Stabilization Parametric Region of Distributed PID Controllers for General First-Order Multi-Agent Systems With Time Delay

The stabilization problem of distributed proportional-integral-derivative (PID) controllers for general first-order multi-agent systems with time delay is investigated in the paper. The closed-loop multi-input multi-output (MIMO) framework in frequency domain is firstly introduced for the multi-agent system. Based on the matrix theory, the whole system is decoupled into several subsystems with respect to the eigenvalues of the Laplacian matrix. Considering that the eigenvalues may be complex numbers, the consensus problem of the multi-agent system is transformed into the stabilizing problem of all the subsystems with complex coefficients. For each subsystem with complex coefficients, the range of admissible proportional gains is analytically determined. Then, the stabilizing region in the space of integral gain and derivative gain for a given proportional gain value is also obtained in an analytical form. The entire stabilizing set can be determined by sweeping proportional gain in the allowable range. The proposed method is conducted for general first-order multi-agent systems under arbitrary topology including undirected and directed graph topology. Besides, the results in the paper provide the basis for the design of distributed PID controllers satisfying different performance criteria. The simulation examples are presented to check the validity of the proposed control strategy.