Stability radii of infinite-dimensional systems subjected to unbounded stochastic perturbations

In this paper, we use the framework of stability radii to study the robust stability of linear deterministic systems on real Hilbert spaces which are subjected to unbounded stochastic perturbations. First, we establish an existence and uniqueness theorem of the solution of the abstract equation describing the system. Then we characterize the stability radius in terms of a Lyapunov equation or equivalently in terms of the norm of an input-output operator.     

A Riccati equation approach to maximizing the stability radius of a linear system by state feedback under structured stochastic Lipschitzian perturbations

The object of this paper is to maximize the stability radius of a linear state space system by state feedback under Lipschitzian structured stochastic perturbations. The supermanl achievable stability radius is characterized via the resolution of a parametrized Riccati equation and a matrix inequality. The dependence on the parameters is investigated and the limiting behaviour is examined. An example illustrating the results is treated at the end of the paper.     

H∞ control and filtering with initial uncertainty for infinite dimensional systems

The H_∞-control problem with a non-zero initial condition for infinite dimensional systems is considered The initial conditions are assumed to be in some subspace. First the H_∞ problem with full information is considered and necessary and sufficient conditions for the norm of an input-output operator to be less than a given number are obtained, The characterization of all admissible controllers is also given. This result is then used to solve the general H^∞ control problem and the filtering problem with initial uncertainty. The filtering problem on finite horizon involves the estimate of the state at final time. The set of all suboptimal filters is given both on finite and infinite horizons.     

Stability of linear stochastic delay differential equations with infinite Markovian switchings

This paper investigates the stability of linear stochastic delay differential equations with infinite Markovian switchings. Some novel exponential stability criteria are first established based on the generalized It formula and linear matrix inequalities. Then, a new sufficient condition is proposed for the equivalence of 4 stability definitions, namely, asymptotic mean square stability, stochastic stability, exponential mean square stability with conditioning, and exponential mean square stability. In particular, our results generalize and improve some of the previous results. Finally, two examples are given to illustrate the effectiveness of the proposed results.     

Detectability and observability of discrete-time stochastic systems and their applications

This paper studies detectability and observability of discrete-time stochastic linear systems. Based on the standard notions of detectability and observability for time-varying linear systems, corresponding definitions for discrete-time stochastic systems are proposed which unify some recently reported detectability and exact observability concepts for stochastic linear systems. The notion of observability leads to the stochastic version of the well-known rank criterion for observability of deterministic linear systems. By using these two concepts, the discrete-time stochastic Lyapunov equation and Riccati equations are studied. The results not only extend some of the existing results on these two types of equation but also indicate that the notions of detectability and observability studied in this paper take analogous functions as the usual concepts of detectability and observability in deterministic linear systems. It is expected that the results presented may play important roles in many design problems in stochastic linear systems.     

Stability analysis and control for switched stochastic delayed systems

This paper investigates almost sure exponential stability and feedback stabilization for switched time‐delay systems with nonlinear stochastic perturbations. The main contributions of this paper are threefold: (i) based on the non‐convolution type multiple Lyapunov functionals and the mathematical induction approach, a mean‐square exponential stability condition for nonlinear stochastic switched systems is first established, such that the obtained average dwell time does not rely on any given decay rate; (ii) by using the method developed in part (i) and the stochastic analysis techniques and limit methods in probability, an almost sure exponential stability criterion for switched delayed systems with nonlinear stochastic uncertainties is presented, and then a state feedback controller for the systems under consideration is designed; and (iii) when certain assumptions are made on the nonlinear stochastic perturbations, the results in this paper are further improved by relaxing some conditions. The effectiveness of the proposed method is demonstrated by three illustrative examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Delay-range-dependent stability for uncertain stochastic systems with interval time-varying delay and nonlinear perturbations

This paper considers the problem of robust stability for a class of uncertain stochastic systems with interval time-varying delay under nonlinear perturbations. A new delay-dependent method for robust stability of the systems is proposed. The innovation of the method includes employment of a tighter integral inequality and construction of an appropriate type of Lyapunov–Krasovskii functional. The restriction used to bound some trace term in the existing methods is also removed. The resulting criterion derived from this method has advantages over previous ones in that it has less conservatism and enlarges the scope of application. The reduction in conservatism of the proposed criterion is attributed to a method to estimate the upper bound on the stochastic differential of the Lyapunov–Krasovskii functional without neglecting any useful terms in the delay-dependent stability analysis. On the basis of the estimation and by utilizing free-weighting matrices, new delay-range-dependent stability criterion is established in terms of linear matrix inequality. Finally, numerical examples are provided to show the effectiveness and reduced conservatism of the proposed method.     

Existence and stability of impulsive stochastic coupled systems in infinite dimensions

This paper considers the exponential stability of a class of infinite-dimensional impulsive stochastic coupled systems. With the help of generalized Itô's formula for the mild solution of infinite-dimensional systems, we avoid limiting the domain of the mild solution. Then we use the combination of the Lyapunov function and graph theory to construct the Lyapunov function of the systems; the criteria of $p$-th moment exponential stability are obtained, which is related to the average impulsive interval $T_a$ and the connectivity of impulsive stochastic systems. In addition, noting that the existence may be affected by impulsive effects and stochastic perturbations, using the graph theory and the principle of contraction mapping, we get the condition that guarantees the existence and uniqueness, which is also related to the structure of the networks. Finally, we consider the stability of impulsive stochastic coupled heat equations and neural networks with reaction diffusion and give some numerical simulations to verify the theoretical results.     

Laguerre series approximation of infinite dimensional systems

Laguerre-Fourier approximations of stable systems are shown to exhibit many desirable properties for various classes of infinite dimensional systems. Specifically, time domain supremum and L¹ norm convergence results, and frequency domain H^∞ norm convergence results, are given for Laguerre-Fourier approximations. It is also shown that the theory of Laguerre polynomials solves explicitly the problem of determining Laguerre-Fourier approximations for a large class of delay systems. Furthermore, it is believed that these results are important for the study of orthonormal series identification as a general technique for identification of infinite dimensional systems.     

Stability and stabilization of nonlinear discrete‐time stochastic systems

This paper mainly studies the locally/globally asymptotic stability and stabilization in probability for nonlinear discrete‐time stochastic systems. Firstly, for more general stochastic difference systems, two very useful results on locally and globally asymptotic stability in probability are obtained, which can be viewed as the discrete versions of continuous‐time Itô systems. Then, for a class of quasi‐linear discrete‐time stochastic control systems, both state‐ and output‐feedback asymptotic stabilization are studied, for which, sufficient conditions are presented in terms of linear matrix inequalities. Two simulation examples are given to illustrate the effectiveness of our main results.     

Stability analysis and synthesis for linear impulsive stochastic systems

This paper presents a general framework for analyzing stability of linear impulsive stochastic systems (LISSs). Some simple mean square stability criteria for the three types of LISSs are firstly derived by analyzing an equivalent system. By exploring the hybrid characteristics of impulsive systems, the novel quasi‐periodic composite polynomial Lyapunov function and the time‐varying discretized Lyapunov function are developed, which leads to unified dwell‐time–based criteria for mean square stability and almost sure stability of LISSs without imposing the stability condition on continuous‐ and discrete‐time dynamics. Next, based on the established stability criteria, the synthesis problem of state‐feedback controller is solved. The computational complexity and the comparison with existing results on the deterministic systems are discussed. Finally, numerical examples are provided to illustrate the usefulness of the proposed results.     

离散广义系统稳定性分析与控制的Lyapunov方法

利用Lyapunov方法,研究离散广义系统稳定性分析与控制问题.得到了离散广义 系统正则、具有因果关系且渐近稳定的等价条件;还给出了相关的鲁棒稳定性分析与镇定方 法.     

Nonstationary discrete-time deterministic and stochastic control systems with infinite horizon

This article is about nonstationary nonlinear discrete-time deterministic and stochastic control systems with Borel state and control spaces, possibly noncompact control constraint sets, and unbounded costs. The control problem is to minimise an infinite-horizon total cost performance index. Using dynamic programming arguments we show that, under suitable assumptions, the optimal cost functions satisfy optimality equations, which in turn give a procedure to find optimal control policies.     

广义线性系统的鲁棒稳定性分析和综合

研究了广义线性系统的鲁棒稳定性分析与综合问题,基于广义线性系统的Ｌｙａｐｕｎｏｖ方程和Ｒｉｃｃａｔｉ方程,给出了相关的鲁棒稳定性分析与综合方法。     

Robust stabilization of infinite dimensional systems by finite dimensional controllers

The problem of robustly stabilizing an infinite dimensional system with transfer function G, subject to an additive perturbation Δ is considered. It is assumed that: G ε ₀(σ) of systems introduced by Callier and Desoer [3]; the perturbation satisfies |W₁ΔW₂| < ε, where W₁ and W₂ are stable and minimum phase; and G and G + Δ have the same number of poles in ⁺. Now write W₁GW₂=G₁ + G₁, where G₁ is rational and totally unstable and G₂ is stable. Generalizing the finite dimensional results of Glover [12] this family of perturbed systems is shown to be stabilizable if and only if ε σ_min (G^*₁)( = the smallest Hankel singular value of G^*₁). A finite dimensional stabilizing controller is then given by where 2 is a rational approximation of G₂ such that

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) and K₁ robustly stabilizes G₁ to margin ε. The feedback system (G, K) will then be stable if |W₁ΔW₂| _∞< ε − Δ.     

Null controllability of an infinite dimensional SDE with state- and control-dependent noise

We consider a controlled stochastic linear differential equation with state- and control-dependent noise in a Hilbert space H. We investigate the relation between the null controllability of the equation and the existence of the solution of “singular” Riccati operator equations. Moreover, for a fixed interval of time, the null controllability is characterized in terms of the dual state. Examples of stochastic PDEs are also considered.