Robust Performance Analysis of Cooperative Control Dynamics via Integral Quadratic Constraints

We study cooperative control dynamics with gradient based forcing terms. As a specific example, we focus on source-seeking dynamics with vehicles embedded in an unknown scalar field with a subset of agents having gradient information. As interaction mechanisms, formation control dynamics and flocking dynamics are considered. We leverage the framework of $\alpha$-integral quadratic constraints to obtain convergence rate estimates whenever exponential stability can be achieved. The communication graph and the interaction potential are assumed to be time-invariant and uncertain. Sufficient conditions take the form of linear matrix inequalities independent of the size of network. A derivation (purely in time-domain) of the so-called \textit{hard} Zames-Falb $\alpha$-IQCs involving general non-causal higher order multipliers is given along with a suitably adapted parameterization of the multipliers to the $\alpha$-IQC setting. The time-domain arguments facilitate a straightforward extension to linear parameter varying systems. Numerical examples illustrate the application of the theoretical results.