Neural Exponential Stabilization of Control-affine Nonlinear Systems

TL;DR

This work addresses the problem of achieving exponential stabilization for control-affine nonlinear systems with certifiable safety certificates. It develops a co-synthesis framework that learns both a neural controller and a Neural Contraction Metric (NCM) within the Control Contraction Metrics (CCM) theory, transforming the intractable infinite-dimensional CCM conditions into tractable finite inequalities via element-wise bounds. A novel free Parametrization of NCMs ensures positive definiteness and PDE-satisfaction by design, while contraction-enforcing regularizers based on the Gershgorin disk theorem enable stable closed-loop behavior; when training loss vanishes, formal verification of the learned NCM and stabilization is guaranteed. The method scales to high-dimensional systems, demonstrated by experiments on set-point stabilization and region-of-attraction expansion, including pendulum benchmarks and an under-actuated system with pre-stabilization, showing that the NN controller can achieve exponential convergence and enlarge the ROC. Overall, the paper provides a practical, scalable framework for neural certificates in nonlinear control with rigorous guarantees and verifiable contraction properties.

Abstract

This paper proposes a novel learning-based approach for achieving exponential stabilization of nonlinear control-affine systems. We leverage the Control Contraction Metrics (CCMs) framework to co-synthesize Neural Contraction Metrics (NCMs) and Neural Network (NN) controllers. First, we transform the infinite-dimensional semi-definite program (SDP) for CCM computation into a tractable inequality feasibility problem using element-wise bounds of matrix-valued functions. The terms in the inequality can be efficiently computed by our novel algorithms. Second, we propose a free parametrization of NCMs guaranteeing positive definiteness and the satisfaction of a partial differential equation, regardless of trainable parameters. Third, this parametrization and the inequality condition enable the design of contractivity-enforcing regularizers, which can be incorporated while designing the NN controller for exponential stabilization of the underlying nonlinear systems. Furthermore, when the training loss goes to zero, we provide formal guarantees on verification of the NCM and the exponentional stabilization under the NN controller. Finally, we validate our method through benchmark experiments on set-point stabilization and increasing the region of attraction of a locally pre-stabilized closed-loop system.

Figures (4)

• Figure 1: Closed-loop response for 20 different initial conditions demonstrating stabilization at $(\pi/4,0)$.
• Figure 2: Evolution of closed-loop trajectories of inverted pendulum stabilized at $(2.0,0)$ for 20 arbitrary initial conditions.
• Figure 3: The phase-portrait of the closed-loop system with the trained NN controller. All the trajectories initialized in the neighborhood of the desired equilibrium point are converging exponentially fast.
• Figure 4: Response of nonlinear system \ref{['eq:sys_example2']} with NCM and LQR control to initial state $x(0) = [0.5, 0.5, 0.5]^\top$ (left) and $x(0) = [10, 10, 10]^\top$ (right). This exhibits the locally optimal and increased region of attraction of the NCM controller.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Proposition 1
• Example 1: PDE \ref{['eq:pde']} with $m=1$
• Remark 1: Computational complexity of Algorithm \ref{['alg']}
• Proposition 2
• Lemma 1