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New Results on Decentralized \({\fancyscript{H}}_{\infty }\) Fuzzy Filtering Design for Continuous-Time Large-Scale Nonlinear Systems with Time-Varying Delay

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Abstract

This paper studies the problem of decentralized \({\fancyscript{H}}_{\infty }\) fuzzy filtering design for a class of continuous-time large-scale nonlinear systems with time-varying delay. The considered large-scale system consists of several nonlinear subsystems with interconnections and state delays. Each nonlinear subsystem is represented by a Takagi–Sugeno (T–S) model, and the state delay of each subsystem is assumed to be of an interval-like time-varying form. Attention is focused on designing a decentralized fuzzy filter such that the resulting filtering error system is asymptotically stable with a guaranteed \({\fancyscript{H}}_{\infty }\) disturbance attenuation level. We firstly propose a two-term approximation method to transform the filtering error system into an interconnected formulation and reformulate the problem of decentralized \({\fancyscript{H}}_{\infty }\) fuzzy filtering design in the context of input–output (IO) stability. Then, based on a Lyapunov–Krasovskii functional (LKF) combined with the scaled small gain (SSG) theorem, less conservative results are presented for the decentralized \({\fancyscript{H}}_{\infty }\) fuzzy filtering design in terms of linear matrix inequalities (LMIs). Finally, two examples are provided to illustrate the effectiveness of the proposed method.

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Acknowledgments

The authors would like to thank the Editor-in-Chief, the Associate Editor, and anonymous reviewers for their constructive comments which helped to greatly improve the presentation of this paper. The authors would also like to thank Prof. Huijun Gao and Prof. Jianbin Qiu from Harbin Institute of Technology for the helpful discussions on this work.

Author information

Authors and Affiliations

  1. Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, 150080, People’s Republic of China

    Zhixiong Zhong, Guanghui Sun & Jinyong Yu

  2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Tasawar Hayat & Fuad Alsaadi

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  1. Zhixiong Zhong
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  2. Guanghui Sun
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Correspondence to Zhixiong Zhong.

Additional information

This work was supported by the Self-Planned Task (NO. SKLRS201402C) of State Key Laboratory of Robotics and Systems (HIT), the National Natural Science Foundation of China (61374031), the Program for New Century Excellent Talents in University (NCET-12-0147), the Harbin Special Funds for Technological Innovation Research (2014RFQXJ067) and the Alexander von Humboldt Foundation of Germany.

Appendix

Appendix

Lemma 4

[6] For any constant positive symmetric matrix \(M\in \mathfrak {R}^{n\times n},M^{T}=M>0,\) scalars \(d_{2}>d_{1}\ge 0,\) the following inequality holds

$$\begin{aligned} \left( \int _{d_{1}}^{d_{2}}x\left( t\right) \mathrm{d}t\right) ^{T}M\left( \int _{d_{1}}^{d_{2}}x\left( t\right) \mathrm{d}t\right) \le \left( d_{2} -d_{1}\right) \int _{d_{1}}^{d_{2}}x^{T}\left( t\right) Mx\left( t\right) \mathrm{d}t. \end{aligned}$$

Lemma 5

[9] For any constant positive semidefinite symmetric matrix \(W\in \mathfrak {R}^{n\times n},W^{T}=W\ge 0,\) two positive integers \(n_{2}\) and \(n_{1}\) satisfying \(n_{2}\ge n_{1}\ge 1,\) the following inequality holds

$$\begin{aligned} \left( \sum \limits _{k=n_{1}}^{n_{2}}x\left( k\right) \right) ^{T}W\left( \sum \limits _{k=n_{1}}^{n_{2}}x\left( k\right) \right) \le \left( n_{2}-n_{1}+1\right) \sum \limits _{k=n_{1}}^{n_{2}}x^{T}\left( k\right) Wx\left( k\right) . \end{aligned}$$

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Zhong, Z., Sun, G., Yu, J. et al. New Results on Decentralized \({\fancyscript{H}}_{\infty }\) Fuzzy Filtering Design for Continuous-Time Large-Scale Nonlinear Systems with Time-Varying Delay. Circuits Syst Signal Process 35, 527–552 (2016). https://doi.org/10.1007/s00034-015-0081-x

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