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Robust Nonlinear H ∞ State Feedback Control of Polynomial Discrete-Time Systems: An Integrator Approach

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Abstract

This paper investigates the problem of designing a nonlinear H ∞ state feedback controller for polynomial discrete-time systems with norm-bounded uncertainties. In general, the problem of designing a controller for polynomial discrete-time systems is difficult, because it is a nonconvex problem. More precisely, in general, its Lyapunov function and control input are not jointly convex. Hence, it cannot be solved by semidefinite programming. In this paper, a novel approach is proposed, where an integrator is incorporated into the controller structure. In doing so, a convex formulation of the controller design problem can be rendered in a less conservative way than the available approaches. Furthermore, we establish the interconnection between robust H ∞ control of polynomial discrete-time systems with norm-bounded uncertainties and H ∞ control of scaled polynomial discrete-time systems. This establishment allows us to convert the robust H ∞ control problems to H ∞ control problems. Then, based on the sum of squares (SOS) approach, sufficient conditions for the existence of a nonlinear H ∞ state feedback controller are given in terms of solvability of polynomial matrix inequalities (PMIs), which can be solved by the recently developed SOS solvers. A tunnel diode circuit is used to demonstrate the validity of this integrator approach.

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Acknowledgements

The authors would like to express their sincere gratitude to Mr. Faiz Rasool, Mr. Mathias Krug, and Mr. Leo for an outstanding discussion regarding this topic. This project has been supported by University Grant (PJP/2013/FKEKK(10A)/S01177 of Universiti Teknikal Malaysia Melaka.

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Authors and Affiliations

  1. Faculty of Electronics and Computer Engineering, Universiti Teknikal Malaysia Melaka, Hang Tuah Jaya, 76100 Durian Tunggal, Malacca, Malaysia

    Shakir Saat & Zahriladha Zakaria

  2. Department of Electrical and Computer Engineering, The University of Auckland, Private Bag 92019, Auckland City, New Zealand

    Sing Kiong Nguang

  3. Department of Electrical Engineering, Yuan Ze University, No. 135, Far-Eastern Road, Chung-Li, Tao-Yuan, Taiwan, Republic of China

    Chih-Min Lin

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  1. Shakir Saat
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  2. Sing Kiong Nguang
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  3. Chih-Min Lin
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  4. Zahriladha Zakaria
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Correspondence to Shakir Saat.

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Saat, S., Nguang, S.K., Lin, CM. et al. Robust Nonlinear H ∞ State Feedback Control of Polynomial Discrete-Time Systems: An Integrator Approach. Circuits Syst Signal Process 33, 331–346 (2014). https://doi.org/10.1007/s00034-013-9645-9

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  • DOI: https://doi.org/10.1007/s00034-013-9645-9

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