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Data-Driven Computation of Polytopic Invariant Sets for Noisy Nonlinear Systems

Sahand Kiani, Constantino M. Lagoa

TL;DR

This work tackles robust, invariant-set design for constrained noisy nonlinear systems without an explicit model by learning from data. It constructs a consistency set of all system parameters compatible with measured data and a known noise bound, then extends a DC-function based contractiveness condition to guarantee that the resulting polytopic set is $\lambda$-contractive for all compatible models. The authors present a tractable convex-optimization algorithm to maximize the largest contractive polytope inside the state constraints and demonstrate the approach on a numerical example, including a randomized enlarging step to reduce conservatism. The framework provides a model-agnostic tool for safe controller design and MPC terminal-set synthesis in data-rich environments.

Abstract

This paper presents a data-driven framework for computing robust, convex polytopic contractive set for constrained noisy nonlinear systems where an analytical model is not available. Our approach utilizes a finite set of collected noisy measurements to construct a polytopic set that bounds all possible system parameters compatible to available information. Based on previous results, we contribute to provide a sufficient condition for a set to be contractive using Difference of Convex functions for a noisy nonlinear system, while the model is not available. Robustness with respect to unknown model of system is guaranteed by requiring that the computed contractive set is invariant for all possible system models that are compatible with the noisy measurements. We present a tractable, optimization-based algorithm that implements this condition to compute the largest possible contractive set within the state constraint set for the unknown, noisy nonlinear system which is subjected to both state and input constraints. The effectiveness of the proposed methodology is demonstrated with a numerical example.

Data-Driven Computation of Polytopic Invariant Sets for Noisy Nonlinear Systems

TL;DR

This work tackles robust, invariant-set design for constrained noisy nonlinear systems without an explicit model by learning from data. It constructs a consistency set of all system parameters compatible with measured data and a known noise bound, then extends a DC-function based contractiveness condition to guarantee that the resulting polytopic set is -contractive for all compatible models. The authors present a tractable convex-optimization algorithm to maximize the largest contractive polytope inside the state constraints and demonstrate the approach on a numerical example, including a randomized enlarging step to reduce conservatism. The framework provides a model-agnostic tool for safe controller design and MPC terminal-set synthesis in data-rich environments.

Abstract

This paper presents a data-driven framework for computing robust, convex polytopic contractive set for constrained noisy nonlinear systems where an analytical model is not available. Our approach utilizes a finite set of collected noisy measurements to construct a polytopic set that bounds all possible system parameters compatible to available information. Based on previous results, we contribute to provide a sufficient condition for a set to be contractive using Difference of Convex functions for a noisy nonlinear system, while the model is not available. Robustness with respect to unknown model of system is guaranteed by requiring that the computed contractive set is invariant for all possible system models that are compatible with the noisy measurements. We present a tractable, optimization-based algorithm that implements this condition to compute the largest possible contractive set within the state constraint set for the unknown, noisy nonlinear system which is subjected to both state and input constraints. The effectiveness of the proposed methodology is demonstrated with a numerical example.

Paper Structure

This paper contains 11 sections, 1 theorem, 22 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Key Contribution
  4. Preliminary Results
  5. Difference of Convex Functions
  6. $\lambda$-Contractiveness for Nonlinear Systems
  7. Problem Statement
  8. Data-Driven Computation of Robust Polytopic Control Invariant Set
  9. Randomized Method for Enlarging $\lambda$-Contractive Set
  10. Simulation Results
  11. Conclusion

Key Result

Theorem 1

Let Assumptions assump1--assump4 hold. Consider a polytope $\Omega = \{x \in \mathbb{R}^n : Hx \le \mathbf{1}\} \subseteq X$, and let $Q = \{\theta^{*,1}, \dots, \theta^{*,q}\}$ be the set of vertices of the consistency set. If there exist control actions $u^p = u^p_v \in U$ defined for each vertex holds for all polytope vertices $p \in \mathbb{N}_{n_v}$, all polytope facets $i \in \mathbb{N}_{n_

Figures (2)

  • Figure 1: Data-driven computation of largest contractive set (purple solid line) within the state constraint set.
  • Figure 2: Data-driven computation of largest contarctive set using randomized enlarging method.

Theorems & Definitions (14)

  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 3
  • Remark 4
  • Definition 5
  • Theorem 1
  • ...and 4 more