Study On Markov Jump Singular Systems Of Incomplete Transition Probabilities

Markov jump system is a hybrid system with characteristics of continuouslyvariable system states and discrete events with Markov jump law. Singular system canprecisely describe many complicated dynamic systems. As a kind of singular systemwith Markov jump parameters, Markov jump singular system (MJSS) is widely appliedto aerospace engineering, network control systems, economics, bioengineering, etc.In practical systems, it is difficult to get complete transition probabilities of MJSSsfor reasons such as feasibility, complexity and measurement cost. So there areimportant theoretical significances and extensive application values for this dissertationto study on MJSSs with incomplete transition probabilities. The problems such asadmissibility, stabilization, H control are investigated in this dissertation as follows:1. The admissibility of continuous-time MJSSs with incomplete transitionprobabilities (TPs) is discussed. Two admissibility criteria are expanded from normalcriteria to be suitable for continuous-time MJSSs with mode-dependent singularmatrices. One admissibility criterion for continuous-time MJSSs with incomplete TPsis presented in the form of strict linear matrix inequalities (LMIs). Another lessconservative admissibility criterion for continuous-time MJSSs with uncertainties inTPs is proposed by means of slack matrices. Simulation examples are given to verifythe effectiveness of the results.2．The stabilization of continuous-time MJSSs is investigated and two statefeedback controllers for the systems with complete TPs and mode-dependent singularmatrices are designed firstly. On account of the transitive property of inequality, twostate feedback controllers are designed to ensure that the continuous-time MJSSs withincomplete TPs are stochastically admissible. At last, a more flexible mode-dependentcontroller with free-connection weighting matrices is designed to ensure that thecontinuous-time MJSSs with uncertainties in TPs are stochastically admissible.Simulation examples are used to illustrate the validity of the proposed methods.3． The admissibility of discrete-time MJSSs with partially unknown TPs isinvestigated and three admissibility criteria are proposed. Firstly, a sufficient conditionfor discrete-time MJSSs with mode-dependent singular matrices and incomplete TPs tobe stochastically admissible is discussed. Secondly, a criterion without equationrestriction is deduced by variables substitution method and can be realized conveniently. Thirdly, a less conservative criterion based on equivalent substitution isproposed to deal with the unknown TPs. Simulation examples are used to verify theeffectiveness of the results.4． The stabilization of discrete-time MJSSs with partially unknown TPs isinvestigated and three state feedback controllers are designed to ensure that theclosed-loop systems are stochastically admissible. Firstly, a normal state feedbackcontroller is given for the stabilization of discrete-time MJSSs with partially unknownTPs. Secondly, a more generalized state feedback controller on account of matrixvariables substitution and slack matrices is designed with less restrict conditions.Thirdly, a less conservative state feedback controller is given to preserve theinformation of unknown parts of TPs. When the state variables can not be obtained, anoutput feedback controller is also designed to ensure that the closed-loop systems withpartially unknown TPs are stochastically admissible. Simulation examples are given toverify the effectiveness of the proposed methods.5．The H control problem of continuous-time MJSSs with incomplete TPs isinvestigated. First of all, two sufficient conditions for continuous-time MJSSs withpartially unknown TPs and bounded TPs are proposed respectively to ensure that thesystems are stochastically admissible and satisfy a prescribed H performance level.Secondly, an H state feedback controller in terms of strict LMIs is designed to ensurethat the closed-loop systems are stochastically admissible and satisfy a prescribed H performance level. Thirdly, a less conservative design method of H state feedbackcontroller is proposed by means of free-connection weighting matrices. Simulationexamples are given to verify the effectiveness of the proposed H control methods.