On the robust stability of time-varying linear systems

This paper considers the robust stability of time-varying linear systems described by a linear differential equation whose coefficients vary inside given intervals and with restricted magnitudes of the rates of change of the coefficients. This problem can be considered as a generalization of the Kharitonov problem, which is in turn a generalization of the Hurwitz problem, and it was formulated as an open problem in [7]. To solve this problem Lyapunov theory is used where a Lyapunov function is obtained (using characteristics of positive real functions [2], [3]) which is multiaffine in the polynomial coefficients. With this Lyapunov function extreme point results are obtained. The structure of the Lyapunov matrix as well as the structure of the conditions for the solution of a robust positive real function problem are characterized. A second approach based on the critical stability conditions is also suggested but the Lyapunov matrix thus obtained is no longer in general multiaffine in the parameters. Examples of low order systems are given. The resulting stability conditions are only sufficient.