On Robust Hurwitz Polynomials

In this note, Kharitonov's theorem on robust Hunvitz poljmomials is simplified for low-order polynomials. Specifically, for n = 3, 4, and 5, the number of polynomials required to check robust stability is one, two, and three, respectively, instead of four. Furthermore, it is shown that for n > 6, the number of polynomials for robust stability checking is necessarily four, thus further simplification is not possible. The same simplifications arise in robust Schur polynomials by using the bilinear transformation. Applications of these simplifications to two-dimensional polynomials as well as to robustness for single parameters are indicated. Since Kharitonov's results were first published [I], several papers have been devoted to extending and interpreting this work. In particular, Bose [2] advanced a network theoretic proof to Kharitonov's theorem. Bialas and Garloff [3] have extended the theorem to robustness as a function of a single parameter. Bose and Zeheb [4] have extended the robusmess results to Schur polynomials. Bose, Jury and Zeheb have simplified the above results to determining only sufficient robustness conditions. A full discrete-time parallel of Kharitonov's results has not yet been obtained. However, some results in this direction were recently published by Hollot and Bartlen [6]. As further background results, it is pertinent to mention the works of Guiver and Bose 171, and Barmish [8]. The above indicates the importance of Kharitonov's theorem in its particular applications which undoubtedly will motivate further research in the future. The contents of this paper are devoted to Kharitonov's theorem where it is shown that for low-order polynomials some simplifications arise. Specifically, instead of having to test four polynomials for stability, in order to deduce robust stability, results show that for robust stability checking of polynomials of degree n = 3,4 ,5 , one requires the testing of only one, two, or three polynomials, respectively. Furthermore, for n 2 6, one normally requires the testing of all the four polynomials, as shown by Kharitonov, thus further simplification is not possible. We demonstrate the latter result by invoking the necessary and differing sufficiency conditions for stability of Lipatov and Sokolov [9]. By using the bilinear transformation as discussed by Bose and Zeheb Manuscript received July 1, 1986; revised April 13, 1987. Paper recommended by Associate Editor, R. Sivan. B. D. 0 . Anderson is with the D e p m e n t of Systems Engineering, Australian National University, Canberra, Australia. E. I. Jury is with the Department of Electrical and Computer Engineering, University of Miami, Coral Gables, FL 33124. M. Mansour is u,ith the Institute of Automatic Control. ETH Zurich, Swimrland. IEEE Log Number 8716154.