Optimal Output Synchronization of Euler–Lagrange Systems With Uncertain Time-Varying Quadratic Cost Functions

In this article, we study the optimal output synchronization problem (OOSP) for uncertain networked Euler-Lagrange (EL) systems. Specifically, the system outputs are expected to be synchronized at the solution of an uncertain distributed time-varying quadratic optimization problem, where each local time-varying cost function includes uncertain parameters. From a centralized perspective, we first develop a controller with adaptive control gains to guide the output of a double-integrator system toward the time-varying optimal solution. By employing the modified average estimators, we extend the centralized design to a distributed implementation to address the OOSP for uncertain EL systems. Using matrix trace properties and composite Lyapunov analysis, we prove that the system outputs can asymptotically converge to the desired time-varying optimal solution. Two examples are used to verify the proposed designs.