Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Data-driven input-to-state stabilization with respect to measurement errors

Hailong Chen, Andrea Bisoffi, Claudio De Persis

TL;DR

This work tackles enforcing input-to-state stability with respect to measurement errors for polynomial input-affine systems using data gathered from open-loop experiments. By characterizing all dynamics consistent with noisy data through set-membership ideas and then formulating data-driven SOS conditions, it designs a state-feedback controller, an ISS Lyapunov function, and comparison functions that certify ISS for all dynamics in an overapproximation of the data set. The key contribution is a practical, automated SOS-based design that remains robust to data uncertainty and relies only on structural knowledge and noise bounds, with a two-step alternating optimization to cope with bilinearity. The approach is validated on a numerical example and shows promise for data-driven event-triggered control applications, offering a path to ISS-certified, communication-efficient nonlinear control without exact model identification.

Abstract

We consider noisy input/state data collected from an experiment on a polynomial input-affine nonlinear system. Motivated by event-triggered control, we provide data-based conditions for input-to-state stability with respect to measurement errors. Such conditions, which take into account all dynamics consistent with data, lead to the design of a feedback controller, an ISS Lyapunov function, and comparison functions ensuring ISS with respect to measurement errors. When solved alternately for two subsets of the decision variables, these conditions become a convex sum-of-squares program. Feasibility of the program is illustrated with a numerical example.

Data-driven input-to-state stabilization with respect to measurement errors

TL;DR

This work tackles enforcing input-to-state stability with respect to measurement errors for polynomial input-affine systems using data gathered from open-loop experiments. By characterizing all dynamics consistent with noisy data through set-membership ideas and then formulating data-driven SOS conditions, it designs a state-feedback controller, an ISS Lyapunov function, and comparison functions that certify ISS for all dynamics in an overapproximation of the data set. The key contribution is a practical, automated SOS-based design that remains robust to data uncertainty and relies only on structural knowledge and noise bounds, with a two-step alternating optimization to cope with bilinearity. The approach is validated on a numerical example and shows promise for data-driven event-triggered control applications, offering a path to ISS-certified, communication-efficient nonlinear control without exact model identification.

Abstract

We consider noisy input/state data collected from an experiment on a polynomial input-affine nonlinear system. Motivated by event-triggered control, we provide data-based conditions for input-to-state stability with respect to measurement errors. Such conditions, which take into account all dynamics consistent with data, lead to the design of a feedback controller, an ISS Lyapunov function, and comparison functions ensuring ISS with respect to measurement errors. When solved alternately for two subsets of the decision variables, these conditions become a convex sum-of-squares program. Feasibility of the program is illustrated with a numerical example.
Paper Structure (9 sections, 3 theorems, 35 equations, 3 figures)

This paper contains 9 sections, 3 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Auxiliary results for the sequel
  5. Motivation for ISS with respect to the measurement error based on event-triggered control
  6. Data-based problem formulation
  7. Main result: learning an ISS Lyapunov Function from data
  8. Numerical example
  9. Conclusion

Key Result

Lemma 1

Consider $\alpha \colon \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ defined as $\alpha(r):=\sum_{k=1}^{N}c_{k}r^{2k}$. The function $\alpha$ is class $\mathcal{K}_{\infty}$ if the scalars $c_1,c_2,\dots,c_{N}$ satisfy $c_1\geq0,c_2\geq0,\dots,c_{N}\geq0$ and $c_1+c_2+\dots+c_N>0$.

Figures (3)

  • Figure 1: Event-triggered control scheme.
  • Figure 2: Data-collection experiment for the considered polynomial system.
  • Figure 3: Evolution of the state (top) and comparison functions (bottom).

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1: sontag2008input
  • Definition 2: sontag2008input
  • Remark 1
  • Theorem 1: Data-driven noisy ISS