Reach-avoid Analysis for Stochastic Discrete-time Systems

The main problem with the soft-computing algorithms is a determination of their parameters. The tuning rules are very general and need experiments during a trial and error method. The equations describing the bat algorithm have the form of difference equations, and the algorithm can be treated as a stochastic discrete-time system. The behaviour of this system depends on its dynamic and preservation stability conditions. The paper presents the stability analysis of the bat algorithm described as a stochastic discrete-time state-space system. The observability and controllability analyses were made in order to verify the correctness of the model describing the dynamic of BA. Sufficient conditions for stability are derived based on the Lyapunov stability theory. They indicate the recommended areas of the location of the parameters. The analysis of the position of eigenvalues of the state matrix shows how the different values of parameters affect the behaviour of the algorithm. They indicate the recommended area of the location of the parameters. Simulation results confirm the theory-based analysis.

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.

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.

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.

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.

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.

On the general Kalman filter for discrete time stochastic fractional systems

This paper describes a theoretical framework for the design of controllers to satisfy probabilistic safety specifications for partially observable discrete time stochastic hybrid systems. We formulate the problem as a partial information stochastic optimal control problem, in which the objective is to maximize the probability that the state trajectory remains within a given safe set in the hybrid state space, using observations of the history of inputs and outputs. It is shown that this optimal control problem, which has a multiplicative payoff structure, is equivalent to a terminal payoff problem when the state space is augmented with a binary random variable capturing the safety of past state evolution. This allows us to derive a sufficient statistic for the probabilistic safety problem as a set of Bayesian filtering equations updating a conditional distribution on the augmented state space, as well as an abstract dynamic programming algorithm for computing the maximal probability of safety and an optimal control policy.

In this paper, we focus on discrete-time stochastic systems modelled by nonlinear stochastic difference equations and propose robust abstractions for verifying probabilistic linear temporal specifications. The current literature focuses on developing sound abstraction techniques for stochastic dynamics without perturbations. However, soundness thus far has only been shown for preserving the satisfaction probability of certain types of temporal-logic specification. We present constructive finite-state abstractions for verifying probabilistic satisfaction of general {\omega}-regular linear-time properties of more general nonlinear stochastic systems. Instead of imposing stability assumptions, we analyze the probabilistic properties from the topological view of metrizable space of probability measures. Such abstractions are both sound and approximately complete. That is, given a concrete discrete-time stochastic system and an arbitrarily small L1-perturbation of this system, there exists a family of finite-state Markov chains whose set of satisfaction probabilities contains that of the original system and meanwhile is contained by that of the slightly perturbed system. A direct consequence is that, given a probabilistic linear-time specification, initializing within the winning/losing region of the abstraction system can guarantee a satisfaction/dissatisfaction for the original system. We make an interesting observation that, unlike the deterministic case, point-mass (Dirac) perturbations cannot fulfill the purpose of robust completeness.

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.

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

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

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this technical note, the stability for time-varying discrete-time stochastic linear systems, with possibly unbounded trajectories, is associated to the existence of the long-run average cost criteria. Under controllability and observability, the stochastic system is stable in the sense that its state tends asymptotically to the origin in the mean. Further conditions related to a lower bound on the long-run average cost, or with periodic time-varying systems, provides uniform second moment stability. </para>

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.