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Practical Considerations for Implementing Robust-to-Early Termination Model Predictive Control

Mohsen Amiri, Mehdi Hosseinzadeh

TL;DR

This work addresses the challenge of implementing MPC under limited computing resources by extending Robust-to-Early Termination (REAP) to discrete time. It develops a discretized primal-dual gradient framework, where the KKT parameter $\\sigma(\\tau|t)$ is adaptively tuned to maintain invariant constraint sets and ensure convergence to the high-fidelity solution even when computation is prematurely terminated. Theoretical guarantees are provided for discrete-time REAP’s convergence and anytime feasibility, and the approach is validated via extensive simulations and real-world experiments on a Parrot Bebop 2 drone, demonstrating zero constraint violations and suboptimality below 0.3% relative to a powerful MPC. A MATLAB toolbox, DiscreteREAP, accompanies the work to facilitate adoption and experimentation in resource-constrained control scenarios.

Abstract

Model Predictive Control (MPC) is widely used to achieve performance objectives, while enforcing operational and safety constraints. Despite its high performance, MPC often demands significant computational resources, making it challenging to implement in systems with limited computing capacity. A recent approach to address this challenge is to use the Robust-to-Early Termination (REAP) strategy. At any time instant, REAP converts the MPC problem into the evolution of a virtual dynamical system whose trajectory converges to the optimal solution, and provides guaranteed sub-optimal and feasible solution whenever its evolution is terminated due to limited computational power. REAP has been introduced as a continuous-time scheme and its theoretical properties have been derived under the assumption that it performs all the computations in continuous time. However, REAP should be practically implemented in discrete-time. This paper focuses on the discrete-time implementation of REAP, exploring conditions under which anytime feasibility and convergence properties are maintained when the computations are performed in discrete time. The proposed methodology is validated and evaluated through extensive simulation and experimental studies.

Practical Considerations for Implementing Robust-to-Early Termination Model Predictive Control

TL;DR

This work addresses the challenge of implementing MPC under limited computing resources by extending Robust-to-Early Termination (REAP) to discrete time. It develops a discretized primal-dual gradient framework, where the KKT parameter is adaptively tuned to maintain invariant constraint sets and ensure convergence to the high-fidelity solution even when computation is prematurely terminated. Theoretical guarantees are provided for discrete-time REAP’s convergence and anytime feasibility, and the approach is validated via extensive simulations and real-world experiments on a Parrot Bebop 2 drone, demonstrating zero constraint violations and suboptimality below 0.3% relative to a powerful MPC. A MATLAB toolbox, DiscreteREAP, accompanies the work to facilitate adoption and experimentation in resource-constrained control scenarios.

Abstract

Model Predictive Control (MPC) is widely used to achieve performance objectives, while enforcing operational and safety constraints. Despite its high performance, MPC often demands significant computational resources, making it challenging to implement in systems with limited computing capacity. A recent approach to address this challenge is to use the Robust-to-Early Termination (REAP) strategy. At any time instant, REAP converts the MPC problem into the evolution of a virtual dynamical system whose trajectory converges to the optimal solution, and provides guaranteed sub-optimal and feasible solution whenever its evolution is terminated due to limited computational power. REAP has been introduced as a continuous-time scheme and its theoretical properties have been derived under the assumption that it performs all the computations in continuous time. However, REAP should be practically implemented in discrete-time. This paper focuses on the discrete-time implementation of REAP, exploring conditions under which anytime feasibility and convergence properties are maintained when the computations are performed in discrete time. The proposed methodology is validated and evaluated through extensive simulation and experimental studies.
Paper Structure (18 sections, 3 theorems, 31 equations, 11 figures, 1 table)

This paper contains 18 sections, 3 theorems, 31 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem statement
  3. MPC Formulation
  4. Terminal Cost Weight $Q_N$
  5. Terminal Constraint Set $\Omega$
  6. Final MPC Problem
  7. Robust-to-Early Termination MPC
  8. Goal
  9. Discrete-Time Implementation of REAP
  10. Theoretical Properties
  11. DiscreteREAP Toolbox
  12. Simulation and Experimental Results
  13. Simulation Analysis and Comparison Study
  14. Constraints on Control Inputs and States
  15. Suboptimality Assessment
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\hat{\mathbf{u}}(\tau-1|t)\in\mathcal{D}_{\hat{\mathbf{u}}}$ and $\hat{\lambda}(\tau-1|t)\in\mathcal{D}_{\hat{\lambda}}$. Let the KKT parameter $\sigma(\tau|t)$ be selected such that the following condition is satisfied: where $\sigma_i^{\hat{\mathbf{u}}}(\tau|t)$ and $\sigma_i^{\hat{\lambda}}(\tau|t)$ are as in eq:Sigmas, $[\nabla_{\hat{\lambda}} \mathcal{B}(x(t), r,\hat{\mathbf{u}}(\tau-1|

Figures (11)

  • Figure 1: Comparison of the Continuous and discrete implementation of REAP regarding constraint violation.
  • Figure 2: Geometric illustration of how to determine the KKT parameter $\sigma(\tau|t)$ to maintain the invariance of the set $\mathcal{D}_{\hat{\mathbf{u}}}$.
  • Figure 3: Geometric illustration of how to determine the KKT parameter $\sigma(\tau|t)$ to maintain the invariance of the set $\mathcal{D}_{\hat{\lambda}}$.
  • Figure 4: Overview of the experimental setup utilized to perform the experimental evaluation.
  • Figure 5: Time-profile of states and control inputs for all five cases.
  • ...and 6 more figures

Theorems & Definitions (11)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 1 more