A time domain approach to model validati

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

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A Time-Domain Approach to Model Validation
Kameshwar Poolla, Pramod Khargonekar, Fellow, ZEEE,
Ashok Tikku, James Krause, Member, IEEE, and Krishan Nagpal

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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.

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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.

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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

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)

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

sequences

if and only if

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Y'Y 5 y2u'u

Similarly, 1:
1:

=

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:

+

(4.2)

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

zyx

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

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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

Given sequences
{yo,

y1,...,yl-l

,

U

=

{ U O , ul,

E RP}

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

*, *, . ..)

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

llAlli2

*, *, .. .).

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

*, *,-..I

= (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

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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

I -7

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.

,

[XI

52

53

x4

-I
x5]1 I
-I

I

I

-I

0

0 1
U
= [O 0

61

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

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

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

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... 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,

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llHlliooI Y.

0

+

+

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.

(5.5)

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

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

ii = (Go,

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.

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

llAIli2

5 1)
(5.1)

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

MAINRESULTS

ALTI-w
= {linear time-invariant

: d; E

y = Mou+ AWu with A E A.

IlAlliCo I Y

i , . * . , ~ - i ,

1-1

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, ~

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

is feasible.

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

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

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

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zyxwvut
zyxwvutsr
zyxwvu
zyxwvu
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

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)

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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
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.

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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:

zyxwvutsr

Does there exist q E mD, a E O,, b E
such that

LP(
is feasible?

C],

Ob

APPENDIX

)

Rb - V u - q, y

PROOFS

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:

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zyxwv

Does there exist q E R Z D ,a E O,, b E
such that
1 l K k ~ b- V U

- qllm

E])lo

5 711nk

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

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

Ob

for k = 1 , . . . ,

AXB = C

z

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

[

~

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

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zyxwvutsr
zyxwvu
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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

AY^ = A

Finally observe that

[

[w’ - z’] BIB

C‘

Y B=
~ cw.

[“:I

+ 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

zyxwvu

> llBwll

and also

A A ’ ~=

since

= 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,

958

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

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’:
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zyxwvutsrqpon
zyxwvuts

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

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zyxwvutsr
zyxwvutsrq

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.

REFERENCES

[ l ] S. Boyd and L. El Ghaoui, “Method of centers for minimizing generalized eigenvalues,”linear Algebra and App., vol. 188, pp. 63-1 11, Jul.
1993.

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

959

zyxw

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.

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uncertainty,” in Proc. I990 Con$ Dec. Contr., Honolulu, HI, Dec. 1990.
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and dimension for continuous time,” ZEEE Trans. Automat. Contr.,
vol. AC-24, pp. 222-230, 1979.
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Sysr. Contr. Lett., vol. 20, pp. 167-178, 1993.

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.

zyxwvutsrq

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
Best Paper (in the IEEE TRANSACTIONS
ON AUTOMATIC
CONTROL)
award. He
was an Associate Editor of the IEEE TRANSACTIONS
ON AUTOMATIC
CONTROL
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.

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