Stability Analysis of the Bat Algorithm Described as a Stochastic Discrete-Time State-Space System

In this paper, stability analysis of deterministic and stochastic nonlinear discrete-time dynamical systems is considered. It is shown that some difficulties, such as independency or identical distribution of random variable sequence, that may arise in using Lyapunov's and LaSalle's approaches for stability analysis of discrete-time stochastic systems may be overcome by means of fixed point theory. Furthermore, suitable definitions of stability are provided. Ultimately, numerical examples are given in order to show the results.

In this paper we propose definitions of equivalence via stochastic bisimulation and of equivalence of stochastic external behavior for the class of discrete-time stochastic nonlinear control systems with possibly degenerate distributed disturbances. The notions are inspired by the analogue notions that exist for probabilistic chains and for behavioral systems, respectively. Connections between the two notions and also with stochastic reachability are discussed. For the subclass of discrete-time stochastic linear control systems with multivariate possibly degenerate normal distributed disturbances, geometric necessary and sufficient conditions for checking stochastic bisimulation equivalence and stochastic external behavior equivalence are derived. The proposed results nicely extend the theory of bisimulation equivalence and equivalence of external behavior known for non-deterministic control systems to a stochastic setting.

The problem of mean square bounded-input bounded-output(BIBO) stability is investigated for a class of discrete-time stochastic control systems with time delays and non-linear perturbations. In this paper, a special point δ in the time delay's variation interval is introduced, and the variation interval is divided into two subintervals. Then, by defining a special Lyapunov-Krasovskii functional and checking its variation in the two subintervals, respectively, some novel delay-dependent stability criteria for the discrete-time stochastic control systems are derived. These conditions are expressed in the forms of linear matrix inequalities(LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Meanwhile, this paper provides a new method for studying discrete-time stochastic mean square BIBO stability. Finally, a numerical example is given to illustrate the validity of the main results.

This paper is concerned with the security control problem with quadratic cost criterion for a class of discrete-time stochastic nonlinear systems subject to deception attacks. A definition of security in probability is adopted to account for the transient dynamics of controlled systems. The purpose of the problem under consideration is to design a dynamic output feedback controller such that the prescribed security in probability is guaranteed while obtaining an upper bound of the quadratic cost criterion. First of all, some sufficient conditions with the form of matrix inequalities are established in the framework of the input-to-state stability in probability. Then, an easy-solution version on above inequalities is proposed by carrying out the well-known matrix inverse lemma to obtain both the controller gain and the upper bound. Furthermore, the main results are shown to be extendable to the case of discrete-time stochastic linear systems. Finally, two simulation examples are utilized to illustrate the usefulness of the proposed controller design scheme.

Stochastic discrete-time systems, i.e., discrete-time dynamic systems subject to stochastic disturbances, are an essential modelling tool for many engineering systems, and reach-avoid analysis is able to guarantee safety (i.e., via avoiding unsafe sets) and progress (i.e., via reaching target sets). In this paper we study the reach-avoid problem of stochastic discrete-time systems over open (i.e., not bounded a priori) time horizons. The stochastic discrete-time system of interest is modeled by iterative polynomial maps with stochastic disturbances, and the problem addressed is to effectively compute an inner approximation of its p-reach-avoid set. The p-reach-avoid set collects those initial states that give rise to a bundle of trajectories which with probability being larger than p eventually hits a designated set of target states while remaining inside a set of safe states before the first hit. The computation of the p-reach-avoid set is first reduced to the computation of a corresponding strict p-super-level set and is then inner-approximated by solving a semi-definite programming problem obtained from a relaxation of the definition of the super-level set. Two examples demonstrate the proposed approach.

This paper is concerned with reduced-order /sub /spl infin// filtering of stochastic systems. Based on linear matrix inequality (LMI) technique, a new design method is proposed for the reduced-order filtering of stochastic linear systems. The method is derived from decomposing the key matrix in LMIs which determines the order of designed filters. Different from the existing methods, the proposed method first minimizes the upper bound of the key matrix and then eliminates its near-zero eigenvalues, which results in a simpler, more direct and reliable design procedure. The method is applicable to both continuous and discrete time stochastic systems. Its effectiveness is illustrated by an example.

This paper is concerned with a class of discrete-time stochastic systems with periodic coefficients and multiplicative noise. By use of an invariant-subspace approach, a simplified criterion is derived for detectability of the concerned systems. This spectral criterion can be verified by only examining the spectra of monodromy operator associated with real positive semi-definite matrices. Based on the proposed detectability criterion, a mean square small gain theorem is shown to supply a sufficient condition which guarantees the stability of single-loop interconnected stochastic periodic systems.

This paper focuses on investigating the pth exponential ultimate boundedness of a class of discrete-time stochastic delay systems and its application in neural networks. By employing a novel method that combines inequality techniques and reductio ad absurdum, several sufficient conditions for the pth exponential ultimate boundedness of the addressed systems are derived. This method not only circumvents the hardship of constructing Lyapunov functional but also yields simpler criteria for exponential ultimate boundedness criteria. As applications, the criteria for boundedness are utilised in neural networks. Finally, the validity of the theoretical results is verified by numerical examples.

This paper presents the stabilization problem of a class of discrete-time stochastic linear systems with time delay using an event-triggered control(ETC) strategy. For the general linear systems, we propose a new event triggering mechanism and deduce sufficient conditions for mean square exponentially stabilizability(MSES) and almost sure exponentially stabilizability(ASES) via ETC. The results show that the stability of the system depends on the system parameters (time delay, noise, event trigger threshold parameter, and unstable characteristic value of the original system). When the feedback control is disturbed by low-intensity noise, the unstable system can be stable in the mean square. Finally, the validity of the theoretical results is illustrated through a simulation example.

Since a real system can hardly be modelled by an exact linear deterministic or stochastic system and in this case a small disturbance may cause instability of adaptive algorithms (Egardt, 1980,Riedle et al 1984,Rohrs,1982),it is of great importance to analyse the influence of the unmodelled dynamics contained in the system upon the behavior of the adaptive control system. For recent years there has been a vast amount of research devoted to this issue (Anderson et al,1986,Bitmead et al,1986,Goodwin et al, 1985,Hili et al,1985,Ioannou et al 1984a,b,Ioannou et ai,1985 and Kosut et al 1984). The authors have analysed the robustness of identification and adaptive control algorithms in (Chen & Guo,1986a,b) for disCreteand continuous-time stochastic systems respectively,when the extended least squares (ELS) identification is applied. Roughly speaking,there it is shown that the estimation error and the deviation of the tracking error from its minimum value is small when the unmodelled dynamics is small in a certain sense. In this paper we are concentrated on the robustness issue of identification for the discrete-time stochastic system which consists of a modelled part being a CARMA process and of an unmodelled part On,i.e. the system is described by

In this paper, an H ∞ filter design problem is studied for networked stochastic linear discrete-time systems with mixed uncertainties of random measurement delays, packet dropouts and missing measurements. We model the packet delivery characteristics of the network as a four-state Markov process and note that the results for discrete-time linear systems with Markovian jumping parameters can be applied. We focus on the design problem of a full-order H ∞ filter such that for all possible uncertain measurements the filtering error dynamics is exponentially mean square stable and a prescribed H ∞ performance bound is guaranteed. Necessary and sufficient bilinear matrix inequality (BMI) condition for the synthesis of the aforementioned filter for a discrete-time jump system is presented. Some simulation examples are presented to show the applicability and effectiveness of the proposed approach.

Bayesian optimal control for a non-autonomous stochastic discrete time system

Resilient [formula omitted]-[formula omitted] filtering with dwell-time-based communication scheduling

A piecewise-constant process containing a single jump is observed under noise in the context of discrete time. The conditional density and maximum a posteriori (MAP) estimator of the jump time as well as the Bayes detector of the jump itself are determined using the powerful measure transformation approach. The Bayes detector provides a convenient sequential detection rule for practical on-line implementation. An asymptotic result for the distribution of the MAP estimator's estimation error and the corresponding convergence rate are derived. This result provides a reference measure of optimal performance for jump-time estimators in discrete-time stochastic systems that does not depend on the jump time's prior distribution

Robust stability for discrete-time stochastic systems employing possibly discontinuous control laws is the focus of this paper. Through novel examples, we illustrate the fact that the existence of a continuous stochastic Lyapunov function implies robustness of stability to sufficiently small, state-dependent, strictly causal, worst-case perturbations, irrespective to the continuity of the stabilizing control law. We emphasize the role of strict causality through an example for which a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case perturbations, which are not strictly causal. Finally we illustrate our main result for a stochastic control system which admits a continuous Lyapunov function, but associated with a necessarily discontinuous control law.