PDE Boundary Control of Multi-Input LTI Systems With Distinct and Uncertain Input Delays

Taking advantage of the concept of a transport partial differential equation (PDE) representation of the delayed input, we propose a predictor-feedback control for the stabilization of multi-input linear time-invariant (LTI) systems with distinct input delays. Different from recently published results about predictor feedbacks for multi-input time delay systems that require a perfect knowledge of the length of delays, the LTI plants taken into account in this paper have the uncertain and distinct time delay in each individual input channel. To deal with the problem of the unknown multi-input delays and the consequently unmeasurable distributed input, an adaptive PDE boundary control scheme including an estimation of the unmeasured transport PDE state and an online identification of the unknown delay is developed. The conventional backstepping transformation for the known delay case is also replaced by a unity-rescaled backstepping transformation for the case of delay adaptation. With Lyapunov-based analysis, we show local stability of the delay-adaptive prediction-based closed-loop systems.