March  2021, 11(1): 127-142. 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 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.

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. 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.

[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. 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. 

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

[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.

[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.

[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. ShiH. 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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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.

[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.

[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.

[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.

[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.

[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. 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.

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. 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.

[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. 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. 

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

[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.

[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.

[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. ShiH. 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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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.

[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.

[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.

[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.

[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.

[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. 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.

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
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