[Paper Review] Contraction Metrics in Adaptive Nonlinear Control

Brett T. Lopez, Jean-Jacques Slotine|arXiv (Cornell University)|Dec 31, 2019

Control and Stability of Dynamical Systems30 references18 citations

TL;DR

This paper proposes a novel adaptive nonlinear control framework using contraction metrics to stabilize uncertain nonlinear systems without requiring explicit Lyapunov functions. By leveraging differential analysis of trajectory convergence and parameter-dependent contraction metrics, the method ensures global asymptotic stability for stabilizable systems, including underactuated and unmatched uncertainty cases, with provable convergence via linear matrix inequalities and adaptive laws.

ABSTRACT

Lyapunov stability theory is the bedrock of direct adaptive control. Fundamentally, Lyapunov stability requires constructing a distance-like function which must decrease with time to ensure stability. Feedback linearization, backstepping, and sum-of-squares optimization are common approaches for constructing such a distance function, but require the system to possess certain inherent/structural properties or involves solving a non-convex optimization problem. These restrictions/complexities arise because Lyapunov stability theory relies on constructing an explicit distance function. This work uses contraction metrics to derive an adaptive controller for stabilizable nonlinear systems by constructing a distance-like function differentially rather than explicitly. Because stabilizability is in fact equivalent to the existence of a contraction metric, the proposed approach is significantly more general than available results in the literature. In particular, the method can be applied to underactuated systems. More broadly, it can also be used in transfer learning where a feedback controller has been carefully learned for a nominal system, but needs to remain effective in the presence of significant but structured variations in parameters. Simulation results illustrate the approach.

Motivation & Objective

• To overcome the limitations of traditional Lyapunov-based adaptive control methods that require explicit construction of control Lyapunov functions.
• To develop a general adaptive control framework applicable to stabilizable nonlinear systems, including those with matched and extended matched uncertainties.
• To enable application to underactuated systems where systematic adaptive controller synthesis is traditionally difficult.
• To integrate robustness features such as deadzones and parameter bounds directly into the controller design.
• To provide a computationally tractable approach using linear matrix inequalities instead of non-convex sum-of-squares optimization.

Proposed method

• Uses control contraction metrics (CCMs) to define a differential distance function between trajectories, avoiding explicit Lyapunov function construction.
• Derives a controller based on the CCM framework, where the metric depends on estimated parameters and is updated via an adaptive law.
• Introduces a parameter-dependent Riemannian energy function E(x, xd, θ̂) to measure trajectory deviation from a desired path.
• Applies a first-order filter to the energy derivative to handle parameter adaptation dynamics and ensure boundedness.
• Employs an adaptation law dθ̂/dt = -Γφ(x)ᵀM(γ(1),θ̂)γs(1) to update parameter estimates, ensuring stability via virtual system analysis.
• Uses a Lyapunov-like function V on a virtual system to prove boundedness and asymptotic convergence of the energy function to zero.

Experimental results

Research questions

• RQ1 Can contraction metrics be used to design adaptive controllers without requiring explicit control Lyapunov functions?
• RQ2 Is the proposed method applicable to nonlinear systems with unmatched or extended matched parametric uncertainties?
• RQ3 Can the approach ensure global asymptotic stability for underactuated systems using only stabilizability as a condition?
• RQ4 How can robustness features like deadzones and parameter bounds be naturally incorporated into the adaptive control framework?
• RQ5 Does the method avoid the computational complexity of sum-of-squares optimization while maintaining stability guarantees?

Key findings

• The proposed controller ensures global asymptotic stability of the closed-loop system by proving that the Riemannian energy E(x, xd, θ̂) converges to zero as t → ∞.
• The method is applicable to any stabilizable nonlinear system, including underactuated systems, due to the equivalence between stabilizability and the existence of a contraction metric.
• The adaptation law ensures boundedness of the parameter error and asymptotic convergence of the state trajectory to the desired path.
• The virtual system analysis confirms that both the energy and parameter error remain bounded, with the energy decaying exponentially under the given conditions.
• The approach avoids non-convex sum-of-squares optimization by formulating the design conditions as linear matrix inequalities.
• The controller can be extended with robust features such as deadzones and parameter bounds without altering the core stability proof.