×

Root locations of an entire polytope of polynomials: It suffices to check the edges. (English) Zbl 0652.93048

Summary: The presence of uncertain parameters in a state space or frequency domain description of a linear, time-invariant system manifests itself as variability in the coefficients of the characteristic polynomial. If the family of all such polynomials is polytopic in coefficient space, we show that the root locations of the entire family can be completely determined by examining only the roots of the polynomials contained in the explosed edges of the polytope. These procedures are computationally tractable, and this criterion improves upon the presently available stability tests for uncertain systems, being less conservative and explicitly determining all root locations. Equally important is the fact that the results are also applicalbe to discrete-time systems.

MSC:

93D99 Stability of control systems
93C05 Linear systems in control theory
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
52Bxx Polytopes and polyhedra
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] [A1] J. Ackermann, Parameter space design of robust control systems,IEEE Trans. Automat. Control,25 (1980), 1058–1072. · Zbl 0483.93041
[2] J. Ackermann, Design of robust controllers by multi-model methods,Proceedings of the Seventh International Symposium on Mathematical Theory of Networks and Systems, Stockholm, 1985. · Zbl 0563.93019
[3] [B1] B. R. Barmish, Invariance of the strict Hurwitz property for polynomials with perturbed coefficients,IEEE Trans. Automat. Control,29 (1984), 935–936. · Zbl 0549.93046
[4] [BG] S. Bialas and J. Garloff, Convex combinations of stable polynomials,J. Franklin Inst.,319 (1985), 373–377. · Zbl 0562.30009
[5] [B2] N. K. Bose, A system-theoretic approach to stability of sets of polynomials,Contemp. Math.,47 (1985), 25–34.
[6] [C] P. L. Chebyshev,Complete Collected Works, Vol. 3. pp. 307–362, Izd. AN SSSR, Moscow, 1948. · Zbl 0041.48503
[7] [FM] A. T. Fam and J. S. Meditch, A canonical parameter space for linear system design,IEEE Trans. Automat. Control,23 (1978), 454–458. · Zbl 0377.93021
[8] [G1] F. R. Gantmacher,The Theory of Matrices, Chelsea, New York, 1960.
[9] [G2] B. K. Ghosh, Some new results on the simultaneous stabilizability of a family of single input, single output systems,Systems Control Lett.,6 (1985), 39–45. · Zbl 0565.93045
[10] [HB] C. V. Hollot and A. C. Bartlett, Some discrete-time counterparts to Kharitonov’s stability criterion for uncertain systems,IEEE Trans. Automat. Control,31 (1986), 355–356. · Zbl 0592.93050
[11] [HHB] L. Huang, C. V. Hollot, and A. C. Bartlett, Stability of families of polynomials: considerations in coefficient space,Internat. J. Control,45 (1987), 649–660. · Zbl 0618.93060
[12] [K] V. L. Kharitonov, Asymptotic stability of an equilibrium position of a family of systems of linear differential equations,Differentsial’nye Uravneniya,14 (1978), 2086–2088.
[13] [M] A. A. Markov,Collected Works, pp. 78–105, Nauka, Moscow, 1948.
[14] [N] Y. I. Naimark,Stability of Linearized Systems, Leningrad Aeronautical Engineering Academy, Leningrad, 1949.
[15] [R] R. T. Rockafellar,Convex Analysis, Princeton University Press, Princeton, 1972. · Zbl 0224.49003
[16] [S] D. D. Siljak,Nonlinear Systems, The Parameter Analysis and Design, Wiley, New York, 1969. · Zbl 0194.39302
[17] [SBD] C. B. Soh, C. S. Berger, and K. P. Dabke, On the stability of polynomials with perturbed coefficients,IEEE Trans. Automat. Control,30 (1985), 1033–1036. · Zbl 0565.93054
[18] K. H. Wei and B. R. Barmish, On making a polynomial Hurwitz invariant by choice of feedback gains,Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, FL, 1985. · Zbl 0686.93020
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.