The proposed necessary and sufficient conditions for synchronization of identical linear systems can be effectively utilized to design optimal or multi-objective controllers by providing a structured framework for controller tuning. Specifically, these conditions allow for the identification of stabilizers that ensure uniform global exponential stability (UGES) of the synchronization set. By parameterizing the stabilizers based on the conditions outlined in the article, designers can explore a range of controller configurations that meet specific performance criteria, such as convergence rates and robustness to disturbances.
For instance, the conditions related to Lyapunov inequalities and the existence of strict quadratic Lyapunov functions can guide the selection of controller gains that not only stabilize the system but also optimize performance metrics like settling time and overshoot. Furthermore, the ability to extend existing Linear Matrix Inequality (LMI)-based results to undirected graphs enhances the applicability of these conditions in practical scenarios, where multi-objective control problems often arise. By integrating these insights, control engineers can develop controllers that balance multiple objectives, such as minimizing energy consumption while ensuring rapid synchronization among agents.