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optimal design; geometric approach; linear systems; discrete- time systems
In this work, a feedforward dynamic controller is devised in order to achieve H2-optimal rejection of signals known with finite preview, in discrete-time systems. The feedforward approach requires plant stability and, more generally, robustness with respect to parameter uncertainties. On standard assumptions, those properties can be guaranteed by output dynamic feedback, while dynamic feedforward is specifically aimed at taking advantage of the available preview of the signals to be rejected, in compliance with a two- degree-of-freedom control structure. The geometric constraints which prevent achievement of perfect rejection are first discussed. Then, the procedure for the design of the feedforward dynamic compensator is presented. Since the approach proposed in this work is based on spectral factorization via Riccati equation of a real rational matrix function directly related to the original to-be-controlled system, the delays introduced to model the preview of the signals to be rejected do not affect the computational burden intrinsic in the solution of the appropriate algebraic Riccati equation. A numerical example helps to illustrate the geometric constraints and the procedure for the design of the feedforward dynamic unit.
[1] Basile G., Marro G.: Controlled and Conditioned Invariants in Linear System Theory. Prentice Hall, Englewood Cliffs, NJ 1992 MR 1149379 | Zbl 0758.93002
[2] Bittanti S., Laub A. J., (eds.) J. C. Willems: The Riccati Equation. Springer-Verlag, Berlin – Heidelberg 1991 MR 1132048 | Zbl 0734.34004
[3] Chen J., Ren Z., Hara, S., Qiu L.: Optimal tracking performance: Preview control and exponential signals. IEEE Trans. Automat. Control 46 (2001), 10, 1647–1653 MR 1858072 | Zbl 1045.93503
[4] Clements D. J.: Rational spectral factorization using state-space methods. Systems Control Lett. 20 (1993), 335–343 MR 1222397 | Zbl 0772.93002
[5] Colaneri P., Geromel J. C., Locatelli A.: Control Theory and Design: An $RH_2$ and $RH_\infty $ Viewpoint. Academic Press, London 1997
[6] Grimble M. J.: Polynomial matrix solution to the standard $H_2$-optimal control problem. Internat. J. Systems Sci. 22 (1991), 5, 793–806 MR 1102097
[7] Hoover D. N., Longchamp, R., Rosenthal J.: Two-degree-of-freedom $\ell _2$-optimal tracking with preview. Automatica 40 (2004), 1, 155–162 MR 2143984 | Zbl 1035.93026
[8] Hunt K. J., Šebek, M., Kučera V.: Polynomial solution of the standard multivariable $H_2$-optimal control problem. IEEE Trans. Automat. Control 39 (1994), 7, 1502–1507 MR 1283931
[9] Imai H., Shinozuka M., Yamaki T., Li, D., Kuwana M.: Disturbance decoupling by feedforward and preview control. ASME J. Dynamic Systems, Measurements and Control 105 (1983), 3, 11–17 Zbl 0512.93029
[10] Kojima A., Ishijima S.: LQ preview synthesis: Optimal control and worst case analysis. IEEE Trans. Automat. Control 44 (1999), 2, 352–357 MR 1668996 | Zbl 1056.93643
[11] Lancaster P., Rodman L.: Algebraic Riccati Equations. Oxford University Press, New York 1995 MR 1367089 | Zbl 0836.15005
[12] Marro G., Prattichizzo, D., Zattoni E.: A unified setting for decoupling with preview and fixed-lag smoothing in the geometric context. IEEE Trans. Automat. Control 51 (2006), 5, 809–813 MR 2232604
[13] Marro G., Zattoni E.: ${H}_2$-optimal rejection with preview in the continuous-time domain. Automatica 41 (2005), 5, 815–821 MR 2157712 | Zbl 1093.93008
[14] Marro G., Zattoni E.: Signal decoupling with preview in the geometric context: exact solution for nonminimum-phase systems. J. Optim. Theory Appl. 129 (2006), 1, 165–183 MR 2281050 | Zbl 1136.93013
[15] Moelja A. A., Meinsma G.: $H_2$ control of preview systems. Automatica 42 (2006), 6, 945–952 MR 2227597 | Zbl 1117.93327
[16] Vidyasagar M.: Control System Synthesis: A Factorization Approach. The MIT Press, Cambridge, MA 1985 MR 0787045 | Zbl 0655.93001
[17] Šebek M., Kwakernaak H., Henrion, D., Pejchová S.: Recent progress in polynomial methods and polynomial toolbox for Matlab version 2. 0. In: Proc. 37th IEEE Conference on Decision and Control, Tampa 1998
[18] Willems J. C.: Feedforward control, PID control laws, and almost invariant subspaces. Systems Control Lett. 1 (1982), 4, 277–282 MR 0670212 | Zbl 0473.93032
[19] Wonham W. M.: Linear Multivariable Control: A Geometric Approach. Third edition. Springer-Verlag, New York 1985 MR 0770574 | Zbl 0609.93001
[20] Yamada M., Funahashi, Y., Riadh Z.: Generalized optimal zero phase tracking controller design. Trans. ASME – J. Dynamic Systems, Measurement and Control 121 (1999), 2, 165–170
[21] Zattoni E.: Decoupling of measurable signals via self-bounded controlled invariant subspaces: Minimal unassignable dynamics of feedforward units for prestabilized systems. IEEE Trans. Automat. Control 52 (2007), 1, 140–143 MR 2286774
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