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Non-Conservative Data-driven Safe Control Design for Nonlinear Systems with Polyhedral Safe Sets

Amir Modares, Bosen Lian, Hamidreza Modares

TL;DR

This work tackles safe control for discrete-time nonlinear systems with parametric uncertainties and disturbances under polyhedral safety constraints. It introduces a data-based closed-loop representation comprising a learned linear part and a nonlinear remainder, and employs a primal–dual optimization to learn the remainder rather than cancel it, enabling safety with lower conservatism. The authors derive tractable conditions (including a lambda-contractive framework) for ensuring robust invariance of the safe set using data-driven gains, and demonstrate computational efficiency and reduced conservatism relative to prior nonlinear data-driven approaches. A simulation illustrates effective enforcement of safety with online-friendly computations, highlighting practical impact for data-driven safe control of nonlinear systems.

Abstract

This paper presents a data-driven nonlinear safe control design approach for discrete-time systems under parametric uncertainties and additive disturbances. We first characterize a new control structure from which a data-based representation of closed-loop systems is obtained. This data-based closed-loop system is composed of two parts: 1) a parametrized linear closed-loop part and a parametrized nonlinear remainder closed-loop part. We show that using the standard practice or learning a robust controller to ensure safety while treating the remaining nonlinearities as disturbances brings about significant challenges in terms of computational complexity and conservatism. To overcome these challenges, we develop a novel nonlinear safe control design approach in which the closed-loop nonlinear remainders are learned, rather than canceled, in a control-oriented fashion while preserving the computational efficiency. To this end, a primal-dual optimization framework is leveraged in which the control gains are learned to enforce the second-order optimality on the closed-loop nonlinear remainders. This allows us to account for nonlinearities in the design for the sake of safety rather than treating them as disturbances. This new controller parameterization and design approach reduces the computational complexity and the conservatism of designing a safe nonlinear controller. A simulation example is then provided to show the effectiveness of the proposed data-driven controller.

Non-Conservative Data-driven Safe Control Design for Nonlinear Systems with Polyhedral Safe Sets

TL;DR

This work tackles safe control for discrete-time nonlinear systems with parametric uncertainties and disturbances under polyhedral safety constraints. It introduces a data-based closed-loop representation comprising a learned linear part and a nonlinear remainder, and employs a primal–dual optimization to learn the remainder rather than cancel it, enabling safety with lower conservatism. The authors derive tractable conditions (including a lambda-contractive framework) for ensuring robust invariance of the safe set using data-driven gains, and demonstrate computational efficiency and reduced conservatism relative to prior nonlinear data-driven approaches. A simulation illustrates effective enforcement of safety with online-friendly computations, highlighting practical impact for data-driven safe control of nonlinear systems.

Abstract

This paper presents a data-driven nonlinear safe control design approach for discrete-time systems under parametric uncertainties and additive disturbances. We first characterize a new control structure from which a data-based representation of closed-loop systems is obtained. This data-based closed-loop system is composed of two parts: 1) a parametrized linear closed-loop part and a parametrized nonlinear remainder closed-loop part. We show that using the standard practice or learning a robust controller to ensure safety while treating the remaining nonlinearities as disturbances brings about significant challenges in terms of computational complexity and conservatism. To overcome these challenges, we develop a novel nonlinear safe control design approach in which the closed-loop nonlinear remainders are learned, rather than canceled, in a control-oriented fashion while preserving the computational efficiency. To this end, a primal-dual optimization framework is leveraged in which the control gains are learned to enforce the second-order optimality on the closed-loop nonlinear remainders. This allows us to account for nonlinearities in the design for the sake of safety rather than treating them as disturbances. This new controller parameterization and design approach reduces the computational complexity and the conservatism of designing a safe nonlinear controller. A simulation example is then provided to show the effectiveness of the proposed data-driven controller.
Paper Structure (10 sections, 5 theorems, 62 equations)

This paper contains 10 sections, 5 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Formalizing the Nonlinear Data-based Safe Control Design
  3. Problem Formulation
  4. Data-based Representation of the Closed-loop System
  5. Data-based Nonlinear Safe Control using Nonlinearity Minimization
  6. Systems with No Disturbance
  7. Systems with Additive Disturbance
  8. A Computationally-efficient Data-based Nonlinear Safe Control Design
  9. Simulation Results
  10. Conclusion

Key Result

Lemma 1

Consider the system system. Let the input-state collected data be given by data-u and data-x1, and the state data data-x1 are further arranged as data-x-data-xx and data-z. Let the controller be given as contNew. Then, the data-based closed-loop representation of the system becomes where with $G_{K,1} \in \mathbb{R}^{T \times n}$ and $G_{K,2} \in \mathbb{R}^{T \times N}$. Moreover, under Assumpt

Theorems & Definitions (13)

  • Definition 1
  • Remark 1
  • Definition 2
  • Lemma 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Corollary 1
  • Remark 4
  • Theorem 2
  • ...and 3 more