Continuous-time Constrained Funnel Synthesis for Incrementally Quadratic Nonlinear Systems

TL;DR

This paper introduces a convex optimization framework for synthesizing time-varying invariant funnels and feedback control for nonlinear systems with bounded disturbances, using incremental quadratic constraints and continuous-time formulations.

Contribution

It develops a novel continuous-time funnel synthesis method employing DLMI reformulation and convex techniques for constraint satisfaction, applicable to complex nonlinear systems.

Findings

Successfully applied to unicycle control example.

Demonstrated obstacle avoidance with a quadrotor.

Validated the effectiveness of the convex synthesis approach.

Abstract

This paper presents a convex optimization-based framework for synthesizing time-varying controlled invariant funnels and associated feedback control around a given nominal trajectory for nonlinear systems subject to bounded disturbances. Nonlinearities are modeled using incremental quadratic constraints, including Lipschitz, L-smooth, and sector-bounded nonlinearities. Funnel invariance is ensured via a DLMI. Together with pointwise-in-time LMIs for state and input constraints, we formulate a continuous-time funnel synthesis problem. To solve it using numerical optimal control techniques, the DLMI is reformulated into a differential matrix equality (DME) and an LMI, where the DME acts as a funnel dynamics equation. We explore different formulations of these funnel dynamics. Continuous-time constraint satisfaction is addressed through two convex methods: one based on intermediate constraint-checking points, and another using a successive convexification method with subgradients to handle nondifferentiable maximum eigenvalue functions. Theoretical justification is provided for the existence of a measurable and integrable subgradient for the latter. The method is demonstrated on two numerical examples: the control of a unicycle and a 6-degree-of-freedom quadrotor for obstacle avoidance.