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Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach

Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach

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  • Author(s): B.A.G. Genuit 1 ; L. Lu 1, 2 ; W.P.M.H. Heemels 1
    • View affiliations
    • Affiliations: 1: The Hybrid and Networked Systems Group, Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
      2: The State Key Laboratory of Integrated Automation for Process Industries, Northeastern University, Shenyang, People's Republic of China
  • Source: Volume 6, Issue 8, 17 May 2012, p. 1015 – 1028
    DOI:  10.1049/iet-cta.2010.0709 , Print ISSN 1751-8644, Online ISSN 1751-8652
© The Institution of Engineering and Technology
Published

Piecewise affine (PWA) feedback control laws defined on general polytopic partitions, as for instance obtained by explicit model predictive control, will often be prohibitively complex for fast systems. In this work the authors study the problem of approximating these high-complexity controllers by low-complexity PWA control laws defined on more regular partitions, facilitating faster on-line evaluation. The approach is based on the concept of input-to-state stability (ISS). In particular, the existence of an ISS Lyapunov function (LF) is exploited to obtain a priori conditions that guarantee asymptotic stability and constraint satisfaction of the approximate low-complexity controller. These conditions can be expressed as local semidefinite programs or linear programs, in case of 2-norm or 1, ∞-norm-based ISS, respectively, and apply to PWA plants. In addition, as ISS is a prerequisite for our approximation method, the authors provide two tractable computational methods for deriving the necessary ISS inequalities from nominal stability. A numerical example is provided that illustrates the main results.

Inspec keywords: Lyapunov methods; predictive control; approximation theory; feedback; asymptotic stability; mathematical programming

Other keywords: input-to-state stability approach; approximation method; piecewise affine feedback control laws; explicit model predictive control; high-complexity controllers; regular piecewise affine functions; local semidefinite programs; asymptotic stability; PWA; nominal stability; constraint satisfaction; low-complexity PWA control laws; general polytopic partitions; ISS Lyapunov function

Subjects: Optimisation techniques; Optimal control; Control system analysis and synthesis methods; Stability in control theory; Interpolation and function approximation (numerical analysis)

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