A unified framework for asymptotic and transient behavior of linear stochastic systems

doi:10.1016/j.amc.2017.12.023

Applied Mathematics and Computation

Volume 325, 15 May 2018, Pages 31-40

https://doi.org/10.1016/j.amc.2017.12.023 Get rights and content

Abstract

This paper is concerned with a unified framework for asymptotic and transient behavior of stochastic systems. In order to explain this problem explicitly, a concept of mean square (γ, α)-stability is first introduced and two stability criteria are derived. By utilizing an auxiliary definition of mean square (γ, T)-stability, the relations among mean square (γ, α)-stability, mean square (γ, T)-stability and finite-time stochastic stability are established. Subsequently, two new sufficient conditions for the existence of state and output feedback mean square (γ, α)-stabilization controllers are presented in terms of matrix inequalities. A numerical algorithm is given to obtain the relation between γ_min and α. Finally, an example is given to illustrate our results.

Introduction

It is well-known that Lyapunov asymptotic stability(LAS) presents steady state behavior of systems and is very useful for practical applications. However, an asymptotic stable system may have large values of the states. Often these large values can destroy the system, which are not acceptable. For example, large transient voltage will destroy power systems [1]. To deal with these unacceptable transient values, a concept of finite-time stability(FTS) was proposed in [2], which mainly concerns the transient behavior of the system over a fixed finite-time interval and has been widely extended to deterministic linear continuous-time systems [3], [4], [5], discrete-time systems [6], stochastic systems [7], [8], Markov jump systems [9], [10], [11], [12], [13], [14] and so on. Nevertheless, up to now, there is no a unified framework to consider both transient performance and steady state performance.

On the other hand, following the development of stochastic differential equation, Itô stochastic systems have received much attention because of their practical applications, such as signal processing [15], gene networks [16], mathematical finance [17], epidemic model [18], [19]. Therefore, the related stability and control problems have been extensively studied. For example, [20] investigated mean square stability of linear stochastic systems by spectrum technique. In the sequel, literature [21] investigated input-to-state stability for stochastic nonlinear systems with state-dependent switching. [22] investigated stochastic $H_{2} / H_{\infty}$ control of mean-field type for continuous-time systems with state-and disturbance-dependent noise. For some results of robust control on this kind of systems, we refer to the monograph [23]. However, most of the results were only concerned with either transient performance or steady state performance.

Motivated by aforementioned discussions, the unified stability and stabilization problems for asymptotic and transient behavior of stochastic systems are considered in this paper. By using stochastic analysis technology, the stability criteria and some stabilizing conditions are obtained. The contributions of this paper lie in the following three aspects: (1) A concept of mean square (γ, α)-stability is first introduced to put asymptotic and transient behavior of Itô stochastic systems into a unified framework, and two stability criteria are obtained. (2) By utilizing an auxiliary definition of mean square (γ, T)-stability, the relations among (γ, α)-stability, (γ, T)-stability and FTS are established. (3) On the basis of stability criteria obtained, two kinds of unified stabilization controllers (state and output feedback controllers) are designed to make closed-loop systems have the desired transient and steady performance, and a numerical algorithm is given to obtain the relation between γ_min and α.

This paper is organized as follows: Section 2 gives a mean square (γ, α)-stability and its relation to mean square (γ, T)-stability and FTS. Section 3 provides several stability conditions for mean square (γ, α)-stability and mean square (γ, T)-stability and analyzes the relations among these stability conditions. Section 4 is to design state and output feedback mean square (γ, α)-stabilization controllers. A numerical algorithm is given in Section 5. Section 6 employs an example to illustrate the results. Section 7 gives the conclusion.

Notations: X^T stands for transpose of a matrix X. The X > 0 means that X is positive-definite. I_n × n stands for n × n identity matrix. λ_max(X)(λ_min(X)) represents the maximum(minimum) eigenvalue of a matrix X. $E [\cdot]$ denotes the operator of the mathematical expectation. tr(X) is the trace of a matrix X. The notation $∥ x ∥ : = \sqrt{x^{T} x}$ denotes Euclidian 2-norm of a vector x and ‖x‖_Q denotes $\sqrt{x^{T} Q x}$ . The asterisk “*” represents the symmetry term in a matrix and diag{⋅⋅⋅} represents a block-diagonal matrix. The shorthand “wrt” is an abbreviation of “with respect to”. k(X) = λ_max(X)/λ_min(X) stands for condition number of a X > 0.

Section snippets

Preliminaries and definitions

Consider the following linear Itô stochastic system ${\begin{matrix} d x (t) = A x (t) d t + A_{1} x (t) d w (t), \\ x (0) = x_{0} \in R^{n}, \end{matrix}$ where $x (t) \in R^{n}$ is the system state, A, $A_{1} \in R^{n \times n}$ are constant matrices and x₀ is the initial state. Without loss of generality, we assume w(t) to be one-dimensional standard Wiener process defined on the probability space (Ω, $F,$ $F_{t},$ P) with $F_{t}$ = σ{w(s): 0 ≤ s ≤ t}.

Next, a definition of the unified description for asymptotic behavior and transient behavior of system (1) is given.

Definition 1

Given the constants γ > 1 and α

Mean square-(γ, α)-stability analysis

This section first gives a sufficient condition for mean square (γ, α)-stability and a sufficient condition for mean square (γ, T)-stability. In regards to stability conditions, the relations among mean square (γ, α)-stability, mean square (γ, T)-stability and FTSS are shown.

Theorem 2

Given γ > 1 and α > 0, system (1) is mean square (γ, α)-stable if there exists a matrix Q > 0 such that the following inequalities hold: $k (Q) < γ,$ $A^{T} Q + Q A + A_{1}^{T} Q A_{1} < - α Q .$

Proof

Let Q > 0 such that (6) and (7) are satisfied. Defining $V (x$

Mean square (γ, α)-stabilization

Consider the following linear $It \hat{o}$ stochastic controlled system: ${\begin{matrix} d x (t) = [A x (t) + B u (t)] d t + [A_{1} x (t) + B_{1} u (t)] d w (t), \\ y (t) = C x (t), x (0) = x_{0} \in R^{n}, \end{matrix}$ where $u (t) \in R^{m}$ and $y (t) \in R^{p}$ are the control input, measurement output, respectively. B, $B_{1} \in R^{n \times m}$ and $C \in R^{p \times n}$ are constant matrices.

Based on Definition 1, mean square-(γ, α)-stabilization can be given as follows.

Definition 4

Given the constants γ > 1 and α > 0, system (25) is called mean square (γ, α)-stabilizable if there exists a feedback controller u*(t) such that the closed-loop

Numerical algorithm

In this section, an algorithm is given to achieve the relation between the minimum value of γ and α. The following analysis and algorithm for Theorem 5 are first given. Similar algorithm can be applied to Theorem 6.

Analysis: By analyzing (27) and (28), we find that if (27) and (28) have feasible solutions wrt X, Y, σ₁ and σ₂ for given γ and α, then γ_min is existent and α_min = 0. So, we first fix α₁ = 0 and search for γ_min from 1, and then we fix α₂ and search for γ_min from 1 again, and so

Simulation results

This section provides an illustrative example to demonstrate the effectiveness of the proposed results. Consider system (25) with parameters as follows: $A = [\begin{matrix} - 0.1 & 1 \\ 0 & - 0.05 \end{matrix}], A_{1} = [\begin{matrix} 0.2 & - 0.5 \\ 0.01 & 0.12 \end{matrix}], B = [\begin{matrix} 1.0 \\ 0.5 \end{matrix}], B_{1} = [\begin{matrix} 0.2 \\ 0.1 \end{matrix}], C = {[\begin{matrix} - 0.2 \\ - 0.3 \end{matrix}]}^{'} .$ Let the scalars γ = 5 and α = 0.2, x(0) = ${[- 1 1]}^{T}$ . Using Euler–Maruyama method to simulate the standard Brownian motion, one can obtain the curves of ‖x(t)‖² (30 curves) and $E [∥ x (t) ∥^{2}]$ of system (1) in Fig. 1. From Fig. 1, it can be seen that the system (1) is mean square

Conclusion

In this study, a unified framework for asymptotic and transient behavior of stochastic systems has been established. Two criteria to test mean square (γ, α)-stability of a system have been obtained. Furthermore, state feedback and output feedback controllers have been designed for the mean square (γ, α)-stabilization of this class of systems, respectively. In addition, the relations among mean square (γ, α)-stability, mean square (γ, T)-stability and finite-time stability have been established.

Acknowledgments

This work was supported in part by the Natural Science Foundation of CQ under Grant CSTC 2014J-CYJA40004, and the Natural Science Foundation of China under Grant 11471061. Also, the work of Z. Yan was supported by the National Natural Science Foundation of China under Grant no. 61403221 and Project funded by China Postdoctoral Science Foundation (2017M610425).

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