Guaranteed cost control of discrete-time switched saturated systems

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August 2021, 26(8): 4515-4522. doi: 10.3934/dcdsb.2020300

Guaranteed cost control of discrete-time switched saturated systems

Haijun Sun and Xinquan Zhang ^,

School of Information and Control Engineering, Liaoning Shihua University, Fushun, Liaoning 113001, China

^* Corresponding author: Xinquan Zhang

Received August 2019 Revised June 2020 Published August 2021 Early access October 2020

The problem of guaranteed cost control is investigated for a class of discrete-time saturated switched systems. The purpose is to design the switched law and state feedback control law such that the closed-loop system is asymptotically stable and the upper-bound of the cost function is minimized. Based on the multiple Lyapunov functions approach, some sufficient conditions for the existence of guaranteed cost controllers are obtained. Furthermore, a convex optimization problem with linear matrix inequalities (LMI) constraints is formulated to determine the minimum upper-bound of the cost function. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Keywords: Guaranteed cost control, saturated switched systems, upper-bound of the cost function, multiple Lyapunov functions, LMI.

Mathematics Subject Classification: Primary: 93C10, 93C55; Secondary: 93D15.

Citation: Haijun Sun, Xinquan Zhang. Guaranteed cost control of discrete-time switched saturated systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4515-4522. doi: 10.3934/dcdsb.2020300

References:

[1]	A. Benzaouia, O. Akhrif and L. Saydy, Stabilisation and control synthesis of switching systems subject to actuator saturation, Int. J. Syst. Sci., 41 (2010), 397-409. doi: 10.1080/00207720903045791.
[2]	M. S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Contr., 43 (1998), 475-482. doi: 10.1109/9.664150.
[3]	W. H. Chen, J. X. Xu and Z. H. Guan, Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays, IEEE Trans. on Automatic Control, 48 (2003), 2270-2277. doi: 10.1109/TAC.2003.820165.
[4]	D. Cheng, Stabilization of planar switched systems, System & Control Letters, 51 (2004), 79-88. doi: 10.1016/S0167-6911(03)00208-1.
[5]	J. Daafouz, P. Riedinger and C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Trans. Automat. Contr., 47 (2002), 1883-1887. doi: 10.1109/TAC.2002.804474.
[6]	H. Fang, Z. Lin and T. Hu, Analysis of linear systems in the presence of actuator saturation and $L_2$-disturbances, Automatica, 40 (2004), 1229-1238. doi: 10.1016/j.automatica.2004.02.009.
[7]	J. M. Gomes da Silva and S. Tarbouriech, Anti-windup design with guaranteed regions of stability for discrete-time linear systems, Systems & Control Letters, 55 (2006), 184-192. doi: 10.1016/j.sysconle.2005.07.001.
[8]	J. M. Gomes da Silva, D. Limon, T. Alamo and E. F. Camacho, Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints, IEEE Trans. Automat. Contr., 53 (2008), 2367-2372. doi: 10.1109/TAC.2008.2007521.
[9]	T. Hu, Z. Lin and B. M. Chen, Analysis and design for discrete-time linear systems subject to actuator saturation, Systems & Control Letters, 45 (2002), 97-112. doi: 10.1016/S0167-6911(01)00168-2.
[10]	M. Jungers and J. Daafouz, Guaranteed cost certification for discrete-time linear switched systems with a dwell time, IEEE Trans. Automat. Contr., 58 (2013), 768-772. doi: 10.1109/TAC.2012.2211441.
[11]	H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Contr., 54 (2009), 308-322. doi: 10.1109/TAC.2008.2012009.
[12]	L. Liu, Z. Wang, Z. Huang and H. Zhang, Adaptive predefined performance control for MIMO systems with unknown direction via generalized fuzzy hyperbolic model, IEEE Trans. Fuzzy Syst., 25 (2017), 527-542. doi: 10.1109/TFUZZ.2016.2566803.
[13]	L. Lu and Z. Lin, Design of switched linear systems in the presence of actuator saturation, IEEE Trans. Automat. Contr., 53 (2008), 1536-1542. doi: 10.1109/TAC.2008.921021.
[14]	L. Lu and Z. Lin, A switching anti-windup design using multiple Lyapunov functions, IEEE Trans. Automat. Contr., 55 (2010), 142-148. doi: 10.1109/TAC.2009.2033753.
[15]	W. Ni and D. Cheng, Control of switched linear systems with input saturation, Int. J. Syst. Sci., 41 (2010), 1057-1065. doi: 10.1080/00207720903201865.
[16]	W. Rui and J. Zhao, Guaranteed cost control for a class of uncertain delay systems with actuator failures based on switching method, Int. J. Contr. Automat. Syst., 5 (2007), 492-500.
[17]	X. M. Sun, G. P. Liu, D. Rees and W. Wang, Delay-dependent stability for discrete systems with large delay sequence based on switching techniques, Automatica, 44 (2008), 2902-2908. doi: 10.1016/j.automatica.2008.04.006.
[18]	D. Wang, S. Wang, S. Yuan, B. Cai and L. Zhang, Guaranteed performance control of switched linear systems: A differential-Riccati-equation-based approach, Peer-to-Peer Networking and Applications, 12 (2019), 1810-1819. doi: 10.1007/s12083-019-00748-w.
[19]	L. Yu and J. Chu, An LMI approach to guaranteed cost control of linear uncertain time-delay systems, Automatica, 35 (1999), 1155-1159. doi: 10.1016/S0005-1098(99)00007-2.
[20]	X. Zhang, J. Zhao and G. M. Dimirovski, Robust state feedback stabilization of uncertain switched linear systems subject to actuator saturation, Proceedings of the 2010 American Control Conference, (2010), 3269–3274. doi: 10.1155/2015/521636.
[21]	J. Zhang and T. Raïssi, Saturation control of switched nonlinear systems, Nonlinear Analysis: Hybrid Systems, 32 (2019), 320-336. doi: 10.1016/j.nahs.2019.01.005.
[22]	X. Zhang and J. Zhao, Guaranteed cost control of uncertain discrete-time switched linear systems with actuator saturation, The 25th Chinese Control and Decision Conference, (2013), 1341–1345.
[23]	X. Zhang, J. Zhao and G. M. Dimirovski, $L_2$-Gain analysis and control synthesis of uncertain discrete-time switched linear systems with time delay and actuator saturation, Int. J. Contr., 84 (2011), 1746-1758. doi: 10.1080/00207179.2011.625046.
[24]	J. Zhao and D. J. Hill, On stability, and $L_2$-gain and $H_{\infty}$ control for switched systems, Automatica, 44 (2008), 1220-1232. doi: 10.1016/j.automatica.2007.10.011.
[25]	Q. Zheng and F. Wu, Output feedback control of saturated discrete-time linear systems using parameter-dependent Lyapunov functions, Systems & Control Letters, 57 (2008), 896-903. doi: 10.1016/j.sysconle.2007.12.011.
[26]	Z. Zuo, Z. Jia, Y. Wang, H. Zhao and G. Zhang, Guaranteed cost control for discrete-time uncertain systems with saturating actuators, Proceedings of the 2008 American Control Conference, (2008), 3632–3637. doi: 10.1109/ACC.2008.4587057.
[27]	Z. Zuo, L. Liu, Y. Wang, H. Zhao and G. Zhang, Guaranteed cost control of linear time-delay systems with input constraints: The Razumikhin functional approach, Proceedings of the 27th China Control Conference, (2008), 440–443.

show all references

References:

[1]	A. Benzaouia, O. Akhrif and L. Saydy, Stabilisation and control synthesis of switching systems subject to actuator saturation, Int. J. Syst. Sci., 41 (2010), 397-409. doi: 10.1080/00207720903045791.
[2]	M. S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Contr., 43 (1998), 475-482. doi: 10.1109/9.664150.
[3]	W. H. Chen, J. X. Xu and Z. H. Guan, Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays, IEEE Trans. on Automatic Control, 48 (2003), 2270-2277. doi: 10.1109/TAC.2003.820165.
[4]	D. Cheng, Stabilization of planar switched systems, System & Control Letters, 51 (2004), 79-88. doi: 10.1016/S0167-6911(03)00208-1.
[5]	J. Daafouz, P. Riedinger and C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Trans. Automat. Contr., 47 (2002), 1883-1887. doi: 10.1109/TAC.2002.804474.
[6]	H. Fang, Z. Lin and T. Hu, Analysis of linear systems in the presence of actuator saturation and $L_2$-disturbances, Automatica, 40 (2004), 1229-1238. doi: 10.1016/j.automatica.2004.02.009.
[7]	J. M. Gomes da Silva and S. Tarbouriech, Anti-windup design with guaranteed regions of stability for discrete-time linear systems, Systems & Control Letters, 55 (2006), 184-192. doi: 10.1016/j.sysconle.2005.07.001.
[8]	J. M. Gomes da Silva, D. Limon, T. Alamo and E. F. Camacho, Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints, IEEE Trans. Automat. Contr., 53 (2008), 2367-2372. doi: 10.1109/TAC.2008.2007521.
[9]	T. Hu, Z. Lin and B. M. Chen, Analysis and design for discrete-time linear systems subject to actuator saturation, Systems & Control Letters, 45 (2002), 97-112. doi: 10.1016/S0167-6911(01)00168-2.
[10]	M. Jungers and J. Daafouz, Guaranteed cost certification for discrete-time linear switched systems with a dwell time, IEEE Trans. Automat. Contr., 58 (2013), 768-772. doi: 10.1109/TAC.2012.2211441.
[11]	H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Contr., 54 (2009), 308-322. doi: 10.1109/TAC.2008.2012009.
[12]	L. Liu, Z. Wang, Z. Huang and H. Zhang, Adaptive predefined performance control for MIMO systems with unknown direction via generalized fuzzy hyperbolic model, IEEE Trans. Fuzzy Syst., 25 (2017), 527-542. doi: 10.1109/TFUZZ.2016.2566803.
[13]	L. Lu and Z. Lin, Design of switched linear systems in the presence of actuator saturation, IEEE Trans. Automat. Contr., 53 (2008), 1536-1542. doi: 10.1109/TAC.2008.921021.
[14]	L. Lu and Z. Lin, A switching anti-windup design using multiple Lyapunov functions, IEEE Trans. Automat. Contr., 55 (2010), 142-148. doi: 10.1109/TAC.2009.2033753.
[15]	W. Ni and D. Cheng, Control of switched linear systems with input saturation, Int. J. Syst. Sci., 41 (2010), 1057-1065. doi: 10.1080/00207720903201865.
[16]	W. Rui and J. Zhao, Guaranteed cost control for a class of uncertain delay systems with actuator failures based on switching method, Int. J. Contr. Automat. Syst., 5 (2007), 492-500.
[17]	X. M. Sun, G. P. Liu, D. Rees and W. Wang, Delay-dependent stability for discrete systems with large delay sequence based on switching techniques, Automatica, 44 (2008), 2902-2908. doi: 10.1016/j.automatica.2008.04.006.
[18]	D. Wang, S. Wang, S. Yuan, B. Cai and L. Zhang, Guaranteed performance control of switched linear systems: A differential-Riccati-equation-based approach, Peer-to-Peer Networking and Applications, 12 (2019), 1810-1819. doi: 10.1007/s12083-019-00748-w.
[19]	L. Yu and J. Chu, An LMI approach to guaranteed cost control of linear uncertain time-delay systems, Automatica, 35 (1999), 1155-1159. doi: 10.1016/S0005-1098(99)00007-2.
[20]	X. Zhang, J. Zhao and G. M. Dimirovski, Robust state feedback stabilization of uncertain switched linear systems subject to actuator saturation, Proceedings of the 2010 American Control Conference, (2010), 3269–3274. doi: 10.1155/2015/521636.
[21]	J. Zhang and T. Raïssi, Saturation control of switched nonlinear systems, Nonlinear Analysis: Hybrid Systems, 32 (2019), 320-336. doi: 10.1016/j.nahs.2019.01.005.
[22]	X. Zhang and J. Zhao, Guaranteed cost control of uncertain discrete-time switched linear systems with actuator saturation, The 25th Chinese Control and Decision Conference, (2013), 1341–1345.
[23]	X. Zhang, J. Zhao and G. M. Dimirovski, $L_2$-Gain analysis and control synthesis of uncertain discrete-time switched linear systems with time delay and actuator saturation, Int. J. Contr., 84 (2011), 1746-1758. doi: 10.1080/00207179.2011.625046.
[24]	J. Zhao and D. J. Hill, On stability, and $L_2$-gain and $H_{\infty}$ control for switched systems, Automatica, 44 (2008), 1220-1232. doi: 10.1016/j.automatica.2007.10.011.
[25]	Q. Zheng and F. Wu, Output feedback control of saturated discrete-time linear systems using parameter-dependent Lyapunov functions, Systems & Control Letters, 57 (2008), 896-903. doi: 10.1016/j.sysconle.2007.12.011.
[26]	Z. Zuo, Z. Jia, Y. Wang, H. Zhao and G. Zhang, Guaranteed cost control for discrete-time uncertain systems with saturating actuators, Proceedings of the 2008 American Control Conference, (2008), 3632–3637. doi: 10.1109/ACC.2008.4587057.
[27]	Z. Zuo, L. Liu, Y. Wang, H. Zhao and G. Zhang, Guaranteed cost control of linear time-delay systems with input constraints: The Razumikhin functional approach, Proceedings of the 27th China Control Conference, (2008), 440–443.

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