Balanced Realizations of Regime-Switching Linear
Systems
∗
Y.J. Liu
†
G. Yin
‡
Q. Zhang
§
J.B. Moore
¶
March 23, 2006
Abstract
In this work, we construct a framework for balanced realization of linear systems
subject to regime switching modulated by a continuous-time Markov chain with a finite
state space. First, definition of balanced realization is given. Then a ρ-balanced realiza-
tion is developed. When the state space of the Markov chain is large, the computational
effort becomes a real concern. To resolve this problem, we introduce a two-time-scale
formulation and use decomposition/aggregation and averaging techniques to reduce
the computational complexity. Approximation procedures are developed. Numerical
examples are also presented for demonstration.
Key words. balanced realization, continuous-time Markov chain, decomposition, ag-
gregation, time-scale separation.
Brief Title. Balanced Realizations for Switching Systems
∗
The research was supported in part by the National Science Foundation and in part by the National ICT
Australia (NICTA), which is funded by the Australian Government Backing Australia’s Ability Initiative in
part through the Australian Research Council, and in part by the ARC discovery grants A00105829 and
DP0450539.
†
Department of Mathematics, Missouri Southern State University, Joplin, MO 64801, liu-y@mssu.edu
‡
Department of Mathematics, Wayne State University, Detroit, MI 48202, gyin@math.wayne.edu
§
Department of Mathematics, The University of Georgia, Athens, GA 30602, qingz@math.uga.edu
¶
Department of Systems Engineering, Research School of Information Sciences and Engineering, Aus-
tralian National University, Canberra, ACT 0200, Australia, john.moore@anu.edu.au
1
1 Introduction
This work is concerned with balanced realizations (or canonical realizations) of regime-
switching linear systems, in which continuous dynamics and discrete events coexist. The
underlying systems are modulated by a continuous-time Markov chain with state space
M = {1, . . . , m}. The matrix coefficients of the linear systems depend on the states of the
Markov chain. At any given instant t , the Markov chain takes one of the values from M (e.g.,
i). Then the system dynamics are determined by the matrix coefficients associated with the
state i. After a random sojourn time, the Markov chain switches to a new state j 6= i and
stays there until the next jump; at this time the system dynamics are determined by the
matrix coefficients associated with j rather than i. Such random jump linear systems arise
frequently in wireless communications, manufacturing systems, financial engineering, and
other applications, where regime changes are utilized to formulate the random environment.
For some of the recent work on jump linear systems and their applications, we refer the
reader to [10, 11, 21, 22, 23, 28] among others.
Balanced realizations have been studied for finite dimensional linear systems over the
past two decades. In the 1980s, to bridge the gap between minimal realization theory and to
treat the problem of finding lower order approximation, a “canonical” realization was first in-
troduced in the seminal paper [13], where the problem was studied with principal component
analysis of linear systems. The term “balance” was used since the realizations have certain
symmetry between the input and the output maps characterized by the controllability and
observability Grammians. Owing to its importance and its wide range of applicability, bal-
anced realizations have attracted much attention. One of its main applications is in model
reduction. While asymptotic stability of the reduced order systems was studied in [13], error
bounds between the reduced order model and the original system was obtained in [6] in
terms of associated singular values. There have been substantial extensions to time-varying
linear systems. Main existence results concerning balanced realizations are contained in [16]
and [19]. Subsequently, further work in this directions can be found in [8, 9, 14, 15] and
references therein.
In this paper, we treat regime-switching linear systems or Markov jump linear systems.
The systems under consideration are only piecewise deterministic, whereas the time-varying
random switchings result in a much more complex structure. Our focus is on the development
of “canonical realization.” We keep using the term “balanced realization” although our setup
is different from that of the deterministic linear systems studied in the literature. Important
questions concerning such models include: How should the canonical realization be defined
2
in such a setting? Can the desired canonical form of these systems be reached? One of
the main thoughts used in [9] is: In lieu of using exact balancing solutions in the time-
varying case, we construct the so-called ρ-balancing solutions, where ρ grows monotonically
to ∞ and is a natural tracking parameter. [In fact, the symbol µ in lieu of ρ is used in
[9].] That is, to approximate the algebraic problems by dynamic systems with differential
Riccati equations, rather than finding the exact solutions of balancing equations (time-
varying algebraic equations). When regime switching is involved, we must solve not one, but
a system of Riccati equations. Can error bounds of the approximation still be derived in
such cases? In addition, when the number of the states of the Markov chain is large, we face
large-scale systems, for which the amount of computation could be of real concern. How can
we reduce the computation complexity? This paper aims to address these questions and to
contribute to the following:
1. We introduce a novel model for canonical or balanced realizations of systems with
regime switching. The corresponding switching mechanism regulates the moves from
one configuration to another. The resulting system is piecewise deterministic.
2. We show how the balanced realization problem can be carried out for the switching
models and demonstrate how systems with regime switching can be handled. Due to
the time-varying feature, the balancing equation is a system of time-varying equations
to be satisfied at all t. There is no feasible procedures to handle such a task. In
addition, in contrast to [9], another difficulty arises since we have to deal with a cou-
pled system of Riccati equations. As a remedy, we also use ρ-approximated balanced
realizations (solutions of systems of differential Riccati equations) to approximate the
system of algebraic balancing equations. Under such a setting, we derive error bounds
for approximating the solution of the system of balancing equations by that of the
differential system. Our numerical results show that the ρ-approximated balance pre-
serves the robustness and the ability of coping with ill-conditioned controllability or
observability Grammians.
3. We device a strategy for further reducing the computation complexity of the underlying
system to treat large-scale systems. In such a setting, on top of the difficulties men-
tioned in the previous paragraph, the Markov chain has a large state space. It follows
that solving the system of ρ-approximated balanced realization can be a computational
infeasible task due to the inherent large dimensionality. To overcome the difficulties,
we present a methods based on a two-fold approximation, namely, a two-time-scale
3
approximation and a ρ-approximated balance realization. We use decomposition to
take advantage of the natural hierarchical structure of the systems, use an aggregation
procedure to reduce the total number of states under consideration, and use averaging
methods to design approximation strategy.
The rest of the paper is arranged as follows. Section 2 begins with canonical realizations
when a Markov chain is the modulating force. Section 3 is concerned with reducing complex-
ity by means of time-scale separation. Several numerical examples are presented in Section
4 for demonstration purposes. Some further remarks are made in Section 5.
2 Balanced Realizations of Regime-Switching Systems
2.1 Problem Formulation and Preliminary Results
Suppose that α(t) is a continuous-time Markov chain with finite state space M = {1, 2, . . . , m}
and generator
Q = (q
ij
) ∈ R
m×m
such that q
ij
≥ 0 for i 6= j, and
m
X
j=1
q
ij
= 0. (2.1)
Consider the following system
d
dt
x(t) = A(α(t), t)x(t) + B(α(t), t)u(t), x(t
0
) = x
0
,
y(t) = C(α(t), t)x(t),
(2.2)
where the state x(t) ∈ R
n×1
, input u(t) ∈ R
r×1
, and output y(t) ∈ R
m
0
. Note that since a
finite-state Markov chain is used, effectively, (2.2) can be written as
d
dt
x(t) =
m
X
i=1
A(i, t)x(t)I
{α(t)=i}
+
m
X
i=1
B(i, t)u(t)I
{α(t)=i}
, x(t
0
) = x
0
,
y(t) =
m
X
i=1
C(i, t)x(t)I
{α(t)=i}
,
where I
S
is the indicator function of the set S. In what follows, if a square matrix D is
positive definite (resp. nonnegative definite), we often write it as D > 0 (resp. D ≥ 0).
For D
1
∈ R
ι×`
for some ι, ` ≥ 1, D
0
1
denotes its transpose. Throughout the paper, for each
i ∈ M and a suitable function f(t, i), we denote
Qf(t, ·)(i) =
m
X
j=1
q
ij
f(t, j) =
X
j6=i
q
ij
(f(t, j) − f(t, i)) for each i ∈ M.
To proceed, we assume condition (A1) holds.
4
(A1) For each i ∈ M, A(i, ·), B(i, ·), C(i, ·) are bounded and continuously differentiable
functions.
Definition 2.1. For each t ≥ 0 and i ∈ M, a realization (A(i, t), B(i, t), C(i, t)) is said to be
uniformly completely controllable if and only if there is a δ > 0 such that for some positive
L
c
(δ) and U
c
(δ),
∞ > U
c
(δ)I ≥ G
c
(t − δ, t, i) ≥ L
c
(δ)I > 0, (2.3)
where G
c
is the controllability Grammian
G
c
(t − δ, t, i) =
Z
t
t−δ
Φ(t, λ, i)B(i, λ)B
0
(i, λ)Φ
0
(t, λ, i)dλ, (2.4)
and Φ(t, λ, i) is the state transition matrix (see [2, p. 349]) of the equation
dz(t)
dt
= A(i, t)z(t).
Definition 2.2. For each t ≥ 0 and i ∈ M, a realization (A(i, t), B(i, t), C(i, t)) is said to
be uniformly completely observable if and only if there is a δ > 0 such that for some positive
L
o
(δ) and U
o
(δ),
∞ > U
o
(δ)I ≥ G
o
(t, t + δ, i) ≥ L
o
(δ)I > 0, (2.5)
where G
o
is the observability Grammian
G
o
(t, t + δ, i) =
Z
t+δ
t
Φ
0
(λ, t, i)C
0
(i, λ)C(i, λ)Φ(λ, t, i)dλ. (2.6)
As noted in [9, p. 316], if uniform complete observability and controllability hold for δ
then it also holds for all
b
δ > δ. Although the balancing condition depends on δ, the results
to be developed does not depend on the δ selection. For notational simplicity, we suppress
δ and write G
c
(t − δ, t, i) as G
c
(t, i) and G
o
(t, t + δ, i) as G
o
(t, i) in what follows.
Definition 2.3. The system (2.2) is said to have a balanced realization (or canonical realiza-
tion) if there are nonsingular coordinate transformations T (t, i) with P (t, i)
def
= T
0
(t, i)T (t, i) >
0 such that
P (t, i)G
c
(t, i)P (t, i) = G
o
(t, i) + QP (t, ·)(i), i = 1, 2, . . . , m.
(2.7)
Our objective is to find balanced realizations of the regime-switching linear systems. To
simplify the notation, we write (2.7) in a matrix form. To proceed, introduce the following
notation:
P (t) = diag(P (t, 1), . . . , P (t, m)),
G
c
(t) = diag ( G
c
(t, 1), . . . , G
c
(t, m)),
G
o
(t) = diag(G
o
(t, 1), . . . , G
o
(t, m)),
(2.8)
5
