• Previous Article
    Public debt dynamics under ambiguity by means of iterated function systems on density functions
  • DCDS-B Home
  • This Issue
  • Next Article
    On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type
doi: 10.3934/dcdsb.2020300

Guaranteed cost control of discrete-time switched saturated systems

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.

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. BenzaouiaO. 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. ChenJ. 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. DaafouzP. 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. FangZ. 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 SilvaD. LimonT. 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. HuZ. 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. LiuZ. WangZ. 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. SunG. P. LiuD. 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. WangS. WangS. YuanB. 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. ZhangJ. 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. BenzaouiaO. 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. ChenJ. 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. DaafouzP. 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. FangZ. 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 SilvaD. LimonT. 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. HuZ. 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. LiuZ. WangZ. 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. SunG. P. LiuD. 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. WangS. WangS. YuanB. 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. ZhangJ. 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

[1]

Jianfeng Lv, Yan Gao, Na Zhao. The viability of switched nonlinear systems with piecewise smooth Lyapunov functions. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1825-1843. doi: 10.3934/jimo.2020048

[2]

Davide La Torre, Simone Marsiglio, Franklin Mendivil, Fabio Privileggi. Public debt dynamics under ambiguity by means of iterated function systems on density functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021070

[3]

Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021061

[4]

Jaouad Danane. Optimal control of viral infection model with saturated infection rate. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 363-375. doi: 10.3934/naco.2020031

[5]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012

[6]

Guanwei Chen, Martin Schechter. Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed terms. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021124

[7]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

[8]

Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $ H_{\infty} $ control design for active suspension systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021036

[9]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[10]

Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021021

[11]

Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003

[12]

Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252

[13]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038

[14]

Muhammad Aslam Noor, Khalida Inayat Noor. Properties of higher order preinvex functions. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 431-441. doi: 10.3934/naco.2020035

[15]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216

[16]

Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021076

[17]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709

[18]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021035

[19]

Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021022

[20]

Kai Cai, Guangyue Han. An optimization approach to the Langberg-Médard multiple unicast conjecture. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021001

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (56)
  • HTML views (184)
  • Cited by (0)

Other articles
by authors

[Back to Top]