In the car control systems, it is hard to measure some key vehicle states directly and accurately when running on the road and the cost of the measurement is high as well. To address these problems, a vehicle state estimation method based on the kernel principal component analysis and the improved Elman neural network is proposed. Combining with nonlinear vehicle model of three degrees of freedom (3 DOF), longitudinal, lateral, and yaw motion, this paper applies the method to the soft sensor of the vehicle states. The simulation results of the double lane change tested by Matlab/SIMULINK cosimulation prove the KPCAIENN algorithm (kernel principal component algorithm and improved Elman neural network) to be quick and precise when tracking the vehicle states within the nonlinear area. This algorithm method can meet the software performance requirements of the vehicle states estimation in precision, tracking speed, noise suppression, and other aspects.
Electronic control systems are increasingly being applied to modern vehicles to improve its safety, stability, and comfort. Electronic stability program (ESP), four wheel steering (4WS), and active steering control (ASC) improve the handling stability of a vehicle by controlling the body lateral and yawing motion. Active roll control (ARC) and active body control (ABC) improve the driving comfort by controlling the vertical movement and roll stability. The premise condition to realize the particular functions of these electronic control systems is to obtain accurate state parameters of vehicle when car is running on the road. Those parameters include vertical speed, lateral acceleration, yaw velocity, sideslip angle, and so on. Due to the measuring technology and its cost, it is quite a challenge to measure all the important state variables directly. At present, the soft sensor technology based on state estimation has been successfully applied to the estimation of vehicle dynamics and kinematics parameter information. Venhovens and Naab [
Principal component analysis (PCA) is a kind of linear algorithm. It only considers the secondorder statistics and can only extract the linear relation of the data. When nonlinear relation exists among a large number of variables of a nonlinear system, the principal component analysis (PCA) will not meet the requirements any more. While the kernel principal component analysis (KPCA) is a nonlinear extension of the principal component analysis (PCA) [
Literature [
Elman neural network is based on BP network. It improves the network’s processing capability of dynamic information by adding a context node. This thesis proposes a new vehicle state estimation method based on the modified Elman neural network. This method uses error analysis to adjust the gain factors in real time, thereby achieving real time updating of the network structure.
The paper comes up with a soft sensor modeling method based on the kernel principal component analysis and the modified Elman neural network by combining Mahalanobis Distance, the kernel principal component analysis (KPCA), and the modified Elman neural network. When used in the soft sensor of the key states such as the vehicle’s yaw rate, this method is proved to have higher forecasting accuracy in simulation tests and satisfied the software performance requirements of the vehicle states estimator.
The 2DOF model used to analyze the vehicle’s yaw motion and lateral motion is a linear model, which is on the assumption that the vehicle is driving in uniform velocity. But in reality, longitudinal velocity is subject to change at any time and this changing longitudinal velocity has a significant impact on the yaw and lateral motion and constitutes a nonlinear relation among the state variables. This paper established the vehicle state estimation soft sensor model based on the longitudinal, lateral, and yaw 3DOF nonlinear vehicle model. Figure
3DOF vehicle dynamic model.
As is shown in Figure
In the equation,
With the steering wheel angle as control input and the steering wheel to the front wheel angle transmission ratio
Soft sensor technology is also named soft instrument technology [
Block diagram of vehicle state estimation system.
The basic idea of the kernel principal component analysis is to implicitly map the data in input space to feature space through kernel function and then to process the kernel principal component analysis in the feature space. This is a powerful approach of extracting nonlinear features, realizing the reduction of the feature parameters, reducing of data redundancy, and getting the primary feature vectors with low classification error rate and high sensitivity with the driving conditions.
There are many feature parameters that can be used to represent the vehicle status, such as average running time, the average acceleration time, the average deceleration time, the average time of driving at uniform velocity, the average idling time, the average running distance, the maximal driving speed, the average speed, the standard deviation of speed, the maximum vehicle acceleration, the maximum vehicle deceleration, the average acceleration during the acceleration period, the average deceleration during the deceleration period, the standard deviation of the acceleration, and the changing rate of the throttle opening. Firstly, this thesis reduces the feature parameters representing vehicle status via correlation analysis. In other words, it uses sequences
In the equation, “
Sort the above results of operation from highest irrelevance to the lowest and then make correlation analysis on the first 15 parameters and the categories of working condition. Then rate the correlation from high to low, and finally we get the 10 feature parameters as the classification standard for working conditions; they are the maximal driving speed, average speed, the minimum deceleration, the average acceleration, the uniform velocity time ratio, idling time ratio, the average acceleration during the acceleration period, the average deceleration during the deceleration period, the average degree of the throttle opening, and the time when the throttle opening degree is zero.
Through nonlinear mapping
The covariance matrix for the after mapping data
If we set the eigenvalue for the covariance matrix
For all the
When
Substitute (
To avoid complex direct calculations, in the feature space, we define the elements
In the equation,
As
Solve (
In the equation,
The projection of the after mapping data on the feature vectors
In the equation,
This thesis chooses
In the equation,
If the mapped data are nonzero mean, mean value method needs to be used on kernel function
In the equation,
Elman regression neural network is a kind of typical dynamic neural network. Based on the feed forward neural network, it adds one more node, the context node. Through the delay and storage of the output of the hidden node by the context node, the output of the hidden node can selflink to the input of the hidden node, thereby enabling the network to be sensitive to the historical data, increasing the capacity of the network’s processing capability of dynamic information and realizing dynamic modeling.
The characteristics of the existing modified Elman neural network is an added selffeedback connection with a fixed gain as
Modified Elman neural network.
Comparing with the standard Elman network, the modified Elman network not only has a more flexible approximation to high order system essentially but can also achieve the same effect as gained by the standard Elman network when it adopts dynamic BP algorithm training.
The expressions between each layer of the modified Elman neural network are
In the equation, connection weight
By analyzing the equation (
when
when
when
The modified Elman neural network overall error objective function can be expressed as
Calculating the partial derivatives of
In the equation,
Combining the Mahalonobis distance, kernel principal component analysis (KPCA), and improved Elman neural network, after Mahalonobis distance getting rid of the fault data among the sampling data, we can select the kernel principal components using KPCA and establish the neural network.
Specific steps are shown in Figure
Path of double lane change test.
To validate the KPCIENN algorithm to be accurate and reliable, the double lane change test is adopted. Using SUV’s real vehicle parameters, virtual simulation of the sharp double lane change test is based on the ISO388811999 standard with sampling period of 0.025 s. During the test, the vehicle passes quickly through the benchmarks set on the road, and the passage is shown in Figure
Path of double lane change test.
To stimulate the external disturbances that affect the vehicle, suppose the vehicle was affected by an instantaneous crosswinds gust shown in Figure
Crosswind spectrum.
The samples were analyzed by KPCA. The accumulative contribution rate has already reached 90.09% as seen in Table
Cumulative contribution ratio of different principles.
Principle number  Eigenvalues  Difference value  Percent variance (%)  Cumulative contribution ratio (%) 


5.123895  2.960055  47.4027  47.4027 

2.16384  0.56343  21.19649  68.59919 

1.60041  1.006635  15.67728  84.27647 

0.59367  0.2184  5.815851  90.09232 

0.375375  0.088725  3.676731  93.76905 

0.286545  0.128625  2.807074  96.57613 

0.15792  0.05124  1.547177  98.1233 

0.106785  0.04179  1.045749  99.16905 

0.06489  0.03822  0.588857  99.75791 

0.02667  0  0.242095  100 
Figures
Comparison of yaw rate between estimated value and simulated value (a) vehicle states note affected by crosswind (b) vehicle states affected by crosswind.
Comparison of side slip angle between estimated value and simulated value (a) vehicle states note affected by crosswind (b) vehicle states affected by crosswind.
Judging from Figures
This thesis proposed a KPCIENN algorithm which is applicable to the states with abrupt changes and established a soft sensor method for the key states needed in the controlling of the lateral, longitudinal, and yaw stability of the vehicle body. In the sharp double lane change test, the estimated values gained by the KPCIENN algorithm were proved to be highly accurate and satisfied the software performance requirements of the vehicle state estimators. The stimulation results of the Matlab/SIMULINK cosimulation showed that the KPCIENN algorithm can estimate the speed, sideslip angle, and yaw velocity with accuracy and strong real time performance as well as an effective tracking speed and an outstanding property of restraining noises. It is suitable for online estimation of vehicle states. This thesis validated the accuracy and effectiveness of the KPCIENN soft sensor algorithm using real vehicle parameters in stimulated environment. The next step is to use the established soft sensor model and the estimation algorithm in field test through hardware in loop simulation technique, which can further verify its practicability and reliability.
The authors declare no conflict of interests.
All authors contributed in the theory and analysis developed in the manuscript and in finalizing the manuscript. All authors have read and approved the final manuscript.
The present work is supported by Shandong Province Crucial R&D Plan Project, China (no. 2015GGX105008), and Shandong Provincial Science and Technology Development Plan Project, China (no. 2014GGX105001).