Consensus of multi-agent system with reaction-diffusion terms and time delays via output-feedback adaptive boundary control

Multi-agent systems (MASs) are widely used in many necessary fields such as cybersecurity, precision agriculture, satellites, and smart grids. In many practical processes, a proportion of states cannot be obtained directly. Therefore, it is desired to design an observer. This article explores the utilization of output-feedback adaptive control techniques to achieve consensus in MASs based on partial differential equations (PDEs) with reaction-diffusion terms and time delays. First, in dealing with time-invariant delays, an observer according to the Luenberger method is proposed to assess the state of agents. Convergence conditions of the estimated states are obtained by employing Lyapunov functions and Wirtinger inequality. Using the obtained estimated states, a boundary adaptive control strategy is proposed to make the error system consensus. The merit of this control method lies in the fact that communication between agents only occurs at the spatial boundary position, not throughout the entire spatial domain, and the parameters of the system can be constantly changed. Consensus criteria of MASs with time-invariant delays are obtained by using the output-feedback adaptive control, and the results are further generalized to the model with time-varying delays. Lastly, two experimental scenarios are provided to show the practicality of the proposed theories.