On input-to-state stability of min–max nonlinear model predictive control

Lazăr, Mircea; Peña, David Muñoz de la; Heemels, W.P.M.H.; Álamo, T.

doi:10.1016/j.sysconle.2007.06.013

Cited by 143 publications

(104 citation statements)

References 16 publications

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“…(12) Proof: By assuming convexity of S i (·) we have (12) is equivalent to (9) with p i (α, x) as in (11). From Theorem 2.7 it follows that any control law, i.e.…”

Section: Resultsmentioning

confidence: 99%

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Low-complexity polynomial approximation of explicit MPC via linear programming

Kvasnica

Löfberg

Herceg

et al. 2010

Proceedings of the 2010 American Control Conference

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Abstract-This paper addresses the issue of the practical implementation of Model Predictive Controllers (MPC) to processes with short sampling times. Given an explicit solution to an MPC problem, the main idea is to approximate the optimal control law defined over state space regions by a single polynomial of pre-specified degree which, when applied as a state-feedback, guarantees closed-loop stability, constraint satisfaction, and a bounded performance decay. It is shown how to search for such a polynomial by solving a single linear program.

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“…(12) Proof: By assuming convexity of S i (·) we have (12) is equivalent to (9) with p i (α, x) as in (11). From Theorem 2.7 it follows that any control law, i.e.…”

Section: Resultsmentioning

confidence: 99%

“…Theorem 2.5 (Asymptotic/exponential stability [11]): Let X T be a bounded positively invariant set in R nx for the autonomous (closed-loop) system…”

Section: Assumption 24 (Stability Feasibility)mentioning

confidence: 99%

Low-complexity polynomial approximation of explicit MPC via linear programming

Kvasnica

Löfberg

Herceg

et al. 2010

Proceedings of the 2010 American Control Conference

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“…Lemma 1: [30], [31]: Let α 1 , α 2 , γ, σ ∈ K ∞ . Let X with 0 ∈ int(X) be an RPI set with respect to E for system (3) and let V : X → R + be a function with V (0) = 0.…”

Section: Preliminariesmentioning

confidence: 99%

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

Genuit

Heemels

2012

IET Control Theory &Amp; Applications

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Piecewise affine (PWA) feedback control laws defined on general polytopic partitions, as for instance obtained by explicit MPC, will often be prohibitively complex for fast systems. In this work we 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 (SDPs) or linear programs (LPs), 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, we provide two tractable computational methods for deriving the necessary ISS inequalities from nominal stability. A numerical example is provided that illustrates the main results.The authors are with the Hybrid and

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“…Theorem II.4: [13], [17] Let 1; 2; 3 2 K1, 2 K and let be a subset of d . Let with 0 2 int( ) be a RPI set with respect to for system (1) and let V : 0 !…”

Section: B Stability and Input-to-state Stabilitymentioning

confidence: 99%