ON THE STABILITY OF THE DISCRETE TIME JUMP LINEAR
SYSTEM
Adam Czornik
Silesian Technical University, Department of Automatic Control and Robotics
Akademicka Street 16, 44-100 Gliwice, Poland
Aleksander Nawrat
Silesian Technical University, Department of Automatic Control and Robotics
Akademicka Street 16, 44-100 Gliwice, Poland
Keywords:
Jump linear systems, stability, stabilizability.
Abstract:
In this paper we investigate the relationships between individual mode stability and mean square stability of
jump linear system. It is well known that generally stability of a dynamical system in all its modes does not
guarantee stability of the jump linear system defined by all these modes. We present conditions under which
stability of all modes implies the mean square stability of the overall system.
1 INTRODUCTION
Linear dynamical systems with Markovian jumps in
parameter values have recently attracted a great deal
of interest. The main reason is that such systems may
serve as models for a large variety of industrial con-
trol processes. One class of examples is given by non-
linear control plants characterized by linearized mod-
els corresponding to several operating points which
appear due to abrupt changes. Such problems are
typical in control systems of a solar thermal central
receiver (Sworder and Rogers, 1983), electric load
modeling (Malhame and Chong, 1985), aircraft flight
control systems (Moerder et al., 1989) or ship autopi-
lot systems (Astrom and Wittenmark,1989). Markov-
ian jumps in parameter values may also result from
the random failure/repair processes( see e.g. (Siljak,
1980), (Willsky, 1976), (Petkovski, 1987)) and result-
ing fault prone control systems ((Siljak, 1980), (Sil-
jak, 1980), (Swierniak et al., 1998)). Yet another
class of processes with Markov jumps could be met
in flexible manufacturing systems (see e.g. (Boukas
and Haurie, 1990), (Boukas et al., 1995), (Boukas and
Liu, 2001)). Moreover in (Athans, 1987) Athans sug-
gested that such a model setting has also the poten-
tial to become a basic framework in solving control-
related issues in battle management, command, and
communications systems.
This paper deals with the problem of stability of the
linear discrete time dynamical systems with Markov-
ian jumps in the parameters values. This problem has
been investigated in a number of papers (see e. g.
(Fang and Loparo, 2002), (Ji and Chizeck, 1990a),
(Feng et al., 1992), (Li et al., 2000)) not only be-
cause of its theoretical complexity but also because
its broad practical implications. The way of modeling
is to represent the overall system by a number of de-
terministic linear plants and a Markov chain the states
of which called modes define which among these de-
terministic plants describes the dynamics of the sys-
tem in given time. The interesting but well known
property of the jump linear systems (see e.g. (Fang
and Loparo, 2002)) is that deterministic stability of
all system matrices in all modes is neither necessary
nor sufficient condition for mean square stability of
the considered process.
More precisely let us fix an underlying probability
space {Ω, F, P } and consider the linear system with
Markovian jumping parameters
x (n + 1) = A(r(n))x (n) , (1)
or it controlled analogue
x (n + 1) = A(r(n))x (n) + B(r(n))u (n) , (2)
where x(n) ∈ R
l
denotes the state vector, u(n) ∈
R
m
is the control input, r(n) is a Markov chain which
takes values in a finite set S = {1, 2, ..., s} with tran-
sition probability matrix P = [p (i, j)]
i,j∈S
and ini-
tial distribution P (r (0) = i
0
) = 1. Furthermore,
for r(n) = i, A
i
:= A(i) and B
i
:= B(i) are
constant matrices of appropriate sizes. Denote by
x (n, x
0
, i
0
) the solution of (1) with initial condition
x(0) = x
0
and initial distribution P (r (0) = i
0
) = 1
at time n = 0. The control u (n) is assumed to be
75
Czornik A. and Nawrat A. (2006).
ON THE STABILITY OF THE DISCRETE TIME JUMP LINEAR SYSTEM.
In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 75-78
DOI: 10.5220/0001202000750078
Copyright
c
 SciTePress

measurable with respect to the σ−field generated by
{r(0), ..., r(n)} .
In the literature three types of stability of system (1)
are considered: mean square stability, moment sta-
bility and almost sure stability. In this paper we in-
vestigate the mean square stability and stabilizability.
Their formal definitions are as follows.
Definition 1 System (1) is mean square stable( MS
stable), if for all (x
0
, i
0
) ∈ R
l
× S, we have
lim
n−→∞
E kx (n, x
0
, i
0
)k
2
= 0.
In that case we call the pair (A, P ), where A =
(A
1
, ..., A
s
) mean square stable. System (2) is mean
square stabilizable (MS stabilizable) if there exists a
feedback control u(n) = K(r(n))x(n) such that the
resulting closed loop system is mean square stable.
In that case we call (A, B, P ) MS stabilizable, where
B = (B
1
, ..., B
s
) .
The natural question which we have mentioned at
beginning is how the individual properties of matrices
A
i
or pairs of matrices (A
i
, B
i
) such as deterministic
stability or deterministic stabilizability (controllabil-
ity) are related to the MS stability of (1) or MS stabi-
lizability of (2).
It is known that deterministic stability of all ma-
trices A
i
, i ∈ S is neither necessary nor sufficient
for MS of (1) (see (Ji and Chizeck, 1990b), Examples
6.1). However the deterministic stability of each ma-
trices
p
p(i, i)A
i
is necessary for MS stability (see
Theorem 2.1 in (Ji and Chizeck, 1990b)). It is also
known that individual deterministic stability of each
pair (A
i
, B
i
) is neither sufficient nor necessary (see
Example 6.2 in (Ji and Chizeck, 1990b)) for MS sta-
bilizability of (2). Moreover the individual determin-
istic controllability of each pair (A
i
, B
i
) is not suffi-
cient for MS stabilizability of (2) (see Example 6.2 in
(Ji and Chizeck, 1990b)).
It is also known that if the switching signal r in (1)
is deterministic and all matrices A
i
, i ∈ S are stable,
then it is possible to ensure the stability of the sys-
tem by switching sufficiently slowly. This means that
instability arises in (1) as a result of rapid switching.
Given this basic fact, a natural and obvious method to
ensure the stability of (1) is to somehow constrain the
rate at which switching takes place. This idea has ap-
peared in many studies on time varying systems (see,
(Guo and Rugh, 1995), (Ilchmann et al., 1987)).
In this paper we consider the system (1) with matri-
ces A
i
, i ∈ S being stable and investigate the problem
how to characterize the transition probabilities P for
which the system is MS stable. We also propose cer-
tain new sufficient conditions for MS stability of (1)
in terms of spectra of matrices A
i
and basing on this
result we propose new sufficient conditions for MS
stabilizability of (2) given in terms of individual pair
of matrices (A
i
, B
i
) . Finally we present a procedure
for MS stabilization of system (2) by pole placement
of pairs (A
i
, B
i
) .
2 STABILITY OF MATRICES A
I
VERSUS MS STABILITY
The necessary and sufficient conditions for mean
square stability of (1) are given by the following the-
orem (see, (Costa and Fragoso, 1993) ).
Theorem 1 The system (1) is mean square stable if
and only if there exists a positive definite solution P
i
,
i ∈ S of the following coupled Lyapunov inequality
P
i
− A
′
i


X
j∈S
p
ij
P
j


A
i
> 0.
For a symmetric matrix X of size n by n denote
λ
1
(X), ..., λ
n
(X) the eigenvalues arranged such that
λ
1
(X) ≥ ... ≥ λ
n
(X). We have the following theo-
rem.
Theorem 2 If matrices A
i
, i ∈ S are such that
λ
1
(A
i
A
′
i
) < 1, and there exists a sequence of pos-
itive numbers (x
i
)
i∈S
such that
x
i
λ
1
(A
i
A
′
i
)
>
X
j∈S
j6=i
p
ij
x
j

1 − λ
1

A
j
A
′
j

, i ∈ S. (3)
Then the system (1) is mean square stable.
Proof. Define the following n by n matrices
Q
i
= x
i
· I.
If λ
1
(A
i
A
′
i
) < 1, i ∈ S then by inequality
|λ
1
(A
i
)| ≤
q
λ
1
(A
i
A
′
i
),
(see (Weyl, 1949)) |λ
1
(A
i
)| < 1, matrices A
i
are
stable and consequently discrete Lyapunov equation
P
i
− A
′
i
P
i
A
i
= Q
i
, (4)
has a positive definite solution P
i
. Moreover the as-
sumption that λ
1
(A
i
A
′
i
) < 1 implies that the solu-
tion has the following upper bound (see (Yasuda and
Hirai, 1979))
λ
1
(P
i
) ≤
λ
1
(Q
i
)
1 − λ
1
(A
i
A
′
i
)
.
Using this inequality with the fact that λ
1
(Q
j
) = x
j
we have
λ
1
(P
j
) ≤
x
j
1 − λ
1

A
j
A
′
j

.
ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
76

Therefore
X
j∈S
j6=i
p
ij
P
j
≥
X
j∈S
j6=i
p
ij
x
j
1 − λ
1

A
j
A
′
j

· I
and
A
′
i




X
j∈S
j6=i
p
ij
P
j




A
i
≥




X
j∈S
j6=i
p
ij
x
j
1 − λ
1

A
j
A
′
j





λ
1
(A
i
A
′
i
) .
Using the last inequality together with (4) we have
P
i
− A
′
i


X
j∈S
p
ij
P
j


A
i
= P
i
− p
ii
A
′
i
P
i
A
i
−
A
′
i




X
j∈S
j6=i
p
ij
P
j




A
i
≥
P
i
− A
′
i
P
i
A
i
− A
′
i




X
j∈S
j6=i
p
ij
P
j




A
i
≥≥
x
i
· I − A
′
i




X
j∈S
j6=i
p
ij
P
j




A
i




x
i
− λ
1
(A
i
A
′
i
)
X
j∈S
j6=i
p
ij
x
j
1 − λ
1

A
j
A
′
j





· I
and from the assumption 3 we conclude that
P
i
− A
′
i


X
j∈S
p
ij
P
j


A
i
> 0,
and on the basis of Theorem 1, (1) is MS stable.
Using the Theorem 2 with x
i
= 1, i ∈ S and ob-
serve that
X
j∈S
j6=i
p
ij

1 − λ
1

A
j
A
′
j

≤




X
j∈S
j6=i
p
ij




max
j∈S
j6=i
1
1 − λ
1

A
j
A
′
j

!
=
(1 − p
ii
)
1
min
j∈S
j6=i

1 − λ
1

A
j
A
′
j

=
1 − p
ii
1− max
j∈S
j6=i
λ
1

A
j
A
′
j

we obtain the following Corollary.
Corollary 3 If matrices A
i
, i ∈ S are such that
λ
1
(A
i
A
′
i
) < 1, and
λ
1
(A
i
A
′
i
) + max
j∈S
j6=i
λ
1

A
j
A
′
j

− 1
λ
1
(A
i
A
′
i
)
< p
ii
, i ∈ S.
(5)
Then the system (1) is mean square stable.
In the next theorem we will propose a design proce-
dure for MS stabilization of system (2) under the addi-
tional constrains on the transition rates matrix. How-
ever it is worth to notice another possible application
of this result. Sometimes (see (Ji and Chizeck, 1989)
for details) it is reasonable to consider the system (2)
with matrix P which is separately controlled. In that
case the Theorem 2 may be used to find the values of
P for which the system is MS stable.
As a straightforward consequence of Theorem 2
and the definition of MS stabilizability we have the
following result.
Theorem 4 Suppose that for system (2) there ex-
ist positive numbers (x
i
)
i∈S
and feedback matrices
(K
i
)
i∈S
, such that
x
i
λ
1

e
A
i
e
A
′
i

>
X
j∈S
j6=i
p
ij
x
j

1 − λ
1

e
A
j
e
A
′
j

, i ∈ S,
where
e
A
i
= A
i
+ K
i
B
i
, then (A, B, P ) is MS stabi-
lizable and (K
i
)
i∈S
is a stabilizable feedback.
3 CONCLUSIONS
We have discussed links between individual mode sta-
bility (stabilizability) and MS stability (MS stabiliz-
ability) of jump linear system. The main result states
that if the individual mode are stable then the jump
linear system is mean square stable provided that the
probabilities p
ii
are close to 1. It means that it is
possible to ensure the stability of the jump system
by switching sufficiently slowly between individual
modes. Another consequence of the main result is a
procedure to stabilize the jump linear system by pole
placement of system matrices for all modes.
ON THE STABILITY OF THE DISCRETE TIME JUMP LINEAR SYSTEM
77

ACKNOWLEDGEMENTS
The work has been supported by KBN grant No 0
T00B 029 29 and 3 T11A 029 028.
REFERENCES
K. J. Astrom and B. Wittenmark, Adaptive Control,
Addison-Wesley, 1989.
M. Athans, Command and control (C2) theory: A challenge
to control science, IEEE Transactions on Automatic
Control, vol. 32, 286-293, 1987.
E. K. Boukas and A. Haurie, Manufacturing flow control
and predictive maintenance : stochastic control ap-
proach, IEEE Transactions on Automatic Control, vol.
35, 1024-1031, 1990.
E. K. Boukas, Q. Zhang and G. Yin, Robust production
and maintenance planning in stochastic manufactur-
ing systems, IEEE Transaction on Automatic Control,
vol. 40, 1098-1102, 1995.
E. K. Boukas, Z. K. Liu, Production and maintenance con-
trol for manufacturing systems, IEEE Transactions on
Automatic Control, vol. 46, no. 9, pp. 1455-1460,
2001.
O. L.V. Costa, M. D. Fragoso, Stability results for discrete-
time linear systems with Markovian jumping parame-
ters, Journal of Mathematical Analysis and Applica-
tions, vol. 179, No. 1, pp. 154-178, 1993.
Y. Fang, A. Loparo, Stabilization of continuous-time jump
linear systems, IEEE Transactions on Automatic Con-
trol, vol. 47, pp. 1596-1603, 2002.
X. Feng, K. A. Loparo, Y. Ji, H. J. Chizeck, Stochastic sta-
bility properties of jump linear sytems, IEEE Transac-
tions on Automatic Control, vol. 37, pp. 38-53, 1992.
D. Guo, W. Rugh, A stability result for linear parameter-
varying systems, Systems and Control Letters, vol. 24,
pp. 1-5, 1995.
A. Ilchmann, D. Owens, Pratzel-Wolters, Sufficient condi-
tions for stability of linear time-varying systems, Sys-
tems and Control Letters, vol. 9, pp. 157-163, 1987.
Y. Ji, H.J. Chizeck, Controllability, stability, and
continuous-time Markovian jump linear quadratic
control, IEEE Transactions on Automatic Control,
vol. 35, pp. 777-788, 1990.
Y. Ji, H. Chizeck, Optimal quadratic control of jump linear
systems with seperately controlled transition probabil-
ities, International Journal of Control, vol 49, no. 2,
481-491, 1989.
Y. Ji, H. Chizeck, Jump linear quadratic Gaussian con-
trol: steady-state solution and testable conditions,
Control-Theory and Advanced Technology, Vol. 6,
No.3, pp.289-319, 1990.
Z. G. Li, Y. C. Soh, C. Y. Wen, Sufficient conditions for
almost sure stability of jump linear systems, IEEE
Transactions on Automatic Control, vol. 45, no. 7, pp.
1325-1329, 2000.
R. Malhame and C. Y. Chong, Electric load model synthe-
sis by diffusion approximation of a high-order hybrid-
state stochastic system, IEEE Transactions on Auto-
matic Control, vol. 30, pp. 854-860, 1985.
D. D. Moerder, N. Halyo, J. R. Broussard and A. K.
Caglayan, Application of precomputed control laws
in a reconfigurable aircraft flight control system, Jour-
nal of Guidance, Control and Dynamics, vol. 12, 324-
333,1989
D. B. Petkovski, Multivariable control system design : a
case study of robust control of nuclear power plants,
Fault Detection and Reliability, vol. 9, 239-246, 1987.
D. D. Siljak, Reliable control using multiple control sys-
tems, International Journal on Control, vol. 31, no 2,
pp. 303-329, 1980.
D. D. Sworder and R. O. Rogers, An LQ- solution to a con-
trol problem associated with solar thermal central re-
ceiver, IEEE Transactions on Automatic Control, vol.
28, pp. 971-978, 1983.
D. D. Siljak, Reliable control using multiple control sys-
tems, International Journal of Control, vol. 31, no. 2,
303-329, 1980.
A. Swierniak, K. Simek and E. K. Boukas, Intelligent ro-
bust control of fault tolerant linear systems, in : H. E.
Rauch (ed.), Artificial Intelligence in Real-Time Con-
trol, Pergamon, 245-249, 1998.
K. Yasuda, K. Hirai, Upper and lower bounds on the so-
lution of the algebraic Riccati equation, IEEE Trans-
actions on Automatic Control, vol. 24, pp. 483-487,
1979.
H. Weyl, Inequalities between the two kinds of eigenvalues
of linear transformation, Proc. Nat. Acad. Sci., vol. 35,
pp. 408-411, 1949.
A . Willsky, A survey of design methods for failure detec-
tion in dynamics systems, Automatica, vol. 12, 601-
611, 1976.
ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
78