タイムスケール分割による2レベル制御

抄録

This paper proposes a method for designing control systems based on the decomposition of time scale, and discusses its property by the singular perturbation theory. The system treated in the paper is a multivariable linear time-invariant system consisting of subsystems S1 and S2 such that S1: x=A₁₁x+A₁₂z+B₁u₁, S2: z=A₂₁x+A₂₂z+B₂u₂.The following procedure is proposed to regulate the state x. i) Find the controllers' gains K₁, K₂, so that S1 with z=K₁x, u₁=K₂x is stable and satisfies the pre-assiged specification, i.e., it satisfies x*=(A₁₁+A₁₂K₁+B₁K₂)x*=Ã₁x*, with Ã₁ stable, and also find K such that Re λ_i [B₂K]<0. ii) And then construct the feedback system by applying u₁=K₂x and u₂=(1/ε)K(z-K₁x), where ε>0. Such a design method is achieved if (A₁₁, [A₁₂ B₁]) is stabilizable and rank B₂=dim z. It is shown that the feedback system is stable and the trajectory x tends to x* as ε→0. As the second part of the paper, the case where the gains K₁, K₂, and K are determined by the solutions of LQR problems is considered. It is shown that such gains give a suboptimal solution to the LQR problem of the total system with an appropriate cost function. Based on the LQR theory, the robust stability is discussed. Finally, a numerical example is given for illustration.