Root neighborhoods, generalized lemniscates, and robust stability of dynamic systems

Abstract

A root neighborhood (or pseudozero set) of a degree-n polynomial p(z) is the set of all complex numbers that are the roots of polynomials whose coefficients differ from those of p(z), under a specified norm in $${\mathbb{C}^{n+1}}$$, by no more than a fixed amount $${\epsilon}$$. Root neighborhoods corresponding to commonly used norms are bounded by higher-order algebraic curves called generalized lemniscates. Although it may be neither convenient nor useful to derive their implicit equations, such curves are amenable to graphical analysis by means of simple contouring algorithms. Root neighborhood methods offer advantages over alternative approaches (the Kharitonov theorems and their generalizations) for investigating the robust stability of dynamic systems with uncertain parameters, since they offer valuable insight concerning which roots of the characteristic polynomial will become unstable first, and the relative importance of parameter variations on the root locations—and hence speed and damping of the system response. We derive a generalization of root neighborhoods to the case of polynomial coefficients having an affine linear dependence on a set of complex uncertainty parameters, bounded under a general weighted norm, and we discuss their applications to robust stability problems. The methods are illustrated by several computed examples.