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LMI tools for eventually periodic systems. (English) Zbl 1106.93327
Summary: This paper is focused on the concept of an eventually periodic linear discrete-time system. We derive a necessary and sufficient analysis condition for checking open-loop stability and performance of such systems, and use this to derive exact controller synthesis conditions given eventually periodic plants. All the conditions derived are provided in terms of semi-definite programming problems. The motivation for this work is controlling nonlinear systems along prespecified trajectories, notably those which eventually settle down into periodic orbits and those with uncertain initial states.

MSC:
93C55 Discrete-time control/observation systems
93B40 Computational methods in systems theory (MSC2010)
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[1] Ball, J.A.; Gohberg, I.; Kaashoek, M.A., Nevanlinna-Pick interpolation for time-varying input-output mapsthe discrete case, Oper. theory: adv. appl., 56, 1-51, (1992) · Zbl 0747.93052
[2] Bittanti, S.; Colaneri, P.; De Nicolao, G., The periodic Riccati equation, () · Zbl 0656.93067
[3] Boyd, S.; Elgaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory, (1994), SIAM Philidelphia, PA
[4] Didinsky, G.; Basar, T., Design of minimax controllers for linear systems with nonzero initial states under specified information structures, Internat. J. robust nonlinear control, 2, 1, 1-30, (1992) · Zbl 0756.93023
[5] Dullerud, G.E.; Lall, S.G., A new approach to analysis and synthesis of time-varying systems, IEEE trans. automat. control, 44, 1486-1497, (1999) · Zbl 1136.93321
[6] M. Farhood, Control of nonstationary LPV systems, M.Sc. Thesis, Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign. · Zbl 1369.93244
[7] Gahinet, P.; Apkarian, P., A linear matrix inequality approach to H∞ control, Internat. J. robust nonlinear control, 4, 421-448, (1991) · Zbl 0808.93024
[8] Green, M.; Limebeer, D.J.N., Linear robust control, (1995), Prentice-Hall Englewood Cliffs, NJ
[9] Halanay, A.; Ionescu, V., Time-varying discrete linear systems, (1994), Birkhäuser Basel · Zbl 0804.93036
[10] Iglesias, P.A., An entropy formula for time-varying discrete-time control systems, SIAM J. control optim., 34, 1691-1706, (1996) · Zbl 0931.93049
[11] Iwasaki, T.; Skelton, R.E., All controllers for the general H∞ control problemlmi existence conditions and state space formulas, Automatica, 30, 1307-1317, (1994) · Zbl 0806.93017
[12] Khargonekar, P.P.; Nagpal, K.M.; Poolla, K.R., H∞ control with transients, SIAM J. control optim., 29, 1373-1393, (1991) · Zbl 0738.93022
[13] Packard, A., Gain scheduling via linear fractional transformations, Systems control lett., 22, 79-92, (1994) · Zbl 0792.93043
[14] C.L. Pirie, G.E. Dullerud, Robust controller synthesis for uncertain time-varying systems, Proceedings of the American Control Conference, 2000. · Zbl 0998.93015
[15] Yacubovich, V.A., A frequency theorem for the case in which the state and control spaces are Hilbert spaces with an application to some problems of synthesis of optimal controls, Sibirskii mat. zh., 15, 639-668, (1975), (English translation in Siberian Mathematics Journal)
[16] Y. Zhou, Monotonicity and finite escape time of solutions of the discrete-time Riccati equation, Proceedings of the Fifth European Control Conference, 1999.
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