J. Cent. South Univ. (2020) 27: 2687-2695

DOI: https://doi.org/10.1007/s11771-020-4491-y

Intelligent prediction on air intake flow of spark ignition engine by a chaos radial basis function neural network

LI Yue-lin(李岳林)^{1, 2}, LIU Bo-fu(刘博夫)¹, WU Gang(吴钢)^{1, 2}, LIU Zhi-qiang(刘志强)^{1, 2},DING Jing-feng(丁景峰)^{1, 2}, ABUBAKAR Shitu³

1. Key Laboratory of Safety and Design and Reliability Technology of Engineering Vehicles in Hunan Province, Changsha 410114, China;

2. College of Automotive and Mechanical Engineering, Changsha University of Science and Technology,Changsha 410114, China;

3. College of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: To ensure the control of the precision of air-fuel ratio (AFR) of port fuel injection (PFI) spark ignition (SI) engines, a chaos radial basis function (RBF) neural network is used to predict the air intake flow of the engine. The data of air intake flow is proved to be multidimensionally nonlinear and chaotic. The RBF neural network is used to train the reconstructed phase space of the data. The chaos algorithm is employed to optimize the weights of output layer connection and the radial basis center of Gaussian function in hidden layer. The simulation results obtained from Matlab/Simulink illustrate that the model has higher accuracy compared to the conventional RBF model. The mean absolute error and the mean relative error of the chaos RBF model can reach 0.0017 and 0.48, respectively.

Key words: intake air flow; spark ignition engine; chaos; RBF neural network

Cite this article as: LI Yue-lin, LIU Bo-fu, WU Gang, LIU Zhi-qiang, DING Jing-feng, ABUBAKAR Shitu. Intelligent prediction on air intake flow of spark ignition engine by a chaos radial basis function neural network [J]. Journal of Central South University, 2020, 27(9): 2687-2695. DOI: https://doi.org/10.1007/s11771-020-4491-y.

1 Introduction

Due to the growing shortage of energy [1-3] and energy crisis in the world, many countries have been paying more attention to adopting measures to decrease energy waste and emissions [4-7] from engines. An spark ignition (SI) engine is considered as a dynamic, multivariable, highly nonlinear and delayed time- varying system. To decrease fuel consumption and control emissions from the engines [8, 9], some advanced combustion [10-13] and after treatment technology [14-16] have been used in direct-injection (DI) engines and these previous pieces of literature concluded that the air-fuel ratio (AFR) has a great effect on engine performance and emissions, and realizing a perfect AFR value according to the measured air intake flow data is a key issue for injecting the corresponding perfect fuel mass [17, 18].

The successful applications of artificial neural network in the combustion and emission control of the engines [19-21] provide a new way to solve these problems. It is well known that traditional ANN model is oversimplified so that the actual characteristics of biological neurons cannot be well represented [22-25]. Considering that chaos has ergodicity and is extremely sensitive to the initial value, chaos neural networks [26-28] will be a more accurate prediction method for air-fuel ratio control of the engines.

In this work, in order to improve the prediction accuracy of the air intake flow of spark-ignition engine, a novel prediction model of intake airflow of spark-ignition engines is established by a chaos radial basis function neural network, and the simulation results obtained from Matlab/Simulink illustrate that the model has higher accuracy compared to the conventional RBF model.

2 Data processing of intake air flow based on chaos theory

2.1 Phase space rebuilding

The phase space rebuilding theory implies that the initial system’s phase space structure is reversely constructed by one-dimension time sequences. The basic idea is that any component development in the system is determined by the interplaying others. These related components’ information is usually hidden in the evolution of each component so that a group of parts time series data is used to extract and restore the original law of the system. This law is a high-dimensional space under a trajectory.

Then the reconstructed phase space can preserve some characteristics of the power system, such as the attractor dimension. In the reconstructed R^m space on the trajectory, it maintains the original power system to maintain differential homeomorphism. Therefore, the phase space reconstruction theory provides the prerequisite for the study time series.

Because the dynamics of the intake manifold of the petrol engines is nonlinear, the intake system is much of a multidimensional nonlinearity. The inducted air is highly multidimensional and nonlinear because its airflow prediction problem can be transformed into phase space in a short evolution process. The required inducted airflow constitutes the intake system’s time sequence data. The initial multidimensional nonlinear chaos system was rebuilt by the phase space reconstruction and prediction was made due to the rebuilt phase space. Chaos identification of inducted airflow time series can discriminate the multidimensional nonlinear chaotic properties of petrol engine intake and passage system.

Assuming the time series of the induction air flow of the gasoline engine is {q(t_i)}:

             (1)

where t₀ is the original time, △t is the sampling time interval.

After the apposite time lag coefficient τ and embedded dimension m are chosen, the initial time series {q(t_i)} can be extended into one phase distribution of m-dimensional phase space:

   (2)

where τ =k·△t, and k is an integer.

The m-dimensional space’s phase point consists of every column, and space point with m components space points n=N-(m-1)τ. The evolutionary trajectory of the m-dimensional phase space was described by the connection between the n phase points. The initial system and reconstructed phase space are topological equivalence.

2.2 Chaos recognition

Methods to determine chaos time sequences include Lyapunov exponent, power spectrum, associated dimension D₂, K entropy and so on.

In this work, the maximum Lyapunov exponent method is used. If the exponent is more than zero, the time series is chaotic. Based on the C—C method, an improved algorithm of maximum Lyapunov exponent is obtained. This method could find the maximum Lyapunov exponent.

The calculation steps are expressed as follows:

Firstly, the mean period of the time sequence is calculated by carrying out a Fourier transform approach on the time sequences of the gasoline engines induction airflow. Hence the time series’ average frequency is equal to the energy weighted average frequency and is reciprocal to the mean cycle. Secondly, the C—C arithmetic is applied to calculating τ, m and the time window τ_ω helps to accomplish the reconstruction. The following equations describe the calculation procedures as follows:

1) The average length of the time series is calculated by

                 (3)

where r_j=j·σ/2 (j=1, 2, 3, 4), σ represents the standard deviation.

                      (4)

                     (5)

It can be found that the results are comprehensive and responsive to the relativity of time series data.

2) The C—C method is applied to seeking for the first minimum of Eq. (4) automatically, which means to find out the independent first partial maximum of the time series. At this point, the time delay C—C method corresponds to the first partial maximum. Meanwhile, the S_cor(t)_min is found to be the first entire maximum time window of the time series τ_ω=t·τ_s (where τ is the sampling interval of the time series).

Therefore, the embedding m-dimension can be computed by:

                            (6)

According to the Taken’s embedding theorem, the value of time lag coefficient τ will not be limited. In practice, time lag coefficient τ should not be too large or too small.

Therefore, τ=2 is chosen. Based on the calculation, it is found that τ_ω is equal to 10, m is equal to 8 and the mean frequency of the time sequence is 0.0616.

Obviously, the reciprocal of the frequency is the mean period of 16.2337. The largest Lyapunov exponent is 0.0327, which is a little more than 0, indicating that the time sequences of the induction airflow are chaos characteristics and can achieve short term forecast.

According to the time lag coefficient τ and embedding m-dimension solved from the last step, the adjacent point Q_J of every point q_j of the phase space is reconstructed by Eq. (7) and the transient separation is limited.

             (7)

where ω=T/△t, △t is the sampling cycle of the sequence.

Next step is to calculate the adjacent point Q_J corresponds to the distance D_j(i) after ith discrete time steps according to every point q_j in the reconstructed phase space |q_j, j=1, 2, …, N|.

 (8)

Then presuming that the adjacent point of the i-th point in the phase space is approximately diverged at the maximum Lyapunov exponent rate which is d_i(j)=C_ie^λi(j)△t, where C_i is the origin detachment distance constant. After taking the logarithm of both sides of the equation, the result is expressed as:

        (9)

Equation (9) stands for a group approximately parallel to the lines whose slope is λ₁. Afterward, the least square method is applied to achieve the fitting of the maximum Lyapunov index λ₁:

                    (10)

where p is the number of nonzero d_i(j).

3 Intelligent prediction of intake air flow

3.1 Intelligent prediction model development

3.1.1 RBF neural network model

RBF neural network could promote the performance of the controller effectively when the system has great uncertainty. In terms of control, the neural network adaptive law can be derived by the Lyapunov method, and the stability and convergence of the whole closed-loop system could be ensured by adjusting the adaptive weight. The results show that the RBF neural network could approximate any nonlinear function under a compact set and arbitrary precision.

As shown in Figure 2, RBF neural network is a three-layer network with one hidden layer. The input layer and the hidden layer have a nonlinear relationship, but there is a linear relationship between the hidden layer and the output layer. The neuron activation function of the hidden layer consists of one radial basis function. Each of the hidden layer node includes a central vector c, c and the input parameter vector x have the same dimension, and the Euclidean distance between them is defined as ||x(n)-c_i||. The output of the hidden layer consists of the nonlinear activation function:

          (11)

Figure 1 Neural network structure

where r is a positive scalar representing the width of the Gaussian basis function; and m is the number of nodes of the hidden layer. The output of the network is implemented by the following weighting function:

  (12)

where ω is the output layer weights and n is the number of outputs.

3.1.2 Chaos RBF neural network model

Logistic map chaos model is used to generate chaotic variables:

                      (13)

The evolution rule of chaotic model of Logistic map is x₀, x₁=F(x₀), x₂=F(x₁)=F(F(x₀))= F²(x₀), …, x_n=F(x_n-1)=Fⁿ(x₀).

Let x∈X, the set {x, f(x), f ²(x), …, f ^k(x), …} is generated by function F(k, x)=f ^k(x).

The Lyapunov exponent of the Logistic map is:

   (14)

When the exponent λ is maximized, μ=4, so λ_max=ln2=0.6931, which is positive, so the mapping formula is completely chaotic, the system inputs x_k∈(0, 1), the output value is ergodic in this interval.

A novel chaos RBF NN model is proposed by integrating chaos majorization algorithm with RBF neural network, as shown in Figure 2. It is utilized for the prediction of the induction flow of gasoline engine. Based on the principle of the phase space reconstruction, the data of the original induction air flow after normalization are reconstructed. The chaos neural network model is used to train and forecast the rebuilt data. Logistic map chaotic model is used to get the chaos variable of ω_i and c_i, so that it can reach the global optimal quickly and speed up the convergence of RBF neural network. The chaotic method is used to train the network to minimize the objective function J.

3.2 Intelligent prediction model application

3.2.1 Experimental data acquisition

In this study, the experiments are designed based on orthogonal method. The test engine is HL495Q four cylinders EFI gasoline engine, whose displacement is 2.84 L, calibration power and corresponding speed is 76.8 kW and 4000 r/min, respectively, the idle speed is 720 r/min. The test selects the CW260 eddy current dynamometer. The hardware platform consists of the host computer and PCL-818 data acquisition card. The intake pressure sensors (CY-YD-200A) and the flow mass sensors (D5400) are installed in the inlet pipe. The entire system structure is shown in Figure 3.

Figure 2 Diagram of chaos RBF prediction model of intake air flow

Figure 3 Diagram of data acquisition

For the purpose of recovering the simulation signal completely, at the same time, according to Shannon’s sample law, the sampling frequency was set to 100 Hz. The engine transition conditions are caused by throttling. The rapid opening and closing are typically engine transient conditions for analysis and control of the effect of verification. Due to the engine limit, throttle opening cannot do random fluctuations test in the whole range of throttling using a phased test. Therefore, the experiment is divided into two stages: acceleration and deceleration. Acceleration condition experimental process steps are as follows: set the experimental test time to 5 s. Make the throttle speed at different levels (within 1 s, within 2 s, within 3 s, within 4 s, within 5 s) from the idle position of 85% opening degree. Meanwhile, measure the throttle angle, intake manifold pressure and airflow signal, and the experimental data of 5×500 groups were recorded. Similarly, deceleration condition experimental process steps are as follows: make the throttle speed at different levels (within 1, 2, 3, 4, 5 s) start from 85% opening to the idle position, then collect 5×500 groups of experimental data.

3.2.2 Intelligent prediction model validation

The 5×500 sets of data collected by the bench test were normalization. Take four-fifth of them (within 1, 2, 3, 4, 5 s) as the training samples of RBF neural network flow forecasting model. Other 1×500 sets of data were used as test samples for the prediction model. The RBF neural network’s input layer has 6 nodes, the hidden layer has 10 nodes, with 1 output layer node, the learning steps are selected as 4000, and the allowed deviation is 0.001. Figure 4 shows the RBF neural network model’s prediction results obtained after training.

Figure 4 Prediction results of RBF neural network output model in acceleration condition

In the light of the aforesaid chaos method, the originally collected data is firstly normalized, and then the phase space is rebuilt. The reconstructed 6×600 groups data were used for the training and verification of the neural network data, among which 5×600 groups data were used as the training sample, and the remaining 1×600 groups data were used for the verification. The RBF NN structure is constructed based on the reconstruction of phase space. The empirical parameters in RBF-NN are determined by the optimization based on chaos theory.

Performance index function is initial as shown in Eq. (15), learning step is set as 2000, and the allowed error is 0.01. The radial basis function is chosen as the transfer function of the hidden layer, and the output is a linear function. The network is created by applying the RBF neural network function in the SIMULINK software. After training, RBF network performance was tested using 500 group data to test normalization. Figure 5 shows chaos neural network model based on the predicted results obtained after training.

,

                     (15)

If the predictive ability between the models cannot be compared between the previous predicted data, then the deviation of the measured values of different network models can be compared.

Figure 5 Prediction results of chaos RBF NN output model in acceleration condition

Figures 6 and 7 show the difference between the measured intake air flow for the mean value model and the actual measured values under acceleration/deceleration conditions. As the traditional average model uses some empirical formulas and estimated parameters, and there are dynamic charge and discharge phenomenon in engine intake manifold system. As a result, mean value model cannot accurately represent the actual flow in the transition conditions.

Figures 8 and 9 revealed the outcome of the calculation RBF neural network and the practical measured values under acceleration/deceleration conditions. RBF neural network can be extremely similar with any nonlinear function that can overcome the air flow sensor response hysteresis. However, RBF neural network is not good at the nonlinear mapping, so its input and output training indices are a little weaker. In general, this model can reflect the trend of the induction air flow in the transient condition.

Figure 6 Prediction results of mean value model in acceleration condition

Figure 7 Prediction results of mean value model in acceleration condition

Figure 8 Prediction results of RBF neural network model in acceleration condition

Figure 9 Prediction results of RBF neural network model in deceleration condition

In acceleration condition, chaos RBF neural network model predicted the value of intake flow and the actual value is shown in Figure 10, in deceleration condition, chaos RBF model predicted value of intake flow and actual value is shown in Figure 11. At run time, the chaotic optimization of RBF neural network has faster convergence speed. The reason for this precise match in the presence of data also includes improved approximation performance of the network.

Figure 10 Prediction results of chaos RBF neural network model in acceleration condition

Figure 11 Prediction results of chaos RBF neural network model in deceleration condition

For the purpose of comparing the prediction property of the three models further, the mean square error, mean absolute error and mean relative error were adopted as the index to judge the performance of the models. The error comparison of acceleration condition and deceleration condition models are listed in Table 1, respectively. Both the acceleration and the deceleration conditions, have shown clear comparison of the predictive performance of the three models.

By analyzing the data in Table 1, the mean absolute error of chaos RBF neural network model is 14.98 times smaller than the average absolute error of the mean value model. Therefore, the chaos RBF neural network model is better than the traditional average model used before. Besides, the mean relative error of the chaos RBF neural network model is an order of magnitude smaller than the average relative error of the RBF NN model. In addition, chaos RBF neural network training and forecast of the whole process is 26 s less than RBF neural network.

4 Conclusions

1) The induction flow time series is proved to be multidimensional nonlinear, chaotic, and short-term predictable.

2) The RBF neural network is combined with the forecast of chaotic time series to forecast the engine induction flow. The errors of chaotic RBF network model are smaller than other models, explaining its predictive performance to have been greatly improved, and has faster convergence rate at run time.

3) The chaos RBF neural network model can overcome the existence of the delay in the intake flow rate of the sensor, and lay the foundation for improving the control precision of AFR in transient condition.

Table 1 Data in acceleration condition

Contributors

LI Yue-lin provided the concept and edited the draft of manuscript. LIU Bo-fu and LIU Zhi-qiang developed the intelligent prediction model. WU Gang and DING Jing-feng performed the application of the model in intake air flow prediction. Shitu ABUBAKAR edited the draft of manuscript.

Conflict of interest

LI Yue-lin, LIU Bo-fu, WU Gang, LIU Zhi-qiang, DING Jing-feng and Shitu ABUBAKAR declare that they have no conflict of interest.

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(Edited by FANG Jing-hua)

中文导读

基于混沌径向基函数神经网络的汽油机进气流量的智能预测

摘要：为了确保进气道喷射燃料汽油机的空燃比的控制精度，本文使用混沌径向基函数神经网络来预测进气流量。由于进气流量时间序列被证明是多维非线性和混沌的，于是选用径向基函数神经网络用于训练初始数据的重构相空间。然后，利用混沌算法定义输出层连接权重和隐层的高斯函数径向基中心，加快网络的收敛速度。通过Matlab/Simulink软件进行仿真，结果表明该模型可以获得比单纯的径向基函数神经网络模型更高的预测精度。对于实验与仿真数据，混沌径向基函数神经网络模型预测值的平均绝对误差达到0.001715，平均相对误差达到0.48205。

关键词：进气流量；汽油机；混沌；径向基函数神经网络

Foundation item: Project(51176014) supported by the National Natural Science Foundation of China, Project(2016JJ2003) supported by Natural Science Foundation Project of Hunan Province, China; Project(KF1605) supported by Key Laboratory of Safety Design and Reliability Technology of Engineering Vehicle in Hunan Province, China

Received date: 2019-04-23; Accepted date: 2019-07-08

Corresponding author: WU Gang, PhD, Associate Professor; Tel: +86-18569077512; E-mail: wug1999@163.com; ORCID: https://orcid. org/0000-0002-4883-6806