Linear Quadratic Nash Game of Stochastic Singular Time-Delay Systems with Multiple Decision Makers

1 Introduction

Singular systems, also known as descriptor systems, generalized state-space systems or implicit systems, are described by differential-algebraic equations. Singular systems have been extensively studied over the past decades due to the fact that they can describe a great many natural phenomena in physical systems such as microelectronic circuits, economics, demography and so on (see, e.g., [1–3]). A great number of fundamental notions and results in control and system theory based on state-space systems have been extended successfully to singular systems (see, e.g., [4–8] and the references therein).

In the past few decades, stochastic systems governed by Itô’s differential equations have received a great deal of research attention [9, 10]. Although a variety of results for optimal control of linear stochastic systems have been reported, the dynamic games of such systems have received comparatively little attention. Moreover, to the best of our knowledge, stochastic Nash games for singular time-delay systems have not been fully investigated. Since delays appear in many practical plants, the design of such strategy is an important issue that remains open.

This paper is concerned with the problem of Nash game of stochastic singular time-delay systems with multiple decision makers. In terms of a set of linear matrix inequalities, we present a sufficient condition for the existence of both Nash strategies and the upper bound of the cost function for the first time. Moreover, in order to demonstrate the effectiveness of the proposed theory, stochastic H₂∕H_∞ control with multiple decision makers is discussed as an immediate application.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, the main results are given, which generalizes the results of [11]. Section 4 discusses the stochastic H₂∕H_∞ control with multiple decision makers by using the obtained results. Section 5 ends this paper with some comments.

Notations: Throughout this paper, unless otherwise specified, we will employ the following notations. ( Ω, ℱ, {ℱ_t}_t≥0,ℙ) is a complete probability space with Ω being a sample space, ℱ being a σ-field, {ℱ_t}_t≥0 being a natural filtration and ℙ being a probability measure. A^T is the transpose of a matrix A; A^–1is inverse of a matrix A; A > 0 means that A is positive definite; I_n denotes the n × n identity matrix; rank(A) denotes the rank of A; deg (det(sI – A)) denotes degree of determinant sI – A.Rⁿ is the n-dimensional Euclidean space; E[·] denotes the expectation operator.

2 Preliminary Results

Consider the following time-invariant stochastic singular time-delay systems

where x(t) ∊ Rⁿ is the state vector; u(t) ∊ R^m is the control vector; E is a known singular matrix with rank(E) = r ≤ n; w_p(t) ∊ R is a one-dimensional standard Wiener process defined in the filtered probability space (Ω,ℱ, {ℱ_t}_t≥0, ℙ)· Without loss of generality, it is assumed that w_r(t) and w_s(t) are mutually independent for all r, s = 1, 2, ⋯ , M, and E[w(t)w^T(t)] = I_M, where w(t) := [w₁(t), w₂(t), ⋯ , w_M(t)]^t. Here, the scalar δ > 0 is the time delay of the system. φ(t) is a real-valued initial function. A, A₁, B, A_p are given real matrices of suitable sizes.

Let us consider the following stochastic linear quadratic control problem subject to (1):

The following definitions are similar with Definition 3.1 which was introduced in [12].

3 Main Results

In this section, we will utilize the obtained results of stochastic linear quadratic optimal control to derive the results of stochastic Nash games.

4 Application to Stochastic H₂∕H_∞ Control

Over the last decade, stochastic control problems governed by Itô’s differential equation have attracted considerable research interest. Recently, stochastic linear quadratic and H_∞ control problems with state- and control-dependent noise have been investigated (see, e.g., [13, 14]). They have received much attention and have been widely used in various fields. In particular, the stochastic H₂∕H_∞ control with state dependent noise and state, control and disturbance-dependent noise have been addressed (see, e.g., [10, 15, 16]), but up to present, stochastic H₂∕H_∞ control with multiple decision makers have not been reported, and the design of such strategy is an issue that remains to be considered.

Now, we apply the above proposed theory to solve some problems related to stochastic H₂∕H_∞ control with multiple decision makers.

Consider the following stochastic controlled system with state-dependent noise, which involve N-decision makers

where DiTDi=Imi,C=[C1TC2T⋯CNT]T,x(t)∈RRn,z(t)∈RRm,v(t)∈RRl,ui(t)∈RRmi stand for the system state, controlled output, exogenous disturbance signal and i-th control input, respectively. All coefficient matrices are assumed to be real constant. Given a disturbance attenuation level γ > 0, define performance functions

and

The infinite-time horizon stochastic H₂∕H_∞ control with multiple decision makers of system (20) can be stated as follows.

5 Conclusions

In the present paper, we have dealt with the Nash game for stochastic singular time-delay systems with multiple decision makers in infinite-time horizon. In terms of a set of linear matrix inequalities, sufficient condition for the existence of Nash strategies is given for the first time. Moreover, the infinite-time horizon stochastic H₂/H_∞ control with multiple decision makers are treated by using these obtained results.