Robust Stabilization of Multivariable High Gain Feedback Systems

Abstract

In this paper we consider a design problem of multivariable high gain feedback systems with robust stability.
High gain feedback control has many advantages for system performances, while it was reported that plant perturbations often cause instability of high gain feedback systems. Hence we discuss a robust stabilization problem of high gain feedback systems.
The plant is assumed to belong to the multiplicative output perturbation class M(P₀, r)={P:P=(I+L)P₀, ||L(jω)||≤|r(jω)|, ^∀ω, where L doesn't change the number of unstable poles}. Here P₀ denotes the m×p nominal plant and r denotes, the bound of perturbations. The system contains high gains (not necessarily linear gains) in each feedback loop.
We say a system to be robust positive-real if PC(I+PC)^-1 remains stable and positive-real for all plant P belonging to M(P₀, r), where C denotes a compensator. Obviously RPR (Robust Positive-Realness) guarantees the stability of the system for all P belonging to M(P₀, r) and for all nonlinear gains (while D.C. gains≥1).
We assume rank P₀=m, ψ(N₀) and ψ(D₀) are coprime, where P₀=N₀D^-10 is a right coprime factorization over the ring of stable real rational matrices and ψ(Q) denotes the largest invariant factor of Q. Moreover we assume the roots of |r(jω)|²-1=0 are finitely many and their multiplicities≤2.
Under these conditions there exists a proper compensator which attains RPR and rank P₀C=m if and only if the following two conditions are satisfied:
1) ψ(N₀) has no finite zeros in the open R.H.P. and has no multiple jω-axis zeros (including j∞), and2)|r(jω)|{=0, for jω-axis zeros of ψ(N₀), <1, for jω-axis zeros of ψ(D₀), ≤1, elsewhere.
A numerical example is given in order to show that RPR copes with the perturbations which cause instability for LQ optimal control system.