Geometrical Interpretation of Feasibility of the Lyapunov Inequalities and Controller Gain Minimization for a Class of Polytopic Systems

摘要

In this work an efficient geometric method is proposed to obtain feasible Lyapunov stability based linear matrix inequalities for a class of polytopic systems. The transformation of continuous models into a convex tensor product form is discussed first, and it is showed how its vertices influence the feasibility. It is shown that a lower variance of vertex tensor forms a tighter convex hull increasing the chance to solve the inequalities, and derive smaller controller gains. The evaluation of the variance of vertices, the nuclear norm of their three-way tensor is calculated. When the common Lyapunov function cannot be achieved, tighter geometric representation or parameter space division into sectors can ensure the existence of stabilizing controller gains, as its is interpreted as minimizing the nuclear norm of the vertex tensor. Furthermore, an engineering stabilization problem is also presented where numerical simulation case study confirmed the findings.