American Institute of Mathematical Sciences

2015, 5(1): 11-23. doi: 10.3934/naco.2015.5.11

Delay-range dependent $H_\infty$ control for uncertain 2D-delayed systems

 1 College of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China, China 2 Department of Chemical & Biochemical Engineering, College of Chemistry & Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China 3 Department of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

Received  December 2014 Revised  March 2015 Published  March 2015

This paper proposes a delay-range dependent method to solve a two-dimensional (2D) stabilization and $H_\infty$ control problem for a class of uncertain delayed systems described by the Roessor model with a range delay. By using a new 2D Lyapunov-Krasovskii function and introducing a differential inequality to the difference Lyapunov functional for 2D systems, sufficient delay-range dependent conditions for the existence of the proposed feedback controller scheme are established in terms of linear matrix inequalities (LMIs), which depend on both the difference between the upper and lower delay bounds and the upper delay bound of the interval time-varying delay. By solving these LMIs, the 2D law is explicitly formulated, together with an adjustable robust $H_\infty$ performance level. The analysis of the application in the thermal process demonstrates the effectiveness of the proposed controller.
Citation: Li-Min Wang, Jing-Xian Yu, Jia Shi, Fu-Rong Gao. Delay-range dependent $H_\infty$ control for uncertain 2D-delayed systems. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 11-23. doi: 10.3934/naco.2015.5.11
References:
 [1] S. F. Chen, Stability analysis for 2-D systems with interval time-varying delays and saturation nonlinearities, Signal Process, 90 (2010), 2265-2275. [2] S. F. Chen and I. K. Fong, Delay-dependent robust stability and stabilization of two-dimensional with state-delayed systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 16 (2009), 1-17. [3] S. F. Chen, Delay-dependent stability for 2D systems with time-varying delay subject to state saturation in the Roesser model, Applied Mathematics and Computation, 216 (2010), 2613-2622. doi: 10.1016/j.amc.2010.03.104. [4] A. Dhawan and H. Kar, Optimal guaranteed cost control of 2-D discrete uncertain systems: An LMI approach, Signal Processing, 87 (2007), 3075-3085. [5] A. Dhawan and H. Kar, An LMI approach to robust optimal guaranteed cost control of 2-D discrete systems described by the Roesser model, Signal Processing, 90 (2010), 2648-2654. [6] A. Dhawan and H. Kar, An improved LMI-based criterion for the design of optimal guaranteed cost controller for 2-D discrete uncertain systems, Signal Processing, 91 (2011), 1032-1035. [7] A. Dhawa and H. Kar, LMI-based criterion for the robust guaranteed cost control of 2-D systems described by the Fornasini-Marchesini second model, Signal Process, 87 (2007), 479-488. [8] C. Du, L. Xie and C. Zhang, H∞ control and robust stabilization of two-dimensional system in Roesser models, Automatica, 37 (2001), 205-211. doi: 10.1016/S0005-1098(00)00155-2. [9] C. Du and L. Xie, Stability analysis and stabilization of uncertain two-dimensional discrete systems: an LMI approach, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 46 (1999), 1371-1374. [10] Z. X. Duan, Z. R Xiang and H. R Karimi, Delay-dependent exponential stabilization of positive 2D switched state-delayed systems in the Roesser model, Information Sciences, 272 (2014), 173-184. doi: 10.1016/j.ins.2014.02.121. [11] L. El Ghaoui, F. Oustry and M. AitRami, A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE Transactions on Automatic Control, 42 (1997), 1171-1176. doi: 10.1109/9.618250. [12] X. P. Guan, Z. Y. Lin and G. R. Duan, Robust guaranteed cost control for 2-D discrete systems, IEE Proc. Control Theory Appl., 148 (2001), 355-361. [13] T. Hinamoto, Stability of 2-D discrete systems described by the Fornasini-Marchesini second model, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 44 (1997), 254-257. doi: 10.1109/81.557373. [14] T. Kaczorek, Two-Dimensional Linear System, Berlin, Germany: Springer-Verlag, 1985. [15] Y. S. Moon, P. Park, W. H. Kwon and Y. S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, International Journal of control, 74 (2001), 1447-1455. doi: 10.1080/00207170110067116. [16] W. Paszke, K. Galkowski, E. Rogers and D. H. Owens, H-infinity and guaranteed cost control of discrete linear repetitive processes, Linear Algebra Appl., 412 (2006), 93-131. doi: 10.1016/j.laa.2005.01.037. [17] W. Paszke, J. Lam, K. Galkowski, S. Xu and Z. Lin, Robust stability and stabilisation of 2-D discrete state-delayed systems, Syst. Control Lett., 51 (2004), 278-291. doi: 10.1016/j.sysconle.2003.09.003. [18] W. Paszke, J. Lam, K. Galkowski, S. Xu and E. Rogers, H∞ control of 2-D linear state-delayed systems, in The 4th IFAC Workshop on Time-Delay Systems (France), September, (2003), 8-10. [19] D. Peng, X. Guan and C. Long, Robust output feedback guaranteed cost control for 2-D uncertain state delayed systems, Asian J. Control, 9 (2004), 470-474. doi: 10.1111/j.1934-6093.2007.tb00436.x. [20] D. Peng and X. Guan, Output feedback H∞ control for 2D state-delayed systems, Circuits Syst. Signal Process, 28 (2009), 147-167. doi: 10.1007/s00034-008-9074-3. [21] L. M. Wang, S. Y. Mo, D. H. Zhou and F. R. Gao, Robust design of feedback integrated with iterative learning control for batch processes with uncertainties and interval time-varying delays, J. Process Control, 21 (2011), 987-996. [22] L. M. Wang, S. Y. Mo, D. H. Zhou, F. R. Gao and X. Chen, Robust delay dependent iterative learning fault-tolerant control for batch processes with state delay and actuator failures, J. Process Control, 22 (2012), 1273-1286. [23] L. M. Wang, S. Y. Mo, D. H. Zhou, F. R. Gao and X. Chen, Delay-range-dependent robust 2D iterative learning control for batch processes with state delay and uncertainties, J. Process Control, 23 (2013), 715-730. [24] L. M. Wang, X. Chen and F. R. Gao, Delay-range-dependent robust BIBO stabilization of 2D discrete delayed systems via LMI approach, In Proceedings of 19th IFAC World Congress, (2014), 10994-10999. [25] L. Wu, P. Shi, H. Gao and C. Wang, H∞ mode reduction for two-dimensional discrete state-delayed systems, IEE Proc. Vis. Image Signal Process, 156 (2006), 769-784. [26] J. M. Xu and L. Yu, Delay-dependent guaranteed cost control for uncertain 2-D discrete systems with state delay in the FM second model, Journal of the Franklin Institute, 346 (2009), 159-174. doi: 10.1016/j.jfranklin.2008.08.003. [27] J. M. Xu and L. Yu, Delay-dependent robust H∞ control for uncertain 2-D discrete state-delay systems in the second FM model, Multi-dimensional Syst. Signal Process, 20 (2009), 333-349. [28] S. Ye, W. Wang and Y. Zou, Robust guaranteed cost control for class of two-dimensional discrete systems with shift-delays, Multi-dimensional Syst. Signal Process, 20 (2009), 297-307. doi: 10.1007/s11045-008-0063-2. [29] S. X. Ye, J. Z. Li and J. Yao, Robust H∞ control for a class of 2-D discrete delayed systems, ISA Transactions, 53 (2015), 1456-1462. [30] K. W. Yu and C. H. Lien, Stability criteria for uncertain neutral systems with interval time-varying delays, Chaos, Solitons and Fractals, 38 (2008), 650-657. doi: 10.1016/j.chaos.2007.01.002.

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References:
 [1] S. F. Chen, Stability analysis for 2-D systems with interval time-varying delays and saturation nonlinearities, Signal Process, 90 (2010), 2265-2275. [2] S. F. Chen and I. K. Fong, Delay-dependent robust stability and stabilization of two-dimensional with state-delayed systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 16 (2009), 1-17. [3] S. F. Chen, Delay-dependent stability for 2D systems with time-varying delay subject to state saturation in the Roesser model, Applied Mathematics and Computation, 216 (2010), 2613-2622. doi: 10.1016/j.amc.2010.03.104. [4] A. Dhawan and H. Kar, Optimal guaranteed cost control of 2-D discrete uncertain systems: An LMI approach, Signal Processing, 87 (2007), 3075-3085. [5] A. Dhawan and H. Kar, An LMI approach to robust optimal guaranteed cost control of 2-D discrete systems described by the Roesser model, Signal Processing, 90 (2010), 2648-2654. [6] A. Dhawan and H. Kar, An improved LMI-based criterion for the design of optimal guaranteed cost controller for 2-D discrete uncertain systems, Signal Processing, 91 (2011), 1032-1035. [7] A. Dhawa and H. Kar, LMI-based criterion for the robust guaranteed cost control of 2-D systems described by the Fornasini-Marchesini second model, Signal Process, 87 (2007), 479-488. [8] C. Du, L. Xie and C. Zhang, H∞ control and robust stabilization of two-dimensional system in Roesser models, Automatica, 37 (2001), 205-211. doi: 10.1016/S0005-1098(00)00155-2. [9] C. Du and L. Xie, Stability analysis and stabilization of uncertain two-dimensional discrete systems: an LMI approach, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 46 (1999), 1371-1374. [10] Z. X. Duan, Z. R Xiang and H. R Karimi, Delay-dependent exponential stabilization of positive 2D switched state-delayed systems in the Roesser model, Information Sciences, 272 (2014), 173-184. doi: 10.1016/j.ins.2014.02.121. [11] L. El Ghaoui, F. Oustry and M. AitRami, A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE Transactions on Automatic Control, 42 (1997), 1171-1176. doi: 10.1109/9.618250. [12] X. P. Guan, Z. Y. Lin and G. R. Duan, Robust guaranteed cost control for 2-D discrete systems, IEE Proc. Control Theory Appl., 148 (2001), 355-361. [13] T. Hinamoto, Stability of 2-D discrete systems described by the Fornasini-Marchesini second model, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 44 (1997), 254-257. doi: 10.1109/81.557373. [14] T. Kaczorek, Two-Dimensional Linear System, Berlin, Germany: Springer-Verlag, 1985. [15] Y. S. Moon, P. Park, W. H. Kwon and Y. S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, International Journal of control, 74 (2001), 1447-1455. doi: 10.1080/00207170110067116. [16] W. Paszke, K. Galkowski, E. Rogers and D. H. Owens, H-infinity and guaranteed cost control of discrete linear repetitive processes, Linear Algebra Appl., 412 (2006), 93-131. doi: 10.1016/j.laa.2005.01.037. [17] W. Paszke, J. Lam, K. Galkowski, S. Xu and Z. Lin, Robust stability and stabilisation of 2-D discrete state-delayed systems, Syst. Control Lett., 51 (2004), 278-291. doi: 10.1016/j.sysconle.2003.09.003. [18] W. Paszke, J. Lam, K. Galkowski, S. Xu and E. Rogers, H∞ control of 2-D linear state-delayed systems, in The 4th IFAC Workshop on Time-Delay Systems (France), September, (2003), 8-10. [19] D. Peng, X. Guan and C. Long, Robust output feedback guaranteed cost control for 2-D uncertain state delayed systems, Asian J. Control, 9 (2004), 470-474. doi: 10.1111/j.1934-6093.2007.tb00436.x. [20] D. Peng and X. Guan, Output feedback H∞ control for 2D state-delayed systems, Circuits Syst. Signal Process, 28 (2009), 147-167. doi: 10.1007/s00034-008-9074-3. [21] L. M. Wang, S. Y. Mo, D. H. Zhou and F. R. Gao, Robust design of feedback integrated with iterative learning control for batch processes with uncertainties and interval time-varying delays, J. Process Control, 21 (2011), 987-996. [22] L. M. Wang, S. Y. Mo, D. H. Zhou, F. R. Gao and X. Chen, Robust delay dependent iterative learning fault-tolerant control for batch processes with state delay and actuator failures, J. Process Control, 22 (2012), 1273-1286. [23] L. M. Wang, S. Y. Mo, D. H. Zhou, F. R. Gao and X. Chen, Delay-range-dependent robust 2D iterative learning control for batch processes with state delay and uncertainties, J. Process Control, 23 (2013), 715-730. [24] L. M. Wang, X. Chen and F. R. Gao, Delay-range-dependent robust BIBO stabilization of 2D discrete delayed systems via LMI approach, In Proceedings of 19th IFAC World Congress, (2014), 10994-10999. [25] L. Wu, P. Shi, H. Gao and C. Wang, H∞ mode reduction for two-dimensional discrete state-delayed systems, IEE Proc. Vis. Image Signal Process, 156 (2006), 769-784. [26] J. M. Xu and L. Yu, Delay-dependent guaranteed cost control for uncertain 2-D discrete systems with state delay in the FM second model, Journal of the Franklin Institute, 346 (2009), 159-174. doi: 10.1016/j.jfranklin.2008.08.003. [27] J. M. Xu and L. Yu, Delay-dependent robust H∞ control for uncertain 2-D discrete state-delay systems in the second FM model, Multi-dimensional Syst. Signal Process, 20 (2009), 333-349. [28] S. Ye, W. Wang and Y. Zou, Robust guaranteed cost control for class of two-dimensional discrete systems with shift-delays, Multi-dimensional Syst. Signal Process, 20 (2009), 297-307. doi: 10.1007/s11045-008-0063-2. [29] S. X. Ye, J. Z. Li and J. Yao, Robust H∞ control for a class of 2-D discrete delayed systems, ISA Transactions, 53 (2015), 1456-1462. [30] K. W. Yu and C. H. Lien, Stability criteria for uncertain neutral systems with interval time-varying delays, Chaos, Solitons and Fractals, 38 (2008), 650-657. doi: 10.1016/j.chaos.2007.01.002.
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