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 A_{a} is the open area, λ_{CD} is the throttle discharge coefficient, P_{a} is the upstream pressure, i.e. ambient pressure in this case, R is the ideal gas constant, T_{a} is the upstream temperature, p_{m} is the downstream pressure and ṁ_{a} is the air mass flow through the throttle.

λ_{CD} is then tuned to calibrate (1) at each operating speed and load condition against a measured quantity. To calibrate (1) with λ_{CD} , optimisation tools can be used to adjust λ_{CD} at each iteration so that the value of the calculated mass flow rate ṁ_{a} is very close to the measured mass flow rate which is defined as ṁ_{aMeasured}. 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 λ_{CD} 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 λ_{CD}.

**1.2 Calibration of mass flow model with λ _{CD} using control design **

We define the error between measured air mass and calculated air mass as:

where e_{air} is the error. A proportional controller can be designed such that

where K_{p1} is a positive scalar. Substituting (3) into (1), (1) becomes

If we choose an appropriate control gain for K_{p1} , m_{a} can be controlled to follow m_{aMeasured} and hence ṁ_{a} follows ṁ_{aMeasured}, i.e. e_{air}→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 T_{m} is the manifold air temperature, V_{m} is the intake manifold volume, and ṁ_{β} is the mass flow into the cylinder.

The mass flow into the cylinder can be modelled as

where λ_{l}(·) is the volumetric efficiency of the intake ports and valves, i.e. λ_{l}(ω_{e},p_{m})=m_{β}⁄(ρ_{m}V_{d}), ρ_{m} is the air density in the intake manifold, V_{d} is the engine displacement, ω_{e} is the engine speed, λ is the air fuel ratio, i.e. λ=ṁ_{β}/(σ_{0}ṁ_{φ}), σ_{0} 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 p_{out} is the exhaust manifold pressure and V_{c} is the clearance volume, κ, γ_{0}, γ_{1} and γ_{2} are tuning parameters. We have chosen γ_{0} as the tuning parameter and γ_{1} and γ_{2} are fixed.

**2.2 Calibration of intake manifold pressure with γ _{0} using control design **

We define the pressure error as

A proportional controller can be designed such that

where K_{p2} is a positive scalar. Substituting (7) into (6) and (6) into (5), equation (4) becomes

Because the values of RT_{m}ṁ_{a}/V_{m} and λ_{lp}/V_{m}·(γ_{1}ω_{e}+γ_{2}ω_{e}^{2})V_{d}ω_{e}/(4π(1+1/(λσ_{0}))) are much smaller than p_{m} in SI units in (8) at steady state condition, (8) can be approximated by

By choosing an appropriate control gain K_{p2} , (9) converges and e_{p}→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 p_{m} (blue) and the measured intake manifold pressure p_{mMeasured}(red). Subplot 2 shows the calculated mass flow rate through the throttle ṁ_{a} (blue) and the measured mass flow rate through the throttle ṁ_{aMeasured} (red). Both subplots show that the calculated values are very close estimations of the measured values. Subplot 3 and 4 show the values of γ_{0} and λ_{CD}, which are obtained from equations (7) and (3) with K_{p2}=5, K_{p1}=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 γ_{0} and λ_{CD} 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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