On explicit suboptimal LQR with state and input constraints

Abstract-A model predictive control law is given by the solution to a parametric optimization problem that can be precomputed offline and provides an explicit map from state to control input. In this paper, an algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second order interpolets. The resulting interpolation is shown to be everywhere feasible and continuous. Further, tests to confirm stability and to compute a bound on the performance loss are introduced. Since the controller approximation is built on a grid hierarchy, convergence to a stabilizing control law is guaranteed and the evaluation of the control law in real-time systems is naturally fast and runs in a bounded logarithmic time. Two examples are provided; A two-dimensional example with an evaluation speed of 31 ns and a four-dimensional example with an evaluation speed of 119 ns.

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“…for all R ∈ R k do 7: Check Stability by Theorem (2) and Feasibility by Lemma (3) for current region R 8:…”

Section: B Online Computationmentioning

confidence: 99%

A Multiresolution Approximation Method for Fast Explicit Model Predictive Control

Summers

Jones

Lygeros

et al. 2011

IEEE Trans. Automat. Contr.

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“…Algorithms have been developed by (Bemporad et al 2000b, Johansen et al 2000b, Tøndel et al 2001 for constructing polyhedral partitions of the state space that explicitly defines the PWL functionẑ * (x). Below, we give a simplified description of the algorithm, while a complete description and analysis that also covers degeneracy and infeasibility is found in (Tøndel et al 2001):…”

Section: Multi-parametric Quadratic Programmingmentioning

confidence: 99%

“…Recently, several algorithms for computing explicit solutions to constrained linear model predictive control (MPC) problems have been reported (Bemporad et al 1999, Bemporad et al 2000b, Seron et al 2000, Bemporad et al 2000a, Tøndel et al 2001, Johansen et al 2000b, Johansen et al 2000a. Their main motivation is that an explicit solution avoids the need for real-time optimization, and may therefore open new application areas where MPC has not traditionally been used due to the need for high sampling rates or software reliability issues.…”

Section: Introductionmentioning

confidence: 99%

“…In (Johansen et al 2000b, Johansen et al 2000a a different solution approach is taken, starting with the Hamilton-Jacobi-Bellman equation for the optimal control problem. The solution strategy allows suboptimality and complexity reduction to be introduced by pre-determining a small number of sampling in-1 Email: Tor.Arne.Johansen@itk.ntnu.no stants when the active set is allowed to change on the horizon.…”

Section: Introductionmentioning

confidence: 99%

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Complexity Reduction in Explicit Linear Model Predictive Control

Tøndel

Johansen

2002

IFAC Proceedings Volumes

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Explicit piecewise linear (PWL) state feedback solutions to constrained linear model predictive control (MPC) problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the components of the state vector. This allows MPC to be implemented via a PWL function evaluation without real-time optimization. The main drawback of this approach is dramatic increase in off-line computational complexity and number of regions in the state space partition as the number of states, inputs and constraints increases. Here we study two approaches to complexity reduction. First, we consider input trajectory parameterization. Second, we develop a search tree that allows PWL function evaluation to be implemented in real time with low computational complexity.

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“…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.…”

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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On explicit suboptimal LQR with state and input constraints

Cited by 39 publications

References 11 publications

A Multiresolution Approximation Method for Fast Explicit Model Predictive Control

A Multiresolution Approximation Method for Fast Explicit Model Predictive Control

Complexity Reduction in Explicit Linear Model Predictive Control

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

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