Robust Stability via Sign-Definite Decomposition

This paper considers the problem of robust stability of a polynomial family whose coefficients are polynomial functions of the parameters of interests. The problem occurs in the design of a fixed order or fixed structure multivariable feedback controller , parametrized by a real design parameter vector , for a plant , containing a vector of uncertain parameters. The characteristic polynomials of such systems often contain coefficients which depend polynomially on and . Using results on sign-definite decomposition, a new stability test is developed that gives a sufficient condition for Hurwitz stability of the family of closed loop systems that result when and vary over prescribed boxes. This test is reminiscent of Kharitonov´s Theorem, even though the family of polynomials considered here is certainly not restricted to be interval or even convex. Moreover the test does reduce to Kharitonov´s Theorem for the special case of interval polynomials. Using this criterion recursively and modularly, sets of controllers that stabilize the family of uncertain plants are determined.