Proceedings ArticleDOI

# Model reduction of behavioural systems

15 Dec 1993-pp 3652-3657

Abstract: We consider model reduction of uncertain behavioural models. Machinery for gap-metric model reduction and multidimensional model reduction using linear matrix inequalities is extended to these behavioural models. The goal is a systematic method for reducing the complexity of uncertain components in hierarchically developed models which approximates the behavior of the full-order system. This paper focuses on component model reduction that preserves stability under interconnection. >
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FP8
-
5:lO
Proceedlnga
of the 32nd Confaronce
on Declalon and Control
San Antonlo,
Taxer
*
December
1993
Model Reduction
of
Behavioural Systems
Carolyn Beck'
Abstract
We consider model reduction of
uiicertain
behavioural models. Ma-
chinery for gap-metric model reduction and n~ultidimensional model
reduction using Linear Matrix Inequalities is extended to these be-
havioural models. The goal is
a
systematic method for reducing the
complexity of uncertain components
in
hierarchically developed mod-
els which approximates the behavior
of
the full-order system. This
paper focuses on component model reduction that preserves stability
under interconnection.
1
Introduction
In this paper we consider the problem of reducing uncertain be-
havioural systems of the type proposed by D'Andrea and Paganini
[5].
The motivation for this problem comes from the desire to reduce
the complexity of separate component models in
a
system in order to
reduce the complexity of tlie full system. Unfortunately, the critical
issue of what coiisitutes
a
good approximation of
a
component is prob-
lem specific and depends in detail
on
tlie rest of the system to which the
component is connected,
as
well as
on
the performance requirements
on that system. For example, in the standard plant/controller feed-
back system, approximating either tlie plant or controller may change
a
stable feedback system to an unstable one if the system is not robust
to the approximation error. More generally, any approximation made
to a component may result in large subsequent differences in the full
system, including instability, depending
on
the system to which it is
connected.
While it is impossible to gnarantee without further assumptions
that any properties of the full system will be preserved
if
a
component
is approximated, we can add some reasonable and mild assumptions
about the properties of the interconnection that will allow
us
to guar-
antee,
for
example, that stability will be maintained. This allows
us
to develop an order reduction methodology for behavioral models that
is general, natural, and does not depeiid
on
detailed knowledge
of
tlie
rest of the system. While it is conservative when compared to what
might be possible if the entire system is considered, it has the advan-
tage of being applied purely at tlie coniponent level. Our methodology
provides guaranteed upper error bounds, and maintains robust stabil-
ity of interconnected systems
if
the error is less than a certain stability
radius. It has the additional appealing interpretation
of
approximating
component behaviors
as
subspaces
of
Lz.
The results in this paper build
on
a
number
of
results from model
reduction
[[7], 181, [13], [17],
[18],
[25], [2G]],
particularly balanced trun-
cation and its recent extension to multi-dimensional (MD) and/or un-
certain input-output
(IO)
systems with guaranteed upper bounds
on
the error in the Q-norm
[25].
Recently, this method has been used to
determine necessary and sufficient conditions for obtaining minimal re-
alizations of MD/uncertain systems in the &-norm
[2].
Unfortunately,
the systems considered in these methods consist of only one compo-
nent which is modelled
as
a
linear fractional transformation (LFT)
on an uncertainty/frequency structure. We exteiid these methods to
develop model reduction tccliniques for iirterconriected behavioural
sys-
tem components.
After defining
our notation, we give
a
brief review of the be-
havioural system framework and the gap metric
in
section
2.
We
*Electrical Engineering,
MIS
116-81,
Caliioroia
Iustitute
01
Technology,
'Control
and Dynamical Systen~s,
M/S
11G-81,
California Institute
of
Technol-
Pasadena,
CA
911?5
ogy,
Pasadena, CA 91125
John
Doyle'
then discuss model reduction and robust stability properties for gen-
eral
1D
behavioural systems in section
3.
Relevant model reduction
results for standard
IO
systems are reviewed in conjuction with be-
havioural system model reduction discussions. In section
4,
we extend
these techniques to behavioural system models which contain uncer-
tainty. We show that if uncertain behavioural system components are
reduced such that the resulting error is less than
a
specified stability
radius then the stability of the interconnected reduced system implies
stability of the interconnected full system. We present one solution
method to this reduction problem, via solution of
a
set of coupled lin-
ear matrix inequalities (LMIs) in section
5.
Solutions to these LMIs
may be thought
of
as
sufficient conditions, and we discuss the sev-
eral sources
of
potential conservativeness and possibilities for reducing
them. Throughout this paper, we try to relate the concepts involved
with behavioural systems to those of standard 1D feedback systems.
2
Background
We first define the notation used in this paper, and then present rel-
evant background material in behavioural system representations and
the gap metric.
For
a
more extensive treatment
of
these subjects, see
references
[27],[1], [11],[10]
and
[24].
2.1
Notation
The notation we use is
as
follows.
1-z1,
'H,
denote the Hardy spaces of
possibly vector- or matrix-valued functions with analytic continuation
on the unit disc, and
Lz, L,
the corresponding Lebesgue spaces
of
functions square integrable and essentially bounded, respectively, on
the unit circle, each with norms
11
*
112,
I(.
Itm.
RE,
and
RL,
are the
subspaces
of
1.1,
and
C,
whose elements are rational functions. We
represent the integers by
2,
the time shift operator by
z-l,
and the
identity matrix by
I,
where the dimensions will be assumed to be clear
from the context,
or
will otherwise be stated. The maximum singular
value of
A
is denoted
ir(A),
and
A'
denotes the adjoint.
2.2
Behavioural System Representations
The most striking feature of the representation of dynamical systems
in the behavioural framework,
as
proposed by Willems
[27],
is
the fact
that there are no explicit inputs and outputs. Instead, the system
is viewed as
a
phenomenon
to be modelled, which produces elements
that are referred to
as
outcomes.
From
a
mathematical model for the
phenomenon we can determine
a
set,
t?,
of
possible outcomes, which
is called the
behaviour
of
the model. In particular, if dynamical sys-
tems are considered in this context, then the phenomenon produces
outcomes which are functions of time. A dynamical system is defined
in
a
behavioural framework as follows
[27].
Definition
1 A
dynamical syslem
C
is
a
triple
C
=
(I",
W,
B),
with
T
E
Z
the time
axis,
W
ihe signal spccce, and
B
C
WT
the behaviour.
Here,
WT
represents the set of all maps from T to W, and W
=
RP.
In
order to incorporate uncertainty into our models, we adapt
the
output
nulling
representation defined by Weiland
[28]
to describe
1D behavioural systems.
A
brief summary of this type of behavioural
system representation follows.
For
more details, see
[5]
aid
(281.
CVe
consider the equations:
p=
Ax+Bw
0
=
Cx
-I-
Dw
where
A,
B,
C,
D
are constant, finite dimensional matrices and
2,
w
E
Cz.
The vector dimensions of
z
and
w
will not be specified unless
pertinent to the discussion.
01 91 -221 6/93/$3.00 0 1993 IEEE 3652 The behaviour of a dynamical system C is then characterized in this framework by This system has representation matriz R = [ 1, which we hence- forth denote by R = {A, B,C, D}. We define a frequency/uncertainty parameter A E A, where we assume A is an operator on I22 with the following structure: Typically, we define 61 = z-l and the remaining 6i as uncertainties or perturbations to the system. Uncertainty is then incorporated into our behavioural system model in a linear fractional manner by setting z = Ap. Thus, for a given A E A, we explicitly write the LFT on R and A as: assuming the inverse of (I - AA) exists. (In (51 this LFT is denoted by S(A, R)). We define such a system behaviourally as follows. Definition 2 An uncertain (LTI) dynamical system is a pummeter- izedfamily {E, : A E A} of dynamic systems denoted by the quadruple CA = (Z, RV, B, A) whose behaviour can be expressed by This behauiouml representation is denoted (A, R), and is culled a Gen- eralized Output Nulling (GON) representation. A GON representation is called regular if D is surjective, dependent if (A R) is not surjective V A E A, and singular if it is neither regular nor dependent [5]. We want to reduce the behavioural representation matrix, R, with guaranteed error bounds such that if the interconnected system is sta- ble with the reduced representation matrix, R,, then the intercon- nected system is stable with the full representation matrix, R. Addi- tionally, we would like the behaviours described by R and R, to be close to each other. 2.3 The Gap Metric The results in this paper for 1D behavioural systems with no uncer- tainty (i.e., A = %-'I) are essentially equivalent to existing results using the gap metric. We present a general review of the gap which includes the gap metrics of both [lo] and [24], and discuss relevant robust stability properties of these metrics. We note that although existing gap metric results are developed for continuous time systems, the identical results for discrete systems also hold. The gap may be defined between subspaces, behaviours, and IO systems using normalized coprime factors. We begin by considering the gap between two closed subspaces SI and S2 of a IIilbert space N, B = {w E Lz I 3x E La satisfying (1)) A = {diag [&Iq,,. . ..6,1q,], 6, : Lc~ H Lz} (2) (3) A* R = D +CA(I - AA)-'B B = {w E L2 : (A * R)w = 0 for any A E A} (4) (5) defined as 61((S1,&) = IIKs, - Hs211 9 where Hs, is the orthogond projection on S1. Note that h,(&,&) is a metric, and satisfies For the purpose of exploring model reduction methods, we will be re- quired only to consider problems where the gaps between subspaces are strictly less than 1 (for the general case see [lo] and [24] and references therein). In this case, we can use the directed gap, defined as 0 5 61((Sl,S?) 5 1. to derive the following alternative expression for 6,(Sl,SZ) < 1 671(S,,S*) = a',(Sl,S,, = &(S2,Sl). In particular, consider the typical plant/controller feedback con- figuration. The plant and controller are modelled as LFTs on the fre- quency structure A = z-lI, with resulting transfer functions denoted by E and C. As in [lo] and (241, we denote a normalized right coprime factorization (rc8 of pi by [Ni, Mi], and a normalized left coprime fac- torization (lcn by [fii, ai]. Similarly, [Ne, M,] and [NC, will denote normalized right and left coprime factorizations of a controller C. We write Recal! that [fii,[4i] is a normalized lcf of Pi if, and o?ly if, i) Pi = MTINi, ii) Gi E N-, iii) there exists an I' E H, : GiY = I (the coprimeness condition) and iv) GiG: = I (the normalization con- dition). Similar conditions can be given for nocmalized rch. Such factorizations always exist, with deg(Gi) = deg(Gi) = deg(P,). Note that GiGi = 17-1; = 0, thus [ Gi @ ] and [ K i? ] are unitary. The RHo-graph Qi of Pi is the closed subspace of 7i2 consisting of all pairs w = (U, y) such that y - Piu = 0 or Gi = GiNz = {w E 71, : Giw - = 0) (7) The ?&-gap between two systems, PI and Pz, is defined accordingly as the distance between their respective ?&graphs, Q1 and G2. The &-gap is defined similarly. The formula for the ?&-gap metric derived in 1111 is For details see [lo] and the references therein. Of particular relevance to the problem we consider is the u-gap, defined by Vinnicombe [24]. Vinnicombe defines the u-gap in such a way that, provided 6,(Pl,Pz) < 1, it is equal to the Lz-gap and is defined as 6,(Pi,P2) = ~L~(G~TG~) = IlGzGillm The last expression follows from the definition of fcz(Q1,Gz) and the fact that [Gi 6;] is unitary. Here we are using Gi to denote L2 sub- spaces. The assumption in this paper that all of the gaps are strictly less than 1 is justited by the fact tha: we obtain approximations to GI by a lower order Ga that satisfies ((GI - G&, < 6 < 1. Since it is easily shown that we may always assume that all gaps are strictly less than 1. This greatly simplifies the discussion without incurring any loss of generality. 2.4 Approximation in the Gap As in [15], [lo], and [24], for a plant P and controller C connected in a standard feedback configuration, we consider the matrix transfer function np,c = [ f. ] (I - CW' [I Cl , and define the associated genemlized stubility margin as bp,C = ~~IIp,c~~~l if np,C is stable, and I,, = 0 otherwise. Using the nor- malized rcf and lcfrepresentations in (G), IIp,c = G(kG')-'k, thus, b,, = ~~(LG)-l~J~l. For this problem setup, both the 'Hz-gap and the u-gap have the fol- lowing property ([10],[24]): Property 1: Given a nominal plant PI and a controller C then: IIp,,c is stable for ull plants, Pz, satisfying 6.( PI, P2) 5 p if, and only if, P < bp,#C. This property tells us that any plant at a distance less than P from the nominal will be stabilized by any controller stabilizing the nominal plant with a stability margin of p. The v-gap also has the additional property: Property 2: Given a nominal plant PI, a perturbed plant Pa, and a number P < bopl(P) then: IIp2,c is stable for all controllers, C, satisfying P < bp,,C if, and only if, 6,(P1, P2) 5 P, where bopt(P) := supc b,c, the optimal stability radius. This second property says that any plant at a distance greater than p from the nominal will be destabilized by some controller which stabilizes the nominal with a stability margin of at least P. Addition- ally, it can be shown that the performance change in replacing plant PO by Pl is bounded by 64Pg,JJ1). These properties make a,(., .) def- initely most excellent, and ideal for a priori model reduction of the plant, since we need only assume that the controller eventually used will have reasonable generalized stability margin. 3653 2.5 Balanced Tkuncation Model Reduction and the Gap Metric We briefly summarize balanced model reduction methods and state re- sulting upper error bounds. Consider the standard IO system described by an LFT on the frequency structure A = z-'I and R = {A, B, C, D}. This system is considered balanced if there exists a diagonal matrix C = diag[ulIl,u212,. . .,unIn] with ul > uz > . . . > U, 2 0 such that ACA'-C+BB' =Oand A'CA--C+C'C =O. Theentriesui,ofC, are called the Hankel singular values of the system, and Ii are used to indicate that the multiplicity of ai may be greater than 1. Assuming R is balanced, partition R and C into the following subblocks: Ar AIZ Br R=[AE;' 2$1
E=[?
;z]
E,
=
diag[Ulrl,.
.
.
,
gklk]
>
0
and
=
diUg[Uk+iIk+l,.
. .
,
gnIn]
2
0
Then truncate
R
to
R,
=
{Ar,
B,,C,,D}.
Bounds for the error result-
ing from this truncation
as
derived in
[7], [8],
and
[25]
are
n
ut+l
I
IIA
*
R
-
A,
*
R.11,
52
(Ti.
(8)
i=k+I
For
a
system represented by
a
normalized
rcf,
G,
a
balanced trun-
cation reduction method with an upper error bound in the
graph
metric
has been computed by Meyer
[17].
Balancing and truncating the
rcf
representation
G
results in
a
reduced, normalized
rcf,
G,
with
6(P,P,)
=
i;f
IIG
-
GrQII,
5
2
2
ui
where
6(-,.)
denotes the graph metric and
U;
are the Hankel singular
values of the representation for
C:.
The graph metric and the gap
metric are closely related,
[ll],
and bounds
for
approximation in the
gap metric have
also
been computed
[12].
Additionally, we note that
Weiland develops
a
more general notion of balanced representations in
the behavioural framework, along with an upper bound for balanced
truncation which is equivalent to that in
(8)
[as].
In view of the properties of the
6,(.,.)
we would like to model
reduce using this
as
our metric, but currently no such method with
associated bounds exists.
It
is widely accepted among the cognoscenti
that
our
best option is to use
6v(Po,pi)
I
IlGo-GiIlm
and make the norm on the right sindl using Ilankel norm approxima-
tion
or
balanced truncation
191.
See
[U]
for inore details.
3
Model Reduction
of
1D
Behavioural
Sys-
i=k+l
tems
In order to more readily connect results in the gap metric and
1D
behavioural system representations in anticipatio?
of
generalizing
to
include uncertainty, we will henceforth denote
G
=
(A
*
R),
and
IC
=
(A
*
F),
where
R
and
F
are representation matrices for com-
ponents in an interconnected behavioural system and A is the fre-
quency/uncertainty set for tlie systcni. Note that we abuse the nota-
tion here,
as
(A*
R)
is not meant to represent tlie system at one specific
A,
but instead represents the system
as
an
operator and
A
represents
the set
A.
This notation is used throughout this paper. We assume
R
and
F
are regular representations. Given one component,
D
c
L2,
of an interconnected behavioural system, where is represented by
(A,
R),
we
want to approxiinate
G
by
a
simpler representation. One
natural approach is to seek approximations to the beliavior itself as
a
subspace in
La,
which would be similar to approximations in the
Lz-
gap
or
v-gap. We cannot approximate these subspaces directly, but we
can determine approximation bounds using error bounds on
G
as
sug-
gested by the v-gap properties. As in the gap theory, approximation
of behaviours can be related directly to properties of interconnections.
Consider the behavioural system described by the interconnection
of two behavioural components defiiied as the subspaces
G
and
K
of
Lz.
We adopt much of tlie gap notation but do not assume that
0
and
K
are the graphs of IO operators. Assume we have normalized transfer
functions
G
and
G
such that
G
=
GL?
=
{U
CL,
:Gw
=
0)
(9)
and
[G
G*]
is unitary. If we consider nornialized behavioural de-
scriptions, robust stability criteria are easily constructed which are
analogous to robustness in the gap metric. We construct these crite-
ria in such
a
manner that they reduce to the robust stability criteria
in tlie gap metric given in
[lo]
for the standard feedback configura-
tion with components
P
and
C.
The general definition
for
normalized
behavioural representations we use throughout this paper is
as
follows:
Definition
3
A
behaviouml representation
(A
*
R)
is
normalized if
(A
*
R)(
A
*
R)'
=
I
In the
1D
case we also can find
G
such that
[G
e]
is unitary. We choose
the term
normalized
to correspond with the
IO
definition
of
systems
represented by normalized coprime factors. This concept is the same
as
that
of
coisometn'c
defined by Weiland
128).
Weiland shows that,
as
in the standard
case,
if we are given
a
behavioural representation
matrix
R
=
{A,
8,
C,
D} which is not normalized, we can compute an
equivalent normalized representation by solving an algebraic Ricatti
equation (Theorem
3.215,
[28]),
which is equivalent to the normalization
methods developed for solving
H,
optiinal control problems in
[SI.
The interconnected system we will consider
is
Gw=o;
kv=o
(10)
w+v=n
where
n
represents noise injected at the interconnection. We will
as-
sume that
a
well-formed interconnection involves the maps from
n
to
w
and
v
being bounded. Basically,
as
we don't know in detail how our
component will be connected, it is reasonable to expect that if noise
is
injected at the interconnection, tlien this noise will not be greatly
amplified. In the standard feedback configuration considered in the
gap case, this has
a
clear interpretation
as
(IIIp,Cllm.
The implications
of this assumption are less clear in tlie behavioural case.
For
example,
it excludes interconnections that yield singular representations. Nev-
ertheless, we make this assuniption on interconnections
as
a
reasonable
starting point.
3.1
Reduction
of
Behavioural Components
The behavioural system representation allows us to perform model
reduction and robustness analysis
for
more general system descriptions
than the standard
IO
setup. Consider the interconnected behavioural
system described by
(10).
We form the input/output relations
quite readily by computing
w
=
II~,~I~,
v
=
n-
ir,c
-11
Note that in the standard feedback setting !/llG,F//,
-I
=
bp,c.
We
assume tlie above inverses exist and are stable,
111
which case we say the
interconnected behavioural system is noiiiinally stable. Additionally,
this assumption implies that
6,
as
a
matrix, lias more columns than
rows and is therefore guaranteed to have
a
kernel, but
&
may not.
Directly applying the balanced truncation model reduction
method previously described to
R
=
{A,
U,
C,
D}
results in
a
reduced
representation matrix
R,
=
{Ar,
U,,
C,,
0,)
and corresponding
A,
such that
IlG
-
G,llw
is guarantced to be bounded by some value, say
c.
Our first objective is to state conditions under which behavioural
systems are robust to such reductioq
so
we first consider the be-
havioural system described by
(10)
with additive uncertainty. That
is, suppose
6
is perturbed to
G
+
A, where A,
E
Al.
While we
are most interested in the case where
A,
represents approximation
error, it is also possible that
A1
represeuts unstructured uncertainty
which is possibly time-varying. The following robust stability lemmas
for behavioural system representations are trivial extensions
of
corre-
sponding gap results, but arc stated for the purpose of generalization
to uncertain behavioural systems. A sketch of tlie proof for Lemma
1
is given,
as
the same nwtliod
of
proof can be used in the uncertain
case.
Lemma
1
Suppose the interconnected ~ehoviouval system described by
(10)
is
nominally stable, and
AG
E
A,
with
llA~ll
5
c,
then the
corresponding
perturkd
system
(6
+
AC)W
=
0;
fin
=
0
3654

w+v=n
is
stable
for
all
llAcll
5
E
ifl
Proof:
Substituting
v
=
n
-
w,
our uncertain behavioural system inter-
connection equations can be written in matrix form
as
([
:I+
[
?I)-=
[;In
._
-
Rewriting this in
a
transfer function form from
n
to
w
gives
w=
(I+[
ql[
?])-I[
;I-'[
:In
Thus, stability is guaranteed
iff
0
We can then prove the following theorem.
Theorem
1
Suppose(Al*R1)
=
GI
is iiormalizedand
IIIIcl,~II,
<
$. If the behavioural system give? by (10) with GI is stable, then it is stable when e, is replaced by any Gz with the property In particular, suppose Gz represents the nc$nal system and
GI
the reduced system. Then we can normalize
Gz
and truncate using
Meyer's algorithm, giving us
a
normalized
GI.
We then want to find
the smallest dimension
GI
satsfying
llGl
-
G211m
5
I~&,,RII,
,
to sat-
isfy the assumptions of Theorem
1.
This theorem follows immediately
from Lemma
1
and the following Lemma, the proof of which is very
straightforward and therefore not presented here.
Lemma
2
Suppose
-1
is normalized and
IlIIzll
<
$, then The above theorem can be compared to similar theorems using the v-gap. It gives sufficient conditions for safely reducing a component model, but is potentially conservative. A less conservative theorem, which follows immediately from Vinnicombe's results [24], is Theorem 2 Suppose GI is normulized und IlII~,,nll, < f. If the behaviouml system given by (10) with GI is stuble, then it is stable when GI is replaced by any Gz with the properly IIC, - G:21Im < 1 and 6r2(Gi,Gz) < E The test hLL.(G1,&) < 6 is the least conservative possible if the only information given about the interconnection is that IIII~,,~ 11, < f. Unfortunately we can't use SL,(ql,Gz) directly, so we will have to be content with model reduction based on the bound 6r2(G1,&) 5 llGl - GZllm. This approach appears to be effective in the gap case, and can be extended in the next section to behavioral systems with uncertainty. 4 Behavioural Systems with Uncertainty In an attempt to develop a model reduction method for uncertain behavioural systems, we first consider balanced truncation model re- duction for uncertain IO systems. We would like to extend this method to behavioural systems described by a representation matrix R = {A, B,C, D} and frequency/uncertainly structure A as given in (2). We present a set of sufficient conditions on the error resulting from reducing uncertain behavioural realizations which, if satisfied, guarantee stability of the resulting interconnected system. To determine the robustness of behavioural system stability to model reduction, we must have some measure of the error incurred by such a reduction, and a precise notion of stability for such a system. We utilize the following definitions of Q-stability, and the Q-norm for this purpose. Definition 4 The uncertuin system represented by (R, A) is said to be Q-stable if there exists a non-sinyular matrix T such that TA = AT VA and a(TAT-') < 1. Analogous to the definition of &-stability of uncertain behavioural systems, we define the following &-norm by which to measure the model reduction error. Definition 5 The Q-norm of a system representation (R, A) is given by IlA * RJlq = inf 7 : 3T such that I ([ 3: 1) < 1) (12) { where TA = AT. For 1D systems, the Q-norm is the same as the H, norm. For uncertain systems, Q-stability and performance are necessary and suf- ficient for robust stability and performance, when A represents arbi- trary linear operators. For the repeated 6 case considered here, the proof of this involves a generalization [22] that directly combines ex- isting results [16], [23], [21]. For input-output systems, the importance of stability and robustness is clear and we will use Q-stability and per- formance for our final interconnected systems. This allows our A to be time-varying (and even nonlinear) but would be conservative if A had additional structure, such as time-invariance. For behavioural representations, the use of a stable representa- tion matrix R is perhaps less fundanlental but is very convenient for manipulation and computation. Stable R generalizes the use of sta- ble coprime factor representations for input-output systems and then norms can be used to define generalizations of normalized coprime fac- tors. These generalizations allow us to compare robust stability results formulated in the gap metric and provide for a natural representation and measure of error iii the bchsvioural framework. 4.1 A brief review of model reduction results for MD/uncertain IO systems is presented. The reader is referred to 1251 and 121 for full details. We again consider a system represented as an LFT on a A structure. In this case the frequency/uncertainty set A is defined as in (2). Gener- ally, one Si represents the system frequency variable, e.g., z-l, and the remaining 6i are arbitrary operators on .Cz representing uncertainty. If all 6i represent frequency variables, reduction corresponds to state or- der reduction, as in the 1D case. If 6i represent uncertainty, reduction corresponds to simplification of the uncertainty descriptions. As in the preceeding discussions, we consider only similarity trans- formations which commute with the A structure, to which we refer M allowable transformations. Thus, an allowable transformation T has block-diagonal structure. The definition of balanced we use differs slightly from the standard definition. Non-strict inequalities are used rather than equalities in the Lyapunov equations as allowable solutions to the equalities may not exist for uncertain systems. Definition 6 : A italizalion {A, B, C, D} is balanced if 3C = diay[ulIl,uJ2,. . ., o,,I,,] > 0 such that Model Reduction Results for Uncertain Systems AEA' - C + BU' 5 0 and A'CA- C +c'C 5 0 We discuss reduction of a 2D system for notational convenience. The stated results hold for any number of blocks. We assume the system representation, is balanced, Q-stable, and is partitioned, along with C, as follows: A.. - E,, = diag[u,lI,l,. . .,U,&,] > 0 and G, = diag[o'(b,+l~l,(r.,+l,, . . . ,U,",L",I > 0 where u,~ 2 u,~ 2 ... 1 U ,,,,. The balanced truncation model reduction theorem is M follows: 3655 1 Ail, A12, BI, [ CI. c2. D Theorem 3 If we truncate R to R, = A21, A22. Ba, (and A to A, with corresponding partitions), then R, is balanced, A, * R, is Q-stable, and i=l j=b,+l Thus, if Etl Ey;k,+l uij is small, then we can reduce the system order and the resulting error in t.he Q-norm of the system is small. In the 1D case previously discussed, if this error is less than 1111211-1, then the reduced coniponent can be used in the interconnected system without causing instability. This robust stability condition generalizes to the uncertain case. In addition to the reduction theory presented, necessary and suf- ficient conditions for exact reducibility of uncertain systems in the Q-norm have recently been found (21 which are also applicable to be- havioural system representations. We summarize these results, without proof, in the following. Proofs are given in 121. Theorem 4 Let R = {A, B, C, D} be a &-stable system represen- tation matrix with uncertainty/frequency structure A E A, with dim(A) = n. Then there exists (A?,R,), with dim(A,) = k < n, such that II(A * R - Ar * R,)llQ I c if, und ody if, there exist allowable X 2 0 and Y 2 0 satisfying (i) AYA' - Y + BB' 5 0 (ii) A'XA - X + C'C 5 0 (iii) XY 2 r21, with the lowest n - k eigenvulues of XY being equal to €2 where c 2 0. For E > 0, the proof of Theorem 4 follows directly from applica- tion of machinery developed by Pacliard [20]. The exact reducibility case, that is, the c = 0 case, is more involved. In proving sufficiency in Theorem 4 for c = 0, we actually construct a reduced system real- ization, R,, by balancing and truncating R. Alternatively, in proving necessity for E = 0, Rr may be any system matrix as long as it is of smaller dimension. Additionally, &-stability of G is not required for the necessity proof. We can apply Theorem 4 to uncertain behavioural system representations in the same way that we apply the basic model reduction results. 4.2 Robust Stability of Interconnections Obviously, we would like to niaintain as many similarities as possible between the MD/uncertain case and the 1D case. 111 particular, we would like the uncertain behavioural system representations to be nor- malized. However, in the uncertain case, there is no guarantee that an allowable solution to the required Ecatti equations exists which yields a normalized realization. In fact, such a solution most likely does not exist. Thus, rather than attempting to find a normalized realization, we utilize the following concepts of expansiveness and contractiveness. Definition 7 The behaviouml syslerri representalion, (A, R), is a- contractive (expansive) if, for all A E A such that A*A = I, (A * I()( A * Ry 5 [ 2) a?I If (I = 1 then we simplify the terminology to just contractive or ex- pansive. Equivalently, a representation (A, R) is a-contractive for all unitary A if and is a-expansive for all unitary A if TAT-' there exists allowable T : a where E(.) refers to the smallest non-zero singular vdue. Note that in the above definitions, we evaluate expansiveness and contractiveness only for unitary A. While in general this gives only a subset of the behaviors, it will include those that are worst-case for stability and performance of the interconnected system, because for any L2 stability or performance condition, unitary A are always worst- case for operator A. This is not true, of course, for A that include real parametric uncertainty, which is iiot considered in this paper. We require that the realization we reduce be contractive and a- expansive (for a < l), in order that Lemmas 1 and 2, and Theorem 1 generalize to the uncertain case with as little additional conserva- tiveness in the stability margins as possible. This is discussed in more detail following the statements of Lemma 3 and Theorem 5. The result and proof of Lemma 1 hold for the uncertain case ex- actly as stated, using the Q-norm for the system matrices and the appropriate operator norm for A,. As in the 1D case, Theorem 5 follows immediately from Lemmas 1 and 3. Theorem 5 Suppose (AI, RI) is a-expansive with a < 1 and IIIIR,,FIIQ <$.
If
the
behavioural system given by
(10)
with
(Al,
RI)
is
stable, then it is stable when (AI, RI) is replaced by any (Az, Rz) with
the property
11A1*
RI
-
A2
*
&IlQ
I
~6
Lemma
3
Suppose (R,A) is a-expansive wtth
0
<
a
<
1
and
The proof of Lemma
3
relies on the showing that the following
inequalities hold.
The relations between the norms above provide some insight
as
to
why we want
our
realizations
(A, R)
to be contractive, and
(I
<
1
as
close to
1
as
possible. Given contractiveness we can show
That
is,
we can determine a lower bound which provides information
on the conservativeness
of
the allowed niodel reduction error.
5
LMI
Solutions
for
Behavioural
Model
Re-
duction
Using the robust stability results formulated
for
behavioural systems in
the preceeding section, and the model reduction tecliiiiques discussed
in section
4.1, we arrive at
a
set of coupled LMI conditions which,
if satisfied, provide
a
robustly stable model reduction method for the
behavioural system framework. These
LMI
conditions are derived by
determining if either the system representation we are given,
or
an
equivalent
system representation, satisfies
a
set
of
norm bounds and Q-
stability requirements. We first discuss equivalence of diflerent system
representations for behavioural systems.
5.1
Equivalent System Representations
To
apply the model reduction tccliniqucs and robust stability analysis
previously described, stability and a-espusiveness (with
a
<
1)
of
representations,
(A,
R),
are required. Additionally, we want this rep-
resentation to be contractive. If the giveii representation matrix,
R
is
not stable, a-expansive and contractive, we determine
if
an
equivalent
representation exists which docs satisfy these constraints, where we use
the following notion of equivalence:
Definition
8
Given a representation (A,
R)
for a belravioural system,
an eqzriualent representation is dejined
by
any representation matrix
R
such that for each A
6
A
{W
:
(A
*
R)u
=
0)
I
{I3
:
(A
*
&)I3
=
0)
Predictably, transforming the system iliatrices of
R
by an allowable
similarity transformation rcsults
in
aii equivalent representation,
as
does multiplying the system matrices
C
aiid
D
on
the left by
a
constant
nonsingular matrix. Less obvious
is
the use of what is known
as
output
injection in the standard
IO
case, that is, using
a
constant matrix,
L,
to construct
a
new representation
R
=
{(A
+
LC),
(B
+
LD),
C,
D}.
Recall that using an output nulling representation implies that the
output is always
0,
thus, adding
L
*
0
=
LCs(k)
+
LDw
to
p
does not
affect the behaviour, resulting in an equivalent representation.
For
1D systems representations, all equivalent representations are
obtained by output injection, similarity transformations and trunca-
tions. Whether we obtain all equivalent MD/uncertain system repre-
sentations by these same methods remains
a
topic of current research.
3656

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18 Nov 1994
Abstract: We present model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation (LFT) on a repeated scalar uncertainty structure. These reduction methods can be interpreted either as doing state order reduction for multi-dimensionalsystems, or as uncertainty simplification in the case of uncertain systems, and are based on finding solutions to a pair of Linear Matrix Inequalities (LMIs). A related necessary and sufficient condition for the exact reducibility of stable uncertain systems is also presented.

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• ...These new results, which were first noted in [ 4 ], are based on machinery presented in [18]....

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Carolyn L. Beck1Institutions (1)
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Abstract: The emphasis of this thesis is on the development of systematic methods for reducing the size and complexity of uncertain system models. Given a model for a large complex system, the objective of these methods is to find a simplified model which accurately describes the physical system, thus facilitating subsequent control design and analysis. Model reduction methods and realization theory are presented for uncertain systems represented by Linear Fractional Transformations (LFTs) on a block diagonal uncertainty structure. A complete generalization of balanced realizations, balanced Gramians and balanced truncation model reduction with guaranteed error bounds is given, which is based on computing solutions to a pair of Linear Matrix Inequalities (LMIs). A necessary and sufficient condition for exact reducibility of uncertain systems, the converse of minimality, is also presented. This condition further generalizes the role of controllability and observability Gramians, and is expressed in terms of singular solutions to the same LMIs. These reduction methods provide a systematic means for both uncertainty simplification and state order reduction in the case of uncertain systems, but also may be interpreted as state order reduction for multi-dimensional systems. LFTs also provide a convenient way of obtaining realizations for systems described by rational functions of several noncommuting indeterminates. Such functions arise naturally in robust control when studying systems with structured uncertainty, but also may be viewed as a particular type of description for a formal power series. This thesis establishes connections between minimal LFT realizations and minimal linear representations of formal power series, which have been studied extensively in a variety of disciplines, including nonlinear system realization theory. The result is a fairly complete development of minimal realization theory for LFT systems. General LMI problems and solutions are discussed with the aim of providing sufficient background and references for the construction of computational procedures to reduce uncertain systems. A simple algorithm for computing balanced reduced models of uncertain systems is presented, followed by a discussion of the application of this procedure to a pressurized water reactor for a nuclear power plant.

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• ...Alternatively, a coprime factorization approach could be used; an initial attempt at this is discussed in [4]....

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• ...These new results, which were first noted in [4], are based on technical machinery presented in [54] and [56]....

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##### References
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Journal ArticleDOI
B. Moore1Institutions (1)
Abstract: Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations of X_{c}, X_{\bar{o}} , the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.

4,861 citations

Journal ArticleDOI
Keith Glover1Institutions (1)
Abstract: The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by Ĝ(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions Ĝ(s) + F(s) that minimize where Ĝ(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.

2,903 citations

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Andrew Packard1, John Doyle2Institutions (2)
TL;DR: A tutorial introduction to the complex structured singular value (μ) is presented, with an emphasis on the mathematical aspects of μ.
Abstract: A tutorial introduction to the complex structured singular value (μ) is presented, with an emphasis on the mathematical aspects of μ. The μ-based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using Linear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state space methods.

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Journal ArticleDOI
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Abstract: A self-contained exposition is given of an approach to mathematical models, in particular, to the theory of dynamical systems. The basic ingredients form a triptych, with the behavior of a system in the center, and behavioral equations with latent variables as side panels. The author discusses a variety of representation and parametrization problems, in particular, questions related to input/output and state models. The proposed concept of a dynamical system leads to a new view of the notions of controllability and observability, and of the interconnection of systems, in particular, to what constitutes a feedback control law. The final issue addressed is that of system identification. It is argued that exact system identification leads to the question of computing the most powerful unfalsified model. >

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### "Model reduction of behavioural syst..." refers background or methods in this paper

• ...For a more extensive treatment of these subjects, see references [ 27 ],[1], [11],[10] and [24]....

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• ...The most striking feature of the representation of dynamical systems in the behavioural framework, as proposed by Willems [ 27 ], is the fact that there are no explicit inputs and outputs....

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• ...A dynamical system is defined in a behavioural framework as follows [ 27 ]....

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