Functional Stochastic Modeling of Empirical Knock Sensor Signals

Abstract

Knock behaves a cyclically random process but also excites deterministic knock resonant behavior within any given cycle. While individual instances of the resonant response are readily acquired, the stochastic / cyclic variations of such signals (which also reflect the underlying knock process) are harder to quantify. In this work, a more complete model of this process is developed, capturing both the cyclic variability in the knock response as well as its functional resonant behavior. A recently developed alignment process is first used to evaluate the characteristic ensemble mean knock ‘signature’ of the data. A functional linearization about this ensemble mean is then used to decompose and model cyclic variations in terms of small variations in amplitude, frequency and phasing of the signal. The model is fitted to the data, encapsulating the stochastic variation of the knock signal within the stochastic variation of the estimated parameters. A residual analysis used to assess the goodness of fit as a function of crank angle, and the distribution and covariance of the estimated parameters is discussed.