# Article

 Title: Continuous-time periodic systems in $H_2$ and $H_\infty$. Part I: Theoretical aspects (English) Author: Colaneri, Patrizio Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 36 Issue: 2 Year: 2000 Pages: [211]-242 Summary lang: English . Category: math . Summary: The paper is divided in two parts. In the first part a deep investigation is made on some system theoretical aspects of periodic systems and control, including the notions of $H_2$ and $H_\infty$ norms, the parametrization of stabilizing controllers, and the existence of periodic solutions to Riccati differential equations and/or inequalities. All these aspects are useful in the second part, where some parametrization and control problems in $H_2$ and $H_\infty$ are introduced and solved. (English) Keyword: periodic system Keyword: Riccati differential equation MSC: 34H05 MSC: 49N20 MSC: 93B36 MSC: 93C15 idZBL: Zbl 1249.93097 idMR: MR1760025 . Date available: 2009-09-24T19:32:18Z Last updated: 2015-03-26 Stable URL: http://hdl.handle.net/10338.dmlcz/135345 . Related article: http://dml.cz/handle/10338.dmlcz/135354 . Reference: [1] Bittanti S.: Deterministic and stochastic linear periodic systems.In: Time Series and Linear Systems (S. Bittanti, ed.), Springer Verlag, Berlin 1986, pp. 141–182 MR 0897824 Reference: [2] Bittanti S., Colaneri P., Nicolao G. De: The periodic Riccati equation.In: The Riccati equation (S. Bittanti, A. J. Laub, J. C. Willems, eds.), Springer Verlag, Berlin 1990, pp. 127–162 MR 1132054 Reference: [3] Bolzern P., Colaneri P.: The periodic Lyapunov equation.SIAM J. Matrix Analysis Appl. 4 (1998), 499–512 MR 0964664 Reference: [4] Doyle J. C., Glover K., Khargonekaar P. P., Francis B. A.: State–space solutions to standard $H_2$ and $H_{\infty }$ control problems.IEEE Trans. Automat. Control 34 (1989), 831–846 MR 1004301, 10.1109/9.29425 Reference: [5] Mita T., Zhi L. Kang, Ohushi S.: Correction of the FI result in $H_\infty$ control and parametrization of $H_\infty$ state feedback controllers.IEEE Trans. Automat. Control 38 (1993), 343–347 MR 1206827, 10.1109/9.250489 Reference: [6] Colaneri P., Geromel J. C., Locatelli A.: Control Systems Design – a $H_2$ and $H_\infty$ Viewpoint.Academic Press, New York 1997 Reference: [7] Bamieh B., Pearson J. B.: A general framework for linear periodic systems with application to $H_\infty$ sampled-data control.IEEE Trans. Automat. Control 37 (1992), 418–435 MR 1153103, 10.1109/9.126576 Reference: [8] Colaneri P.: Hamiltonian systems and periodic Riccati equations in $H_2$ and $H_{\infty }$ analysis and control of linear periodic systems.In: Proc. 30th Conference on Decision and Control, Brighton 1991, pp. 1914–1919 Reference: [9] Shayman M. A.: On the phase portrait of the matrix Riccati equation arising from the periodic control problem.SIAM J. Control Optim. 23 (1985), 27–32 Zbl 0578.93051, MR 0798057, 10.1137/0323045 Reference: [10] Xie L., Souza C. E. De, Fragoso M. D.: $H_{\infty }$ filtering for linear periodic systems with parameter uncertainty.Systems Control Lett. 17 (1991), 343–350 MR 1136535, 10.1016/0167-6911(91)90133-Y Reference: [11] Colaneri P., Souza C. De: The $H_\infty$ control problem for continuous-time linear periodic systems.In: Proc. of the $2^{\mathrm {nd}}$ IFAC Workshop on Systems Structure and Control, Prague 1992, pp. 292–295 Reference: [12] Limeeberg D. J., Anderson B. D. O., Khargonekar P. P., Green M.: A game theoretical approach to $H_\infty$ control of time-varying systems.SIAM J. Control Optim. 30 (1992), 262–283 MR 1149068, 10.1137/0330017 Reference: [13] Ravi R., Nagpal K. M., Khargonekaar P. P.: $H_\infty$ control of linear time-varying systems: a state-space approach.SIAM J. Control Optim. 29 (1991), 1394–1413 MR 1132188, 10.1137/0329071 Reference: [14] Chapellat H., Dahleh M., Bhattacharrya S. P.: Structure and optimality of multivariable periodic controllers.IEEE Trans. Automat. Control 38 (1993), 1300–1303 Zbl 0784.93052, MR 1235270, 10.1109/9.233174 Reference: [15] Feintuch A., Khargonekar P., Tannenbaum A.: On the minimization problem for linear time–varying periodic controller.SIAM J. Control Optim. (1986), 1076–1085 MR 0854072, 10.1137/0324064 Reference: [16] Basar T., Bernhard P.: $H_\infty$ Optimal Control and Related Minimax Design Problems.Birkhäuser, Basel 1991 MR 1096294 Reference: [17] Colaneri P.: Continuous–time periodic systems in $H_2$ and $H_{\infty }$ – Part II: State–feedback problems.Kybernetika 36 (2000), No. 3 (to appear) MR 1773508 .

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