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Learning Neural Network Controllers with Certified Robust Performance via Adversarial Training

Neelay Junnarkar, Yasin Sonmez, Murat Arcak

Abstract

Neural network (NN) controllers achieve strong empirical performance on nonlinear dynamical systems, yet deploying them in safety-critical settings requires robustness to disturbances and uncertainty. We present a method for jointly synthesizing NN controllers and dissipativity certificates that formally guarantee robust closed-loop performance using adversarial training, in which we use counterexamples to the robust dissipativity condition to guide training. Verification is done post-training using alpha,beta-CROWN, a branch-and-bound-based method that enables direct analysis of the nonlinear dynamical system. The proposed method uses quadratic constraints (QCs) only for characterization of non-parametric uncertainties. The method is tested in numerical experiments on maximizing the volume of the set on which a system is certified to be robustly dissipative. Our method certifies regions up to 78 times larger than the region certified by a linear matrix inequality-based approach that we derive for comparison.

Learning Neural Network Controllers with Certified Robust Performance via Adversarial Training

Abstract

Neural network (NN) controllers achieve strong empirical performance on nonlinear dynamical systems, yet deploying them in safety-critical settings requires robustness to disturbances and uncertainty. We present a method for jointly synthesizing NN controllers and dissipativity certificates that formally guarantee robust closed-loop performance using adversarial training, in which we use counterexamples to the robust dissipativity condition to guide training. Verification is done post-training using alpha,beta-CROWN, a branch-and-bound-based method that enables direct analysis of the nonlinear dynamical system. The proposed method uses quadratic constraints (QCs) only for characterization of non-parametric uncertainties. The method is tested in numerical experiments on maximizing the volume of the set on which a system is certified to be robustly dissipative. Our method certifies regions up to 78 times larger than the region certified by a linear matrix inequality-based approach that we derive for comparison.

Paper Structure

This paper contains 24 sections, 2 theorems, 20 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Background and Problem Setup
  4. $\alpha$,$\beta$-CROWN
  5. Dissipativity
  6. Quadratic Constraints
  7. Main Results
  8. Verification using $\alpha$,$\beta$-CROWN
  9. Verification using LMIs
  10. Methodology
  11. Controller and Certificate
  12. Storage Function
  13. Controller
  14. QC Multipliers and Supply-Rate Scale
  15. Training Procedure
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be a set of QCs, $V: \mathbb{R}^{n} \to \mathbb{R}_{\geq 0}$ be such that $V(0) = 0$, and $\rho > 0$. Then eq:diss-sys is robustly dissipative with respect to a supply rate $s(d, e)$ on $(\Omega_{V,\rho}, B_{\overline{d}}^{n_d}, \mathbf{\Delta}(\mathcal{M}))$ if there exist $M_\tex for all $x \in \Omega_{V,\rho}, w \in \mathbb{R}^{n_w}, d \in B_{\overline{d}}^{n_d}$ where $z = (G

Figures (2)

  • Figure 3: Certified regions projected onto the plant state $(\theta, \dot\theta)$ for (a) the $\ell_2$-gain bound experiment and (b) the robust stability experiment under sector-bound uncertainty ($\alpha = 0.25$). Four methods are compared: LMI baseline (purple, dotted), before training (blue), training only the storage function with a fixed controller (green), and jointly training both controller and storage function (orange).
  • Figure 4: Uncertainty structure: Plant input is sum of control $\tilde{u}$ and $w = \Delta(\tilde{u})$, where $\Delta$ is uncertain and satisfies $\|w\| \leq \alpha \|\tilde{u}\|$ pointwise in time.

Theorems & Definitions (8)

  • Definition 1: Robust Dissipativity
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3: Tight bounding boxes