Dynamic modelling of diesel engine emissions using the parametric Volterra series

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The design of powertrain controllers relies on the availability of data-driven models of the emissions formation from internal-combustion engines. Typically these are in the form of tables or statistical regression models based on data obtained from stabilised experiments. However, as the complexity of engine systems increases, the number of experiments required to obtain the effects of each actuator becomes large. In addition, the models are only valid under stable operating conditions and do not give any information as to dynamic behaviour. In this paper, the use of the Volterra series (dynamic polynomial models) calculated from dynamic measurements is presented as an alternative to the steady-state models. Dynamic measurements of gaseous exhaust emissions were taken for a 2.0 l automotive diesel engine installed on a transient engine dynamometer. Sinusoidally based excitations were used to vary the engine speed, the load, the main injection timing, the exhaust gas recirculation valve position and the fuel injection pressure. Volterra models calculated for nitrogen oxide and carbon dioxide emissions presented high levels of fit with R 2 values of 0.85 and 0.91 respectively and normalised r.m.s. error values of 6.8% and 6.6% respectively for a cold-start New European Driving Cycle. Models for carbon monoxide and total hydrocarbon emissions presented poorer levels of fit (normalised r.m.s. errors of 26% and 17% respectively), with difficulties in obtaining the high non-linearities of the measured data, notably for very high emission levels.
Original languageEnglish
Pages (from-to)164-179
Number of pages16
JournalProceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering
Issue number2
Early online date24 Oct 2013
Publication statusPublished - 1 Feb 2014


  • diesel engine emissions
  • engine control systems
  • engine dynamics
  • engine testing
  • diesel engine design and development


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