results have been developed in this con- text, e.g., [I] consider the systems with bounded disturbances, 4 consider the systems with nonparametric uncertainties, and

12]

PI

consider the systems with both nonparametric uncertain- ties and disturbances. However, all the results mentioned above are purely for discrete-time systems.

In this paper, we formally formulate an on-line set-membership identification problem for linear time-invariant single-input single- output continuous-time systems with nonparametric uncertain- ties and disturbances. The goals of this problem are to on-line find a parameter set estimate which always contains the param- eter set of the true system, and to reduce monotonically the es- timation error with respect to time. To solve this problem, we use the available input-output timed data to delineate a posterior parameter set which accounts for a priori knowledge of the non- parametric uncertainty and the disturbance. Then, we derive an ellipsoidal set t o bound the intersection of the prior parameter set and the posterior parameter set. Finally, we propose an efficient algorithm to minimize the volume of the ellipsoidal-bounding set. Here, we establish some notations and definitions that will be used throughout the paper. Let

&

and Rq denote the set of non- negative real numbers and q dimensional real vectors, respective- ly. Let lzll :=

Ci

Izil, 1x12 := (E,

~ f ) ' / ~ ,

and

lzlco

:= maxi

Iz;I

denote the 1-norm, 2-norm, and m-norm, respectively, in Rq s-

pace. For any t E

&

and any signal x :

&

+ Rq, we define some norms on I as

Let A be a linear time-invariant continuous-time system with proper stable transfer matrix A(s). Let ha be the impulse re- sponse of A and define induced Lz norm on A as

Let

M

be a square matrix, then we denote the determinant of M

by det(M), the symmetric positive definiteness of

M

by M

>

0,

and the symmetric positive semidefiniteness of M by

M 2

0.

'2

Problem Formulation

The problem is t o on-line identify a model set of a system G suitable for (adaptive) robust control design by using the avail- able input-output timed data {U(T),Y(~)}:=~,

t

E

&,

which are generated by a single-input single-output system described by y = Gu

+

d, where U is an applied input, y is the observed out- put, and d is the disturbance.

The model set

M

is defined as follows:

M

:= { y = Ge,au

+

d : Ge,a E

E,

d E

V }

where

E

:= {Go.A : 0 E

so,

I I A I I

5

11

V

:= { d = AvWvv : llAvll

5

1, llvllm

5

1)

SO is known and is referred to as the prior parameter set, A V is an uncertain proper stable system, and WV is a known proper stable system. The transfer function of Ge,a is characterized by a coprime factor form as follows:

where

De(s) = sn

+

an-1sn-l

+

...

+

a0 Ne(s) = 1,s"'

+

bm-lSm-*

+

...

+

bo

0 = [ a n w l , .

. .

, a o , b,,

. . . ,

bOlT

with A ( s ) being a Hurwitz polynomial with degree deg(A)

2

n

2

m. A is an uncertain proper stable system and W is a known proper stable system.

Throughout this paper the following assumption is made. ( A l ) G E Q and d E

V.

G E

D

means that there exists an Ge.,a. E

Q

such that G =

GO.,^..

We call this specific 0' a valid parameter for representing the system. Because the valid parameter for a system may not be unique, i t may seem ambiguous as t o which valid parameter should be identified. Thus, the problem formulated here is to identify the valid parameter set 0' :=

{e*

: G = GO.,&. E

Q}

To proceed, we have t o impose some explicit assumptions concerning

0' and So.

A2: L e t 0 * = [a:-, ,...,

a & K

,...,

e]'.

F o r e a c h k = O

,...,

n -

1, the sign of u; is known and is the same for all 0' E 0'. Without loss of generality, the sign a; is assumed positive.

A3: So

:=,

(0 : (0 - d(0))TP-'(O)(O - d(0))

5

l}, where q = m

+

n

+

1, O(0) and matrix P ( 0 ) are known, and P ( 0 )

>

0.

To Summarize: The on-line set-membership identification prob-

lem is precisely formulated as follows.

Given: ( i ) A(s), W ( s ) , in, n, P(O), and d(0) for which it is

known that G E

0 ,

(ii) Wv for which it is known that d E

V ,

and jii) { u ( T ) ,

Y(T))J,~,

t. E

&.

j t n d : an on-line algorithm Atlp",o which maps the given in-

formation into an ellipsoid in

Rq,

namely,

01 91 -221 6/93/$3.00 0 1993 IEEE

88

st

:=

{0

: (0 - B ( t ) ) T P - ' ( t ) ( o - d ( t ) )

5

1) which satisfies

0'

c St, aud

5

L ( S f L ) , V t ,

2

t ,

2

0 (1) where L denotes Lebesgue measure 011 Rq. In other words, the el- lipsoid St contains the valid parameter set 0' and the estimation error L ( S t

\

0') is monotonically decreasing.

3

Main Results

We first use the available input-output timed data to delin- eate a posterior parameter set which accounts for a priori knowl- edge of the nonpararnet.ric iincertainty and the disturbance as follows.

L e m m a 1 Under assurnpfions ( A I ) and (AZ), f o r any

t E

Ro, we have 0'

c

O f whfre

x := ( W N U , -CV,yIT, ?j := F,y

and the transfer functzon of system F k 2s defined by F k ( s ) :=

s k / A ( s ) , for all k = 0 , .

.

.

,

n

Then, we derive an ellipsoidal set to bound the intersection of the posterzor parameter set Of, and the prior parameter set St,-]

which is defined i n the following Lemma.

L e m m a 2 Under assumpttons (A 1) and (AZ), zf P(t,-l)

>

0,

0'

c

St,-, :=

{Q

:

(Q

- d ( t t - l ) ) T P - ' ( t , - l ) ( O - B ( L - 1 ) )

I

1 )

h.l(t,) :=

l ' ( 4 ( T ) d T ( T )

- 2$(T)qT(T))dT

2

0 (2)

then, for any CY E Ro, 0'

c

{ S t , - ]

n

O f , }

c

where

=

{ e

:

(e

-

e(t,))Tr-l(tm

-

k ) )

I

-&,)I

The next issue is how to choose a . We propose that a should be chosen to minimize the volume of the ellipsoid O;-,,t, or, e- quivalently, det(y(ti)r(t,)). To find the minimizer a,, we need to decompose real matrix ,%[(ti) into dyadic form as follows:

e

~ 4 ( t t ) = q k ( t i ) q : ( t i ) (4)

k = l

where & = rank(M(t;)) and q k ( t i ) = x:"(t,)ek(t,), where e c ( t ; )

is an eigenvector associated with a positive eigenvalue & ( t i ) of

M ( t i ) for all IC = 1.. . .

.e.

T h e o r e m 1 Under the statemenf of Lemma 2, then d e t ( y ( t , ) r ( t , ) ) = f ( a ) d e t ( P ( t i - I ) )

where f a ) i s a rational junction of a with finite order and can

Initialization: q = m

+

n

+

1, po = 1, and A. = P ( t i - l ) . Step 1: Decompose M ( t ; ) into the form i n

(4).

be calcu

I

ated b y the following steps:

Recursion: For

k

= 1 , . .

. ,&.

Step 2: pk = ( 1

+

a q T ( t , ) A k - l q k ( t , ) ) p k - i

E n d Recursion: I f k

<

P.

increment

k

and return to Step 4.

Step 3: Ak = A k - l - ( C Y A k - l q k ( t j ) q ~ ( t i ) A k - l ) / ( l + a ~ ~ ( t i ) A k - l ~ k ( t i ) )

Thus the value of a , say C Y $ , whzch mznzmzzes d e t ( y ( t , ) r ( t , ) ) , can l e analytzcally found as C Y , = arg min,,A f ( a ) ! where A =

( 0 ) U { C Y

>

0 : $ f ( C Y ) = 0 ) .

A l g o r i t h m : (Set- A4em bersh ip Identification, A

1

Eo)

In this algorithm, the required signals

4,

q2,

7, z are construct-

ed and/or calculated on-line, and they are sampled at some time sequence { t k } i n order to calculate an estimate S, of the valid parameter set at real time t . The estimate St is calculated as follows:

Initialization: to = 0, U ( t 0 ) = U(O), and P ( t 0 ) = P ( 0 ) .

Recursion: For i = 1 , 2, 3,

. ~ .

Step 1: Sample signals 4, $, 7 , z at time t i .

Step 2: If ( 2 ) is not satisfied, let P ( t j ) = P ( t , - l ) , & t i ) =

Step 3: Find a minimizer a, as described in Theorem 1 and Step 4: Update &ti) according to (3).

Step 5: If y(tt)

5

0, go to Step 9 , else, continue. Step 6: Update P ( t , ) according to P ( t t ) = y ( t ; ) r ( t , )

Step 7 : Let t,+l = t and

d(t,-l), and go to Step 7, else, continue.

let CY = C Y , .

Step 5: Increment z and retur? to Step 1.

Step 9: If r(tl) = 0, let S , = { O ( t , ) } for all t E [ t z , CO) and then

stop. Otherwise give a massage that assumptions ( A I ) , (A2), or

(A3) are not consistent with the true system.

T h e o r e m 2 Under ass7imptions ( A l ) , (A2), and (A3), the es- timate St calculated by the algorithm A t ( z o satisfies (1).

R e m a r k : The parameter sets we have developed so far assume a coprime factor form of nonparametric uncertainty with bounded induced L z norm. This is not a necessary restriction as the sets could also have been developed for other nonparametric uncer- tainty forms and other norms, e.g., the additive form, the multi- plicative form, the induced L" norm, the

H Z

norm, or the

H"

norm.

4

Conclusion

In this paper, we formulate and solve from an ellipsoidal- bounding standpoint an on-line set-membership identification prob- lem for continuous-time systems which have nonparametric un- certainties and disturbances. A concrete on-line algorithm is specified for this problem. In spite of the nonparametric un- certainty and disturbance, this on-line algorithm guarantees that the valid parameter set for representing the true system is always contained in the parameter set estimate and the estimation error is monotonically decreasing with respect to time.

References

[I] E. Fogel and Y.F. Huaiig, "On the Value of Information in System Identification-Bounded Noise Case", Automatica,

No. 2, 19S2.

[2] R.L. Kosut, M.K. Lau, and S. Boyd, "Set-Membership Iden- tification of Systems with Parametric and Nonparametric Uncertainty", I E E E Trans. AC, No. 7, 1992.

[3] B. Wahlberg and L. Ljung, "Hard Frequency-Domain Model Error Bounds from Least-Squares Like Identification Tech- niques", I E E E Trans. AC, No. 7, 1992.

[4] R.C. Younce and C.E. Rohrs, "Identification with Nonpara-

metric Uncertainty", I E E E Trans.

AC,

No. 6 , 1992. 89