On parametric Hurwitz stability margin of real polynomials
The paper deals with the problem of determining Hurwitz stability of a ball of polynomials defined by a weighted lp norm in the coefficient space where p is an arbitrary positive integer including infinity. The solution of the case when the weights are supposed to be the same for coefficient being above and below its nominal value corresponding to symmetric ball has been given by Tsypkin and Polyak. However, sometimes it seems to be useful to have a possibility to consider these weights as different, resulting in the asymmetric ball. This is, for example, the situation where the weights express our level of confidence that the real value of a coefficient lies in some interval. Such approach is used if the value of a coefficient is estimated by an expert. Solution of the problem is based on frequency domain plot in the complex plane and on applying the Zero Exclusion Theorem. The main idea consists in separation of the original problem into four subproblems and using an appropriate coordinate transformation which makes the value set independent of frequency. This transformation makes it possible to move the relative value set into the origin of the complex plane and to easily formulate the necessary and sufficient condition of Hurwitz stability of asymmetric ball of polynomials with prescribed radius or determine the maximum radius preserving stability. The whole graphical procedure consists of four plots instead of one, needed in the symmetric case.
Bibliogr. 12 poz.
- ANDERSON, B.D.O, JURY, E.I. and MANSOUR, M. (1987) On robust Hurwitz polynomials. IEEE Trans. Automat. Control 32, 909-913.
- BARMISH, B.R. (1994) New Tools for Robustness of Linear Systems. Macmillan Publishing Company, New York.
- BHATTACHARYYA, S.P., CHAPELLAT, H. and KEEL, L.H. (1995) Robust Control: The Parametric Approach. Prentice-Hall, Inc., New Jersey.
- BONDIA, J. and PICÓ, J. (2003a) Analysis of linear systems with fuzzy parametric uncertainty. Fuzzy sets and systems 135, 81-121.
- BONDIA, J. and PICÓ, J. (2003b) A geometric approach to robust performance of parametric uncertain systems. Int. Journal of Robust and Nonlinear Control 13, 1271-1283.
- DASGUPTA, S. (1988) Kharitonov’s theorem revisited. Systems and Control Letters 11, 381-384.
- DASGUPTA, S., PARKER, P.J., ANDERSON, B.D.O., KRAUS, F.J. and MANSOUR, M. (1991) Frequency domain conditions for the robust stability of linear and nonlinear systems. IEEE Trans. Circuit Systems 38, 389-397.
- KHARITONOV, V.L. (1978) Ob asymptoticheskoi ustoichivosti polozheniya ravnovesiya semeistva sistem lineynych differentsialnykh uravnenii. Differentsialnyje uravneniya 14, 2086-2088.
- KRAUS, F.J., ANDERSON, B.D.O. and MANSOUR, M. (1988) Robust Schur polynomial stability and Kharitonov’s theorem. International Journal of Control 47, 1213-1225.
- MANSOUR, M. (1994) On robust stability of linear systems. Systems and Control Letters 22, 137-143.
- MANSOUR, M., KRAUS, F.J. and ANDERSON, B.D.O. (1989) Strong Kharitonov theorem for discrete systems. In: M. Milanese, R. Tempo and A. Vicino, eds., Robustness in Identification and Control. Plenum Publishing, New York.
- TSYPKIN, Y.Z., and POLYAK, B.T. (1991) Frequency domain criteria for lp-robust stability of continuous linear systems. IEEE Trans. Automat. Control 36, 1464-1469.