Finite-Time Stability Tools for Control and Estimation

Abstract

This survey presents some existing and new results on analysis and design of finite-time and fixed-time converging systems. Two main groups of approaches for analysis/synthesis of this kind of convergence are considered: based on Lyapunov functions and the theory of homogeneous systems. The focus is put on the dynamics described by ordinary differential equations, time-delay models and partial differential equations. Some popular control and estimation algorithms, which posses accelerated converge rates, are reviewed. The issues of discretization of finite-/fixed-time converging systems are discussed.

Adaptive state observation of linear time‐varying systems with delayed measurements and unknown parameters

“…the proof of claim (i) is completed invoking (10) and (12). Claim (ii) is established multiplying (11) by  ⊤ (t) and using (7b) and (10). Finally, the proof of boundedness of the signals 𝜁(t), Υ(t), and 𝜒(t) follows immediately from Assumptions 2 and 3 and boundedness of v(t) and φ(t).…”

“…the proof of claim (i) is completed invoking (10) and (12). Claim (ii) is established multiplying (11) by  ⊤ (t) and using (7b) and (10).…”

Averaging method for the stability analysis of strongly nonlinear mechanical systems

“…Lemma 2 [Efimov and Polyakov, 2021] Let f : R n → R be a continuously differentiable homogeneous function of order σ ∈ R, then…”

“…Let us note that, in numerous models of mechanical systems, forces of viscous friction and restoring forces are approximated by homogeneous functions with homogeneity orders higher than one (see, e.g., [Gendelman and Lamarque, 2005, Kovacic and Zukovic, 2017, Efimov and Polyakov, 2021 and the references therein).…”

“…Remark 2 It is worth mentioning that in the case where ν = 2µ µ+1 the system (1) is homogeneous of the order µ−1 2 with respect to the dilation (1; µ+1 2 ) (see [Efimov and Polyakov, 2021]).…”