\(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) fuzzy filters with memory for Takagi-Sugeno discrete-time systems. (English) Zbl 1423.93193
Summary: This paper is concerned with the problem of \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) fuzzy filter design for Takagi-Sugeno (T-S) discrete-time systems. The novelty of the proposed method, in the context of T-S fuzzy systems, is the inclusion of an arbitrary number of past information (states and measurements) in the structure of the filter, producing a T-S fuzzy filter with memory. Linear matrix inequality relaxations based on multi-polynomial fuzzy Lyapunov functions are proposed for the filter design, providing filters that attain lower bounds for the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) performance criteria, when compared to other approaches available in the fuzzy literature that employ memoryless filters. The advantages of the proposed approach are illustrated by numerical examples.
MSC:
| 93C42 | Fuzzy control/observation systems |
| 93C55 | Discrete-time control/observation systems |
| 93D20 | Asymptotic stability in control theory |
| 93E11 | Filtering in stochastic control theory |
Keywords:
fuzzy filtering; fuzzy system models; \(\mathcal{H}_\infty\) and \(\mathcal{H}_2\) performance; linear matrix inequalitiesReferences:
| [1] | Takagi, T.; Sugeno, M., Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern., SMC-15, 1, 116-132 (1985) · Zbl 0576.93021 |
| [2] | Tanaka, K.; Wang, H., Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach (2001), John Wiley & Sons: John Wiley & Sons New York, NY |
| [3] | Kalman, R. E., A new approach to linear filtering and prediction problems, J. Dyn. Syst. Meas. Control Trans. ASME, 82, 35-45 (1960) |
| [4] | Anderson, B. D.O.; Moore, J. B., Optimal Filtering (1979), Prentice-Hall: Prentice-Hall Englewood, NJ · Zbl 0688.93058 |
| [5] | El Sayed, A.; Grimble, M. J., A new approach to the \(H_\infty\) design of optimal digital linear filters, IMA J. Math. Control Inf., 6, 2, 233-251 (1989) · Zbl 0678.93063 |
| [6] | Grimble, M. J., \(H_\infty\) design of optimal linear filters, (Byrnes, C. I.; Martin, C. F.; Saeks, R. E., Linear Circuits, Systems and Signal Processing: Theory and Applications (1988), North-Holland: North-Holland Amsterdam, The Netherlands), 533-540 · Zbl 0697.93060 |
| [7] | Wang, H. O.; Tanaka, K.; Griffin, M. F., An approach to fuzzy control of nonlinear systems: stability and design issues, IEEE Trans. Fuzzy Syst., 4, 1, 14-23 (1996) |
| [8] | Tanaka, K.; Ikeda, T.; Wang, H. O., Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs, IEEE Trans. Fuzzy Syst., 6, 2, 250-265 (1998) |
| [9] | Kiriakidis, K.; Grivas, A.; Tzes, A., Quadratic stability analysis of the Takagi-Sugeno fuzzy model, Fuzzy Sets Syst., 98, 1, 1-14 (1998) · Zbl 0929.93024 |
| [10] | Tseng, C.-S., Robust fuzzy filter design for nonlinear systems with persistent bounded disturbances, IEEE Trans. Syst. Man Cybern., Part B, Cybern., 36, 4, 940-945 (2006) |
| [11] | Sala, A.; Ariño, C., Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya’s theorem, Fuzzy Sets Syst., 158, 24, 2671-2686 (2007) · Zbl 1133.93348 |
| [12] | Montagner, V. F.; Oliveira, R. C.L. F.; Peres, P. L.D., Convergent LMI relaxations for quadratic stabilizability and \(H_\infty\) control for Takagi-Sugeno fuzzy systems, IEEE Trans. Fuzzy Syst., 17, 4, 863-873 (2009) |
| [13] | Xie, X.-P.; Liu, Z.-W.; Zhu, X.-L., An efficient approach for reducing the conservatism of LMI-based stability conditions for continuous-time T-S fuzzy systems, Fuzzy Sets Syst., 263, 71-81 (2015) · Zbl 1361.93030 |
| [14] | Tanaka, K.; Hori, T.; Wang, H. O., A multiple Lyapunov function approach to stabilization of fuzzy control systems, IEEE Trans. Fuzzy Syst., 11, 4, 582-589 (2003) |
| [15] | Bernal, M.; Sala, A.; Jaadari, A.; Guerra, T. M., Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions, Fuzzy Sets Syst., 185, 1, 5-14 (2011) · Zbl 1238.93086 |
| [16] | Zhou, S.; Lam, J.; Xue, A., \(H_\infty\) filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach, Fuzzy Sets Syst., 158, 2, 180-193 (2007) · Zbl 1110.93034 |
| [17] | Zhang, J.; Xia, Y., New LMI approach to fuzzy \(H_\infty\) filter designs, IEEE Trans. Circuits Syst. II, Express Briefs, 56, 9, 739-743 (2009) |
| [18] | Ding, B., Homogeneous polynomially nonquadratic stabilization of discrete-time Takagi-Sugeno systems via nonparallel distributed compensation law, IEEE Trans. Fuzzy Syst., 18, 5, 994-1000 (2010) |
| [19] | Tognetti, E. S.; Oliveira, R. C.L. F.; Peres, P. L.D., Selective \(H_2\) and \(H_\infty\) stabilization of Takagi-Sugeno fuzzy systems, IEEE Trans. Fuzzy Syst., 19, 5, 890-900 (2011) |
| [20] | Zhang, H.; Xie, X.; Tong, S., Homogenous polynomially parameter-dependent \(H_\infty\) filter designs of discrete-time fuzzy systems, IEEE Trans. Syst. Man Cybern., Part B, Cybern., 41, 5, 1313-1322 (2011) |
| [21] | Choi, D. J.; Park, P., \(H_\infty\) state-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent Lyapunov functions, IEEE Trans. Fuzzy Syst., 11, 2, 271-278 (2003) |
| [22] | Kim, S. H.; Park, P., \(H_\infty\) state-feedback control design for fuzzy systems using Lyapunov functions with quadratic dependence on fuzzy weighting functions, IEEE Trans. Fuzzy Syst., 16, 6, 1655-1663 (2008) |
| [23] | Guerra, T. M.; Kruszewski, A.; Bernal, M., Control law proposition for the stabilization of discrete Takagi-Sugeno models, IEEE Trans. Fuzzy Syst., 17, 3, 724-731 (2009) |
| [24] | Tognetti, E. S.; Oliveira, R. C.L. F.; Peres, P. L.D., \(H_\infty\) and \(H_2\) nonquadratic stabilisation of discrete-time Takagi-Sugeno systems based on multi-instant fuzzy Lyapunov functions, Int. J. Syst. Sci., 46, 1, 76-87 (2015) · Zbl 1317.93163 |
| [25] | Lee, D. H.; Joo, Y. H., Extended robust \(H_2\) and \(H_\infty\) filter design for discrete time-invariant linear systems with polytopic uncertainty, Circuits Syst. Signal Process., 33, 2, 393-419 (2014) |
| [26] | Frezzatto, L.; de Oliveira, M. C.; Oliveira, R. C.L. F.; Peres, P. L.D., Robust \(H_\infty\) filter design with past output measurements for uncertain discrete-time systems, Automatica, 71, 151-158 (2016) · Zbl 1343.93085 |
| [27] | Oliveira, R. C.L. F.; Peres, P. L.D., Parameter-dependent LMIs in robust analysis: characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations, IEEE Trans. Autom. Control, 52, 7, 1334-1340 (2007) · Zbl 1366.93472 |
| [28] | Cao, Y.; Frank, P. M., Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach, IEEE Trans. Fuzzy Syst., 8, 2, 200-211 (2000) |
| [29] | Daafouz, J.; Bernussou, J., Poly-quadratic stability and \(H_\infty\) performance for discrete systems with time varying uncertainties, (Proceedings of the 40th IEEE Conference on Decision and Control. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, USA (2001)), 267-272 |
| [30] | Agulhari, C. M.; Oliveira, R. C.L. F.; Peres, P. L.D., Robust LMI parser: a computational package to construct LMI conditions for uncertain systems, (Proceedings of the XIX Brazilian Conference on Automation. Proceedings of the XIX Brazilian Conference on Automation, Campina Grande, PB, Brazil (2012)), 2298-2305 |
| [31] | Duan, Z. S.; Zhang, J. X.; Zhang, C. S.; Mosca, E., Robust \(H_2\) and \(H_\infty\) filtering for uncertain linear systems, Automatica, 42, 11, 1919-1926 (2006) · Zbl 1130.93422 |
| [32] | Lacerda, M. J.; Oliveira, R. C.L. F.; Peres, P. L.D., Robust \(H_2\) and \(H_\infty\) filter design for uncertain linear systems via LMIs and polynomial matrices, Signal Process., 91, 5, 1115-1122 (2011) · Zbl 1222.93127 |
| [33] | Frezzatto, L.; Lacerda, M. J.; Oliveira, R. C.L. F.; Peres, P. L.D., Robust \(H_2\) and \(H_\infty\) memory filter design for linear uncertain discrete-time delay systems, Signal Process., 117, 322-332 (2015) |
| [34] | Löfberg, J., YALMIP: a toolbox for modeling and optimization in Matlab, (Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design. Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design, Taipei, Taiwan (2004)), 284-289 |
| [35] | Andersen, E. D.; Andersen, K. D., The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm, (Frenk, H.; Roos, K.; Terlaky, T.; Zhang, S., High Performance Optimization. High Performance Optimization, Applied Optimization, vol. 33 (2000), Springer: Springer US), 197-232 · Zbl 1028.90022 |
| [36] | Ding, D.-W.; Yang, G.-H., Fuzzy filter design for nonlinear systems in finite-frequency domain, IEEE Trans. Fuzzy Syst., 18, 5, 935-945 (2010) |
| [37] | Li, W.; Xu, Y.; Li, H., Robust \(\ell_2 - \ell_\infty\) filtering for Takagi-Sugeno fuzzy systems with norm-bounded uncertainties, Discrete Dyn. Nat. Soc., 2013, 1-8 (2013) · Zbl 1264.93129 |
| [38] | Lee, J.-W., On uniform stabilization of discrete-time linear parameter-varying control systems, IEEE Trans. Autom. Control, 51, 10, 1714-1721 (2006) · Zbl 1366.93579 |
| [39] | Zhou, S. S.; Ren, W.; Li, J., Generalized \(H_2 / H_\infty\) filtering for discrete-time nonlinear system with T-S fuzzy models, (Proceedings of the 6th International Conference on Fuzzy Systems and Knowledge Discovery, Vol. 6. Proceedings of the 6th International Conference on Fuzzy Systems and Knowledge Discovery, Vol. 6, Tianjin, China (2009)), 530-534 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
