American Institute of Mathematical Sciences

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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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Cheng, Stabilization of planar switched systems, System & Control Letters, 51 (2004), 79-88.  doi: 10.1016/S0167-6911(03)00208-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Cheng, Stabilization of planar switched systems, System & Control Letters, 51 (2004), 79-88.  doi: 10.1016/S0167-6911(03)00208-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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