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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

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
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