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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

 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:
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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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