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Controller synthesis for input-state data with measurement errors

Andrea Bisoffi, Lidong Li, Claudio De Persis, Nima Monshizadeh

TL;DR

This work develops data-driven state-feedback controllers for discrete-time LTI systems when measurements are corrupted by errors-in-variables. By modeling the measurement errors with energy or instantaneous bounds, the authors derive LMIs that certify robust stabilization of all data-consistent system matrices via a common Lyapunov function, yielding a gain $K = Y P^{-1}$. The energy-bound result provides a necessary and sufficient condition, while the instantaneous-bound result offers a practical sufficient condition, both facilitating controller design without exact knowledge of $(A^{\star},B^{\star})$. Numerical experiments on a distillation-column model show that instantaneous-bound LMIs are often feasible and less conservative than converting instantaneous bounds to an energy bound, offering concrete guidance on data requirements and performance implications.

Abstract

We consider the problem of designing a state-feedback controller for a linear system, based only on noisy input-state data. We focus on input-state data corrupted by measurement errors, which, albeit less investigated, are as relevant as process disturbances in applications. For energy and instantaneous bounds on these measurement errors, we derive linear matrix inequalities for controller design where the one for the energy bound is equivalent to robust stabilization of all systems consistent with the noisy data points via a common Lyapunov function.

Controller synthesis for input-state data with measurement errors

TL;DR

This work develops data-driven state-feedback controllers for discrete-time LTI systems when measurements are corrupted by errors-in-variables. By modeling the measurement errors with energy or instantaneous bounds, the authors derive LMIs that certify robust stabilization of all data-consistent system matrices via a common Lyapunov function, yielding a gain . The energy-bound result provides a necessary and sufficient condition, while the instantaneous-bound result offers a practical sufficient condition, both facilitating controller design without exact knowledge of . Numerical experiments on a distillation-column model show that instantaneous-bound LMIs are often feasible and less conservative than converting instantaneous bounds to an energy bound, offering concrete guidance on data requirements and performance implications.

Abstract

We consider the problem of designing a state-feedback controller for a linear system, based only on noisy input-state data. We focus on input-state data corrupted by measurement errors, which, albeit less investigated, are as relevant as process disturbances in applications. For energy and instantaneous bounds on these measurement errors, we derive linear matrix inequalities for controller design where the one for the energy bound is equivalent to robust stabilization of all systems consistent with the noisy data points via a common Lyapunov function.
Paper Structure (9 sections, 4 theorems, 53 equations, 3 figures)

This paper contains 9 sections, 4 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Matrix elimination result
  5. Problem formulation
  6. Controller design for energy bound
  7. Controller design for instantaneous bound
  8. Numerical results
  9. Conclusions

Key Result

Proposition 1

Consider matrices $E \in \mathbb{R}^{n_1 \times n_2}$, $F\in\mathbb{R}^{n_1 \times n_3}$, $G \in \mathbb{R}^{n_3 \times n_3}$ with $G = G^\top \succeq 0$. Then,

Figures (3)

  • Figure 1: Scheme of data collection experiment for input-state measurements.
  • Figure 2: Closed loop of \ref{['sys_x_star']} and $u = K x$, where $K$ is designed by \ref{['rob_contr_probl_LMI_inst']}.
  • Figure 3: Ratio of $n_{\textup{feas}}/20 \in [0,1]$ as a function of $(T,\log_{10}(\theta))$ for \ref{['rob_contr_probl_LMI']} (top) and \ref{['rob_contr_probl_LMI_inst']} (bottom), with "good" regions in yellow.

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof