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Robust stabilization of polytopic systems via fast and reliable neural network-based approximations

Filippo Fabiani, Paul J. Goulart

TL;DR

The paper tackles fast, certified stabilization of linear polytopic systems by approximating traditional stabilizing controllers with ReLU neural networks. It develops an offline MILP-based certificate that exactly computes the worst-case approximation error $\bar{e}_\alpha$ and the Lipschitz constant of the error, ensuring uniform ultimate boundedness of the closed-loop when using the NN surrogate. The analysis covers both vertex-based and selection-based controllers, characterizes the surrogate mappings as continuous and piecewise-affine, and provides practical guidance on training, network complexity, and safe deployment via projection or augmentation. This yields a principled route to implement high-speed, hardware-friendly surrogate controllers with formal guarantees for constrained, uncertain systems.

Abstract

We consider the design of fast and reliable neural network (NN)-based approximations of traditional stabilizing controllers for linear systems with polytopic uncertainty, including control laws with variable structure and those based on a (minimal) selection policy. Building upon recent approaches for the design of reliable control surrogates with guaranteed structural properties, we develop a systematic procedure to certify the closed-loop stability and performance of a linear uncertain system when a trained rectified linear unit (ReLU)-based approximation replaces such traditional controllers. First, we provide a sufficient condition, which involves the worst-case approximation error between ReLU-based and traditional controller-based state-to-input mappings, ensuring that the system is ultimately bounded within a set with adjustable size and convergence rate. Then, we develop an offline, mixed-integer optimization-based method that allows us to compute that quantity exactly.

Robust stabilization of polytopic systems via fast and reliable neural network-based approximations

TL;DR

The paper tackles fast, certified stabilization of linear polytopic systems by approximating traditional stabilizing controllers with ReLU neural networks. It develops an offline MILP-based certificate that exactly computes the worst-case approximation error and the Lipschitz constant of the error, ensuring uniform ultimate boundedness of the closed-loop when using the NN surrogate. The analysis covers both vertex-based and selection-based controllers, characterizes the surrogate mappings as continuous and piecewise-affine, and provides practical guidance on training, network complexity, and safe deployment via projection or augmentation. This yields a principled route to implement high-speed, hardware-friendly surrogate controllers with formal guarantees for constrained, uncertain systems.

Abstract

We consider the design of fast and reliable neural network (NN)-based approximations of traditional stabilizing controllers for linear systems with polytopic uncertainty, including control laws with variable structure and those based on a (minimal) selection policy. Building upon recent approaches for the design of reliable control surrogates with guaranteed structural properties, we develop a systematic procedure to certify the closed-loop stability and performance of a linear uncertain system when a trained rectified linear unit (ReLU)-based approximation replaces such traditional controllers. First, we provide a sufficient condition, which involves the worst-case approximation error between ReLU-based and traditional controller-based state-to-input mappings, ensuring that the system is ultimately bounded within a set with adjustable size and convergence rate. Then, we develop an offline, mixed-integer optimization-based method that allows us to compute that quantity exactly.
Paper Structure (20 sections, 5 theorems, 41 equations, 4 figures, 3 tables)

This paper contains 20 sections, 5 theorems, 41 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Traditional controllers for polytopic systems: advantages and disadvantages
  3. Proposed approach and related work
  4. Summary of contributions
  5. On the approximation of traditional controllers for polytopic systems
  6. Vertex-based control laws
  7. Selection-based controller
  8. Technical preliminaries
  9. networks are continuous mappings
  10. On the ultimate boundedness of polytopic systems with neural network controllers
  11. Geometric characterization of controllers for polytopic systems
  12. Certifying the approximation quality of ReLU-based controllers
  13. Constraints satisfaction, network complexity and discussion
  14. On the input constraints satisfaction
  15. Structural fulfilment
  16. ...and 5 more sections

Key Result

proposition 1

Suppose that $(b,\rho)$ are chosen so that $b \in (0,1)$ and $\rho \in (\lambda, 1)$. There exists a computable parameter $\zeta > 0$ such that, if $\bar{e}_{\alpha} < \zeta$, then the system in eq:polytopic with contr0oller $u(x) = \Phi_{\textnormal{NN}}(x)$ is ultimately bounded in $b \mathcal{S}

Figures (4)

  • Figure 1: Two-dimensional triangulation of a symmetric C-polytope $\mathcal{S}$ into a sequence of eight simplices, $\mathcal{T}^{(h)}$, each one formed by two vertices $\{x^{(h)}_{v}\}^2_{v = 1}$, and the origin.
  • Figure 2: Three-dimensional schematic representation of the first part of the proof of Theorem \ref{['th:PWA']}.
  • Figure 3: Pictorial representation of the proof of Theorem \ref{['th:PWA']}. For any $x \in \mathcal{S}^{(h)}$ (shaded red area, top left figure) \ref{['eq:online_control_2']} reduces to an , which provides an affine optimal solution in the given $x$ over a polyhedral partition of the whole of $\mathcal{S}$ (coloured regions, top right figure). Overall, this leads to a polyhedral partition of each sector, $\mathcal{S}^{(h)} \cap \mathcal{R}_{\mathcal{A}^{(h)}}$ (coloured regions inside the polyhedral sector, bottom figure), where $\mathcal{R}_{\mathcal{A}^{(h)}}$ denotes the critical region of states $x$ associated with the set of active constraints $\mathcal{A}^{(h)} \subseteq \{1,\ldots,pM+\ell\}$, $\mathcal{R}_{\mathcal{A}^{(h)}} \coloneqq \left\{ x \in \mathcal{S}\ \left| \ \mathcal{A}^{(h)}(x) = \mathcal{A}^{(h)} \right. \right\}$ and $\mathcal{A}^{(h)}(x)$ as defined in \ref{['eq:active_constraints']}.
  • Figure 4: Examples of partitions characterizing the mapping $\Phi(\cdot)$ as defined in \ref{['eq:online_control_1']}. The dashed black lines denote different polyhedral sectors of $\mathcal{S}$, in turn partitioned into further polyhedral regions (coloured areas).

Theorems & Definitions (14)

  • definition 1
  • definition 2
  • proposition 1
  • proof
  • theorem 1
  • Example 1
  • theorem 2
  • proof
  • theorem 3
  • proof
  • ...and 4 more