International Journal of Automotive Technology

, Volume 19, Issue 5, pp 771–781 | Cite as

PID Controller Modelling and Optimization in Cr Systems with Standard and Reduced Accumulators

  • Alessandro Ferrari
  • Antonio Mittica
  • Pietro Pizzo
  • Zhiru Jin


The proportional-integrative-derivative (PID) controller and the pressure control valve of a Common Rail system are modelled by taking into account electronic, electrical, hydraulic and mechanical aspects. A fully predictive model of the injection apparatus is realized and validated by means of comparison with experimental data. The effects of the PID parameters on the injection system dynamics are illustrated and discussed on the basis of model results, which refer to steadystate and transient working conditions. The influence of the accumulator size on the rail pressure time history is investigated when the rail volume is dramatically reduced (up to 2.5 cm3). In particular, the effect of the large rail pressure drop that occurs at the end of the main injection for Minirail layout solutions is examined when after injections are implemented. An objective is to try to determine possible suitable values of the PID controller parameters and of the pressure-sensor sampling-frequency for rails of reduced size.

Key words

PID control Common rail Fuel injection system modelling Accumulation volume Proportional Integrative and Derivative parts 




restricted flow-area


geometric constant of the magnetic circuit


common rail


diameter of the valve seat; derivative part


energizing time




system transfer function; mass flow-rate


electric current; integrative part






proportional part


pressure control valve


proportional-integrative-derivative control


pulse width modulation








spring stiffness


valve armature mass


engine speed



x, ,

valve displacement, velocity and acceleration


valve-seat cone semi-angle


valve damping coefficient


pressure drop


flux-force angle


flow coefficient



after injection


derivative part


flux forces


integrative part




proportional part




main injection


nominal value


pressure control




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Copyright information

© The Korean Society of Automotive Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Alessandro Ferrari
    • 1
  • Antonio Mittica
    • 1
  • Pietro Pizzo
    • 1
  • Zhiru Jin
    • 1
  1. 1.Energy DepartmentPolitecnico di TorinoTorinoItaly

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