System Decoupling And Pole Assignment Problems And Indefinite Least Squares Problem

The following problems are studied in this thesis:1.The decoupling and pole assignment problems of the right invertiblesystem {C,A,B}.We deduce several rank relations related to the matrices of the system {C,A,B}to obtain the necessary and sufficient condition for the system being right invertible,and propose a new canonical decomposition of the right invertible system.From thisdecomposition,we study the Smith form of the matrix pencil P(s)=(?)to find out the finite zeros and infinite zeros of P(s),the range of the ranks of P(s) fors∈C,and the controllability of the right invertible system.We also apply this canonical decomposition to study decoupling and prescriptpole assignment problems of the right invertible system {C,A,B}.Some new resultsabout the triangular decoupling upon the row permutation,row by row decoupling andassociated pole assignment problems are deduced.2.The indefinite least squares (ILS)problem and the equality con-strained indefinite least squares (ILSE) problem.At first,perturbation analysis for indefinite least squares (ILS) problem and equal-ity constrained indefinite least squares (ILSE) problem are studied.By defining a newkind of weighted generalized inverse,the solutions and the perturbation bounds of thetwo problems are derived.Krylov subspace methods are considered currently to be among the most im-portant iterative techniques available for solving linear equations and standard leastsquares (LS) problems.In this thesis,three kinds of Krylov subspace methods areapplied to solve the ILS problem.The results are different than they worked on theLS problem.The lower bidiagonalization method which is very efficient for solving theLS problem is quite unstable for solving the ILS problem.Several numerical experi-ments are shown to compare the performances of the three algorithms and illustrateour results.The hyperbolic Householder elimination method is proposed to solve the ILSEproblem.The solution of ILSE is the limit of the solution of the corresponding uncon-strained weighted indefinite least squares problem (WILS).Based on this observation,we derive a type of elimination method by applying the hyperbolic QR method to aboveWILS problem and taking the limit analytically.Theoretical analysis show that the method obtained is forward stable under a reasonable assumption.We illustrate ourresults with numerical tests.3.The rank-constrained Hermitian nonnegative-definite least squaressolutions to the matrix equation AXA~H=B.We discuss rank-constrained Hermitian nonnegative-definite least squares solu-tions to the matrix equation AXA~H=B,in which conditions that B is Hermitian andnonnegative-definite and the matrix equation is consistent may not hold.We derivethe rank range and expression of these least squares solutions.