# zbMATH — the first resource for mathematics

Complete characterization of strictly positive real regions and robust strictly positive real synthesis method. (English) Zbl 0966.93042
The authors consider the robust synthesis problem for strictly positive real transfer functions. The concept of weak strictly positive real regions is introduced and its properties are discussed. A new effective algorithm for robust strictly positive real synthesis is proposed. Illustrative numerical examples are provided to show the effectiveness of this algorithm.

##### MSC:
 93B50 Synthesis problems 93D10 Popov-type stability of feedback systems 93D21 Adaptive or robust stabilization
Full Text:
##### References:
 [1] Kalman, R. E., Lyapunov functions for the problem of Lur’e in automatic control, Proc. Nat. Acad. Sci. (USA), 1963, 49: 201. · Zbl 0113.07701 [2] Popov, V. M., Hyperstability of Automatic Control Systems, New York: Springer-Verlag, 1973. · Zbl 0276.93033 [3] Desoer, C. A., Vidyasagar, M., Feedback Systems: Input-Output Properties, San Diego: Academic Press, 1975. · Zbl 0327.93009 [4] Anderson, B. D. O., Moore, J. B., Linear Optimal Control, New York: Prentice Hall, 1970. [5] Landau, Y. D., Adaptive Control: The Model Reference Approach, New York: Marcel Dekker, 1979. · Zbl 0475.93002 [6] Bhattacharyya, S. P., Chapellat, H., Keel, L. H., Robust Control: The Parametric Approach, New York: Prentice Hall, 1995. · Zbl 0838.93008 [7] Barmish, B. R., New tools for Robustness of Linear Systems, New York: MacMillan Publishing Company, 1994. · Zbl 1094.93517 [8] Bartlett, A. C., Hollot, C. V., Huang, L., Root locations for an entire polytope of polynomial: It suffices to check the edges, Math. Contr. Signals Syst., 1988, 1: 61. · Zbl 0652.93048 [9] Barmish, B. R., Kang, I. H., Extreme point results for robust stability of interval plants: Beyond first order compensators, Automatica, 1992, 28: 1169. · Zbl 0763.93066 [10] Rantzer, A., Stability conditions for polytopes of polynomials, IEEE Trans. Automat. Contr., 1992, AC-37: 79. · Zbl 0747.93064 [11] Dasgupta, S., Bhagwat, A. S., Conditions for designing strictly positive real transfer functions for adaptive output error identification, IEEE Trans. Circuits Syst., 1987, CAS-34: 731. [12] Chapellat, H., Dahleh, M., Bhattacharyya, S. P., On robust nonlinear stability of interval control systems, IEEE Trans. Automat. Contr., 1991, AC-36: 59. · Zbl 0722.93054 [13] Wang, L., Huang, L., Finite verification of strict positive realness of interval rational functions, Chinese Science Bulletin, 1991, 36: 262. [14] Hollot, C. V., Huang, L., Xu, Z. L., Designing strictly positive real transfer function families: A necessary and sufficient condition for low degree and structured families, Proc. of the International Symposium on Mathematical Theory of Networks and Systems (eds. Kaashoek, M. A., Van Schuppen, J. H., Ran, A. C. M.), Boston, Basel, Berlin: Birkhäuser, 1989, 215–227. · Zbl 0735.93057 [15] Patel, V. V., Datta, K. B., Classification of units inH and an alternative proof of Kharitonov’s theorem, IEEE Trans. Circuits Syst.: Part I, 1997, CAS-44: 454. · Zbl 0892.93031 [16] Anderson, B. D. O., Dasgupta, S., Khargonekar, P., et al., Robust strict positive realness: characterization and construction, IEEE Trans. Circuits Syst., 1990, CAS-37: 869. · Zbl 0713.93048 [17] Betser, A., Zeheb, E., Design of robust strictly positive real transfer functions, IEEE Trans. Circuits Syst. Part I, 1993, CAS-40: 573. · Zbl 0800.93377 [18] Yu, W., Huang, L., Necessary and sufficient conditions for robust SPR stabilization of low-order systems, Chinese Science Bulletin, 1998, 43: 2275. [19] Wang, L., On strict positive realness of multilinearly parametrized interval systems, Science in China, Ser. E, 1998, 41(5): 552. · Zbl 0917.93055 [20] Marquez, H. J., Agathoklis, P., On the existence of robust strictly positive real rational functions, IEEE Trans. Circuits Syst., Part I, 1998, CAS-45: 962. · Zbl 0980.93051 [21] Yu, W., Wang, L., Some remarks on the definition of strict positive realness of transfer functions, in Proc. of Chinese Conference of Decision and Control, (ed. Zhang, S. Y.), Shenyang: Northeastern University Press, 1999, 135–139. [22] Wang, L., Huang, L., Interval-polynomial stability theory and its applications in testing the strict positive realness of interval transfer functions, IMA Journal of Mathematical Control and Information, 1996, 13(1): 19. · Zbl 0854.93111 [23] Huang, L., Hollot, C. V., Xu, Z. L., Robust analysis of strictly positive real function set, Preprints of the Second Japan-China Joint Symposium on Systems Control Theory and Its Applications (ed. Kimura, H.), Osaka: Osaka University Press, 1990, 210–220. [24] Yu Wensheng, Robust strictly positive real synthesis and robust stability analysis, PhD Thesis (in Chinese), Peking University Beijing, 1998. [25] Wang, L., Huang, L., Robust stability of discrete systems and some results on the root distribution of polynomials, Chinese Science Bulletin, 1990, 35(24): 1859. [26] Wang, L., Unified approach to robust performance of a class of transfer functions with multilinearly correlated perturbations, Journal of Optimization Theory and Applications, 1998, 96(3): 709. · Zbl 0908.93022 [27] Yang, L., Hou, X., Zeng, Z., Complete discrimination system for polynomials, Science in China (in Chinese), Ser. E., 1996, 26: 424. · Zbl 0866.68104 [28] Yang, L., Zhang, J., Hou, X., Nonlinear Algebraic Equations and Machine Proving (in Chinese), Shanghai: Shanghai Science and Education Press, 1996. [29] Yang, L., Xia, B. C., Explicit criterion to determine the number of positive roots of a polynomial, MM-Preprints, 1997, 15: 134–145.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.