Discrete-Time Distributed Optimization for Linear Uncertain Multi-Agent Systems

The distributed optimization algorithm proposed by J. Wang and N. Elia in 2010 has been shown to achieve linear convergence for multi-agent systems with single-integrator dynamics. This paper extends their result, including the linear convergence rate, to a more complex scenario where the agents have heterogeneous multi-input multi-output linear dynamics and are subject to external disturbances and parametric uncertainties. Disturbances are dealt with via an internal-model-based control design, and the interaction among the tracking error dynamics, average dynamics, and dispersion dynamics is analyzed through a composite Lyapunov function and the cyclic small-gain theorem. The key is to ensure a small enough stepsize for the convergence of the proposed algorithm, which is similar to the condition for time-scale separation in singular perturbation theory.