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Simultaneous LQ control of a set of LTI systems using constrained generalized sampled-data hold functions. (English) Zbl 1111.93024

Summary: Sampled-data control of a set of continuous-time LTI systems is considered. It is assumed that a predefined guaranteed continuous-time quadratic cost function, which is, in fact, the sum of the performance indices for all systems, is given. The main objective here is to design a decentralized periodic output feedback controller with a prespecified form, e.g., polynomial, piecewise constant, exponential, etc., which minimizes the above mentioned guaranteed cost function. This problem is first formulated as a set of matrix inequalities, and then by using a well-known technique, it is reformulated as a LMI problem. The set of linear matrix inequalities obtained provides necessary and sufficient conditions for the existence of a decentralized optimal simultaneous stabilizing controller with the prespecified form (rather than a general form). Moreover, an algorithm is presented to solve the resultant LMI problem. Finally, the efficiency of the proposed method is demonstrated in two numerical examples.

MSC:

93B52 Feedback control
93D15 Stabilization of systems by feedback
93B05 Controllability
93C15 Control/observation systems governed by ordinary differential equations

References:

[1] Blondel, V.; Gevers, M., Simultaneous stabilizability of three linear systems is rationally undecidable, Mathematics of Control, Signals, and Systems, 6, 2, 135-145 (1993) · Zbl 0792.93109
[2] Cao, Y. Y.; Lam, J., A computational method for simultaneous LQ optimal control design via piecewise constant output feedback, IEEE Transactions on Systems, Man, and Cybernetics, 31, 5, 836-842 (2001)
[3] Cao, Y. Y.; Sun, Y. X.; Lam, J., Simultaneous stabilization via static output feedback and state feedback, IEEE Transactions on Automatic Control., 44, 6, 1277-1282 (1999) · Zbl 0955.93044
[4] Chammas, A. B.; Leondes, C. T., On the design of linear time-invariant systems by periodic output feedback: Part I, discrete-time pole placement, International Journal of Control, 27, 885-894 (1978) · Zbl 0388.93021
[5] Davison, E. J.; Chang, T. N., Decentralized stabilization and pole assignment for general proper systems, IEEE Transactions on Automatic Control, 35, 6, 652-664 (1990) · Zbl 0800.93448
[6] Feuer, A.; Goodwin, G. C., Generalized sample hold functions-frequency domain analysis of robustness, sensitivity, and intersample difficulties, IEEE Transactions on Automatic Control, 39, 5, 1042-1047 (1994) · Zbl 0814.93047
[7] Fonte, C.; Zasadzinski, M.; Bernier-Kazantsev, C.; Darouach, M., On the simultaneous stabilization of three or more plants, IEEE Transactions on Automatic Control, 46, 7, 1101-1107 (2001) · Zbl 1002.93049
[8] Howitt, G. D.; Luus, R., Control of a collection of linear systems by linear state feedback control, International Journal of Control, 58, 79-96 (1993) · Zbl 0777.93040
[9] Hyslop, G. L.; Schattler, H.; Tarn, T. J., Descent algorithms for optimal periodic output feedback control, IEEE Transactions on Automatic Control, 37, 12, 1893-1904 (1992) · Zbl 0783.93042
[10] Jia, Y.; Ackermann, J., Condition and algorithm for simultaneous stabilization of linear plants, Automatica, 37, 9, 1425-1434 (2001) · Zbl 0990.93098
[11] Kabamba, P. T., Control of linear systems using generalized sampled-data hold functions, IEEE Transactions on Automatic Control, 32, 9, 772-783 (1987) · Zbl 0627.93049
[12] Kabamba, P. T.; Yang, C., Simultaneous controller design for linear time-invariant systems, IEEE Transactions on Automatic Control, 36, 1, 106-111 (1991) · Zbl 0745.93046
[13] Lavaei, J., & Aghdam, A. G. (2006). High-performance simultaneous stabilizing periodic feedback control with a constrained structure. Proceedings of the 2006 American control conference (pp. 839-844). Minneapolis, USA.; Lavaei, J., & Aghdam, A. G. (2006). High-performance simultaneous stabilizing periodic feedback control with a constrained structure. Proceedings of the 2006 American control conference (pp. 839-844). Minneapolis, USA.
[14] Miller, D. E.; Rossi, M., Simultaneous stabilization with near optimal LQR performance, IEEE Transactions on Automatic Control, 46, 10, 1543-1555 (2001) · Zbl 1022.93040
[15] Tarn, T. J., & Yang, T., 1988. Simultaneous stabilization of infinite-dimensional systems with periodic output feedback. Linear circuit systems and signals processing: Theory and application (pp. 409-424).; Tarn, T. J., & Yang, T., 1988. Simultaneous stabilization of infinite-dimensional systems with periodic output feedback. Linear circuit systems and signals processing: Theory and application (pp. 409-424). · Zbl 0675.93056
[16] Toker, O., & Özbay, H. (1995). On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. Proceedings of the 1995 American control conference (pp. 2525-2526). Seattle, Washington.; Toker, O., & Özbay, H. (1995). On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. Proceedings of the 1995 American control conference (pp. 2525-2526). Seattle, Washington.
[17] Vidyasagar, M.; Viswanadham, N., Algebraic design techniques for reliable stabilization, IEEE Transactions on Automatic Control, 27, 5, 1085-1095 (1982) · Zbl 0496.93044
[18] Wang, S. H., Stabilization of decentralized control systems via time-varying controllers, IEEE Transactions on Automatic Control, 27, 3, 741-744 (1982) · Zbl 0478.93043
[19] Werner, H., 1996. An iterative algorithm for suboptimal periodic output feedback control. UKACC international conference on control (pp. 814-818).; Werner, H., 1996. An iterative algorithm for suboptimal periodic output feedback control. UKACC international conference on control (pp. 814-818).
[20] Youla, D.; Bongiorno, J.; Lu, C., Single-loop feedback stabilization of linear multivariable dynamical plants, Automatica, 10, 2, 159-173 (1974) · Zbl 0276.93036
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