Robust Performance Analysis of Source-Seeking Dynamics with Integral Quadratic Constraints

TL;DR

This paper develops a novel framework using integral quadratic constraints to analyze the performance and convergence of source-seeking vehicle dynamics, including linear, non-minimum phase, and LPV models, with reduced conservatism.

Contribution

It introduces a new derivation of Zames-Falb IQCs for $eta$-integral quadratic constraints and extends the analysis to LPV vehicle models, improving convergence rate estimates.

Findings

Reduced conservatism in convergence estimates for vehicle dynamics.

Extension of IQC framework to LPV and non-linear vehicle models.

Demonstrated benefits on quadrotor and LPV examples.

Abstract

We analyze the performance of source-seeking dynamics involving either a single vehicle or multiple flocking-vehicles embedded in an underlying strongly convex scalar field with gradient based forcing terms. For multiple vehicles under flocking dynamics embedded in quadratic fields, we show that the dynamics of the center of mass are equivalent to the dynamics of a single agent. We leverage the recently developed framework of $\alpha$-integral quadratic constraints (IQCs) to obtain convergence rate estimates. We first present a derivation of \textit{hard} Zames-Falb (ZF) $\alpha$-IQCs involving general non-causal multipliers based on purely time-domain arguments and show that a parameterization of the ZF multiplier, suggested in the literature for the standard version of the ZF IQCs, can be adapted to the $\alpha$-IQCs setting to obtain quasi-convex programs for estimating convergence rates. Owing to the time-domain arguments, we can seamlessly extend these results to linear parameter varying (LPV) vehicles possibly opening the doors to non-linear vehicle models with quasi-LPV representations. We illustrate the theoretical results on a linear time invariant (LTI) model of a quadrotor, a non-minimum phase LTI plant and two LPV examples which show a clear benefit of using general non-causal dynamic multipliers to drastically reduce conservatism.