The Numerical Solution And The Least Squares Solution Of The Discrete Lyapunov Matrix Equation

In the field of the automatic and power systems, we usually require solving thecorresponding Lyapunov matrix equation and Sylvester matrix equation or Riccati matrixequation. Therefore, it is very important for solving these matrix equations. But in many cases,it is difficult to obtain the exact solution of the matrix equations, so it can only use numericalmethods to obtain the approximate solution.In this paper, we establish three new numerical methods to solve the discrete Lyapunovmatrix equation, so that we obtain the numerical solution of the matrix equation and have acertain nature of the least squares solution. The main tasks are as follows:1) We have established a new numerical algorithm for solving the discrete Lyapunovmatrix equation. This new iterative algorithm is called about transform DLF algorithm. Thismethod transforms the coefficient matrix of the discrete Lyapunov matrix equation to theFrobenius form of a matrix by similar elementary transformation. And then establish anumerical method of solving the matrix equation, which the coefficient matrix is the Frobeniusform. Finally, through anti approximately obtained the approximate solution of the matrixequation. The advantage of this method os that the coefficient matrix is reduced to theFrobenius form after the operation, the amount of computation and storage are greatly reduced,improving the operation accuracy of the algorithm.2) Study the center self-adjoint and least squares solution of the discrete Lyapunov matrixequation X AXB C. And obtain the best approximation of the solution. Establish theappropriate algorithm, and conduct the theoretical analysis and numerical calculation. Similarly,consider the least squares solution of centrosymmetric solution and center symmetric solutionof the matrix equation.3) Finally, obtain a numerical method for solving large discrete Lyapunov matrix equationby the reduction method. Like Krylov subspace method did find a projection matrix modelreduction, resulting in a relatively small size of the control system. First, reduced high-orderSylvester matrix equation. Then use existing methods, such as low-level Schur method forsolving the Sylvester matrix equation. And finally obtain the numerical solution of the originallarge matrix equations.