# Article

 Title: $\ell^1$-optimal control for multirate systems under full state feedback  (English) Author: Aubrecht, Johannes Author: Voulgaris, Petros G. Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 35 Issue: 5 Year: 1999 Pages: [555]-586 Summary lang: English . Category: math . Summary: This paper considers the minimization of the $\ell ^\infty$-induced norm of the closed loop in linear multirate systems when full state information is available for feedback. A state-space approach is taken and concepts of viability theory and controlled invariance are utilized. The essential idea is to construct a set such that the state may be confined to that set and that such a confinement guarantees that the output satisfies the desired output norm conditions. Once such a set is computed, it is shown that a memoryless nonlinear controller results, which achieves near-optimal performance. The construction involves the solution of several finite linear programs and generalizes to the multirate case earlier work on linear time-invariant (LTI) systems. MSC: 93B36 MSC: 93B52 MSC: 93C35 idMR: MR1728468 . Date available: 2009-09-24T19:28:02Z Last updated: 2012-06-06 Stable URL: http://hdl.handle.net/10338.dmlcz/135308 . Reference: [1] Aubin J. P.: Viability Theory.Birkhäuser, Boston 1991 MR 1134779 Reference: [2] Aubin J. P., Cellina A.: Differential Inclusions.Springer–Verlag, New York 1984 Zbl 0538.34007, MR 0755330 Reference: [3] Dahleh M. A., Voulgaris P. G., Valavani L. S.: Optimal and robust controllers for periodic and multirate systems.IEEE Trans. Automat. Control AC–37 (1992), 1, 90–99 Zbl 0747.93028, MR 1139618 Reference: [4] Diaz–Bobillo I. J., Dahleh M. A.: State feedback $\ell ^1$-optimal controllers can be dynamic.Systems Control Lett. 19 (1992), 2, 245–252 MR 1178920 Reference: [5] Diaz–Bobillo I. J., Dahleh M. A.: Minimization of the maximum peak-topeak gain: the general multiblock problem.IEEE Trans. Automat. Control 38 (1993), 10, 1459–1482 MR 1242894 Reference: [6] Frankowska H., Quincampoix M.: Viability kernels of differential inclusions with constraints: Algorithm and applications.J. Math. Systems, Estimation, and Control 1 (1991), 3, 371–388 MR 1151310 Reference: [7] Meyer D. G.: A parametrization of stabilizing controllers for multirate sampled–data systems.IEEE Trans. Automat. Control 5 (1990), 2, 233–236 Zbl 0705.93031, MR 1038429 Reference: [8] Meyer D. G.: A new class of shift–varrying operators, their shift–invariant equivalents, and multirate digital systems.IEEE Trans. Automat. Control 35 (1990), 429–433 MR 1047995 Reference: [9] Meyer D. G.: Controller parametrization for time–varying multirate plants.IEEE Trans. Automat. Control 35 (1990), 11, 1259–1262 MR 1074895 Reference: [10] Quincampoix M.: An algorithm for invariance kernels of differential inclusions.In: Set–Valued Analysis and Differential Inclusions (A. B. Kurzhanski and V. M. Veliov, eds.). Birkhäuser, Boston 1993, pp. 171–183 Zbl 0794.49005, MR 1269813 Reference: [11] Quincampoix M., Saint–Pierre P.: An algorithm for viability kernels in Holderian case: Approximation by discrete dynamical systems.J. Math. Systems, Estimation, and Control 5 (1995), 1, 1–13 MR 1646282 Reference: [12] Shamma J. S.: Nonlinear state feedback for $\ell ^1$ optimal contro.Systems Control Lett. 21 (1993), 265–270 Zbl 0798.93030, MR 1241404 Reference: [13] Shamma J. S.: Optimization of the $\ell ^\infty$-induced norm under full state feedback.To appear. Summary in: Proceedings of the 33rd IEEE Conference on Decision and Control, 1994 Reference: [14] Shamma J. S., Tu K.–Y.: Set–valued observers and optimal disturbance rejection.To appear Zbl 0958.93013, MR 1669978 Reference: [15] Stoorvogel A. A.: Nonlinear ${\mathcal L}_1$ optimal controllers for linear systems.IEEE Trans. Automat. Control 40 (1995), 4, 694–696 MR 1324862 .

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