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Fast Stochastic MPC using Affine Disturbance Feedback Gains Learned Offline

Hotae Lee, Francesco Borrelli

TL;DR

The proposed MPC approach achieves comparable control performance in terms of Region of Attraction and average closed-loop costs to classical MPC optimizing over disturbance feedback policies, while delivering a 10-fold improvement in computational speed.

Abstract

We propose a novel Stochastic Model Predictive Control (MPC) for uncertain linear systems subject to probabilistic constraints. The proposed approach leverages offline learning to extract key features of affine disturbance feedback policies, significantly reducing the computational burden of online optimization. Specifically, we employ offline data-driven sampling to learn feature components of feedback gains and approximate the chance-constrained feasible set with a specified confidence level. By utilizing this learned information, the online MPC problem is simplified to optimization over nominal inputs and a reduced set of learned feedback gains, ensuring computational efficiency. In a numerical example, the proposed MPC approach achieves comparable control performance in terms of Region of Attraction (ROA) and average closed-loop costs to classical MPC optimizing over disturbance feedback policies, while delivering a 10-fold improvement in computational speed.

Fast Stochastic MPC using Affine Disturbance Feedback Gains Learned Offline

TL;DR

The proposed MPC approach achieves comparable control performance in terms of Region of Attraction and average closed-loop costs to classical MPC optimizing over disturbance feedback policies, while delivering a 10-fold improvement in computational speed.

Abstract

We propose a novel Stochastic Model Predictive Control (MPC) for uncertain linear systems subject to probabilistic constraints. The proposed approach leverages offline learning to extract key features of affine disturbance feedback policies, significantly reducing the computational burden of online optimization. Specifically, we employ offline data-driven sampling to learn feature components of feedback gains and approximate the chance-constrained feasible set with a specified confidence level. By utilizing this learned information, the online MPC problem is simplified to optimization over nominal inputs and a reduced set of learned feedback gains, ensuring computational efficiency. In a numerical example, the proposed MPC approach achieves comparable control performance in terms of Region of Attraction (ROA) and average closed-loop costs to classical MPC optimizing over disturbance feedback policies, while delivering a 10-fold improvement in computational speed.

Paper Structure

This paper contains 18 sections, 1 theorem, 27 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Scenario Approach
  4. Offline Sampling approach for Extracting feature disturbance feedback gains
  5. Constraint matrix from dynamics and offline samples
  6. SVD based Feature Extraction
  7. Truncated feedback policy
  8. Probabilistic Properties
  9. Probabilistic scaling based set approximation
  10. Probabilistic Scaling Approach
  11. Adjusted Probabilistic Scaling Approach
  12. Online MPC with principal feature policy and approximate sets
  13. Construct the online MPC problem with offline components
  14. Implementation details
  15. Simulation Results
  16. ...and 3 more sections

Key Result

proposition 1

Assume that the same sample set from the truncation process of Sec. sec:feature_truncation is employed for computing the scaling factor for the truncated variables. Then, all $z^{\mathrm{recon}}_t$ reconstructed from $z^{\mathrm{trun}}_t \in S(\gamma^\star)$, is contained within $\mathcal{Z}_\epsilo

Figures (4)

  • Figure 1: Comparisons of ROA with three control policies
  • Figure 2: Comparisons of ROA with three control policies for the less tight scenarios with $s_0^{\mathrm{env}}=21$
  • Figure 3: (a) Comparison of the closed-loop costs over 50 trials, (b) Gap distance graph during 20 time steps with $s_0^{\mathrm{env}}=20$
  • Figure 4: (a) Comparison of the closed-loop costs over 50 trials, (b) Gap distance graph during 20 time steps with $s_0^{\mathrm{env}}=21$

Theorems & Definitions (8)

  • remark 1
  • remark 2
  • remark 3
  • definition 1: $\epsilon$-Chance Constrained Set of $z_t$
  • definition 2: Sampled Set of truncated $z_t^{\mathrm{trun}}$
  • proposition 1: Probabilistic Scaling with Truncation
  • proof
  • remark 4