Abstract.
In this paper, we study the problem of robust H∞ controller design for uncertain continuous-time systems with variance and D-stability constraints. The parameter uncertainties are allowed to be unstructured but norm-bounded. The aim of this problem is the design of an output feedback controller such that, for all admissible uncertainties, the closed-loop poles be placed within a specified disk, the H∞ norm bound constraint on the disturbance rejection attenuation be guaranteed, and the steady-state variance for each state of the closed-loop system be no more than the prescribed individual upper bound, simultaneously. A parametric design method is exploited to solve the problem addressed. Sufficient conditions for the existence of the desired controllers are derived by using the generalized inverse theory. The analytical expression of the set of desired controllers is also presented. It is shown that the obtained results can be readily extended to the dynamic output feedback case and the discrete-time case.
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References
1. A. Ben-Israel and T. N. E. Greville, Generalized inverses: theory and applications. Wiley, New York (1974).
2. W. J. Chang and H. Y. Chung, A study of H∞ norm and variance-constrained design using dynamic output feedback for linear discrete systems. Int. J. Control 57 (1993), 473–484.
3. J. H. Chou, Pole-assignment robustness in a specified disk. Systems Control Lett. 16 (1991), 41–44.
4. E. G. Collins (Jr.) and R. E. Skelton, A theory of state covariance assignment for discrete systems. IEEE Trans. Automat. Control 32 (1987), 35–41.
5. K. Furuta and S. B. Kim, Pole assignment in a specified disk. IEEE Trans. Automat. Control 32 (1987), 423–427.
6. K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error bounds. Int. J. Control 39 (1984), 1115–1193.
7. K. M. Grigoriadis and R. E. Skelton, Minimum-energy covariance controllers. Automatica 33 (1997), 569–578.
8. W. M. Haddad and D. S. Bernstein, Controller design with regional pole constraints. IEEE Trans. Automat. Control 37 (1992), 54–69.
9. W. M. Haddad, D. S. Bernstein, and D. Mustafa, Mixed-norm H2/H∞ regulation and estimation: the discrete-time case. Systems Control Lett. 16 (1991), 235–248.
10. H. Y. Horng, J. H. Chou, and I. R. Horng, Robustness of eigenvalue clustering in various regions of the complex plane for perturbed systems. Int. J. Control 57 (1993), 1469–1483.
11. A. Hotz and R. E. Skelton, Covariance control theory. Int. J. Control 46 (1987), 13–32.
12. Y. T. Juang and K. H. Chen, Robust pole-assignment of linear dynamic systems. Control Theory Adv. Technology 5 (1989), 67–74.
13. R. E. Skelton and T. Iwasaki, Liapunov and covariance controllers. Int. J. Control 57 (1993), 519–536.
14. R. E. Skelton, T. Iwasaki, and K. M. Grigoriadis, A unified algebraic approach to linear control design. Taylor & Francis, Basingstoke (1997).
15. R. E. Skelton, J. H. Xu, and K. Yasuda, On the freedom in covariance control. Int. J. Control 59 (1994), 1567–1578.
16. S. C. Tsay, I. K. Fong, and T. S. Kuo, D-stability analysis for discrete optimal regulator. Control Theory Adv. Technology 6 (1990), 237–246.
17. E. Yaz, R. E. Skelton, and E. Grigoriadis, Robust regional pole assignment with output feedback. Proc. IEEE Conf. on Decision and Control (1993), 3009–3013.
18. R. K. Yedavalli and Y. Liu, H∞ control with regional stability constraints. Automatica 31 (1995), 611–615.
19. Z. Wang, Robust state estimation for perturbed systems with error variance and circular pole constraints: the discrete-time case. Int. J. Control 73 (2000), 303–311.
20. Z. Wang and K. J. Burnham, LMI approach to output feedback control for linear uncertain systems with D-stability constraints. J. Optim. Theory Appl. 113 (2002), No. 2, 357–372.
21. Z. Wang, X. Chen, and Z. Guo, Controller design for continuous systems with variance and circular pole constraints. Int. J. Systems Sci. 26 (1995), 1249–1256.
22. Z. Wang and B. Huang, Robust H2/H∞ filtering for linear systems with error variance constraints. IEEE Trans. Signal Process. 48 (2000), 2463–2467.
23. Z. Wang, B. Huang, and P. Huo, Sampled-data filtering with error covariance assignment, IEEE Trans. Signal Process. 49 (2001), 666–670.
24. Z. Wang and H. Unbehauen, Robust H∞ control for systems with time-varying parameter uncertainty and variance constraints. Cybern. Systems Int. J. 31 (2000), 175–191.
25. Y. Wang, L. Xie, and C. E. de Souza, Robust control of a class of uncertain nonlinear systems. Systems Control Lett. 19 (1992), 139–149.
26. J. C. Willems, Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Automat. Control 16 (1971), 621–634.
27. L. Xie, Output feedback H∞ control of systems with parameter uncertainty. Int. J. Control 63 (1996), 741–750.
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2000 Mathematics Subject Classification. 93E15, 93B36, 93B55.
This work was partially supported by the EPSRC under Grant GR/S27658/01, the Nuffield Foundation under Grant NAL/00630/G, and the Alexander von Humboldt Foundation of Germany.
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Wang, Z., Ho, D. Output Feedback Robust H∞ Control with D-Stability and Variance Constraints: Parametrization Approach. J Dyn Control Syst 11, 263–280 (2005). https://doi.org/10.1007/s10883-005-4174-x
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DOI: https://doi.org/10.1007/s10883-005-4174-x