The Constraint Solution And Numerical Algorithm Of Nonlinear Matrix Equation In Control System

In solid mechanics, parameter identifcation and automatic control felds,many theory and practical problems can often be converted into studying relatedmatrix properties and related matrix equations. In control system, discussingstability and controllability can often be changed to studying the constraint so-lution of related nonlinear matrix equations. This thesis discusses the constraintsolution and numerical algorithm of Riccati matrix equations in control system.This thesis frstly shows numerical estimates of the solution for the contin-uous algebraic Riccati matrix equation (CARE); then obtains upper and lowermatrix bounds for algebraic Riccati matrix equations and coupled algebraic Ric-cati matrix equations; further, by applying the derived bounds, discusses theexistence uniqueness condition and fxed iterative algorithm of the solution forthe discrete (coupled) algebraic Riccati matrix equation (DARE).In chapter one, some background knowledge and recent works for Riccatimatrix equations, and some basic symbols with defnitions are introduced in thispaper.In chapter two, when the system matrix is (non) positive semi-defnite, thisthesis discusses upper and lower bounds on eigenvalue sum and trace for the so-lution of the CARE by using majorization inequalities and classical eigenvalueinequalities, through converting some problems into discussing related quadraticinequality, combining H¨lder (Cauchy-Schwarz) inequality and inequality tech-niques. Numerical examples illustrate the efectiveness.In chapter three, by constructing the equivalent form of the CARE, utiliz-ing eigenvalue and singular value inequalities of matrix’s sum with product, thisthesis proposes new lower and upper matrix bounds even series lower boundsfor the solution of this equation. Then, this thesis shows new lower and uppermatrix bounds for the solution of this equation using the properties of matrixSchur complement to construct the equivalent form of the DARE, in terms ofmatrix inequalities. Further, applying fxed point theorem and the derived ma-trix bounds, the fxed point iterative algorithm of the solution for the DARE ispresented. Finally, corresponding numerical examples are given to illustrate theefectiveness of our results.In chapter four, new upper and lower matrix bounds for the solution of thecontinuous coupled algebraic Riccati equation (CCARE) are established by using some classical matrix identities to construct the equivalent form of this equation,utilizing the properties of-matrix and nonnegative matrix to solve linear in-equalities. Then, this thesis proves that the derived upper bounds improve andextend some recent results. In addition, this thesis proposes iterative algorithmsto obtain tighter upper matrix bounds of the CCARE, and numerical examplesshow the efectiveness of the derived results. Finally, with the aid of the equivalentcondition of-matrix and the equivalent form of the discrete coupled algebraicRiccati equation (DCARE), new upper and lower matrix bounds for the solutionof this equation are developed.In chapter fve, this thesis proposes several existence uniqueness conditionsof the solution for the DCARE applying the relative properties of Frobeniusnorm and norm space, combining the characteristic of compact operator andfxed point theorem, using Cauchy-Schwarts inequality and the derived upperand lower bounds. Further, in the case of the precision permitted, several fxedpoint iterative algorithms of the solution for the DCARE are showed applyingthe characteristic of Cauchy sequence and triangle inequality, utilizing mathe-matical induction and the defnition of matrix sequence convergence. Finally,corresponding numerical examples are given to illustrate the efectiveness.