Slow Invariant Manifold of Laser with Feedback

Previous studies have demonstrated, experimentally and theoretically, the existence of slow–fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant manifold analytical equation of such a laser model and also highlights its symmetries if any exist. This equation and its graphical representation in the phase space enable, on the one hand, discriminating the slow evolution of the trajectory curves from the fast one and, on the other hand, improving our understanding of this slow–fast regime.

Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

Output-Feedback Sliding Mode Control for Permanent Magnet Synchronous Motor Servo System Subject to Unmatched Disturbances

This paper aims to investigate the speed regulation problem for permanent magnet synchronous motor (PMSM) servo systems subject to unknown load torque disturbances. The proposed method utilizes sliding mode control (SMC), invariant manifold theory, and disturbance observation technique. In the PMSM servo systems, the unknown load torques will affect the control performance to a large extent, which is unmatched. In addition, compared with full-state measurement, the output-feedback framework is easy to implement and reduces the sensor costs. However, it is difficult to handle unmatched disturbance and unmeasured states simultaneously. To this end, this paper specifically combines the sliding mode control theory with the invariant manifold theory and puts forward an output-feedback disturbance rejection control method. The key idea is that the unmatched disturbance in the PMSM servo systems is transformed into matched one by taking advantage of the invariant manifold, which is different from existing results. The transformation maintains most of dynamics of the PMSM system for control design, which improves the accuracy. In addition, an extended state observer is designed to estimate the current and lumped disturbance simultaneously; then, the output-feedback SMC method is proposed by introducing the estimations. Besides, the switching gain in the proposed sliding mode controller can change with estimation errors adaptively, and the chattering reduces. Simulation results on a PMSM system validate the effectiveness of the proposed control strategy.

Oseledets Splitting and Invariant Manifolds on Fields of Banach Spaces

We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an application, we prove an invariant manifold theorem for nonlinear cocycles acting on measurable fields of Banach spaces.