- Email: [email protected]
Stability analysis of time-varying discrete stochastic systems with multiplicative noise and state delays
Accepted Manuscript
Stability analysis of time-varying discrete stochastic systems with multiplicative noise and state delays Xiushan Jiang, Senping Tian, Tianliang Zhang, Weihai Zhang PII: DOI: Reference:
S0016-0032(18)30449-6 10.1016/j.jfranklin.2018.06.034 FI 3535
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
23 November 2017 15 June 2018 18 June 2018
Please cite this article as: Xiushan Jiang, Senping Tian, Tianliang Zhang, Weihai Zhang, Stability analysis of time-varying discrete stochastic systems with multiplicative noise and state delays, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.06.034
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Stability analysis of time-varying discrete stochastic systems with
Xiushan Jiang1 , 1
Senping Tian1∗,
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multiplicative noise and state delays Tianliang Zhang1 ,
Weihai Zhang2
School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, Guangdong Province, P. R. China
2 College
of Electrical Engineering and Automation, Shandong University of Science and Technology,
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Qingdao 266590, Shandong Province, P. R. China
Abstract
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The aim of this paper is to study the stability of discrete stochastic time-delayed systems with multiplicative noise, where the coefficients are assumed to be time-varying with a general time-varying rate or a small time-varying rate. Firstly, by the Kronecker algebra theory and H-representation technique, the exponential stability of the stochastic system with common time-varying coefficients is investigated by the spectral approach. It is shown that the time-varying stochastic systems with state delays is exponentially stable in mean square sense if and only if its corresponding generalized spectral radius is less than one. Secondly, under definite conditions, by applying the so-called “frozen” technique, it is shown that the stability of a “frozen” system implies that of the corresponding slowly time-varying system.
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Keywords. Stability, discrete time-delay systems, stochastic systems, time-varying coefficients, slowly time-varying coefficients.
Introduction
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Stochastic systems play a remarkable role in a wide range of applications involving economics, finance and mechanics and have been extensively studied during the past several decades; see, e.g., [14, 15, 16, 21, 22, 32] . In addition, stochastic time-delay system is also one of the fundamental research topics in control theory which has received an increasing attention in recent years. Time-delay phenomenon is very common in many engineering systems such as fault diagnosis processes, chemical processes, hybrid systems and so on. To mention a few, the problems of stochastic time-delay systems such as stability analysis, robust control and some important applications can be found in the past literature; see, e.g., [12, 10, 26, 39, 40] and the references therein. Stability is one of the most important notions in modern control theory, which is a necessary prerequisite to make the system behave well. For an unstable open-loop system, in practical engineering, one is required to design a stabilizing controller in many control problems such as optimal H2 control, adaptive control, H∞ control, mixed H2 /H∞ control and so on. So, various stability concepts for linear stochastic ∗ Corresponding
author. Email: [email protected]
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systems has been studied widely by many researchers in recent years, such as exponential stability [8, 9, 15] , finite-time stability [6, 7] , almost sure stability [4, 5, 15, 28] , global asymptotical stability [13, 30] , asymptotic mean square stability [29] and critical stability [31] . For linear time-varying systems, the property of stability is of increasing interest in recent years and considerable research effort has been taken. Different kinds of methods have been adopted to find less conservative criteria of stability. It can be remarked that, in spite of time-invariant systems or time-varying systems, the Lyapunov function method serves as a main technique for most existing works about the stability analysis, but finding suitable Lyapunov functions is still a difficult task; see [2, 24, 35, 36, 37] and so on. Another method is to investigate special cases of time-varying systems by decomposing the system matrix of a linear time-varying system into two parts, one is a constant matrix and the other one is a time-varying derivation, which satisfies certain conditions, see [11, 27] . Unlike most previous works, in this paper, we will obtain the stability criteria of linear stochastic discrete systems with time-varying coefficients and time-delays by using the spectral technique and “frozen” technique, respectively. It is well-known that, for linear time-invariant deterministic system xk+1 = Axk , the stability can be determined in terms of the locations of the eigenvalues of the system matrix A, while the stability of a linear time-varying system xk+1 = Ak xk is not directly available by the eigenvalue placement methodology [25] . In order to formulate a similar stability criterion by means of eigenvalues in time-varying systems, [18] generalized the concept of spectral radius by imposing some special restrictions on the coefficient matrix sequence {Ak }k∈Z + and proved the relationship between the spectral radius and the exponential stability. In addition, [20] proposed several new formulas for computing the generalized spectral radius. Due to widespread applications of stochastic systems, The spectrum concept and spectrum assignment were proposed in [29] by the associated Lyapunov operators, the latter is similar to the pole placement in modern control theory, we refer the reader to [32] for stability analysis of stochastic timeinvariant systems. Up to now, there is no analogous stability criteria for linear stochastic time-varying systems. To fill in this gap, our first contribution of this paper is to present spectral criteria for a kind of stochastic discrete time-varying systems with state delays and general time-varying coefficients based on generalized spectral radius method. Different from the stochastic time-invariant case, the generalized spectral radius is associated with the following standard discrete time-varying linear system x(k + 1) = Θ(Hn2 ×(n(n+1)/2) (k), A(k), C(k))x(k).
(1)
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In this process, H-representation technique plays a vital role, which has been introduced in [38] and got important applications in [39, 40] . In this paper, we will establish the connection between stability in mean square sense of stochastic systems and stability of deterministic systems by the H-representation technique. The second contribution of this paper is to obtain a stability criterion for slowly time-varying stochastic systems by “frozen” technique. This technique has been extensively studied in deterministic cases; see [1, 3, 8, 17, 19, 23] . Here, we generalize this “frozen” technique in deterministic systems to discrete time-varying stochastic systems with state delays: ( x(k + 1) = A(k)x(k) + A0 (k)x(k − τ ) + C(k)x(k)w(k) + C0 (k)x(k − τ )w(k), (2) x(k) = ϕ(k), k = 0, −1, −2, · · · , −τ. In system (2), we define the state transition matrix as Φ(k, s), which is generated by the coefficient matrices A(k), A0 (k), C(k) and C0 (k). The “frozen” technique will allow us to involve neither Lyapunov 2
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functions in some situations nor stability assumptions about the state transition matrix Φ(k, s). Concretely speaking, the frozen technique can be described as follows. If p ∈ N + is any fixed integer, then the following stochastic discrete-time autonomous system x(k + 1) = A(p)x(k) + A0 (p)x(k − τ ) + [C(p)x(k) + C0 (p)x(k − τ )]w(k)
(3)
Preliminaries
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is taken as a particular case of time-varying stochastic system (2) with its time dependence “frozen” at time p. In this paper, it is shown that if each “frozen” system is stable or exponentially stable in mean square (ESMS) and the coefficient change rate of system (2) is small enough, then our considered system is stable in some senses. Besides, we also consider the time-delay phenomena in our system, which arise frequently in many practical applications. One major stream of researching time-delay systems is to find a corresponding strict Lyapunov function, which can often be transformed into a Lyapunov-Krasovskii functional. Since we apply spectral approach and “frozen” technique to research stability problem in our system, we use two different methods to deal with time-delays. In essence, those two methods can be summarized as to expand the state dimensions. There are still some differences on those two methods. Compared with the first method which simply transform a system with time-delays into an augmented system without delays, the seconde method would reduce the conservativeness by retaining some characters of time-delays. In another hand, we have to point out that the system dimensional space would be highter after expanding the state dimensions which makes the calculation inconvenient in practical application. So we use the H-representation technique to reduce the dimension of system matrix. The paper is organized in the following way. In Section 2, some adequate preliminaries and definitions are presented. We state our main results in Section 3. Firstly, we propose an useful notion of generalized spectral radius which can be applied to analyze the operator sequences for our considered systems. In addition, a stability criterion is given in terms of the spectral radius. Secondly, we give new sufficient conditions for the exponential stability in mean square sense of the considered systems by “frozen technique”. Section 4 concludes this paper. For convenience, we adopt the following notation throughout this paper: AT : the transpose of a matrix or vector A. Rl : the l-dimensional real vector space with the usual inner product h·, ·i. R: the set of all real numbers. Rm×n : the vector space of all m × n real matrices. Sn : the set of all n × n symmetric matrices. E(·): the mathematical expectation operator. N := {0, 1, 2, . . .}, N−τ := {0, −1, −2, · · · , −τ } and N + := {1, 2, . . .}. (0 − 1)-matrix: all matrix elements are either 0 or 1.
Consider the following discrete-time time-varying stochastic system with state delay x(k + 1) = A0 (k)x(k) + C0 (k)x(k)w(k) +[Aτ (k)x(k − τ ) + Cτ (k)x(k − τ )w(k)], k ∈ N + , x(k) = ϕ(k) ∈ Rn , k ∈ N−τ ,
(4)
where x(k) ∈ Rn and u(k) ∈ Rnu are the system state and control input, respectively. τ is a positive integer that denotes the upper bound of time-delay. For the given deterministic initial condition ϕ(k) ∈ Rn , k ∈ N−τ , the corresponding solution process is denoted by x(k). {w(k), k ∈ N } is the onedimensional independent white noise process which is defined on the complete probability space (Ω, F, P). 3
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Ft := σ(w(s) : 0 ≤ s ≤ k) is the natural filtering generated by w(·). Besides, the initial states ϕ(k)k∈N−τ are independent of the process w(k). We assume Ew(k) = 0, Ew(k)w(s) = δks , where δks = 1 for k = s, δks = 0 for s 6= k, i.e., δks is a Kronecker function. {A0 (k), C(k), Aτ (k), Cτ (k)}k∈N are the time-varying matrix sequences with the appropriate dimensions. The following concepts of stability and exponential stability will be used in formulating the main results of the paper.
Ekx(k)k2 < ε, ∀k ∈ N + .
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Definition 2.1 The zero solution of system (4) is stable or Lyapunov stable if for every ε < 0 and every k0 ∈ N + , there is a number δ > 0(depending on ε and k0 ) such that every solution of this system with initial condition kϕ(k)k < δ for k ∈ N−τ satisfies the condition (5)
Definition 2.2 System (4) is said to be exponentially stable in mean square (ESMS) if there exist β ≥ 1, λ ∈ (0, 1) such that for any 0 ≤ k0 < k < +∞, there holds
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E k x(k) k2 ≤ βλk−k0 max k ϕ(s) k2 −τ ≤s≤0
(6)
for any solution x(k) of system (4) with the initial condition x(k)k∈N−τ = ϕ(k) ∈ Rn .
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As we have stated before, H-representation technique is an important tool for connecting a stochastic system with its related deterministic system. So we are in a position to introduce the H-representation concept and some properties which is developed in [38] according to our considered system. For any 2 time-varying matrix X(·) = [x(·)ij ]n×n ∈ Sn . vec[X(·)] : Rn×n → Rn ×1 denotes the vectorization of the matrix sequence X(·) generated by stacking the rows of X(·) into a column vector and vecs[X(·)] : n(n+1) Rn×n → R 2 ×1 is the column vector which is generated by all upper diagonal elements of X(·). That is,
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→ − X (·) = vec[(X)(·)] = (x11 , x12 , · · · , x1n , x21 , · · · , x2n , · · · , xn1 , · · · , xnn )T , e = vecs[(X)(·)] = (x11 , x12 , · · · , x1n , x22 , · · · , x2n , · · · , xn−1,n−1 , xn−1,n , xnn )T . X(·)
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It is not difficult to verify that, for x ∈ Rn , there exists
vec(X) = vec[xxT ] = x ⊗ x.
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The relationship between vec(X(·)) and vecs(X(·)) can be determined by the following lemma. n(n+1) 2 Lemma 2.1 [38] For any X ∈ Sn , there exists a unique transfer matrix H n2 , n(n+1) ∈ Rn × 2 , 2 such that → − e X(·) . (i) X (·) = H n2 , n(n+1) 2 T 2 n(n+1) (ii) H n2 , n(n+1) H n , is invertible. 2 2
Remark 2.1 The matrix H n2 , n(n+1) is a column full rank (0 − 1)-matrix, which is called an H2 e representation matrix and is used to eliminate repeated elements in X(·). 4
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We introduce H-representation technique in detail by the following unforced delay-free system ( x(k + 1) = A0 (k)x(k) + C0 (k)x(k)w(k), x(0) = x0 ∈ Rn .
(7)
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Set X(k) = E[x(k)x(k)T ] ∈ Sn . Based on the fact of that x(k) and w(k) are independent and x(k)x(k)T is Fk−1 -measurable, X(k) with k ∈ N satisfies ( X(k + 1) = A0 (k)X(k)A0 (k)T + C0 (k)X(k)C0 (k)T , (8) X(0) = x(0)x(0)T ∈ Sn , k ∈ N . The following Kronecker matrix property will be adopted throughout this paper.
Lemma 2.2 [34] For matrices A, B, C, D with suitable dimensions, A ⊗ C denotes the Kronecker product of A and C, which has the following properties: (AC ⊗ BD) = (A ⊗ B)(C ⊗ D),
vec(ABC) = (A ⊗ C T )vec(B).
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(A ⊗ C)T = AT ⊗ C T ,
By means of Lemma 2.2, (8) is equivalent to ( vec(X(k + 1)) = (A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k))vec(X(k)), vec(X(0)) = vec(X0 ) = vec(x(0)x(0)T ) ∈ Sn , k ∈ N .
(9)
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Then (9) can be turned into a standard deterministic system by H-representation matrix. Lemma 2.3
where
e + 1) = Θ H n2 , n(n+1) , A0 (k), C0 (k) X(k), e X(k 2 n(n+1) e e0 ∈ R 2 , k ∈ N , X(0) =X
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(
(11)
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Θ H n2 , n(n+1) , A0 (k), C0 (k) 2 h i−1 := H T n2 , n(n+1) H n2 , n(n+1) H T n2 , n(n+1) 2 2 2 ×(A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k))H n2 , n(n+1) . 2
(10)
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e 2 Next, we give a result about the relationship between E kx(k)k in system (7) and X(k)
in system (10).
e 2 Lemma 2.4 E kx(k)k in system (7) and X(k)
in system (10) satisfy (i)
(ii)
2
E kx(k)k ≤
√
e n X(k)
.
rn + 1
e 2 X(k) E kx(k)k .
≤ 2
5
(12)
(13)
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e Proof. As shown above, X(k)
= vecs{E[x(k)x(k)T ]} . Note that 2
E kx(k)k
e 2
X(k) =
Then
e 2
X(k)
=
n X
2
n X
=
i=1
X
E|xi (k)|
!2
,
(14)
2
(15)
(E(xi (k)xj (k))) .
1≤i≤j≤n
X
(E|xi (k)|2 )2 +
i=1
≥
2
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(E(xi (k)xj (k)))
1≤i
n X
1 (E|xi (k)| ) ≥ n i=1 2 2
n X i=1
E|xi (k)|
!2
2
.
(16)
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So, combining equation (14) and inequality (16), we can get that the relationship (12) holds. For proving the second result in this lemma, we use Cauchy-Schwarz inequality to get X 2 (E(xi (k)xj (k))) 1≤i
≤
X
2
1≤i
1 2
(E|xi (k)xj (k)|) ≤
X
1≤i
X
1≤i
E|xi (k)|2 E|xj (k)|2
(E|xi (k)| ) + (E|xj (k)|2 )2 .
2 2
(17)
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≤
Then, substituting (17) into (16), the following inequality holds
≤
n X
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≤
(E|xi (k)|2 )2 +
i=1
n X
(E|xi (k)|2 )2 +
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e 2
X(k)
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≤
i=1
n
n−1 n−1 1X i(E|xi (k)|2 )2 (E|xn (k)|2 )2 + 2 2 i=1 n
n−1X (E|xi (k)|2 )2 2 i=1
n+1X (E|xi (k)|2 )2 . 2 i=1
(18)
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(13) is derived from (18) immediately. Therefore, the proof is complete.
e 2 Remark 2.2 The relationship between X(k)
and E kx(k)k shown in Lemma 2.4 means that the stability property in mean square sense of the original system (7) are equivalent to the asymptotic stability of the standard deterministic system (10).
3
3.1
Stability Analysis Spectrum method for time-varying systems with general time-varying rate coefficients
It is well-known that a linear time-varying discrete-time system without multiplicative noise is stable if and only if the spectral radius of the operator associated with the system is less than one; see [18] . In this 6
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section, we will extend the previous results [18, 37, 38] to linear discrete time-varying stochastic systems with state delays and multiplicative noise by the method of augmented systems. Our considered system in this subsection is shown as (k)x(k) + C0 (k)x(k)w(k) x(k + 1) = A0P m (19) + τ =1 [Aτ (k)x(k − τ ) + Cτ (k)x(k − τ )w(k)], k ∈ N , x(k) = ϕ(k), k ∈ N−m . By introducing an n(m + 1)-dimensional column vector x ¯(k)k∈N as T x ¯(k) = x(k)T , x(k − 1)T , . . . , x(k − m)T ,
(20)
system (19) can now be written as the form of an equivalent augmented stochastic system without delays such as x ¯(k + 1) = A(k)¯ x(k) + C(k)w(k)¯ x(k) (21) T
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C(k) =
C0 (k) C1 (k) · · · 0 0 ··· .. .. .. . . . 0 0 ···
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with the same initial state x ¯(0) = [ϕ(0), ϕ(−1), · · · , ϕ(−m)] , where A0 (k) A1 (k) · · · Am−1 (k) Am (k) I 0 ··· 0 0 A(k) = .. .. .. .. .. . . . . . 0 0 ··· I 0 Cm−1 (k) Cm (k) 0 0 .. .. . . 0 0
,
(22)
.
Obviously, A(k) ∈ Rn(m+1)×n(m+1) , C(k) ∈ Rn(m+1)×n(m+1) . Now we will reduce the system dimension , A, C by H-representation technique introduced above. Recalling the definition of Θ H n2 , n(n+1) 2
Θ H n2 (m + 1)2 , n(m+1)(mn+n+1) , A(k), C(k) 2 h i−1 T 2 2 n(m+1)(mn+n+1) 2 2 n(m+1)(mn+n+1) := H n (m + 1) , H n (m + 1) , 2 2 ×H T n2 (m + 1)2 , n(m+1)(mn+n+1) 2 ×(A(k) ⊗ A(k) + C(k) ⊗ C(k))H n2 (m + 1)2 , n(m+1)(mn+n+1) , 2 2 2 n(m+1)(mn+n+1) Θ H n (m + 1) , 2 h i−1 T 2 2 n(m+1)(mn+n+1) 2 2 n(m+1)(mn+n+1) := H n (m + 1) , H n (m + 1) , 2 2 ×H T n2 (m + 1)2 , n(m+1)(mn+n+1) . 2
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where
(23)
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in Lemma 2.3, setting X(k) = E[¯ x(k)¯ x(k)T ] ∈ Sn(m+1) , system (21) leads to e + 1) = Θ H n2 (m + 1)2 , n(m+1)(mn+n+1) , A(k), C(k) X(k), e X(k 2 e e0 X(0) =X n(m+1)(mn+n+1) 2 = Θ H n2 (m + 1)2 , n(m+1)(mn+n+1) vec[(¯ x(0)¯ x(0)T )] ∈ R , k ∈ N, 2
To elucidate our transformed standard system (23), we present an example as follows: 7
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Example 3.1 Consider the following stochastic system:
h
iT
x(k) x(k − 1) "
A(k) =
k 1
3k 0
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, then we get
#
,
C(k) =
In the transformed system,
x2 (k) e X(k) = E x(k)x(k − 1) , x2 (k − 1)
"
2k 0
4k 0
#
H(4, 3) =
1 0 0 0
0 1 1 0
0 0 0 1
.
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Set x ¯(k) =
x(k + 1) = kx(k) + 2kx(k)w(k) + 3kx(k − 1) + 4kx(k − 1)w(k).
,
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5k 2 Θ (H(4, 3), A(k), C(k)) = 11k 2 25k 2
2k 3k 0
1 0 . 0
(26)
Definition 3.1 [33] System (23) is said to be exponentially stable (ES) if there exist β ≥ 1, λ ∈ (0, 1) such that for any 0 ≤ k0 < k < +∞, there holds
e
e (27)
X(k; k0 ) ≤ βλk−k0 X 0
M
e e0 . for any solution X(k) of system (23) with the initial value X
Lemma 3.1 The equilibrium of system (23) is exponentially stable implies that stochastic system (19) is ESMS.
E k¯ x(k)k
2
2
=
m X n X i=0 j=1
CE
≤ n(m + 1)
AC
e 2
X(k)
2
E|xj (k − i)|2
PT
ED
e Proof. Using the same method as in Lemma 2.4, system state X(k) in (23) and the augmented state x ¯(k) in (20) are shown as
X
= =
m X n X i=0 j=1
X
0≤f ≤g≤m 1≤i≤j≤n m X X f =0 1≤i≤j≤n
E|xj (k − i)|2
2
, 2
[Exi (k − f )xj (k − g)] 2
[Exi (k − f )xj (k − f )] +
≤ z1 + z2 + z3 + z4 ,
where z1 =
m X n X i=0 j=1
z2 =
m X
X
f =0 1≤i
(28)
X
0≤f
E|xj (k − i)|2
2
8
n
2
[Exi (k − f )xj (k − g)]
,
m
2
(Exi (k − f )xj (k − f )) ≤
X
2 n − 1 XX E|xj (k − i)|2 , 2 i=0 j=1
(29)
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z3 = z3 =
X
X
0≤g
m
2
(Exi (k − f )xj (k − g)) ≤
m
(Exi (k − f )xi (k − g))2 ≤
n
2 m(n − 1) X X E|xj (k − i)|2 , 2 i=0 j=1 n
2 m XX E|xj (k − i)|2 . 2 i=0 j=1
(30)
Then, we have
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p
e 2 (31) E k¯ x(k)k ≤ n(m + 1) X(k)
, r
mn + n + 1
e 2 E k¯ x(k)k . (32)
X(k) ≤ 2
e 2 Inequalities (31) and (32) mean that E k¯ x(k)k and X(k)
have the same order and this completes the proof.
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Lemma 3.2 [29] System (19) is ESMS if there exist β ≥ 1, λ ∈ (0, 1) such that for any 0 ≤ k0 < k < +∞, e k0 , X e0 ) k in system (23) satisfies k X(k;
e
e k−k0 e0 ) , (33)
X(k; k0 , X
≤ X 0 βλ i.e.,
k−1 Y
i=k0
!
n(m + 1)(mn + n + 1)
e k−k0 e0 Θ H n2 (m + 1)2 , . X , A(i), C(i)
≤ X 0 βλ
2
(34)
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ED
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Remark 3.1 Our aim in this subsection is to transform the study of ESMS of the stochastic system (19) into that of exponential stability of the deterministic system (23). Although, in (22), there are a lot of zeros in A(k) and C(k), system (22) is still a discrete stochastic system for C(k) 6= 0, k ∈ N . If we do not use the H-representation technique, we, of course, can also obtain a deterministic difference equation on vec(X(k)) corresponding to system (23), which is an n2 (m + 1)2 -dimensional system larger than the n(m+1)(nm+n+1) -dimensional system (23), and which will increase the computational complexity if we use 2 the difference equation on vec(X(k)) instead of (23). [18]
to
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The following definition generalizes the spectral radius of deterministic time-varying systems stochastic time-varying systems.
AC
Definition 3.2 Consider system (23). Set the time-varying sequence n(m + 1)(mn + n + 1) {M (k)}k∈N := Θ H n2 (m + 1)2 , , A(k), C(k) . 2 k∈N We define
1
(k−k 0)
k−1
Y
ρ (M ) := lim sup sup M (i)
k→∞ k0 ∈N
(35)
i=k0
as the generalized spectral radius of the system sequence {M (k)}k∈N .
We now need to define another numerical quantity δ(M ) as
k−1 ( )
Y
+ k−k0 δ(M ) := inf λ ∈ R | ∃ β ≥ 1, s.t. M (i) ≤ βλ
i=k0
9
k∈N
,
(36)
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which is called the power bound of {M (k)}k∈N . Next we state and prove our main result. Theorem 3.1 Assume that kM (k)k < ∞ for all k ∈ N . Then, the following two results hold: (i) The power bound and the generalized spectral radius of {M (k)}k∈N are identical, i.e., ρ (M ) = λ ⇔ δ(M ) = λ, λ ∈ R+ .
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(ii) System (19) is ESMS if and only if the generalized spectral radius of {M (k)}k∈N is less than one. Proof. Necessity part of (i): Assume that ρ (M ) = λ. Then, according to the limit definition, there exists ks ∈ N + , such that 1
ks −k 0 s −1 kY
sup M (i) ≤ (λ + ε) (37)
k0 ∈N + i=k0
i=k0
AN US
holds for any ε > 0. Similarly, for all k ∈ Nk+s , we have
k−1
Y
M (i) ≤ (λ + ε)k−k0 .
(38)
M
Let µ(M (k)) denote the least upper bound of kM (k)kk∈N . We know that µ(M (k)) is bounded and set ( k ) µ(M (k)) β := sup . (39) (λ + ε) k∈{0,1,··· ,(ks −1)}
ED
So β is a finite number with β ≥ 1 and for k ∈ {0, 1, . . . , (ks − 1)}, it can be shown that
k−1
Y
e k−k0 e0 . M (i)X
≤ X 0 β(λ + ε)
(40)
i=k0
CE
PT
Considering (38), inequality (40) also holds for all k ∈ Nk+s . Hence, δ(M ) ≤ λ = ρ(M ). Sufficiency part: When δ(M ) = λ, by (36), it yields that, for any ε > 0, there exists β ≥ 1 such that
k−1
Y
e k−k0 e sup M (i)X0 ≤ X . (41) 0 β(λ + ε)
k0 ∈N i=k0
AC
Hence, we have
k−1
Y
sup M (i) ≤ β(λ + ε)k−k0 ,
k0 ∈N
(42)
i=k0
which further leads to 1
k−k
k−1
0 1 Y
≤ lim sup β k−k0 (λ + ε) = (λ + ε). lim sup sup M (i)
k→∞ k0 ∈N k→∞
(43)
i=k0
(43) implies that ρ(M ) ≤ λ = δ(M ). The proof (i) is complete. Based on Definition 2.2 and Lemma 3.2 about ESMS, (ii) can be directly obtained from Theorem 3.1-(i). 10
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i=k0
3.2
[18]
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Remark 3.2 Generalized spectral radius ρ (M ) has the following equivalent definitions; see 1
(k−k 0)
k−1 Y
, M (i) ρ1 (M ) : = lim sup
k→∞ k0 ∈N i=k0 1
(k−k ) 0
k−1 Y
, ρ2 (M ) : = inf sup M (i)
k→∞ k0 ∈N i=k0 1
(k−k ) 0
k−1
Y . M (i) ρ3 (M ) : = lim sup
k→∞, k0 ∈N
Frozen Method for Slowly Time-Varying Systems
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Below, we adopt a less conservative method[40] to deal with system (4), where in system (4), we assume ∞ that the matrix sequences {A0 (k)}∞ k=0 and {C0 (k)}k=0 vary sufficiently slowly, that is, there exists positive constant q, such that _ kA0 (k) − A0 (j)k kC0 (k) − C0 (j)k ≤ q|k − j| (44) holds for all k ≥ j ≥ 0. By defining an useful auxiliary function L(k, x(k)) = x(k)T (E + E T )x(k), where E ∈ Sn and is a (0 − 1)-matrix, we have 1, x(k + 1))} = 12 E{x(k + 1)T (E + E T )x(k + 1)} = x(k) [A0 (k)T EA0 (k) + C0 (k)T EC0 (k)]x(k) +x(k − τ )T [Aτ (k)T EAτ (k) + Cτ (k)T ECτ (k)]x(k − τ ) +[x(k − τ )T Aτ (k)T (E + E T )A0 (k) + x(k − τ )T Cτ (k)T (E + E T )C0 (k)]x(k).
(45)
M
1 2 E{L(k + T
Define (ij)
(ij)
ED
Eij = [esl ]n×n , esl = δis δjl , s, l, i, j = 1, 2, ..., n
(46)
PT
and then we adopt the following method to unify the system dimension: T T T T T T r(k, xk , xk−τ , E) = [x(k − τ ) Aτ (k) (E + E )A0 (k) + x(k − τ ) Cτ (k) (E + E )C0 (k)]x(k), R(k, xk , xk−τ ) = (r(k, xk , xk−τ , E11 ), · · · , r(k, xk , xk−τ , E1n ), · · · , 2 T r(k, xk , xk−τ , En1 ), · · · , r(k, xk , xk−τ , Enn )) ∈ Rn .
CE
Based on (45), (46) and Lemma 2.2, we obtain
AC
[vec(E)]T E[x(k + 1) ⊗ x(k + 1)] (47) = [vec(E)]T [A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k)]E[x(k) ⊗ x(k)] T +[vec(E)] [Aτ (k) ⊗ Aτ (k) + Cτ (k) ⊗ Cτ (k)]E[x(k − τ ) ⊗ x(k − τ )]vec(E) + Er(k, xk , xk−τ , E).
Substituting E = E11 , E12 , · · · , Enn into system (47) leads to E[x(k + 1) ⊗ x(k + 1)] = [A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k)]E[x(k) ⊗ x(k)] +[Aτ (k) ⊗ Aτ (k) + Cτ (k) ⊗ Cτ (k)]E[x(k − τ ) ⊗ x(k − τ )] + ER(k, xk , xk−τ ).
(48)
Now, according to H-representation matrix introduced in Lemmas 2.1 and 2.3, we are in a position to n(n+1) reduce the dimension of system (48). Set x e(k) = vecx(x(k) ⊗ x(k)) ∈ R 2 , we have Ex e(k + 1) = Θ H n2 , n(n+1) , A (k), C (k) Ex e(k) 0 0 2 (49) 2 n(n+1) e xk , xk−τ ), , Aτ (k), Cτ (k) E x e(k − τ ) + E R(k, +Θ H n , 2 11
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where h i−1 Θ H n2 , n(n+1) H T n2 , n(n+1) , A0 (k), C0 (k) := H T n2 , n(n+1) H n2 , n(n+1) 2 2 2 2 , ×(A0 (k)T ⊗ A0 (k)T + C0 (k)T ⊗ C0 (k)T )H n2 , n(n+1) 2 h i−1 n(n+1) n(n+1) n(n+1) n(n+1) Θ H n2 , 2 , Aτ (k), Cτ (k) := H T n2 , 2 H n2 , 2 H T n2 , 2 ×(Aτ (k)T ⊗ Aτ (k)T + Cτ (k)T ⊗ Cτ (k)T )H n2 , n(n+1) , 2 i−1 h e xk , xk−τ ) := H T n2 , n(n+1) H n2 , n(n+1) R(k, xk , xk−τ ). R(k, H T n2 , n(n+1) 2 2 2
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(50)
We give a two-dimensional system example to show the process of transforming system (4) into system (49).
"
0.5 0 C0 (k) = −0.5 sin k −0.5 " # x1 (k − 1) xτ (k) = . x2 (k − 1)
Then
0.5 0 0 0.5 sin2 k
M
A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k) =
0.0025 0 0 0
ED
PT
Cτ (k) ⊗ Cτ (k) =
−0.25 sin k 0 −0.25 sin2 k 0.5 sin k
0 0.0025 0 0
,
Cτ (k) =
−0.25 sin k −0.25 sin2 k 0 0.5 sin k
0 0 0 0 0.0025 0 0 0.0025
12
"
,
0.05 0
0.25 sin2 k −0.25 sin k −0.25 sin k 0.5
0.05x1 (k)x1 (k − 1) −0.05 sin kx1 (k)x1 (k − 1) − 0.05x2 (k)x2 (k − 1) 0.05x1 (k)x2 (k − 1) −0.05 sin kx1 (k)x2 (k − 1) − 0.05x2 (k)x2 (k − 1)
AC
CE
ER(k, xk , xk−τ ) =
#
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Example 3.2 For system (4), we take " # −0.5 0.5 sin k A0 (k) = , −0.5 sin k −0.5 " # x1 (k) Aτ (k) = 0, x(k) = , x2 (k)
.
0 0.05
,
#
,
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Θ H n2 , n(n+1) 2
Θ H
n2 , n(n+1) 2
x21 (k) ˜(k) = E x1 (k)x2 (k) , , Ex x22 (k) ,
, A0 (k), C0 (k) =
0.5 0 0.5 sin2 k
0.0025 , Aτ (k), Cτ (k) = 0 0
−0.5 sin k −0.25 sin2 k sin k 0 0.025 0
0.25 sin2 k −0.25 sin k , 0.5
0 0 , 0.0125
0.05x1 (k)x1 (k − 1) ˜ xk , xk−τ ) = R(k, −0.025(sin k)x1 (k)x1 (k − 1) − 0.025x2 (k)x2 (k − 1) + 0.025x1 (k)x2 (k − 1) . −0.05(sin k)x1 (k)x2 (k − 1) − 0.05x2 (k)x2 (k − 1)
AN US
get
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Using H-representation technique, we x21 (k + 1) Ex ˜(k + 1) = E x1 (k + 1)x2 (k + 1) x22 (k + 1) x21 (k − 1) Ex ˜(k − 1) = E x1 (k − 1)x2 (k − 1) x22 (k − 1)
Now, we are ready to establish our main results in this subsection. First of all, we give the following lemma.
M
Lemma 3.3 There exists constants i > 0, i = 1, 2, such that the growth condition
e
˜(k)k + 2 sup kE x ˜(k − m)k
E R(k, xk , xk−τ ) ≤ 1 kE x
ED
holds.
(51)
m∈[−τ,0]
Proof. Using the following fact, the proof of this lemma is obvious.
PT
Ex2i (k − τ ) ≤
sup m∈[−τ,0]
Ex2i (k − m),
(52)
AC
CE
Taking y(k) = E x ˜(k) and y(k − τ ) = E x ˜(k − τ ), system (4) can be turned into y(k + 1) = Θ H n2 , n(n+1) , A0 (k), C0 (k) y(k) 2 e xk , xk−τ ), k ∈ N + , +Θ H n2 , n(n+1) , A (k), C (k) y(k − τ ) + E R(k, τ τ 2 n(n+1) y(k) = ϕ(k) e = H T n2 , n(n+1) (ϕ(k) ⊗ ϕ(k)) ∈ R 2 , k ∈ N−τ . 2
For simplicity, we denote n(n + 1) n(n + 1) Θ0 (k) = Θ H n2 , , A0 (k), C0 (k) , Θτ (k) = Θ H n2 , , Aτ (k), Cτ (k) . 2 2 In system (52), the characterization of the slowly time-varying coefficients is shown by Θ0 (k). Remark 3.3 In what follows, we can find that the stability/exponential stability of stochastic system (4) -dimensional deterministic system (52). If we use the can be transformed into the study of an n(n+1) 2 augmented system method as done in (23), then an n(2n + 1)-dimensional system is to be used, which is larger more than (52) in dimension, and will greatly increase computational difficulty. 13
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The following proposition is obvious. Proposition 3.1 If the deterministic system (52) is stable/exponentially stable, then the stochastic system (4) is also stable/exponentially stable in mean square sense.
Theorem 3.2 Consider system (52). Denote
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According to Proposition 3.1, we present the following result.
∞ X
Γ1 = sup Θk0 (v) < ∞, Γ2 = sup kΘτ (k)k , ~ = max kϕ(k)k e sup Θk0 (v) < ∞. k,v≥0
−τ ≤k≤0
k≥0
k=0
Suppose (52) satisfies the following two conditions: 1. There is a % = %(q) > 0 such that
holds for all k ≥ j ≥ 0. 2. ℵ = ℵ(Θ0 (k), Θτ ) =
∞ X
(%k + 1 + 2 + Γ2 ) sup Θk0 (v) < 1.
k=0
(54)
(55)
v≥0
M
Then, the zero solution of system (52) is stable.
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kΘ0 (k) − Θ0 (j)k ≤ %|k − j|
(53)
v≥0
ED
Proof. For the slowly time-varying system (4), system coefficients A0 (k) and C0 (k) satisfy condition (44). By using H-representation technique, in system (52), we have
PT
kΘ0 (k) − Θ0 (j)k
= (H T H)−1 H T [A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k)] H − (H T H)−1 H T [A0 (j) ⊗ A0 (j) + C0 (j) ⊗ C0 (j)] H
T −1 T ≤ (H H) H kA0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k) − A0 (j) ⊗ A0 (j) − C0 (j) ⊗ C0 (j)k kHk ≤ %(q)|k − j|,
e xk , xk−τ ), k ∈ N + , y(k + 1) − Θ0 (s) = (Θ0 (k) − Θ0 (s)) y(k) + Θτ (k)y(k − τ ) + E R(k,
AC
where
CE
where H = H n2 , n(n+1) and k ≥ j ≥ 0. From system (52), we have 2
(56)
n(n + 1) , A0 (s), C0 (s) Θ0 (s) = Θ H n2 , 2
and s is a fixed nonnegative integer. Using the variation of constants formula, we get y(l + 1) = Θl+1 e + 0 (s)ϕ(0)
l X j=0
h i e Θl−j 0 (s) (Θ0 (j) − Θ0 (s)) y(j) + Θτ (j)y(j − τ ) + E R(j, xj , xj−τ ) .
(57)
By taking s = l, (57) yields that y(l + 1) = Θl+1 e + 0 (l)ϕ(0)
l X j=0
h i e Θl−j 0 (l) (Θ0 (j) − Θ0 (l)) y(j) + Θτ (j)y(j − τ ) + E R(j, xj , xj−τ ) . 14
(58)
ACCEPTED MANUSCRIPT
Applying Lemma 3.3 and condition (54) , we obtain l
X
l−j
kϕ(0)k sup Θl+1 e + (l) (l)
Θ 0 0
l∈N
j=0
· (% |l − j| + 1 ) ky(j)k + kΘτ (j)k ky(j − τ )k + 2 ≤
0≤k≤l
l∈N
k=0
l X
k
Θ0 max ky(j)k (kΘτ (k)k + 2 ) + −τ ≤j≤k
∞ X
l+1
sup Θ0 (l) kϕ(0)k e + max ky(k)k (%k + 1 ) sup Θk0 (v) 0≤k≤l
l∈N
k=0
0≤k≤l
k≥0
∞ X
k≥0
0≤k≤l
l∈N
M
−τ ≤k≤0
k≥0
ED
sup Θk0 (v)
v≥0 k=0 ∞ X
e + max ky(k)k sup Θl+1 0 (l) kϕ(0)k
+(sup kΘτ (k)k + 2 ) max kϕ(k)k e
So
sup Θk0 (v)
v≥0 k=0 ∞ X
+(sup kΘτ (k)k + 2 ) max kϕ(k)k e −τ ≤k≤0
v≥0
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+(sup kΘτ (k)k + 2 ) max ky(k)k
≤
ky(j − m)k
l X
e sup Θl+1 + max ky(k)k (%k + 1 ) Θk0 (l) 0 (l) kϕ(0)k k=0
≤
sup −τ ≤m≤0
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ky(l + 1)k ≤
(%k + 1 + 2 + sup kΘτ (k)k) sup Θk0 (v)
k=0 ∞ X
k=0
k≥0
v≥0
sup Θk0 (v) . v≥0
ky(l + 1)k ≤ Γ1 kϕ(0)k e + ℵ max ky(k)k + (Γ2 + 2 )~, 0≤k≤l
(59)
PT
where Γ1 , Γ2 and ℵ are shown as in (53). (59) yields that
max ky(k)k ≤ Γ1 kϕ(0)k e + ℵ max ky(k)k + (Γ2 + 2 )~.
k∈N +
k∈N
(60)
CE
Let ky(k)k < kϕ(0)k e (k ∈ N + ), then system (52) is stable. If the inequality ky(k)k ≥ kϕ(0)k e (k ∈ N + ), then we have max+ ky(k)k = max ky(k)k, k∈N
k∈N
AC
which combines the condition (55) giving
So it follows that
max ky(k)k ≤ (Γ1 kϕ(0)k e + (Γ2 + 2 )~) (1 − ℵ)−1 .
(61)
ky(k)k ≤ [Γ1 kϕ(0)k e + (Γ2 + 2 )~] (1 − ℵ)−1 , k ∈ N .
(62)
k∈N
This proves the stability of the zero solution of system (52).
Corollary 3.1 If the conditions in Theorem 3.2 are satisfied, then the zero solution of system (52) is exponentially stable. 15
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Proof. To establish the exponential stability of the zero solution of system (52), we define a new variable Ex eλ (k) = E x e(k)λk (63) with a constant λ > 1. Substituting (63) into system (52), we get
(64)
CR IP T
where
ˆ 0 (k)E x ˆ τ (k)E x ˆ xk,λ , xk−τ,λ ), Ex eλ (k + 1) = Θ eλ (k) + Θ eλ (k − τ ) + E R(k, ˆ Θ0 (k) = λΘ0 (k), ˆ Θτ (k) = λΘτ (k), ˆ e λ−k xk,x ,x E R(k, xk,λ , xk−τ,λ ) = λk+1 E R(k, ). k,λ k−τ,λ sup
kE x ˜λ (k − m)k,
AN US
The growth condition (51) yields that
ˆ
˜λ (k)k + ˆ2
E R(k, xk,λ , xk−τ,λ ) ≤ ˆ1 kE x
m∈[−τ,0]
where ˆ1 = λ1 , ˆ2 = λτ +1 2 . Similar computations to the proof of Theorem 3.2, it follows that E x ˜ı (k) is a bounded function. So, by equation (63), the system considered is exponentially stable, which also means that system (4) is exponentially stable in mean square sense. The proof is completed.
Conclusion
M
4
PT
ED
This paper has discussed the stability analysis of time-varying stochastic systems with time-delays. For general linear time-varying stochastic systems with time-delays, the augmented system method has been used to transform the original system into a delay-free system. We have applied the operator spectrum technique and “frozen” method to present some new criteria on stability and exponential stability in mean square sense. The stabilization and other robust stabilization properties about this kind of systems still need to further study.
CE
Acknowledgments
AC
This work was supported by the National Natural Science Foundation of China (Nos. 61374104, 61773170), and the Natural Science Foundation of Guangdong Province of China (2016A030313505).
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[38] W. Zhang and B. S. Chen, H-representation and applications to generalized Lyapunov equations and linear stochastic systems, IEEE Trans. Autom. Control 57 (12) (2012) 3009–3022. [39] X. Zhao and F. Deng, Moment stability of nonlinear stochastic systems with time-delays based on H−representation technique, IEEE Trans. Autom. Control 59 (3) (2014) 814–819.
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[40] X. Zhao and F. Deng, Moment stability of nonlinear discrete stochastic systems with time-delays based on H−representation technique, Automatica 50 (2014) 530–536.
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Stability analysis of time-varying discrete stochastic systems with multiplicative noise and state delays Xiushan Jiang, Senping Tian, Tianliang Zhang, Weihai Zhang PII: DOI: Reference:
S0016-0032(18)30449-6 10.1016/j.jfranklin.2018.06.034 FI 3535
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
23 November 2017 15 June 2018 18 June 2018
Please cite this article as: Xiushan Jiang, Senping Tian, Tianliang Zhang, Weihai Zhang, Stability analysis of time-varying discrete stochastic systems with multiplicative noise and state delays, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.06.034
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Stability analysis of time-varying discrete stochastic systems with
Xiushan Jiang1 , 1
Senping Tian1∗,
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multiplicative noise and state delays Tianliang Zhang1 ,
Weihai Zhang2
School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, Guangdong Province, P. R. China
2 College
of Electrical Engineering and Automation, Shandong University of Science and Technology,
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Qingdao 266590, Shandong Province, P. R. China
Abstract
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The aim of this paper is to study the stability of discrete stochastic time-delayed systems with multiplicative noise, where the coefficients are assumed to be time-varying with a general time-varying rate or a small time-varying rate. Firstly, by the Kronecker algebra theory and H-representation technique, the exponential stability of the stochastic system with common time-varying coefficients is investigated by the spectral approach. It is shown that the time-varying stochastic systems with state delays is exponentially stable in mean square sense if and only if its corresponding generalized spectral radius is less than one. Secondly, under definite conditions, by applying the so-called “frozen” technique, it is shown that the stability of a “frozen” system implies that of the corresponding slowly time-varying system.
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Keywords. Stability, discrete time-delay systems, stochastic systems, time-varying coefficients, slowly time-varying coefficients.
Introduction
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Stochastic systems play a remarkable role in a wide range of applications involving economics, finance and mechanics and have been extensively studied during the past several decades; see, e.g., [14, 15, 16, 21, 22, 32] . In addition, stochastic time-delay system is also one of the fundamental research topics in control theory which has received an increasing attention in recent years. Time-delay phenomenon is very common in many engineering systems such as fault diagnosis processes, chemical processes, hybrid systems and so on. To mention a few, the problems of stochastic time-delay systems such as stability analysis, robust control and some important applications can be found in the past literature; see, e.g., [12, 10, 26, 39, 40] and the references therein. Stability is one of the most important notions in modern control theory, which is a necessary prerequisite to make the system behave well. For an unstable open-loop system, in practical engineering, one is required to design a stabilizing controller in many control problems such as optimal H2 control, adaptive control, H∞ control, mixed H2 /H∞ control and so on. So, various stability concepts for linear stochastic ∗ Corresponding
author. Email: [email protected]
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systems has been studied widely by many researchers in recent years, such as exponential stability [8, 9, 15] , finite-time stability [6, 7] , almost sure stability [4, 5, 15, 28] , global asymptotical stability [13, 30] , asymptotic mean square stability [29] and critical stability [31] . For linear time-varying systems, the property of stability is of increasing interest in recent years and considerable research effort has been taken. Different kinds of methods have been adopted to find less conservative criteria of stability. It can be remarked that, in spite of time-invariant systems or time-varying systems, the Lyapunov function method serves as a main technique for most existing works about the stability analysis, but finding suitable Lyapunov functions is still a difficult task; see [2, 24, 35, 36, 37] and so on. Another method is to investigate special cases of time-varying systems by decomposing the system matrix of a linear time-varying system into two parts, one is a constant matrix and the other one is a time-varying derivation, which satisfies certain conditions, see [11, 27] . Unlike most previous works, in this paper, we will obtain the stability criteria of linear stochastic discrete systems with time-varying coefficients and time-delays by using the spectral technique and “frozen” technique, respectively. It is well-known that, for linear time-invariant deterministic system xk+1 = Axk , the stability can be determined in terms of the locations of the eigenvalues of the system matrix A, while the stability of a linear time-varying system xk+1 = Ak xk is not directly available by the eigenvalue placement methodology [25] . In order to formulate a similar stability criterion by means of eigenvalues in time-varying systems, [18] generalized the concept of spectral radius by imposing some special restrictions on the coefficient matrix sequence {Ak }k∈Z + and proved the relationship between the spectral radius and the exponential stability. In addition, [20] proposed several new formulas for computing the generalized spectral radius. Due to widespread applications of stochastic systems, The spectrum concept and spectrum assignment were proposed in [29] by the associated Lyapunov operators, the latter is similar to the pole placement in modern control theory, we refer the reader to [32] for stability analysis of stochastic timeinvariant systems. Up to now, there is no analogous stability criteria for linear stochastic time-varying systems. To fill in this gap, our first contribution of this paper is to present spectral criteria for a kind of stochastic discrete time-varying systems with state delays and general time-varying coefficients based on generalized spectral radius method. Different from the stochastic time-invariant case, the generalized spectral radius is associated with the following standard discrete time-varying linear system x(k + 1) = Θ(Hn2 ×(n(n+1)/2) (k), A(k), C(k))x(k).
(1)
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In this process, H-representation technique plays a vital role, which has been introduced in [38] and got important applications in [39, 40] . In this paper, we will establish the connection between stability in mean square sense of stochastic systems and stability of deterministic systems by the H-representation technique. The second contribution of this paper is to obtain a stability criterion for slowly time-varying stochastic systems by “frozen” technique. This technique has been extensively studied in deterministic cases; see [1, 3, 8, 17, 19, 23] . Here, we generalize this “frozen” technique in deterministic systems to discrete time-varying stochastic systems with state delays: ( x(k + 1) = A(k)x(k) + A0 (k)x(k − τ ) + C(k)x(k)w(k) + C0 (k)x(k − τ )w(k), (2) x(k) = ϕ(k), k = 0, −1, −2, · · · , −τ. In system (2), we define the state transition matrix as Φ(k, s), which is generated by the coefficient matrices A(k), A0 (k), C(k) and C0 (k). The “frozen” technique will allow us to involve neither Lyapunov 2
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functions in some situations nor stability assumptions about the state transition matrix Φ(k, s). Concretely speaking, the frozen technique can be described as follows. If p ∈ N + is any fixed integer, then the following stochastic discrete-time autonomous system x(k + 1) = A(p)x(k) + A0 (p)x(k − τ ) + [C(p)x(k) + C0 (p)x(k − τ )]w(k)
(3)
Preliminaries
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is taken as a particular case of time-varying stochastic system (2) with its time dependence “frozen” at time p. In this paper, it is shown that if each “frozen” system is stable or exponentially stable in mean square (ESMS) and the coefficient change rate of system (2) is small enough, then our considered system is stable in some senses. Besides, we also consider the time-delay phenomena in our system, which arise frequently in many practical applications. One major stream of researching time-delay systems is to find a corresponding strict Lyapunov function, which can often be transformed into a Lyapunov-Krasovskii functional. Since we apply spectral approach and “frozen” technique to research stability problem in our system, we use two different methods to deal with time-delays. In essence, those two methods can be summarized as to expand the state dimensions. There are still some differences on those two methods. Compared with the first method which simply transform a system with time-delays into an augmented system without delays, the seconde method would reduce the conservativeness by retaining some characters of time-delays. In another hand, we have to point out that the system dimensional space would be highter after expanding the state dimensions which makes the calculation inconvenient in practical application. So we use the H-representation technique to reduce the dimension of system matrix. The paper is organized in the following way. In Section 2, some adequate preliminaries and definitions are presented. We state our main results in Section 3. Firstly, we propose an useful notion of generalized spectral radius which can be applied to analyze the operator sequences for our considered systems. In addition, a stability criterion is given in terms of the spectral radius. Secondly, we give new sufficient conditions for the exponential stability in mean square sense of the considered systems by “frozen technique”. Section 4 concludes this paper. For convenience, we adopt the following notation throughout this paper: AT : the transpose of a matrix or vector A. Rl : the l-dimensional real vector space with the usual inner product h·, ·i. R: the set of all real numbers. Rm×n : the vector space of all m × n real matrices. Sn : the set of all n × n symmetric matrices. E(·): the mathematical expectation operator. N := {0, 1, 2, . . .}, N−τ := {0, −1, −2, · · · , −τ } and N + := {1, 2, . . .}. (0 − 1)-matrix: all matrix elements are either 0 or 1.
Consider the following discrete-time time-varying stochastic system with state delay x(k + 1) = A0 (k)x(k) + C0 (k)x(k)w(k) +[Aτ (k)x(k − τ ) + Cτ (k)x(k − τ )w(k)], k ∈ N + , x(k) = ϕ(k) ∈ Rn , k ∈ N−τ ,
(4)
where x(k) ∈ Rn and u(k) ∈ Rnu are the system state and control input, respectively. τ is a positive integer that denotes the upper bound of time-delay. For the given deterministic initial condition ϕ(k) ∈ Rn , k ∈ N−τ , the corresponding solution process is denoted by x(k). {w(k), k ∈ N } is the onedimensional independent white noise process which is defined on the complete probability space (Ω, F, P). 3
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Ft := σ(w(s) : 0 ≤ s ≤ k) is the natural filtering generated by w(·). Besides, the initial states ϕ(k)k∈N−τ are independent of the process w(k). We assume Ew(k) = 0, Ew(k)w(s) = δks , where δks = 1 for k = s, δks = 0 for s 6= k, i.e., δks is a Kronecker function. {A0 (k), C(k), Aτ (k), Cτ (k)}k∈N are the time-varying matrix sequences with the appropriate dimensions. The following concepts of stability and exponential stability will be used in formulating the main results of the paper.
Ekx(k)k2 < ε, ∀k ∈ N + .
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Definition 2.1 The zero solution of system (4) is stable or Lyapunov stable if for every ε < 0 and every k0 ∈ N + , there is a number δ > 0(depending on ε and k0 ) such that every solution of this system with initial condition kϕ(k)k < δ for k ∈ N−τ satisfies the condition (5)
Definition 2.2 System (4) is said to be exponentially stable in mean square (ESMS) if there exist β ≥ 1, λ ∈ (0, 1) such that for any 0 ≤ k0 < k < +∞, there holds
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E k x(k) k2 ≤ βλk−k0 max k ϕ(s) k2 −τ ≤s≤0
(6)
for any solution x(k) of system (4) with the initial condition x(k)k∈N−τ = ϕ(k) ∈ Rn .
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As we have stated before, H-representation technique is an important tool for connecting a stochastic system with its related deterministic system. So we are in a position to introduce the H-representation concept and some properties which is developed in [38] according to our considered system. For any 2 time-varying matrix X(·) = [x(·)ij ]n×n ∈ Sn . vec[X(·)] : Rn×n → Rn ×1 denotes the vectorization of the matrix sequence X(·) generated by stacking the rows of X(·) into a column vector and vecs[X(·)] : n(n+1) Rn×n → R 2 ×1 is the column vector which is generated by all upper diagonal elements of X(·). That is,
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→ − X (·) = vec[(X)(·)] = (x11 , x12 , · · · , x1n , x21 , · · · , x2n , · · · , xn1 , · · · , xnn )T , e = vecs[(X)(·)] = (x11 , x12 , · · · , x1n , x22 , · · · , x2n , · · · , xn−1,n−1 , xn−1,n , xnn )T . X(·)
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It is not difficult to verify that, for x ∈ Rn , there exists
vec(X) = vec[xxT ] = x ⊗ x.
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The relationship between vec(X(·)) and vecs(X(·)) can be determined by the following lemma. n(n+1) 2 Lemma 2.1 [38] For any X ∈ Sn , there exists a unique transfer matrix H n2 , n(n+1) ∈ Rn × 2 , 2 such that → − e X(·) . (i) X (·) = H n2 , n(n+1) 2 T 2 n(n+1) (ii) H n2 , n(n+1) H n , is invertible. 2 2
Remark 2.1 The matrix H n2 , n(n+1) is a column full rank (0 − 1)-matrix, which is called an H2 e representation matrix and is used to eliminate repeated elements in X(·). 4
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We introduce H-representation technique in detail by the following unforced delay-free system ( x(k + 1) = A0 (k)x(k) + C0 (k)x(k)w(k), x(0) = x0 ∈ Rn .
(7)
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Set X(k) = E[x(k)x(k)T ] ∈ Sn . Based on the fact of that x(k) and w(k) are independent and x(k)x(k)T is Fk−1 -measurable, X(k) with k ∈ N satisfies ( X(k + 1) = A0 (k)X(k)A0 (k)T + C0 (k)X(k)C0 (k)T , (8) X(0) = x(0)x(0)T ∈ Sn , k ∈ N . The following Kronecker matrix property will be adopted throughout this paper.
Lemma 2.2 [34] For matrices A, B, C, D with suitable dimensions, A ⊗ C denotes the Kronecker product of A and C, which has the following properties: (AC ⊗ BD) = (A ⊗ B)(C ⊗ D),
vec(ABC) = (A ⊗ C T )vec(B).
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(A ⊗ C)T = AT ⊗ C T ,
By means of Lemma 2.2, (8) is equivalent to ( vec(X(k + 1)) = (A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k))vec(X(k)), vec(X(0)) = vec(X0 ) = vec(x(0)x(0)T ) ∈ Sn , k ∈ N .
(9)
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Then (9) can be turned into a standard deterministic system by H-representation matrix. Lemma 2.3
where
e + 1) = Θ H n2 , n(n+1) , A0 (k), C0 (k) X(k), e X(k 2 n(n+1) e e0 ∈ R 2 , k ∈ N , X(0) =X
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(
(11)
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Θ H n2 , n(n+1) , A0 (k), C0 (k) 2 h i−1 := H T n2 , n(n+1) H n2 , n(n+1) H T n2 , n(n+1) 2 2 2 ×(A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k))H n2 , n(n+1) . 2
(10)
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e 2 Next, we give a result about the relationship between E kx(k)k in system (7) and X(k)
in system (10).
e 2 Lemma 2.4 E kx(k)k in system (7) and X(k)
in system (10) satisfy (i)
(ii)
2
E kx(k)k ≤
√
e n X(k)
.
rn + 1
e 2 X(k) E kx(k)k .
≤ 2
5
(12)
(13)
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e Proof. As shown above, X(k)
= vecs{E[x(k)x(k)T ]} . Note that 2
E kx(k)k
e 2
X(k) =
Then
e 2
X(k)
=
n X
2
n X
=
i=1
X
E|xi (k)|
!2
,
(14)
2
(15)
(E(xi (k)xj (k))) .
1≤i≤j≤n
X
(E|xi (k)|2 )2 +
i=1
≥
2
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(E(xi (k)xj (k)))
1≤i
n X
1 (E|xi (k)| ) ≥ n i=1 2 2
n X i=1
E|xi (k)|
!2
2
.
(16)
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So, combining equation (14) and inequality (16), we can get that the relationship (12) holds. For proving the second result in this lemma, we use Cauchy-Schwarz inequality to get X 2 (E(xi (k)xj (k))) 1≤i
≤
X
2
1≤i
1 2
(E|xi (k)xj (k)|) ≤
X
1≤i
X
1≤i
E|xi (k)|2 E|xj (k)|2
(E|xi (k)| ) + (E|xj (k)|2 )2 .
2 2
(17)
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≤
Then, substituting (17) into (16), the following inequality holds
≤
n X
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≤
(E|xi (k)|2 )2 +
i=1
n X
(E|xi (k)|2 )2 +
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e 2
X(k)
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≤
i=1
n
n−1 n−1 1X i(E|xi (k)|2 )2 (E|xn (k)|2 )2 + 2 2 i=1 n
n−1X (E|xi (k)|2 )2 2 i=1
n+1X (E|xi (k)|2 )2 . 2 i=1
(18)
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(13) is derived from (18) immediately. Therefore, the proof is complete.
e 2 Remark 2.2 The relationship between X(k)
and E kx(k)k shown in Lemma 2.4 means that the stability property in mean square sense of the original system (7) are equivalent to the asymptotic stability of the standard deterministic system (10).
3
3.1
Stability Analysis Spectrum method for time-varying systems with general time-varying rate coefficients
It is well-known that a linear time-varying discrete-time system without multiplicative noise is stable if and only if the spectral radius of the operator associated with the system is less than one; see [18] . In this 6
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section, we will extend the previous results [18, 37, 38] to linear discrete time-varying stochastic systems with state delays and multiplicative noise by the method of augmented systems. Our considered system in this subsection is shown as (k)x(k) + C0 (k)x(k)w(k) x(k + 1) = A0P m (19) + τ =1 [Aτ (k)x(k − τ ) + Cτ (k)x(k − τ )w(k)], k ∈ N , x(k) = ϕ(k), k ∈ N−m . By introducing an n(m + 1)-dimensional column vector x ¯(k)k∈N as T x ¯(k) = x(k)T , x(k − 1)T , . . . , x(k − m)T ,
(20)
system (19) can now be written as the form of an equivalent augmented stochastic system without delays such as x ¯(k + 1) = A(k)¯ x(k) + C(k)w(k)¯ x(k) (21) T
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C(k) =
C0 (k) C1 (k) · · · 0 0 ··· .. .. .. . . . 0 0 ···
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with the same initial state x ¯(0) = [ϕ(0), ϕ(−1), · · · , ϕ(−m)] , where A0 (k) A1 (k) · · · Am−1 (k) Am (k) I 0 ··· 0 0 A(k) = .. .. .. .. .. . . . . . 0 0 ··· I 0 Cm−1 (k) Cm (k) 0 0 .. .. . . 0 0
,
(22)
.
Obviously, A(k) ∈ Rn(m+1)×n(m+1) , C(k) ∈ Rn(m+1)×n(m+1) . Now we will reduce the system dimension , A, C by H-representation technique introduced above. Recalling the definition of Θ H n2 , n(n+1) 2
Θ H n2 (m + 1)2 , n(m+1)(mn+n+1) , A(k), C(k) 2 h i−1 T 2 2 n(m+1)(mn+n+1) 2 2 n(m+1)(mn+n+1) := H n (m + 1) , H n (m + 1) , 2 2 ×H T n2 (m + 1)2 , n(m+1)(mn+n+1) 2 ×(A(k) ⊗ A(k) + C(k) ⊗ C(k))H n2 (m + 1)2 , n(m+1)(mn+n+1) , 2 2 2 n(m+1)(mn+n+1) Θ H n (m + 1) , 2 h i−1 T 2 2 n(m+1)(mn+n+1) 2 2 n(m+1)(mn+n+1) := H n (m + 1) , H n (m + 1) , 2 2 ×H T n2 (m + 1)2 , n(m+1)(mn+n+1) . 2
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where
(23)
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in Lemma 2.3, setting X(k) = E[¯ x(k)¯ x(k)T ] ∈ Sn(m+1) , system (21) leads to e + 1) = Θ H n2 (m + 1)2 , n(m+1)(mn+n+1) , A(k), C(k) X(k), e X(k 2 e e0 X(0) =X n(m+1)(mn+n+1) 2 = Θ H n2 (m + 1)2 , n(m+1)(mn+n+1) vec[(¯ x(0)¯ x(0)T )] ∈ R , k ∈ N, 2
To elucidate our transformed standard system (23), we present an example as follows: 7
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Example 3.1 Consider the following stochastic system:
h
iT
x(k) x(k − 1) "
A(k) =
k 1
3k 0
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, then we get
#
,
C(k) =
In the transformed system,
x2 (k) e X(k) = E x(k)x(k − 1) , x2 (k − 1)
"
2k 0
4k 0
#
H(4, 3) =
1 0 0 0
0 1 1 0
0 0 0 1
.
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Set x ¯(k) =
x(k + 1) = kx(k) + 2kx(k)w(k) + 3kx(k − 1) + 4kx(k − 1)w(k).
,
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5k 2 Θ (H(4, 3), A(k), C(k)) = 11k 2 25k 2
2k 3k 0
1 0 . 0
(26)
Definition 3.1 [33] System (23) is said to be exponentially stable (ES) if there exist β ≥ 1, λ ∈ (0, 1) such that for any 0 ≤ k0 < k < +∞, there holds
e
e (27)
X(k; k0 ) ≤ βλk−k0 X 0
M
e e0 . for any solution X(k) of system (23) with the initial value X
Lemma 3.1 The equilibrium of system (23) is exponentially stable implies that stochastic system (19) is ESMS.
E k¯ x(k)k
2
2
=
m X n X i=0 j=1
CE
≤ n(m + 1)
AC
e 2
X(k)
2
E|xj (k − i)|2
PT
ED
e Proof. Using the same method as in Lemma 2.4, system state X(k) in (23) and the augmented state x ¯(k) in (20) are shown as
X
= =
m X n X i=0 j=1
X
0≤f ≤g≤m 1≤i≤j≤n m X X f =0 1≤i≤j≤n
E|xj (k − i)|2
2
, 2
[Exi (k − f )xj (k − g)] 2
[Exi (k − f )xj (k − f )] +
≤ z1 + z2 + z3 + z4 ,
where z1 =
m X n X i=0 j=1
z2 =
m X
X
f =0 1≤i
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X
0≤f
E|xj (k − i)|2
2
8
n
2
[Exi (k − f )xj (k − g)]
,
m
2
(Exi (k − f )xj (k − f )) ≤
X
2 n − 1 XX E|xj (k − i)|2 , 2 i=0 j=1
(29)
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z3 = z3 =
X
X
0≤g
m
2
(Exi (k − f )xj (k − g)) ≤
m
(Exi (k − f )xi (k − g))2 ≤
n
2 m(n − 1) X X E|xj (k − i)|2 , 2 i=0 j=1 n
2 m XX E|xj (k − i)|2 . 2 i=0 j=1
(30)
Then, we have
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p
e 2 (31) E k¯ x(k)k ≤ n(m + 1) X(k)
, r
mn + n + 1
e 2 E k¯ x(k)k . (32)
X(k) ≤ 2
e 2 Inequalities (31) and (32) mean that E k¯ x(k)k and X(k)
have the same order and this completes the proof.
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Lemma 3.2 [29] System (19) is ESMS if there exist β ≥ 1, λ ∈ (0, 1) such that for any 0 ≤ k0 < k < +∞, e k0 , X e0 ) k in system (23) satisfies k X(k;
e
e k−k0 e0 ) , (33)
X(k; k0 , X
≤ X 0 βλ i.e.,
k−1 Y
i=k0
!
n(m + 1)(mn + n + 1)
e k−k0 e0 Θ H n2 (m + 1)2 , . X , A(i), C(i)
≤ X 0 βλ
2
(34)
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Remark 3.1 Our aim in this subsection is to transform the study of ESMS of the stochastic system (19) into that of exponential stability of the deterministic system (23). Although, in (22), there are a lot of zeros in A(k) and C(k), system (22) is still a discrete stochastic system for C(k) 6= 0, k ∈ N . If we do not use the H-representation technique, we, of course, can also obtain a deterministic difference equation on vec(X(k)) corresponding to system (23), which is an n2 (m + 1)2 -dimensional system larger than the n(m+1)(nm+n+1) -dimensional system (23), and which will increase the computational complexity if we use 2 the difference equation on vec(X(k)) instead of (23). [18]
to
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The following definition generalizes the spectral radius of deterministic time-varying systems stochastic time-varying systems.
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Definition 3.2 Consider system (23). Set the time-varying sequence n(m + 1)(mn + n + 1) {M (k)}k∈N := Θ H n2 (m + 1)2 , , A(k), C(k) . 2 k∈N We define
1
(k−k 0)
k−1
Y
ρ (M ) := lim sup sup M (i)
k→∞ k0 ∈N
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i=k0
as the generalized spectral radius of the system sequence {M (k)}k∈N .
We now need to define another numerical quantity δ(M ) as
k−1 ( )
Y
+ k−k0 δ(M ) := inf λ ∈ R | ∃ β ≥ 1, s.t. M (i) ≤ βλ
i=k0
9
k∈N
,
(36)
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which is called the power bound of {M (k)}k∈N . Next we state and prove our main result. Theorem 3.1 Assume that kM (k)k < ∞ for all k ∈ N . Then, the following two results hold: (i) The power bound and the generalized spectral radius of {M (k)}k∈N are identical, i.e., ρ (M ) = λ ⇔ δ(M ) = λ, λ ∈ R+ .
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(ii) System (19) is ESMS if and only if the generalized spectral radius of {M (k)}k∈N is less than one. Proof. Necessity part of (i): Assume that ρ (M ) = λ. Then, according to the limit definition, there exists ks ∈ N + , such that 1
ks −k 0 s −1 kY
sup M (i) ≤ (λ + ε) (37)
k0 ∈N + i=k0
i=k0
AN US
holds for any ε > 0. Similarly, for all k ∈ Nk+s , we have
k−1
Y
M (i) ≤ (λ + ε)k−k0 .
(38)
M
Let µ(M (k)) denote the least upper bound of kM (k)kk∈N . We know that µ(M (k)) is bounded and set ( k ) µ(M (k)) β := sup . (39) (λ + ε) k∈{0,1,··· ,(ks −1)}
ED
So β is a finite number with β ≥ 1 and for k ∈ {0, 1, . . . , (ks − 1)}, it can be shown that
k−1
Y
e k−k0 e0 . M (i)X
≤ X 0 β(λ + ε)
(40)
i=k0
CE
PT
Considering (38), inequality (40) also holds for all k ∈ Nk+s . Hence, δ(M ) ≤ λ = ρ(M ). Sufficiency part: When δ(M ) = λ, by (36), it yields that, for any ε > 0, there exists β ≥ 1 such that
k−1
Y
e k−k0 e sup M (i)X0 ≤ X . (41) 0 β(λ + ε)
k0 ∈N i=k0
AC
Hence, we have
k−1
Y
sup M (i) ≤ β(λ + ε)k−k0 ,
k0 ∈N
(42)
i=k0
which further leads to 1
k−k
k−1
0 1 Y
≤ lim sup β k−k0 (λ + ε) = (λ + ε). lim sup sup M (i)
k→∞ k0 ∈N k→∞
(43)
i=k0
(43) implies that ρ(M ) ≤ λ = δ(M ). The proof (i) is complete. Based on Definition 2.2 and Lemma 3.2 about ESMS, (ii) can be directly obtained from Theorem 3.1-(i). 10
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i=k0
3.2
[18]
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Remark 3.2 Generalized spectral radius ρ (M ) has the following equivalent definitions; see 1
(k−k 0)
k−1 Y
, M (i) ρ1 (M ) : = lim sup
k→∞ k0 ∈N i=k0 1
(k−k ) 0
k−1 Y
, ρ2 (M ) : = inf sup M (i)
k→∞ k0 ∈N i=k0 1
(k−k ) 0
k−1
Y . M (i) ρ3 (M ) : = lim sup
k→∞, k0 ∈N
Frozen Method for Slowly Time-Varying Systems
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Below, we adopt a less conservative method[40] to deal with system (4), where in system (4), we assume ∞ that the matrix sequences {A0 (k)}∞ k=0 and {C0 (k)}k=0 vary sufficiently slowly, that is, there exists positive constant q, such that _ kA0 (k) − A0 (j)k kC0 (k) − C0 (j)k ≤ q|k − j| (44) holds for all k ≥ j ≥ 0. By defining an useful auxiliary function L(k, x(k)) = x(k)T (E + E T )x(k), where E ∈ Sn and is a (0 − 1)-matrix, we have 1, x(k + 1))} = 12 E{x(k + 1)T (E + E T )x(k + 1)} = x(k) [A0 (k)T EA0 (k) + C0 (k)T EC0 (k)]x(k) +x(k − τ )T [Aτ (k)T EAτ (k) + Cτ (k)T ECτ (k)]x(k − τ ) +[x(k − τ )T Aτ (k)T (E + E T )A0 (k) + x(k − τ )T Cτ (k)T (E + E T )C0 (k)]x(k).
(45)
M
1 2 E{L(k + T
Define (ij)
(ij)
ED
Eij = [esl ]n×n , esl = δis δjl , s, l, i, j = 1, 2, ..., n
(46)
PT
and then we adopt the following method to unify the system dimension: T T T T T T r(k, xk , xk−τ , E) = [x(k − τ ) Aτ (k) (E + E )A0 (k) + x(k − τ ) Cτ (k) (E + E )C0 (k)]x(k), R(k, xk , xk−τ ) = (r(k, xk , xk−τ , E11 ), · · · , r(k, xk , xk−τ , E1n ), · · · , 2 T r(k, xk , xk−τ , En1 ), · · · , r(k, xk , xk−τ , Enn )) ∈ Rn .
CE
Based on (45), (46) and Lemma 2.2, we obtain
AC
[vec(E)]T E[x(k + 1) ⊗ x(k + 1)] (47) = [vec(E)]T [A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k)]E[x(k) ⊗ x(k)] T +[vec(E)] [Aτ (k) ⊗ Aτ (k) + Cτ (k) ⊗ Cτ (k)]E[x(k − τ ) ⊗ x(k − τ )]vec(E) + Er(k, xk , xk−τ , E).
Substituting E = E11 , E12 , · · · , Enn into system (47) leads to E[x(k + 1) ⊗ x(k + 1)] = [A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k)]E[x(k) ⊗ x(k)] +[Aτ (k) ⊗ Aτ (k) + Cτ (k) ⊗ Cτ (k)]E[x(k − τ ) ⊗ x(k − τ )] + ER(k, xk , xk−τ ).
(48)
Now, according to H-representation matrix introduced in Lemmas 2.1 and 2.3, we are in a position to n(n+1) reduce the dimension of system (48). Set x e(k) = vecx(x(k) ⊗ x(k)) ∈ R 2 , we have Ex e(k + 1) = Θ H n2 , n(n+1) , A (k), C (k) Ex e(k) 0 0 2 (49) 2 n(n+1) e xk , xk−τ ), , Aτ (k), Cτ (k) E x e(k − τ ) + E R(k, +Θ H n , 2 11
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where h i−1 Θ H n2 , n(n+1) H T n2 , n(n+1) , A0 (k), C0 (k) := H T n2 , n(n+1) H n2 , n(n+1) 2 2 2 2 , ×(A0 (k)T ⊗ A0 (k)T + C0 (k)T ⊗ C0 (k)T )H n2 , n(n+1) 2 h i−1 n(n+1) n(n+1) n(n+1) n(n+1) Θ H n2 , 2 , Aτ (k), Cτ (k) := H T n2 , 2 H n2 , 2 H T n2 , 2 ×(Aτ (k)T ⊗ Aτ (k)T + Cτ (k)T ⊗ Cτ (k)T )H n2 , n(n+1) , 2 i−1 h e xk , xk−τ ) := H T n2 , n(n+1) H n2 , n(n+1) R(k, xk , xk−τ ). R(k, H T n2 , n(n+1) 2 2 2
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(50)
We give a two-dimensional system example to show the process of transforming system (4) into system (49).
"
0.5 0 C0 (k) = −0.5 sin k −0.5 " # x1 (k − 1) xτ (k) = . x2 (k − 1)
Then
0.5 0 0 0.5 sin2 k
M
A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k) =
0.0025 0 0 0
ED
PT
Cτ (k) ⊗ Cτ (k) =
−0.25 sin k 0 −0.25 sin2 k 0.5 sin k
0 0.0025 0 0
,
Cτ (k) =
−0.25 sin k −0.25 sin2 k 0 0.5 sin k
0 0 0 0 0.0025 0 0 0.0025
12
"
,
0.05 0
0.25 sin2 k −0.25 sin k −0.25 sin k 0.5
0.05x1 (k)x1 (k − 1) −0.05 sin kx1 (k)x1 (k − 1) − 0.05x2 (k)x2 (k − 1) 0.05x1 (k)x2 (k − 1) −0.05 sin kx1 (k)x2 (k − 1) − 0.05x2 (k)x2 (k − 1)
AC
CE
ER(k, xk , xk−τ ) =
#
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Example 3.2 For system (4), we take " # −0.5 0.5 sin k A0 (k) = , −0.5 sin k −0.5 " # x1 (k) Aτ (k) = 0, x(k) = , x2 (k)
.
0 0.05
,
#
,
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Θ H n2 , n(n+1) 2
Θ H
n2 , n(n+1) 2
x21 (k) ˜(k) = E x1 (k)x2 (k) , , Ex x22 (k) ,
, A0 (k), C0 (k) =
0.5 0 0.5 sin2 k
0.0025 , Aτ (k), Cτ (k) = 0 0
−0.5 sin k −0.25 sin2 k sin k 0 0.025 0
0.25 sin2 k −0.25 sin k , 0.5
0 0 , 0.0125
0.05x1 (k)x1 (k − 1) ˜ xk , xk−τ ) = R(k, −0.025(sin k)x1 (k)x1 (k − 1) − 0.025x2 (k)x2 (k − 1) + 0.025x1 (k)x2 (k − 1) . −0.05(sin k)x1 (k)x2 (k − 1) − 0.05x2 (k)x2 (k − 1)
AN US
get
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Using H-representation technique, we x21 (k + 1) Ex ˜(k + 1) = E x1 (k + 1)x2 (k + 1) x22 (k + 1) x21 (k − 1) Ex ˜(k − 1) = E x1 (k − 1)x2 (k − 1) x22 (k − 1)
Now, we are ready to establish our main results in this subsection. First of all, we give the following lemma.
M
Lemma 3.3 There exists constants i > 0, i = 1, 2, such that the growth condition
e
˜(k)k + 2 sup kE x ˜(k − m)k
E R(k, xk , xk−τ ) ≤ 1 kE x
ED
holds.
(51)
m∈[−τ,0]
Proof. Using the following fact, the proof of this lemma is obvious.
PT
Ex2i (k − τ ) ≤
sup m∈[−τ,0]
Ex2i (k − m),
(52)
AC
CE
Taking y(k) = E x ˜(k) and y(k − τ ) = E x ˜(k − τ ), system (4) can be turned into y(k + 1) = Θ H n2 , n(n+1) , A0 (k), C0 (k) y(k) 2 e xk , xk−τ ), k ∈ N + , +Θ H n2 , n(n+1) , A (k), C (k) y(k − τ ) + E R(k, τ τ 2 n(n+1) y(k) = ϕ(k) e = H T n2 , n(n+1) (ϕ(k) ⊗ ϕ(k)) ∈ R 2 , k ∈ N−τ . 2
For simplicity, we denote n(n + 1) n(n + 1) Θ0 (k) = Θ H n2 , , A0 (k), C0 (k) , Θτ (k) = Θ H n2 , , Aτ (k), Cτ (k) . 2 2 In system (52), the characterization of the slowly time-varying coefficients is shown by Θ0 (k). Remark 3.3 In what follows, we can find that the stability/exponential stability of stochastic system (4) -dimensional deterministic system (52). If we use the can be transformed into the study of an n(n+1) 2 augmented system method as done in (23), then an n(2n + 1)-dimensional system is to be used, which is larger more than (52) in dimension, and will greatly increase computational difficulty. 13
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The following proposition is obvious. Proposition 3.1 If the deterministic system (52) is stable/exponentially stable, then the stochastic system (4) is also stable/exponentially stable in mean square sense.
Theorem 3.2 Consider system (52). Denote
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According to Proposition 3.1, we present the following result.
∞ X
Γ1 = sup Θk0 (v) < ∞, Γ2 = sup kΘτ (k)k , ~ = max kϕ(k)k e sup Θk0 (v) < ∞. k,v≥0
−τ ≤k≤0
k≥0
k=0
Suppose (52) satisfies the following two conditions: 1. There is a % = %(q) > 0 such that
holds for all k ≥ j ≥ 0. 2. ℵ = ℵ(Θ0 (k), Θτ ) =
∞ X
(%k + 1 + 2 + Γ2 ) sup Θk0 (v) < 1.
k=0
(54)
(55)
v≥0
M
Then, the zero solution of system (52) is stable.
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kΘ0 (k) − Θ0 (j)k ≤ %|k − j|
(53)
v≥0
ED
Proof. For the slowly time-varying system (4), system coefficients A0 (k) and C0 (k) satisfy condition (44). By using H-representation technique, in system (52), we have
PT
kΘ0 (k) − Θ0 (j)k
= (H T H)−1 H T [A0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k)] H − (H T H)−1 H T [A0 (j) ⊗ A0 (j) + C0 (j) ⊗ C0 (j)] H
T −1 T ≤ (H H) H kA0 (k) ⊗ A0 (k) + C0 (k) ⊗ C0 (k) − A0 (j) ⊗ A0 (j) − C0 (j) ⊗ C0 (j)k kHk ≤ %(q)|k − j|,
e xk , xk−τ ), k ∈ N + , y(k + 1) − Θ0 (s) = (Θ0 (k) − Θ0 (s)) y(k) + Θτ (k)y(k − τ ) + E R(k,
AC
where
CE
where H = H n2 , n(n+1) and k ≥ j ≥ 0. From system (52), we have 2
(56)
n(n + 1) , A0 (s), C0 (s) Θ0 (s) = Θ H n2 , 2
and s is a fixed nonnegative integer. Using the variation of constants formula, we get y(l + 1) = Θl+1 e + 0 (s)ϕ(0)
l X j=0
h i e Θl−j 0 (s) (Θ0 (j) − Θ0 (s)) y(j) + Θτ (j)y(j − τ ) + E R(j, xj , xj−τ ) .
(57)
By taking s = l, (57) yields that y(l + 1) = Θl+1 e + 0 (l)ϕ(0)
l X j=0
h i e Θl−j 0 (l) (Θ0 (j) − Θ0 (l)) y(j) + Θτ (j)y(j − τ ) + E R(j, xj , xj−τ ) . 14
(58)
ACCEPTED MANUSCRIPT
Applying Lemma 3.3 and condition (54) , we obtain l
X
l−j
kϕ(0)k sup Θl+1 e + (l) (l)
Θ 0 0
l∈N
j=0
· (% |l − j| + 1 ) ky(j)k + kΘτ (j)k ky(j − τ )k + 2 ≤
0≤k≤l
l∈N
k=0
l X
k
Θ0 max ky(j)k (kΘτ (k)k + 2 ) + −τ ≤j≤k
∞ X
l+1
sup Θ0 (l) kϕ(0)k e + max ky(k)k (%k + 1 ) sup Θk0 (v) 0≤k≤l
l∈N
k=0
0≤k≤l
k≥0
∞ X
k≥0
0≤k≤l
l∈N
M
−τ ≤k≤0
k≥0
ED
sup Θk0 (v)
v≥0 k=0 ∞ X
e + max ky(k)k sup Θl+1 0 (l) kϕ(0)k
+(sup kΘτ (k)k + 2 ) max kϕ(k)k e
So
sup Θk0 (v)
v≥0 k=0 ∞ X
+(sup kΘτ (k)k + 2 ) max kϕ(k)k e −τ ≤k≤0
v≥0
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+(sup kΘτ (k)k + 2 ) max ky(k)k
≤
ky(j − m)k
l X
e sup Θl+1 + max ky(k)k (%k + 1 ) Θk0 (l) 0 (l) kϕ(0)k k=0
≤
sup −τ ≤m≤0
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ky(l + 1)k ≤
(%k + 1 + 2 + sup kΘτ (k)k) sup Θk0 (v)
k=0 ∞ X
k=0
k≥0
v≥0
sup Θk0 (v) . v≥0
ky(l + 1)k ≤ Γ1 kϕ(0)k e + ℵ max ky(k)k + (Γ2 + 2 )~, 0≤k≤l
(59)
PT
where Γ1 , Γ2 and ℵ are shown as in (53). (59) yields that
max ky(k)k ≤ Γ1 kϕ(0)k e + ℵ max ky(k)k + (Γ2 + 2 )~.
k∈N +
k∈N
(60)
CE
Let ky(k)k < kϕ(0)k e (k ∈ N + ), then system (52) is stable. If the inequality ky(k)k ≥ kϕ(0)k e (k ∈ N + ), then we have max+ ky(k)k = max ky(k)k, k∈N
k∈N
AC
which combines the condition (55) giving
So it follows that
max ky(k)k ≤ (Γ1 kϕ(0)k e + (Γ2 + 2 )~) (1 − ℵ)−1 .
(61)
ky(k)k ≤ [Γ1 kϕ(0)k e + (Γ2 + 2 )~] (1 − ℵ)−1 , k ∈ N .
(62)
k∈N
This proves the stability of the zero solution of system (52).
Corollary 3.1 If the conditions in Theorem 3.2 are satisfied, then the zero solution of system (52) is exponentially stable. 15
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Proof. To establish the exponential stability of the zero solution of system (52), we define a new variable Ex eλ (k) = E x e(k)λk (63) with a constant λ > 1. Substituting (63) into system (52), we get
(64)
CR IP T
where
ˆ 0 (k)E x ˆ τ (k)E x ˆ xk,λ , xk−τ,λ ), Ex eλ (k + 1) = Θ eλ (k) + Θ eλ (k − τ ) + E R(k, ˆ Θ0 (k) = λΘ0 (k), ˆ Θτ (k) = λΘτ (k), ˆ e λ−k xk,x ,x E R(k, xk,λ , xk−τ,λ ) = λk+1 E R(k, ). k,λ k−τ,λ sup
kE x ˜λ (k − m)k,
AN US
The growth condition (51) yields that
ˆ
˜λ (k)k + ˆ2
E R(k, xk,λ , xk−τ,λ ) ≤ ˆ1 kE x
m∈[−τ,0]
where ˆ1 = λ1 , ˆ2 = λτ +1 2 . Similar computations to the proof of Theorem 3.2, it follows that E x ˜ı (k) is a bounded function. So, by equation (63), the system considered is exponentially stable, which also means that system (4) is exponentially stable in mean square sense. The proof is completed.
Conclusion
M
4
PT
ED
This paper has discussed the stability analysis of time-varying stochastic systems with time-delays. For general linear time-varying stochastic systems with time-delays, the augmented system method has been used to transform the original system into a delay-free system. We have applied the operator spectrum technique and “frozen” method to present some new criteria on stability and exponential stability in mean square sense. The stabilization and other robust stabilization properties about this kind of systems still need to further study.
CE
Acknowledgments
AC
This work was supported by the National Natural Science Foundation of China (Nos. 61374104, 61773170), and the Natural Science Foundation of Guangdong Province of China (2016A030313505).
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