Poincaré maps modeling and local orbital stability analysis of discontinuous piecewise affine periodically driven systems

Abstract

This paper presents a methodology to study the local stability of periodic orbits (orbital stability) in switched discontinuous piecewise affine (DPWA) periodically driven multiple-input multipleoutput (MIMO) systems. The switched system of interest has a bilinear state space representation where the controller inputs are binary signals taking values in the set {0,1}. These systems are characterized by a set of affine differential equations together with switching rules to commute between them. These switching rules are described by switching functions that are periodic in time and linear in state. The methodology is based on obtaining a discrete time model (Poincaré map), its steady state operation points, and its Jacobian matrix. This provides a powerful tool for studying their stability and to predict some kind of instability phenomena that the system can undergo like subharmonic oscillations. The proposed approach is applied to a power electronic circuit which toggles among six different system equations with five switching boundaries within a switching cycle.

Analysis of Bifurcation Behavior of a Piecewise Linear Vibrator with Electromagnetic Coupling for Energy Harvesting Applications

“…We will present here three different methods. The first approach will be based on a Poincaré map function [El Aroudi et al, 2007]. The second approach is the Floquet theory ( [Aizerman and Gantmakher, 1958]) together with Fillipov methods for systems with discontinuous vector field [Fillipov, 1988], [Leine and Nijemeijer, 2004], [Giaouris et al, 2008].…”

“…By calculating each term in (31), the Jacobian matrix J can be obtained in a straightforward manner (see [El Aroudi et al, 2007] for more details). Evaluating this Jacobian matrix in a fixed point and computing its corresponding eigenvalues λ i would give us the stability of its underlying periodic orbit.…”

A Time-Domain Asymptotic Approach to Predict Saddle-Node and Period Doubling Bifurcations in Pulse Width Modulated Piecewise Linear Systems

“…A particular class of switched systems are those characterized by linear differential equations between switching events. These systems are called therefore piecewise linear (PWL) or piecewise affine (PWA) systems [2]. Most of the PWL systems studied in the literature are characterized by switching among linear subsystems when certain time-varying and T − periodic boundaries in the state space are reached.…”

Bifurcation Analysis of Current Mode Control ?uk DC-DC Converter

“…They are able to present nonlinear phenomena like bifurcations and chaos [1]. Different nonlinear phenomena were discovered in different DC-DC converters under different control strategies [2,3,4].…”