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Stability and Constrained Control of a Class of Discrete-Time Fuzzy Positive Systems with Time-Varying Delays

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Abstract

This paper deals with the stability of nonlinear discrete-time positive systems with time-varying delays represented by the Takagi–Sugeno (T–S) fuzzy model. The time-varying delays in the systems can be unbounded. Sufficient conditions of stability which are not relevant to the magnitude of delays are derived by a solution trajectory. Based on the stability results, the problems of controller design via the parallel distributed compensation (PDC) scheme are solved. The control is under the positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. Constrained control is also considered, further requiring that the state trajectory of the closed-loop system be bounded by a prescribed boundary if the initial condition is bounded by the same boundary. The stability results and control laws are formulated as linear matrix inequalities (LMIs) and linear programs (LPs). A numerical example and a real plant are studied to demonstrate the application of the proposed methods.

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Abbreviations

A⪰0::

All entries of matrix A are nonnegative

A⪯0::

All entries of matrix A are nonpositive

A≻0::

All entries of matrix A are positive

A≺0::

All entries of matrix A are negative

R n::

n-dimensional nonnegative vector space

\(\mathbf{{R}}_{+}^{n}\)::

n-dimensional positive vector space

\(\mathcal{R}^{n}\)::

n-dimensional real vector space

\(\mathcal{R}^{n\times m}\)::

The set of all real matrices of (n×m) dimension

\(\mathcal{N}\)::

The set of the natural numbers

\(\mathcal{N}_{0}\)::

\(0 \cup\mathcal{N}\)

∥x∥::

l ∞ norm of vector \(x \in\mathcal {R}^{n}\), i.e., ∥x∥=max{|x 1|,…,|x n |}

⌈x⌉::

Smallest integer greater than or equal to real number x

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Acknowledgements

The work described in this paper was partially supported by the National Natural Science Foundation of China (Grant No. 60904004), the GRF: CityU 112809 of Hong Kong SAR Government, the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2009J020), and grants from Chengdu Administration of Science and Technology (11DXYB212JH-027 and 11DXYB212JH-02).

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Authors and Affiliations

  1. Centre for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, P.R. China

    Yanbing Mao & Hongbin Zhang

  2. School of Statistics, Southwestern University of Finance & Economics, Chengdu, 610074, P.R. China

    Yan Qin

  3. Department of Systems Engineering & Engineering Management, City University of Hong Kong, Kowloon, Hong Kong

    Chuangyin Dang

Authors
  1. Yanbing Mao
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  2. Hongbin Zhang
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  3. Yan Qin
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  4. Chuangyin Dang
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Correspondence to Yanbing Mao.

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Mao, Y., Zhang, H., Qin, Y. et al. Stability and Constrained Control of a Class of Discrete-Time Fuzzy Positive Systems with Time-Varying Delays. Circuits Syst Signal Process 32, 889–904 (2013). https://doi.org/10.1007/s00034-012-9471-5

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  • DOI: https://doi.org/10.1007/s00034-012-9471-5

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