Stability
Ah&met--In this paper we characterize the boundary af(B) of the, image f(B) of a box B in R" under a nonlinear mapping f :%"' + C. We generalize results recently reported bv Polvak and Koean (1993) 1Necessarv and Sufficient Conditions for Robust Stability' of Muhiaffine Systems. Mathematics Research Report 93-06, University of Maryland Baltimore County] for multiaffine mappings, and provide computationally tractable necessary and sufficient robust stability conditions for quasipolynomials with interval coefficients and interval delays. A numerical stability verification for a quasipolynomial family with two interval delays is presented. 1. Introduction Stability conditions for time-delay systems are of great importance for industrial applications. Delays often occur in the transmission of information or material between different parts of a system. Transportation systems, communication systems, chemical processing systems, metallurgical processing systems, environmental systems and power systems are examples of time-delay systems (see e.g. Malek-Zavarei and Jamshidi (1987) and Step&r (1989)). The mathematical formulation of a time-delay system results in a system of delay-differential equations. Any mathematical model of an engineering system possesses the unavoidable inaccuracy. The existence of the inaccuracies implies that the analysis of stability and performance as well as system design, based on a nominal model only, may not be meaningful in applications.
The stability analysis of a time-delay system is based on investigation of the root location region for the characteristic quasipolynomial. A fundamental result concerning stability of a quasipolynomial is found in Pontryagin (1955). A significant research effort has been devoted to robust stability criterion for quasipolynomial families. Fu et al. (1989) generalized the celebrated Edge Theorem to quasipolynomial families with 'constant' delays and coefficients depending affinely on parameters. Barmish and Shi (1989) investigated robust stability of quasipolynomial families with coefficients depending affinely on parameters and 'interval' delays. An application of a frequency domain technique reduces the original robust stability problem to a global minimization problem in a finite-dimensional Euclidean space. The latter problem, in general, is very difficult to solve. An efficient algorithm for handling the minimization problem remains to be an open problem. Tsypkin and Fu *