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.