Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

TL;DR

The paper surveys contraction theory as a unifying, differential stability framework for nonlinear, time-varying systems and demonstrates how contraction metrics can be constructed via convex LMIs to yield explicit incremental robustness. It develops CV-STEM and CCM approaches for jointly designing contraction metrics and control laws, tying them to bounded-real and KYP/LMI theory to guarantee $\mathcal{L}_2$-robustness. It then extends these ideas to learning-based and data-driven control by introducing Neural Contraction Metrics (NCM/NSCM) and adaptive contractions (aNCM), providing exponential-tracking bounds in the presence of learning errors and disturbances. The tutorial further shows how contraction theory enables safe, robust motion planning (LAG-ROS), adaptive control, and model-free learning, and it connects these to practical tools like geodesics, Riemannian metrics, and Lipschitz-regularized networks. Overall, the work offers a principled, scalable framework for provable stability and robustness in learning-enabled control for deterministic, stochastic, and data-driven settings, with broad implications for robotics and autonomous systems.

Abstract

Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit the systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is, therefore, to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks.

Figures (13)

• Figure 1: Lyapunov theory and contraction theory, where $V$ is a Lyapunov function and $\delta z = \Theta (x,t)\delta x$ for $M(x,t)=\Theta(x,t)\Theta(x,t)^{\top}\succ 0$ that defines a contraction metric (see Theorem \ref{['Thm:contraction']}).
• Figure 2: Block diagram of CV-STEM.
• Figure 3: Illustration of NCM ($x$: system state; $M$: positive definite matrix that defines optimal contraction metric; $x_i$ and $M_i$: sampled $x$ and $M$; $\hat{x}$: estimated system state; $y$: measurement; and $u$: system control input). Note that the target trajectory $(x_d,u_d)$ is omitted in the figure for simplicity.
• Figure 4: Lorenz oscillator state estimation error smoothed using a $15$-point moving average filter in Example \ref{['ex:lorenz']} (left), and spacecraft motion $(p_x,p_y)$ on a planar field in Example \ref{['ex:sc_robust_control']} (right).
• Figure 5: Rocket model (angle of attack $\varphi$, pitch rate $q$).

Theorems & Definitions (141)

• Definition 2.1
• Lemma 2.1
• proof
• Theorem 2.1
• proof
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Example 2.1
• Example 2.2