In a previous blog post, Linearized mean value engine model-based idle control, a mean value air path model was presented. That model was used for an idle control application. In other applications a mean value air path model can be used as a pressure or mass flow rate source representing the intake manifold dynamics and coupled to a swept volume model such as what is used in crank angle resolved applications. This blog post describes how to generate and tune calibration parameters in a mean value engine air path model using control design. The method is more computationally efficient than using optimisation tools for such an application. In the following text, the air path model is recalled.
- Mean value model for intake mass flow and its calibration
1.1 Mass flow model
The mass flow through the throttle can be approximated as
where Aa is the open area, λCD is the throttle discharge coefficient, Pa is the upstream pressure, i.e. ambient pressure in this case, R is the ideal gas constant, Ta is the upstream temperature, pm is the downstream pressure and ṁa is the air mass flow through the throttle.
λCD is then tuned to calibrate (1) at each operating speed and load condition against a measured quantity. To calibrate (1) with λCD , optimisation tools can be used to adjust λCD at each iteration so that the value of the calculated mass flow rate ṁa is very close to the measured mass flow rate which is defined as ṁaMeasured. The number of iterations for convergence depends on the convergence rate and the initial value defined. This method involves a large amount of simulations and the choice of the initial value is also very crucial and could involve an amount of trial and error. A more efficient way to tune λCD by using a control method is presented below which only simulates the model once and does not depend so heavily on the initial value of λCD.
1.2 Calibration of mass flow model with λCD using control design
We define the error between measured air mass and calculated air mass as:
where eair is the error. A proportional controller can be designed such that
where Kp1 is a positive scalar. Substituting (3) into (1), (1) becomes
If we choose an appropriate control gain for Kp1 , ma can be controlled to follow maMeasured and hence ṁa follows ṁaMeasured, i.e. eair→0.
- Mean value model for intake manifold pressure and its calibration
2.1 Intake manifold pressure model
The manifold pressure can be modelled as
where Tm is the manifold air temperature, Vm is the intake manifold volume, and ṁβ is the mass flow into the cylinder.
The mass flow into the cylinder can be modelled as
where λl(·) is the volumetric efficiency of the intake ports and valves, i.e. λl(ωe,pm)=mβ⁄(ρmVd), ρm is the air density in the intake manifold, Vd is the engine displacement, ωe is the engine speed, λ is the air fuel ratio, i.e. λ=ṁβ/(σ0ṁφ), σ0 is approximately 14.67 for a typical RON 95 gasoline engine running at a stochiometric air fuel ratio. We can approximate the volumetric efficiency of the intake ports and valves as the product of a function of the pressure and a function of speed:
where pout is the exhaust manifold pressure and Vc is the clearance volume, κ, γ0, γ1 and γ2 are tuning parameters. We have chosen γ0 as the tuning parameter and γ1 and γ2 are fixed.
2.2 Calibration of intake manifold pressure with γ0 using control design
We define the pressure error as
A proportional controller can be designed such that
where Kp2 is a positive scalar. Substituting (7) into (6) and (6) into (5), equation (4) becomes
Because the values of RTmṁa/Vm and λlp/Vm·(γ1ωe+γ2ωe2)Vdωe/(4π(1+1/(λσ0))) are much smaller than pm in SI units in (8) at steady state condition, (8) can be approximated by
By choosing an appropriate control gain Kp2 , (9) converges and ep→0.
- Simulation results
Figure 1 shows the simulation results using control design as described above. Subplot 1 in figure 1 shows the calculated intake manifold pressure pm (blue) and the measured intake manifold pressure pmMeasured(red). Subplot 2 shows the calculated mass flow rate through the throttle ṁa (blue) and the measured mass flow rate through the throttle ṁaMeasured (red). Both subplots show that the calculated values are very close estimations of the measured values. Subplot 3 and 4 show the values of γ0 and λCD, which are obtained from equations (7) and (3) with Kp2=5, Kp1=20.
In this application, i.e. mean value model of an engine air path, it is possible to use control design to calculate the values required for tuning the parameters γ0 and λCD by stabilising the error dynamics to zero asymptotically. Thus an optimisation tool which results depend on an initial guess of the tuning parameter values and which uses time consuming simulation iterations is no longer required. It is very common to have a large number of steady state conditions that are required to be calibrated. In this application, the control based calibration method is thus a much more efficient and robust method.
Written by: Xiaoran Han – Project Engineer
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