doi: 10.3934/naco.2020020

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

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 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.

Citation: Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 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.  Google Scholar

[2]

E. Boukas, Control of Singular Systems with Random Abrupt Changes, Springer, Berlin, 2008.  Google Scholar

[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.  Google Scholar

[4]

L. Dai, Singular Control Systems, Springer, Berlin, 1989. doi: 10.1007/BFb0002475.  Google Scholar

[5]

M. Deistler, Singular arma systems: A structure theory, Numerical Algebra, Control and Optimization, 9 (2019), 383-391.  doi: 10.3934/naco.2019025.  Google Scholar

[6]

Y. DongJ. 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.  Google Scholar

[7]

G. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. doi: 10.1007/978-1-4419-6397-0.  Google Scholar

[8]

Z. Feng and P. Shi, Admissibilization of singular interval-valued fuzzy systems, IEEE Transactions on Fuzzy Systems, 25 (2016), 1765–1776. Google Scholar

[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.  Google Scholar

[10]

Z. GuanJ. 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.   Google Scholar

[11] L. Huang, Linear Algebra in System and Control Theory, Science Press, Beijing, 1984.   Google Scholar
[12]

B. JiangY. KaoC. 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.  Google Scholar

[13]

B. JiangY. KaoH. 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.  Google Scholar

[14]

S. MarirM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

P. ShiH. Wang and C. Lim, Network-based event-triggered control for singular systems with quantizations, IEEE Transactions on Industrial Electronics, 63 (2015), 1230-1238.   Google Scholar

[18]

Y. WangY. XiaH. 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.  Google Scholar

[19]

J. Willems, Dissipative dynamical systems, Part I: General theory, Archive for Rational Mechanics and Analysis, 45 (1972), 321-351.  doi: 10.1007/BF00276493.  Google Scholar

[20]

Y. XiaE. BoukasP. 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.  Google Scholar

[21]

S. XieL. 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.  Google Scholar

[22]

H. XuK. 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.  Google Scholar

[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.  Google Scholar

[24]

S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer, Berlin, 2006.  Google Scholar

[25]

M. YangY. WangJ. 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.  Google Scholar

[26]

X. YangX. 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.  Google Scholar

[27]

J. YaoZ. GuanG. 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.  Google Scholar

[28]

H. ZhangZ. 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.  Google Scholar

[29]

Q. ZhangL. LiX. 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.  Google Scholar

[30]

Y. ZhangY. HeM. 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.  Google Scholar

[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.  Google Scholar

[32]

G. ZhuangQ. MaB. ZhangS. 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.  Google Scholar

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.  Google Scholar

[2]

E. Boukas, Control of Singular Systems with Random Abrupt Changes, Springer, Berlin, 2008.  Google Scholar

[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.  Google Scholar

[4]

L. Dai, Singular Control Systems, Springer, Berlin, 1989. doi: 10.1007/BFb0002475.  Google Scholar

[5]

M. Deistler, Singular arma systems: A structure theory, Numerical Algebra, Control and Optimization, 9 (2019), 383-391.  doi: 10.3934/naco.2019025.  Google Scholar

[6]

Y. DongJ. 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.  Google Scholar

[7]

G. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. doi: 10.1007/978-1-4419-6397-0.  Google Scholar

[8]

Z. Feng and P. Shi, Admissibilization of singular interval-valued fuzzy systems, IEEE Transactions on Fuzzy Systems, 25 (2016), 1765–1776. Google Scholar

[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.  Google Scholar

[10]

Z. GuanJ. 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.   Google Scholar

[11] L. Huang, Linear Algebra in System and Control Theory, Science Press, Beijing, 1984.   Google Scholar
[12]

B. JiangY. KaoC. 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.  Google Scholar

[13]

B. JiangY. KaoH. 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.  Google Scholar

[14]

S. MarirM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

P. ShiH. Wang and C. Lim, Network-based event-triggered control for singular systems with quantizations, IEEE Transactions on Industrial Electronics, 63 (2015), 1230-1238.   Google Scholar

[18]

Y. WangY. XiaH. 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.  Google Scholar

[19]

J. Willems, Dissipative dynamical systems, Part I: General theory, Archive for Rational Mechanics and Analysis, 45 (1972), 321-351.  doi: 10.1007/BF00276493.  Google Scholar

[20]

Y. XiaE. BoukasP. 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.  Google Scholar

[21]

S. XieL. 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.  Google Scholar

[22]

H. XuK. 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.  Google Scholar

[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.  Google Scholar

[24]

S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer, Berlin, 2006.  Google Scholar

[25]

M. YangY. WangJ. 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.  Google Scholar

[26]

X. YangX. 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.  Google Scholar

[27]

J. YaoZ. GuanG. 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.  Google Scholar

[28]

H. ZhangZ. 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.  Google Scholar

[29]

Q. ZhangL. LiX. 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.  Google Scholar

[30]

Y. ZhangY. HeM. 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.  Google Scholar

[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.  Google Scholar

[32]

G. ZhuangQ. MaB. ZhangS. 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.  Google Scholar

Figure 1.  The Markov process
Figure 2.  The state trajectories of the open-loop system (when $ \omega(t) = 0 $)
Figure 3.  The state trajectories of the closed-loop system (when $ \omega(t) = 0 $)
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
$ \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 & 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 & 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 & Continuous Dynamical Systems - S, 2020  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 & Continuous Dynamical Systems - S, 2020  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 & Optimization, 2020  doi: 10.3934/naco.2020011

[6]

Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889

[10]

Yanqing Liu, Yanyan Yin, Kok Lay Teo, Song Wang, Fei Liu. Probabilistic control of Markov jump systems by scenario optimization approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1447-1453. doi: 10.3934/jimo.2018103

[11]

Peng Cheng, Feng Pan, Yanyan Yin, Song Wang. Probabilistic robust anti-disturbance control of uncertain systems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020076

[12]

Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems - A, 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 & Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435

[14]

Yueyuan Zhang, Yanyan Yin, Fei Liu. Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020105

[15]

N. U. Ahmed. Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach. Evolution Equations & Control Theory, 2012, 1 (2) : 235-250. doi: 10.3934/eect.2012.1.235

[16]

Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016

[17]

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

[18]

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

[19]

Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082

[20]

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 & Optimization, 2020, 10 (3) : 345-354. doi: 10.3934/naco.2020006

 Impact Factor: 

Metrics

  • PDF downloads (29)
  • HTML views (213)
  • Cited by (0)

Other articles
by authors

[Back to Top]