The Solution Bounds And Iteration Algorithms Of The Algebraic Riccati Matrix Equation And Their Application

In automatic control , engineering calculation, solid mechanics, parameter identification and biological engineering fields, it relates to the design, control and optimization of the control system. In the design of the control system, stability,controllability and observability must be considered seriously in order to insure that the resulting products can meet all their previous performance index. Ex-ploring these features can often be converted to solve the related matrix equations.Especially, some important properties , such as the optimal control and the stabil-ity ,can often be converted to the solution and the solution bounds of the Riccati matrix equation in control system. This thesis discusses the solution bounds and the iteration algorithms of the Riccati matrix equations in control system. And according to the bounds, we give their specific applications in the redundant opti-mal control.In chapter one, we introduce some background knowledge and recent works for the Riccati matrix equations, some recent works about the redundant optimal control, and some basic symbols with definitions.In chapter two, by using some inequality techniques and majorization inequal-ities, we propose new upper bounds of the solution of the continuous algebraic Ric-cati equation (CARE), which improve some recent results. In redundant optimal control, by utilizing these upper bounds, some sufficient conditions are presented to strictly decrease feedback controller gain when we increase the columns of the input matrix. The superiority of our new results is demonstrated by some numer-ical examples.In chapter three, by using the properties of the inverse matrix of M-matrix,the eigenvalues inequality and the majorization inequalities, new upper bounds of the solution of the continuous coupled algebraic Riccati equation (CCARE) axe derived. when the continuous coupled algebraic Riccati equation (CCARE) de-grades into the continuous algebraic Riccati equation (CARE), the derived upper bounds of CARE also improve some recent results.In chapter four, by constructing the equivalent form of the discrete coupled algebraic Riccati equation (DCARE), applying some inequality techniques and the properties of symmetric matrices, we consider the coupled term as a whole and derive the solution bounds of the discrete coupled algebraic Riccati equation,which improve some recent results. Further, using the derived matrix bounds,the relative properties of Frobenius norm and a fixed point theorem, we discuss the existence uniqueness condition of solution of the DCARE. By the definition of matrix sequence convergence and the Cauchy sequences, we design the fixed point iteration algorithms for the solution of the DCARE. Finally, numerical examples are given to illustrate the obtained theory.In chapter five, when the restrictive conditions of the results are stronger than the fourth chapter, by constructing the equivalent form of the DCARE, utilizing the properties of nonnegative matrix, inequality techniques and the properties of the inverse matrix of M-matrix to solve matrix inequalities, we propose the solu-tion bounds of the DCARE, which is better than bounds of the fourth chapter.Then, using the derived matrix bounds, Cauchy-Schwarz inequality and a fixed point theorem, we discuss the existence uniqueness condition of solution of the DCARE. Further, we design the fixed point iteration algorithms for the solution of the DCARE. The superiority and the effectiveness of our new results are demon-strated by corresponding numerical examples.