Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability

Luís Tiago Paiva; Fernando A. C. C. Fontes

doi:10.3934/dcdsb.2019098

Article Contents

2019, Volume 24, Issue 5: 2335-2364. Doi: 10.3934/dcdsb.2019098

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Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability

Systec–ISR, Faculdade de Engenharia, Universidade do Porto, 4200-465, Porto, Portugal

Received: December 31, 2017

Revised: December 31, 2018

Early access: March 2019

Published: March 2019

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Abstract

This article addresses the problem of controlling a constrained, continuous–time, nonlinear system through Model Predictive Control (MPC). In particular, we focus on methods to efficiently and accurately solve the underlying optimal control problem (OCP). In the numerical solution of a nonlinear OCP, some form of discretization must be used at some stage. There are, however, benefits in postponing the discretization process and maintain a continuous-time model until a later stage. This is because that way we can exploit additional freedom to select the number and the location of the discretization node points.We propose an adaptive time–mesh refinement (AMR) algorithm that iteratively finds an adequate time–mesh satisfying a pre–defined bound on the local error estimate of the obtained trajectories. The algorithm provides a time–dependent stopping criterion, enabling us to impose higher accuracy in the initial parts of the receding horizon, which are more relevant to MPC. Additionally, we analyze the conditions to guarantee closed–loop stability of the MPC framework using the AMR algorithm. The numerical results show that the proposed AMR strategy can obtain solutions as fast as methods using a coarse equidistant–spaced mesh and, on the other hand, as accurate as methods using a fine equidistant–spaced mesh. Therefore, the OCP can be solved, and the MPC law obtained, faster and/or more accurately than with discrete-time MPC schemes using equidistant–spaced meshes.

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Mathematics Subject Classification: Primary: 49J15; Secondary: 65L50, 93D20, 93C57, 93C10.

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Figure 1. Illustration of the multi–level adaptive time–mesh refinement strategy

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Figure 2. Illustration of the extended (time–dependent) time–mesh refinement strategy with different refinement thresholds

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Figure 3. Illustration of the extended time–mesh refinement algorithm for MPC

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Figure 4. Construction of the (extended) admissible control $ {\bf{\tilde u}} $ with $ \Pi = \{t_k\}_{k \in \mathbb{N}} $, $ t_k = k \delta $, and with $ \pi_r = \{s_i\}_{i \in 0, 1, \ldots N_r} $, $ s_i = i \delta/2 $

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Figure 5. Car–like system geometry

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Figure 6. Pathwise state constraints (13) for (P_CP)

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Figure 7. Optimal path computed in the initial coarse mesh

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Figure 8. Discretization error estimate in the initial coarse mesh

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Figure 9. Optimal path computed in the final mesh $ \pi_{\rm{AMR}} $

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Figure 10. Optimal trajectory and control

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Figure 11. Discretization error in the coarse mesh and the MPC refining levels

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Figure 12. Path resulting from the AMR–MPC scheme

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Figure 13. Trajectory and control resulting from the AMR–MPC scheme

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Table 1. Results for problem (P_CP) solved in each time-mesh

$ \pi_j $	$ N_j $	$ \Delta t_j $	$ I_j $	$ \left\|\left\|\varepsilon_{\bf{x}}^{(j)}\right\|\right\|_\infty $	CPU time (s)
$ \pi_j $	$ N_j $	$ \Delta t_j $	$ I_j $		Solver	$ \varepsilon_{\bf{x}} $
$ \pi_0 $	21	$ 0.5 $	42	$ 1.0016{\rm{E}}^{-4} $	$ 0.9816 $	$ 0.0563 $
$ \pi_1 $	82	$ 1/54 $	42	$ 3.3801{\rm{E}}^{-7} $	$ 0.7061 $	$ 0.0642 $
$ \pi_{\rm{AMR}} $	82	$ 1/54 $	84	$ 3.3801{\rm{E}}^{-7} $	$ 1.6877 $	$ 0.1205 $

$ \pi_{\rm{F}} $	541	$ 1/54 $	403	$ 4.0358{\rm{E}}^{-7} $	$ 13.2473 $	$ 0.4675 $

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Table 2. Results for each MPC and AMR iterations

MPC Iter	AMR Iter	$ N_j $	$ \Delta t_j $	$ I_j $	$ \left\|\left\|\varepsilon_{\bf{x}}^{(j)}\right\|\right\|_\infty $	CPU time (s)
MPC Iter	AMR Iter	$ N_j $	$ \Delta t_j $	$ I_j $		Solver	$ \varepsilon_{\bf{x}} $
	$ \pi_{0} $	21	0.5	$ 42 $	$ 1.002{\rm{E}}^{-4} $	$ 0.982 $	$ 0.0563 $

1	$ \pi_{1} $	21	0.5	$ 8 $	$ 1.002{\rm{E}}^{-4} $	$ 0.105 $	$ 0.0156 $
	$ \pi_{2} $	52	0.0625	$ 22 $	$ 3.525{\rm{E}}^{-6} $	$ 0.344 $	$ 0.0374 $
	$ \pi_{\rm{AMR}} $	52	0.0625	$ 30 $	$ 3.525{\rm{E}}^{-6} $	$ 0.449 $	$ 0.0530 $

2	$ \pi_{1}=\pi_{\rm{AMR}} $	31	0.0625	$ 11 $	$ 3.525{\rm{E}}^{-6} $	$ 0.1564 $	$ 0.0230 $

3	$ \pi_{1}=\pi_{\rm{AMR}} $	21	0.5	$ 11 $	$ 2.042{\rm{E}}^{-7} $	$ 0.1639 $	$ 0.0139 $

4	$ \pi_{1}=\pi_{\rm{AMR}} $	21	0.5	$ 7 $	$ 4.321{\rm{E}}^{-7} $	$ 0.0936 $	$ 0.0126 $

5	$ \pi_{1}=\pi_{\rm{AMR}} $	21	0.5	$ 7 $	$ 4.515{\rm{E}}^{-7} $	$ 0.0912 $	$ 0.0123 $

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