Solutions To The Sylvester Matrix Equation And Its Application To The Switched Systems

The generalised Sylvester matrix equation ??EVFBWAV, has an wide application in linear systems. Many problem, such as eigenstructure assignment and observe design, can be solved usually by solving to this equation. When IE ?, the equation becomes ??VFBWAV, and it plays a vital role in many problems in conventional linear systems.The matrix equation ??CVFAV is the classical matrix equation studied in the usual matrix theory, and it also has a wide range of applications in the control theory.In this paper, two kinds of matrix equations are given. We transform the original equation into a set of equations, through which the constraint of matrix in the original equation becomes clear. And then we can be mainly focus on the matrix V to get the solution. In one way, the equation is transformed to a special linear Sylvester matrix equation ??CEVFAV. In this case, a matrix polynomial method for solving the matrix equation ??CVFAV of the nonsingular systems, which is the usual linear system, is given. An alternative approach makes the internal structure of the matrix more clear. By this way, we find that the computation can be greatly reduced. And it is pointed out that when the matrix F has a special structure, the equation can be solved more directly and effectively when F is an upper triangular matrix.Switched systems began to be people’s attention from the beginning of the last century 90’s. In recent years, many results have been obtained in the research on the stability of switched systems. In this paper, based on a classical stability criterion, the switched system is stabilized by state feedback. The original problem which is needed to consider the Lie algebra solvable problem, is transformed into a problem of solving the matrix equation through taking the idea of matrix similarity. The matrix equation problem is solved by the method given in this paper, and the state feedback stabilization of the switched system is realized.The example computation and Matlab simulation results show that the mothod to solve the matrix equation used in the paper is simple and easy, and is different from the original mothods.The application of this method to so lve the problem of feedback stabilization of switched systems is fast and effective, moreover, a great improvement on the original method was made.