A note on delay robustness for homogeneous systems with negative degree

Abstract

Robustness with respect to delays is discussed for homogeneous systems with negative degree. It is shown that if homogeneous system with negative degree is globally asymptotically stable at the origin in the delay-free case then the system is globally asymptotically stable with respect to a compact set containing the origin independently of delay. The possibility of applying the result for local analysis of stability for not necessary homogeneous systems is analyzed. The theoretical results are supported by numerical examples.

Sliding mode control design using canonical homogeneous norm

“…be feasible with respect to X ∈ R n×n , Y ∈ R 1×n . If a function N ∶ R n ⧵ {0} → R is continuous, positive definite, N ∈ H d (R n ), deg H d (N) = 1, and it satisfies the inequalities (21) with P = X −1 and 0 < min ≤ max , then the closed-loop system (18), (19) with…”

“…Nonlinear homogeneous differential equations/inclusions form an important class of control systems. () They appear as local approximations() or set‐valued extensions() of nonlinear systems and include models of process control, nonholonomic systems, mechanical models with frictions, etc.…”

A distributed finite-time observer for linear systems

“…Nevertheless, following [28]- [31], since under (13) the error system (10) is homogeneous of negative degree and globally finite-time stable, then the distributed observer (9) is robust with respect to additive disturbances in the right-hand side of (8), measurement noises and delays in the output channel.…”

On simple scheme of finite/fixed-time control design

“…In addition, the proposed controllers homogenize the system that implies robustness abilities to external perturbations and time-delays (see, for example, Bernuau, Polyakov, Efimov, & Perruquetti, 2013;Efimov, Polyakov, Perruquetti, & Richard, 2016;Zimenko, Efimov, Polyakov, & Perruquetti, 2017).…”