Abstract
The filtering problem is among the fundamental issues in control and signal processing. Several approaches such as H 2 optimal filtering and H ∞ optimal filtering have been developed to address this issue. While the optimal H 2 filtering problem has been extensively studied in the past for linear systems, to the best of our knowledge, it has not been studied for bilinear systems. This is indeed surprising, since bilinear systems are important class of nonlinear systems with well-established theories and applications in various fields. The problem of H 2 optimal filtering for both discrete-time and continuous bilinear systems is addressed in this paper. The filter design problem is formulated in the optimization framework. The problem for the discrete-time case is expressed in terms of linear matrix inequalities which can be efficiently solved. The results are used for the optimal filtering of a bilinear model of an electro-hydraulic drive.
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References
Anderson, B.D.O., Moore, J.B.: Optimal Filtering. Prentice Hall, Englewood Cliffs (1979)
Basar, T., Bernhard, P.: H ∞-Optimal Control and Related Minimax Design Problems: a Dynamic Game Approach. System and Control: Foundations and Applications. Birkhauser, Boston (1995)
Chen, C.-T.: Hybrid approach for dynamic model identification of an electro-hydraulic parallel platform. Nonlinear Dyn. 67, 695–711 (2012)
D’Alessandro, P., Isidori, A., Ruberti, A.: Realization and structure theory of bilinear dynamic systems. SIAM J. Control 12, 517–535 (1974)
Dorissen, H.T.: Canonical forms for bilinear systems. Syst. Control Lett. 13, 154–160 (1989)
Gao, H., Meng, X., Chen, T.: A new design of robust H 2 filters for uncertain systems. Syst. Control Lett. 57, 585–593 (2008)
Geromel, Jose C.: Optimal linear filtering under parameter uncertainty. IEEE Trans. Signal Process. 47, 168–175 (1999)
Juang, J.-N.: Continuous-time bilinear system identification. Nonlinear Dyn. 39, 79–94 (2005)
Palhares, R.M., Peres, P.L.D.: Optimal filtering schemes for linear discrete-time systems: a linear matrix inequality approach. Int. J. Inf. Syst. Sci. 29, 587–593 (1998)
Petersen, I.R., Mcfarlane, D.C.: Optimal guaranteed cost filtering for uncertain discrete-time linear systems. In: Proceedings of IFAC Symposium on Robust Control Design, Rio de Janeiro, Brazil, pp. 329–334 (1994)
Petersen, I.R., McFarlane, D.C.: Optimal guaranteed cost filtering for uncertain discrete-time linear systems. Int. J. Robust Nonlinear Control 6, 267–280 (1996)
Scheidl, R., Manhartsgruber, B.: On the dynamic behavior of servo-hydraulic drives. Nonlinear Dyn. 17, 247–268 (1998)
Schwartz, H., Ingenbleek, R.: Observing the state of hydraulic drives via bilinear approximated models. Control Eng. Pract. 2, 61–64 (1994)
Shaker, H.R., Tahavori, M.: Control reconfigurability of bilinear hydraulic drive systems. In: International Conference on Fluid Power and Mechatronics, pp. 477–480 (2011)
Svoronos, S., Stephanopoulos, G., Aris, R.: Bilinear approximation of general non-linear dynamic systems with linear inputs. Int. J. Control 31, 109–126 (1980)
Tuan, H.D., Apkarian, P., Nguyen, T.Q.: Robust and reduced-order filtering: new LMI-based characterizations and methods. IEEE Trans. Signal Process. 49, 2975–2984 (2001)
van de Wouw, N., Nijmeijer, H., van Campen, D.H.: A Volterra Series approach to the approximation of stochastic nonlinear dynamics. Nonlinear Dyn. 27, 397–409 (2002)
Zhang, L., Lam, J.: On H 2 model reduction of bilinear systems. Automatica 38, 205–216 (2002)
Zhang, L., Lam, J., Huang, B., Yang, G.H.: On gramians and balanced truncation of discrete-time bilinear systems. Int. J. Control 76, 414–427 (2003)
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Shaker, H.R. H 2 optimal filtering for bilinear systems. Nonlinear Dyn 70, 999–1005 (2012). https://doi.org/10.1007/s11071-012-0506-z
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DOI: https://doi.org/10.1007/s11071-012-0506-z