Explicit Prediction-Based Control for Linear Difference Equations With Distributed Delays

This letter presents a prediction-based control law for linear difference equations subject to a distributed state delay and a pointwise input delay. We propose to use a prediction-based control to overcome the instability potentially related to the distributed delay. We obtain an explicit formulation of the controller, depending only on the state and input history and involving integral kernels, which are the solutions to recursive Volterra equations. In view of future delay-sensitivity analysis, we develop an alternative approach to prove closed-loop stability, recasting the input delay as a transport Partial Differential Equation. In an analog manner to the stability analysis methodology developed for linear Delay Differential Equations, we propose a backstepping transformation to map the closed-loop system to a distributed-delay free target system. Simulation results underline the efficiency of the proposed control design.