Second-Order Sliding-Mode Leader-Follower Consensus for Networked Uncertain Diffusion PDEs with Spatially Varying Diffusivity

The primary concern of the present chapter is to address the distributed leader-following consensus tracking problem for a network of agents governed by uncertain diffusion PDEs with the Neumann-type boundary actuation and uncertain spatially varying diffusivity. Except for the “leader” agent that generates the reference profile to be tracked, all remaining agents, called “followers,” are required to asymptotically track the infinite-dimensional time-varying leader state. The dynamics of the follower agents are affected by smooth boundary disturbances unbounded in magnitude and with a bounded derivative. The proposed local interaction rule is developed by assuming that only neighboring collocated boundary sensing is available, and it consists of a nonlinear sliding-mode-based protocol. The performance and stability properties of the resulting infinite-dimensional networked system are then formally studied by means of Lyapunov analysis. The analysis demonstrates the global exponential stability of the resulting error boundary-value problem in a suitable Sobolev space. The effectiveness of the developed control scheme is supported by simulation results.