Design of multivariable feedback systems with infinite gain margin and decoupling (Q1064302)

The paper discusses the problem of designing a linear feedback compensator for linear time-invariant systems that attains the infinite gain margin (IGM) under uncertainties of feedback gains. Thus, consider a feedback system whose forward path consists of an \(m\times r\) matrix P(s) of rational functions and an \(r\times m\) compensator C(s) and whose feedback path consists of a diagonal matrix \(K=diag(k_ i)\) of constant gains. The problem is to find a C such that the closed-loop system is stable for all \(k_ i\geq 1\), \(1\leq i\leq m\). The main result of the paper is the following theorem: Let P be strictly proper with rank P\(=m\), and let \(P=ND^{-1}\) be a right coprime factorization. Suppose that the largest invariant factors, \(\psi_ 0\) and \(\psi_ N\) of D and N, respectively, are coprime. Under these conditions if \(\psi_ N\) is unimodular then there is a C that attains IGM. Conversely, if such a C exists and in addition PC is an invertible diagonal matrix, then \(\psi_ N\) is unimodular. An extension of the theorem to the case when the feedback loop consists of certain types of nonlinearities is also given.