Predictor-Feedback Prescribed-Time Stabilization of LTI Systems With Input Delay

This article first deals with the problem of prescribed-time stability of linear systems without delay. The analysis and design involve the Bell polynomials, the generalized Laguerre polynomials, the Lah numbers, and a suitable polynomial-based Vandermonde matrix. The results can be used to design a new controller-with time-varying gains-ensuring prescribed-time stabilization of controllable linear time-invariant (LTI) systems. The approach leads to similar results compared to Holloway et al. 2019, but offers an alternative and compact control design (especially for the choice of the time-varying gains). Based on the preliminary results for the delay-free case, we then study controllable LTI systems with single input and subject to a constant input delay. We design a predictor feedback with time-varying gains. To achieve this, we model the input delay as a transport partial differential equation (PDE) and build on the cascade PDE-ordinary differential equation setting (inspired by Krstic 2009) so as the design of the prescribed-time predictor feedback is carried out based on the backstepping approach, which makes use of time-varying kernels. We guarantee the bounded invertibility of the backstepping transformation, and we prove that the closed-loop solution converges to the equilibrium in a prescribed time. A simulation example illustrates the results.