Design of High Gain Feedback System with Robust Stability

Abstract

This paper discusses the problem of designing high gain feedback systems with robust stability.
Concerned with disturbance attenuation and/or transient response, it is known that a high gain feedback control is of great worth. However, to make a high gain feedback scheme more reliable, it is important to consider the robust stability problem against plant perturbation.
In this paper we suppose that the feedback system has a nonlinear feedback gain k(y)/y≥1 as a high gain element, and has the SISO plant with uncertainty of the class of M(p₀, r) ={p:p=(1+l)p₀, |l(jω)|≤|r(jω)|, ∀ω∈R, l does not change the number of unstable poles}, where p₀ and l denote the nominal plant and perturbation, respectively, and are proper real rational functions.
We call a triple (p₀, c, r) is robustly positive real if pc/(1+pc) is stable and positive real for all plant in M(p₀, r). It is obvious, from passivity theorem, that the robust positive reality of (p₀, c, r) guarantees the stability of the feedback system for all nonlinear gain k(y)/y≥1 and for all perturbation in M(p₀, r). Hence we only deal with the problem of designing a compensator which achieves robust positive realness.
The main results of this paper are essentially stated as follows. A given triple (p₀, c, r) is robustly positive real if and only if the following conditions hold.
1) c stabilizes p₀, 2) |r(jω)||a₀(jω)|<1, ∀ω∈R_e and 3) |r(jω)||t₀(jω)|+|1-t₀(jω)|≤1, ∀ω∈R, where a₀:=p₀c/(1+p₀c), t₀:=2p₀c/(1+2p₀c).
As for the realizability of compensators, we show the following result. Suppose that the roots of |r(jω)|²-1=0 are finitely many and are with multiplicities ≤2. Then for a given pair (p₀, r) there exists a compensator that attains robust positive realness if and only if the followings are satisfied:
1) p₀ has neither finite zeros in the open right-half plane nor, multiple jω-axis zeros (including j^∞), 2) |r(jω)|{=0, for jω-axis zero of p₀, <1, for jω-axis pole of p₀, ≤1, elsewhere.
It should be noted that the modeling error must be less than or equal to 100% and, in particular, must be zero at the jω-axis zeros of p₀ for achieving the robust positive realness.