Stability and Stabilization of Discrete-Time Semi-Markov Jump
Linear Systems via Semi-Markov Kernel Approach
Lixian Zhang, Yusong Leng, and Patrizio Colaneri, Fellow, IEEE
Abstract—This technical note is concerned with the problems
of stability and stabilization for a class of discrete-time semi-
Markov jump linear systems (S-MJLSs). The discrete-time semi-
Markov k ernel (SMK) is introduced, wher e the probability density
function of sojourn-time is dependent on both current and next
system mode. As a consequence, different types of distributions
and/or different parameters in a same type of distribution of
sojourn-time, depending on the target mode towards which the
system jumps, can coexist in each mode of a SMK. The underlying
S-MJLSs are therefore more general than those considered in ex-
isting studies. A new stability concept generalizing the traditional
mean-square stability is proposed such that numerically testable
criteria on the basis of SMK are obtained. Numerical examples are
presented to illustrate the v alidity and advantage of the developed
theoretical results.
Index Terms—Mean-square stability, semi-Markov jump linear
systems, semi-Markov kernel, sojourn-time.
I. I
NTRODUCTION
The past decades have seen a great advance in theories and ap-
plications of Markov jump linear systems (MJLSs). The systems
can effectively model dynamical processes involved with stochastic
switching (generally autonomous) subject to a Markov chain. Typ-
ical examples include fault-tolerant control systems where abrupt
faults occur randomly and networked control systems where network-
induced communication imperfections vary in a stochastic way, see
for example [1] and [2]. The primary concern of MJLSs is the stability
analysis that is relatively challenging due to the hybrid nature of the
system. Up to date, quite a few important stability notions, stochastic
stability, mean-square stability (MSS), almost sure stability and so on,
are persistently utilized and many results are rather fundamental in
the field of stochastic switching systems, see for example, [3] and [4].
Other issues including control, estimation, model reduction of MJLSs
and the underlying systems with various complex dynamics, time-
delays, uncertainties, positiveness, etc., have also been intensively
studied, see for example, [5]–[7] and the references therein.
However, as pointed out in [8]–[12], although Markov processes or
Markov chain do have the ability in describing the mode switching in
Manuscript received April 24, 2014; revised March 4, 2015; accepted
May 20, 2015. Date of publication June 1, 2015; date of current version
January 26, 2016. This work was supported in part by the National Natural
Science Foundation of China under Grant 61322301, the National Natural
Science Foundation of Heilongjiang under Grants F201417 and JC2015015,
and the Fundamental Research Funds for the Central Universities, China
HIT.BRETIII.201211 and HIT.BRETIV.201306. Recommended by Associate
Editor C. Belta.
L. Zhang and Y. Leng are with the School of Astronautics, Harbin Institute of
Technology, Harbin 150080, China (e-mail: lixianzhang@hit.edu.cn; yusong-
leng@hit.edu.cn).
P. Colaneri is with the Dipartimento di Elettronica, Informazione e Bioingeg-
neria, Politecnico di Milano and CNR-IEIIT, 20133 Milano, Italy (e-mail:
colaneri@elet.polimi.it).
Color versions of one or more of the figures in this paper are available
online.
many practical applications, they cannot cover all the scenarios yet.
A key restriction in the underlying MJLSs is that the sojourn-time
(the interval between two consecutive jumps) of each subsystem is
subject to exponential distribution (geometric distribution in discrete-
time domain, respectively). To relax the restriction, the concepts of
nonhomogeneous Markov chain and semi-Markov chain, where the
transition probabilities (TPs) are time-varying and memory, respec-
tively, have been introduced in the control community, and studies on
the corresponding systems have been gradually launched. As for the
former case that is relatively tractable, the systems with dwell-time
1
switching TPs [10], piecewise homogeneous TPs [11], and polytopic
time-varying TPs [12], etc., have been proposed within the past few
years, and various methodologies have been explored for different
situations. Such proposals have provided a basic foundation for further
studies on the systems with nonhomogeneous TPs.
On the other hand, the developed theories on semi-Markov jump
linear systems (S-MJLSs) are far away from maturity yet, although
the systems have been investigated since 1960s, see for example,
[8], [9], [13] and [14]. The inherent difficulties mainly lie in how
the probability density function (PDF) information of the sojourn-
time can be completely used in deriving criteria of stability analysis
and control synthesis, and further, how the obtained criteria can be
numerically tested. With the assumption that the PDF is dependent
on the current system mode, the stabilization problem for a class
of continuous-time S-MJLSs has been addressed in [8] by solving
a set of coupled algebraic Riccati equations. Further improvements
achieved in [9] offer a framework under which the control problems
can be solved by the techniques of linear matrix inequalities (more
conveniently to be checked), with aprioriinformation of the upper
and lower bounds of a PDF. However, it is noted that almost all the
existing results presume that each mode possesses a single distribution
of sojourn-time with certain parameters in a semi-Markov chain. It
is very likely that the types of distributions and/or the parameters in
a same type of distribution of sojourn-time can be different for each
mode, depending on the target mode to which the system jumps from
the current mode.
Motivated by the above observations, in this technical note, we aim
at addressing the problems of stability and stabilization for a class of
discrete-time S-MJLSs. There are two main contributions. First,the
concept of discrete-time semi-Markov kernel (SMK) is introduced,
and the PDF of sojourn-time in the SMK is allowed to depend on both
the current and next system mode. As a result, different parameters
in a same type of distribution or even different types of distributions
of sojourn-time can be simultaneously considered in each mode of
a SMK depending on the target mode, and the underlying S-MJLSs
are therefore more general than those studied previously. Second,a
new stability concept called σ-error mean square stability (σ-MSS) is
proposed, where σ is used to characterize the degree of “approximation
error” of σ-MSS to MSS (without the error the underlying system
holds MSS). The stability concept generalizes the traditional MSS,
1
A term commonly used in the context of nondeterministic switched systems,
where dwell-time is generally no greater than the running time (or sojourn-time
called in this technical note) of an activated subsystem.
Fig. 1. Illustration of stochastic processes R
n
, k
n
and S
n
(M =3).
based on which the numerically testable stability and s tabilization
criteria that explicitly contains the PDF information of sojourn-time
can be obtained.
Notations: In this technical note, R
n
denotes the n-dimensional
Euclidean space; ·refers to the Euclidean vector norm; R
+
and Z
+
denote the set of non-negative real numbers and set of non-negative
integers, respectively; R
[s
1
,s
2
]
, Z
≥s
1
and Z
[s
1
,s
2
]
denote the sets {k ∈
R
+
|s
1
≤ k ≤ s
2
}, {k ∈ Z
+
|k ≥ s
1
} and {k ∈ Z
+
|s
1
≤ k ≤ s
2
},re-
spectively. For notation (Ψ, F, Pr), Ψ represents the sample space,
F is the σ-algebra of subsets of the sample space and Pr is the
probability measure on F. C
1
denotes the space of continuously
differentiable functions, and a function κ :[0, ∞) → [0, ∞) is said to
be of class K
∞
if it is continuous, strictly increasing, unbounded, and
κ(0) = 0. In addition, diag{···} stands for a block-diagonal matrix
and diag
(n)
{X} a n × n block-diagonal matrix where all diagonal
entries are X. Symbol ∗ is used as an ellipsis for the terms that are
introduced by symmetry.
II. P
RELIMINARIES AND PROBLEM FORMULATION
Fix the complete probability space (Ψ, F, Pr) and consider the
following discrete-time stochastic switching systems:
x(k +1)=A(r
k
)x(k)+B(r
k
)u(k) (1)
where x(k) ∈ R
n
, u(k) ∈ R
n
u
are the system state and control in-
put, respectively. {r
k
}
k∈Z
+
is a stochastic process, considered to
be a semi-Markov chain, which takes values in a finite set I
Δ
=
{1, 2,...,M}, and governs the switching among M system modes.
For r
k
= i ∈I, the pair of matrices of the ith system mode is
denoted by (A
i
,B
i
), which are real known matrices with appropriate
dimensions.
To introduce the semi-Markov chain formally, we shall recall two
concepts on Markov renewal chain and semi-Markov kernel, for which
three following stochastic processes are first needed:
i) The stochastic process {R
n
}
n∈Z
+
taking values in I,whereR
n
is the index of system mode at the nth jump;
ii) The stochastic process {k
n
}
n∈Z
+
taking values in Z
+
,wherek
n
denotes the time at the nth jump. It is noted that k
0
=0and k
n
increases monotonically with n;
iii) The stochastic process {S
n
}
n∈Z
+
taking values in Z
+
,where
S
n
= k
n
− k
n−1
, ∀n ∈ Z
≥1
denotes the sojourn-time of mode
R
n−1
between the (n − 1)th jump and nth jump, and S
0
=0.
These stochastic processes are illustrated in Fig. 1, and more details
can be found in [15] and the references therein.
Definition 1 [15]: The stochastic process {(R
n
,k
n
)}
n∈Z
+
is
said to be a discrete-time homogeneous Markov renewal chain
(MRC) if for any j ∈I, τ ∈ Z
+
and n ∈ Z
+
, Pr(R
n+1
= j, S
n+1
=
τ|R
0
,...,R
n
= i; k
0
,...,k
n
)= Pr(R
n+1
= j, S
n+1
= τ |R
n
= i)
=Pr(R
1
= j, S
1
= τ |R
0
= i).
Then, let π
ij
(τ)
Δ
=Pr(R
n+1
=j, S
n+1
=τ |R
n
= i),∀i, j ∈I,
∀τ ∈Z
+
, the matrix Π(τ)=[π
ij
(τ)]
i,j∈I
is called discrete-time semi-
Markov kernel (SMK), where π
ij
(τ) ∈ R
[0,1]
and
∞
τ =0
j∈I
π
ij
(τ)=1 with π
ij
(0) = 0. In addition, from [15], {R
n
}
n∈Z
+
is
called the embedded Markov chain (EMC) of MRC {(R
n
,k
n
)}
n∈Z
+
,
and the TPs matrix Θ=[θ
ij
]
i,j∈I
of {R
n
}
n∈Z
+
is defined by
θ
ij
Δ
=Pr(R
n+1
= j|R
n
= i), ∀i, j ∈Iwith θ
ii
=0.
With the above concepts, the definition of semi-Markov chain is
given as below.
Definition 2 [15]: Consider a MRC {(R
n
,k
n
)}
n∈Z
+
. The chain
{r
k
}
k∈Z
+
is said to be a semi-Markov chain (SMC) associated with
MRC {(R
n
,k
n
)}
n∈Z
+
,ifr
k
= R
N(k)
, ∀k ∈ Z
+
,whereN(k)
Δ
=
max{n ∈ Z
+
|k
n
≤ k}.
It is worth noting that the difference between EMC and SMC lies in
that the stochastic variable varies with jump instant k
n
in the former,
whereas with the sampling instant k in the latter. From Definition 2,
it is straightforward that the evolution of SMC is generated by the
SMK Π(τ) that is dependent on sojourn-time τ . Thus, the knowledge
on the probability density function (PDF) of sojourn-time is required
to characterize a SMC. In this technical note, the PDF depending on
both the current and next system mode is considered and defined as
ω
ij
(τ)
Δ
=Pr(S
n+1
= τ |R
n+1
= j, R
n
= i), ∀i, j ∈I,∀τ ∈ Z
+
.As
a consequence
π
ij
(τ)=
Pr(R
n+1
=j, R
n
=i)
Pr(R
n
= i)
Pr(R
n+1
=j, S
n+1
=τ,R
n
=i)
Pr(R
n+1
= j, R
n
= i)
= θ
ij
ω
ij
(τ). (2)
Remark 1: Letting the PDF that only depends on the current system
mode be denoted as f
i
(τ)=Pr(S
n+1
= τ |R
n
= i), ∀i ∈I, ∀τ ∈
Z
+
, it yields that f
i
(τ)=
j∈I
π
ij
(τ)=
j∈I
θ
ij
ω
ij
(τ). It can be
observed that different ω
ij
(τ) may lead to a same f
i
(τ).Inalmostall
the previous studies on S-MJLSs, f
i
(τ) is used to obtain the sojourn-
time-dependent transition probability at mode i, say, denoted as λ
i
(τ),
and further the TPs λ
ij
(τ) by multiplying θ
ij
. Therefore the PDF
f
i
(τ) can be only one type, but ω
ij
(τ) that is used to form SMK in
this technical note can be r
k+1
dependent, having different types of
distributions or different parameters in a same type of distribution for
any i ∈I. Therefore, the PDF considered in the technical note is more
specific and capable of describing the corresponding SMC accurately
rather than f
i
(τ).
Throughout the technical note, the cumulative density function
(CDF) of sojourn-time for the ith system mode, ∀i ∈I, is denoted
as F
i
(τ)=Pr(S
n+1
≤ τ |R
n
= i)=
τ
l=0
j∈I
π
ij
(l), and it is
assumed that ω
ij
(0) = F
i
(0) = 0 without loss of generality. Now,
to present the purposes of this technical note more precisely, the
following stability definitions are required.
Definition 3: Consider a discrete-time stochastic switching nonlin-
ear system x
k+1
= f(x
k
,r
k
),wherer
k
is a certain stochastic process
governing the system switching and taking values in I. The system
is said to be mean-square stable if, for any initial conditions x
0
∈
R
n
,r
0
∈I, the following holds:
lim
k→∞
E
x(k)
2
|
x
0
,r
0
=0. (3)
Remark 2: In Definition 3, when r
k
is a Markov chain and the
underlying system is linear, we can typically find the corresponding
version of Definition 3 in the literature of MJLSs, see for example,
[10] and [16]. Nonetheless, one drawback of Definition 3 is that it
is established allowing for the random sojourn-time to be any length
(even infinity), regardless of the fact that the practical sojourn-time
is generally finite. Therefore, letting T
i
max
denote the upper bound of
sojourn-time for the ith mode of system (1), we generalize the MSS in
this technical note to the following concept.
Definition 4: System (1) with u(k) ≡ 0,issaidtobeσ-error mean-
square stable if, for any initial conditions x
0
∈ R
n
,r
0
∈Iand the up-
per bound of sojourn-time T
i
max
∈Z
≥1
, ∀i∈I, the following holds:
lim
k→∞
E
x(k)
2
|
x
0
,r
0
,S
n+1
≤T
i
max
|
R
n
=i
=0. (4)
Further, σ is defined as
σ
Δ
=
i∈I
ln
F
i
T
i
max
. (5)
Remark 3: Note that in Definition 4, σ varies with T
i
max
(σ decreases if all T
i
max
are increased). In view of this, σ is actu-
ally capable to characterize the degree of “approximation error” of
σ-MSS to MSS. Particularly, when T
i
max
→∞, ∀i ∈I, which im-
plies that F
i
(T
i
max
) → 1, then, σ → 0 and accordingly the σ-MSS
approximates to MSS without any error.
Then, the objectives in this technical note are to derive the σ-MSS
criterion for system (1), and to design a state-feedback stabilizing con-
troller guaranteeing the σ-MSS of the resulting closed-loop system.
The mode-dependent controller is considered here with the form
u(k)=K
i
x(k), ∀r
k
= i ∈I (6)
where K
i
is the controller gain to be determined.
III. M
AIN RESULTS
In this section, the numerically testable stability and stabilization
criteria for S-MJLSs will be developed. A result on MSS of stochastic
switching nonlinear systems is first given as below for later use.
Lemma 1: Consider a discrete-time stochastic switching nonlinear
system x
k+1
= f(x
k
,r
k
),wherex
k
and r
k
denote the system state
and mode index, respectively. The switching instants are denoted by
k
0
,k
1
,...,k
s
,... with k
0
=0. The system is mean-square stable,
if there exist a set of C
1
functions V (x
k
,r
k
):R
n
→ R and three
class K
∞
functions α
1
,α
2
,α
3
, such that for any initial conditions
x
0
∈ R
n
,r
0
∈Iand a given finite h
i
> 0, ∀r
k
s
= i ∈I
α
1
(x
k
) ≤ V (x
k
,r
k
s
) ≤ α
2
(x
k
) (7)
V (x
k
,r
k
s
) ≤ h
i
V (x
k
s
,r
k
s
) ,k∈ Z
(k
s
,k
s+1
]
(8)
E
V
x
k
s+1
,r
k
s+1
|
x
0
,r
0
−V(x
k
s
,r
k
s
)≤−α
3
(x
k
s
) . (9)
Proof: It follows from the proof of [16, Theorem 1] that (9)
ensures
∞
s=0
E[α
3
(x
k
s
)]|
x
0
,r
0
≤E[V (x
0
,r
0
)]< ∞which implies
lim
s→∞
E[α
3
(x
k
s
)]|
x
0
,r
0
=0.Sinceα
3
∈K
∞
, lim
s→∞
E[x
k
s
2
]|
x
0
,r
0
=0holds. On the other hand, taking mathematical
expectations at both sides of (7), (8), we have E[α
1
(x
k
)]|
x
0
,r
0
≤
h
i
E[V (x
k
s
,r
k
s
)]|
x
0
,r
0
≤ h
i
E[α
2
(x
k
s
)]|
x
0
,r
0
.Ass →∞, k →∞,
therefore lim
k→∞
E[α
1
(x
k
)]|
x
0
,r
0
≤h
i
lim
s→∞
E[α
2
(x
k
s
)]|
x
0
,r
0
,
which implies (3).
Then, the following theorem gives a criterion of σ-MSS for un-
forced S-MJLSs.
Theorem 1: Consider S-MJLS (1) with u(k) ≡ 0 andagivenfinite
constant h
i
> 0. If, ∀i ∈I, there exist T
i
max
∈ Z
≥1
and matrices P
i
0 such that ∀t ∈ Z
[1,T
i
max
]
A
t
i
P
i
A
t
i
− h
i
P
i
≺ 0 (10)
T
i
max
τ =1
A
τ
i
P
i
(τ)A
τ
i
− P
i
≺ 0 (11)
where P
i
(τ)
Δ
=
j∈I
π
ij
(τ)P
j
/η
i
with η
i
Δ
=
T
i
max
τ =1
j∈I
π
ij
(τ),
then the system is σ-error mean-square stable.
Proof: Construct the Lyapunov function as V
i
(x
k
)
Δ
=
V (x
k
,R
n
)|
R
n
=i
= x
k
P
i
x
k
, ∀i ∈I,whereP
i
satisfies (10) and (11).
First, it is straightforward that
inf
i∈I
{λ
min
(P
i
)}x
k
2
≤ V
i
(x
k
) ≤ sup
i∈I
{λ
max
(P
i
)}x
k
2
(12)
Fig. 2. Illustration of Theorem 1 for M =3.In(k, V
i
)-coordinate, solid
line shows the real evolution of Lyapunov function, and in (τ, V
i
)-coordinate,
dashed line illustrates a possible evolution of the Lyapunov function starting
from a fixed mode at a certain jumping instant; circles initiate V
1
, squares
initiate V
2
and triangles initiate V
3
.
where λ
min
(P
i
) (respectively, λ
max
(P
i
)) denotes the minimal (re-
spectively, maximal) eigenvalue of P
i
. In addition, for the case R
n
=
i, the following is ensured by (10) ∀t ∈ Z
[1,T
i
max
]
V
i
(x
k
n
+t
) − h
i
V
i
(x
k
n
)=x
k
n
A
t
i
P
i
A
t
i
− h
i
P
i
x
k
n
< 0. (13)
On the other hand, for R
n
= i, R
n+1
= j, letting the sojourn-time
k
n+1
− k
n
be denoted by τ, it follows from (11) that
E
V
j
x
k
n+1
|
x
0
,r
0
,S
n+1
≤T
i
max
|
R
n
=i
− V
i
x
k
n
= x
k
n
⎡
⎣
T
i
max
τ=1
j∈I
π
ij
(τ)A
τ
i
P
j
A
τ
i
/η
i
− P
i
⎤
⎦
x
k
n
≤−λ
min
⎛
⎝
−
T
i
max
τ=1
A
τ
i
P
i
(τ)A
τ
i
+ P
i
⎞
⎠
x
k
n
2
≤−β
x
k
n
2
(14)
where η
i
is defined in (11) and β
Δ
=inf
i∈I
{λ
min
(−
T
i
max
τ =1
A
τ
i
P
i
(τ)A
τ
i
+ P
i
)}. Then, by (12), (13), (14) and Lemma 1, it
follows that S-MJLS (1) is mean-square stable provided that the upper
bound of sojourn-time is T
i
max
, i.e., (4) holds, thus the σ-MSS of the
system is guaranteed.
Note that the Lyapunov function of each mode is not necessarily
monotonically decreasing (h
i
can be greater than 1), as shown in (10).
An illustration of Theorem 1 for M =3 is given in Fig. 2, where
the system at k
n
is supposed to run in mode 1. In (τ,V
i
)-coordinate,
V
2
(x
k
n
+τ
) and/or V
3
(x
k
n
+τ
) (the corresponding hollow square or
triangle) can be greater than V
1
(x
k
n
) (the solid circle in (k, V
i
)-
coordinate). However, as long as the expectation
¯
V
i
(x
k
n+1
) is lower
than V
i
(x
k
n
), the value of Lyapunov function at jumping instants
will tend to zero in stochastic sense, forcing V
i
(x
k
) → 0, despite the
allowable increase of Lyapunov function associated with mode 1 to
some extent.
Remark 4: Combining with Remark 3, one can conclude from
Theorem 1 that the unforced S-MJLSs will be mean-square stable
when T
i
max
→∞, ∀i ∈I, but obviously the resulting conditions
(10), (11) can not be numerically tested. Therefore, a finite T
i
max
is
necessitated in Theorem 1, which corresponds to the proposed σ-MSS
with σ>0.
Remark 5: In Theorem 1, the SMK Π(τ ) is directly utilized to
establish the stability criterion. In fact, Π(τ) can be first used to obtain
the memory TPs (if denoted by λ
ij
(τ)), and the following stability
criterion can be arrived at:
j∈I
λ
ij
(τ)A
i
P
j
(τ)A
i
− P
i
(τ − 1) ≺ 0 (15)
where P
i
(τ) is a time-varying Lyapunov matrix and τ ∈ Z
≥1
.Note
that when λ
ij
(τ) ≡ λ
ij
(the corresponding PDF of sojourn-time is
subject to geometric distribution with parameter μ
i
, i.e., ω
i1
(τ)=
···= ω
iM
(τ)=μ
i
(1 − μ
i
)
τ −1
Δ
= ω
i
(τ))andP
i
(τ) ≡ P
i
, (15) re-
duces to the usual stability criterion for MJLSs. It can be readily proved
(cf. [4]) that (15) is a necessary and sufficient condition for the MSS
of the unforced system (1). However, (15) is not numerically testable
even conservatively setting P
i
(τ) ≡ P
i
. Additional approximation
techniques such as taking bounds for λ
ij
(τ) are needed to make (15)
be time-invariant, cf. [9]. Besides, (15) aims at MSS that is not “scaled”
compared with σ-MSS.
Remark 6: It is also worth mentioning that if the sojourn-time is
subject to “exponentially modulated periodic (EMP)” distribution (cf.
[17]), say ω
i
(τ)=¯ω
i
(τ)(1 − μ
i
)
τ −1
,where¯ω
i
(τ) satisfies ¯ω
i
(τ)=
¯ω
i
(τ + T ) with T being the period, then the TPs obtained from SMK
will be periodically time-varying. The corresponding criterion in (15)
will be periodic and testable accordingly, i.e., only a finite set of P
i
(v),
v ∈ Z
[1,T ]
is required. The benefit in this scenario therefore suggests
that the EMP distribution can be considered to approximate to those
general “exponential-like” distributions of sojourn-time, for the sake
of a finite number of conditions.
Though the stability criterion explored in Theorem 1 can be used for
analysis of the σ-MSS of S-MJLSs, the requirements (10), (11) impose
a significant difficulty in deriving tractable conditions of controller
design due to the existence of the power of A
i
. To circumvent the
difficulty, certain techniques will be further explored below to obtain
sufficient conditions for Theorem 1.
Theorem 2: Consider the S-MJLS (1) with u(k) ≡ 0 and a given
finite constant h
i
> 0. If, ∀i ∈I, there exist T
i
max
∈ Z
≥1
and a
set of matrices O
i
(t, m), ∀t ∈ Z
[1,T
i
max
]
, ∀m ∈ Z
[0,t]
with O
i
Δ
=
O
i
(t, t) 0 and O
i
(τ,n), ∀τ ∈ Z
[1,T
i
max
]
, ∀n ∈ Z
[0,τ −1]
such that
∀t ∈ Z
[1,T
i
max
]
, ∀m ∈ Z
[0,t−1]
and ∀n ∈ Z
[0,T
i
max
−1]
A
i
O
i
(t, m +1)A
i
− O
i
(t, m) ≺ 0 (16)
O
i
(t, 0) − h
i
O
i
≺ 0 (17)
T
i
max
τ=n+1
A
i
O
i
(τ,n +1)A
i
−O
i
(τ,n)
≺ 0 (18)
T
i
max
τ=1
O
i
(τ, 0) − O
i
≺ 0 (19)
where O
i
(l, l)
Δ
=
j∈I
π
ij
(l)O
j
/η
i
with η
i
defined in (11), then the
unforced S-MJLS is σ-error mean-square stable.
Proof: It follows from (16) that
t−1
m=0
A
m
i
(A
i
O
i
(t, m +
1)A
i
− O
i
(t, m))A
m
i
≺ 0 and accordingly
A
t
i
O
i
(t, t)A
t
i
− O
i
(t, 0) ≺ 0. (20)
By (17), (20), and bearing in mind O
i
= O
i
(t, t), it can be obtained
that A
t
i
O
i
A
t
i
≺ h
i
O
i
. Further, setting the positive definite matrix
O
i
= P
i
,wehaveA
t
i
P
i
A
t
i
− h
i
P
i
≺ 0.
Likewise, (18) ensures that
T
i
max
−1
n=0
A
n
i
[
T
i
max
τ =n+1
[A
i
O
i
(τ,n +
1)A
i
−O
i
(τ,n)]]A
n
i
≺ 0, which is equivalent to
T
i
max
τ =1
τ −1
n=0
A
n
i
[A
i
O
i
(τ,n +1)A
i
−O
i
(τ,n)]A
n
i
≺ 0 and implies
T
i
max
τ=1
A
τ
i
O
i
(τ,τ)A
τ
i
−O
i
(τ, 0)
≺ 0. (21)
Combining (19) and (21), we have
T
i
max
τ =1
A
τ
i
O
i
(τ,τ)A
τ
i
− P
i
≺
0. Then setting the positive definite matrix O
i
(τ,τ)=P
i
(τ) gives
rise to
T
i
max
τ =1
A
τ
i
P
i
(τ)A
τ
i
− P
i
≺ 0. Thus, by Theorem 1, it can
be concluded that the system is σ-error mean-square stable and this
completes the proof.
Then, based on Theorem 2, the existence conditions of mode-
dependent stabilizing controller (6) for S-MJLS (1) are presented in
the following theorem.
Theorem 3: Consider the S-MJLS (1) with a given finite con-
stant h
i
> 0. If, ∀i ∈I, there exist T
i
max
∈ Z
≥1
and a set of matri-
ces H
i
(t, m), ∀t ∈ Z
[1,T
i
max
]
, ∀m ∈ Z
[0,t]
with H
i
Δ
= H
i
(t, t) 0
and H
i
(n), ∀n ∈ Z
[0,T
i
max
]
,Z
i
,U
i
such that ∀t ∈ Z
[1,T
i
max
]
, ∀m ∈
Z
[0,t−1]
and ∀n ∈ Z
[0,T
i
max
−1]
H
i
(t, m +1)− Z
i
− Z
i
A
i
Z
i
+ B
i
U
i
∗−H
i
(t, m)
≺ 0 (22)
H
i
(t, 0) − h
i
H
i
≺ 0 (23)
H−Z−Z
0(A
i
Z
i
+ B
i
U
i
)L
i
(n +1)
∗H
i
(n +1)− Z
i
− Z
i
(A
i
Z
i
+ B
i
U
i
)L
i
(n +1)
∗∗ −H
i
(n)
≺ 0 (24)
H
i
(0) − H
i
≺ 0 (25)
where A
i
Δ
= diag
(M)
{A
i
}, B
i
Δ
= diag
(M)
{B
i
}, Z
i
Δ
= diag
(M)
{Z
i
},
U
i
Δ
= diag
(M)
{U
i
}, H
Δ
= diag{H
1
,H
2
,...,H
M
}, Z
Δ
=
diag{Z
1
,Z
2
,...,Z
M
}, L
i
(n)=I, ∀n ∈ Z
[1,T
i
max
−1]
with
L
i
(T
i
max
)=0,andL
i
(n)
Δ
=[
i1
(n)I,
i2
(n)I,...,
iM
(n)I]
with
ij
(n)
Δ
=
π
ij
(n)/η
i
and η
i
defined in (11), then a mode-
dependent controller of form (6) can be obtained to guarantee the
σ-MSS of the resulting closed-loop system. Moreover, the admissible
controller gain is given by K
i
= U
i
Z
−1
i
.
Proof: Letting
¯
O
i
(l)
Δ
=
T
i
max
τ =l+1
O
i
(τ,l), ∀l ∈ Z
[0,T
i
max
−1]
and
¯
O
i
(T
i
max
)
Δ
=0, we can rewrite (18) and (19) as
¯
O
i
(0) − O
i
≺0 (26)
A
i
O
i
(n+1,n+1)A
i
+A
i
¯
O
i
(n+1)A
i
−
¯
O
i
(n)≺0. (27)
Consider (27), by Schur complement, it yields that
−O 0 OA
i
L
i
(n +1)
∗−
¯
O
i
(n +1)
¯
O
i
(n +1)A
i
∗∗ −
¯
O
i
(n)
≺ 0 (28)
where A
i
, L
i
(n) are defined in (24) and O
Δ
= diag{O
1
,O
2
,...,
O
M
}. Performing a congruence transformation to (28) by diag{O
−1
V,
¯
O
i
(n +1)
−1
V
i
,I},whereV
Δ
= diag{V
1
,V
2
,...,V
M
}, and since
(O−V)
O
−1
(O−V) 0 ensures O−V−V
−V
O
−1
V,we
can obtain
O−V−V
0 V
A
i
L
i
(n +1)
∗
¯
O
i
(n +1)− V
i
− V
i
V
i
A
i
∗∗−
¯
O
i
(n)
≺ 0. (29)
Then, apply the congruence transformation to (24) by diag{V,V
i
,V
i
},
it gives (29) while replacing A
i
by A
i
+ B
i
K
i
and setting Z
i
Δ
= V
−1
i
,
U
i
Δ
= K
i
V
−1
i
, H = V
−1
OV
−1
, H
i
(n)
Δ
= V
−1
i
¯
O
i
(n)V
−1
i
.
On the other hand, performing a congruence transformation to (25)
by V
i
, we have (26) accordingly. Then, it implies that (24) and (25)
guarantee (18) and (19) in Theorem 2. Same techniques can be applied
to the case that (22) and (23) ensures (16) and (17). Thus, it can be
concluded from Theorem 2 that (22)–(25) can guarantee the σ-MSS
of the resulting closed-loop system with the admissible controller gain
given by K
i
= U
i
Z
−1
i
.
Remark 7: In Theorems 1 and 3, for a given set of h
i
,theT
i
max
can be maximized while achieving a minimal σ-error and ensuring
a feasible solution of (10), (11), and (22)–(25), for stability and
stabilization problems, respectively. Besides, note also that if h
i
> 1,
the corresponding Lyapunov function in the ith mode can raise to
some extent, which implies that mode i can be unstable/unstabilizable
with τ ≤ T
i
max
. Nevertheless, it can be conjectured that if such an
unstable/unstabilizable mode is expected to run longer (i.e., a larger
T
i
max
), then probably a much larger h
i
(h
i
1) is required, which
will result in a huge energy growth.
TABL E I
T
HE OPTIMAL T
i
max
AND THE MINIMAL σ-E RROR FOR GIVEN DIFFERENT h
i
, i =1,2
Fig. 3. 100 realizations of state response of the system when generating different random jumping sequences. In (b) and (d), the sequences are further subject
to T
1
max
=5,T
2
max
=7and T
1
max
=4,T
2
max
=3, respectively. (a) α =1.07 (not MSS). (b) α =1.07 (σ-MSS with σ-error =0.09). (c) α =1.15 (not
MSS). (d) α =1.15 (σ-MSS with σ-error =0.27).
IV. NUMERICAL EXAMPLES
In this section, both a MJLS and a S-MJLS will be provided to show
the validity and advantage of the obtained theoretical results.
Example 1: (MJLS) Consider an unforced MJLS with two modes
A
1
= α
−0.36 0.69
−1.81 1.97
A
2
= α
0.34 0.62
−0.37 1.36
where α>0 characterizes the distance of system eigenvalues and unit
circle. The EMC is an anti-identity matrix, and the sojourn-time is
subject to geometric distribution with ω
1
(τ)=0.4(1 − 0.4)
τ −1
and
ω
2
(τ)=0.5(1 − 0.5)
τ −1
.
Based on the usual criterion of MSS (the required TPs matrix can be
straightforwardly obtained from ω
1
(τ) and ω
2
(τ)), it can be readily
checked that the system is not mean-square stable when α ≥ 1.07.
Turning to Theorem 1, given different h
i
, we can maximize the upper
bound of sojourn-time for each mode (denoted as [T
i
max
]
opt
, i =1,2)
in achieving a minimal σ-error of approximation to MSS, as computed
in Table I. It can be seen that when α>1.27, no matter how large the
set of h
i
is given, a finite σ-error for the MSS cannot be guaranteed (the
corresponding system does not hold σ-MSS). As for 1.07 ≤ α ≤ 1.27,
however, different minimal σ-error can be found for different h
i
.
Particularly, the smaller the α is assigned, the larger the [T
i
max
]
opt
can
be found, and thus the smaller the minimal σ-error can be achieved.
Consider α =1.07, x
0
=[1 1]
, Fig. 3(a) illustrates the state re-
sponse when randomly generating 100 realizations of jumping se-
quences satisfying the geometric distribution, and Fig. 3(b) presents
the case where the jumping sequences are further subject to T
1
max
=
5,T
2
max
=7. The scenario of α =1.15 is also given in Fig. 3(c)
and (d) (T
1
max
=4,T
2
max
=3). It can be observed that although the
two systems are not mean-square stable, they are σ-error mean-
square stable with different approximation σ-error to be 0.09 and
0.27, respectively. Both Table I and Fig. 3 therefore demonstrate that
the proposed σ-MSS is of the advantage of being well-scaled rather
than MSS.
Example 2: (S-MJLS) Consider a dynamic system with possible
failures in both structure and actuator as
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
A
1
=
−0.36 0.69
−1.81 1.97
B
1
=
−0.1
0.1
(Normal Case)
A
2
=
0.34 0.62
−0.37 1.36
B
2
=
0.1
0.1
(Slight Fault)
A
3
=
0.34 0.62
−0.37 1.36
B
3
=
0
0
(Serious Fault (unworkable actuator))
The switching among the three modes is governed by a SMC, where
the SMK is computed by (2) with Θ=[00.70.3; 0.400.6; 0.50.50]
and, ∀i, j ∈{1, 2, 3}
[ω
ij
(τ )]
=
⎡
⎣
0
0.6
τ
·0.4
10−τ
·10!
(10−τ )!τ!
0.4
τ
·0.6
10−τ
·10!
(10−τ )!τ!
0.9
(τ −1)
2
−0.9
τ
2
0
0.5
10
·10!
(10−τ )!τ!
0.4
(τ −1)
1.3
−0.4
τ
1.3
0.3
(τ −1)
0.8
−0.3
τ
0.8
0
⎤
⎦
·
Note from ω
ij
(τ) that the “hybrid” PDF of sojourn-time in mode
1 and mode 3 are considered as Bernoulli distribution and Weibull
distribution, respectively, with different parameters when the target
modes are different, and the two types of distributions are supposed
to coexist in mode 2.
First, it is checked that the open-loop system is not σ-error mean-
square stable for any h
i
> 0 by Theorem 1. Applying Theorem 3,
the desired mode-dependent controllers can be designed such that
the resulting closed-loop systems are σ-error mean-square stable for
certain sets of h
i
.Also,[T
i
max
]
opt
can be further obtained to achieve
a minimal σ-error for the MSS of the system and ensuring a feasible
solution of the controller. It can be verified that no matter how large h
1
and h
2
are assigned, no feasible controllers can be obtained if h
3
< 1.
Thus, consider h
3
≥ 1 and change h
1
,h
2
, the different [T
i
max
]
opt
can be obtained. Fig. 4(a)–(c) present the two sets of [T
i
max
]
opt
that
vary with h
2
∈ R
[0.001,0.005]
and h
3
∈ R
[1.1,1.5]
, for given h
1
to be
0.01 and 0.001, respectively [the corresponding σ-error, which can be
computed by (5), is shown in Fig. 4(d)].
Then, setting h
1
=0.01,h
2
=0.005 and h
3
=1.5, the upper
bound of sojourn-time can be optimized as [T
1
max
]
opt
=[T
2
max
]
opt
=
9 and [T
3
max
]
opt
=3, which guarantee the minimal σ-error to be
0.0467. Given the initial condition x
0
=[−0.51]
, Fig. 5 shows
the 100 realizations of the state response of the closed-loop system
for randomly generating jumping sequences satisfying (the required
T
i
max
) T
1
max
= T
2
max
=9and T
3
max
=3. It can be observed that the
designed controller is valid in the presence of an unstabilizable mode
and despite the coexistence of “hybrid” distributions of sojourn-time
in a same SMK. In addition, as shown in Fig. 4(d), when fixing
T
1
max
= T
2
max
=9,theσ-error can be smaller if further increasing
T
3
max
, which seemingly can be attained by raising h
i
. However, it can
be checked that even h
i
=10
7
can not give rise to an increase from
T
3
max
=3to T
3
max
=4, which verifies the conjecture in Remark 7 that
only a proper sojourn-time of the unstabilizable mode can be allowed.
