Guaranteed Characterization of Capture Basins of Nonlinear State-Space Systems

Delanoue, Nicolas; Jaulin, Luc; Hardouin, Laurent; Lhommeau, Mehdi

doi:10.1007/978-3-540-85640-5_20

Cited by 4 publications

(5 citation statements)

References 8 publications

(8 reference statements)

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“…Researchers started to apply viability theory on control technology since 1990 . Related works compute the viability kernels and the capture basins for systems without uncertainties, whereas the works of Zarch et al handled fault tolerance evaluation. Most of them compute the viability kernels and the capture basins by reachability analysis, and the analytic methods are mainly categorized to Hamilton‐Jacobi equations, zonotopes, and interval methods.…”

Section: Introductionmentioning

confidence: 99%

“…Most of them compute the viability kernels and the capture basins by reachability analysis, and the analytic methods are mainly categorized to Hamilton‐Jacobi equations, zonotopes, and interval methods. The algorithms of zonotope have been adopted in the works of Zarch et al, whereas tools of interval methods have been taken in related works, and comparatively, Hamilton‐Jacobi equations have been used in other works . As both control and uncertain parametric variables, in viability theory, are regarded as parameters in a parameterized dynamic system, prior to 1994, discriminating kernel proposed in the work of Cardaliaguet et al has distinguished them.…”

Section: Introductionmentioning

confidence: 99%

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Computation of discriminating kernel and robust capture basin with regulation map by interval methods

Peng

Stancu

Dang

2019

Intl J Robust & Nonlinear

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This paper proposes a novel algorithm that characterizes the robust capture basin and the discriminating kernel for constrained nonlinear systems with uncertainties based on viability theory. For nonlinear systems with constrained inputs and bounded uncertainties, the viability kernel is the largest set of states possessing a possibility to be viable in a set, and the capture basin is the largest set of states possessing a possibility to reach a target in a finite time, and keeping viable in a set before reaching the target. However, in the viability theory, both control and uncertainty in a parameterized system are considered as parameters:the discriminating kernel and the proposed robust capture basin link viability theory with robust control, which take both control and uncertainties into account. For the constrained uncertain nonlinear systems, the discriminating kernel is the largest set of states that is robust invariant in a set with proper control, and the robust capture basin is the largest set of states reaching their target in finite time with proper control despite of uncertainties and keeping viable in a set before reaching the target. Furthermore, we map all the states to optimal regulatory control such that the systems are regulated by a regulation map. To compute the robust capture basin and the discriminating kernel, we use interval methods to provide guaranteed solutions. The proposed algorithms in this paper approximate an outer approximation of the minimum reachable target and inner approximations of the robust capture basin and the discriminating kernel in a guaranteed way.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computation of discriminating kernel and robust capture basin with regulation map by interval methods

Peng

Stancu

Dang

2019

Intl J Robust & Nonlinear

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“…Extending the results of [30,31], in this chapter we introduce a constructive algorithm for the approximation of an explicit receding horizon NMPC control law. We approximate the optimal control law by adaptive interpolation using second order interpolets, while concurrently verifying feasibility and stability of the resulting feedback system via the computation of an inner approximation of the capture basin (see, e.g., [12]). In contrast to the capture basin computational method considered in [12,31], we develop a mechanism for computing the capture basin using zonotopes [22,11, 33] and DC programming [3] that significantly reduces the complexity of the combined function approximation and verification procedure.…”

Section: Introductionmentioning

confidence: 99%

“…We approximate the optimal control law by adaptive interpolation using second order interpolets, while concurrently verifying feasibility and stability of the resulting feedback system via the computation of an inner approximation of the capture basin (see, e.g., [12]). In contrast to the capture basin computational method considered in [12,31], we develop a mechanism for computing the capture basin using zonotopes [22,11,33] and DC programming [3] that significantly reduces the complexity of the combined function approximation and verification procedure. Using zonotopes and DC programming rather than interval analysis [25,6] additionally leads to an approximate control law with less storage requirements and a larger verifiable region of attraction.…”

Section: Introductionmentioning

confidence: 99%

A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

Raimondo

Riverso

Summers

et al. 2012

Distributed Decision Making and Control

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In this chapter an algorithm for nonlinear explicit model predictive control is presented. A low complexity receding horizon control law is obtained by approximating the optimal control law using multiscale basis function approximation. Simultaneously, feasibility and stability of the approximate control law is ensured through the computation of a capture basin (region of attraction) for the closed-loop system. In a previous work, interval methods were used to construct the capture basin (feasible region), yet this approach suffered due to slow computation times and high grid complexity.In this chapter, we suggest an alternative to interval analysis based on zonotopes. The suggested method significantly reduces the complexity of the combined function approximation and verification procedure through the use of DC (difference of convex) programming, and recursive splitting. The result is a multiscale function approximation method with improved computational efficiency for fast nonlinear explicit model predictive control with guaranteed stability and constraint satisfaction. IntroductionThis chapter proposes a method of approximate explicit model predictive control (MPC) for nonlinear systems. While it is possible to compute the optimal control law offline for a limited number of cases (e.g., affine or piecewise affine dynamics [5,20,29]), it is in general necessary to approximate, and therefore validation techniques are required for the resulting approximate closed-loop system. In this chapter, we present a new technique for approximation and certification of stability and recursive feasibility for explicit NMPC controllers, in which the control law is precomputed and verified offline in order to speed online computation. The control law is approximated via an adaptive interpolation using second order interpolets, which results in an extremely fast online computation time and low data storage. The resulting suboptimal closed-loop system is verified by computing an inner approximation of the capture basin and an algorithm is proposed that iteratively improves the approximation where needed in order to maximize the size of the capture basin. The key novelty of this chapter is the use of difference of convex (DC) programming and zonotope approximation in order to significantly improve both the computational performance and efficacy of the calculation of the capture basin.Methods for the approximation of explicit solutions of nonlinear model predictive control (NMPC) problems have been addressed recently by various authors (e.g., see [8,19]). In [8], the authors compute an approximate control lawũ(x) with a bound on the controller approximation error ( u * (x) −ũ(x) ), from which performance and stability properties are derived using set membership (SM) function approximation theory. In [19] the authors use multiparametric nonlinear programming to compute an explicit approximate solution of the NMPC problem defined on an orthogonal structure of the state-space partition. An additional example of the approximat...

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Section: Introductionmentioning

confidence: 99%

Fast explicit nonlinear model predictive control via multiresolution function approximation with guaranteed stability

Summers

Raimondo

Jones

et al. 2010

IFAC Proceedings Volumes

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In this paper an algorithm for nonlinear explicit model predictive control is introduced based on multiresolution function approximation that returns a low complexity approximate receding horizon control law built on a hierarchy of second order interpolets. Feasibility and stability guarantees for the approximate control law are given using reachability analysis, where interval methods are used to construct a capture basin (feasible region). A constructive algorithm is provided that combines adaptive function approximation with interval methods to build a receding horizon control law that is suboptimal, yet with a region of guaranteed feasibility and stability. The resulting control law is built on a grid hierarchy that is fast to evaluate in real-time systems.

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Guaranteed Characterization of Capture Basins of Nonlinear State-Space Systems

Cited by 4 publications

References 8 publications

Computation of discriminating kernel and robust capture basin with regulation map by interval methods

Computation of discriminating kernel and robust capture basin with regulation map by interval methods

A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

Fast explicit nonlinear model predictive control via multiresolution function approximation with guaranteed stability

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