Linearised Mean Value Engine Model-based Idle Control

March 10th, 2017 by Mahdieh Mehrabi

Written by: Xiaoran Han – Project Engineer

1. Introduction

PID controllers have been widely used for control design due to their simplicities and non-model based design. However, they lack robustness with respect to uncertainties and external disturbances. This blog post describes a method for developing a mean value model of intake manifold pressure and engine speed for model based idle controller design which can be applied to mean value and crank angle resolved engine models. This post shows the results of a mean value nonlinear model and compares these with the result of crank angle resolved engine model. A linearised model is then derived for controller design. Validation results show the derived nonlinear and linear model can accurately catch the dynamics of Dymola crank angle resolved engine model at the operating point at which linear model is derived.

2. Model development

2.1 Manifold model

Mass flow through throttle can be approximated as


where L1 is the open area, L2 is the pressure upstream, L3is the ideal gas constant, L4 is the temperature upstream, L5 is the pressure downstream and L6 is the air mass flow through the throttle.

Manifold pressure can be modelled as


where L7 is the manifold temperature, L8 is the volume of intake manifold, and L9 is the mass flow into cylinder.

Mass flow into the cylinder can be modelled as


where L10 is the volumetric efficiency of intake ports and valves, i.e. L11, L12 is air density in intake manifold, L13 is the engine displacement volume, L14 is the engine speed, L15is air fuel ratio, i.e. L16, L17 is approximately 14.67 for typical RON 95 gasoline. Approximation of the volumetric efficiency of intake ports and valves


where L18 is the exhaust manifold pressure and L19 is clearance volume, L20 and L21 are tuning parameters.

2.2 Engine speed model

Engine brake torque can be modelled as


where L22 is the brake mean effective pressure, L23 is fuel mean effective pressure, i.e. brake mean effective pressure that an engine with an efficiency of 1 would produce with the fuel mass L24 burnt per engine cycle, L25 and L26 are energy losses related to friction and gas exchange, L27 represents the thermodynamic properties of the engine (related to indicated or inner mean pressure), L28 heating energy of the fuel, L29 mass of fuel, L30 are tuning parameters and L31 is transport delay approximated by L32.

Engine speed with a constant inertia L33 is


where L34 is load torque and can be expressed as


2.3 Linearisation of nonlinear manifold and speed model

The combined and simplified manifold pressure and speed dynamics assuming volumetric efficiency and air/fuel ratio are set to 1

8 and 9

To simplify the notation, the following abbreviations are introduced


Using these definitions, the combined nonlinear system can be written compactly as


Linearising the equation (11) at equilibrium L35 defined by


yields a second order linear system whose state space description




When delay is zero or is neglected then (13) becomes




The following section will show validation of the nonlinear model (11) and linearised model (13) in simulation.

3. Model validation

For idle speed control we are interested in the engine speed around 700 to 800 rpm, in this case nonlinear model (11) is linearised around this operating point at 710 rpm. Tuning parameters are chosen as L36.3 in (4) and L37.3 for L38.3 in (5). In Figure 1, below, results from crank angle resolved Dymola engine model, nonlinear model (11) and linear model (15) are compared. Throttle opening is kept at a constant angle and a step load torque is applied to the engine shaft at 20 seconds. Manifold pressure for crank angle resolved engine model and nonlinear model have similar behaviour before and after the disturbance. Pressure from linear model stays at its equilibrium before the disturbance and then deviates from its equilibrium because of step disturbance from load. Engine speeds from all three models have similar values before the disturbance with acceptable margin of difference after the occurrence of disturbances.

Figure 1

Figure 1: manifold pressure, engine speed and load torque from crank angle resolved Dymola engine model,

nonlinear model (11) and linear model (15) with a constant throttle opening.

Validation result shows dynamics of the engine at its linearised operating point can be accurately represented by its nonlinear and linear model. Model based controller design can then begin to be considered based on nonlinear or linear model. Next blog post will show a robust sliding model controller design and its result for engine idle speed control.


[1]. L.Guzzella and C.H. Onder, Introduction to modeling and control of internal combustion engine systems, Springer-Verlag Berlin Heidelberg 2004.


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