Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control

Hui Lv; Xing'an Wang

doi:10.3934/naco.2020020

Previous Article
NACO Home
This Issue
Next Article
A PID control method based on optimal control strategy

March 2021, 11(1): 127-142. doi: 10.3934/naco.2020020

Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control

Hui Lv ^1,, and Xing'an Wang ^2,,

1.	Key Laboratory of Advanced Design, and, Intelligent Computing, Ministry of Education, School of Software, Dalian University, Dalian, 116622, China
2.	College of Environmental and Chemical Engineering, Dalian University, Dalian, 116622, China

^* Corresponding author: lh8481@tom.com; wangxingan@dlu.edu.cn

Received May 2019 Revised October 2019 Published March 2021 Early access March 2020

Fund Project: The first author is supported by National Natural Science Foundation of China (No.61802040), Natural Science Foundation of Liaoning Province (No.20180551241) and High-level Talent Innovation Support Program of Dalian City (No.2018RQ75). The second author is supported by National Key Laboratory for Precision Hot Processing of Metals(No.614290903061808) and Doctoral Starting Foundation of Dalian University (No.2017QL024)

This paper investigates the problem of dissipative control for a class of uncertain singular Markovian jump systems. Different from the traditional control strategy, a derivative gain and impulsive control part are added in the proposed controller. A linearization approach via congruence transformations is proposed to solve the feedback design problem. In addition, the derived results contain $ H_{\infty} $ and passive control as special cases. Finally, examples are provided to illustrate the effectiveness and applicability of the proposed methods.

Keywords: Dissipative control, Uncertain singular Markovian jump systems, Impulsive control.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

References:

[1]	M. Aliyu and E. Boukas, $H_{\infty}$ filtering for nonlinear singular systems, IEEE Transactions on Circuits and Systems I: Regular Papers, 59 (2012), 2395-2404. doi: 10.1109/TCSI.2012.2189038.
[2]	E. Boukas, Control of Singular Systems with Random Abrupt Changes, Springer, Berlin, 2008.
[3]	B. Brogliato, R.Lozano, B. Maschke and O. Egeland, Dissipative Systems Analysis and Control: Theory and Applications, Springer, New York, 2000. doi: 10.1007/978-3-030-19420-8.
[4]	L. Dai, Singular Control Systems, Springer, Berlin, 1989. doi: 10.1007/BFb0002475.
[5]	M. Deistler, Singular arma systems: A structure theory, Numerical Algebra, Control and Optimization, 9 (2019), 383-391. doi: 10.3934/naco.2019025.
[6]	Y. Dong, J. Sun and Q. Wu, $H_{\infty}$ filtering for a class of stochastic Markovian jump systems with impulsive effects, International Journal of Robust and Nonlinear Control, 18 (2008), 1-13. doi: 10.1002/rnc.1194.
[7]	G. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. doi: 10.1007/978-1-4419-6397-0.
[8]	Z. Feng and P. Shi, Admissibilization of singular interval-valued fuzzy systems, IEEE Transactions on Fuzzy Systems, 25 (2016), 1765–1776.
[9]	Z. Feng and P. Shi, Sliding mode control of singular stochastic Markov jump systems, IEEE Transactions on Automatic Control, 62 (2017), 4266-4273. doi: 10.1109/TAC.2017.2687048.
[10]	Z. Guan, J. Yao and D. Hill, Robust $H_{\infty}$ control of singular impulsive systems with uncertain perturbations, IEEE Transactions on Circuits and Systems II: Express Briefs, 52 (2005), 293-298.
[11]	L. Huang, Linear Algebra in System and Control Theory, Science Press, Beijing, 1984.
[12]	B. Jiang, Y. Kao, C. Gao and X. Yao, Passification of uncertain singular semi-Markovian jump systems with actuator failures via sliding mode approach, IEEE Transactions on Automatic Control, 62 (2017), 4138-4143. doi: 10.1109/TAC.2017.2680540.
[13]	B. Jiang, Y. Kao, H. Karimi and C. Gao, Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates, IEEE Transactions on Automatic Control, 63 (2018), 3919-3926. doi: 10.1109/tac.2018.2819654.
[14]	S. Marir, M. Chadli and D. Bouagada, New admissibility conditions for singular linear continuous-time fractional-order systems, Journal of the Franklin Institute, 354 (2017), 752-766. doi: 10.1016/j.jfranklin.2016.10.022.
[15]	E. Medina and D. Lawrence, State feedback stabilization of linear impulsive systems, Automatica, 45 (2009), 1476-1480. doi: 10.1016/j.automatica.2009.02.003.
[16]	I. Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems & Control Letters, 8 (1987), 351-357. doi: 10.1016/0167-6911(87)90102-2.
[17]	P. Shi, H. Wang and C. Lim, Network-based event-triggered control for singular systems with quantizations, IEEE Transactions on Industrial Electronics, 63 (2015), 1230-1238.
[18]	Y. Wang, Y. Xia, H. Shen and P. Zhou, SMC design for robust stabilization of nonlinear Markovian jump singular systems, IEEE Transactions on Automatic Control, 63 (2017), 219-224. doi: 10.1109/tac.2017.2720970.
[19]	J. Willems, Dissipative dynamical systems, Part I: General theory, Archive for Rational Mechanics and Analysis, 45 (1972), 321-351. doi: 10.1007/BF00276493.
[20]	Y. Xia, E. Boukas, P. Shi and J. Zhang, Stability and stabilization of continuous-time singular hybrid systems, Automatica, 45 (2009), 1504-1509. doi: 10.1016/j.automatica.2009.02.008.
[21]	S. Xie, L. Xie and D. Souza, Robust dissipative control for linear systems with dissipative uncertainty, International Journal of Control, 70 (1998), 169-191. doi: 10.1080/002071798222352.
[22]	H. Xu, K. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38 (2008), 1419-1422. doi: 10.1109/TSMC.1972.4309113.
[23]	J. Xu and J. Sun, Finite-time stability of linear time-varying singular impulsive systems, IET Control Theory and Applications, 4 (2010), 2239-2244. doi: 10.1049/iet-cta.2010.0242.
[24]	S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer, Berlin, 2006.
[25]	M. Yang, Y. Wang, J. Xiao and Y. Huang, Robust synchronization of singular complex switched networks with parametric uncertainties and unknown coupling topologies via impulsive control, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 4404-4416. doi: 10.1016/j.cnsns.2012.03.021.
[26]	X. Yang, X. Li and J. Cao, Robust finite-time stability of singular nonlinear systems with interval time-varying delay, Journal of the Franklin Institute, 355 (2018), 1241-1258. doi: 10.1016/j.jfranklin.2017.12.018.
[27]	J. Yao, Z. Guan, G. Chen and D. Ho, Stability, robust stabilization and $H_{\infty}$ control of singular-impulsive systems via impulsive control, Systems & Control Letters, 55 (2006), 879-886. doi: 10.1016/j.sysconle.2006.05.002.
[28]	H. Zhang, Z. Guan and G. Feng, Reliable dis sipative control for stochastic impulsive systems, Automatica, 44 (2008), 1004-1010. doi: 10.1016/j.automatica.2007.08.014.
[29]	Q. Zhang, L. Li, X. Yan and S. Spurgeon, Sliding mode control for singular stochastic Markovian jump systems with uncertainties, Automatica, 79 (2017), 27-34. doi: 10.1016/j.automatica.2017.01.002.
[30]	Y. Zhang, Y. He, M. Wu and J. Zhang, Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices, Automatica, 47 (2011), 79-84. doi: 10.1016/j.automatica.2010.09.009.
[31]	J. Zhao and D. Hill, Dissipative theory for switched systems, IEEE Transactions on Automatic Control, 53 (2008), 941-953. doi: 10.1109/TAC.2008.920237.
[32]	G. Zhuang, Q. Ma, B. Zhang, S. Xu and J. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time delays, Systems & Control Letters, 114 (2018), 1-10. doi: 10.1016/j.sysconle.2018.02.004.

show all references

References:

[1]	M. Aliyu and E. Boukas, $H_{\infty}$ filtering for nonlinear singular systems, IEEE Transactions on Circuits and Systems I: Regular Papers, 59 (2012), 2395-2404. doi: 10.1109/TCSI.2012.2189038.
[2]	E. Boukas, Control of Singular Systems with Random Abrupt Changes, Springer, Berlin, 2008.
[3]	B. Brogliato, R.Lozano, B. Maschke and O. Egeland, Dissipative Systems Analysis and Control: Theory and Applications, Springer, New York, 2000. doi: 10.1007/978-3-030-19420-8.
[4]	L. Dai, Singular Control Systems, Springer, Berlin, 1989. doi: 10.1007/BFb0002475.
[5]	M. Deistler, Singular arma systems: A structure theory, Numerical Algebra, Control and Optimization, 9 (2019), 383-391. doi: 10.3934/naco.2019025.
[6]	Y. Dong, J. Sun and Q. Wu, $H_{\infty}$ filtering for a class of stochastic Markovian jump systems with impulsive effects, International Journal of Robust and Nonlinear Control, 18 (2008), 1-13. doi: 10.1002/rnc.1194.
[7]	G. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. doi: 10.1007/978-1-4419-6397-0.
[8]	Z. Feng and P. Shi, Admissibilization of singular interval-valued fuzzy systems, IEEE Transactions on Fuzzy Systems, 25 (2016), 1765–1776.
[9]	Z. Feng and P. Shi, Sliding mode control of singular stochastic Markov jump systems, IEEE Transactions on Automatic Control, 62 (2017), 4266-4273. doi: 10.1109/TAC.2017.2687048.
[10]	Z. Guan, J. Yao and D. Hill, Robust $H_{\infty}$ control of singular impulsive systems with uncertain perturbations, IEEE Transactions on Circuits and Systems II: Express Briefs, 52 (2005), 293-298.
[11]	L. Huang, Linear Algebra in System and Control Theory, Science Press, Beijing, 1984.
[12]	B. Jiang, Y. Kao, C. Gao and X. Yao, Passification of uncertain singular semi-Markovian jump systems with actuator failures via sliding mode approach, IEEE Transactions on Automatic Control, 62 (2017), 4138-4143. doi: 10.1109/TAC.2017.2680540.
[13]	B. Jiang, Y. Kao, H. Karimi and C. Gao, Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates, IEEE Transactions on Automatic Control, 63 (2018), 3919-3926. doi: 10.1109/tac.2018.2819654.
[14]	S. Marir, M. Chadli and D. Bouagada, New admissibility conditions for singular linear continuous-time fractional-order systems, Journal of the Franklin Institute, 354 (2017), 752-766. doi: 10.1016/j.jfranklin.2016.10.022.
[15]	E. Medina and D. Lawrence, State feedback stabilization of linear impulsive systems, Automatica, 45 (2009), 1476-1480. doi: 10.1016/j.automatica.2009.02.003.
[16]	I. Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems & Control Letters, 8 (1987), 351-357. doi: 10.1016/0167-6911(87)90102-2.
[17]	P. Shi, H. Wang and C. Lim, Network-based event-triggered control for singular systems with quantizations, IEEE Transactions on Industrial Electronics, 63 (2015), 1230-1238.
[18]	Y. Wang, Y. Xia, H. Shen and P. Zhou, SMC design for robust stabilization of nonlinear Markovian jump singular systems, IEEE Transactions on Automatic Control, 63 (2017), 219-224. doi: 10.1109/tac.2017.2720970.
[19]	J. Willems, Dissipative dynamical systems, Part I: General theory, Archive for Rational Mechanics and Analysis, 45 (1972), 321-351. doi: 10.1007/BF00276493.
[20]	Y. Xia, E. Boukas, P. Shi and J. Zhang, Stability and stabilization of continuous-time singular hybrid systems, Automatica, 45 (2009), 1504-1509. doi: 10.1016/j.automatica.2009.02.008.
[21]	S. Xie, L. Xie and D. Souza, Robust dissipative control for linear systems with dissipative uncertainty, International Journal of Control, 70 (1998), 169-191. doi: 10.1080/002071798222352.
[22]	H. Xu, K. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38 (2008), 1419-1422. doi: 10.1109/TSMC.1972.4309113.
[23]	J. Xu and J. Sun, Finite-time stability of linear time-varying singular impulsive systems, IET Control Theory and Applications, 4 (2010), 2239-2244. doi: 10.1049/iet-cta.2010.0242.
[24]	S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer, Berlin, 2006.
[25]	M. Yang, Y. Wang, J. Xiao and Y. Huang, Robust synchronization of singular complex switched networks with parametric uncertainties and unknown coupling topologies via impulsive control, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 4404-4416. doi: 10.1016/j.cnsns.2012.03.021.
[26]	X. Yang, X. Li and J. Cao, Robust finite-time stability of singular nonlinear systems with interval time-varying delay, Journal of the Franklin Institute, 355 (2018), 1241-1258. doi: 10.1016/j.jfranklin.2017.12.018.
[27]	J. Yao, Z. Guan, G. Chen and D. Ho, Stability, robust stabilization and $H_{\infty}$ control of singular-impulsive systems via impulsive control, Systems & Control Letters, 55 (2006), 879-886. doi: 10.1016/j.sysconle.2006.05.002.
[28]	H. Zhang, Z. Guan and G. Feng, Reliable dis sipative control for stochastic impulsive systems, Automatica, 44 (2008), 1004-1010. doi: 10.1016/j.automatica.2007.08.014.
[29]	Q. Zhang, L. Li, X. Yan and S. Spurgeon, Sliding mode control for singular stochastic Markovian jump systems with uncertainties, Automatica, 79 (2017), 27-34. doi: 10.1016/j.automatica.2017.01.002.
[30]	Y. Zhang, Y. He, M. Wu and J. Zhang, Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices, Automatica, 47 (2011), 79-84. doi: 10.1016/j.automatica.2010.09.009.
[31]	J. Zhao and D. Hill, Dissipative theory for switched systems, IEEE Transactions on Automatic Control, 53 (2008), 941-953. doi: 10.1109/TAC.2008.920237.
[32]	G. Zhuang, Q. Ma, B. Zhang, S. Xu and J. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time delays, Systems & Control Letters, 114 (2018), 1-10. doi: 10.1016/j.sysconle.2018.02.004.

Figure 1. The Markov process

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. The state trajectories of the open-loop system (when $ \omega(t) = 0 $)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. The state trajectories of the closed-loop system (when $ \omega(t) = 0 $)

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. Minimum attenuation level calculated by different methods

$ \gamma_{min} $	$ \rho=-0.5 $	$ \rho=0 $	$ \rho=0.5 $	$ \rho=1 $	$ \rho=1.5 $
Theorem 10 ([28])	1.2704	1.5821	2.2548	4.4628	————
Corollary 3.4 ($ K_{ei}=0 $)	1.1724	1.1738	1.1743	1.1749	1.1753
Corollary 3.4 ($ K_{ei}\neq0 $)	1.1040	1.1050	1.1059	1.1067	1.1074

Table Options

Download as excel

$ \gamma_{min} $	$ \rho=-0.5 $	$ \rho=0 $	$ \rho=0.5 $	$ \rho=1 $	$ \rho=1.5 $
Theorem 10 ([28])	1.2704	1.5821	2.2548	4.4628	————
Corollary 3.4 ($ K_{ei}=0 $)	1.1724	1.1738	1.1743	1.1749	1.1753
Corollary 3.4 ($ K_{ei}\neq0 $)	1.1040	1.1050	1.1059	1.1067	1.1074

[1]	Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control and Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019
[2]	M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365
[3]	Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368
[4]	Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1591-1605. doi: 10.3934/dcdss.2020413
[5]	Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011
[6]	Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4665-4681. doi: 10.3934/dcds.2015.35.4665
[7]	Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275
[8]	Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1
[9]	Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889
[10]	Peng Cheng, Feng Pan, Yanyan Yin, Song Wang. Probabilistic robust anti-disturbance control of uncertain systems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2441-2450. doi: 10.3934/jimo.2020076
[11]	Yanqing Liu, Yanyan Yin, Kok Lay Teo, Song Wang, Fei Liu. Probabilistic control of Markov jump systems by scenario optimization approach. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1447-1453. doi: 10.3934/jimo.2018103
[12]	Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164
[13]	C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial and Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435
[14]	N. U. Ahmed. Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach. Evolution Equations and Control Theory, 2012, 1 (2) : 235-250. doi: 10.3934/eect.2012.1.235
[15]	Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial and Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016
[16]	Elena K. Kostousova. On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques. Conference Publications, 2015, 2015 (special) : 723-732. doi: 10.3934/proc.2015.0723
[17]	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
[18]	Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial and Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082
[19]	Yuan Li, Ruxia Zhang, Yi Zhang, Bo Yang. Sliding mode control for uncertain T-S fuzzy systems with input and state delays. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 345-354. doi: 10.3934/naco.2020006
[20]	Yueyuan Zhang, Yanyan Yin, Fei Liu. Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3013-3026. doi: 10.3934/jimo.2020105

Impact Factor:

Tools

Metrics

Tools

Article outline

Figures and Tables

Impact Factor:

Tools

Figures and Tables

Impact Factor:

American Institute of Mathematical Sciences

Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors