In this section the PEM and SEM are employed to identify a simulation model of a real engine system, rather than of a known system where the model structure and true parameter values are available. The purpose is to demonstrate the effectiveness and compatibility of SEM on parameter estimation of a black box model and also exhibit the SEM in an industrial application.
The virtual engine (RT model) is considered as the real system. The objective of the
experiment is to develop a torque model and aλmodel which will be used to design controllers
by offline approaches and therefore simulation models are favoured. A 3×2 MIMO model is
identified by time series data collected from the virtual engine and validated by sets of engine data as well. Assuming each model output is not affected by the other output, the MIMO
model can be divided into two 3×1 MISO models. The inputs are selected to be injection
fuel mass (u1), spark advance (u2) and throttle angle (u3) and the type of inputs is uniformly
CHAPTER 5. SELECTION OF PARAMETER ESTIMATION METHODS 114 selected for system identification in order to maximize the data information and improve the model accuracy. However in the following sections we only discuss the benefit of the proposed SEM estimation method in parameter estimation therefore a set of common UDRN signals is employed. The amplitudes of inputs are constrained as follows:
0mg < u1 < 35mg (5.12)
5◦ < u2 < 30◦
5◦ < u3 < 20◦
As stated in Section 2.3, the sample time should be selected according to the rise time of the output. In this experiment the sample time is 0.03 sec for both MISO models. Since the sample time is small, the data length should be long enough to capture the dynamics of the system. In this chapter the objective is to develop a better estimation method for simulation models so that the methodology on the selection of data length is not discussed here. The data length of input is selected as 2000 which represent the data recorded in 60 sec. The structure of the torque model is taken as follows:
y1(t) = θ1(1) +θ1(2)u1(t−1) +θ1(3)u1(t−2) +θ1(4)u1(t−3) (5.13)
+θ1(5)u2(t−1) +θ1(6)u3(t−1) +θ1(7)u3(t−2) +θ1(8)y1(t−1)
Using the same identification signal, parameters estimated by PEM and SEM are listed in Table 5.3.
Table 5.3: Estimated parameters of torque model by PEM and SEM
θ1(1) θ1(2) θ1(3) θ1(4) θ1(5) θ1(6) θ1(7) θ1(8)
P EM -4.5 4.56 -5.19 0.94 0.067 1.14 -0.73 0.87
SEM -2.57 -6.46 14.76 -8.2 0.046 -0.043 0.36 0.92
Two simulation models are established by using θP EM and θSEM respectively. The
identification signal is applied to these two models and simulated outputs are recorded and compared with the system output in order to evaluate the model accuracy. The output fitness of models is shown in Table 5.4, where the SEM provides a considerable improvement in both
MSE andR2.
Table 5.4: Validation results of torque model
MSE R2
MP EM 113.97 77.78%
MSEM 76.08 83.72%
CHAPTER 5. SELECTION OF PARAMETER ESTIMATION METHODS 115 structure is given by:
y2(t) = θ2(1) +θ2(2)u1(t−1) +θ2(3)u1(t−2) +θ2(4)u2(t−1) (5.14)
+θ2(5)u3(t−3) +θ2(6)y2(t−1) +θ2(7)u2(t−1)y2(t−1) +θ2(8)u1(t−1)u3(t−1)
Estimated parameters and validation results are shown in Table 5.5 and Table 5.6.
Table 5.5: Estimated parameters of λmodel by PEM and SEM
θ1(1) θ1(2) θ1(3) θ1(4) θ1(5) θ1(6) θ1(7) θ1(8)
P EM 0.41 0.6 -0.6 0.0017 0.0046 0.58 -0.0001 -7.29×10−05
SEM 0.49 0.77 -0.77 0.00088 0.0058 0.46 -0.00056 -0.0002
Table 5.6: Validation results of λmodel
MSE R2
MP EM 0.0174 97.68%
MSEM 0.0144 97.7%
Compared to the model of torque, the values ofR2 of theλmodel by PEM and SEM are
very high and similar. However, the values of MSE still indicate the superiority of the SEM. Generally for a model of good quality, the improvement in estimation accuracy derived by the SEM might be limited. Moreover the objective function of the numerical optimization in the SEM can be selected flexibly, not necessarily to be the squared error between simulated output and system output, according to a specific requirement of the model quality. The estimated model thus has a superior performance in that aspect than using the PEM. This numerical minimization is also favoured since an analytical solution of the objective function is not always available.
The MIMO simulation model is then validated by 10 other sets of signals collected from
the virtual engine and the result of averaged MSE andR2 are shown in Table 5.7
Table 5.7: Validation results of MIMO model
MSE R2
MP EM(T orque) 154.97 73.68% MSEM(T orque) 115.90 79.62%
MP EM(λ) 0.0155 97.15%
MSEM(λ) 0.0131 97.39%
Based on the results of the tests discussed in this chapter, the accuracy of models es- timated the SEM is always better than the PEM method therefore the benefit of the SEM method in parameter estimation for simulation models is proved. For the use in industrial applications, the model accuracy should be further improved by other DoE methodologies such as optimal input design and model structure selection.
CHAPTER 5. SELECTION OF PARAMETER ESTIMATION METHODS 116
5.6
Conclusions
Features of prediction models and simulation models and their practical applications are discussed. Appropriate parameter estimation methods for each type of model are introduced accordingly. An example of LS estimation of the prediction model is demonstrated and also used for identification of the simulation model. The proposed SEM minimizes a quadratic scalar function of output error, which is similar to PEM, nevertheless the estimated output is purely determined by the input and simulated output. The SEM is found to give more accurate parameter estimation than traditional PEM if the intended use of the estimated model is for simulation while the PEM has the drawback of neglecting the possible error accumulation.
The SEM is firstly implemented to an identification of a known torque model which is derived from experimental data from the real engine. In the process of identification and statistic validation, the superior performance of this method is fully displayed by both mea- surement criteria. Another application of a black box modelling, the virtual engine identifi- cation is given subsequently in which the SEM leads to a remarkably improved identification and validation result of the MIMO engine model. It indicates that the SEM can be utilized for the estimation of simulation models in practical applications rather than in purely ideal situations.
In a general practice where the selected simulation model structure has both input and output regressors, it is recommended to start with the LS method for the initial values followed by a SEM estimation.
Chapter 6
Static Calibration and Controller
Design
6.1
Introduction
In recent years, the design of control system for modern IC engines is one of the most important steps in the process of engine development. To satisfy the legislative demands of environmental protection and the requirements of manufactures and customers, the major purposes of engine control is to lower the emissions and minimize the fuel consumption with a satisfying engine performance. Because of the nonlinearity of engines and complexity of operating conditions, static look-up table based feedforward controllers are still widely used to realize the control objectives. The whole operating region is represented by a grid of operating points and static calibrations are carried out at each operating point so as to obtain the steady-state settings of related engine calibration parameters. Static maps are thereby formed by the optimal settings of calibration parameters obtained in experimental steady-state testing and utilized to control the engine by the engine management system [3, 2, 102, 103].
The following chapter describes a basic static calibration on the virtual engine for con- strained fuel optimization. Firstly the procedure and targets of the calibration are explained and corresponding settings of the RT model are given. The selection of a reasonable operating region according to the simplified virtual engine is discussed. The process of finding optimal settings at an operating point are illustrated and a static map is obtained by testing over the operating region. The effectiveness of the map is then validated on the virtual engine. Because of its known efficiency, the static map is used as the basis of comparison to the dynamic map developed in the next chapter in order to assess the effectiveness of dynamic model based calibration.
CHAPTER 6. STATIC CALIBRATION AND CONTROLLER DESIGN 118