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In modern engine control applications, there is a distinct trend towards model-based control schemes. There are various reasons for this trend: Physical models allow deeper insights compared to heuristic functions, controllers can be designed faster and more accurately, and the possibility of obtaining an automated application scheme for the final engine to be controlled is a significant advantage. Another reason is that if physical effects can be separated, higher order models can be applied for different subsystems. This is in contrast to heuristic functions where the determination of the various maps poses large problems and is thus only feasible for low order models. One of the most important parts of an engine management system is the air-to-fuel control. The catalytic converter requires the mean air-to-fuel ratio to be very accurate in order to reach its optimal conversion rate. Disturbances from the active carbon filter and other additional devices have to be compensated.

Modern model-based control schemes and their application on different engines need mathematical models for the various dynamic subsystems of interest. Here, the fuel path of an SI engine is investigated. When the engine speed and the throttle angle are kept constant, the fuel path is excited only by the fuel injected. Taking the NO concentration of the exhaust gas as a measure for the air/fuel ratio, models are derived for the wall-wetting dynamics, the gas mixture, as well as for the air/fuel ratio sensor. When only the spark advance is excited, the gas flow dynamics can be studied. A very fast NO measurement device is used as reference. Its time constant is below the segment time of one single cylinder (180° crank angle for a 4-cylinder engine), therefore its dynamics are much faster than the time constants of the systems investigated. A model structure considering the muliplexing effects of the discrete operation of an engine is given for the fuel path of a BMW 1.8 liter engine.

The dynamics of the fuel-path subsystem of an SI engine, between fuel injection command signal and measured air-to-fuel ratio, is modeled approximately by a series connection of a first-order low-pass filter and a time delay element. The three parameters involved in this approximation, i.e., the time constant and the gain factor of the low-pass filter as well as the time delay, depend on the operating point of the engine. In order to design a gain-scheduled controller for the entire operating range of the engine, the parameters are identified for a number of operating points. For the automation of the parameter identification of all operating points desired, an on-line identification based on the recursive least-squares method is used. The algorithm for the decision of whether to increase or decrease the integer part of the current estimated time delay, which is a multiple of the sampling period, is based on an estimation of the fractional part of the time delay at each point.