Stabilization of stochastic singular nonlinear hybrid systems

This paper deals with the control of the class of singular nonlinear stochastic hybrid systems. Under some appropriate assumptions, results on stochastic stability and stochastic stabilization are developed. Two state feedback controllers (linear and nonlinear) that stochastically stabilize the class of systems we are considering are designed. LMI sufficient conditions are developed to compute the gains of these controllers.     

Robust$${\cal {H}}_{\infty}$$ Stabilization of Stochastic Hybrid Systems with Wiener Process

This paper deals with the class of uncertain continuous-time linear stochastic hybrid systems with Wiener process. The uncertainties that we are considering are of the norm-bounded type. The robust stochastic stabilization problem is treated. LMIs based sufficient conditions are developed to design the state feedback controller that robustly and stochastically stabilizes the studied class of systems and at the same time rejects a disturbance of desired level. The minimum disturbance rejection is also determined. A numerical example is provided to show the validity of the proposed results.     

Guaranteed Cost Control for a Class of Uncertain Stochastic Impulsive Systems with Markovian Switching

Abstract

This article is concerned with the problem of guaranteed cost control for a class of uncertain stochastic impulsive systems with Markovian switching. To the best of our knowledge, it is the first time that such a problem is investigated for stochastic impulsive systems with Markovian switching. For an uncontrolled system, the conditions in terms of certain linear matrix inequalities (LMIs) are obtained for robust stochastical stability and an upper bound is given for the cost function. For the controlled systems, a set of LMIs is developed to design a linear state feedback controller which can stochastically stabilize the class of systems under study and guarantee the given cost function to have an upper bound. Further, an optimization problem with LMI constraints is formulated to minimize the guaranteed cost of the closed-loop system. Finally, a numerical example is provided to show the effectiveness of the proposed method.     

On State Feedback Stabilization of Singular Systems with Random Abrupt Changes

This paper deals with the class of continuous-time singular linear systems with random abrupt changes. The state feedback stabilization and its robustness for this class of systems with norm-bounded uncertainties are tackled. Sufficient conditions for designing either a stabilizing controller or a robust stabilizing controller are developed in the LMI setting. The developed sufficient conditions are used to synthesize the state feedback controller that guarantees that either the nominal system or the uncertain system is piecewise regular, impulse free and stochastically stable or robust stochastically stable. The research of this author was supported by NSERC, Grant RGPIN36444-02.     

Non-fragile robust stabilization and control for uncertain stochastic nonlinear time-delay systems

This paper deals with the problem of non-fragile robust stabilization and H_∞ control for a class of uncertain stochastic nonlinear time-delay systems. The parametric uncertainties are real time-varying as well as norm bounded. The time-delay factors are unknown and time-varying with known bounds. The aim is to design a memoryless non-fragile state feedback control law such that the closed-loop system is stochastically asymptotically stable in the mean square and the effect of the disturbance input on the controlled output is less than a prescribed level for all admissible parameter uncertainties. New sufficient conditions for the existence of such controllers are presented based on the linear matrix inequalities (LMIs) approach. Numerical example is given to illustrate the effectiveness of the developed techniques.     

Improved results for stochastic stabilization of a class of discrete-time singular Markovian jump systems with time-varying delay

In this paper, the problem of stochastic stabilization for a class of discrete-time singular Markovian jump systems with time-varying delay is investigated. By using the Lyapunov functional method and delay decomposition approach, improved delay-dependent sufficient conditions are presented, which guarantee the considered systems to be regular, causal and stochastically stabilizable. Finally, some numerical examples are provided to illustrate the effectiveness of the obtained methods.     

Robust stability and stabilization of linear stochastic systems with Markovian switching and uncertain transition rates

This paper is concerned with the robust stabilization problem for a class of linear uncertain stochastic systems with Markovian switching. The uncertain stochastic system with Markovian switching under consideration involves parameter uncertainties both in the system matrices and in the mode transition rates matrix. New criteria for testing the robust stability of such systems are established in terms of bi-linear matrix inequalities (BLMIs), and sufficient conditions are proposed for the design of robust state-feedback controllers. A numerical example is given to illustrate the effectiveness of our results.     

On Robust stability of singular systems with random abrupt changes

This paper deals with the class of continuous-time singular linear systems with Markovian switching. Sufficient conditions on stochastic stability and robust stochastic stability are developed in the LMI setting. The developed sufficient conditions are used to check if either the nominal or the uncertain systems are regular, impulse-free and stochastically stable or robust stochastically stable.     

Robust Stability and Stabilization of a Class of Nonlinear Itô-Type Stochastic Systems via Linear Matrix Inequalities

This article presents a new approach to robust quadratic stabilization of nonlinear stochastic systems. The linear rate vector of a stochastic system is perturbed by a nonlinear function, and this nonlinear function satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of stochastic systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The new formulation provides a suitable setting for robust stabilization of nonlinear stochastic systems where the underlying deterministic systems satisfy the generalized matching conditions. Our sufficient conditions are written in matrix forms, which are determined by solving linear matrix inequalities (LMIs), which have significant computational advantage over any other existing techniques. Examples are given to demonstrate the results.     

Exponential stability for stochastic reaction-diffusion BAM neural networks with time-varying and distributed delays

In this paper we study the stability for a class of stochastic bidirectional associative memory (BAM) neural networks with reaction-diffusion and mixed delays. The mixed delays considered in this paper are time-varying and distributed delays. Based on a new Lyapunov-Krasovskii functional and the Poincaré inequality as well as stochastic analysis theory, a set of novel sufficient conditions are obtained to guarantee the stochastically exponential stability of the trivial solution or zero solution. The obtained results show that the reaction-diffusion term does contribute to the exponentially stabilization of the considered system. Moreover, two numerical examples are given to show the effectiveness of the theoretical results.     

Robust stochastic stability and H∞ performance for a class of uncertain impulsive stochastic systems

In this paper, we discuss the problem of robust stochastic stability and H_∞ performance for a class of uncertain impulsive stochastic systems under sampled measurements. The parameter uncertainties are assumed to be time-varying and value-bounded. We give a sufficient condition in terms of certain linear matrix inequalities (LMIs) to guarantee the uncertain impulsive stochastic system to be robustly stochastically stable. Furthermore, we discuss a stochastically stable filter, using the locally sampled measurements, which ensures both the stochastic stability and a prescribed level of H_∞ performance for the filtering error system for all admissible uncertainties. We give a sufficient condition for the existence of such a filter and an explicit expression of a desired filter if relevant conditions are satisfied.     

Delay-Dependent Criteria for Robust Stabilization of Markovian Switching Networks with Time-Varying Delay

Abstract

A problem of feedback stabilization of hybrid systems with time-varying delay and Markovian switching is considered. Delay-dependent sufficient conditions for stability based on linear matrix inequalities (LMI's) for stochastic asymptotic stability is obtained. The stability result depended on the mode of the system and of delay-dependent. The robustness results of such stability concept against all admissible uncertainties are also investigated. This new delay-dependent stability criteria is less conservative than the existing delay-independent stability conditions. An example is given to demonstrate the obtained results.     

Stabilization of Passive Nonlinear Stochastic Differential Systems by Bounded Feedback

Abstract

The purpose of this paper is to give a systematic method for global asymptotic stabilization in probability of nonlinear control stochastic differential systems the unforced dynamics of which are Lyapunov stable in probability. The approach developed in this paper is based on the concept of passivity for nonaffine stochastic differential systems together with the theory of Lyapunov stability in probability for stochastic differential equations. In particular, we prove that, as in the case of affine in the control stochastic differential systems, a nonlinear stochastic differential system is asymptotically stabilizable in probability provided its unforced dynamics are Lyapunov stable in probability and some rank conditions involving the affine part of the system coefficients are satisfied. Furthermore, for such systems, we show how a stabilizing smooth state feedback law can be designed explicitly. As an application of our analysis, we construct a dynamic state feedback compensator for a class of nonaffine stochastic differential systems.     

Dynamic programming in stochastic control of systems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.     

Diffusion approximation for signaling stochastic networks

This paper introduces an unified approach to diffusion approximations of signaling networks. This is accomplished by the characterization of a broad class of networks that can be described by a set of quantities which suffer exchanges stochastically in time. We call this class stochastic Petri nets with probabilistic transitions, since it is described as a stochastic Petri net but allows a finite set of random outcomes for each transition. This extension permits effects on the network which are commonly interpreted as “routing” in queueing systems. The class is general enough to include, for instance, G-networks with negative customers and triggers as a particular case. With this class at hand, we derive a heavy traffic approximation, where the processes that drive the transitions are given by state-dependent Poisson-type processes and where the probabilities of the random outcomes are also state-dependent. The objective of this approach is to have a diffusion approximation which can be readily applied in several practical problems. We illustrate the use of the results with some numerical experiments.     

Improved H∞ performance analysis of uncertain Markovian jump systems with overlapping time‐varying delays

In this article, we study the problem of robust H_∞ performance analysis for a class of uncertain Markovian jump systems with mixed overlapping delays. Our aim is to present a new delay‐dependent approach such that the resulting closed‐loop system is stochastically stable and satisfies a prescribed H_∞ performance level χ. The jumping parameters are modeled as a continuous‐time, finite‐state Markov chain. By constructing new Lyapunov‐Krasovskii functionals, some novel sufficient conditions are derived to guarantee the stochastic stability of the equilibrium point in the mean‐square. Numerical examples show that the obtained results in this article is less conservative and more effective. The results are also compared with the existing results to show its conservativeness. © 2016 Wiley Periodicals, Inc. Complexity 21: 460–477, 2016     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Control of stochastic chaos using sliding mode method

Stabilizing unstable periodic orbits of a deterministic chaotic system which is perturbed by a stochastic process is studied in this paper. The stochastic chaos is modeled by exciting a deterministic chaotic system with a white noise obtained from derivative of a Wiener process which eventually generates an Ito differential equation. It is also assumed that the chaotic system being studied has some model uncertainties which are not random. The sliding mode controller with some modifications is used for stochastic chaos suppression. It is shown that the system states converge to the desired orbit in such a way that the error covariance converges to an arbitrarily small bound around zero. As some case studies, the stabilization of 1-cycle and 2-cycle orbits of chaotic Duffing and Φ⁶${\Phi }^6$ Van der Pol systems is investigated by applying the proposed method to their corresponding stochastically perturbed systems. Simulation results show the effectiveness of the method and the accuracy of the statements proved in the paper.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

On New Multivariate Probability Distributions and Stochastic Processes with System Reliability and Maintenance Applications

Two distinct methods for construction of some interesting new classes of multivariate probability densities are described and applied. As common results of both procedures three n-variate pdf classes are obtained. These classes are considered as generalizations of the class of univariate Weibullian, gamma, and multivariate normal pdfs. An example of an application of the obtained n-variate pdfs to the problem of modeling the reliability of multicomponent systems with stochastically dependent life-times of their components is given. Obtaining sequences over n = 2, 3, ... of consistent n-variate pdfs, that obey a relatively simple common pattern, for each n, allows us to extend some of the constructions from random vectors to discrete time stochastic processes. Application of one, so obtained, class of highly non-Markovian, but still sufficiently simple, stochastic processes for modeling maintenance of systems with repair, is presented. These models allow us to describe and analyze repaired systems with histories of all past repairs.