The novel sufficient conditions of almost sure exponential stability for semi-Markov jump linear systems

Systems & Control Letters 137 (2020) 104622

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The novel sufficient conditions of almost sure exponential stability for semi-Markov jump linear systems✩ Bao Wang a,b ,1 , Quanxin Zhu b,c ,

∗,1

a

School of Mathematics and Physics, Xuzhou Institute of Technology, Xuzhou 221000, Jiangsu, China MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, Hunan, China c Key Laboratory of Applied Statistics and Data Science, Hunan Normal University, College of Hunan Province, Changsha, 410081, China b

article

info

Article history: Received 21 May 2019 Received in revised form 28 December 2019 Accepted 5 January 2020 Available online xxxx Keywords: Semi-Markov jump linear system Stationary distribution Strong law of large numbers Linear matrix inequality

a b s t r a c t This paper discusses the almost sure exponential stability problem for semi-Markov jump linear systems. By comprehensively utilizing the coupled Lyapunov matrices and the ergodic property of semi-Markov switching process, the novel sufficient stability conditions for the considered systems are obtained, which are expressed in terms of linear matrix inequalities and the probability structure of semi-Markov switching process. Finally, an example is given to illustrate the effectiveness of our results. © 2020 Elsevier B.V. All rights reserved.

1. Introduction A randomly switched system is a hybrid dynamical system consisting of a set of subsystem modes and a random switching process that orchestrates these modes. The stability analysis problem of such systems has been extensively studied, see [1–18]. When the switching process is modelled as a Markov process, the corresponding system is called the Markov jump system. In a Markov jump system, the sojourn time of each mode is a random variable following an exponential distribution. The memoryless property of exponential distribution leads to a fact that the transition rates between different modes of the considered system are constants, which is difficult to be satisfied for many practical systems. For example, in a fault tolerant control system, the transition rate function of switching process is described as a bathtub curve (see [19]), such system cannot be modelled as a Markov jump system. By relaxing the probability distribution of sojourn time in each mode from exponential distribution to the more general case, the Markov process can be generalized to the semi-Markov process. ✩ This work was jointly supported by the National Natural Science Foundation of China (61773217), Hunan Provincial Science and Technology Project Foundation, China (2019RS1033), the Scientific Research Fund of Hunan Provincial Education Department, China (18A013), Hunan Normal University, China National Outstanding Youth Cultivation Project (XP1180101), the Construct Program of the Key Discipline in Hunan Province, China and Cultivating Fund Project of Xuzhou Institute of Technology, China (XKY2018121). ∗ Corresponding author. E-mail address: [email protected] (Q. Zhu). 1 The authors claim that two authors contribute equally to this article. https://doi.org/10.1016/j.sysconle.2020.104622 0167-6911/© 2020 Elsevier B.V. All rights reserved.

The randomly switched system with semi-Markov switching is referred to as a semi-Markov jump system. The research on semi-Markov jump system has received much attention in recent years. For example, in [20–22], the phase-type distribution was introduced to depict the sojourn time of each mode in switching process, and the stochastic stability conditions of the corresponding semi-Markov jump system have been obtained. In [23,24], the sojourn time of each mode in switching process was assumed to follow Weibull distribution. These results on semi-Markov jump system all heavily rely on the special distribution of sojourn time of each mode in semi-Markov switching process, and do not involve the almost sure stability issue, which motivates our study in this paper. This paper is devoted to studying the almost sure exponential stability (ASE-stability) problem for a class of semi-Markov jump linear systems (SMJLSs). In our results, the sojourn time of each mode in semi-Markov switching process can follow arbitrary probability distribution. There are two main innovations in this paper. (1) The switching process is modelled as an ergodic semiMarkov process, the probability structure of switching process plays an important role on the ASE-stability of the considered systems. (2) The only restriction of semi-Markov switching process in this paper is the boundness of the second moments of sojourn times in semi-Markov switching process. Thus, our results are more feasible and applicable than some published results. The remainder of this paper is organized as follows: In Section 2, we introduce the formal definition and the ergodic property of semi-Markov process. In Section 3, by utilizing the method of time dependent coupled Lyapunov matrices and the ergodic property of semi-Markov process, the novel sufficient conditions

2

B. Wang and Q. Zhu / Systems & Control Letters 137 (2020) 104622

of ASE-stability for a class of SMJLSs are obtained, which generalize and improve the ones in Theorem 1 of [25]. In Section 4, an example is presented to illustrate the validity of our results. Section 5 gives the conclusion of this paper. Notations. In this note, R denotes the real number set. ⋂Z denotes the integer set. For any subset A of R, ZA denotes Z A. Rn denotes the n-dimensional Euclidean space. Rn×m denotes the set of n × m real matrices. For a matrix X , let X T be the transpose of X and Her(X ) = X + X T . For a square matrix X , X > 0 (respectively, X < 0) means that X is positive (respectively, negative) definite. λmin (X ) (respectively, λmax (X )) denotes the minimum (respectively, maximum) eigenvalue of X . I(·) denotes the indicator function. The triple (Ω , F , P ) refers to the probability space. E(·) and Var(·) denote the mathematical expectation and variance with respect to P , respectively.

occurrences of state i on (0, t ], Nij (t) be the switching occurrences from state i to state j on (0, t ], Si (t) be the total sojourn time of staying at state i on∑(0, t ]. In addition, Nij (t) = pij Ni (t), N(t) = ∑ i∈I Ni (t) and t = i∈I Si (t). By Eq. 16.13 and Eq. 16.14 of [28], we know that Si (t) lim = E(τi ), a.s., ∀i ∈ I . (6) t →∞ Ni (t) and lim

t →∞

Ni (t) N(t)

=˜ πi , a.s., ∀i ∈ I .

(7)

In this paper, a continuous-time SMJLS on a complete probability space (Ω , F , P ) can be described as the following form: dx(t) = Ar(t) x(t)dt + Gr(t) x(t)dB(t)

(8)

on t ≥ 0 with the initial value x(0) = x0 ∈ R , in which {r(t), t ≥ 0} is a semi-Markov process with state space I , x(t) ∈ Rn is the state vector, and B(t) is a scalar Brownian motion. For each i ∈ I , Ai , Gi ∈ Rn×n . Let x(t , x0 ) be the state of system (8) at time t starting from the initial value x(0) = x0 . It is easy to see that x(t , 0) ≡ 0. We give the following lemma to describe an important property of system (8). n

2. Preliminaries In this section, we first introduce the following three stochastic processes (see [26]): 1. {rn }n∈Z≥0 with state space I = {1, 2, . . . , M }: rn denotes the index of mode at the nth jump; 2. {tn }n∈Z≥0 : tn denotes the time instant at the nth jump with t0 = 0; 3. {sn }n∈Z≥0 is an independent sequence, sn := tn+1 − tn denotes the sojourn time of mode rn between the nth jump and the n + 1-th jump. Next, we give the formal definition of semi-Markov process and some related notions. Definition 2.1 (See [27]). The stochastic process {(rn , tn )}n∈Z≥0 is a homogeneous Markov renewal chain (MRC), if for any i, j ∈ I , h ≥ 0 and n ∈ Z≥0 , P (rn+1 = j, sn ≤ h|rn = i, tn , rk , tk , k ∈ Z[0,n−1] )

= P (rn+1 = j, sn ≤ h|rn = i) = P (r1 = j, s0 ≤ h|r0 = i).

(1)

Definition 2.2 (See [27]). Consider a Markov renewal chain {(rn , tn )}n∈Z≥0 . 1. The chain {rn }n∈Z≥0 is called the embedded Markov chain (EMC) of the MRC {(rn , tn )}n∈Z≥0 , and the transition probability (TP) matrix P = [pij ]M ×M of {rn }n∈Z≥0 , in which pij = P (rn+1 = j|rn = i) with pii = 0 for any n ∈ Z≥0 . 2. {r(t), t ≥ 0} is the semi-Markov process (SMP) associated with MRC {(rn , tn )}n∈Z≥0 , if r(t) = rn , t ∈ [tn , tn+1 ), ∀n ∈ Z≥0 .

(2)

Lemma 2.3.

The state x(t , x0 ) of system (8) obeys

P (x(t ; x0 ) ̸ = 0 on t ≥ 0) = 1, ∀x0 ̸ = 0.

(9)

That is, almost all the sample paths of system (8) starting from a nonzero state will never reach the zero. The proof of this lemma is similar to the proof of Lemma 2.1 in [17], so we omit it. At the last part of this section, we give the formal definition of ASE-stability for system (8). Definition 2.4. The system (8) is almost surely exponentially stable, if for any x0 ∈ Rn , lim sup t →∞

1 t

ln |x(t ; x0 )| < 0, a.s.

(10)

3. Main results In this section, we will obtain the novel sufficient conditions of ASE-stability for system (8) in the semi-Markov switching case and the Markov switching case, respectively. In the following theorem, the switching process {r(t), t ≥ 0} is a semi-Markov process with stationary distribution π .

It can be seen that EMC {rn }n∈Z≥0 can be denoted as {r(tn )}n∈Z≥0 . For each i ∈ I , we denote τi be the sojourn time of each visiting mode i. Throughout this paper, we assume that for any i ∈ I ,

Theorem 3.1. For given constants βi ∈ R and γi ∈ R, i ∈ I , if there exist matrices Pi,1 > 0, Pi,2 > 0, i ∈ I , and constants µij > 0, i, j ∈ I , such that following conditions

E(τi2 ) < ∞,

Her(Pi,1 Ai − γi Pi,1 Gi ) +

(3)

and for any n ∈ Z≥0 , sn = τr(tn ) , a.s.

+GTi Pi,1 Gi + (4)

We further assume that EMC {r(tn )}n∈Z≥0 is ergodic with stationary distribution ˜ π = (˜ π1 , ˜ π2 , . . . , ˜ πM ), and SMP {r(t), t ≥ 0} is ergodic with stationary distribution π = (π1 , π2 , . . . , πM ), there is a fact that (see Eq. 16.10 of [28])

πi = ∑

˜ πi E(τi ) , i ∈ I. πj E(τj ) j∈I ˜

(5)

For any t > 0, we denote N(t) be the total switching occurrences of {r(t), t ≥ 0} on (0, t ], for any i, j ∈ I , Ni (t) be the switching

1 E(τi )

1 E(τi )

2

γi2 Pi,1

(Pi,2 − Pi,1 ) < βi Pi,1 ,

Her(Pi,2 Ai − γi Pi,2 Gi ) +

+GTi Pi,2 Gi +

1

1 2

(11)

γi2 Pi,2

(Pi,2 − Pi,1 ) < βi Pi,2 ,

(12)

Pj,1 ≤ µij Pi,2 , j ̸ = i,

(13)

∑ ] ∑ [ j∈I pij ln µij πi β i + ≤ 0, E(τi )

(14)

i∈I

are satisfied. Then system (8) is ASE-stable.

B. Wang and Q. Zhu / Systems & Control Letters 137 (2020) 104622 −

3

Proof. It is clear that x(t , x0 ) ≡ 0 when x0 = 0. Thus, system (8) is ASE-stable when x0 = 0. Fixing any x0 ̸ = 0, we write x(t , x0 ) = x(t) for simplicity. We consider the following time dependent coupled Lyapunov function:

By (13) and (16), for any n ∈ Z≥1 , we have

V (t , x(t))

ln V (tn , x(tn ))

= xT (t)Pr(t),1 x(t) ( ) t − tN(t) T + x (t) Pr(t),2 − Pr(t),1 x(t), τr(t)

≤ ln µr(tn− )r(tn ) + ln V (tn− , x(tn− )) (15)

For any t ∈ [tk , tk+1 ) and let r(tk ) = i, k ∈ Z≥0 , i ∈ I , (15) reduces to the following form: V (t , x(t))

= xT (t)Pi,1 x(t) +

t − tk

τi

(

)

xT (t) Pi,2 − Pi,1 x(t),

(16)

where τi is the sojourn time of each visiting mode i, i ∈ I . Thus, for any t ∈ [tk , tk+1 ), k ∈ Z≥0 , Itoˆ formula can be rewritten as dV (t , x(t))

= LV (t , x(t), r(t))dt + HV (t , x(t), r(t))dB(t),

(17)

in which LV (t , x(t), r(t))

[

T

(

= x (t) Her Pr(t),1 Ar(t) + 1

(Pr(t),2 − Pr(t),1 )Ar(t)

τr(t)

+

τr(t)

≤ ln V (tn−1 , x(tn−1 )) + ln µr(tn−1 )r(tn ) (∫ t ∫ tn− )[ LV (t , x(t), r(t)) + + V (t , x(t)) tn−1 tn ⏐ ⏐2 ] 1 ⏐ HV (t , x(t), r(t)) ⏐ ⏐ ⏐ dt − ⏐ ⏐ 2 V (t , x(t)) ] (∫ t ∫ tn− )[ HV (t , x(t), r(t)) dB(t). + + V (t , x(t)) tn tn−1

≤ ln V (0, x0 ) +

(∫ +

(19)

d ln V (t , x(t))

=

LV (t , x(t), r(t))

⏐2 ] ⏐ dt ⏐

1 ⏐ HV (t , x(t), r(t)) ⏐

⏐ − ⏐⏐ 2 ]

V (t , x(t)) V (t , x(t)) [ HV (t , x(t), r(t)) + dB(t). V (t , x(t))

(20)

ln V (tn− , x(tn− ))

= ln V (tn−1 , x(tn−1 )) ⏐ ⏐2 ] ∫ tn− [ LV (t , x(t), r(t)) 1 ⏐ HV (t , x(t), r(t)) ⏐ ⏐ ⏐ dt + − ⏐ ⏐ V (t , x(t)) 2 V (t , x(t)) tn−1

(24)

ln µr(tk )r(tk+1 )

k=0 n−1 tk−+1

t

+

∑∫

)[

LV (t , x(t), r(t))

(25)

k=0

For any t ∈ [tk , tk+1 ) and r(tk ) = i, k ∈ Z≥0 , i ∈ I , by denoting

√

√

t − tk x(t), x2 (t) = x(t), τi τi ηT (t) = (xT1 (t), xT2 (t)), Pi = diag {Pi,1 , Pi,2 }, ˜ Pi = diag {Pi,2 − Pi,1 , Pi,2 − Pi,1 } x1 (t) =

tk+1 − t

Ξ i = diag {Her(Pi,1 Ai ) + GTi Pi,1 Gi , Her(Pi,2 Ai ) + GTi Pi,2 Gi }, ∆i = {Pi,1 Gi , Pi,2 Gi },

(

LV (t , x(t), i) = ηT (t) Ξ i +

ln V (t , x(t))

Similarly, for any n ∈ Z≥1 ,

∈

(18) and (19) can be denoted as the following simple expressions:

Thus, for any t ∈ [tn , tn+1 ), n ∈ Z≥0 , we have

= ln V (tn , x(tn )) ⏐ ⏐2 ] ∫ t[ LV (t , x(t), r(t)) 1 ⏐ HV (t , x(t), r(t)) ⏐ ⏐ ⏐ dt + − ⏐ ⏐ V (t , x(t)) 2 V (t , x(t)) tn ] ∫ t[ HV (t , x(t), r(t)) + dB(t). V (t , x(t)) tn

n−1 ∑

V (t , x(t)) tn k=0 tk ⏐ ⏐2 ] 1 ⏐ HV (t , x(t), r(t)) ⏐ ⏐ dt − ⏐⏐ ⏐ 2 V (t , x(t)) )[ (∫ t ∑ ] − ∫ n−1 tk+1 HV (t , x(t), r(t)) + + dB(t). V (t , x(t)) tk tn

)]

By Lemma 2.3, we know that for any x0 ̸ = 0, x(t) will never reach zero with probability one. Thus, we can compute the differential of function ln V (t , x(t)) for any t ∈ [tk , tk+1 ), k ∈ Z≥0 as the following form

[

(23)

ln V (t , x(t))

(18)

x(t).

(22)

Substituting (22) and (23) into (21) yields that for any t [tn , tn+1 ), n ∈ Z≥1 ,

[ ( = xT (t) Her Pr(t),1 Gr(t) (Pr(t),2 − Pr(t),1 )Gr(t)

dB(t).

= ln µr(tn−1 )r(tn ) + ln V (tn− , x(tn− )).

HV (t , x(t), r(t))

t − tk

V (t , x(t))

tn − 1

and

+

HV (t , x(t), r(t))

]

ln V (t , x(t))

GTr(t) Pr(t),1 Gr(t)

(Pr(t),2 − Pr(t),1 ) + τr(t) ] t − tk T Gr(t) (Pr(t),2 − Pr(t),1 )Gr(t) x(t), + τr(t)

+

tn

[

By utilizing the mathematical induction method, for any t ∈ [tn , tn+1 ), n ∈ Z≥1 , we obtain that

)

t − tk

∫

1

τi

) ˜ Pi η(t),

HV (t , x(t), i) = ηT (t)∆i η(t),

(26) (27)

then for any t ∈ [tk , tk+1 ) and r(tk ) = i, k ∈ Z≥0 , i ∈ I , there exists a constant γi ∈ R, i ∈ I , such that (21)

LV (t , x(t), i)

⏐ ⏐2 1 ⏐ HV (t , x(t), i) ⏐ ⏐ − ⏐⏐ 2 V (t , x(t)) ⏐

V (t , x(t)) LV (t , x(t), i)

HV (t , x(t), i) 1 − γi + γi2 V (t , x(t)) 2 ( ) ηT (t) Ξ i + τ1 ˜ Pi η(t) i ηT (t)∆i η(t) 1 2 = − γi T + γi T η (t)Pi η(t) η (t)Pi η(t) 2

≤

V (t , x(t))

4

B. Wang and Q. Zhu / Systems & Control Letters 137 (2020) 104622

[

ηT (t) Ξ i − γi ∆i + =

η

]

By (34) and (35), we have

˜ + 1 γ 2 Pi η(t) 2 i

1 P E(τi ) i

T (t)P i

η(t)

i∈I

xT (t)Pi,1 x(t) +

t −tk T x (t)(P

τi

i,2

− Pi,1 )x(t)

.

(28)

[ ηT (t) Ξ i − γi ∆i +

N(t)

lim

t

t →∞

< βi .

(29)

∑n

k=0

ln µr(tk )r(tk+1 )

xT (t)P

+

∑

r(t),1 x(t)

+

t −tN(t) T x (t)(Pr(t),2 τ r(t)

n→∞

(30)

= lim

n→∞

=˜ πi − HV (t , x(t), r(t))

]

V (t , x(t))

dB(t)

(31)

is a local continuous martingale with M(0) = 0. By (27), for any t ∈ [tk , tk+1 ) and r(tk ) = i, k ∈ Z≥0 , i ∈ I , we have

⏐ ⏐ ⏐ ⏐ ⏐ HV (t , x(t), i) ⏐2 ⏐ ηT (t)∆i η(t) ⏐2 2 ⏐ ⏐ =⏐ ⏐ ⏐ V (t , x(t)) ⏐ ⏐ ηT (t)Pi η(t) ⏐ ≤ ρ1 ,

(32)

ρ1 := max

max{|λmin (∆i )|, |λmax (∆i )|}

λmin (Pi )

i∈I

M(t)

t

t →∞

lim

t

t →∞

= ∑ = ∑

lim

⏐ ∫ t⏐ ⏐ HV (t , x(t), r(t)) ⏐2 2 ⏐ ⏐ ⏐ V (t , x(t)) ⏐ dt ≤ ρ1 t . 0

tk−+1

[(

∑n

k=0

− lim

I(r(tk ) = i)sk

nE(τi ) Si (tn+1 ) Ni (tn+1 ) n→∞

1 E(τi )

lim

n→∞

Ni (tn+1 ) N(tn+1 )

= 0. a.s..

(39)

(40)

+ 1.

(41)

×

1

1

−

)

τr(t) E(τr(t) ) xT (t)(Pr(t),2 − Pr(t),1 )x(t)

tk

(33)

xT (t)Pr(t),1 x(t) +

t −tk T x (t)(Pr(t),2 τ r(t)

]

− Pr(t),1 )x(t)

dt ,

(42)

= lim ∑

Xk =

Sj (t) Nj (t) Nj (t) N(t)

= lim

t →∞

Ni (t) Si (t) Si (t)

t

(43)

∑

Xki .

(44)

i∈I

j∈I Sj (t)

Si (t) Ni (t) Ni (t) N(t)

limt →∞

Xki = I(r(tk ) = i)Xk , which implies that for any k ∈ Z≥0 ,

Si (t)

˜ πi E(τi ) = πi , a.s. πj E(τj ) j∈I ˜

t

E(τi )

]

and

t →∞

Ni (t)

−

N(tn ) E(τi )˜ πi

E(τr(t ) ) k

sk E(τi )

mini∈I {λmin (Pi,1 ), λmin (Pi,2 )}

∫

(34)

and t →∞

Ni (tn )

Xk :=

= 0, a.s.

limt →∞ j∈I

n

]

sk

−

For any k ∈ Z≥0 and i ∈ I , we denote

By (5), (6) and (7), we can obtain that for any i ∈ I , Si (t)

(38)

⏐ ⏐ ⏐ ⏐ xT (t)(Pr(t),2 − Pr(t),1 )x(t) ⏐ ⏐ t −tk T ⏐ xT (t)P x (t)(Pr(t),2 − Pr(t),1 )x(t) ⏐ r(t),1 x(t) + τ r(t) ⏐ T ⏐ ⏐ η (t)˜ Pr(t) η(t) ⏐ ⏐ ≤ ρ2 , = ⏐⏐ T η (t)Pr(t) η(t) ⏐

Then, it follows from Theorem 3.4 of [29] that lim

dt .

For any t ∈ [tk , tk+1 ) and k ∈ Z≥0 , we have

ρ2 =

} ,

which implies that for any t > 0,

⟨M(t), M(t)⟩ =

E(τr(t) )

where maxi∈I {λmax (Pi,1 ), λmax (Pi,2 )}

where

{

n I(r(tk ) = i)

k=0

n→∞

where

0

∑n

= lim

− Pr(t),1 )x(t)

βi Si (t) + M(t),

M(t) =

I(r(tk ) = i) 1 −

k=0

dt

τr(tk )

∑n = lim

]

sk

n[

n→∞

)

i∈I

∫ t[

[

= lim

E(τr(t) )

−

]

1

Yki

n

k=0 I(r(tk ) = i)

xT (t)(Pr(t),2 − Pr(t),1 )x(t)

×

τr(t)

∑n

k=0

−

1

By (4), (6) and (7), we can obtain that for any i ∈ I , lim

τr(t)

[

Yki := I(r(tk ) = i)

N(t)−1

0

tk−+1

∫

n→∞

1

(37)

i∈I

tk

ln V (t , x(t))

∑

∑ πi < ∞, a.s., E(τi )

=

For any k ∈ Z≥0 and i ∈ I , we denote

] ˜ + 1 γ 2 Pi η(t) 2 i

1 P E(τi ) i

Substituting (28) and (29) into (25), we can obtain that for any t > 0,

+

(36)

{t → ∞}.

ηT (t)Pi η(t)

1

βi πi , a.s.,

i∈I

which implies that the event {N(t) → ∞} is equivalent to

By (11) and (12), we can obtain that

∫ t [(

∑

and

xT (t)(Pi,2 − Pi,1 )x(t)

≤ ln V (0, x0 ) +

=

t

t →∞

[ ] 1 1 + − τi E(τi ) ×

βi Si (t)

∑ lim

=

πi E(τi )

, a.s..

(35)

By (38) and (40), we can get that for any n ∈ Z≥0 and i ∈ I , ⏐∑ ⏐ ⏐ ⏐ ∑ n [ ⏐ n ⏐ ⏐1 ⏐ ⏐ ⏐≤⏐ − 1 ⏐× X I(r(tk ) = i) ki ⏐ ⏐ ⏐ τi E(τi ) ⏐ k=0 k=0 ⏐ ] ∫ t− ⏐ ⏐ k+1 ⏐ xT (t)(Pr(t),2 − Pr(t),1 )x(t) ⏐ ⏐dt t − tk T ⏐ xT (t)P x (t)(Pr(t),2 − Pr(t),1 )x(t) ⏐ tk r(t),1 x(t) + τ r(t) ⏐∑ ⏐ ⏐ ⏐ n ⏐1 ⏐ n ⏐ 1 ⏐∑ ⏐ ⏐ ⏐, ≤ ρ2 ⏐⏐ − I(r(t ) = i)s = ρ Y (45) k k 2 ki ⏐ ⏐ ⏐ τi E(τi ) k=0

k=0

B. Wang and Q. Zhu / Systems & Control Letters 137 (2020) 104622

that is

which together with (39) implies that for any i ∈ I ,

∑n lim

[∫ t (

Xki

k=0

= 0. a.s.,

n

n→∞

(46)

combining with (44), we can obtain that

∑n

k=0

lim

∑n

lim

]

Xki

k=0

n

n→∞ i∈I

Xk =

k=0

<

N0 ∑

N(t) ∑

Xk +

= 0, a.s.,

(47)

Xk

k=N0

Xk + ε N(t), a.s.

(48)

N0

tk−+1

N0

Xk ≤ ρ2

k=0

∑∫

(

1 sk

tk

k=0

τr(t)

r(t),2

dt

t

− Pr(t),1 )x(t)

By utilizing (37) again, we can obtain that lim

k=0

ln µr(tk )r(tk+1 )

∑

∑t

∑

∑

t →∞ i∈I

= lim

j∈I ,j̸ =i

t →∞ j∈I

Nij (t) ln µij

t pij Ni (t) ln µij

= lim t →∞ t ∑ πi ∑ = pij ln µij , a.s.. E(τi )

(57)

j∈I

i∈I

+

lim sup 1

E(sk )

(49)

k=0

N(t)

sk =

k=0

∑

N(t)

sk −

k=0

∑

sk .

(50)

k=N0 +1

By (6), (7) and (37), we have

∑N(t)

k=0 sk

N(t)

∑

i∈I

=

Si (t)

N(t) ∑ Si (t) Ni (t) ∑ Ni (t) N(t)

i∈I

→

˜ πi E(τi ) < ∞, a.s.,

(58)

i∈I

For any N(t) > N0 , we have N0

t

∑ ) ∑ ( j∈I pij ln µij < πi β i + , a.s., E(τi )

dt

0 ∑ ρ2 sk . mini∈I {E(τi )}

ln V (t , x(t))

t →∞

)

N

≤ ρ2 N0 +

=

t −tN(t) T x (t)(P

]/

Substituting (33), (36), (56) and (57) into (30), we have

By (4), (40) and (42), we have

∑

E(τr(t) )

(56)

i∈I

k=0

∑

)

xT (t)(Pr(t),2 − Pr(t),1 )x(t)

∑N(t)−1

k=0

N0 ∑

τr(t)

0

1

−

= 0, a.s..

which implies that for any ε ∑ > 0, there exists an integer N0 > 0, m such that for any m > N0 , k=0 Xk < mε, a.s. Thus, for any N(t) > N0 , N(t) ∑

t →+∞

1

xT (t)Pr(t),1 x(t) +

n→∞

=

lim

×

Xk

n ∑[

5

(51)

i∈I

which together with (15) and (14) implies that (10) holds. Thus, we complete the proof of this theorem. Remark 3.2. 1. The conditions (11) and (12) furnish the quantitative estimate of the stability degree of each subsystem mode. These two conditions were first presented in Theorem 1 of [25]. Our Theorem 3.1 manifests that these conditions can be applicable to the SMJLSs with stochastic case. 2. The condition (13) is a standard condition, which is often used in switched systems literature (see [30]). In this paper, we can let λmax {Pi,2 } , ∀i , j ∈ I . (59) µij = λmin {Pj,1 } 3. By denoting

and

µi = max{µij }, i ∈ I ,

∑N(t)

k=N0 +1 sk

N(t)

the condition (14) can be guaranteed by

∑N(t)

k=N0 +1 sk

=

·

N(t) − N0

N(t) − N0 N(t) ∑N(t) s N(t) − N0 k = k=0 · N(t) N(t) ∑ → ˜ πi E(τi ) < ∞, a.s.,

] ∑ [ ln µi πi β i + ≤ 0. E(τi )

(52)

which together with (50) and (51) implies that when N(t) → ∞, k=0 sk

N(t)

→ 0, a.s.

(53)

Thus, by (48), (49) and (53), when N(t) → ∞, we have

∑N(t)

k=0

Xk

N(t)

→ 0, a.s.,

(54)

lim

t →∞

k=0

t

Xk

= 0, a.s.,

This inequality is composed of the average sojourn times E(τi ), i ∈ I and the stationary distribution π of semi-Markov switching process, which implies that the proper switching can stabilize the whole system even if some subsystem modes are unstable. In order to better illustrate the relationship between our conditions and the ones in some published results, we give the following corollaries. Corollary 3.3. For given constants βi ∈ R and γi ∈ R, i ∈ I , if there exist matrices Pi > 0, i ∈ I and constants µij > 0, i, j ∈ I , such that (14) and the following inequalities Her(Pi Ai − γi Pi Gi ) + GTi Pi Gi +

which together with (37) implies that

∑N(t)

(61)

i∈I

i∈I

∑N0

(60)

j∈I

1 2

γi2 Pi < βi Pi ,

Pj ≤ µij Pi , j ̸ = i, (55)

are satisfied. Then system (8) is ASE-stable.

(62) (63)

6

B. Wang and Q. Zhu / Systems & Control Letters 137 (2020) 104622

Proof. For any i ∈ I , by letting Pi = Pi,1 = Pi,2 , the inequalities (11)–(13) of Theorem 3.1 reduce to the inequalities (62) and (63), respectively. Thus, by Theorem 3.1, we complete the proof. Corollary 3.4. For given constants βi ∈ R and κi > 0, i ∈ I , if there exist matrices Pi > 0 and constants µi > 0, i ∈ I , such that (61), (63) and the following inequalities Her(Pi Ai ) + GTi Pi Gi < βi Pi ,

(64)

Her(Pi Gi ) > κi Pi (< −κi Pi ),

(65)

are satisfied. Then system (8) is ASE-stable. Proof. For any i ∈ I , letting γi = (65) that Her(Pi Ai − γi Pi Gi ) + GTi Pi Gi +

1 2

κi 2

, then it follows from (64) and

γi2 Pi < βi Pi .

(66)

Thus, by Corollary 3.3, we complete the proof. If we further assume that for each i ∈ I , the sojourn time τi of semi-Markov process {r(t), t ≥ 0} follows the exponential distribution with mean E(τi ) = λ1 , then this switching process i reduces to Markov process and its generator matrix reduces to a constant matrix Q = (qij )M ×M , in which qij = λi pij , j ̸ = i,

(67)

and qii = −λi , i, j ∈ I .

(68)

Next, we let Gi ≡ 0, i ∈ I , then system (8) reduces to the deterministic case with the following form: dx(t) = Ar(t) x(t)dt , t ≥ 0.

(69)

Corollary 3.5 (Theorem 1 of [25]). For given constants βi ∈ R, i ∈ I , if there exist matrices Pi,l > 0, i ∈ I , l = 1, 2 and constants µij > 0, i, j ∈ I , such that the condition (13) of Theorem 3.1 and the following inequalities Her(Pi,1 Ai ) + λi (Pi,2 − Pi,1 ) < βi Pi,1 ,

(70)

Her(Pi,2 Ai ) + λi (Pi,2 − Pi,1 ) < βi Pi,2 ,

(71)

) ∑ ( ∑ πi β i + qij ln µij ≤ 0,

(72)

j∈I ,j̸ =i

i∈I

are satisfied. Then system (69) with Markov switching is ASE-stable. Proof. For any i ∈ I , we choose γi = 0 and let E(τi ) = λ1 , Gi = 0. i Then, the conditions (11) and (12) of Theorem 3.1 reduce to (70) and (71), respectively. By (67), the condition (14) of Theorem 3.1 reduces to (72). Thus, from Theorem 3.1 we see that the proof is completed. 4. An example In this section, we give an example to show the validity of our results. Fig. 1. Simulation results of modes (73)–(75).

Example. Consider the system (8) with the following three modes:

( A1 =

( A2 =

−1 2

( A3 =

−0.8 0.5

−2 1

−1.2

( ) −1 0 , G1 = , 0 0 0.2 ) ( ) −2 −1 2 , G2 = , 1 −1 1 ) ( ) −2.5 −1 0 , G3 = . 0

)

0

1

(73) (74) (75)

Fig. 1 shows 100 realizations of sample paths of modes (73)–(75), respectively. It can be observed that modes (73) and (75) are stable and mode (74) is unstable. The condition (14) implies that the probability structure of semi-Markov switching process plays an important role on the stability of the whole system.

B. Wang and Q. Zhu / Systems & Control Letters 137 (2020) 104622

7

Firstly, we discuss the effect of the average sojourn times E(τi ), i = 1, 2, 3 on the stability of the whole system. We let the TP matrix of EMC {r(tn )}n∈Z≥0 be

⎛

0

P =⎝

1 2 1 2

1 2 1 2

1 2

0 1 2

⎞ ⎠,

(76)

0

by the formula ˜ π =˜ π P, we obtain that the stationary distribution of EMC {r(tn )}n∈Z≥0 is ˜ π = ( 31 , 13 , 13 ). Thus, the condition (14) has the following form:

] 3 3 [ ∑ 1∑ ln µij ≤ 0. βi E(τi ) + 2

i=1

(77)

j=1

By assuming Pi,1 = Pi,2 , i = 1, 2, 3 and letting γ1 = γ2 = −0.5, γ3 = 1, β1 = β3 = −0.2, β2 = 5, the LMIs (11) and (12) have the feasible solutions

(

1.218 1.514

(

0.3389 −0.0523

(

0.6989 0.0241

P1,1 = P1,2 = P2,1 = P2,2 = P3,1 = P3,2 =

1.514 6.1611

)

,

−0.0523 1.6449 ) 0.0241 . 1.6476

)

,

According to (59) in Remark 3.2, we have µ12 = 2.0686, µ13 = 2.0832, µ21 = 19.5425, µ23 = 4.8937 and µ31 = 9.4270, µ32 = 2.3589. Substituting these values into (77), we have E(τ1 ) + E(τ3 ) − 25E(τ2 ) > 22.8080.

(78)

It implies that if the average sojourn times of stable modes (73) and (75) are large enough and the average sojourn time of unstable mode (74) is small enough, the corresponding semi-Markov switching process can stabilize the whole system. We can choose the proper distributions of τ1 , τ2 and τ3 satisfying inequality (78), for example

τ1 ∼ Weibull(1, 1/4), τ2 ∼ Weibull(1, 1), τ3 ∼ Weibull(1, 1/4).

(79)

By Theorem 3.1, the whole system that consists of modes (73)–(75) can be stabilized by a semi-Markov switching process satisfying (76) and (79). (a) of Fig. 2 shows 100 realizations of sample path of the whole system switched by a semi-Markov switching satisfying (76) and (79). It can be observed that the whole system can be stabilized by the designed semi-Markov switching process. Secondly, we discuss the effect of stationary distribution π of semi-Markov switching process {r(t), t ≥ 0} on the stability of the whole system. We let the sojourn times of modes (73)–(75) be

τ1 ∼ Weibull(5, 3), τ2 ∼ Exp(1/10), τ3 ∼ Weibull(5, 3).

(80)

It is obtained that E(τ1 ) = E(τ2 ) = E(τ3 ) = 10, and the condition (61) in Remark 3.2 has the following form: ln µ1

[ ] ln µ2 ˜ π1 β1 + +˜ π2 β 2 + E(τ1 ) E(τ2 ) [ ] ln µ3 +˜ π3 β 3 + ≤ 0, E(τ3 ) [

]

(81)

(

28.3932 −14.1554

−14.1554 24.5541

)

,

(

28.7833 −7.9388

−7.9388 6.3186

)

(

22.1742 −4.4581

−4.4581 49.9120

)

(

19.3927 −2.7263

(

23.8881 5.4102

(

13.2403 7.1209

P1,2 = P2,1 = P2,2 = P3,1 = P3,2 =

where ˜ π = (˜ π1 , ˜ π2 , ˜ π3 ) is the stationary distribution of EMC {r(tn )}n∈Z≥0 . By letting γ1 = γ2 = γ3 = 0.5 and β1 = β3 = −0.5, β2 = 0.5, the LMIs (11) and (12) have the feasible solutions P1,1 =

Fig. 2. Simulation results of the whole system switched by the proper semi-Markov switching process.

, ,

−2.7263 , 39.0038 ) 5.4102 , 27.5928 ) 7.1209 . 14.4267

)

By (59) and (60) in Remark 3.2, we have µ1 = 12.7709, µ2 = 2.1429, µ3 = 7.5674. Thus, the inequality (81) can be guaranteed by

˜ π1 + ˜ π3 − ˜ π2 ≥ 0.1524.

(82)

We let TP matrix P be (76), which can ensure that (82) holds. (b) of Fig. 2 shows 100 realizations of sample path of the whole

8

B. Wang and Q. Zhu / Systems & Control Letters 137 (2020) 104622

system switched by a semi-Markov switching process satisfying (80) and (82). It can be observed that the whole system can be stabilized by the designed semi-Markov switching process. 5. Conclusion The ASE-stability analysis problem for a class of SMJLSs has been studied in this paper. The corresponding sufficient conditions are given in terms of LMIs and the main elements in the probability structure of semi-Markov process. Our results generalize and improve Theorem 1 of [25] to the SMJLS case. Finally, we present an example to show the validity of our results. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, New York, 1990. [2] D. Chatterjee, D. Liberzon, Stability analysis and stabilization of randomly switched systems, in: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 2643–2648. [3] D. Chatterjee, D. Liberzon, On stability of randomly switched nonlinear systems, IEEE Trans. Automat. Control 52 (12) (2007) 2390–2394. [4] D. Chatterjee, D. Liberzon, Stabilizing randomly switched systems, SIAM J. Control Optim. 49 (5) (2008) 2008–2031. [5] C. Zhu, G. Yin, Q. Song, Stability of random-switching systems of differential equations, Quart. Appl. Math. 67 (2) (2009) 201–220. [6] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl. 79 (1) (1999) 45–67. [7] X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. [8] F. Zhu, Z. Han, J. Zheng, Stability analysis of stochastic differential equations with Markovian switching, Systems Control Lett. 61 (12) (2012) 1209–1214. [9] C.G. Yuan, J. Lygeros, Stabilization of a class of stochastic differential equations with Markovian switching, Systems Control Lett. 54 (9) (2005) 819–833. [10] Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with lévy noise and Markov switching, Internat. J. Control 90 (8) (2017) 1703–1712. [11] Q. Zhu, Pth moment exponential stability of impulsive stochastic functional differential equations with Markovian switching, J. Franklin Inst. B 351 (7) (2014) 3965–3986.

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