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.     

On value functions for impulsive control of piecewise-deterministic processes

The paper deals with value functions for optimal stopping and impulsive control for piecewise-deterministic processes with discounted cost. The associated dynamic programming equations are variational and quasi-variational inequalities with integral and first-order differential terms The technique used is to approximate the value functions for an optimal stopping (impulsive control. switching control) problem for a piecewise-deterministic process by value functions for optimal stopping (impulsive control, switching control) problems for Feller piecewise-deterministic processes     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

Impulsive control for one-side Lipschitz nonlinear MASs under semi-Markovian switching topologies with partially unknown transition probabilities

This article addresses the consensus problem of impulsive control for the multi-agent systems under uncertain semi-Markovian switching topologies. Considering the control and information exchanging cost in the implementation of multi-agent systems, an impulsive control protocol is developed not only to relieve the network burden but address the consensus problem. In addition, globally Lipschitz condition, as required in many existing literatures, is not needed in this article, so we introduce one-side Lipschitz condition to loosen the constraint of Lipschitz constant and widen the range of nonlinear application. According to cumulative distribution functions and Lyapunov functional, sufficient criteria are derived for the mean square consensus of multi-agent systems. It is shown that the impulsive sequence is not only inconsistent with switching sequence but also mode-dependent. Finally, simulation results are given to validate the superiority of the theoretical results.     

Impulsive consensus of one-sided Lipschitz nonlinear multi-agent systems with Semi-Markov switching topologies

Many real systems involve not only parameter changes but also sudden variations in environmental conditions, which often causes unpredictable topologies switching. This paper investigates the impulsive consensus problem of the one-sided Lipschitz nonlinear multi-agent systems (MASs) with Semi-Markov switching topologies. Different from the existing modeling methods of the Markov chain, the Semi-Markov chain is adopted to describe this kind of randomly occurring changes reasonably. To cope with the communication and control cost constraints in the multi-agent systems, the distributed impulsive control method is applied to address the leader–follower consensus problem. Beyond that, to obtain a wider nonlinear application range, the one-sided condition is delicately developed to the controller design, and the results are different from the ones obtained in the traditional method with the Lipschitz condition (note that the existing results are usually only applicable to the case with small Lipschitz constant). Based on the characteristics of cumulative distribution functions, the theory of Lyapunov-like function and impulsive differential equation, the asymptotically mean square consensus of multi-agent systems is maintained with the proposed impulsive control protocol. Finally, an explanatory simulation is presented to validate the correctness of the proposed approach conclusively.     

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.     

Impulsive consensus of multi-agent systems with stochastically switching topologies

In this paper, the impulsive consensus problem for multi-agent systems is investigated. The purpose of this paper is to provide a valid consensus protocol that overcomes the difficulty caused by stochastically switching structures via impulsive control. Some sufficient conditions of almost sure consensus are proposed when the switching structures are the independent process or the Markov process. It is shown that the sum-zero rows of matrix play a key role in achieving group consensus. Furthermore, simulation examples are provided to illustrate and visualize the effectiveness of these results.     

Consensus seeking in multi-agent systems via hybrid protocols with impulse delays

This paper studies the consensus problem of multi-agent systems with both fixed and switching topologies. A hybrid consensus protocol is proposed to take into consideration of continuous-time communications among agents and delayed instant information exchanges on a sequence of discrete times. Based on the proposed algorithms, the multi-agent systems with the hybrid consensus protocols are described in the form of impulsive systems or impulsive switching systems. By employing results from matrix theory and algebraic graph theory, some sufficient conditions for the consensus of multi-agent systems with fixed and switching topologies are established, respectively. Our results show that, for small impulse delays, the hybrid consensus protocols can solve the consensus problem if the union of continuous-time and impulsive-time interaction digraphs contains a spanning tree frequently enough. Simulations are provided to demonstrate the effectiveness of the proposed consensus protocols.     

有限时区上的最优转换和停止问题

讨论了有限时区上的最优转换和停止问题,它是一类同时具备脉冲控制和最优停止特征的最优控制问题.问题的最优值以及最优转换和停止决策可以由具有混合障碍的多维反射倒向随机微分方程的解来刻画.接着考虑了形式更一般的反射倒向随机微分方程并证明了方程解的存在唯一性.     

有限时区上的最优转换和停止问题

讨论了有限时区上的最优转换和停止问题,它是一类同时具备脉冲控制和最优停止特征的最优控制问题.问题的最优值以及最优转换和停止决策可以由具有混合障碍的多维反射倒向随机微分方程的解来刻画.接着考虑了形式更一般的反射倒向随机微分方程并证明了方程解的存在唯一性.     

Computational Method for Time-Optimal Switching Control

An efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time-optimal switching control of nonlinear systems with a single control input. The problem is formulated in the arc times space, arc times being the durations of the arcs. A feasible switching control, or as a special case bang-bang control, is found using the STC method previously developed by the authors to get from an initial point to a target point with a given number of switchings. Then, by means of constrained optimization techniques, the cost being considered as the summation of the arc times, a minimum-time switching control solution is obtained. Example applications of the TOS algorithm involving second-order and third-order systems are presented. Comparisons are made with a well-known general optimal control software package to demonstrate the efficiency of the algorithm.     

Infinite horizon optimal control problems with multiple thermostatic hybrid dynamics

We study an optimal control problem for a hybrid system exhibiting several internal switching variables whose discrete evolutions are governed by some delayed thermostatic laws. By the dynamic programming technique we prove that the value function is the unique viscosity solution of a system of several Hamilton-Jacobi equations, suitably coupled. The method involves a contraction principle and some suitably adapted results for exit-time problems with discontinuous exit cost.     

A decomposition approach for optimal control problems with integer innerpoint state constraints

A large class of optimal control problems for hybrid dynamic systems can be formulated as mixed‐integer optimal control problems (MIOCPs). A decomposition approach is suggested to solve a special subclass of MIOCPs with mixed integer inner point state constraints. It is the intrinsic combinatorial complexity of the discrete variables in addition to the high nonlinearity of the continuous optimal control problem that forms the challenges in the theoretical and numerical solution of MIOCPs. During the solution procedure the problem is decomposed at the inner time points into a multiphase problem with mixed integer boundary constraints and phase transitions at unknown switching points. Due to a discretization of the state space at the switching points the problem can be decoupled into a family of continuous optimal control problems (OCPs) and a problem similar to the asymmetric group traveling salesman problem (AGTSP). The OCPs are transcribed by direct collocation to large‐scale nonlinear programming problems, which are solved efficiently by an advanced SQP method. The results are used as weights for the edges of the graph of the corresponding TSP‐like problem, which is solved by a Branch‐and‐Cut‐and‐Price (BCP) algorithm. The proposed approach is applied to a hybrid optimal control benchmark problem for a motorized traveling salesman. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Optimal Control of a Train

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn–Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem.     

Stability analysis of a reduced model of the lac operon under impulsive and switching control

The study of the stability is an important motif in biological systems. Stability plays a significant role in some of the basic processes of life. In this paper, we provide a theoretical method for analyzing the exponential stability of a reduced model of the lac operon under the impulsive and switching control. Under the hybrid control, the reduced model becomes a nonlinear hybrid impulsive and switching system, and we get the sufficient conditions of its exponential stability. Finally, we give examples to illustrate the effectiveness of our method.     

Stability of hybrid stochastic differential systems with switching and time delay

This article introduces a hybrid stochastic differential system with impulsive, switching and time-delay. Some stability criteria of p-moment global asymptotical stability, p-moment global exponential stability and mean square stability of this system are derived by using switching Lyapunov function approach, Itô formula, impulsive differential inequality method, and linear matrix equality techniques. Three examples are presented to demonstrate the efficiency of the obtained results.     

Finite-time stability and stabilization of nonlinear stochastic hybrid systems

This paper deals with the problem of finite-time stability and stabilization of nonlinear Markovian switching stochastic systems which exist impulses at the switching instants. Using multiple Lyapunov function theory, a sufficient condition is established for finite-time stability of the underlying systems. Furthermore, based on the state partition of continuous parts of systems, a feedback controller is designed such that the corresponding impulsive stochastic closed-loop systems are finite-time stochastically stable. A numerical example is presented to illustrate the effectiveness of the proposed method.     

Exponential stabilization of nonlinear systems under saturated control involving impulse correction

This paper studies the problem of locally exponential stabilization (LES) of nonlinear systems under a class of hybrid control in the framework of actuator saturation, where the limitation of actuator saturation on both continuous state feedback control and impulsive control are fully considered. Based on impulsive control theory and differential inclusion approach, some sufficient conditions for LES are derived, where a novel set inclusion relation is proposed to handle the double saturation nonlinearities. Different from the existing results that each part of saturated hybrid control (SHC) is required to stabilize the system individually, our results relax the requirement by making full use of the correction effect of the impulse. Moreover, the maximum of estimation of domain of attraction is obtained by a convex optimal problem and corresponding algorithm. The result is applied to the robustness for a class of nonlinear systems. Finally, the validity of the results is shown by two examples and their simulations, where the synchronization problem of Chua’s oscillator is illustrated in the framework of actuator saturation.     

Optimal Control of a Stochastic Hybrid System with Discounted Cost

We address the optimal control problem of a very general stochastic hybrid system with both autonomous and impulsive jumps. The planning horizon is infinite and we use the discounted-cost criterion for performance evaluation. Under certain assumptions, we show the existence of an optimal control. We then derive the quasivariational inequalities satisfied by the value function and establish well-posedness. Finally, we prove the usual verification theorem of dynamic programming.     

Stabilization of complex network with hybrid impulsive and switching control

This paper studies the asymptotic stability properties of a class of complex dynamical networks under a hybrid impulsive and switching control. By utilizing the concept of impulsive control and the stability results for impulsive systems, some new criteria for global and local stability are established for this model. Some numerical examples and simulations are included to illustrate the effectiveness of the theoretical results.