Using control design for calibrating Mean Value Engine air path models

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.

1. 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.

2. 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_mV_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/(σ₀ṁ_φ), 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 p_out is the exhaust manifold pressure and V_c 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 K_p2 is a positive scalar. Substituting (7) into (6) and (6) into (5), equation (4) becomes

Because the values of RT_mṁ_a/V_m and λ_lp/V_m·(γ₁ω_e+γ₂ω_e²)V_dω_e/(4π(1+1/(λσ₀))) 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.

3. 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 γ₀ 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 γ₀ 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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