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In this work the authors present a model to simulate the in-cylinder pressure oscillations due to knock. Pressure oscillations are predicted by the explicit integration of a Partial Differential Wave Equation (PDWE) similar, in its structure, to the so-called “Equation of Telegraphy”. This equation differs mainly from the classical wave formulation for the presence of a loss term. The general solution of such equation is obtained by the Fourier method of variables separation. The integration space is a cylindrical acoustic cavity whose volume is evaluated at the knock onset. The integration constants are derived from the boundary and initial conditions. A novel approach is proposed to derive the initial condition for the derivative of the oscillating component of pressure. It descends, conceptually, from the integration of the linearized relation between the derivative of pressure versus time and the expansion velocity of burned gas.

Idle Speed Control plays a crucial role to reduce fuel consumption that turns in both a direct economic benefit for customers and CO\d reduction particularly important to tackle the progressive global environmental warming. Typically, control strategies available in the automotive literature solve the idle speed control problem acting both on the throttle position and the spark advance, while the Air-Fuel Ratio (AFR), that strongly affects the indicated engine torque, is kept at the stoichiometric value for the sake of emission reduction. Gasoline Direct Injection (GDI) engines, working lean and equipped with proper mechanisms to reduce NOx emissions, overcome this limitation allowing the AFR to be used for the idle speed regulation.