Research On Brocket Problem

In feedback control, it is necessary to select the feedback-benefit matrices of the objects to control the systems in order to assign poles, and simultaneously to assure that the automatic control systems are asymptotically stable. Although increasing the benefit of feedback control systems can decrease the steady error, the systems may become unstable when the benefit leads greatly. It is important to study the scopes in which the stabilizing matrices can take on the value to stabilize the systems. Brocket Problem is that what the conditions are for a matrix to exist such that the system is asymptotically stable. The case that the stabilizing matrices are a constant matrix or a periodic matrix has been studied a lot. Boikov has given the method and constraints for obtaining the stabilizing matrices of time-dependent continuous linear systems. Making use of his method, one can obtain stabilizing matrices for non-periodic systems, but the constraints are strict and sometimes it is difficult to compute the stabilizing matrices. In this paper, new constraints are given, which improve Boikov's constraints. Under these constraints, stabilizing matrices exist such that the systems are asymptotically stable. Another new method, symmetrization, is also given in this paper. Its advantages are discussed. Furthermore, this paper studies the Brocket Problem for discrete systems, and gives two methods for constructing the stabilizing matrices of time-dependent and time-independent linear discrete systems. The advantages and disadvantages of these methods are analyzed.