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

• Published in: IEEE transactions on automatic control Vol. 67; no. 6; pp. 2784 - 2799
• Main Authors: Espitia, Nicolas, Perruquetti, Wilfrid
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Automatic; Backstepping; Backstepping control design; Combinatorial analysis; Control design; Control systems design; Delay; delay systems; Delays; Engineering Sciences; Feedback; infinite-dimensional systems; Kernel; Linear systems; Ordinary differential equations; Partial differential equations; Polynomials; prescribed-time convergence; Stability analysis; Time-varying systems; prescribed-time convergence; delay systems; Infinite dimensional systems; backstepping control design
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2021.3093527

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.