Indefinite Optimal Control Of Discrete-Time Singular Markov Jump Systems

Discrete-time singular Markov jump systems(SMJSs)are Markov jump systems(MJSs)with discrete-time singular subsystems,which have spread applications in aircraft model,power systems and dynamic Leontief model of multi-sector economy.Linear quadratic(LQ)optimal control is a kind of vital tool to solve the problems in numerous fields,such as finance and control theory,where the indefinite LQ problem refers to the optimal control problem in which the weight matrices of the quadratic cost function contain negative eigenvalues.Due to the non-causality,random switching and impulsiveness during switching in the structure of discrete-time SMJSs,investigating the indefinite LQ problem is more challenging.In this thesis,under the form of discrete-time SMJSs,the indefinite LQ problem for discrete-time square and rectangular SMJSs,discrete-time stochastic singular Markov jump systems(SSMJSs)with multiple state-dependent noises,as well as the robust indefinite LQ problem for discrete-time uncertain SMJSs are studied,respectively.The main academic contributions are summarized as follows.·The indefinite LQ optimal control problems for discrete-time SMJSs on finite and infinite horizon are addressed,where the weight matrices for state and control of quadratic cost function are both indefinite.Based on some variable rank,equality and inequality of matrices as well as the equivalent transformation of state-control pair,the indefinite LQ problem for SMJSs is equivalently transformed into standard LQ problem for MJSs,which guarantees the existence of unique semi-positive definite solution of the generalized stochastic algebraic Riccati equation.Meanwhile,the optimal control and non-negative optimal cost value are acquired,and the resulting optimal closed-loop system is regular,casual and stochastically stable.The practical of rotating masses can illustrate the theoretical results.This part was published in《Journal of the Franklin Institute》(358:8993-9022,2021).·The indefinite LQ optimal control problems for discrete-time rectangular SM JSs with additive noise on finite and infinite horizon are investigated.On finite horizon,the indefinite LQ problem for rectangular SMJSs is equivalently turned into another new indefinite LQ problem for MJSs.The quadratic cost function in the new kind of indefinite LQ problem contains the state,control and noise terms,and sufficient and necessary conditions are given for the solvability of the transformed indefinite LQ problem.Then,the corresponding sufficient and necessary conditions are derived to acquire the unique optimal control and non-negative optimal cost value.Besides,on infinite horizon,several sufficient and necessary conditions,in terms of rank for the coefficient matrices of rectangular SMJSs,are provided to ensure that the dynamic part of the optimal closed-loop system has a unique solution.The suficient and necessary condition of stochastic stability of the difference subsystem for the optimal closed-loop system is proposed.This part was submitted to(IEEE Transactions on Systems,Man,and Cybernetics:Systems》and has been modified.·The LQ optimal control problem with indefinite weight matrices for discretetime SSMJSs involving multiple state-dependent noises is solved.By using the Moore-Penrose generalized inverse of matrices and the equivalent transformation of restrict systems,the indefinite LQ problem for SSMJSs is equivalently converted into the indefinite LQ problem for MJSs.Under the assumption of stabilizability for MJSs,the condition is proposed for the solvability of the indefinite LQ problem,and the unique optimal control and the non-negative optimal cost value are acquired.Besides,the necessary and sufficient conditions in terms of strict linear matrix inequalities(LMIs)are established to ensure that SSMJSs are causal and mean-square stable.Moreover,an efficient iterative algorithm is proposed to verify the mean-square stabilizability of SSMJSs by solving LMI optimization problem.The dynamic Leontief model of a multi-sector economy is used to show the feasibility of the given algorithm.·The robust indefinite LQ problem for discrete-time uncertain SMJSs is studied.A new quadratic cost function,which combines penalty function and weight matrices,is introduced in order to transform the optimization problem constrained by uncertain system into an unconstrained optimization problem.First,the robust indefinite LQ problem for uncertain SMJSs is transformed into the robust LQ problem for uncertain MJSs with positive definite weight matrices.The transformed robust LQ problem is settled by the robust least-squares method,and the condition of existence of robust LQ optimal regulator and the analytic form of the optimal regulator are obtained.On infinite horizon,the optimal state feedback is given,which can guarantee the regularity,causality and stochastic stability of the corresponding optimal closed loop system and eliminate uncertain parameters.A practical example of DC motor is used to verify the validity of the conclusions in this chapter.