A recursive algorithm is derived to compute the generalized frequency response functions for a large class of non-linear integro-differential equations. Applications to Duffing's equation and a modified Van der Pol model are discussed.

The characterization of knock intensity distributions can provide useful insights into the process and help to improve knock control system designs. In this paper, an extensive statistical analysis is performed on knock intensity data recorded under a broad range of operating conditions. First, the critical issue of whether the data exhibit any cycle-to-cycle correlations is investigated, and it is shown that knock intensity closely approximates a cyclically independent random process. The study then focuses on the variation of knock intensity distributions with operating condition, and on the quantification of these distributions using simple scalar measures. The relationship between knock event distributions and knock intensity distributions is also investigated, and it is shown that knock event data are binomially distributed regardless of the underlying knock intensity distribution. This supports ongoing efforts to exploit binomial probability theory in knock event simulation and controller design.

Analytical and graphical methods of interpreting generalised frequency response functions for non-linear systems are derived. It is shown that non-linear phenomena can be classified into intra-kernel and inter-kernel interference and that worst-case responses can be computed. The results are illustrated using several discrete- and continuous-time non-linear systems.

Knock intensity behaves as a random process which may be characterized using simple scalar metrics such as the mean and variance, or (more commonly) by the probability of knock events. However, such measures discard much of the information present in the signal. Several researchers have therefore sought to obtain a more complete characterization of the process by fitting parametric log-normal or gamma distribution models to knock intensity distributions. The present study extends this work both in terms of the range of engine operating conditions considered and in terms of the evaluation of the goodness of fit between two different models and the experimental data. In particular, new and arguably more application-appropriate measures of the goodness of fit provide a clearer assessment of the performance of the models, and a like-for-like comparison of log-normal and gamma distribution model forms demonstrates that the log-normal model better characterizes the experimental data used in this study.

The transient response of a catalytic converter to fluctuations in exhaust gas composition has a significant impact on tailpipe emissions. Advanced emission control strategies therefore need to incorporate a model for such behavior, which must also be sufficiently simple for practical implementation in-vehicle. To this end, a variety of semi-empirical models have been developed, including most recently a number of oxygen “storage-dominated” models. In this paper a new storage-dominated model is developed, which includes for the first time the effects of space velocity. The parameters of model may be estimated using the invariant embedding method.