On the Stability of Distributed PID Controllers for Multi-agent Systems with Two Inputs and Two Outputs

Based on the multi-delay and the strong coupling for multivariable systems, we aim to determine the stabilizing region of distributed two-loop proportional-integral-derivative (PID) controllers for a multi-agent system with a two-input two-output multivariable system as an individual agent. First, we decompose the multi-agent system into several subsystems with respect to the eigenvalues of the Laplacian matrix, which reduces the complexity of the system. Thus, the stability problem of the whole multi-agent system is transformed into that of the subsystems. Then, by introducing an equivalent transfer function (ETF), we further decouple the decomposed subsystems into independent single-input single-output systems with time delay. Based on the Hermite-Biehler theorem, the range of the admissible proportional gains (k_p) for each equivalent single loop is analytically derived. For each value of k_p in the entire range, the stabilizing region of the subsystem in the space of integral gain (k_i) and derivative gain (k_d) is determined, and the linear programming characteristic of the stabilizing (k_i, k_d) region is obtained. By solving the intersection of the stabilizing region for all subsystems, the stabilizing region of a distributed PID controller for the multi-agent system can be determined. The simulation results verify the accessibility, simplicity, and effectiveness of the proposed method.