A multiscale approximation scheme for explicit model predictive control with stability, feasibility, and performance guarantees

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

doi:10.1109/cdc.2009.5400583

Cited by 12 publications

(36 citation statements)

References 19 publications

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“…In addition, stability and constraint satisfaction are guaranteed a priori (as mentioned earlier) without needing convexity requirements on the value function corresponding to the original high-complexity controller as e.g. in [20]. These major advantages are obtained by requiring the original high-complexity controller to satisfy the ISS property, which under certain conditions is inherited from nominal stability of the high-complexity closed-loop system.…”

Section: Introductionmentioning

confidence: 99%

“…As advocated in [16], [20], [22], [24]- [27], it can be very beneficial for the eventual (circuit) implementation to use canonical PWA controllers that are based on regular partitions using e.g. regular simplices [20], [22], [24]- [27] or (multiscale) hypercubes [16].…”

Section: B Motivationmentioning

confidence: 99%

“…regular simplices [20], [22], [24]- [27] or (multiscale) hypercubes [16]. Essentially, any desired polytopic shape can be chosen in our approximation method that follows.…”

Section: B Motivationmentioning

confidence: 99%

“…Besides, no a priori stability guarantees are provided, only an a posteriori MILPbased stability test. The authors of [16] overcome the latter issues in [20], where an approximation method for constrained linear systems is proposed, which also uses a multiscale structure as in [16], but now based on barycentric interpolation. Bounds guaranteeing stability and constraint satisfaction are obtained a priori.…”

Section: Introductionmentioning

confidence: 99%

“…The approximate control law is defined on a regular (possibly multiscale) partition, which can be chosen in a desirable way, e.g. based on multiscale rectangles as in [16] or regular simplices as in [20], [22], [24]- [27]. Due to the choice of a regular partition, the resulting regular PWA functions inherently result in efficient, low-complexity on-line implementations in terms of both computational effort and memory requirements.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: B Motivationmentioning

confidence: 99%

“…regular simplices [20], [22], [24]- [27] or (multiscale) hypercubes [16]. Essentially, any desired polytopic shape can be chosen in our approximation method that follows.…”

Section: B Motivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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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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Approximate model predictive building control via machine learning

et al. 2018

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A multiscale approximation scheme for explicit model predictive control with stability, feasibility, and performance guarantees

Cited by 12 publications

References 19 publications

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

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

Approximate model predictive building control via machine learning

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