ON A DETECTABILITY CONCEPT OF
DISCRETE-TIME INFINITE MARKOV JUMP LINEAR
SYSTEMS
⋆
Eduardo F. Costa
∗
Jo˜ ao B. R. do Val
∗,1
Marcelo D. Fragoso
∗∗
∗
UNICAMP - FEEC, Depto. de Telem´ atica, C.P. 6101,
13081-970, Campinas, SP, Brazil
∗∗
LNCC/CNPq, Av. Getulio Vargas 333, Quitandinha,
25651-070, Petr´ opolis, RJ, Brazil
Abstract: This paper introduces a concept of detectability for discrete-time infinite
Markov jump linear systems that relates the stochastic convergence of the output with
the stochastic convergence of the state. It is shown that the new concept generalizes a
known stochastic detectability concept and, in the finite dimension scenario, it is reduced
to the weak detectability concept. It is also shown that the detectability concept proposed
here retrieves the well known property of linear deterministic systems that observability
is stricter than detectability.
Keywords: Stochastic jump processes, Markov parameters, Markov models, systems
concepts, observability.
1. INTRODUCTION
This paper is concerned with the discrete-time Infinite
Markov jump linear system (MJLS) defined in a fixed
stochastic basis (Ω, F , (F
k
), P ) by
Ψ :
x(k + 1)= A
θ(k)
x(k), k ≥ 0,
y(k)= C
θ(k)
x(k), x(0)= x
0
, θ(0)= θ
0
(1)
where x and y are the state and the output variables,
respectively. The mode θ is the state of an underlying
discrete-time Markov chain Θ = {θ(k); k ≥ 0} taking
values in S = {1, 2,... } and having a stationary tran-
sition probability matrix P =[ p
ij
], i, j ∈ Z. θ
0
∈ S is a
random variable for which μ
i
= P(θ
0
= i), i ∈ S , and
x
0
is a second order random variable. It is assumed
that matrices A
i
and C
i
, i ∈ S , belong respectively
⋆
Research supported in part by FAPESP, Grant 98/13095-8, by
CNPq, Grant 300721/86-2(RN) and by the PRONEX Grant 015/98
’Control of Dynamical Systems’
1
Corresponding author. Email: jbosco@dt.fee.unicamp.br Fax:
55-19-3289 1395
to the collections of real matrices A =(A
1
, A
2
,... ),
dim(A
i
)= n × n, and C =( C
1
, C
2
,... ), dim( C
i
)= q ×
n, for which sup
i∈S
‖A
i
‖ < ∞ and sup
i∈S
‖C
i
‖ < ∞. We
also assume that x(k) and θ(k) are observed at each
time instant k.
When one deals with system Ψ, the usual detectability
concept is the stochastic detectability (S-detectability),
which is a dual concept of stochastic stabilizability;
see (Costa and Fragoso, 1995) in the same setting of
this paper, or (Fragoso and Baczynski, 2001) in the
continuous time case, or (Costa, 1995) and (Morozan,
1995) in the finite dimension case. However, the S-
detectability concept presents the drawbacks pointed
out in the sequel.
Consider the weak observability (W-observability)
concept that follows from the extension of the finite
state space case, see Section 4. It appears in (Costa
and do Val, n.d.b), (Costa and do Val, 2001) and
(Morozan, 1995), and it is more general than other
observability concepts for MJLS, like the ones appear-
Copyright © 2002 IFAC
15th Triennial World Congress, Barcelona, Spain