## Abstract

In this paper closed-form conditions for predicting the boundary of period-doubling (PD) bifurcation or saddle-node (SN) bifurcation in a class of PWM piecewise linear systems are obtained from a time-domain asymptotic approach. Examples of switched system considered in this study are switching dc-dc power electronics converters, temperature control systems and hydraulic valve control systems among others. These conditions are obtained from the steady-state discrete-time model using an asymptotic approach without resorting to frequency-domain Fourier analysis and without using the monodromy or the Jacobian matrix of the discrete-time model as it was recently reported in the existing literature on this topic. The availability of such design-oriented boundary expressions allows to understand the effect of the different parameters of the system upon its stability and its dynamical behavior.

## Highlights

• IntroductionThe system equations are linear and time-invariant and can be solved in closed-form

• For each phase, the system equations are linear and time-invariant and can be solved in closed-form

• In [6], the transformation from the Fourier frequency-domain to the timedomain is based on elementary partial fraction decomposition after defining some elementary cases of the system transfer function in the s−domain and listing them in the form of tables. This transfer function cannot be directly defined for systems with A2 A1 making the approach only applicable for a limited class of Pulse Width Modulation (PWM) systems like the ones considered in [6]. In particular those that can be be formulated in the form of a linear system and a square-wave siganl generated by a comparator like the PWM process

## Summary

Introduction

The system equations are linear and time-invariant and can be solved in closed-form. After obtaining the Jacobian or monodromy matrix, critical boundary conditions for some singularities like saddlenode (SN) bifurcation or period-doubling (PD) can be obtained by imposing in the characteristic equation in the zdomain that one of the eigenvalues is equal to +1 or −1 respectively [4] Another approach recently used in [6] for locating these boundaries which was wrongly called harmonic balance is by expanding the feedback signal into a. In [6], the transformation from the Fourier frequency-domain to the timedomain is based on elementary partial fraction decomposition after defining some elementary cases of the system transfer function in the s−domain and listing them in the form of tables This transfer function cannot be directly defined for systems with A2 A1 making the approach only applicable for a limited class of PWM systems like the ones considered in [6]. Some concluding remarks are drawn in the last section

Asymptotic methods for bifurcation boundary prediction
Conclusions
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