Design of adaptive output feedback synchronizing controllers for networked PDEs with boundary and in-domain structured perturbations and disturbances

This work considers a class of networked distributed parameter systems with structured perturbations and disturbances. These systems are assumed to have identical unknown structured perturbations and unknown disturbances but different initial conditions. The multi-task controllers aim to adaptively compensate both structured perturbation and disturbance effects. Additionally, to follow a virtual leader by ensuring the asymptotic convergence of each of the networked states to the leader's state. Finally, to synchronize in the sense of the convergence of the pairwise state errors. A four-part controller is proposed to address all tasks and provides many new elements for control of networked spatially distributed systems. The adaptive estimates of both the disturbance and the structured perturbation terms include a consensus term in their adaptive laws which provide the first coupling of the networked systems and aims at providing a weak version of persistence of excitation. The consensus protocol included in the synchronization component of the controller addresses the communication burden by transmitting output signals to its communicating systems instead of entire states and further, adapts the synchronization weights in proportion to the pairwise state disagreement, as a means to minimize the control effort. An abstract theoretical framework is established which handles a wide class of infinite dimensional systems including PDEs with both in-domain and boundary control and observation, and is conducive to well-posedness and stability analysis. Using the proposed multi-component controllers, the convergence of the networked states to the leader's state is established using Lyapunov stability arguments for infinite dimensional systems, along with the boundedness of all signals. The well-posedness of all networked closed loop systems is shown by using established results on an analytic semigroup approach. A numerical example involving five networked partial differential equations with boundary observation, control, and disturbances and structured perturbations at the boundary, is presented and which provides insights on the control and synchronization of networked PDE systems.