Robust stability under a class of nonlinear parametric perturbations

This paper considers the robust stability verification of linear time-invariant systems admitting a class of nonlinear parametric perturbations. The general setting is one of determining the closed-loop stability of systems whose open-loop transfer functions consist of powers, products, and ratios of polytopes of polynomials. Apart from this general setting, two special cases of independent interest are also considered. The first special case concerns uncertainties in the open-loop gain and real poles and zeros, while the second special case treats uncertainties in the open-loop gain and complex poles and zeros. In light of the zero exclusion principle, robust stability is equivalent to zero exclusion of the value sets of the system characteristic function (a value set consists of the values of the characteristic functions at a fixed frequency). The main results of this paper are as follows. 1) The value set of the characteristic function at each fixed frequency is determined by the edges and some frequency-dependent internal line segments. 2) Consequently, Hurwitz invariance verification simplifies to that of checking certain continuous scalar functions for avoidance of the negative real axis. 3) For the case of real zero-pole-gain variations, the critical lines are all frequency independent, and therefore, the determination of the robust stability is even simpler. 4) For the case of complex zero-pole-gain variations, the critical internal lines are shown to be either frequency independent or to be confined in certain (two-dimensional) planes or (three-dimensional) boxes