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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

 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

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.
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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [28] R. Riaza, Differential-Algebraic Systems: Analytical Aspects And Circuit Applications,, World Scientific, (2008). doi: 10.1016/0098-1354(88)85052-X. [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. [30] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control,, IEEE Transactions on Systems, 15 (1985), 116. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [28] R. Riaza, Differential-Algebraic Systems: Analytical Aspects And Circuit Applications,, World Scientific, (2008). doi: 10.1016/0098-1354(88)85052-X. [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. [30] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control,, IEEE Transactions on Systems, 15 (1985), 116. [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. [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. [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. [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. [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. [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. [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.
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