Control Of Semi-markov Jump Linear Systems Under Complex Switching Dynamics

In practical systems,such as electric circuits,mechanical systems,vehicles and aircrafts,communication networks,and biological systems,stochastic jumps can occur between system modes,which are usually caused by abrupt changes in external surroundings and internal parameters or structures.As an important means of describing jump systems,semi-Markov jump systems have gained wide recognition,in which the transition probabilities rely on the past jump sequences,that is,have the “memory” property.The model of semi-Markov jump systems can be utilized to describe various practical systems,such as multiple-bus systems,reward systems,mechatronic systems,and ecological systems.For the analysis of semi-Markov jump linear systems,the semi-Markov kernel approach is powerful,in which the probability distributions of sojourn time depending on both the current and the next modes are contained.With the aid of the semi-Markov kernel and the upper bound of sojourn time for each mode,stability and mode-dependent stabilization conditions can be obtained with the value of Lyapunov function to be allowably increasing.However,in previous studies,it is quite ideal to hypothesize the semi-Markov process to be homogeneous,with completely known semi-Markov kernel,without lower bounds of sojourn time,and without delays in the mode switchings of controllers,etc.Hence,it is of great significance to study the modeling,analysis and control issues of semi-Markov jump linear systems with these complex switching dynamics systematically and profoundly.The thesis aims at studying the fundamental stability analysis and control synthesis issues for discrete-time semi-Markov jump linear systems with complicated switching dynamics,including lower-bounded sojourn time,switching delays of controller mode,incomplete sojourn and transition information,and nonhomogeneous semi-Markov process.In the framework that the Lyapunov functions are not necessarily monotonically decreasing,the semi-Markov kernel approach is employed to derive the analysis and synthesis criteria.Main results are summarized as follows:Firstly,the thesis introduces the backgrounds of the areas related to this study,introduces some basic conceptions,properties,approaches,reviews the literature on some recent studies,and lists some practical examples of systems with stochastic mode jumps.Then,the thesis expounds the motivations and significances of studying semi-Markov jump linear systems with complex switching dynamics.Afterwards,the stability analysis and stabilizing control issues are investigated for semi-Markov jump linear systems with both the upper and lower bounds of sojourn time,and correspondingly,the concept of σ-error mean square stability is put forward for the sojourn time with the two bounds.By virtue of the constructed Lyapunov function that not only depends on the current system mode but also on the elapsed time the system has been in the current mode,numerically testable stability criteria are developed for the semiMarkov jump linear system;and with certain techniques eliminating powers of matrices,the corresponding stabilization criteria are put forward.Finally,the effectiveness of the proposed control method and the superiority of allowing for the lower bound of sojourn time are validated via a numerical example and a practical example of a direct current motor,respectively.Then,the thesis establishes the stability and stabilization conditions for semi-Markov jump linear systems with constant delays in the switchings of controller mode.The adopted Lyapunov function depends on the modes of both the system and the controller as well as the time since the occurrence of the last mode jump.Numerically testable stability criteria are developed on the basis of the new proposed σ-error mean-square stability that integrates the weights of all the system modes.By virtue of certain techniques that can eliminate the terms containing products of matrices,a desired mode-dependent stabilizing controller is designed such that the closed-loop system is σ-error mean-square stable by allowing using mode-unmatched controllers.The obtained theoretical results are applied to the control problem of one joint of a space robot manipulator to demonstrate the effectiveness,applicability and superiority of the proposed control strategy as well as the necessity of considering the mode-switching delays in the designed controller.The next issue is concerned with the stability analysis and control synthesis for semiMarkov jump linear systems with incomplete sojourn and transition information.Motivated by the fact that the statistic characteristics of sojourn time and mode transitions are difficult to acquire,the sojourn-time probability mass functions and the transition probabilities for jump instants are considered to be partially accessed.The model relaxes the conventional hypothesis that all the probability mass functions and transition probabilities are completely known,and thus is more general.Numerically testable mean-square stability criteria are established for semi-Markov jump linear systems with unknown available probability mass functions and/or transition probabilities,and the existence conditions of the desired stabilizing controller are developed to guarantee the mean-square stability of the resulting closed-loop systems.The theoretical results are testified by several numerical examples and a practical example of space robot manipulator,to demonstrate the effectiveness,superiority and applicability of the developed control methodology.Furthermore,the mean-square stability analysis and stabilization problems are addressed for semi-Markov jump linear systems with fragmentary probabilistic distributions of sojourn time.The probabilistic distribution information of sojourn time for a certain sojourn-time probability mass function is incomplete while the transition information is partly available.Stability and stabilization criteria are developed by leveraging all the known semi-Markov kernel information.The results here are not only more widely applicable but also less conservative than those only utilize the semi-Markov kernel elements corresponding to completely known sojourn-time probability mass functions.The effectiveness and the superiority of our established theoretical results are exemplified by a numerical example and an RLC circuit model.Additionally,the stability analysis and stabilizing controller design methods are proposed for two classes of discrete-time nonhomogeneous semi-Markov jump linear systems via the convex polytopic approach.The first class is concerned with the sojourn-time probability mass functions that are independent of jump instants,while the other considers the existence of probability mass functions of sojourn time depending on jump instants.Certain techniques are developed for both classes of systems by transforming their timevarying semi-Markov kernels into two convex polytopic forms that are more tractable,respectively.By constructing the polytopic quadratic Lyapunov functions depend on both the current mode and the elapsed time in the current mode,the derived stability and stabilization criteria can be numerically tested with lower conservativeness.Finally,the proposed control strategies are applied to an illustrative application of automotive electronic throttle valve to show the effectiveness of the theoretical results.The results also reveal the significance of considering the inhomogeneity of semi-Markov jump linear systems and the importance of the constructed Lyapunov function.