Institute for Systems Research

In this paper, participation factors are revisited, resulting in new definitions. The aim ofthese definitions is to achieve a conceptual framework that doesn't hinge on anyparticular choice of initial condition. The initial condition is modeled as an uncertainquantity, which can be viewed either in a set-valued or a probabilistic setting.If the initial condition uncertainty obeys a symmetry condition, the new definitionsare found to reduce to the original definition of participation factors.

Bothpitchfork bifurcation and transcritical bifurcation are addressed.The results obtained forpitchfork bifurcations apply to general nonlinear models smoothin the state and the control. For transcritical bifurcations,the results require the system to be affine in the control.

Using this approach, both current mode control and discontinuous conduction modecan be handled systematically in a unified framework,making the modeling for these cases simpler than with the use of averaging.The results of this paper are similar to the results of Tymerski,but they are presented in a simpler manner tailored to facilitate immediate application to specific circuits.

It is shown howsampling the output at certain instants improves the obtained phase response.Frequency responses obtained from the sampled-data model aremore accurate than those obtained from various averaged models. In addition, a new ("lifted")continuous-time switching frequency-dependent model of the power stage isderived from the sampled-data model. Detailed examples illustrate themodeling tools presented here, and also provide a means of comparingresults obtained from the sampled-data approach with those obtainedfrom averaging.

General models are presented in a unified and simple manner, while removingsimplifying approximations present in previous work. These models applyboth for current mode control and voltage mode control.The general models are nonlinear. They are used toobtain {it analytical} linearized models, which are in turn employedto obtain local stability results.

Detailed examplesillustrate the modeling and analysis in the paper,and point to situations in which the sampled-data approachgives results superior to alternate methods.For instance, it is shownthat the sampled-data approach will reliablypredict the (local) stability of aconverter for which averaging or simulation predicts instability.