Delay effects on the limit cycling behavior in resonant inverters with state feedback

Abstract

Bifurcations of limit cycles in an H-bridge LC resonant inverter are reexamined taking into account a time delay in the switching transition. The analysis is accomplished by means of a model of the inverter that compresses, in only three parameters, all the elements associated to both series and parallel topologies of the inverter, to the parasitic effects, to the state feedback control, and to the switching time delay. Emphasis is made in the deviation of preexisting bifurcations without delay and the new ones arising when the time delay is taken into account. It is shown from the analysis and numerical simulations that the delay can degrade the quality of oscillations and even inhibit them, but it is also demonstrated that to some extent, this drawback can be compensated by an appropriate state feedback.

Dynamics of a time-delayed relay system

“…This situation was also considered in Ref. [6] in an investigation of switch-mode power converters. Using numerical case studies, the authors found periodic orbits as well as smooth and discontinuity induced bifurcations similar to our results.…”

“…In the latter case, trajectories are guaranteed to cross the switching manifold such that solutions of DDE (6) are necessarily oscillatory. In the former case, there are initial conditions that result in nonoscillatory solutions to DDE (6), ones that approach the fixed point without ever crossing the switching manifold. In this paper, we focus on oscillatory solutions and will exclude such initial conditions from consideration.…”

“…In this section, we discuss periodic solutions to the delayed relay system (6) and present an overview of their parameter regions of stability and bifurcations. Periodic solutions and their bifurcations are analytically calculated and characterized using a set of finite-dimensional maps, which are defined and analyzed in Sec.…”

“…With respect to the delay τ , this periodic solution has a discrete frequency ν > ν (for a four-symbol symmetric solution ν = ν + 2n). In terms of the dimensionless DDE (6), this mapping implies x ν (t| , Q) = x ν (t| , Q). Here, x ν (t| , Q) denotes a periodic solution of DDE (6) with frequency ν and x ν (t| , Q) the corresponding ν -frequency solution of DDE (6) with the parameter changed to = + ω c nP.…”

“…Aside from control applications, nonsmooth DDEs arise as descriptions of naturally occurring systems for which nonlinearities are well approximated by functions that take on discrete values [1,2]. Due to their importance, nonsmooth DDEs have been the focus of much recent attention [1][2][3][4][5][6][7].…”