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Article: Stability analysis and control design for 2-D fuzzy systems via basis-dependent Lyapunov functions
| Title | Stability analysis and control design for 2-D fuzzy systems via basis-dependent Lyapunov functions |
|---|---|
| Authors | |
| Keywords | 2-D system Basis-dependent Lyapunov function Control design Fuzzy system Stability analysis |
| Issue Date | 2013 |
| Publisher | Springer New York LLC. The Journal's web site is located at http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0923-6082 |
| Citation | Multidimensional Systems And Signal Processing, 2013, v. 24 n. 3, p. 395-415 How to Cite? |
| Abstract | This paper investigates the problem of stability analysis and stabilization for two-dimensional (2-D) discrete fuzzy systems. The 2-D fuzzy system model is established based on the Fornasini-Marchesini local state-space model, and a control design procedure is proposed based on a relaxed approach in which basis-dependent Lyapunov functions are used. First, nonquadratic stability conditions are derived by means of linear matrix inequality (LMI) technique. Then, by introducing an additional instrumental matrix variable, the stabilization problem for 2-D fuzzy systems is addressed, with LMI conditions obtained for the existence of stabilizing controllers. Finally, the effectiveness and advantages of the proposed design methods based on basis-dependent Lyapunov functions are shown via two examples. © 2011 The Author(s). |
| Persistent Identifier | http://hdl.handle.net/10722/147107 |
| ISSN | 2023 Impact Factor: 1.7 2023 SCImago Journal Rankings: 0.499 |
| ISI Accession Number ID | |
| References | Boyd S., ElGhaoui L., Feron E., Balakrishnan V. (1994) Linear matrix inequalities in systems and control theory. SIAM, Philadelphia, PA doi: 10.1137/1.9781611970777 Chen C. W., Tsai J. S. H., Shieh L. S. (1999) Two-dimensional discrete-continuous model conversion. Circuits, Systems and Signal Processing 18: 565–585 doi: 10.1007/BF01269917 Choi D., Park P. (2003) H ∞ state-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent Lyapunov functions. IEEE Transaction on Fuzzy Systems 11: 271–278 doi: 10.1109/TFUZZ.2003.809903 de Oliveira M. C., Geromel J. C., Bernussou J. (2002) Extended H 2 and H ∞ norm characterizations and controller parametrizations for discrete-time systems. International Journal of Control 75: 666–679 doi: 10.1080/00207170210140212 Du C., Xie L. (1999) Stability analysis and stabilization of uncertain two-dimensional discrete systems: an LMI approach. IEEE Transactions on Circuits and Systems (I) 46(11): 1371–1374 doi: 10.1109/81.802835 Du C., Xie L., Soh Y. C. (2000) H ∞ filtering of 2-D discrete systems. IEEE Transactions on Signal Processing 48(6): 1760–1768 doi: 10.1109/78.845933 Du C., Xie L., Soh Y. C. (2001) H ∞ reduced-order approximation of 2-D digital filters. IEEE Transactions on Circuits and Systems (I) 48(6): 688–698 doi: 10.1109/81.928152 Fornasini E., Marchesini G. (1978) Doubly indexed dynamical systems: State-space models and structural properties. Mathematical Systems Theory 12: 59–72 doi: 10.1007/BF01776566 Gao H., Lam J., Xu S., Wang C. (2004) Stabilization and H ∞ control of two-dimensional Markovian jump systems. IMA Journal of Mathematics and Control Information 21(4): 377–392 doi: 10.1093/imamci/21.4.377 Gao H., Lam J., Xu S., Wang C. (2005) Stability and stabilization of uncertain 2-D discrete systems with stochastic perturbation. Multidimensional Systems and Signal Processing 16(1): 85–106 doi: 10.1007/s11045-004-4739-y Gao H., Zhao Y., Lam J., Chen K. (2009) H ∞ fuzzy filtering of nonlinear systems with intermittent measurements. IEEE Transactions on Fuzzy Systems 17(2): 291–300 doi: 10.1109/TFUZZ.2008.924206 Guerra T. M., Vermeiren L. (2004) LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form. Automatica 40(8): 823–829 doi: 10.1016/j.automatica.2003.12.014 Haddad M. W., Bernstein S. D. (1995) Parameter-dependent Lyapunov functions and the Popov criterion in robust analysis and synthesis. IEEE Transactions on Automatic Control 40(3): 536–543 doi: 10.1109/9.376077 Hinamoto T. (1997) Stability of 2-D discrete systems described by the Fornasini–Marchesini second model. IEEE Transactions on Circuits and Systems (I) 44(3): 254–257 doi: 10.1109/81.557373 Kim E., Kim S. (2002) Stability analysis and synthesis for an affine fuzzy control system via LMI and ILMI: Continuous case. IEEE Transactions on Fuzzy Systems 10(3): 391–400 doi: 10.1109/TFUZZ.2002.1006442 Kim E., Lee H. (2000) New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Transactions on Fuzzy Systems 8(5): 523–534 doi: 10.1109/91.873576 Lam J., Zhou S. (2007) Dynamic output feedback H ∞ control of discrete-time fuzzy systems: a fuzzy-basis-dependent Lyapunov function approach. International Journal of Society Systems Science 38(1): 25–37 doi: 10.1080/00207720601042967 Lin Z., Lam J., Galkowski K., Xu S. (2001) A constructive approach to stabilizability and stabilization of a class of nD systems. Multidimensional Systems and Signal Processing 12(3–4): 329–343 doi: 10.1023/A:1011909707499 Liu D. (1998) Lyapunov stability of two-dimensional digital filters with overflow nonlinearities. IEEE Transactions on Circuits and Systems (I) 45(5): 574–577 doi: 10.1109/81.668870 Liu H., Sun F., Hu Y.N. (2005) H ∞ control for fuzzy singularly perturbed systems. Fuzzy Sets and Systems 155: 272–291 doi: 10.1016/j.fss.2005.05.004 Liu X., Zhang Q.L. (2003) New approaches to controller designs based on fuzzy observers for T-S fuzzy systems via LMI. Automatica 39: 1571–1582 doi: 10.1016/S0005-1098(03)00172-9 Lu W. M., Doyle J. C. (1995) H ∞ control of nonlinear systems: A convex characterization. IEEE Transactions on Automatic Control 40(9): 1668–1675 doi: 10.1109/9.412643 Lu W.S. (1994) On a Lyapunov approach to stability analysis of 2-D digital filters. IEEE Transactions on Circuits and Systems (I) 41(10): 665–669 doi: 10.1109/81.329727 Tuan H. D., Apkarian P., Nguyen T. Q. (2002) Robust mixed H 2/H ∞ filtering of 2-D systems. IEEE Transactions on Signal Processing 50(7): 1759–1771 doi: 10.1109/TSP.2002.1011215 Wang H. O., Tanaka K., Griffin M. (1996) An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Transactions on Fuzzy Systems 4(1): 14–23 doi: 10.1109/91.481841 Wu L., Ho D. W. C. (2009) Fuzzy filter design for nonlinear Itô stochastic systems with application to sensor fault detection. IEEE Transactions on Fuzzy System 17: 233–242 doi: 10.1109/TFUZZ.2008.2010867 Wu L., Shi P., Gao H., Wang C. (2008) H ∞ filtering for 2D Markovian jump systems. Automatica 44(7): 1849–1858 doi: 10.1016/j.automatica.2007.10.027 Wu L., Su X., Shi P., Qiu J. (2011) A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B 41: 273–286 doi: 10.1109/TSMCB.2010.2051541 Wu L., Wang Z., Gao H., Wang C. (2007) H ∞ and l 2-l ∞ filtering for two-dimensional linear parameter-varying systems. International Journal of Robust & Nonlinear Control 17(12): 1129–1154 doi: 10.1002/rnc.1169 Xie L., Du C., Soh Y. C., Zhang C. (2002) H ∞ and robust control of 2-D systems in FM second model. Multidimensional Systems and Signal Processing 13: 256–287 doi: 10.1023/A:1015808429836 Xu H., Zou Y., Xu S., Lam J., Wang Q. (2005) H ∞ model reduction of 2-D singular Roesser models. Multidimensional Systems and Signal Processing 16(3): 285–304 doi: 10.1007/s11045-005-1678-1 Yoneyama J. (2006) Robust H ∞ control analysis and synthesis for Takagi-Sugeno general uncertain fuzzy systems. Fuzzy Sets and Systems 57(16): 2205–2223 doi: 10.1016/j.fss.2006.03.020 Zhou S., Lam J., Xue A. (2007) H ∞ filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach. Fuzzy Sets and Systems 158(2): 180–193 doi: 10.1016/j.fss.2006.09.001 Zhou S., Li T. (2005) Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent Lyapunov-Krasovskii function. Fuzzy Sets and Systems 151(1): 139–153 doi: 10.1016/j.fss.2004.08.014 |
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.author | Chen, X | en_HK |
| dc.contributor.author | Lam, J | en_HK |
| dc.contributor.author | Gao, H | en_HK |
| dc.contributor.author | Zhou, S | en_HK |
| dc.date.accessioned | 2012-05-28T08:17:26Z | - |
| dc.date.available | 2012-05-28T08:17:26Z | - |
| dc.date.issued | 2013 | en_HK |
| dc.identifier.citation | Multidimensional Systems And Signal Processing, 2013, v. 24 n. 3, p. 395-415 | en_HK |
| dc.identifier.issn | 0923-6082 | en_HK |
| dc.identifier.uri | http://hdl.handle.net/10722/147107 | - |
| dc.description.abstract | This paper investigates the problem of stability analysis and stabilization for two-dimensional (2-D) discrete fuzzy systems. The 2-D fuzzy system model is established based on the Fornasini-Marchesini local state-space model, and a control design procedure is proposed based on a relaxed approach in which basis-dependent Lyapunov functions are used. First, nonquadratic stability conditions are derived by means of linear matrix inequality (LMI) technique. Then, by introducing an additional instrumental matrix variable, the stabilization problem for 2-D fuzzy systems is addressed, with LMI conditions obtained for the existence of stabilizing controllers. Finally, the effectiveness and advantages of the proposed design methods based on basis-dependent Lyapunov functions are shown via two examples. © 2011 The Author(s). | en_HK |
| dc.language | eng | en_US |
| dc.publisher | Springer New York LLC. The Journal's web site is located at http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0923-6082 | en_HK |
| dc.relation.ispartof | Multidimensional Systems and Signal Processing | en_HK |
| dc.rights | The Author(s) | en_US |
| dc.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | en_US |
| dc.subject | 2-D system | en_HK |
| dc.subject | Basis-dependent Lyapunov function | en_HK |
| dc.subject | Control design | en_HK |
| dc.subject | Fuzzy system | en_HK |
| dc.subject | Stability analysis | en_HK |
| dc.title | Stability analysis and control design for 2-D fuzzy systems via basis-dependent Lyapunov functions | en_HK |
| dc.type | Article | en_HK |
| dc.identifier.openurl | http://www.springerlink.com/link-out/?id=2104&code=M835H4U8U720M51H&MUD=MP | en_US |
| dc.identifier.email | Lam, J:james.lam@hku.hk | en_HK |
| dc.identifier.authority | Lam, J=rp00133 | en_HK |
| dc.description.nature | published_or_final_version | en_US |
| dc.identifier.doi | 10.1007/s11045-011-0166-z | en_HK |
| dc.identifier.scopus | eid_2-s2.0-84878220113 | en_HK |
| dc.identifier.hkuros | 236063 | - |
| dc.relation.references | Boyd S., ElGhaoui L., Feron E., Balakrishnan V. (1994) Linear matrix inequalities in systems and control theory. SIAM, Philadelphia, PA | en_US |
| dc.relation.references | doi: 10.1137/1.9781611970777 | en_US |
| dc.relation.references | Chen C. W., Tsai J. S. H., Shieh L. S. (1999) Two-dimensional discrete-continuous model conversion. Circuits, Systems and Signal Processing 18: 565–585 | en_US |
| dc.relation.references | doi: 10.1007/BF01269917 | en_US |
| dc.relation.references | Choi D., Park P. (2003) H ∞ state-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent Lyapunov functions. IEEE Transaction on Fuzzy Systems 11: 271–278 | en_US |
| dc.relation.references | doi: 10.1109/TFUZZ.2003.809903 | en_US |
| dc.relation.references | de Oliveira M. C., Geromel J. C., Bernussou J. (2002) Extended H 2 and H ∞ norm characterizations and controller parametrizations for discrete-time systems. International Journal of Control 75: 666–679 | en_US |
| dc.relation.references | doi: 10.1080/00207170210140212 | en_US |
| dc.relation.references | Du C., Xie L. (1999) Stability analysis and stabilization of uncertain two-dimensional discrete systems: an LMI approach. IEEE Transactions on Circuits and Systems (I) 46(11): 1371–1374 | en_US |
| dc.relation.references | doi: 10.1109/81.802835 | en_US |
| dc.relation.references | Du C., Xie L., Soh Y. C. (2000) H ∞ filtering of 2-D discrete systems. IEEE Transactions on Signal Processing 48(6): 1760–1768 | en_US |
| dc.relation.references | doi: 10.1109/78.845933 | en_US |
| dc.relation.references | Du C., Xie L., Soh Y. C. (2001) H ∞ reduced-order approximation of 2-D digital filters. IEEE Transactions on Circuits and Systems (I) 48(6): 688–698 | en_US |
| dc.relation.references | doi: 10.1109/81.928152 | en_US |
| dc.relation.references | Fornasini E., Marchesini G. (1978) Doubly indexed dynamical systems: State-space models and structural properties. Mathematical Systems Theory 12: 59–72 | en_US |
| dc.relation.references | doi: 10.1007/BF01776566 | en_US |
| dc.relation.references | Gao H., Lam J., Xu S., Wang C. (2004) Stabilization and H ∞ control of two-dimensional Markovian jump systems. IMA Journal of Mathematics and Control Information 21(4): 377–392 | en_US |
| dc.relation.references | doi: 10.1093/imamci/21.4.377 | en_US |
| dc.relation.references | Gao H., Lam J., Xu S., Wang C. (2005) Stability and stabilization of uncertain 2-D discrete systems with stochastic perturbation. Multidimensional Systems and Signal Processing 16(1): 85–106 | en_US |
| dc.relation.references | doi: 10.1007/s11045-004-4739-y | en_US |
| dc.relation.references | Gao H., Zhao Y., Lam J., Chen K. (2009) H ∞ fuzzy filtering of nonlinear systems with intermittent measurements. IEEE Transactions on Fuzzy Systems 17(2): 291–300 | en_US |
| dc.relation.references | doi: 10.1109/TFUZZ.2008.924206 | en_US |
| dc.relation.references | Guerra T. M., Vermeiren L. (2004) LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form. Automatica 40(8): 823–829 | en_US |
| dc.relation.references | doi: 10.1016/j.automatica.2003.12.014 | en_US |
| dc.relation.references | Haddad M. W., Bernstein S. D. (1995) Parameter-dependent Lyapunov functions and the Popov criterion in robust analysis and synthesis. IEEE Transactions on Automatic Control 40(3): 536–543 | en_US |
| dc.relation.references | doi: 10.1109/9.376077 | en_US |
| dc.relation.references | Hinamoto T. (1997) Stability of 2-D discrete systems described by the Fornasini–Marchesini second model. IEEE Transactions on Circuits and Systems (I) 44(3): 254–257 | en_US |
| dc.relation.references | doi: 10.1109/81.557373 | en_US |
| dc.relation.references | Kaczorek T. (1985) Two-dimensional linear systems. Springer, Berlin, Germany | en_US |
| dc.relation.references | Kim E., Kim S. (2002) Stability analysis and synthesis for an affine fuzzy control system via LMI and ILMI: Continuous case. IEEE Transactions on Fuzzy Systems 10(3): 391–400 | en_US |
| dc.relation.references | doi: 10.1109/TFUZZ.2002.1006442 | en_US |
| dc.relation.references | Kim E., Lee H. (2000) New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Transactions on Fuzzy Systems 8(5): 523–534 | en_US |
| dc.relation.references | doi: 10.1109/91.873576 | en_US |
| dc.relation.references | Lam J., Zhou S. (2007) Dynamic output feedback H ∞ control of discrete-time fuzzy systems: a fuzzy-basis-dependent Lyapunov function approach. International Journal of Society Systems Science 38(1): 25–37 | en_US |
| dc.relation.references | doi: 10.1080/00207720601042967 | en_US |
| dc.relation.references | Lin Z., Lam J., Galkowski K., Xu S. (2001) A constructive approach to stabilizability and stabilization of a class of nD systems. Multidimensional Systems and Signal Processing 12(3–4): 329–343 | en_US |
| dc.relation.references | doi: 10.1023/A:1011909707499 | en_US |
| dc.relation.references | Liu D. (1998) Lyapunov stability of two-dimensional digital filters with overflow nonlinearities. IEEE Transactions on Circuits and Systems (I) 45(5): 574–577 | en_US |
| dc.relation.references | doi: 10.1109/81.668870 | en_US |
| dc.relation.references | Liu H., Sun F., Hu Y.N. (2005) H ∞ control for fuzzy singularly perturbed systems. Fuzzy Sets and Systems 155: 272–291 | en_US |
| dc.relation.references | doi: 10.1016/j.fss.2005.05.004 | en_US |
| dc.relation.references | Liu X., Zhang Q.L. (2003) New approaches to controller designs based on fuzzy observers for T-S fuzzy systems via LMI. Automatica 39: 1571–1582 | en_US |
| dc.relation.references | doi: 10.1016/S0005-1098(03)00172-9 | en_US |
| dc.relation.references | Lu W. M., Doyle J. C. (1995) H ∞ control of nonlinear systems: A convex characterization. IEEE Transactions on Automatic Control 40(9): 1668–1675 | en_US |
| dc.relation.references | doi: 10.1109/9.412643 | en_US |
| dc.relation.references | Lu W.S. (1994) On a Lyapunov approach to stability analysis of 2-D digital filters. IEEE Transactions on Circuits and Systems (I) 41(10): 665–669 | en_US |
| dc.relation.references | doi: 10.1109/81.329727 | en_US |
| dc.relation.references | Lu W.S., Antoniou A. (1992) Two-dimensional digital filters. Marcel Dekker, New York | en_US |
| dc.relation.references | Tuan H. D., Apkarian P., Nguyen T. Q. (2002) Robust mixed H 2/H ∞ filtering of 2-D systems. IEEE Transactions on Signal Processing 50(7): 1759–1771 | en_US |
| dc.relation.references | doi: 10.1109/TSP.2002.1011215 | en_US |
| dc.relation.references | Wang H. O., Tanaka K., Griffin M. (1996) An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Transactions on Fuzzy Systems 4(1): 14–23 | en_US |
| dc.relation.references | doi: 10.1109/91.481841 | en_US |
| dc.relation.references | Wu L., Ho D. W. C. (2009) Fuzzy filter design for nonlinear Itô stochastic systems with application to sensor fault detection. IEEE Transactions on Fuzzy System 17: 233–242 | en_US |
| dc.relation.references | doi: 10.1109/TFUZZ.2008.2010867 | en_US |
| dc.relation.references | Wu L., Shi P., Gao H., Wang C. (2008) H ∞ filtering for 2D Markovian jump systems. Automatica 44(7): 1849–1858 | en_US |
| dc.relation.references | doi: 10.1016/j.automatica.2007.10.027 | en_US |
| dc.relation.references | Wu L., Su X., Shi P., Qiu J. (2011) A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B 41: 273–286 | en_US |
| dc.relation.references | doi: 10.1109/TSMCB.2010.2051541 | en_US |
| dc.relation.references | Wu L., Wang Z., Gao H., Wang C. (2007) H ∞ and l 2-l ∞ filtering for two-dimensional linear parameter-varying systems. International Journal of Robust & Nonlinear Control 17(12): 1129–1154 | en_US |
| dc.relation.references | doi: 10.1002/rnc.1169 | en_US |
| dc.relation.references | Xie L., Du C., Soh Y. C., Zhang C. (2002) H ∞ and robust control of 2-D systems in FM second model. Multidimensional Systems and Signal Processing 13: 256–287 | en_US |
| dc.relation.references | doi: 10.1023/A:1015808429836 | en_US |
| dc.relation.references | Xu H., Zou Y., Xu S., Lam J., Wang Q. (2005) H ∞ model reduction of 2-D singular Roesser models. Multidimensional Systems and Signal Processing 16(3): 285–304 | en_US |
| dc.relation.references | doi: 10.1007/s11045-005-1678-1 | en_US |
| dc.relation.references | Yoneyama J. (2006) Robust H ∞ control analysis and synthesis for Takagi-Sugeno general uncertain fuzzy systems. Fuzzy Sets and Systems 57(16): 2205–2223 | en_US |
| dc.relation.references | doi: 10.1016/j.fss.2006.03.020 | en_US |
| dc.relation.references | Zhou S., Lam J., Xue A. (2007) H ∞ filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach. Fuzzy Sets and Systems 158(2): 180–193 | en_US |
| dc.relation.references | doi: 10.1016/j.fss.2006.09.001 | en_US |
| dc.relation.references | Zhou S., Li T. (2005) Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent Lyapunov-Krasovskii function. Fuzzy Sets and Systems 151(1): 139–153 | en_US |
| dc.relation.references | doi: 10.1016/j.fss.2004.08.014 | en_US |
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| dc.relation.references | Jadbabaie, A. (1999). A reduction in conservatism in stability and L 2 gain analysis of Takagi-Sugeno fuzzy systems via linear matrix inequalities. In Proceedings of the 14th IFAC triennial world congress (pp. 285–289). Beijing, China. | en_US |
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| dc.identifier.spage | 395 | en_HK |
| dc.identifier.epage | 415 | en_HK |
| dc.identifier.eissn | 1573-0824 | en_US |
| dc.identifier.isi | WOS:000316871800001 | - |
| dc.publisher.place | United States | en_HK |
| dc.description.other | Springer Open Choice, 28 May 2012 | en_US |
| dc.identifier.scopusauthorid | Chen, X=54399871500 | en_HK |
| dc.identifier.scopusauthorid | Lam, J=7201973414 | en_HK |
| dc.identifier.scopusauthorid | Gao, H=7402971422 | en_HK |
| dc.identifier.scopusauthorid | Zhou, S=7404166480 | en_HK |
| dc.identifier.citeulike | 10054816 | - |
| dc.identifier.issnl | 0923-6082 | - |