Boundary adaptive synchronization of networked PDEs with adaptive parameter estimators

This work considers the synchronization control of networked PDEs that have boundary observation and control. The PDEs are assumed identical, with information exchange dictated by a complete directed graph. Parametric uncertainties entering in the boundary, result in structured perturbation terms that may render the open loop system unstable. Additionally constant disturbance terms entering at the boundary are impeding state regulation. To address these parametric uncertainties, a precompensator is utilized to stabilize each networked PDE. Another component of the controller provides an on-line cancellation of the parametric uncertainties by attempting to cancel them through their adaptive estimate, and to attenuate the effects of boundary disturbances, their adaptive estimates are utilized in the controller. To provide a robustness in the adaptive estimation of the parametric uncertainties and the boundary disturbances, the standard adaptive laws are complemented with a consensus protocol term that penalizes the pairwise disagreement of the adaptive estimates. Finally, to enforce synchronization of all networked PDEs, a synchronization controller is included which utilizes only the outputs of all communicating PDE systems weighted by adaptive synchronization gains. These gains, derived from Lyapunov redesign methods, include a consensus term for added robustness and to enforce convergence. Simulation studies of five networked diffusion PDEs with boundary observation and control are included.