Zepeng Gao， Zheng Liu， Sizhong Chen， Hongbin Ren， Zechao Li and Yong Chen

（Automobile Research Institute, School of Mechanical Engineering, Beijing Institute of Technology,Beijing 100081, China）

Abstract: A method of extracting and detecting vehicle stability state characteristics based on entropy is proposed. The vehicle’s longitudinal and lateral dynamics models are established for complex driving and maneuver conditions. The corresponding state observer is designed by adopting the moving horizon estimation algorithm, which realizes the observation of the vehicle stability state considering the global state information. Meanwhile, the Shannon entropy is modified to approximate entropy, and the approximate entropy value of the observed vehicle state is calculated. Furthermore, the optimal controller is designed to further validate the reliability of the entropy value as the reference of control system. Simulation results demonstrate that this method can quickly detect the instability state of the system during the process of vehicle driving, which provides a reference for risk prediction and active control.

Key words: vehicle stability state；state observer；moving horizon estimation；Shannon entropy；approximate entropy

An advanced vehicle stability control system relies on accurate vehicle state information[1],but in complex terrain environments and maneuver conditions[2], the state variables that reflect vehicle stability are difficult to gather directly through the sensors[3]. Therefore, the design and application of the corresponding observer becomes the necessary way to obtain vehicle state information in real time[4]. Ma developed a new quasi-stationary method to predict the normal tire force of a heavy truck on a sloped road[5].Fan corrected the accuracy of INS/GPS integrated navigation for the traditional observation model[6]. Hashemi proposed a corner-based estimator, which was robust to model uncertainties[7].Jung employed the interacting multiple model filter method to avoid the chattering responses induced by direct switching between different dynamic models and improve the observation accuracy[8].

However, the above research on vehicle handling stability rarely involves the vertical stochastic excitation induced by road roughness.The high-frequency vibration of stochastic excitation will make the state variable oscillate frequently, and this transient information may lead to the corresponding wrong intervention and control measures.

On the basis of considering extreme maneuver conditions and stochastic road excitation,this paper adopts improved Shannon entropy to extract the characteristics of vehicle state[9]. The entropy value is obtained by processing the sample point sequence, which reflects the state and characteristics of sample points in the subsequence for a period of time, and its probability of instantaneous cataclysm is very small. The time interval of the subsequence is very short for the driver, but it is enough for the control system to take corresponding auxiliary control measures. Therefore, it is of great significance to current state identification and future state prediction of the vehicle system[10]. As the feedback signal, it is more conducive for the control system to determine the future vehicle state and take appropriate intervening control measures.

The innovation of this paper is summarized as follows:

① The influence of road excitation on vehicle handling and stability is considered in the process of state observation.

② An observation algorithm based on moving horizon estimation adopts a period of timedomain information, thus improving the observation accuracy of vehicle state information.

③ A Characteristics extraction method based on entropy calculation can obtain vehicle stability information through the entropy value of the vehicle state, which is conducive to further active intervention control.

1 Vehicle Dynamics Modeling

The longitudinal and lateral dynamic models required for the modeling are shown in Fig.1.

Fig. 1 Vehicle dynamics model

The calculation formula of the vehicle’s longitudinal tire force is

whereTt,ijandTb,ijare the wheel ’s driving torque and braking torque, respectively,μis the road’s adhesion coefficient,Rwis the wheel’s rolling radius,Iω,ijis the moment of inertia,ωi,jis the wheel’s angular speed, andFz,ijis the tire’s dynamic load.

Tire lateral force can be calculated as where the calculation method of the tires’ sideslip angles is whereaandbare distances from the vehicle’s CG to the front and rear axles, respectively, andBis the wheelbase.

The longitudinal and lateral dynamic motion equations of the vehicle are expressed as

wheremis the vehicle’s mass,δis the wheel’s steering angle,Fwis air resistance,θandφare the pitch and roll angles of the vehicle body, respectively, andIxx,Iyy, andIzzare the moments of inertia around theX-axis,Y-axis, andZ-axis, respectively.

According to the vertical dynamic model of the vehicle shown in Fig.2 the dynamic load of the tire can be obtained as

wherewkandvkare the model noise and measurement noise, respectively, which correspond to the matricesPandQ, respectively, in the simulation process.

The system input, state variables, and system output are respectively:

2 State Observation of Vehicle System

In order to effectively utilize the global state information of the vehicle system, a moving horizon estimation (MHE) algorithm is employed to observe the vehicle state. The MHE algorithm includes the constraints in the time domain and the system’s nonlinear characteristics[11]. It needs to make use of the data sequence of lengthNto solve the optimization problem online, and the schematic diagram of the algorithm is as shown in Fig.4.

Fig. 4 MHE algorithm principle

The observer of the dynamic system is designed as

wherews,wi, andviare the disturbance values,and the initial system state is set to=x0.

The constraint conditions corresponding to

3 Vehicle State Characteristics Extraction based on Entropy

Claude Elwood Shannon introduced entropy from thermodynamics into information theory in 1948 to realize the quantification of information in different data sets[12]. This concept of entropy is called Shannon entropy, which is adopted to quantify the total amount of uncertainty in the whole probability distribution, and its calculation equation is

In order to make it more suitable for online calculation and optimization, it is necessary to convert Shannon entropy into approximate entropy for calculation[13]. Assuming that the dimension of the system’s data sequence matrixX={xij} isn, the data’s sample points are divided intomepoints as a subsequence, from which (n-me+1) subsequence segments can be obtained as

The distance between theith subsequenceX(i) and the other (n-q) subsequence segments isd[X(i),X(j)], and the maximum distance between the corresponding data points corresponding to the two subsequences is

After setting the threshold valueγand calculating (n-m) times, if the distance between the two sequences is less thanγ, then the two subsequences are considered to be similar, and the similarity probability between this subsequence and other sequence segments can be calculated as

4 Optimal Control Based on Body Attitude Stability

To extract vehicle state characteristics, the optimal control can be adopted to further validate the effectiveness of the extracted characteristics[14]. As regards to the stability control of the vehicle body ’s attitude, the control problem based on entropy value can be converted into a quadratic optimal solution problem. The evaluation index of the controller needs to consider the influence of the dynamic deflection and relative velocity of the suspension, the roll angle, and the roll angular velocity of the vehicle body. The design of the optimal controller can be found in Ref.[14]. Therefore, the objective function is obtained as

whereQ0andrare the output and control vector weighting coefficient matrices respectively.The optimal control forceFc,ijof the active suspension can be obtained by solving the Riccati equation. When the optimal control forceFc,ijis applied to the suspension system, the attitude of the vehicle body can be controlled in complex road environments.

5 Analysis of Simulation Results

The simulation is realized by the joint simulation platform of CarSim-Matlab/Simulink, in which the vehicle model is an E-class SUV, and the necessary model parameters are shown in Tab.1. The modeling method of stochastic road excitation can be found in Ref.[15]. The maneuver conditions used in the simulation process are as follows: the driver makes an emergency turn on a B-class road bank at a speed of 30 km/h within 0–4 s, then the same maneuver is performed on a D-class road bank at a speed of 30 km/h within 5–8 s. The road adhesion coefficient is set toμ=0.85 and the simulation step size is set to 0.001 s. It is worth noting that in order to simplify the research process, the combine-slip influence is not considered in the simulation process. The steering wheel angle changes as shown in Fig.5. The driver turns the steering wheel to 300° in the forward direction within 1.4–1.6 s, turns it to 300° in the reverse direction within 1.8–2.2 s after holding for 0.2 s, and then returns to the horizontal direction within 2.4–2.6 s after holding for 0.2 s. The handling behavior of driver within 4–8 s is similar to that within 0–4 s.

Tab. 1 Dynamic model parameters

Fig. 5 Steering wheel angle of driver

5.1 Observation results of MHE algorithm

The system measurement output from CarSim and the vehicle state observed by MHE algorithm are shown in following Figs.6–9.

Fig. 6 Observation results of roll angle

Fig. 7 Observation results of roll rate

According to Fig.6, there is still an error between the MHE observation results and the actual vehicle body’s roll angle, but the maximum error is not more than 0.4° and its corresponding error rate is less than 6%, whose observation accuracy is within the tolerance range. It can be seen from Fig.7 that the accuracy of the observation results of the vehicle’s roll angle rate is also greater than 90% of the actual output, which is acceptable.

Figs.8-9 show that the oscillation frequencies of the vehicle’s lateral acceleration and yaw rate are low between 0–4 s due to the small road excitation. On the contrary, the oscillation amplitude and frequency of these two state variables are violent with the increasing of stochastic excitation between 5 s–8 s. Meanwhile, the observation results show that the MHE algorithm can realize the real-time tracking of the actual system output with high observation in order to consider the global information.

Fig. 8 Observation results of lateral acceleration

Fig. 9 Observation results of yaw rate

However, the characteristic changes of the system’s state variables obtained by the observer are not obvious. In the actual control system, the wrong control instructions may be generated due to changes in high-frequency and transient state information. In order to make the control objective more specific and the time interval clearer,the concept of entropy is employed to identify the characteristics of the state variables, so that they can be more suitable for the actual control system. During the calculation process of entropy, the length of each subsequence is set tome=200; thus, the entropy of observed vehicle states reflecting the system stability can be obtained as shown in Fig.10.

Fig. 10 Entropy of observation results

System stability tends to decrease with the increase of entropy. Fig.10 clearly exhibits the entropy value of different state variables in different intervals. In the interval of 1 400–2 400 iterations, the entropy values of the vehicle body’s roll angle and yaw rate are greater than other state variables. Therefore, it is necessary for the control system to intervene and adjust it to ensure the handling stability of the vehicle. It should be noted that the roll angle rate of the vehicle body increases correspondingly within 2 000–2 200 iterations, so it is necessary to control its size through the suspension system, so as to ensure good attitude stability of the vehicle body.

5.2 Optimization results of the optimal controller

The optimal controller is mainly designed to optimize the vehicle body ’s attitude stability through the active force output of active suspension. The state responses of the vehicle body’s roll angle and its rate are displayed in Figs.11–12.

Fig. 11 Comparison of roll angle response

Fig. 12 Comparison of roll rate response

Fig. 13 Entropy of state variable of controlled system

It can be seen from Figs.11–12 that under the optimization of the optimal control force, the amplitude of the roll angle and its rate are obviously reduced, and the body’s attitude has also been effectively optimized during the process of emergency steering. Fig.13 intuitively indicates the entropy change of the vehicle state in this process. The attitude of the vehicle has been improved to some extent, so the amplitude responses of the corresponding roll angle and its rate have been reduced compared to those in Fig.9.However, due to the lack of corresponding control measures for yaw rate, the corresponding entropy value still fluctuates greatly, which also shows that the entropy can represent the characteristics of vehicle state and be applied to the control system.

6 Conclusions

Aiming to create vehicle stability in a complex environment, a dynamic model considering the longitudinal, lateral and roll motion of the vehicle is established, and a vehicle state observer is designed by using the moving horizon optimization algorithm, thus realizing the real-time observation of the corresponding variables. In order to make the observation results more suitable for online optimization and control of the active control system, different vehicle state entropy values are calculated after Shannon entropy is modified to approximate entropy. Subsequently, the optimal controller is designed.

The observation results suggest that the MHE algorithm can satisfy the demand of high observation accuracy when considering the global state information, which is conducive to further active vehicle intervention and control.Meanwhile, the entropy calculation results explicitly reflect the entropy change of the system state in each iteration interval, which provides an effective reference for the stability control system by defining the control objectives in the corresponding interval. The optimization results further demonstrate that the control measures based on entropy can improve and optimize vehicle performance accurately and effectively.