Controller Design For Nonhomogeneous It(?) Markovian Jump Stochastic Systems With Its Applications

Markovian jump linear systems are widely used to describe many physical systems,such as robot systems,spacecraft systems,network communication systems,which are subject to frequent structural changes.In addition,the external disturbances caused by a variety of environmental factors are inevitable in practice when the aforementioned application systems are working.Over the past few decades,It(?) Markovian jump stochastic systems become a natural and reasonable choice to model such systems.Thus,the control problems associated with the It(?) Markovian jump stochastic systems have received extensive attention.It is well known that the transition rates in the jump process determine the behavior of the Markovian jump systems.In previous literatures,the investigation on Markovian jump systems,including the It(?) Markovian jump stochastic systems,was always based on the assumption that the transition rates are completely known and time-invariant.Under this assumption,the Markov process in Markovian jump systems is called as a homogeneous Markov process.However,the transition rates of the Markovian jump systems may be not completely known and time-invariant in some practical applications.For this reason,the investigation on the Markovian jump linear systems with partly unknown and/or timevarying transition rates is necessary both in theory and in practice.Due to the preceding reasons,the investigation on the stability analysis and stabilization of the It(?) Markovian jump stochastic systems with partly unknown and time-varying transition rates become a main topic of this dissertation.Besides,the effectiveness of the presented theoretical results are illustrated by the model reference tracking controller design for the spacecraft trajectory tracking control problem.The detailed results are as follows:First,the stability analysis and stabilization of the It(?) Markovian jump stochastic systems with partly unknown transition rates are investigated.By fully using the properties of the transition rates,two new stochastic stability criteria are developed for the It(?) Markovian jump stochastic systems with partly unknown transition rates via coupled Lyapunov matrix equations or linear matrix inequalities(LMIs).The main advantage of the proposed stability conditions is that the total number of LMIs in the presented stability criteria is much less than that in some existing results.Therefore,these stability criteria are more convenient in practical applications.Secondly,the problems of stability and stabilization of a class of It(?) Markovian jump stochastic systems with time-varying transition rates are investigated.It is assumed that the transition rates of the considered systems are time-varying but time-invariant within some small intervals.In this case,the Markovian process is called as finite piecewise homogeneous Markovian process.Further,the variations of transition rate matrices in the finite set are considered as two special cases: arbitrary variation and stochastic variation.For these two cases,the stability conditions of the piecewise homogeneous It(?) Markovian jump stochastic systems are given in terms of the LMIs.Moreover,two novel stability criteria are developed for the considered systems by using the existence of the unique positive definite solution of the corresponding coupled Lyapunov matrix equations.Further,two state-feedback controllers are designed via LMIs to stabilize the considered systems.In addition,by combining the preceding results,the stability and stabilization of the It(?) Markovian jump stochastic systems with time-varying and partly unknown transition rates are investigated.Besides,the iterative technique is studied for solving the extended coupled Lyapunov matrix equations arising from the stability analysis of the It(?) nonhomogeneous Markovian jump stochastic systems.By using the idea of the successive over relaxation(SOR)technique for the ordinary linear equations,a novel implicit iterative algorithm is proposed.Under zero initial conditions,the monotonicity and boundedness of the matrix sequences generated by the proposed algorithm are obtained.Based on the obtained properties,it is shown that under zero initial conditions the presented algorithm converges to the unique solution of the considered equations if the associated nonhomogeneous It(?) Markovian jump stochastic systems are stochastically stable.In addition,a necessary and sufficient condition is also given for the convergence of the developed algorithm with arbitrary initial conditions.Besides,by utilizing the same idea,a novel SOR implicit iterative algorithm is given for solving the general coupled Lyapunov equations.Some convergence results are shown for this algorithm with arbitrary initial conditions.Especially,for the case of 2 subsystems the optimal relaxation parameter is given such that the presented algorithm has the fastest convergence rate.Thirdly,the linear quadratic optimal control problem is investigated for the It(?) nonhomogeneous Markovian jump stochastic systems with infinite time horizon.Based on the extended coupled Riccati matrix equation,the necessary and sufficient conditions for the stochastic stabilization of the considered systems with time-varying transition rates are given.Based on the obtained results,the necessary and sufficient conditions are established for the existence of the solutions to the infinite time quadratic optimal control problem of the considered systems.To obtain the solution of the extended coupled Riccati matrix equations,two weighted iterative algorithms are constructed to solve the considered equations.The main feature of the proposed iterative algorithms is that the estimate for the unknown variable is updated by using not only the information in the last step but also the information in the current step.Due to this,a tuning parameter can be appropriately chosen to improve the convergence performance of the iterative algorithms.It is shown that under properly initial conditions the sequences generated by the presented algorithms monotonically converge to the unique positive definite solutions of the extended coupled Riccati matrix equations.Some numerical examples are given to show the superiority of the developed algorithms.Finally,the model reference tracking controllers are designed for It(?) nonhomogeneous Markovian jump stochastic systems under the background of the spacecraft trajectory tracking control problem.By considering the stochastic thruster faults of the chaser spacecraft,the spacecraft trajectory tracking control system is described by a class of It(?) nonhomogeneous Markovian jump stochastic systems.Further,the spacecraft tracking problem is converted to a model reference tracking control problem of It(?) Markovian jump stochastic systems.By using the second part of the preceding results,a state feedback controller is designed for the spacecraft hovering task via LMIs.By applying the third part of the preceding results,a linear quadratic optimal controller is given for the spacecraft flying around task via coupled Riccati matrix equations.These results constitute an approach of applying the theoretical results to practical engineering problems.