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New bounds for robust stability of continuous and discrete-time systems under parametric uncertainty. (English) Zbl 0872.93065
The problem of robust stability of linear state-space models has been an active area of research for some time. For the cases of both structured and unstructured parameteric uncertainty involving state-space models, results exist for continuous as well as discrete-time systems. The uncertain parameters then describe the perturbation in either the open-loop system matrix or the closed-loop system matrix by entering the uncertainty matrix linearly.
When all matrices of a state-space model are perturbed and output feedback is applied, then the system matrix of the closed-loop system contains product-terms of the uncertain parameters. Sufficient conditions for stability have been derived under certain restrictions imposed on the uncertain parameters.
Here, a new approach based on the direct method of Liapunov is presented for systems where all the state-space matrices are perturbed by uncertain parameters. The proposed approach gives comparable results (or improved in some cases) to previous methods. Further, the results derived for the static output feedback case, can also be applied to the dynamic output feedback case by augmenting the system.
The theorems established by analysis for the discrete-time case, have previously been used successfully for synthesis studies. Synthesis results for the cases of structured perturbations in all system matrices based on $$H_{\infty}$$ techniques have also been published, where specific information on the uncertainty bounds is not provided. This paper, however, presents an analysis technique where simple ways are given to compute the stability bounds for the uncertain parameters.
The paper seems mathematically sound, and I did not discover any obvious errors.

##### MSC:
 93D09 Robust stability
Full Text:
##### References:
 [1] B. R. Barmish: New Tools for Robustness of Linear Systems. Macmillan Publishing Company, New York 1994. · Zbl 1094.93517 [2] D. S. Bernstein, W. M. Haddad: Robust stability and performance analysis for linear dynamic systems. IEEE Trans. Automat. Control 34 (1989), 7, 751-758. · Zbl 0689.93020 · doi:10.1109/9.29405 [3] C.E. de Souza M. Fu, L. Xie: $$H_\infty$$ analysis and synthesis of discrete-time systems with time-varying uncertainty. IEEE Trans. Automat. Control 38 (1993), 3, 459-462. · Zbl 0791.93064 · doi:10.1109/9.210145 [4] P. Dorato R. Tempo, G. Muscato: Bibliography on robust control. Automatica 25 (1993), 1, 201-213. · Zbl 0778.93017 · doi:10.1016/0005-1098(93)90183-T [5] M. Fu L. Xie, C. E. de Souza: $$H_\infty$$ control for linear systems with time-varying parameter uncertainty. Control of Uncertain Dynamic Systems (S. P. Bhattacharyya and L. IT Keel, CRC Press, Boca Raton, FL 1991, pp. 63-75. · Zbl 0840.93033 [6] Z. Gao, P. J. Antsaklis: Explicit asymmetric bounds for robust stability of continuous and discrete-time systems. IEEE Trans. Automat. Control 55 (1993), 2, 332-335. · Zbl 0774.93060 · doi:10.1109/9.250486 [7] X. Y. Gu, W. H. Chen: Robust stability analysis for state-space model of discrete-time systems. Proc. of 1991 Amer. Contr. Conf., Boston, MA 1991, pp. 890-891. [8] W. M. Haddad H.-H. Huang, and D. S. Bernstein: Robust stability and performance via fixed-order dynamic compensation: the discrete-time case. Proc. of 1992 Amer. Contr. Conf., Chicago, IL 1992, pp. 66-67. [9] L. H. Keel S. P. Bhattacharyya, J. W. Howze: Robust control with structured perturbations. IEEE Trans. Automat. Control 55 (1988), 1, 68-78. · Zbl 0652.93046 · doi:10.1109/9.362 [10] S. R. Kolla R. K. Yedavalli, J. B. Farison: Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems. Internat. J. Control 50 (1989), 1, 151-159. · Zbl 0684.93021 · doi:10.1080/00207178908953354 [11] I. K. Konstantopoulos, P. J. Antsaklis: Robust Stability of Linear Continuous and Discrete-time Systems under Parametric Uncertainty. Research Report No. ISIS-94-006, ISIS Group at the University of Notre Dame 1994. [12] I. K. Konstantopoulos, P. J. Antsaklis: Robust stabilization of linear continuous systems under parameter uncertainty in all state-space matrices. Proc. of 2nd IEEE Mediterranean Symp. on New Directions in Contr. and Automation, Chania, Crete, Greece 1994, pp. 490-497. [13] I. K. Konstantopoulos, P. J. Antsaklis: Design of Output Feedback Controllers for Robust Stability and Optimal Performance of Discrete-time Systems. Research Report No. ISIS-94-009, ISIS Group at the University of Notre Dame 1994. [14] H. A. Latchman, J. A. Letra: On the computation of allowable bounds for parametric uncertainty. Proc. of 1991 Amer. Contr. Conf., Boston, MA 1991, pp. 867-868. [15] J. Liu, M.A. Zhody: Performance constrainted stabilization problems of system with uncertainties and perturbations. Proc. of 1991 Amer. Contr. Conf., Boston, MA 1991, pp. 3142-3143. [16] X. Niu, J. A. De Abreu-Garcia: Some discrete-time counterparts to continuous-time stability bounds. Proc. of 1991 Amer. Contr. Conf., Boston, MA 1991, pp. 1947-1948. [17] S. Rem P. T. Kabamba, D. S. Bernstein: A guardian map approach to robust stability of linear systems with constant real parameter uncertainty. Proc. of 1992 Amer. Contr. Conf., Chicago, IL 1992, pp. 2649-2652. [18] D. D. Siljak: Parameter space methods for robust control design: a guided tour. IEEE Trans. Automat. Control 34 (1989), 7, 674-688. · Zbl 0687.93033 · doi:10.1109/9.29394 [19] J.-H. Su, I.-K. Fong: Robust stability analysis of linear continuous/discrete-time systems with output feedback controllers. IEEE Trans. Automat. Control 38 (1993), 7, 1154-1158. · Zbl 0788.93072 · doi:10.1109/9.231477 [20] E. Yaz: Deterministic and stochastic robustness measures for discrete systems. IEEE Trans. Automat. Control 33 (1988), 10, 952-955. · Zbl 0661.93028 · doi:10.1109/9.7253 [21] E. Yaz, X. Niu: New robustness bounds for discrete systems with random perturbations. IEEE Trans. Automat. Control 55 (1993), 12, 1866-1870. · Zbl 0792.93027 · doi:10.1109/9.250568 [22] R. K. Yedavali, Z. Liang: Reduced conservatism in stability robustness bounds by state transformation. IEEE Trans. Automat. Control 31 (1986), 9, 863-866. · Zbl 0593.93040 · doi:10.1109/TAC.1986.1104408 [23] R. K. Yedavali: Stability robustness measures under dependent uncertainty. Proc. of 1988 Amer. Contr. Conf., Atlanta, GA 1988, pp. 820-823. [24] K. M. Zhou, P. P. Khargonekar: Stability robustness bounds for linear state space models with structured uncertainty. IEEE Trans. Automat. Control 32 (1987), 7, 621-623. · Zbl 0616.93060 · doi:10.1109/TAC.1987.1104667 [25] K. Zhou P. P. Khargonekar J. Stoustrup, and H. H. Niemann: Robust stability and performance of uncertain systems in state space. Proc. 25th IEEE Conf. Decision Contrl., Tucson, AZ 1992, pp. 662-667.
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