June  2018, 8(2): 475-490. doi: 10.3934/mcrf.2018019

Stability and output feedback control for singular Markovian jump delayed systems

1. 

College of Automation Engineering, Qingdao University of Technology, Qingdao 266555, China

2. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

3. 

Institute of Complexity Science, Qingdao University, Qingdao 266073, China

* Corresponding author: Jian Chen

Received  September 2017 Revised  January 2018 Published  March 2018

Fund Project: The first author is supported by NSF grants 61673227 and 61503222.

This paper is concerned with the admissibility analysis and control synthesis for a class of singular systems with Markovian jumps and time-varying delay. The basic idea is the use of an augmented Lyapunov-Krasovskii functional together with a series of appropriate integral inequalities. Sufficient conditions are established to ensure the systems to be admissible. Moreover, control design via static output feedback (SOF) is derived to achieve the stabilization for singular systems. A new algorithm is built to solve the SOF controllers. Examples are given to show the effectiveness of the proposed method.

Citation: 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
References:
[1]

E. K. BoukasQ. Zhang and G. Yin, Robust production and maintenance planning in stochastic manufacturing systems, IEEE Trans. Automat. Control, 40 (1995), 1098-1102.  doi: 10.1109/9.388692.  Google Scholar

[2]

X. H. ChangJ. H. Park and J. Zhou, Robust static output feedback $ H_{\infty} $ control design for linear systems with polytopic uncertainties, Systems & Control Letters, 85 (2015), 23-32.  doi: 10.1016/j.sysconle.2015.08.007.  Google Scholar

[3]

X. H. Chang and G. H. Yang, New results on output feedback $ H_{\infty} $ control for linear discrete-time systems, IEEE Trans. Autom. Control, 59 (2014), 1355-1359.  doi: 10.1109/TAC.2013.2289706.  Google Scholar

[4]

J. ChenC. LinB. Chen and Q. G. Wang, A Mixed $ H_{\infty } $ and passive control for singular systems with time delay via static output feedback, Applied Mathematics and Computation, 293 (2017), 244-253.  doi: 10.1016/j.amc.2016.08.029.  Google Scholar

[5]

J. ChengJ. H. ParkY. LiuZ. Liu and L. Tang, Finite-time $ H_{\infty } $ fuzzy control of nonlinear Markovian jump delayed systems with partly uncertain transition descriptions, Fuzzy Sets and Systems, 314 (2017), 99-115.  doi: 10.1016/j.fss.2016.06.007.  Google Scholar

[6]

J. E. FengJ. Lam and Z. Shu, Stabilization of Markovian systems via probability rate synthesis and output feedback, IEEE Trans. Autom. Control, 55 (2010), 773-777.  doi: 10.1109/TAC.2010.2040499.  Google Scholar

[7]

K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Boston, MA, USA: Birkhuser, 2003.  Google Scholar

[8]

R. GuoZ. ZhangX. Liu and C. Lin, Existence, uniqueness, and exponential stability analysis for complex-value d memristor-base d BAM neural networks with time delays, Applied Mathematics and Computation, 311 (2017), 100-117.  doi: 10.1016/j.amc.2017.05.021.  Google Scholar

[9]

R. GuoZ. ZhangX. LiuC. LinH. Wang and J. Chen, Exponential input-to-state stability for complex-valued memristor-based BAM neural networks with multiple time-varying delays, Neurocomputing, 275 (2018), 2041-2054.   Google Scholar

[10]

Y. KaoJ. XieL. Zhang and H. R. Karimi, A sliding mode approach to robust stabilisation of Markovian jump linear time-delay systems with generally incomplete transition rates, Nonlinear Analysis: Hybrid Systems, 17 (2015), 70-80.  doi: 10.1016/j.nahs.2015.03.001.  Google Scholar

[11]

S. LongS. Zhong and Z. Liu, $ H_{\infty} $ filtering for a class of singular Markovian jump systems with time-varying delay, Signal Processing, 92 (2012), 2759-2768.   Google Scholar

[12]

S. Long and S. Zhong, Improved results for stochastic stabilization of a class of discrete-time singular Markovian jump systems with time-varying delay, Nonlinear Analysis: Hybrid Systems, 23 (2017), 11-26.  doi: 10.1016/j.nahs.2016.06.001.  Google Scholar

[13]

S. MaC. Zhang and Z. Cheng, Delay-dependent robust $ H_{\infty } $ control for uncertain discrete-time singular systems with time-delay, J. Comput. Appl. Math., 217 (2008), 194-211.  doi: 10.1016/j.cam.2007.01.044.  Google Scholar

[14]

L. J. MirmanO. F. Morand and K. L. Reffett, A qualitative approach to Markovian equilibrium in infinite horizon economies with capital, J. Econom. Theory, 139 (2008), 75-98.  doi: 10.1016/j.jet.2007.05.009.  Google Scholar

[15]

P. ParkW. I. Lee and S. Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Franklin Ins., 352 (2015), 1378-1396.  doi: 10.1016/j.jfranklin.2015.01.004.  Google Scholar

[16]

P. ParkJ. W. Ko and C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47 (2011), 235-238.  doi: 10.1016/j.automatica.2010.10.014.  Google Scholar

[17]

M. ParkO. KwonJ. ParkS. Lee and E. Cha, Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica, 55 (2015), 204-208.  doi: 10.1016/j.automatica.2015.03.010.  Google Scholar

[18]

R. SakthivelaM. JobyK. Mathiyalagan and S. Santra, Mixed $ H_{\infty } $ and passive control for singular Markovian jump systems with time delays, Journal of the Franklin Institute, 352 (2015), 4446-4466.  doi: 10.1016/j.jfranklin.2015.06.017.  Google Scholar

[19]

A. Seuret and F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica, 49 (2013), 2860-2866.  doi: 10.1016/j.automatica.2013.05.030.  Google Scholar

[20]

M. ShenS. YanG. Zhang and J. H. Park, Finite-time $ H_{\infty } $ static output control of Markov jump systems with an auxiliary approach, Applied Mathematics and Computation, 273 (2016), 553-561.  doi: 10.1016/j.amc.2015.10.038.  Google Scholar

[21]

R. Skelton, T. Iwazaki and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, London: Taylor and Francis, 1998.  Google Scholar

[22]

R. C. Tsaur, A fuzzy time series-Markov chain model with an application to forecast the exchange rate between the Taiwan and US dollar, Int. J. Innovative Comput. Inform. Control, 8 (2012), 4931-4942.   Google Scholar

[23]

J. WangH. WangA. Xue and R. Lu, Delay-dependent $ H_{\infty } $ control for singular Markovian jump systems with time delay, Nonlinear Analysis: Hybrid Systems, 8 (2013), 1-12.  doi: 10.1016/j.nahs.2012.08.003.  Google Scholar

[24]

G. WangQ. Zhang and C. Yang, Dissipative control for singular Markovian jump systems with time delay, Optim. Control Appl. Methods, 33 (2012), 415-432.  doi: 10.1002/oca.1004.  Google Scholar

[25]

Z. G. WuJ. H. ParkH. SuB. Song and J. Chu, Mixed $ H_{\infty } $ and passive filtering for singular systems with time delays, Signal Process, 93 (2013), 1705-1711.   Google Scholar

[26]

Z. WuH. Su and J. Chu, $ H_{\infty} $ filtering for singular Markovian jump systems with time delay, Int. J. Robust Nonlinear Control, 20 (2010), 939-957.   Google Scholar

[27]

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

[28]

Y. XueX. ZhangY. Han and M. Shi, A delay-range-partition approach to analyse stability of linear systems with time-varying delays, Int. J. Systems Science, 47 (2016), 3970-3977.  doi: 10.1080/00207721.2016.1169333.  Google Scholar

[29]

H. ZhangQ. Shan and Z. Wang, Stability analysis of neural networks with two delay components based on dynamic delay interval method, IEEE Trans. Neural Netw. Learn. Syst., 28 (2017), 259-267.  doi: 10.1109/TNNLS.2015.2503749.  Google Scholar

[30]

Z. ZhangX. LiuD. ZhouC. LinJ. Chen and H. Wang, Finite-time stabilizability and instabilizability for complex-valued memristive neural networks with time delays, IEEE Transactions on Systems, Man, and Cybernetics: Systems, PP (2017), 1-12.  doi: 10.1109/TSMC.2017.2754508.  Google Scholar

[31]

W. ZhouH. LuC. Duan and M. Li, Delay-dependent robust control for singular discrete-time Markovian jump systems with time-varying delay, Int. J. Robust Nonlinear Control, 20 (2010), 1112-1128.   Google Scholar

show all references

References:
[1]

E. K. BoukasQ. Zhang and G. Yin, Robust production and maintenance planning in stochastic manufacturing systems, IEEE Trans. Automat. Control, 40 (1995), 1098-1102.  doi: 10.1109/9.388692.  Google Scholar

[2]

X. H. ChangJ. H. Park and J. Zhou, Robust static output feedback $ H_{\infty} $ control design for linear systems with polytopic uncertainties, Systems & Control Letters, 85 (2015), 23-32.  doi: 10.1016/j.sysconle.2015.08.007.  Google Scholar

[3]

X. H. Chang and G. H. Yang, New results on output feedback $ H_{\infty} $ control for linear discrete-time systems, IEEE Trans. Autom. Control, 59 (2014), 1355-1359.  doi: 10.1109/TAC.2013.2289706.  Google Scholar

[4]

J. ChenC. LinB. Chen and Q. G. Wang, A Mixed $ H_{\infty } $ and passive control for singular systems with time delay via static output feedback, Applied Mathematics and Computation, 293 (2017), 244-253.  doi: 10.1016/j.amc.2016.08.029.  Google Scholar

[5]

J. ChengJ. H. ParkY. LiuZ. Liu and L. Tang, Finite-time $ H_{\infty } $ fuzzy control of nonlinear Markovian jump delayed systems with partly uncertain transition descriptions, Fuzzy Sets and Systems, 314 (2017), 99-115.  doi: 10.1016/j.fss.2016.06.007.  Google Scholar

[6]

J. E. FengJ. Lam and Z. Shu, Stabilization of Markovian systems via probability rate synthesis and output feedback, IEEE Trans. Autom. Control, 55 (2010), 773-777.  doi: 10.1109/TAC.2010.2040499.  Google Scholar

[7]

K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Boston, MA, USA: Birkhuser, 2003.  Google Scholar

[8]

R. GuoZ. ZhangX. Liu and C. Lin, Existence, uniqueness, and exponential stability analysis for complex-value d memristor-base d BAM neural networks with time delays, Applied Mathematics and Computation, 311 (2017), 100-117.  doi: 10.1016/j.amc.2017.05.021.  Google Scholar

[9]

R. GuoZ. ZhangX. LiuC. LinH. Wang and J. Chen, Exponential input-to-state stability for complex-valued memristor-based BAM neural networks with multiple time-varying delays, Neurocomputing, 275 (2018), 2041-2054.   Google Scholar

[10]

Y. KaoJ. XieL. Zhang and H. R. Karimi, A sliding mode approach to robust stabilisation of Markovian jump linear time-delay systems with generally incomplete transition rates, Nonlinear Analysis: Hybrid Systems, 17 (2015), 70-80.  doi: 10.1016/j.nahs.2015.03.001.  Google Scholar

[11]

S. LongS. Zhong and Z. Liu, $ H_{\infty} $ filtering for a class of singular Markovian jump systems with time-varying delay, Signal Processing, 92 (2012), 2759-2768.   Google Scholar

[12]

S. Long and S. Zhong, Improved results for stochastic stabilization of a class of discrete-time singular Markovian jump systems with time-varying delay, Nonlinear Analysis: Hybrid Systems, 23 (2017), 11-26.  doi: 10.1016/j.nahs.2016.06.001.  Google Scholar

[13]

S. MaC. Zhang and Z. Cheng, Delay-dependent robust $ H_{\infty } $ control for uncertain discrete-time singular systems with time-delay, J. Comput. Appl. Math., 217 (2008), 194-211.  doi: 10.1016/j.cam.2007.01.044.  Google Scholar

[14]

L. J. MirmanO. F. Morand and K. L. Reffett, A qualitative approach to Markovian equilibrium in infinite horizon economies with capital, J. Econom. Theory, 139 (2008), 75-98.  doi: 10.1016/j.jet.2007.05.009.  Google Scholar

[15]

P. ParkW. I. Lee and S. Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Franklin Ins., 352 (2015), 1378-1396.  doi: 10.1016/j.jfranklin.2015.01.004.  Google Scholar

[16]

P. ParkJ. W. Ko and C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47 (2011), 235-238.  doi: 10.1016/j.automatica.2010.10.014.  Google Scholar

[17]

M. ParkO. KwonJ. ParkS. Lee and E. Cha, Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica, 55 (2015), 204-208.  doi: 10.1016/j.automatica.2015.03.010.  Google Scholar

[18]

R. SakthivelaM. JobyK. Mathiyalagan and S. Santra, Mixed $ H_{\infty } $ and passive control for singular Markovian jump systems with time delays, Journal of the Franklin Institute, 352 (2015), 4446-4466.  doi: 10.1016/j.jfranklin.2015.06.017.  Google Scholar

[19]

A. Seuret and F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica, 49 (2013), 2860-2866.  doi: 10.1016/j.automatica.2013.05.030.  Google Scholar

[20]

M. ShenS. YanG. Zhang and J. H. Park, Finite-time $ H_{\infty } $ static output control of Markov jump systems with an auxiliary approach, Applied Mathematics and Computation, 273 (2016), 553-561.  doi: 10.1016/j.amc.2015.10.038.  Google Scholar

[21]

R. Skelton, T. Iwazaki and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, London: Taylor and Francis, 1998.  Google Scholar

[22]

R. C. Tsaur, A fuzzy time series-Markov chain model with an application to forecast the exchange rate between the Taiwan and US dollar, Int. J. Innovative Comput. Inform. Control, 8 (2012), 4931-4942.   Google Scholar

[23]

J. WangH. WangA. Xue and R. Lu, Delay-dependent $ H_{\infty } $ control for singular Markovian jump systems with time delay, Nonlinear Analysis: Hybrid Systems, 8 (2013), 1-12.  doi: 10.1016/j.nahs.2012.08.003.  Google Scholar

[24]

G. WangQ. Zhang and C. Yang, Dissipative control for singular Markovian jump systems with time delay, Optim. Control Appl. Methods, 33 (2012), 415-432.  doi: 10.1002/oca.1004.  Google Scholar

[25]

Z. G. WuJ. H. ParkH. SuB. Song and J. Chu, Mixed $ H_{\infty } $ and passive filtering for singular systems with time delays, Signal Process, 93 (2013), 1705-1711.   Google Scholar

[26]

Z. WuH. Su and J. Chu, $ H_{\infty} $ filtering for singular Markovian jump systems with time delay, Int. J. Robust Nonlinear Control, 20 (2010), 939-957.   Google Scholar

[27]

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

[28]

Y. XueX. ZhangY. Han and M. Shi, A delay-range-partition approach to analyse stability of linear systems with time-varying delays, Int. J. Systems Science, 47 (2016), 3970-3977.  doi: 10.1080/00207721.2016.1169333.  Google Scholar

[29]

H. ZhangQ. Shan and Z. Wang, Stability analysis of neural networks with two delay components based on dynamic delay interval method, IEEE Trans. Neural Netw. Learn. Syst., 28 (2017), 259-267.  doi: 10.1109/TNNLS.2015.2503749.  Google Scholar

[30]

Z. ZhangX. LiuD. ZhouC. LinJ. Chen and H. Wang, Finite-time stabilizability and instabilizability for complex-valued memristive neural networks with time delays, IEEE Transactions on Systems, Man, and Cybernetics: Systems, PP (2017), 1-12.  doi: 10.1109/TSMC.2017.2754508.  Google Scholar

[31]

W. ZhouH. LuC. Duan and M. Li, Delay-dependent robust control for singular discrete-time Markovian jump systems with time-varying delay, Int. J. Robust Nonlinear Control, 20 (2010), 1112-1128.   Google Scholar

Figure 1.  The closed-loop response curves in Example 1
Figure 2.  Jumping modes
Table 1.  Maximun allowable upper bounds of time delay $\tau$ for Example 1
$ \pi_{11} $-0.4-0.55-0.7-0.85-1.00
[26]0.60780.58940.57680.56750.5603
[23]0.63220.61200.59810.58810.5805
[18]0.81810.78150.75970.74730.7377
Corollary 10.98740.93120.89440.86840.8491
$ \pi_{11} $-0.4-0.55-0.7-0.85-1.00
[26]0.60780.58940.57680.56750.5603
[23]0.63220.61200.59810.58810.5805
[18]0.81810.78150.75970.74730.7377
Corollary 10.98740.93120.89440.86840.8491
[1]

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

[2]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[3]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[4]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[5]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[6]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[7]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[8]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[9]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[10]

Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323

[11]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[12]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[13]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[14]

Stefan Ruschel, Serhiy Yanchuk. The Spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[15]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[16]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[17]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[18]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[19]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[20]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (188)
  • HTML views (585)
  • Cited by (15)

Other articles
by authors

[Back to Top]