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Linear Reduced Order Model Predictive Control

Joseph Lorenzetti, Andrew McClellan, Charbel Farhat, Marco Pavone

TL;DR

This article proposes a reduced-order model predictive control (ROMPC) scheme to solve robust, output feedback, constrained optimal control problems for high-dimensional linear systems, and guarantees on robust constraint satisfaction and stability are provided.

Abstract

Model predictive controllers use dynamics models to solve constrained optimal control problems. However, computational requirements for real-time control have limited their use to systems with low-dimensional models. Nevertheless, high-dimensional models arise in many settings, for example discretization methods for generating finite-dimensional approximations to partial differential equations can result in models with thousands to millions of dimensions. In such cases, reduced order models (ROMs) can significantly reduce computational requirements, but model approximation error must be considered to guarantee controller performance. In this work, a reduced order model predictive control (ROMPC) scheme is proposed to solve robust, output feedback, constrained optimal control problems for high-dimensional linear systems. Computational efficiency is obtained by using projection-based ROMs, and guarantees on robust constraint satisfaction and stability are provided. Performance of the approach is demonstrated in simulation for several examples, including an aircraft control problem leveraging an inviscid computational fluid dynamics model with dimension 998,930.

Linear Reduced Order Model Predictive Control

TL;DR

This article proposes a reduced-order model predictive control (ROMPC) scheme to solve robust, output feedback, constrained optimal control problems for high-dimensional linear systems, and guarantees on robust constraint satisfaction and stability are provided.

Abstract

Model predictive controllers use dynamics models to solve constrained optimal control problems. However, computational requirements for real-time control have limited their use to systems with low-dimensional models. Nevertheless, high-dimensional models arise in many settings, for example discretization methods for generating finite-dimensional approximations to partial differential equations can result in models with thousands to millions of dimensions. In such cases, reduced order models (ROMs) can significantly reduce computational requirements, but model approximation error must be considered to guarantee controller performance. In this work, a reduced order model predictive control (ROMPC) scheme is proposed to solve robust, output feedback, constrained optimal control problems for high-dimensional linear systems. Computational efficiency is obtained by using projection-based ROMs, and guarantees on robust constraint satisfaction and stability are provided. Performance of the approach is demonstrated in simulation for several examples, including an aircraft control problem leveraging an inviscid computational fluid dynamics model with dimension 998,930.

Paper Structure

This paper contains 43 sections, 8 theorems, 67 equations, 6 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Full Order Model
  4. Constrained Optimal Control Problem
  5. Reduced Order Model
  6. Reduced Order Model Predictive Control
  7. Reduced Order Controller and State Estimator
  8. Reduced Order Optimal Control Problem
  9. ROMPC Algorithm
  10. Closed-Loop Performance
  11. Closed-Loop Error Dynamics
  12. Closed-Loop Robust Constraint Satisfaction
  13. Closed-Loop Stability
  14. Controller Synthesis
  15. H2-Optimal Controller Synthesis
  16. ...and 28 more sections

Key Result

Lemma 1

Suppose that at time $k_0$ the optimal control problem eq:rompc is feasible and that $H_z\delta_{z,k} \leq \Delta_z$ and $H_u \delta_{u,k} \leq \Delta_u$ for all $k \geq k_0$. Then, under the proposed control scheme the full order system will robustly satisfy the constraints eq:constraints for all $

Figures (6)

  • Figure 1: A block diagram of the ROMPC control scheme, which shows the connection between the simulated ROM and the controlled system (FOM). The optimal control problem is used to control the simulated ROM, and the system (FOM) is driven by the controller to track the simulated ROM.
  • Figure 2: Robust constraint satisfaction can be guaranteed by using error tubes and constraint tightening. The trajectory of the simulated ROM \ref{['eq:rom']}, which is controlled by the OCP \ref{['eq:rompc']} and satisfies the tightened constraints $\bar{z} \in \bar{\mathcal{Z}}$, is shown in pink. The performance variables, $z$, of the full order system \ref{['eq:fom']} are shown in blue, and track $\bar{z}$ with bounded error. The tightened constraint $\bar{\mathcal{Z}}$ guarantees that the entire tube, and thus $z$, satisfies the constraint $z \in \mathcal{Z}$.
  • Figure 3: Plot showing the matrix norm $\lVert E A_\epsilon^t B_\epsilon \rVert$, where $E = [E_z^T, E_u^T]^T$, for different values of $t \in [0,\dots,\tau]$. For each example the value of $\tau$ is the same as presented in Table \ref{['tab:ebounds']}.
  • Figure 4: Simulation results for the supersonic diffuser example in Section \ref{['subsec:supdiff']}. This plot shows the control input for two simulations, one with the ROMPC scheme and another with a reduced order LQR controller.
  • Figure 5: Simulation results for the supersonic diffuser example in Section \ref{['subsec:supdiff']}. This plot shows the performance output for two simulations, one with the ROMPC scheme and another with a reduced order LQR controller.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Lemma 1: Robust Constraint Satisfaction
  • Theorem 1: Robust Stability
  • Corollary 1: Stability
  • Proposition 1
  • Proposition 2
  • Theorem 2: Robust Constraint Satisfaction
  • Remark 1: Practical Considerations for Extremely High-Dimensional Problems
  • Theorem 3: Robust Setpoint Tracking
  • Corollary 2: Setpoint Tracking