Fuzzy Small-Gain Approach for the Distributed Optimization of T–S Fuzzy Cyber–Physical Systems

The distributed optimization problem (DOP) of Takagi–Sugeno (T–S) fuzzy cyber–physical systems is studied under the framework of weight-balanced graphs and quasistrongly connected characteristics. The objective is to drive the outputs of all agents to the optimal solution of a given global objective function regarded as the desired output, based on the partial information of the local objective functions. To this end, distributed optimal coordinators (DOCs) are used to generate optimal solutions of local objective functions that converge to the desired output, and fuzzy reference-tracking controllers are designed to ensure that all agents can track the optimal solutions. As novel technical results, two Lyapunov-based fuzzy input-to-state stability (ISS) small-gain theorems are proposed for the T–S fuzzy interconnected system. Thus, the overall closed-loop system is an interconnected system involving the modules of optimal coordinators and fuzzy tracking controllers with T–S fuzzy subsystems. The fuzzy ISS cyclic-small-gain theorem is applied to analyze the system stability. The DOP of T–S fuzzy cyber–physical systems is solved using the DOCs and fuzzy reference-tracking controllers through the fuzzy small-gain approach. A numerical example is presented to demonstrate the effectiveness and superiority of the proposed method.