An optimal distributed PID-like control for the output containment and leader-following of heterogeneous high-order multi-agent systems

This paper addresses the leader-tracking and the containment control problems for heterogeneous high-order Multi-Agent Systems (MASs) sharing information through a directed communication topology. To solve both the control problems, a fully-distributed Proportional-Integral-Derivative (PID) control strategy is proposed, whose stability is analytically proven by exploiting the regulator equations and the Static Output Feedback (SOF) procedure adapted to the MASs framework. The application of SOF allows recasting the PID control design problem into a state-feedback control design one and, hence, finding the proper values of the proportional, integral and derivative actions via classical state-feedback approaches, such as the Linear-Quadratic-Regulator (LQR) strategy. Numerical simulations confirm the effectiveness of the proposed approach in guaranteeing that each follower tracks the leader behavior in the case of leader-tracking and that each follower converges to the convex hull spanned by the multiple leaders in the case of containment control.