A time-domain approach to model validation - IEEE Xplore

IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. 39, NO. 5, MAY 1994

95 1

A Time-Domain Approach to Model Validation Kameshwar Poolla, Pramod Khargonekar, Fellow, ZEEE, Ashok Tikku, James Krause, Member, IEEE, and Krishan Nagpal model is consistent with the data, we can only conclude that the corresponding model is not contradicted by the given data. This does not mean that the model is a correct description of the physical system. On the other hand, if the model is not consistent with the data, then we can definitively conclude that the model is not a correct representation of the physical system. These observations are motivated by Popper’s analysis of scientific theories [23]. With this caveat, we will continue to use the entrenched term model validation instead of model I. INTRODUCTION invalidation which is perhaps more appropriate. LASSICAL system identification methods deliver a We present computable solutions to the model validation model in which the uncertainty is often characterized problem for a variety of uncertainty models ranging from in terms of noise. In other words, any mismatch between the additive dynamic modeling uncertainty to coprime-factor unmodel and real data is attributed entirely to noise. Whereas this certainty with uncertain parameter values. The uncertainty is suitable for open-loop problems like filtering, estimation, models we treat are employed extensively in robust control and prediction, it is our contention that these models are design and analysis. The solutions are given in terms of cerinappropriate for closed-loop problems like robust feedback tain convex matrix optimization problems constructed directly control. It is also desirable to develop system identification from the input-output data record and the prior information. In methods that deliver models oriented towards available robust some simple situations, we are able to give analytical solutions control methodologies. There has recently been considerable to the problem. As there is a large body of literature on convex research devoted to control-oriented system identification, and optimization problems [l], [lo], [20], [27], our solutions can many papers on robust identification and related subjects have be readily implemented. We would like to mention that convex appeared [71-[91, [121, [13l, [161-[181, 1211, 1291, [301, [321. and linear programming play key roles in related controlImagine that we are given an a priori uncertainty model for a physical plant. This could consist of a nominal plant oriented system identification problems as in [2]. Model validation should be regarded as only one ingredient model together with bounds on the unmodeled dynamics, initial condition uncertainty, and measurement noise. Having of the entire process of obtaining robust control-oriented sysconducted experiments on the physical plant, we also have tem models. Model validation is preceded by system analysis available time-domain input-output data over a finite horizon. and understanding, physical modeling, and identification. If The problem we wish to address is that of determining whether the uncertainty model is invalidated by the input-output data or not the input-output data record is consistent with the record, then it becomes necessary to revisit the identification uncertainty model of the plant. In other words, the problem is step. It may also be necessary to obtain additional data. We do to decide whether the observed data could have been produced not address these issues in this paper. The model validation reby the model for some choice of unmodeled dynamics, initial sults in this paper are also related to questions in robust paramcondition, and measurement noise satisfying the given bounds. eter identification, which have been investigated in [ 121, [ 131 and references cited there. We will not investigate these probThis is called the model validation problem. Now one should bear in mind that we can not really validate lems in this paper and leave them for future studies. Model vala model. All one can do is to say whether or not the model idation problems are very closely related to problems of failure is not invalidated. In other words, if a particular uncertainty detection. These connections will be explored elsewhere. Model validation problems have been previously addressed Manuscript received June 12, 1992; revised April 6, 1993 and July 8, 1993. Recommended by Past Associate Editor, D. S. Bemstein. This work was in some studies. Ljung [15] discusses model validation in supported in part by AFAWAFOSR under Contract F-08635-89-C-0027, by the traditional identification setting. Smith and Doyle [29] NSF Grant ECS 8957461, ECS-9001371, by the Army Research Office under address model validation problems in frequency domain with Contract DAAH04-93-G-0012, and by gifts from Rockwell International. K. Poolla and K. Nagpal are with the Department of Mechanical Engineer- structured uncertainty. They show that the resulting problem ing, University of Califomia, Berkeley, CA 94720 USA. can be converted into a structured singular value ( p ) type A. Tikku is with the Department of Electrical Engineering, University of problem. Califomia, Berkeley, CA 94720 USA. The remainder of the paper is organized as follows. In P. Khargonekar is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122 USA. Section I1 we describe a general formulation for controlJ. Krause is with the Systems and Research Center, Honeywell Inc., oriented model validation problems. In Section I11 we establish Minneapolis, MN 55418 USA. IEEE Log Number 92 16626. notation, and in Section IV we present certain extension

Abstruct-In this paper we offer a novel approach to controloriented model validation problems. The problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input-output data. Our papproach directly uses time-domain input-output data to validate uncertainty models. The algorithms we develop are computationallytractable and reduce to (generally non-differentiable) convex feasibility problems or to linear programming problems. In special cases, we give analytical solutions to these problems.

C

0018-9286/94$04.00 0 1994 IEEE

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theorems that are the basis of our results on model validation. These extension theorems are of independent interest and are. closely related to classical interpolation results as in [14]. Section V contains the principal results of this paper. Here we present algorithms for time-domain model validation for a variety of uncertainty models. Finally, in Section VI we draw conclusions and discuss open issues in this research area. A preliminary version of this paper appears in [22]. 11. PROBLEM FORMULATION Much of control theory is predicated upon mathematical models for physical systems. These models describe how inputs, states, and outputs are related. It is well recognized that one rarely, if ever, has an exact and complete mathematical model of a physical system. Thus, along with a mathematical model, one should also explicitly describe the uncertainty which represents the possible mismatch between the model and the physical system. This uncertainty is both in the signals and the systems. Thus, a complete model must include a nominal model together with descriptions of the signal and the system uncertainty. We will refer to such models as uncertainty models. Such uncertainty models are the starting point for robust control. Model validation problems in the context of robust multivariable control in a frequency domain setting have been studied in depth by Smith and Doyle [28], [29]. Thus, an uncertainty model has four components: a nominal model MO, system uncertainty A, inputs and outputs U , y, and uncertain signal d. A very general class of models can be described by

model. Typically, A,D will be taken to be "balls" of systems and signals in appropriate functions spaces. The signal d includes measurement noise as well as disturbances. We will deal with several specific examples of uncertainty models later in this paper. 2) It should be noted that we can only invalidate models. That is, if the input-output data is consistent with the uncertainty model, we can only conclude that the uncertainty model has not been falsified by the measurements. This does not mean that the uncertain model is a correct description of the physical system. Future measurements may very well invalidate the uncertainty model. 3) In the event we find that the uncertainty model is invalidated by the given input-output data, it becomes necessary to produce a new uncertainty model. This can be done either by adjusting the nominal model MOor by enlarging the sets A,D, or by both. In this paper, we will not treat this problem and will leave it for future work. 111. PRELIMINARIES In this section we establish notation and describe some results on generalized interpolation that we shall use subsequently. For a vector x E R" let

1141; = x'x,

l1~llcc=

mzvlx(i)l

.I:[

For a matrix M E R P X m , let F ( M ) = [X,,,(M'M)]'/z (2.1) denote its largest singular value. For a sequence of vectors U = {u0, u1,.. . , U Z - ~ E R"}, let U E RmzX' denote the Here f will be some function that describes how the modeling associated lower block Toeplitz matrix defined as uncertainty is intertwined with the nominal model. Both MO 0 0 ... 0 U0 and A are taken to be causal systems. To complete the U0 0 ... U 1 uncertainty model description, it is necessary to characterize 0 U = [ : : : ... U 0 the uncertainties A and d. Given sets A,D, it will be assumed Ul-1 Ul-2 Ul-3 ... that A E A and d E D. Thus, the uncertainty model is completely specified by f. MO,A and D through (2.1). In this Let S" denote the set of one-sided sequences with elements paper we shall be exclusively concerned with discrete-time in R". Let z denote the shift operator with action systems. z : S" + S" : ( U o , 211,. . .) --+ (0, U O , . . .). We will assume throughout the paper that the system initial condition is zero. Define the k-step truncation operator 7rk by The general model validation problem can now be described as follows: K k : S m + S m : ( U ( ) , u 1 , " . ) ~ ( U 0 7U 1 7 " ' 7 U k - 1 j 0, o,'.'). ?/ = f

A)

j.

Given input-output measurements U k , yk, k = 0, 1,. . . , Z - 1, the nominal model MO,together with the sets A and D, do there exist A E A and d E D such that the uncertainty model (2.1) is satisfied for IC = 0, 1,. . . , 1 - l? If the answer is negative, then the uncertainty model is said to be invalidated. Otherwise, the uncertainty model is said to be not invalidated.

and let j t k = I - 7rk where I denotes the identity operator on S". An input-output linear system js a linear operator H : S" + S P . This operator will be called causal if for all wl, vz E S"

At this point, a few remarks are in order: 1) The sets A and D capture the mismatch between the nominal model and the physical system. Their sizes reflect the model builder's estimate of the possible discrepancy between the unknown system and the nominal

and will be called time-invariant if it commutes with the shift-operator, i.e.,

q'u1

= 'rk'u2 implies 7 r k H ( V l ) = q H ( w 2 )

H z = zH. Let

Zp

be the Hilbert space of one-sided square-summable

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sequences

if and only if

Y'Y 5 y2u'u

Similarly, 1: 1:

=

I

U

denotes the Banach space of bounded sequences

= (Uo,

Ul,..,):

uk E R", I l ~ l l o o=

s p l l m

< m}.

As usual, ly, 1: can be thought of as subspaces of S". An input-output linear system is said to be 12 stable if H : 17 4 1; and

and is said to be 1,

stable if H : 1:

+

where Y and U are the associated block Toeplitz matrices formed from the sequences U and y, respectively. Remark 4.3: The single-input, single-output case of the above theorem (m = p = 1) has been employed by [34] in the context of control-oriented system identification. In this special case, the result is an immediate consequence of the classical Caratheodory theorem. The multivariable case is significantly more complex and has been proved using a number of different methods [4], [6], [24]. Remark 4.4: In the event that uo # 0, (4.2) may be written as

z[Y(u'U)-$5 y. It is in this form that we shall employ Theorem (4.1). We now present the following extension theorem for linear time-varying operators with the induced 2-norm. Theorem 4.5: Given sequences U = (710, u1, . . . ,ul-1 E R"} and y = {yo, y1,. . ylPl E RP} there exists a stable, causal, linear, time-varying operator A with

1L and

+

Recall that if H is linear shift invariant, then Hi2 coincides with the X, norm of the transfer function of H. IV. EXTENSION THEOREMS

. . . , ulP1 E R"} and y = when does there exist a stable, causal operator A with IlAll I. 1 and such that {yo,

y1,...,yl-l

,

U

=

{ U O , ul,

E RP}

A(U0, U17 .. . Ul-1,

*, *, . ..)

= (Yo, y1,. .. , Pl-1,

*, *, .. .).

Note that the inputs ui and the outputs yi are allowed to be vectors. We shall address this problem for both the induced 2-norm and the induced oo-norm and for both linear time-invariant and linear time-varying operators A. We first consider an extension problem for linear timeinvariant operators with the induced 2-norm. The following result is due to [5] and may also be found in [4l, 161, WI. Theorem 4.1: Given sequences U = {uo, u1, . . . , ul-1 E R"} and y = {yo, y1,.. . , yl-1 E R P } there exists a stable, causal, linear, time-invariant operator A with

llAlli2

*, *,-..I

I -7

= (yo,

Yl,...,Yl-l,

*, *,... )

if and only if IlnYllz

I Yll7QcU112 for

= 1,.. . ,1.

(4.6)

Proofi See the Appendix. 0 Remark4.7: Since condition (4.6) is necessary for any causal extension, it is evident that the above result holds for nonlinear operators also. We now turn our attention to extension theorems for operators A with the induced-oo norm for which we have the following two results. Theorem4.8: Given sequences U = { U O , u ~ , . . . , u ~ -EI R"} and y = {yo, yl, - . . , y l - 1 E R P } there exists a stable, causal, linear, time-invariant operator A with

l l ~ l l i c u5 7 and such that

quo, u l , . . . ,U z - l , *, *, . . .) = (yo, yl,.. . ,yl-l, *, *, . . .) if and only if the following linear programming problem denoted by LP(u, y, y) is feasible: Does there exist a matrix xi E R p x m l such that

[;I

51

x=

20,

25

jl[i] .[il I :-

r I

IlAlli2 5 7

and such that

,

and such that A ( U 0 , Ul,...,UZ-l,

In this section, we treat certain extension problems in which we are given some partial input-output data and wish to know the minimum norm causal operator that could produce this data. The classical Caratheodory [14] problem is an instance of these extension problems. These extension problems will play a central role in our results on control-oriented model validation. Results closely related to these extension theorems may be found in [26]. More precisely, we shall be concerned with the following problem. Given sequences

(4.2)

[XI

52

53

x4

-I x5]1 I -I

I

I

-I

0

0 1 U = [O 0

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954

where b = [yl-l ... y1 yo] and U is the associated Toeplitz matrix formed from the sequence U? Proof: Let (Ho, H I ,.. -) be the impulse response matrix of A and define

... HI HO].

H =

It is evident that the requisite operator A exists if and only if the following least-absolute-deviation problem is feasible

HU = b,

llHlliooI Y.

+

+

and such that

*,

*,.e.)

= (YO, Y I , - . - , Y ~ -*, I , *,.*.I

if and only if Ilrkyllcc

5 'YllRkUllm for /C

= 1, ' . 7 1.

(4.1 1)

Remark 4.12: Since condition (4.1 1) is necessary for any causal extension, it is evident that the above result holds for nonlinear operators also.

v.

i

0

Rp and lld;ll, 5 71) (5.6) which define bounded energy and bounded magnitude disturbance signals respectively. Throughout this section, we shall consider the situation where on applying the input sequence uexpt = E R" to the physical system, we ( U O , U I , . . . , U ~ -U ~;) ; observe the noisy output sequence yexpt = (yo, y1, . . . ,yl-1); y; E Rp. Also, without loss of generality we assume uo # 0, i.e., the input-output experiment is conducted immediately. We first consider a simple instance of model validation. The uncertainty model we consider is that of additive dynamic uncertainty with perfect measurements. More precisely, we hypothesize an uncertainty model of the form : d; E

ALTI-2 = {linear time-invariant operators A with

llAIli2

5 1) (5.1)

ii = (Go,

1)

($07

61,.

$1,'.

IIAllim

5 (5.2)

3 Y ( @ 0 ) 45] y where U and Y are the associated block Toeplitz matrices formed from ii and y respectively. 2) Suppose A = d ~ ~ v - Then, 2 . the above uncertainty model is not invalidated by the observed input-output data if and only if

I yllrkq2 for = 1 , . . . 1. 3

3) Suppose A = ALTI-w. Then, the above uncertainty model is not invalidated by the observed input-output data if and only if the linear program

I 1) (5.3)

. i $1-1) = yexpt - rlMo(Uexpt)-

A ~ ~ men, ~ - the~ above . uncertainty model is not invalidated by the observed input-output data if and only if

ALTV -2 = {linear time-varying operators A with llAll;z

. . , i i l - l ) = rlW(Uexpt)

suppose A =

llrkyll2

A with

(5.8)

Here W is a stable and stably invertible weighting function that captures the fidelity of the nominal model MO at various frequencies. We have the following theorem. Theorem 5.9: Consider the uncertainty model (5.8). Suppose the applied input is uexpt = ( U O , u ~ , - . - , uU;L ~ € ); R" with uo # 0 and the observed output is yeXpt = (yo, yl,.. . ,y ~ - ~ y);; E RP. Define the sequences

5=

In this section we present our principal results on model Validation. We will treat progressively more complex validation problems. In each case, the validation problem reduces to a convex or linear feasibility programming problem which may be solved efficiently by one of several available algorithms. We first define some model uncertainty sets that we shall consider in this section.

1

(5.5)

D, = { d = (do, d l , . .. , dl-1)

MAINRESULTS

ALTI-w = {linear time-invariant

1-1

y = Mou+ AWu with A E A.

IlAlliCo I Y

i , . * . , ~ - i ,

The results in this section apply to any signal uncertainty set D E S P that is convex. Of particular interest are the following signal uncertainty sets

D2 = d = (do, d l , . . . , dl-1) : d; E Rp and C d i d ; 5 r!

This problem can be readily converted to the linear programming problem LP(u, y, 7 ) as in [3], proving the claim. 0 Remark 4.9: Observe that LP(u, y, 7 )is a linear programming problem in 5pmZ variables with 2pmZ pl equality constraints. It is straightforward to verify that the special structure of this problem allows it to be reduced to p decoupled linear programming problems in 5mZ variables with 2mZ 1 equality constraints. Essentially, these p linear programming problems may be viewed as solving the extension problem separately for each of the p output channels of the requisite operator A. At any rate LP(u, y, 7 ) is readily solvable using any of a number of available techniques. The following result is from [ 111. Theorem 4.10: Given sequences U = { U O , ~ 1 . ., . ,'111-1 E R") and y = {yo, 91,.. . ,ylW1E RP) there exists a stable, causal, linear, time-varying operator A with

A(uo, ~

ALTV-~ = {linear time-varying operators A with IlA(li, 5 1). (5.4)

is feasible.

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4) Suppose A = A L T V - ~Then, . the above uncertainty model is not invalidated by the observed input-output data if and only if

Ilmdlllm 5 r l l n G l l m

for IC = 1 , . . . , l .

Proof: Observe that the uncertainty model (5.8) is not invalidated if and only if there exists a A E A such that

A(G,

*,

*,.e.)

= (Y,

*,

*,..e).

We may now invoke the extension theorems (4.1) through (4.10) to establish our result in each of the four cases above.O Remark 5.20: Analogous results can easily be derived for uncertainty models with multiplicative or coprime-factor uncertainty. We now treat model invalidation problems with both measurement noise and additive dynamic modeling errors. Theorem 5.11: Consider the uncertainty model y = Mou

+ AWu + d with A E A, d E D

(5.12)

where D is a convex subset of S P . Suppose the applied input is ueXpt= ( u g l u l , - . . , u l - l ) ; U ; E R" with uo # 0 and the observed output is yexpt = (yo, y1,. . . , ~ l - ~ )y; : E RP. Define the sequences

Y = (YO,

k , . . . , Y l - ~ ) = ~ e x p t-xzMo(uexpt).

Suppose A = A L T I - ~Then, . the above uncertainty model is not invalidated by the observed input-output data if and only if the following convex feasibility problem is solvable: Does there exist q = (40, ql,-..,ql-l) E TID,q; E R P such that

data if and only if the following convex feasibility problem is solvable: Does there exist q = ( 4 0 , q l , - . . , q l - l ) E mD, q; E RP such that

llwc - q11m F 7117rkqlm

for

= 1,.. . , I

Proof: Observe that the uncertainty model (5.12) is not invalidated if and only if there exists a A E A such that

*, . . .) = (Y - 4, *, *, . . .)

A(&, *,

for some q E T ~ DWe . may now invoke the extension theorems (4.1) through (4.10) to establish our result in each of the four cases above. 0 Remark 5.13: Analogous results can easily be derived for uncertainty models with multiplicative or coprime-factor uncertainty. For our final result, we consider a model invalidation problem with measurement noise, unstructured uncertainty, and parametric uncertainty. We specialize to single-input, single-output systems to avoid cumbersome notation. Let t 5 1 and let @ ( z )= 1 + 41z-l . . . + +tz-t be a (given) stable polynomial in 2 - l . Next, define the stable linear time-invariant operators

+

D(z)=

N(z) =

1

a0

+ biz-' + . . . + btz-t a.(.)

+ a1z-1 + . . . + a t . K t

Define bo = 1 and let

F [ ( p - Q)(@c)-;] 5y

c,

where Y , Q are the associated Toeplitz matrices formed from the sequences of vectors G,6, q respectively? Suppose A = d ~ ~ v - Then, 2. the above uncertainty model is not invalidated by the observed input-output data if and only if the following convex feasibility problem is solvable: Does there exist q = (qol ql,...,ql-l) E T ~ Dqi, E R P such that

llvdY - 41112 i Y l I ~ k G l I Zfor IC = 1,..

. I 1

Suppose A = A L T I - ~Then, . the above uncertainty model is not invalidated by the observed input-output data if and only if the following convex feasibility problem is solvable:

Does there exist q = (qo, q1,...,qlPl) E T ~ Dqi, E R P such that

LP(2, Y - 4 , Y ) is feasible? Suppose A = A L T V - ~ . Then, the above uncertainty model is not invalidated by the observed input-output

I

Consider now the uncertainty model described by y =(D

+ AlW,)-'[(N + AzWn)u+ d]

with A = [A, Az] E A, d E D, a E

(5.14)

e,, b E Ob

where D, e,, and Ob are (given) convex subsets of S, Rt, and Rt respectively. Theorem 5.15: Consider the uncertainty model of (5.14). Suppose the applied input is uexpt = ( U O , u l , . . . , u l - l ) ; U ; E R" with u g # 0 and the observed output is yexpt = (yo, y1,. ' . ,yl-1); yi E Rp. Define the sequences ii = (Go,i i l ] . * . , G l & l= ) 7QWn(uexpt) $ = ( Y O , Yl,...,ih-l)

=~lWd(Yexpt)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 5, MAY 1994

956

and define the matrices R , V E R z Xby t

-

To

0

...

T1

TO

...

Proof: Observe that the uncertainty model (5.14) is not invalidated if and only if there exist A = [A, A,] E A, q E T ~ Da, E O,, b E Ob such that

0 0

(D+ A1Wd)Yexpt = ( N + A2Wn)Uexpt + 4.

,

R=

This readily simplifies to the existence of A E A and q E rlD, a E O,, b E Ob such that

We may now invoke the extension theorems (4.1) through (4.10) to establish the our result in each of the four cases 17 above.

V =

VI. CONCLUSION

1) Suppose A = d ~ ~ 1 - 2Then, . the above uncertainty model is not invalidated by the observed input-output data if and only if the following convex feasibility problem is solvable:

Does there exist q E q D , a E O,, b E Ob such that

+ Y ' Y ) - i ]5

a [ ( R B- V A - Q)(i?fi

where U, Y , Q, A, B are the associated Toeplitz matrices formed from the sequences of vectors 6 , y, q, a, b respectively? 2) Suppose A = d~Tv-2.Then, the above uncertainty model is not invalidated by the observed input-output data if and only if the following convex feasibility problem is solvable: Does there exist q E X ~ Da, E O,, b E Ob such that

3) Suppose A = A L T I - ~Then, . the above uncertainty model is not invalidated by the observed input-output data if and only if the following convex feasibility problem is solvable: Does there exist q E mD, a E O,, b E such that

LP(

C],

Ob

APPENDIX

)

Rb - V u - q, y

PROOFS

is feasible? 4) Suppose A = A L T v - ~Then, . the above uncertainty model is not invalidated by the observed input-output data if and only if the following convex feasibility problem is solvable: Does there exist q E R Z D ,a E O,, b E such that 1 l K k ~ b- V U

- qllm

In this paper, we have examined model validation problems for a variety of uncertainty models that employ time-domain experimental data. Each of these problems reduce to convex feasibility problems that are constructed directly from the input-output data and the prior modeling information. In one simple case, we were able to give an analytical solution to the problem. Convex feasibility programming problems such as those encountered above can be solved efficiently upto a thousand variables and constraints using a variety of general purpose algorithms such as interior point methods, ellipsoid methods, etc. [l], [lo], [20], [27]. We are currently investigating the possibility of exploiting the special Toeplitz structures and the structures of the signal uncertainty sets D that arise in these particular problems to provide computationally attractive, perhaps recursive, algorithms for model validation. While we have made some progress, much works remains to be done. We have treated the cases of unstructured uncertainty and parametric uncertainty. The important case of structured unmodeled dynamics uncertainty needs further investigation. In this situation, since the noise and system uncertainties do not enter linearly in the input-output equations, the direct application of our methods fails. This problem has been approached by Smith and Doyle [29] in frequency domain. In this work, we have assumed that the system is relaxed prior to application of the input uexptfor model validation. Further work should also integrate identification techniques with these model validation results.

E])lo

5 711nk

AXB = C is solvable with F ( X ) 5 1 if and only if

Ob

for k = 1 , . . . ,

To establish Theorem (4.5) we shall require the following intermediate lemmas. Lemma 7.1: The linear matrix equation

z

B'B C' o. (7.2) C AA'] Proof: Suppose A X B = C is solvable with F ( X ) 5 1.

[

~

POOLLA er al.: TIME-DOMAIN APPROACH TO MODEL VALIDATION

Observe that

r‘

951

are unitary. Observe that

since the Schur complement I - X’X 2 0. It then follows that

C’’] = [ B I B B’X’A‘] AA AXB AA‘ B‘ 0 I X‘ = [ O A][. I][:

b’b

i‘]”

proving the necessity. Conversely, suppose (7.2) holds. We first establish that the linear matrix equation A X B = C has a solution. Let 3: E N ( B ) . Then the quadratic form

for all vectors y. This forces Cx = 0 proving that N ( B ) C N ( C ) . In a similar fashion it can be shown that R ( C ) C

R ( A ) . These two set containments imply that A X B = C is solvable. Next, let Y be any one solution and define the matrix

Y =~R(A~)Y~R(B). It is clear that Y also solves A X B = C. In point of fact this matrix is a minimum norm solution. We show by contradiction that a(Y) 5 1. Assume therefore that a(?) > 1. Then, there exists a vector v such that llYvllz > 1 1 ~ 1 1 ~ Observe . that Y v E R(A’) and that without loss of generality, we may choose v E R ( B ) . Thus we may write = B w , Y V = A’z, llA’zll

> llBwll

and also

A A ’ ~=

AY^ = A

Finally observe that

[

[w’ - z’] BIB

C‘

Y B= ~ cw.

[“:I

since

+ c’c 5 b’b + a‘a = 1.

It then follows from Lemma (7.1) that the linear matrix equation in C

cb’

+ ZlCV,’ = 0

is solvable with T ( C ) 5 1. Now define G by

G= Notice that G is represented above in singular-value decomposition form. Since .(E) 5 1 it follows that T ( G )= 1. Also, G is block lower triangular because

G12 = cb’

+ Z1CV. = 0.

Finally, observe that

41 [:] =

proving the claim. 0 We are now in a position to present Proof of Theorem 4.5. In the interest of a lucid exposition, we prove the result for square systems, i.e., we will assume that p = m. This proof may be readily modified to include the general case. The necessity is obvious from the fact that A is a causal operator. To prove the sufficiency we must exhibit a block lowertriangular matrix

= w’B’Bw - w’C’z -

z’Cw

= V‘V

-

+ z’AA’z

z‘AA’z < 0

with T ( H ) 5 1 and such that

H=[

U‘]=[ U0

r]

Yo

which contradicts the hypothesis, completing the proof. 0 Lemma 7.3: Given vectors a , b E R” and e, d E RP with ala

2 c’c

and ala

+ b’b = c’c + d’d

w-1

Y1-1

Here the submatrices H i , j are in Rmxmand H is the there exists a block lower triangular matrix G with F(G) 5 1 matrix representation of the operator rlArl. We establish the existence of the desired matrix H by induction on 1. Without and such that loss of generality we take y = 1. The result is trivially true for 1 = 1. Suppose the result holds for 1 = 1, 2, . . . , n. In other words, given any sequences Proof: Without loss of generality, let ala

+ b‘b = c’c + d‘d = 1.

Determine matrices V I ,VZ,21, 2

U=

I

[”b

“1

vi?

2

such that

andY=

[i

4 = ( 4 0 , 4 1 , ~ ~ ~ , 4 n 4n-1) -Z,

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 5, MAY 1994

958

there exists a block lower-triangular matrix G with F(G) and such that

] = [

G[! Qn-1

’:

51

].

rn-1

Define a E R” by

a = J(llw4l; - lJ~n-lYll;)e where e E R” is any unit vector. Define the sequences .iL = ( U O , ~

.

1 , .’

7

U,-2,

5 = (Yo, Y1,. . . ,Yn-2,

Un-1)

a).

Observing that for i = 1,. . . ,n, I l ~ i ~ l2 l a Il7riGll2

we may employ the induction hypothesis to conclude the existence of a lower block-triangular matrix T E R””’”” with F ( T ) 5 1 and such that

Also, choosing b = U,, c = gn-l and d = yn, it follows that a‘a 2 c’c and a‘a

+ b’b = c’c + d’d = 1.

We may thus invoke Lemma (7.3) to conclude the existence of a lower block-triangular matrix G with a(G) 5 1and such that

Define the block lower-triangular matrix

Inm-m 0

[

0

T G][0

0

I”]’

Observe that H is block lower-triangular, F ( H ) 5 1, and that

0

proving the claim. ACKNOWLEDGMENT

The authors gratefully acknowledge an anonymous reviewer for pointing out some relevant literature and an error in an earlier draft of the paper.

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Ashok Tikku received the B.S. and M.S. degrees from the University of Illinois at Urbana in 1989 and 1991, both in electrical engineering. He is currently a Ph.D. student at the University of Califomia at Berkeley. His research interests include system identification and robust control.

Kameshwar Poolla received the B.Tech. degree from the Indian Institute of Technology, Bombay in 1980, and the Ph.D. degree from the University of Florida, Gainesville in 1984, both in electrical engineering. From 1984-1991, he served on the faculty of the Department of Electrical Engineering at the University of Illinois. Urbana. Since then, he has been an Associate. Professor of Mechanical Engineering at the University of Califomia, Berkeley. He has also held visiting- appointments at Honeiwell, McGill, Caltech, and MIT, and has worked as a Field Engineer with Schlumberger AFR, Paris. Dr. Poolla’s research interests include robust multivariable control, adaptive control, time-varying systems, system identification, process control, and image processing. He has been awarded the 1984 Outstanding Dissertation Award from the University of Florida, the 1988 NSF Presidential Young Investigator Award, and the 1993 Hugo Shuck Best Paper Prize.

Pramod P. Khargonekar (M’8 I-SM’9(rF‘93) received the B.Tech, degree in electrical engineering from the Indian of Technology, Bombay, India, in 1977, and the M.S. degree in mathematics and the Ph.D, degree in electrical engineering from the University of Florida, Gainesville, FL, in 1980 and 1981, respectively. From 1981 to 1984, he was with the Department of Electrical Engineenng, University of Florida, and from 1984 to 1989 he was with the Department of Electrical Engineering, University of Minnesota, Minneapolis. In September 1989, he joined The University of Michigan, Ann Arbor, where he is currently Professor of Electrical Engineering and Computer Science. His current research interests include robust control, H2, H,, and H 2 / H , optimal control, sampled-data systems, robust and H, identification, robust adaptive control, time-varying systems, and applications to aeromace control oroblems. Dr. Khargonekar is a recipient of the American Automatic Control Council’s Donald Eckman Award, the NSF Presidential Young Investigator Award, the George Taylor Award from the University of Minnesota, and the Sigma Xi (University of Florida Chapter) Outstanding Research Award. He is a corecipient with Professors J. C. Doyle, B. A. Francis, and K. Glover of the 1991 IEEE W. R. G. Baker Prize Award and the 1990 George Axelby ON AUTOMATIC CONTROL) award. He Best Paper (in the IEEE TRANSACTIONS ON AUTOMATIC CONTROL was an Associate Editor of the IEEE TRANSACTIONS during 1987-1989. He served as the Vice Chair for Invited Sessions for the 1992 American Automatic Control Conference. He is currently an Associate Editor of SIAM Journal on Control and Optimization, Systems and Control Letters, and International Journal of Robust and Nonlinear Control.

James M. Krause (S’85-M’87) was born in Wisconsin in 1958 and received the B.S. degree in EWCS from Marquette University, Milwaukee, WI, in 1981, the S.M. degree in electrical engineeringkomputer science from the Massachusetts Institute of Technology, Cambridge, MA in 1983, and the Ph.D. degree in electrical engineering from the University of Minnesota, Minneapolis in 1987. Since 1983 he has been with the Honeywell Technology Center (previously named the Honeywell Systems and Research Center) Minneapolis, MN, where he has carried out research in estimation and control theory, aerospace and the thermodynamic system control applications, fault tolerant avionics, and advanced software technology. He is currently serving in a technical management capacity as Section Chief of Space and Aviation Guidance and

Krishan M. Nagpal received his education from Indian Institute of Technology, Kanpur and West Virginia University. He is currently with the Department of Electrical and Computer Engineering at University of Iowa.