Stability Of Uncertain Stochastic Systems Research Articles

Robust exponential stability of uncertain stochastic systems with probabilistic time‐varying delays

SummaryThis paper is concerned with the problem of delay‐distribution–dependent robust exponential stability for uncertain stochastic systems with probabilistic time‐varying delays. Firstly, inspired by a class of networked systems with quantization and packet losses, we study the stabilization problem for a class of network‐based uncertain stochastic systems with probabilistic time‐varying delays. Secondly, an equivalent model of the resulting closed‐loop network‐based uncertain stochastic system is constructed. Different from the previous works, the proposed equivalent system model enables the controller design of the network‐based uncertain stochastic systems to enjoy the advantage of probability distribution characteristic of packet losses. Thirdly, by applying the Lyapunov‐Krasovskii functional approach and the stochastic stability theory, delay‐distribution–dependent robust exponential mean‐square stability criteria are derived, and the sufficient conditions for the design of the delay‐distribution–dependent controller are then proposed to guarantee the stability of the resulting system. Finally, a case study is given to show the effectiveness of the results derived. Moreover, the allowable upper bound of consecutive packet losses will be larger in the case that the probability distribution characteristic of packet losses is taken into consideration.

Delay-Dependent Robust Control for Discrete-Time Uncertain Stochastic Systems With Time-Varying Delays

This paper investigates a delay-dependent robust control problem of discrete-time uncertain stochastic systems with delays. The uncertainty considered in this paper is time-varying but norm-bounded, and the delays are considered as interval time-varying case for both state and input. According to the considerations of uncertainty, stochastic behavior, and time delays, the problem considered in this paper is more general than the existing works for uncertain stochastic systems. Via the proposed Lyapunov–Krasovskii function, some sufficient conditions are derived into the extended linear matrix inequality form. Moreover, Jensen inequality and free matrix equation are employed to reduce conservatism of those conditions. Through using the proposed design method, a gain-scheduled controller is designed to guarantee asymptotical stability of uncertain stochastic systems in the sense of mean square. Finally, two numerical examples are provided to demonstrate applicability and effectiveness of the proposed design method.

Probabilistic control of Markov jump systems by scenario optimization approach

This paper addresses the new problem of probabilistic robust stabilization for uncertain stochastic systems by using scenario optimization approach, where the uncertainties are not assumed to be norm-bounded. State feedback controllers are designed to guarantee that the closed-loop system is robust probabilistic stable. The problem of designing the controller gains is formulated and solved as linear matrix inequality (LMI) constraints. Simulation results are presented to illustrate the correctness and usefulness of the controllers designed.

Mean-square practical stability for uncertain stochastic system with additive noise controlled by optimal feedback

Stability concepts addressed in the framework of Lyapunov are not suitable to analyze the stability of stochastic systems with additive noise since it has no equilibrium. The mean-square practical stability is introduced to study the stability of uncertain stochastic systems with additive noise controlled by linear-quadratic optimal feedback. By using Lyapunov functional methods and the comparison principle, criteria on mean-square practical stability of stochastic systems with partially known uncertainties and norm-bounded parameter uncertainties are deduced, respectively. Some numerical examples and simulations are given to illustrate the validity of the theoretical analysis.

New delay-dependent robust exponential stability for uncertain stochastic systems with multiple delays based on extended reciprocally convex approach

New delay-dependent robust exponential stability for uncertain stochastic systems with multiple delays based on extended reciprocally convex approach

New criteria on delayed state feedback stabilization for stochastic systems with time-varying delay

The problems of robust exponential stability in mean square and delayed state feedback stabilization for uncertain stochastic systems with time-varying delay are studied. By using Jensen's integral inequality and combining with the free weighting matrix approach, new delay-dependent stability conditions and delayed state feedback stabilization criteria are obtained in terms of linear matrix inequalities. Meanwhile, the proposed delayed state feedback stabilization criteria are more convenient in application than the existing ones since fewer tuning parameters are involved. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Robust Stabilization for Uncertain Stochastic Systems with Interval Time-varying Delay Using Decomposition Approach

Robust Stabilization for Uncertain Stochastic Systems with Interval Time-varying Delay Using Decomposition Approach

On design of robust reliable [formula omitted] control and input-to-state stabilization of uncertain stochastic systems with state delay

This paper is concerned with the problem of designing a robust reliable H∞ control for uncertain stochastic systems with time delay (USSD) and time-varying norm-bounded parametric uncertainties in the system states. The faulty actuators are considered as a disturbance signal to the system which is augmented with system disturbance input. The main objective of the paper is presenting a state feedback controller that guarantees input-to-state stability (ISS) of the plant and an H∞ norm-bound constraint on the attenuation of the augmented disturbance for all admissible uncertainties and actuator failures. The problem is solved by using Lyapunov functions together with the Razumikhin methodology. To clarify the proposed results, some illustrative examples are presented.

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.

New Delay-Dependent Robust Stability Criterion for Uncertain Stochastic Systems with Time-Varying Delays

The problem of delay-dependent robust stability of uncertain stochastic systems with time-varying delay is discussed in this paper. Based on the Lyapunov-Krasovskii theory and free-weighting matrix technique, new delay-dependent stability criterion is presented. The criterion is in terms of linear matrix inequality (LMI) which can be solved by various available algorithms.

Robust Stabilization of Uncertain Stochastic Systems with Delay in Control Input

This paper discusses the problems of robust stabilization of stochastic systems with parametric uncertainties and time delay in control input. The parametric uncertainties which appear in all system matrices are assumed to be norm bounded. The delay-dependent stabilization condition is derived by taking the relationships between terms in the Leibniz-Newton formula into account. Free-weighting matrices are employed to express these relationship, and the sufficient stabilization condition is formulated in terms of linear matrix inequality (LMI) based on the Lyapunov-Krasovskii theory, which can be solved by LMI toolbox in Matlab

Delay partitioning approach to robust stability analysis for uncertain stochastic systems with interval time-varying delay

This study investigates the problem of delay-dependent robust stability for uncertain stochastic systems with interval time-varying delay. No restrictions on the derivative of the time-varying delay are imposed, although lower and upper bounds of the delay interval are assumed to be known. By decomposing the delay interval into multiple equidistant subintervals, new Lyapunov-Krasovskii functionals are introduced on these intervals, which result in being much less conservative than most existing results in the literature. In addition, the reduction in the conservatism of the proposed stability criteria is also attributed to the free-weighting matrix technique and a newly proposed bounding condition, which is used to estimate the upper bound of stochastic differential of the Lyapunov-Krasovskii functional without neglecting any useful terms. All the stability criteria are formulated in the form of linear matrix inequalities (LMIs), which are dependent on the size of the time delay. Some numerical examples are given to show the less conservatism and applicability of the obtained results. Moreover, when the number of the divided subintervals increases, the corresponding criteria can provide an improvement on the results.

On stability and stabilisation for uncertain stochastic systems with time-delay and actuator saturation

This article concerns the stability analysis and design for uncertain stochastic systems with time-varying delays in state and actuator saturation. The parameter uncertainties belong to a convex polytopic set, and the delays are time varying. A sufficient condition is obtained in terms of a priori designed feedback matrix for determining if a given set is in inside the domain of attraction. Using the linear matrix inequality (LMI) approach, an estimate of the domain of attraction is presented. The problem of designing a state feedback controller such that the domain of attraction is enlarged is formulated through solving an optimisation problem with LMI constraints. A numerical example is given to illustrate the effectiveness of the proposed results.

Delay-dependent Robust Stability for Uncertain Stochastic Systems with Interval Time-varying Delay

This paper is concerned with the stability analysis for uncertain stochastic systems with interval time-varying delay. Improved delay-dependent robust stability criteria of uncertain stochastic systems with interval time-varying delay are proposed without ignoring any terms by considering the relationship among the time-varying delay, its upper bound, and their difference, and using both Itó's differential formula and Lyapunov stability theory. A numerical example is given to illustrate the effectiveness and the benefit of the proposed method.

Robust delayed-state-feedback stabilization of uncertain stochastic systems

Due to time spent in computation and transfer, control input is usually subject to delays. Problems of deterministic systems with input delay have received considerable attention. However, relatively few works are concerned with problems of stochastic system with input delay. This paper studies delayed-feedback stabilization of uncertain stochastic systems. Based on a new delay-dependent stability criterion established in this paper, a robust delayed-state-feedback controller that exponentially stabilizes the uncertain stochastic systems is proposed. Numerical examples are given to verify the effectiveness and less conservativeness of the proposed method.

Delay-dependent robust stability criteria of uncertain stochastic systems with time-varying delay

In this paper, the problems of delay-dependent robust stability for uncertain stochastic systems and their corresponding deterministic systems with time-varying delay are investigated. Based on Lyapunov stability theory combined with linear matrix inequalities (LMIs) techniques, some new delay-dependent stability criteria in terms of LMIs are derived by introducing some free weighting matrices which can be selected properly to lead to much less conservative results. Numerical examples are given to illustrate the effectiveness of the proposed method and an improvement over some existing results in the literature.

Quadratic stabilization for uncertain stochastic systems

This paper discusses the robust quadratic stabilization control problem for stochastic uncertain systems, where the uncertain matrix is norm bounded, and the external disturbance is a stochastic process. Two kinds of controllers are designed, which include state feedback case and output feedback case. The conditions for the robust quadratic stabilization of stochastic uncertain systems are given via linear matrix inequalities. The detailed design methods are presented. Numerical examples show the effectiveness of our results.

Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach

In this paper, the problem of exponential stability in mean square for stochastic systems with multiple delays is investigated. A delay-dependent sufficient condition is derived in terms of linear matrix inequalities (LMIs) by using a descriptor model transformation of the system and by applying Moon's inequality for bounding cross terms. The criteria obtained in this paper can be tested numerically very efficiently using interior point algorithms. An example shows that the proposed methods are less conservative than the other methods.

Comments on "An LMI-based approach for robust stabilization of uncertain stochastic systems with time-varying delays

This paper points out some technical problems in the theorems and proofs in the above paper, and presents their corresponding corrections. In addition, contrary to the claim in the above paper, it is shown that the linear matrix inequality conditions obtained are actually delay independent.

An LMI-based approach for robust stabilization of uncertain stochastic systems with time-varying delays

Based on the linear matrix inequality method, we introduce the robust stability of uncertain linear stochastic differential delay systems with delay dependence. The parameter uncertainty is norm-bounded and the delays are time varying. We then extend the proposed theory to discuss the robust stabilization of uncertain stochastic differential delay systems.