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$\ell^1$-optimal control for multirate systems under full state feedback
Kybernetika, Tome 35 (1999) no. 5, pp. 555-586 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 93B36, 93B52, 93C05, 93C35
Keywords: state-space approach; full state feedback; $\ell^1$ norm; multirate system; near-optimal performance; memoryless nonlinear controller; viability theory 
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Aubrecht, Johannes; Voulgaris, Petros G. $\ell^1$-optimal control for multirate systems under full state feedback. Kybernetika, Tome 35 (1999) no. 5, pp. 555-586. http://geodesic.mathdoc.fr/item/KYB_1999_35_5_a1/

[1] Aubin J. P.: Viability Theory. Birkhäuser, Boston 1991 | MR

[2] Aubin J. P., Cellina A.: Differential Inclusions. Springer–Verlag, New York 1984 | MR | Zbl

[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 | DOI | MR | Zbl

[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

[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 | DOI | MR

[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

[7] Meyer D. G.: A parametrization of stabilizing controllers for multirate sampled–data systems. IEEE Trans. Automat. Control 5 (1990), 2, 233–236 | DOI | MR | Zbl

[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 | DOI | MR

[9] Meyer D. G.: Controller parametrization for time–varying multirate plants. IEEE Trans. Automat. Control 35 (1990), 11, 1259–1262 | DOI | MR

[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 | MR | Zbl

[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

[12] Shamma J. S.: Nonlinear state feedback for $\ell ^1$ optimal contro. Systems Control Lett. 21 (1993), 265–270 | DOI | MR | Zbl

[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

[14] Shamma J. S., Tu K.–Y.: Set–valued observers and optimal disturbance rejection. To appear | MR | Zbl

[15] Stoorvogel A. A.: Nonlinear ${\mathcal L}_1$ optimal controllers for linear systems. IEEE Trans. Automat. Control 40 (1995), 4, 694–696 | DOI | MR