Numerical Algorithms Of Several Class Of Sylvester Matrix Equations In Control Theory

Sylvester equations are the very important equations in control theory and many other fields of engineer.Owing to their important applications,Sylvester equations have attracted considerable attention from many researchers.So the fast and efficient methods for solving the Sylvester equations are hot issues in the study of mathematics.In this thesis,the detailed theoretical analysis and numerical algorithm have been studied for several kinds of matrix equations in control theory,then obtained some satisfactory results.The structure of this thesis is as follows:In the introduction,the sources and applications of several kinds of equations,such as Lyapunov equation?Sylvester equation and Riccati equation are introduced.The development and research status of Sylvester equation are discussed in brief.In Chapter 1,a modified conjugate gradient(MCG)algorithm for solving the minimum-norm Hamiltonian solution of the generalized Sylvester-conjugate equation AXB+CXD =E is studied.Firstly,we introduce the conjugate gradient(CG)algorithm for solving the linear equation and the MCG algorithm for solving the generalized Sylvester-conjugate matrix equation.Then,the algorithm is convergent within finite iterative steps in the absence of round-off error for any initial given Hamiltonian matrix in this section.Fur-thermore,the minimum-norm solution is derived by choosing a special kind of initial matrix,some numerical experiments are given to show that the method is effective.In Chapter 2,a conjugate gradient least squares(CGLS)algorithm is studied for solving the minimum-norm least squares solution of the generalized coupled Sylvester-conjugate matrix equations A1X+B1Y = D1XE1+F1,A2Y+B2X = D2YE2+F2.When the system is consistent,the exact solution is obtained.When the system is inconsistent,the least squares solution is obtained within finite iterative steps in the absence of round-off errors,the numerical experiments are given to show the effectiveness of the algorithm.In Chapter 3,a generalized conjugate direction(GCD)algorithm is considered for solving the generalized coupled Sylvester transpose matrix equations A1XB1+C1YTD1=E1,A2XTB2+C2YD2 = E2 over reflexive(anti-reflexive)matrices.It proves that this algorithm is convergent within finite iterative steps in the absence of round-off errors,a special kind of initial matrix is given to obtain the minimum-norm solution,numerical experiments show that the method is effective.In Chapter 4,on the basis of the HSS iterative method,we establish a general-ized parameterized Hermitian and skew-Hermitian splitting(GPHSS)iteration method for solving matrix equation AXB = C and give a general convergence criterion.By analyzing the correlation properties of spectral radius,the quasi-optimal values of the iteration parameters for the GPHSS method is also obtained.Numerical results validate the feasibility and the efficiency.