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Research Papers

# Robust State Feedback H∞ Control for Discrete-Time Fuzzy System With Random Delays

[+] Author and Article Information
R. Sakthivel

Department of Mathematics,
Sungkyunkwan University,
Suwon 440-746, Republic of Korea
e-mail: krsakthivel@yahoo.com

A. Arunkumar, K. Mathiyalagan

Department of Mathematics,
Anna University Regional Campus,
Coimbatore 641 046, India

Ju H. Park

Department of Electrical Engineering,
Yeungnam University,
Kyongsan 38541, Republic of Korea
e-mail: jessie@ynu.ac.kr

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 28, 2013; final manuscript received March 10, 2017; published online June 5, 2017. Editor: Joseph Beaman.

J. Dyn. Sys., Meas., Control 139(8), 081017 (Jun 05, 2017) (11 pages) Paper No: DS-13-1419; doi: 10.1115/1.4036237 History: Received October 28, 2013; Revised March 10, 2017

## Abstract

This paper investigates the problem of robust stabilization for a class of discrete-time Takagi–Sugeno (TS) fuzzy systems via input random delays in control input. The main objective of this paper is to design a state feedback $H∞$ controller. Linear matrix inequality (LMI) approach together with the construction of proper Lyapunov–Krasovskii functional is employed for obtaining delay dependent sufficient conditions for the existence of robust $H∞$ controller. In particular, the effect of both variation range and distribution probability of the time delay is taken into account in the control input. The key feature of the proposed results in this paper is that the time‐varying delay in the control input not only dependent on the bound but also the distribution probability of the time delay. The obtained results are formulated in terms of LMIs which can be easily solved by using the standard optimization algorithms. Finally, a numerical example with simulation result is provided to illustrate the effectiveness of the obtained control law and less conservativeness of the proposed result.

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## References

Takagi, T. , and Sugeno, M. , 1985, “ Fuzzy Identification of Systems and Its Applications to Modeling and Control,” IEEE Trans. Syst., Man, Cybern., 15(1), pp. 116–132.
Chang, X. H. , and Yang, G. H. , 2011, “ A Descriptor Representation Approach to Observer-Based H Control Synthesis for Discrete-Time Fuzzy Systems,” Fuzzy Sets Syst., 185(2), pp. 38–51.
Kau, S. W. , Lee, H. J. , Yang, C. M. , Lee, C. H. , Hong, L. , and Fang, C. H. , 2007, “ Robust H Fuzzy Static Output Feedback Control of T-S Fuzzy Systems With Parametric Uncertainties,” Fuzzy Sets Syst., 158(2), pp. 135–146.
Zhao, Y. , Gao, H. , Lam, J. , and Du, B. , 2009, “ Stability and Stabilization of Delayed T-S Fuzzy Systems: A Delay Partitioning Approach,” IEEE Trans. Fuzzy Syst., 17(4), pp. 750–762.
Feng, Z. , and Lam, J. , 2013, “ Reliable Dissipative Control for Singular Markovian Systems,” Asian J. Control, 15(3), pp. 901–910.
Liang, J. , Wang, Z. , and Liu, X. , 2012, “ Distributed State Estimation for Uncertain Markov-Type Sensor Networks With Mode-Dependent Distributed Delays,” Int. J. Robust Nonlinear Control, 22(3), pp. 331–346.
Sakthivel, R. , Mathiyalagan, K. , and Anthoni, S. M. , 2012, “ Robust H Control for Uncertain Discrete-Time Stochastic Neural Networks With Time-Varying Delays,” IET Control Theory Appl., 6(9), pp. 1220–1228.
Shen, B. , Wang, Z. , and Liu, X. , 2012, “ Sampled-Data Synchronization Control of Complex Dynamical Networks With Stochastic Sampling,” IEEE Trans. Autom. Control, 57(10), pp. 2644–2650.
Gao, H. , Liu, X. , and Lam, J. , 2009, “ Stability Analysis and Stabilization for Discrete-Time Fuzzy Systems With Time-Varying Delay,” IEEE Trans. Syst., Man, Cybern., B: Cybern., 39(2), pp. 306–317.
Su, X. , Shi, P. , Wu, L. , and Song, Y. D. , 2012, “ A Novel Approach to Filter Design for T-S Fuzzy Discrete-Time Systems With Time-Varying Delay,” IEEE Trans. Fuzzy Syst., 20(6), pp. 1114–1129.
Wu, Z. G. , Shi, P. , Su, H. , and Chu, J. , 2012, “ Reliable H Control for Discrete-Time Fuzzy Systems With Infinite-Distributed Delay,” IEEE Trans. Fuzzy Syst., 20(1), pp. 22–31.
Yang, H. , Shi, P. , Zhang, J. , and Qiu, J. , 2012, “ Robust H Control for a Class of Discrete Time Fuzzy Systems via Delta Operator Approach,” Inf. Sci., 184(1), pp. 230–245.
Thanh, N. T. , and Phat, V . N. , 2012, “ Decentralized H Control for Large-Scale Interconnected Nonlinear Time-Delay Systems via LMI Approach,” J. Process Control, 22(7), pp. 1325–1339.
Sakthivel, R. , Mathiyalagan, K. , and Anthoni, S. M. , 2011, “ Design of a Passification Controller for Uncertain Fuzzy Hopfield Neural Networks With Time-Varying Delays,” Phys. Scr., 84(4), p. 045024.
Sua, Y. K. , Chen, B. , Zhou, Q. , and Lin, C. , 2012, “ Fuzzy Robust H Filter Design for Nonlinear Discrete-Time Systems With Interval Time Delays,” Int. J. Syst. Sci., 43(8), pp. 1568–1579.
Vadivel, P. , Sakthivel, R. , Mathiyalagan, K. , and Thangaraj, P. , 2012, “ Robust Stabilization of Nonlinear Uncertain Takagi-Sugeno Fuzzy Systems by H Control,” IET Control Theory Appl., 6(16), p. 2556.
Xu, S. , and Lam, J. , 2005, “ Robust H Control for Uncertain Discrete Time Delay Fuzzy Systems via Output Feedback Controllers,” IEEE Trans. Fuzzy Syst., 13(1), pp. 82–93.
Bao, H. , and Cao, J. , 2011, “ Delay-Distribution-Dependent State Estimation for Discrete-Time Stochastic Neural Networks With Random Delay,” Neural Networks, 24(1), pp. 19–28. [PubMed]
Lee, T. H. , Park, J. H. , Kwon, O. M. , and Lee, S. M. , 2013, “ Stochastic Sampled-Data Control for State Estimation of Time-Varying Delayed Neural Networks,” Neural Networks, 46, pp. 99–108. [PubMed]
Lee, T. H. , Park, J. H. , Lee, S. M. , and Kwon, O. M. , 2013, “ Robust Synchronization of Chaotic Systems With Randomly Occurring Uncertainties via Stochastic Sampled-Data Control,” Int. J. Control, 86(1), pp. 107–119.
Wu, Z. G. , Park, J. H. , Su, H. , Song, B. , and Chu, J. , 2012, “ Exponential Synchronization for Complex Dynamical Networks With Sampled-Data,” J. Franklin Inst., 349(9), pp. 2735–2749.
Ray, A. , 1994, “ Output Feedback Control Under Randomly Varying Distributed Delays,” Control Dyn., 17(4), pp. 701–711.
Zhang, Y. , Xu, S. , and Zeng, Z. , 2009, “ Novel Robust Stability Criteria of Discrete-Time Stochastic Recurrent Neural Networks With Time Delay,” Neurocomputing, 72(13–15), pp. 3343–3351.
Zhang, J. , Xia, Y. , and Mahmoud, M. S. , 2009, “ Robust Generalised H2 and H Static Output Feedback Control for Uncertain Discrete-Time Fuzzy Systems,” IET Control Theory Appl., 3(7), pp. 865–876.
Lo, J. C. , and Lin, M. L. , 2003, “ Robust H Nonlinear Control via Fuzzy Static Output Feedback,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., 50(11), pp. 1494–1502.

## Figures

Fig. 1

State trajectories of fuzzy system (7) without control when η=1, η=2

Fig. 2

State trajectories of fuzzy system (7) with control when η=1, η=2

Fig. 3

Control trajectories of fuzzy system (7) when η=1, η=2

Fig. 4

Simulation of random variable δ(k) and time‐varying delay τ(k) for nominal model

Fig. 6

State trajectories of the uncertain fuzzy system (6) with control when η=1, η=2

Fig. 7

Control trajectories of the uncertain fuzzy system (6) when η=1, η=2

Fig. 8

Simulation of random variables δ(k) and time‐varying delays τ(k) for the system (6)

Fig. 5

State trajectories of the uncertain fuzzy system (6) without control when η=1, η=2

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