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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. >
Topics: Reduction (complexity) (55%)

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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

Citations
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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.

268 citations


Journal ArticleDOI
Carolyn L. Beck1, John Doyle1, Keith Glover2Institutions (2)
TL;DR: Model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation on a repeated scalar uncertainty structure and a related necessary and sufficient condition for the exact reducibility of stable uncertain systems are presented.
Abstract: Model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure. These methods involve a complete generalization of balanced realizations, balanced Gramians, and balanced truncation model reduction with guaranteed error bounds, based on solutions to a pair of linear matrix inequalities which generalize Lyapunov equations. The resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems.

241 citations


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

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Journal ArticleDOI
Carolyn L. Beck1Institutions (1)
TL;DR: The reduction method proposed is applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured l 2 -induced stability constraint required in the standard nonfactored case.
Abstract: We present a generalization of the coprime factors model reduction method of Meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models. A complete derivation of coprime factorizations for this class of systems is also given. The reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured l 2 -induced stability constraint required in the standard nonfactored case. Reduction error bounds in the l 2 -induced norm of the factorized mapping are given.

55 citations


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  • ...In this case, contractive and expansive coprime factors realizations may be considered instead; see [8] for a discussion relating to this topic....

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Journal ArticleDOI
Mazen Farhood1, Geir E. Dullerud2Institutions (2)
TL;DR: An approach is given for the model reduction of stabilizable and detectable systems, which requires the development and use of coprime factorizations for NSLPV models, and can be explicitly computed using semidefinite programming.
Abstract: This paper focuses on the model reduction of nonstationary linear parameter-varying (NSLPV) systems. We provide a generalization of the balanced truncation procedure for the model reduction of stable NSLPV systems, along with a priori error bounds. Then, for illustration purposes, this method is applied to reduce the model of a two-mass translational system. Furthermore, we give an approach for the model reduction of stabilizable and detectable systems, which requires the development and use of coprime factorizations for NSLPV models. For the general class of eventually periodic LPV systems, which includes periodic and finite horizon systems as special cases, our results can be explicitly computed using semidefinite programming

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DissertationDOI
01 Jan 1997
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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Cites background or methods from "Model reduction of behavioural syst..."

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

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References
More filters

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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.

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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
Jan C. Willems1Institutions (1)
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. >

1,186 citations


"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....

    [...]

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

    [...]


Book
11 Dec 1989
TL;DR: A load dumping vehicle including a frame, a gas turbine engine supported by the frame and a dump body pivotally connected to theframe for movement relative to the frame between a load carrying position and adump position.
Abstract: Preliminaries.- Robust stabilization of uncertain systems.- Robust stabilization of normalized coprime factor plant descriptions.- Reduced order controller design.- A loop shaping design procedure.- Design examples.

917 citations


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