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

Luís Tiago Paiva,,Systec–Isr, Faculdade De Engenharia, Universidade Do Porto, 4200-465, Porto, Portugal ,Fernando A C C Fontes

doi:10.3934/dcdsb.2019098

Luís Tiago Paiva, ,Systec–Isr, Faculdade De Engenharia, Universidade Do Porto, 4200-465, Porto, Portugal + Show 1 more

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https://doi.org/10.3934/dcdsb.2019098

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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.

Highlights

In this work, we address the problem of constructing a sampleddata feedback control law to stabilize a nonlinear continuous–time system using a Model Predictive Control (MPC) technique
An adaptive time–mesh refinement algorithm, to solve the optimal control problems involved in computing the sampled-data MPC law
The main features in the adaptation to an MPC context of the refinement strategy developed for an optimal control problem (OCP) are (a) the pre–defined levels of refinement are time–dependent, which requires a generalization of the initially developed procedure; (b) since similar optimal control problems (OCP’s) are repeatedly solved, the developed procedure projects in the new time–mesh the solution of a previous OCP, to create a warm start; (c) in addition, we show that the MPC with the adaptive time–mesh refinement (AMR) algorithm preserves stability, provided that the design parameters of the controller satisfy a given stability condition

Summary

Introduction

We address the problem of constructing a sampleddata feedback control law to stabilize a nonlinear continuous–time system using a Model Predictive Control (MPC) technique. An adaptive time–mesh refinement algorithm, to solve the optimal control problems involved in computing the sampled-data MPC law. A sampled-data feedback system is obtained whenever the variables in a plant evolve in continuous–time and are controlled using a digital device. This is the most frequent situation in applications with some degree of complexity requiring an advanced control, which is not easy or practical to implement in an analogue device. The resulting sampled-data systems are often addressed, in practice and in the literature, using a simplified discrete–time model. Regarding the MPC literature, in the last decade, frameworks using discrete–time models clearly dominate

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