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Learning-based Event-triggered MPC with Gaussian processes under terminal constraints

Yuga Onoue, Kazumune Hashimoto, Akifumi Wachi

TL;DR

This article analyzes the convergence of the closed-loop system under the event-triggered condition, demonstrating that the system’s state will enter the terminal set within a finite time, assuming small-enough uncertainty in the GP model.

Abstract

Event-triggered control strategy is capable of significantly reducing the number of control task executions without sacrificing control performance. In this paper, we propose a novel learning-based approach towards an event-triggered model predictive control (MPC) for nonlinear control systems whose dynamics are unknown apriori. In particular, the optimal control problems (OCPs) are formulated based on predictive states learned by Gaussian process (GP) regression under a terminal constraint constructed by a symbolic abstraction. The event-triggered condition proposed in this paper is derived from the recursive feasibility so that the OCPs are solved only when an error between the predictive and the actual states exceeds a certain threshold. Based on the event-triggered condition, we analyze the stability of the closed-loop system and show that the finite-time convergence to the terminal set is achieved as the uncertainty of the GP model becomes smaller. Moreover, in order to reduce the uncertainty of the GP model and increase efficiency to find the optimal solution, we provide an overall learning-based event-triggered MPC algorithm based on an iterative task. Finally, we demonstrate the proposed approach through a tracking control problem.

Learning-based Event-triggered MPC with Gaussian processes under terminal constraints

TL;DR

This article analyzes the convergence of the closed-loop system under the event-triggered condition, demonstrating that the system’s state will enter the terminal set within a finite time, assuming small-enough uncertainty in the GP model.

Abstract

Event-triggered control strategy is capable of significantly reducing the number of control task executions without sacrificing control performance. In this paper, we propose a novel learning-based approach towards an event-triggered model predictive control (MPC) for nonlinear control systems whose dynamics are unknown apriori. In particular, the optimal control problems (OCPs) are formulated based on predictive states learned by Gaussian process (GP) regression under a terminal constraint constructed by a symbolic abstraction. The event-triggered condition proposed in this paper is derived from the recursive feasibility so that the OCPs are solved only when an error between the predictive and the actual states exceeds a certain threshold. Based on the event-triggered condition, we analyze the stability of the closed-loop system and show that the finite-time convergence to the terminal set is achieved as the uncertainty of the GP model becomes smaller. Moreover, in order to reduce the uncertainty of the GP model and increase efficiency to find the optimal solution, we provide an overall learning-based event-triggered MPC algorithm based on an iterative task. Finally, we demonstrate the proposed approach through a tracking control problem.

Paper Structure

This paper contains 30 sections, 13 theorems, 85 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gaussian process regression
  4. Transition system
  5. Problem formulation
  6. System description
  7. Goal of this paper
  8. Event-triggered MPC strategy
  9. Learning dynamics with Gaussian processes
  10. Optimal control problem
  11. Event-triggered condition
  12. Proposed event-triggered MPC algorithm
  13. Analysis
  14. Recursive feasibility
  15. Convergence analysis
  16. ...and 15 more sections

Key Result

Lemma 1

Given the training dataset $\mathcal{D}_{N, i} = \{Z_N, Y_{N, i}\}$ for each $f_i$, $i \in \mathbb{N}_{1: n_x}$, and let Assumption rkhsas hold. Then, for all ${x} \in \mathbb{R}^{n_x}$, ${u} \in \mathbb{R}^{n_u}$, and $i \in \mathbb{N}_{1: n_x}$, it follows that $f_i(x, u)\in \mathcal{F}_i(x, u; \m with $\beta_{N, i} = \sqrt{b_i^2 - Y_{N, i}^\top (K_i + \sigma_{w_i}^2 I)^{-1}Y_{N, i} + N}$. $\Box

Figures (2)

  • Figure 1: Tracking trajectories of the robot $({\rm x}(t), {\rm y}(t))$ under the event-triggered MPC (blue solid line) and the symbolic controller (green dashed line), the positions where the event-triggered conditions \ref{['trigger']} are violated (orange cross mark), and the reference trajectories $({\rm x}_r(t), {\rm y}_r(t))$ (red dash-dotted line). At each iteration, the initial state of the robot (blue circle mark) is given randomly from $\mathcal{X}_{\rm init}$ (green rectangle with slashes), and the algorithm proceeds to the next iteration after 40 steps have passed (yellow star mark).
  • Figure 2: (a) and (b) illustrate, at each iteration, the trajectories of the error distance and attitude between the robot \ref{['foldyn']} and the leader \ref{['refdyn']}, respectively. (c) shows the trajectories of the control input evaluated by the control input constraint at each iteration.

Theorems & Definitions (34)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • ...and 24 more