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QSID-MPC: Model Predictive Control with System Identification from Quantized Data

Shahab Ataei, Dipankar Maity, Debdipta Goswami

TL;DR

This work introduces QSID-MPC, a framework that integrates model predictive control with system identification performed on quantized data using dither quantization. It establishes explicit links between quantization resolution $\epsilon$ and identification error, showing $[\hat{A},\hat{B}]$ deviates from the true model by $O(\epsilon)$ in finite data and $O(\epsilon^2)$ in large data, and proves that MPC tracking errors are uniformly ultimately bounded with a bound $\delta(\epsilon)=O(\epsilon^2)$ under favorable data conditions. The theoretical results are validated through numerical experiments on a DC motor with load and a Boeing 747 longitudinal flight model, which demonstrate that increasing the word-length (i.e., decreasing $\epsilon$) reduces identification error and improves MPC performance. The findings provide practical guidelines for deploying data-driven MPC in resource-constrained systems where data must be quantized, highlighting robust performance even under quantization.

Abstract

Least-square system identification is widely used for data-driven model-predictive control (MPC) of unknown or partially known systems. This letter investigates how the system identification and subsequent MPC is affected when the state and input data is quantized. Specifically, we examine the fundamental connection between model error and quantization resolution and how that affects the stability and boundedness of the MPC tracking error. Furthermore, we demonstrate that, with a sufficiently rich dataset, the model error is bounded by a function of quantization resolution and the MPC tracking error is also ultimately bounded similarly. The theory is validated through numerical experiments conducted on two different linear dynamical systems.

QSID-MPC: Model Predictive Control with System Identification from Quantized Data

TL;DR

This work introduces QSID-MPC, a framework that integrates model predictive control with system identification performed on quantized data using dither quantization. It establishes explicit links between quantization resolution and identification error, showing deviates from the true model by in finite data and in large data, and proves that MPC tracking errors are uniformly ultimately bounded with a bound under favorable data conditions. The theoretical results are validated through numerical experiments on a DC motor with load and a Boeing 747 longitudinal flight model, which demonstrate that increasing the word-length (i.e., decreasing ) reduces identification error and improves MPC performance. The findings provide practical guidelines for deploying data-driven MPC in resource-constrained systems where data must be quantized, highlighting robust performance even under quantization.

Abstract

Least-square system identification is widely used for data-driven model-predictive control (MPC) of unknown or partially known systems. This letter investigates how the system identification and subsequent MPC is affected when the state and input data is quantized. Specifically, we examine the fundamental connection between model error and quantization resolution and how that affects the stability and boundedness of the MPC tracking error. Furthermore, we demonstrate that, with a sufficiently rich dataset, the model error is bounded by a function of quantization resolution and the MPC tracking error is also ultimately bounded similarly. The theory is validated through numerical experiments conducted on two different linear dynamical systems.

Paper Structure

This paper contains 9 sections, 3 theorems, 25 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. System Identification
  4. Effects of Dither Quantization on $(\hat{A}, \hat{B})$
  5. Stability of Model Predictive Control (MPC) under Identification Error
  6. Numerical Examples
  7. DC Motor with Load
  8. Boeing 747 Flight Control
  9. Conclusions

Key Result

Theorem 1

Suppose $T\to \infty$ and $\lim_{T\to \infty} \frac{\Psi \Psi^\top}{T}$ is finite and positive definite, then with $\Psi_{\rm uqz}$ being the data matrix $\Psi$ constructed from unquantized data.

Figures (2)

  • Figure 1: Error and phase-portrait profile for DC Motor with Load \ref{['eq: Motor']}: (a) relative error in matrix $A$; (b) relative error in matrix $B$; (c) optimal cost achieved by MPC; (d)--(f) phase portrait from regulation MPC (with model identified from data snapshots quantized by 50 independent dither signal realization) for word lengths $b=2,\ 4, \ 6$ respectively.
  • Figure 2: Error and phase-portrait profile for Boeing 747 longitudinal flight control: (a) relative error in matrix $A$; (b) relative error in matrix $B$; (c) optimal cost achieved by MPC; (d)--(f) phase portrait from regulation MPC (with model identified from data snapshots quantized by 50 independent dither signal realization) for word lengths $b=2,\ 4, \ 6$ respectively.

Theorems & Definitions (6)

  • Theorem 1: large-data regime
  • proof
  • Theorem 2: finite-data regime
  • proof
  • Theorem 3
  • proof