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Asynchronous Decentralized Fuzzy Observer-Based Output Feedback Control of Nonlinear Large-Scale Systems

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Abstract

This paper is concerned with the problem of decentralized robust \({{\mathscr {H}}}_{\infty }\) output feedback control for a class of continuous-time nonlinear large-scale systems based on T–S fuzzy affine models. Based on a common quadratic Lyapunov function and piecewise quadratic Lyapunov functions, some new results are proposed to the asynchronous observer-based controller synthesis for T–S fuzzy affine large-scale systems with unmeasurable premise variables. The observer and controller gains can be attained through solving a set of linear matrix inequalities. Finally, two simulation examples are presented to verify the effectiveness of the proposed approaches.

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Author information

Authors and Affiliations

  1. State Key Laboratory of Robotics and Systems, and Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Wenqiang Ji, Shasha Fu, Hailong Chen & Jianbin Qiu

Authors
  1. Wenqiang Ji
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  2. Shasha Fu
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  3. Hailong Chen
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  4. Jianbin Qiu
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Corresponding author

Correspondence to Jianbin Qiu.

Additional information

This work was supported in part by the Self-Planned Task of State Key Laboratory of Robotics and Systems of Harbin Institute of Technology (No. SKLRS201801A03), the National Natural Science Foundation of China (No. 61873311), and the 111 Project (B16014).

Appendix

Appendix

Lemma 1

[33] For real matricesSandMwith appropriate dimensions, the following inequality

$$\begin{aligned} S^{{\mathrm {T}}}M+M^{{\mathrm {T}}}S\le \varepsilon S^{{\mathrm {T}}}S+\varepsilon ^{-1} M^{{\mathrm {T}}}M \end{aligned}$$

holds for a parameter\(\varepsilon >0\).

Lemma 2

(Tchebyshev’s inequality) For arbitrary vectors\(x_{l}\in {\mathfrak {R}}^{n}\), \(l=1,2,\ldots , N\), the following inequality holds

$$\begin{aligned} \left[ \sum\limits_{l=1}^{N}x_{l}\right] ^{{\mathrm {T}}}\left[ \sum\limits_{l=1}^{N}x_{l}\right] \le N\sum\limits_{l=1}^{N}x^{{\mathrm {T}}}_{l}x_{l}. \end{aligned}$$

Lemma 3

(Projection lemma) [13] Given matrices\({{{\mathcal {W}}}}={{\mathcal {W}}}^{{\mathrm {T}}}\in {\mathfrak {R}}^{n \times n}\), \({{{\mathcal {V}}}}\in {\mathfrak {R}}^{m \times n}\), \({{{\mathcal {U}}}}\in {\mathfrak {R}}^{s \times n}\), the following LMI inequality

$$\begin{aligned} {{{\mathcal {W}}}}+{{{\mathcal {U}}}}^{{\mathrm {T}}} {{{\mathcal {X}}}}^{{\mathrm {T}}} {{{\mathcal {V}}}}+{{{\mathcal {V}}}}^{{\mathrm {T}}}{{{\mathcal {X}}}} {{{\mathcal {U}}}}<0, \end{aligned}$$

is solvable for the variable\({{{\mathcal {X}}}}\)if and only if

$$\begin{aligned} \left\{ \begin{array}{ll} {{{\mathcal {U}}}}^{{\mathrm {T}}}_{\bot }{{{\mathcal {W}}}}{{{\mathcal {U}}}}_{\bot }<0,&\quad \text{ if }\ {{{\mathcal {V}}}}_{\bot }={\mathbf{0 }}, {{{\mathcal {U}}}}_{\bot }\ne {\mathbf{0 }},\\ {{{\mathcal {V}}}}^{{\mathrm {T}}}_{\bot }{{{\mathcal {W}}}}{{{\mathcal {V}}}}_{\bot }<0,&\quad \text{ if }\ {{{\mathcal {U}}}}_{\bot }={\mathbf{0 }}, {{{\mathcal {V}}}}_{\bot }\ne {\mathbf{0 }},\\ {{{\mathcal {U}}}}^{{\mathrm {T}}}_{\bot }{{{\mathcal {W}}}}{{{\mathcal {U}}}}_{\bot }<0,\quad {{{\mathcal {V}}}}^{{\mathrm {T}}}_{\bot }{{{\mathcal {W}}}}{{{\mathcal {V}}}}_{\bot }<0,&\quad \text{ if }\ {{{\mathcal {V}}}}_{\bot }\ne {\mathbf{0 }}, {{{\mathcal {U}}}}_{\bot }\ne {\mathbf{0 }},\\ \end{array}\right. \end{aligned}$$

where\({{{\mathcal {V}}}}_{\bot }\)and\({{{\mathcal {U}}}}_{\bot }\)stand forthe right null spaces of\({{{\mathcal {V}}}}\)and\({{{\mathcal {U}}}}\), respectively.

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Ji, W., Fu, S., Chen, H. et al. Asynchronous Decentralized Fuzzy Observer-Based Output Feedback Control of Nonlinear Large-Scale Systems. Int. J. Fuzzy Syst. 21, 19–32 (2019). https://doi.org/10.1007/s40815-018-0565-5

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