A crank-kinematics-based engine cylinder pressure reconstruction model

Abstract

A new inverse model is proposed for reconstructing steady-state and transient engine cylinder pressure using measured crank kinematics. An adaptive nonlinear time-dependent relationship is assumed between windowed-subsections of cylinder pressure and measured crank kinematics in a time-domain format (rather than in crank-angle domain). This relationship comprises a linear sum of four separate nonlinear functions of crank jerk, acceleration, velocity and crank angle. Each of these four nonlinear functions is obtained at each time instant by fitting separate m-term Chebyshev polynomial expansions, where the total 4 m instantaneous expansion coefficients are found using a standard (overdetermined) linear least-square solution method. A convergence check on the calibration accuracy shows that this initially improves as more Chebyshev polynomial terms are used, but with further increase, the overdetermined system becomes singular. Optimal accuracy Chebyshev expansions are found to be of degree m = 4, using 90 or more cycles of engine data to fit the model. To confirm the model accuracy in predictive mode, a defined measure is used, namely the ‘ calibration peak pressure error’. This measure allows effective a priori exclusion of occasionally unacceptable predictions. The method is tested using varying speed data taken from a three-cylinder direct-injection spark ignition engine fitted with cylinder pressure sensors and a high-resolution shaft encoder. Using appropriately filtered crank kinematics (plus the ‘calibration peak pressure error’), the model produces fast and accurate predictions for previously unseen data. Peak pressure predictions are consistently within 6.5% of target, whereas locations of peak pressure are consistently within ±2.7 °CA. The computational efficiency makes it very suitable for real-time implementation.