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

Summers, Sean; Raimondo, Davide M.; Jones, Colin N.; Lygeros, John; Morari, Manfred

doi:10.3182/20100901-3-it-2016.00275

Cited by 19 publications

(6 citation statements)

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“…For example, the assumption of the knowledge of a global control Lyapunov function as required in directly implies Condition . Other possibilities to verify this assumption can be found in the aforementioned references or are given by suboptimal explicit offline controller design methods, for example, or interval arithmetic methods . Furthermore, assumptions about an upper bound on the optimal cost function in terms of the stage cost similar to Assumption are additionally required in ,Corollary 9 and ,Corollary 10.…”

Section: Preliminaries and Assumptionsmentioning

confidence: 99%

Control over erasure channels: stochastic stability and performance of packetized unconstrained model predictive control

Reble

Quevedo

Allgöwer

2012

Intl J Robust & Nonlinear

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SUMMARYIn this work, we consider the control of discrete‐time constrained nonlinear systems over unreliable packet‐based communication networks. The random packet dropouts are modeled by a two‐state Markov chain, and no acknowledgments of receipt are assumed. To weaken the impact of the packet dropouts, the controller transmits packets containing more than one future control input, and a suitable buffering is applied at the plant actuator side. Because the Markov chain model adopted does not ensure the number of consecutive packet dropouts to be bounded, deterministic stability cannot be guaranteed. Hence, we are interested in stochastic stability of the closed loop instead. We propose an unconstrained model predictive control scheme without additional terminal weighting term for the calculation of the control inputs. Two major advantages of this unconstrained model predictive control scheme can be emphasized. First, to guarantee stochastic stability, we do not require the knowledge of a global control Lyapunov function as terminal cost term but instead only a less restrictive controllability assumption. Second, guaranteed performance bounds on the expected infinite horizon cost of the closed loop can be obtained. Copyright © 2012 John Wiley & Sons, Ltd.

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Section: Preliminaries and Assumptionsmentioning

confidence: 99%

Control over erasure channels: stochastic stability and performance of packetized unconstrained model predictive control

Reble

Quevedo

Allgöwer

2012

Intl J Robust & Nonlinear

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“…Note that, this approach searches the low-complexity control law in the space of continuous PWA functions due to the fact that interpolation techniques of continuous PWA functions is used. In [7], the result of [6] was extended to nonlinear model predictive control (NMPC). An alternative method was proposed in [8], where an approximation procedure for MPC controllers for constrained linear systems is presented using canonical PWA functions based on regular simplices, with guarantees of local optimality and constraint satisfaction.…”

Section: Introductionmentioning

confidence: 99%

Synthesis of low-complexity stabilizing piecewise affine controllers: A control-Lyapunov function approach

Heemels

Bemporad

2011

IEEE Conference on Decision and Control and European Control Conference

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Explicit model predictive controllers computed exactly by multi-parametric optimization techniques often lead to piecewise affine (PWA) state feedback controllers with highly complex and irregular partitionings of the feasible set. In many cases complexity prohibits the implementation of the resulting MPC control law for fast or large-scale system. This paper presents a new approach to synthesize low-complexity PWA controllers on regular partitionings that enhance fast on-line implementation with low memory requirements. Based on a PWA control-Lyapunov function, which can be obtained as the optimal cost for a constrained linear system corresponding to a stabilizing MPC setup, the synthesis procedure for the lowcomplexity control law boils down to local linear programming (LP) feasibility problems, which guarantee stability, constraint satisfaction, and certain performance requirements. Initially, the PWA controllers are computed on a fixed regular partitioning. However, we also present an automatic refinement procedure to refine the partitioning where necessary in order to satisfy the design specifications. A numerical example show the effectiveness of the novel approach.This work was partially supported by the European Commission through project MOBY-DIC "Model-based synthesis of digital electronic circuits for embedded control" (FP7-INFSO-ICT-248858).Liang Lu is currently with the State Key Laboratory of Synthetical

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“…Thus, in this chapter we exploit advances in reachability analysis and adaptive interpolation to construct an approximate explicit control law that encompasses the strengths of the recent works ([8, 19]) while guaranteeing stability and feasibility and preserving a minimal representation of the control law. 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]).…”

Section: Introductionmentioning

confidence: 99%

“…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]).…”

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.…”

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confidence: 99%

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

Cited by 19 publications

References 20 publications

Control over erasure channels: stochastic stability and performance of packetized unconstrained model predictive control

Control over erasure channels: stochastic stability and performance of packetized unconstrained model predictive control

Synthesis of low-complexity stabilizing piecewise affine controllers: A control-Lyapunov function approach

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

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