Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays

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October 2016, 12(4): 1535-1556. doi: 10.3934/jimo.2016.12.1535

Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays

Hongbiao Fan ^1, , Jun-E Feng ^2, and Min Meng ^2,

1.	School of Mathematics Sicences, Dezhou University, Dezhou 253600, China
2.	School of Mathematics, Shandong University, Jinan 250100, China, China

Received December 2014 Revised October 2015 Published January 2016

Abstract
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This paper investigates piecewise observer design for rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Via a series of simple transformations, the considered rectangular descriptor plants are converted into standard ones with multiple time-varying delays. Then, two sufficient delay-dependent conditions for existence of piecewise fuzzy observers are derived based on piecewise Lyapunov functions. Finally, two numerical examples are presented to show the effectiveness of the theoretical results.

Keywords: time-varying delays., piecewise observer, Fuzzy descriptor system, piecewise Lyapunov function.

Mathematics Subject Classification: Primary: 93C10, 93C55; Secondary: 93B0.

Citation: Hongbiao Fan, Jun-E Feng, Min Meng. Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1535-1556. doi: 10.3934/jimo.2016.12.1535

References:

[1]	S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Society for Industrial and Applied Mathematics, (1994). doi: 10.1137/1.9781611970777.fm. Google Scholar
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[6]	M. Darouach, M. Zasadzinski and M. Hayar, Reduced-order observer design for descriptor systems with unknown inputs,, IEEE Transactions on Automatic Control, 41 (1996), 1068. doi: 10.1109/9.508918. Google Scholar
[7]	D. Essawy, Adaptive control of nonlinear systems using fuzzy systems,, Journal of Industrial and Management Optimization, 6 (2010), 861. doi: 10.3934/jimo.2010.6.861. Google Scholar
[8]	G. Feng, Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions,, IEEE Transactions on Fuzzy Systems, 12 (2004), 22. doi: 10.1109/TFUZZ.2003.819833. Google Scholar
[9]	H. Gao and T. Chen, New results on stability of discrete-time systems with time-varying state delay,, IEEE Transactions on Automatic Control, 52 (2007), 328. doi: 10.1109/TAC.2006.890320. Google Scholar
[10]	H. Gao, J. Lam, C. Wang and Y. Wang, Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay,, IEE Proceedings-Control Theory and Applications, 151 (2004), 691. doi: 10.1049/ip-cta:20040822. Google Scholar
[11]	Z. Gao, X. Shi and S. Ding, Fuzzy state/disturbance observer design for T-S fuzzy systems with application to sensor fault estimation,, IEEE Transactions on Systems, 38 (2008), 875. doi: 10.1109/TSMCB.2008.917185. Google Scholar
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[15]	C. Jiang, K. Teo, R. Loxton and G. Duan, A neighboring extremal solution for an optimal switched impulsive control problem,, Journal of Industrial and Management Optimization, 8 (2012), 591. doi: 10.3934/jimo.2012.8.591. Google Scholar
[16]	M. Johansson, A. Rantzer and K.-E. Årzén, Piecewise quadratic stability of fuzzy systems,, IEEE Transactions on Fuzzy Systems, 7 (1999), 713. doi: 10.1109/91.811241. Google Scholar
[17]	D. Koenig, Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation,, IEEE Transactions on Automatic Control, 50 (2005), 212. doi: 10.1109/TAC.2004.841889. Google Scholar
[18]	A. Kumar and P. Daoutidis, Control of Nonlinear Differential Algebraic Equation Systems with Applications to Chemical Processes,, Chapman & Hall/CRC, (1999). doi: 10.1007/978-94-017-3594-0_4. Google Scholar
[19]	F. Li, P. Shi, L. Wu and X. Zhang, Fuzzy-model-based D-stability and non-fragile control for discrete-time descriptor systems with multiple delays,, IEEE Transactions on Fuzzy Systems, 22 (2013), 1019. doi: 10.1109/TFUZZ.2013.2272647. Google Scholar
[20]	X. Liu and Q. Zhang, New approaches to $H_{\infty}$ controller designs based on fuzzy observers for T-S fuzzy systems via LMI,, Automatica, 39 (2003), 1571. doi: 10.1016/S0005-1098(03)00172-9. Google Scholar
[21]	S. Ma and Z. Cheng, Observer design for discrete time-delay singular systems with unknown inputs,, American Control Conference, 6 (2005), 4215. doi: 10.1109/ACC.2005.1470640. Google Scholar
[22]	Y. Ma and G. Yang, Stability analysis for linear discrete-time systems subject to actuator saturation,, Control Theory and Technology, 8 (2010), 245. doi: 10.1007/s11768-010-7261-9. Google Scholar
[23]	S. K. Nguang and P. Shi, $H_{\infty}$ fuzzy output feedback control design for nonlinear systems: An LMI approach,, IEEE Transactions on Fuzzy Systems, 11 (2003), 331. doi: 10.1109/TFUZZ.2003.812691. Google Scholar
[24]	R. Palm and P. Bergsten, Sliding mode observer for a Takagi-Sugeno fuzzy system,, The Ninth IEEE International Conference on Fuzzy Systems, 2 (2000), 665. doi: 10.1109/FUZZY.2000.839072. Google Scholar
[25]	J. Qiu, G. Feng and H. Gao, Static-Output-Feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions,, IEEE Transactions on Fuzzy Systems, 21 (2013), 245. doi: 10.1109/TFUZZ.2012.2210555. Google Scholar
[26]	J. Qiu, G. Feng and H. Gao, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements,, IEEE Transactions on Fuzzy Systems, 20 (2012), 1046. doi: 10.1109/TFUZZ.2012.2191790. Google Scholar
[27]	J. Qiu, G. Feng and H. Gao, Fuzzy-model-based piecewise $H_{\infty}$ static-output-feedback controller design for networked nonlinear systems,, IEEE Transactions on Fuzzy Systems, 18 (2010), 919. doi: 10.1109/TFUZZ.2010.2052259. Google Scholar
[28]	R. Riaza, Differential-Algebraic Systems: Analytical Aspects And Circuit Applications,, World Scientific, (2008). doi: 10.1016/0098-1354(88)85052-X. Google Scholar
[29]	H. Shi, G. Xie and W. Luo, Controllability analysis of linear discrete time systems with time delay in state,, Abstract and Applied Analysis, (2012). doi: 10.1155/2012/490903. Google Scholar
[30]	T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control,, IEEE Transactions on Systems, 15 (1985), 116. Google Scholar
[31]	T. Taniguchi, K. Tanaka, K. Yamafuji and H. Wang, Fuzzy descriptor systems:stability analysis and design via LMIs,, American Control Conference, 3 (1999), 1827. doi: 10.1109/ACC.1999.786165. Google Scholar
[32]	Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold,, Journal of Industrial and Management Optimization, 5 (2009), 467. doi: 10.3934/jimo.2009.5.467. Google Scholar
[33]	Z. Wang, Y. Shen, X. Zhang and Q. Wang, Observer design for discrete-time descriptor systems: An LMI approach,, Systems & Control Letters, 61 (2012), 683. doi: 10.1016/j.sysconle.2012.03.006. Google Scholar
[34]	J. Xiong and J. Lam, Stabilization of linear systems over networks with bounded packet loss,, Automatica, 43 (2007), 80. doi: 10.1016/j.automatica.2006.07.017. Google Scholar
[35]	S. Xu and J. Lam, Robust $H_{\infty}$ control for uncertain discrete-time-delay fuzzy systems via output feedback controllers,, IEEE Transactions on Fuzzy Systems, 13 (2005), 82. doi: 10.1109/TFUZZ.2004.839661. Google Scholar
[36]	Q. Zhang, C. Liu and X. Zhang, Complexity, Analysis and Control of Singular Biological Systems,, Springer, (2012). doi: 10.1007/978-1-4471-2303-3. Google Scholar
[37]	B. Zhu, Q. Zhang and C. Chang, Delay-dependent disspative control for a class of non-linear system via Takagi-Sugeno fuzzy descriptor model with time delay,, IET Control Theory and Applications, 8 (2014), 451. doi: 10.1049/iet-cta.2013.0438. Google Scholar

show all references

References:

[1]	S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Society for Industrial and Applied Mathematics, (1994). doi: 10.1137/1.9781611970777.fm. Google Scholar
[2]	S. Cao, N. W. Rees and G. Feng, Analysis and design of fuzzy control systems using dynamic fuzzy-state space models,, IEEE Transactions on Fuzzy Systems, 7 (1999), 192. doi: 10.1109/91.755400. Google Scholar
[3]	Y. Y. Cao and P. M. Frank, Robust $H_{\infty}$ disturbance attenuation for a class of uncertain discrete-time fuzzy systems,, IEEE Transactions on Fuzzy Systems, 8 (2000), 406. doi: 10.1109/91.868947. Google Scholar
[4]	Q. Chai, L. Ryan, K. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems,, Journal of Industrial and Management Optimization, 9 (2013), 471. doi: 10.3934/jimo.2013.9.471. Google Scholar
[5]	B. S. Chen, C. H. Tseng and H. J. Uang, Mixed $H_{2}/H_{\infty}$ fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach,, IEEE Transactions on Fuzzy Systems, 8 (2000), 249. doi: 10.1109/91.855915. Google Scholar
[6]	M. Darouach, M. Zasadzinski and M. Hayar, Reduced-order observer design for descriptor systems with unknown inputs,, IEEE Transactions on Automatic Control, 41 (1996), 1068. doi: 10.1109/9.508918. Google Scholar
[7]	D. Essawy, Adaptive control of nonlinear systems using fuzzy systems,, Journal of Industrial and Management Optimization, 6 (2010), 861. doi: 10.3934/jimo.2010.6.861. Google Scholar
[8]	G. Feng, Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions,, IEEE Transactions on Fuzzy Systems, 12 (2004), 22. doi: 10.1109/TFUZZ.2003.819833. Google Scholar
[9]	H. Gao and T. Chen, New results on stability of discrete-time systems with time-varying state delay,, IEEE Transactions on Automatic Control, 52 (2007), 328. doi: 10.1109/TAC.2006.890320. Google Scholar
[10]	H. Gao, J. Lam, C. Wang and Y. Wang, Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay,, IEE Proceedings-Control Theory and Applications, 151 (2004), 691. doi: 10.1049/ip-cta:20040822. Google Scholar
[11]	Z. Gao, X. Shi and S. Ding, Fuzzy state/disturbance observer design for T-S fuzzy systems with application to sensor fault estimation,, IEEE Transactions on Systems, 38 (2008), 875. doi: 10.1109/TSMCB.2008.917185. Google Scholar
[12]	T. M. Guerra and L. Vermeiren, LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form,, Automatica, 40 (2004), 823. doi: 10.1016/j.automatica.2003.12.014. Google Scholar
[13]	A. Hmamed, Constrained regulation of linear discrete-time systems with time delay: Delay-dependent and delay-independent conditions,, International Journal of Systems Science, 31 (2000), 529. doi: 10.1080/002077200291109. Google Scholar
[14]	Y. Hosoe and T. Hagiwara, Robust stability analysis based on finite impulse response scaling for discrete-time linear time-invariant systems,, IET Control Theory and Applications, 7 (2013), 1463. doi: 10.1049/iet-cta.2013.0053. Google Scholar
[15]	C. Jiang, K. Teo, R. Loxton and G. Duan, A neighboring extremal solution for an optimal switched impulsive control problem,, Journal of Industrial and Management Optimization, 8 (2012), 591. doi: 10.3934/jimo.2012.8.591. Google Scholar
[16]	M. Johansson, A. Rantzer and K.-E. Årzén, Piecewise quadratic stability of fuzzy systems,, IEEE Transactions on Fuzzy Systems, 7 (1999), 713. doi: 10.1109/91.811241. Google Scholar
[17]	D. Koenig, Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation,, IEEE Transactions on Automatic Control, 50 (2005), 212. doi: 10.1109/TAC.2004.841889. Google Scholar
[18]	A. Kumar and P. Daoutidis, Control of Nonlinear Differential Algebraic Equation Systems with Applications to Chemical Processes,, Chapman & Hall/CRC, (1999). doi: 10.1007/978-94-017-3594-0_4. Google Scholar
[19]	F. Li, P. Shi, L. Wu and X. Zhang, Fuzzy-model-based D-stability and non-fragile control for discrete-time descriptor systems with multiple delays,, IEEE Transactions on Fuzzy Systems, 22 (2013), 1019. doi: 10.1109/TFUZZ.2013.2272647. Google Scholar
[20]	X. Liu and Q. Zhang, New approaches to $H_{\infty}$ controller designs based on fuzzy observers for T-S fuzzy systems via LMI,, Automatica, 39 (2003), 1571. doi: 10.1016/S0005-1098(03)00172-9. Google Scholar
[21]	S. Ma and Z. Cheng, Observer design for discrete time-delay singular systems with unknown inputs,, American Control Conference, 6 (2005), 4215. doi: 10.1109/ACC.2005.1470640. Google Scholar
[22]	Y. Ma and G. Yang, Stability analysis for linear discrete-time systems subject to actuator saturation,, Control Theory and Technology, 8 (2010), 245. doi: 10.1007/s11768-010-7261-9. Google Scholar
[23]	S. K. Nguang and P. Shi, $H_{\infty}$ fuzzy output feedback control design for nonlinear systems: An LMI approach,, IEEE Transactions on Fuzzy Systems, 11 (2003), 331. doi: 10.1109/TFUZZ.2003.812691. Google Scholar
[24]	R. Palm and P. Bergsten, Sliding mode observer for a Takagi-Sugeno fuzzy system,, The Ninth IEEE International Conference on Fuzzy Systems, 2 (2000), 665. doi: 10.1109/FUZZY.2000.839072. Google Scholar
[25]	J. Qiu, G. Feng and H. Gao, Static-Output-Feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions,, IEEE Transactions on Fuzzy Systems, 21 (2013), 245. doi: 10.1109/TFUZZ.2012.2210555. Google Scholar
[26]	J. Qiu, G. Feng and H. Gao, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements,, IEEE Transactions on Fuzzy Systems, 20 (2012), 1046. doi: 10.1109/TFUZZ.2012.2191790. Google Scholar
[27]	J. Qiu, G. Feng and H. Gao, Fuzzy-model-based piecewise $H_{\infty}$ static-output-feedback controller design for networked nonlinear systems,, IEEE Transactions on Fuzzy Systems, 18 (2010), 919. doi: 10.1109/TFUZZ.2010.2052259. Google Scholar
[28]	R. Riaza, Differential-Algebraic Systems: Analytical Aspects And Circuit Applications,, World Scientific, (2008). doi: 10.1016/0098-1354(88)85052-X. Google Scholar
[29]	H. Shi, G. Xie and W. Luo, Controllability analysis of linear discrete time systems with time delay in state,, Abstract and Applied Analysis, (2012). doi: 10.1155/2012/490903. Google Scholar
[30]	T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control,, IEEE Transactions on Systems, 15 (1985), 116. Google Scholar
[31]	T. Taniguchi, K. Tanaka, K. Yamafuji and H. Wang, Fuzzy descriptor systems:stability analysis and design via LMIs,, American Control Conference, 3 (1999), 1827. doi: 10.1109/ACC.1999.786165. Google Scholar
[32]	Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold,, Journal of Industrial and Management Optimization, 5 (2009), 467. doi: 10.3934/jimo.2009.5.467. Google Scholar
[33]	Z. Wang, Y. Shen, X. Zhang and Q. Wang, Observer design for discrete-time descriptor systems: An LMI approach,, Systems & Control Letters, 61 (2012), 683. doi: 10.1016/j.sysconle.2012.03.006. Google Scholar
[34]	J. Xiong and J. Lam, Stabilization of linear systems over networks with bounded packet loss,, Automatica, 43 (2007), 80. doi: 10.1016/j.automatica.2006.07.017. Google Scholar
[35]	S. Xu and J. Lam, Robust $H_{\infty}$ control for uncertain discrete-time-delay fuzzy systems via output feedback controllers,, IEEE Transactions on Fuzzy Systems, 13 (2005), 82. doi: 10.1109/TFUZZ.2004.839661. Google Scholar
[36]	Q. Zhang, C. Liu and X. Zhang, Complexity, Analysis and Control of Singular Biological Systems,, Springer, (2012). doi: 10.1007/978-1-4471-2303-3. Google Scholar
[37]	B. Zhu, Q. Zhang and C. Chang, Delay-dependent disspative control for a class of non-linear system via Takagi-Sugeno fuzzy descriptor model with time delay,, IET Control Theory and Applications, 8 (2014), 451. doi: 10.1049/iet-cta.2013.0438. Google Scholar

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