Indefinite Linear Quadratic Optimal Control For Discrete-Time Uncertain Systems

In engineering practice,since the linear quadratic optimal control model can well simulate the actual problems,and it is relatively easy to be analysed and applied,the researchers usually simplify an optimal control problem as a linear quadratic optimal control problem.In the actual situation,the control systems are often subject to a number of indeterminate factors.If we have enough sample data,we can determine the probability distribution of this kind of indeterminate factor through the method of probability and statistics,which produces the stochastic optimal control problem.However,in many cases,the problems that we have encountered have no ob-servation data or it is unable to get the observation data temporarily.In this case,we have to invite some domain experts to evaluate the belief degree that each event will happen.In order to rationally deal with personal belief degrees,uncertainty theory was founded by Professor Liu Baoding of Tsinghua University in 2007,which is based on normality,duality,subadditivity and product axioms.Based on the uncertainty theory,this thesis studies the indefinite linear quadratic optimal control problem of discrete-time uncertain systems with the expectation val-ue and optimistic value criterion,where the dynamical systems are subject to uncertain noise perturbation.The contents include that the systems are unconstrained,the systems with termi-nal state equality constraints and with process state inequality constraints under the expected value model,as well as indefinite linear quadratic optimal control model with optimistic value criterion.The main innovation of this thesis can be summarized as follows:1.A recurrence equation to solve the problem is first derived by using Bellman's principle of optimality.Then,the necessary conditions for the existence of the state feedback control of the indefinite linear quadratic optimal problem and the problem with cross terms under the expectation value criterion are derived by recurrence equation,and the well-posedness of the problems are discussed.2.The indefinite linear quadratic optimal control problems with terminal state equality constraints and with process state inequality constraints are proposed.We first transform the problems into an equivalent deterministic optimal control problem.Then,some necessary con-ditions for the state feedback optimal control are obtained by using the maximum principle and maximum principle with mixed inequality constraints,respectively.3.Several equivalent conditions are given by constructing deterministic linear quadratic form.In addition,a linear matrix inequality LMI condition is introduced.Then,a sufficient condition for the well-posedness of the indefinite linear quadratic optimal control with cross terms is obtained by the LMI condition.Furthermore,it is shown that the well-posedness of the LQ problem,the solvability of the LQ problem,the feasibility of LMI condition and the solvability of the constrained difference equation are equivalent to each other.4.With the aid of the Bellman's principle of optimality in dynamic programming,a re-currence equation of optimal control problem with optimistic value criterion for discrete-time uncertain systems is obtained.Then,a necessary condition for state feedback optimal control of an indefinite LQ optimal control problem is derived by using the recurrence equation.Further-more,it is shown that the well-posedness of the LQ problem,the solvability of the LQ problem,the feasibility of LMI condition and the solvability of the constrained difference equation are equivalent to each other.