Analysis And Control Of Time-delay Switched Systems

With the development of modern industry, ordinary linear system control methodsand relevant theories can not satisfy the demand of the control accuracy. In order to en-hance the control accuracy, as a kind of hybrid systems, switched systems consisting of afamily of linear time invariant subsystems and a switching law specifying the switchingbetween them are introduced, which have many applications and are widely used in manyfields. Switched systems are expanded from the ordinary linear system state space mod-el, which are not only successfully used in modeling systems with sudden environmentaldisturbances, component failures or repairs and abrupt variations of operating point inpractice but also improve the control accuracy. This thesis, based on previous works ofothers, systematically and deeply investigates the problems of stability analysis and con-trol synthesis for continuous-time and discrete-time switched systems with time delayvia the scaled small gain theorem and parameter-dependent Lyapunov function method.The control problems of switched systems under arbitrary switched signal, controllableswitched signal and Markovian jump switched signal are investigated in this thesis respec-tively. Part of the developed theories are applied to the networked control of mechanicalsystems.The thesis firstly summarizes and analyzes main results in the frontiers of controlsynthesis for time-delay switched systems, and point out that the widely used Lyapunovmethod has limitations in reducing conservativeness. Based on this, the stability conditionof continuous-time and discrete-time switched systems with time delay are proposed byusing the input-output method concerned with the scaled small gain theorem, respectively.The stability conditions presented are significantly less conservative than the main resultsin other references, due to their properly processing to the time-varying delay term. It isworth noting that the scaled small gain theorem, guaranteeing the exponential stability ofthe systems, is also presented, which extends and improves the theory concerned with theinput-output method. As basic ideas, the conditions presented in this part will be extendedin the following discussions.This thesis also investigates the problems of control synthesis for time-delayswitched systems under arbitrary switching signal, controllable switching signal and switching signal with average dwell time, respectively. First, in solving the control prob-lems of continuous-time and discrete-time delayed switched systems, the control strate-gies focusing on continuous-time and discrete-time systems are presented to solve theseproblems, based on the result about stability analysis discussed above, which convertsthe existence condition of admissible controllers into the feasibility of convex problemssubject to linear matrix inequalities. Second, according to the input-output method andthe switching area design approach, the stabilizing switching law existence condition ispresented for time-delay switched systems, which has less conservativeness than most ex-isting results. At last, based on the results about the stability analysis and stabilization ofcontinuous-time switched systems with time delay, the problems of stability analysis and∞control for time-delay switched systems with average dwell time are investigated byusing the model transformation approach and the scaled small gain theorem. The result isformulated in the form of convex matrix inequalities, which is able to tolerate a smallerdwell time or a larger admissible delay bound for the given conditions than most of theapproaches seen in the literature.∞controller existence condition is also presented,which is less conservative than most of existing results, due to the utilization of this novelinput-output method.We investigate the problems of stability analysis and control synthesis of continuous-time and discrete-time Markovian jump time-delay switched systems based on the input-output method and the parameter-dependent Lyapunov function approach. To begin with,according to the characteristic of stochastic systems, the stochastic scaled small gain theo-rem, guaranteeing the stochastic stability of systems, is proposed, which extends the rangeof application of the input-output method and improves the relevant theory of this method.Then, based on the stochastic scaled small gain theorem and parameter-dependent Lya-punov function approach, the stability conditions and the controller existence conditionsconcerned with continuous-time Markovian jump time-delay switched system are pre-sented, which are less conservative than most of existing results due to the utilization ofthe input-output method. What’s more, the stochastic scaled small gain theorem is alsoextended to the discrete-time form, which is used to solve the problems of robust stabilityanalysis and guaranteed cost controller design for discrete-time Markovian jump time-delay switched systems by presenting a robust stability condition and a guaranteed costcontroller existence condition. It is worth noting that the input-output method is first pro- posed for solving the problems of stability analysis and control synthesis of Markovianjump time-delay switched systems.The problems of stability analysis and controller synthesis of Markovian jumpswitched systems with time-varying delay and partially known transition rates are in-vestigated based on the results presented above. According to the success utilizationof the input-output method in solving the stability analysis and the control synthesis oftime-delay switched systems under arbitrary switching signal and Markovian jump time-delay switched systems, the problems of stability analysis and control synthesis of Marko-vian jump switched systems with time-varying delay and partially known transition rates,which are on borderline between both above-mentioned systems, are investigated in thispart. From the new results about Markovian jump time-delay switched systems, the sta-bility condition and stabilization condition are proposed, which are the extension of therelevant control theory concerned with Markovian jump time-delay switched systems. S-ince this kind of systems is more like the practical system, the control problems of themcan guide us to solve control problems of practical systems strongly.The problems of stability analysis and controller synthesis of discrete-time Takagi-Sugeno (T-S) fuzzy systems with time-varying delay are also investigated. The origi-nal system firstly is reformulated as an interconnected form by the model transformationmethod. The lifting method is further employed to convert the forward subsystem to adelay free system, where the main result concerning the stability of the original system isobtained based on the bound real lemma. Furthermore, the state-feedback controller ex-istence condition is presented by the scaled small gain theorem and parameter-dependentLyapunov function approach. Since the input-output method is novel, the stability condi-tion and stabilization condition are both less conservative than existing results.The networked control problems of mechanical systems are investigated based onthe new results above, which has strong application backgrounds in the areas like indus-try manufacture. Aiming at the mechanical systems on the network, we firstly buildsa networked system model considering the influence of packet loss and time delay suffi-ciently. Furthermore, using the input-output method and the idea about solving the controlproblems of Markovian jump time-delay systems, we propose state-feedback control de-sign method based on the scaled small gain theorem and parameter-dependent Lyapunovfunction approach. Then, according to the similar idea about solving the control prob- lems of Markovian jump switched systems with time-varying delay and partially knowntransition probabilities, the networked control problems of mechanical systems with withpartially known packet loss probabilities are investigated, which extends the networkedcontrol theory. The examples illustrate the effectiveness of the proposed method.