Abstract
Switched reluctance motor (SRM) has got great attention in in-wheel motor electric vehicle (IWM-EV), but SRM vertical force, the vertical component of SRM unbalanced radial force, yields SRM vertical vibration and does harm to dynamic performance of IWM-EV. In order to reduce the SRM vertical vibration, electromagnetic active suspension and a linear quadratic Gaussian (LQG) controller were used to suppress the unbalanced radial force in this paper. All the models and the controller were constructed in Matlab/Simulink R2015b. The controller considers five performance indexes: vehicle body acceleration, SRM airgap eccentricity, SRM stator acceleration, suspension dynamic deflections, and tyre deformation. Analytic Hierarchy Process (AHP) was used to calculate the weighted coefficients of performance indexes. Simulations indicate that this electromagnetic active suspension can reduce SRM vertical vibration obviously and improve dynamic performance of IWM-EV.
1. Introduction
Switched reluctance motor (SRM) has achieved good performance due to its remarkable advantages: high starting torque, wide operating speed range, and high efficiency. These advantages endow SRM with great potential on in-wheel motor electric vehicles (IWM-EV). However, asymmetrical magnetic pull caused by airgap eccentricity yields unbalanced radial force; this unbalanced radial force is recognized as one of the main reasons for the SRM vibration [1–5]. To reduce the vibration, scholars have focused on exploring the structure and control strategies of SRM [6–8].
On SRM structure, integrated design and the optimization of SRM components are main methods to reduce SRM vibration, such as multiobjective optimization design method [9], new stator tooth [10], new rotor tooth [11], and new poles ratio [12]. On SRM control strategies, varieties of methods have been proposed, such as torque distribution controller [13], multilevel systematic design method [14], and new motor torque controller [15]. The above studies do reduce the SRM vibration; however, these studies focus only on SRM itself; the negative influence of SRM on vehicle dynamic performance has not been fully considered. As the main component that flexibly connects the wheel and the vehicle body, suspension transmits the force acting between the wheel and the vehicle body. As a kind of suspension system, electromagnetic active suspension can produce controllable active force, which enables the suspension system to dynamically adjust the supporting force according to the vibration of wheel and vehicle body. In the previous work, we analyzed the influence of the SRM unbalanced radial force on comfort and stability of IWM-EV. The conclusion [16, 17] shows SRM vertical force is highly coupled with road excitation and SRM airgap eccentricity, and this coupling yields SRM vertical vibration that does harm to comfort and stability. This study used electromagnetic active suspension and linear quadratic Gaussian (LQG) controller to reduce the SRM vertical vibration and the negative influence on vehicle dynamic performance.
In order to conduct this study, a full IWM-EV model, SRM vertical force model, and control diagram for the electromagnetic active suspension are built in Matlab/Simulink R2015b. Then the weighted coefficients are calculated and LQG controller is completed. Finally, the frequency responses on stochastic roads and time responses on representative roads are analyzed. Conclusion shows that the electromagnetic active suspension can effectively reduce SRM vibration and the negative influence on vehicle dynamic performance.
2. IWM-EV Modeling
Two models were constructed for studying the effects of electromagnetic active suspension. One is SRM vertical force model that reflects the SRM vertical force exerting on the wheel; another is full IWM-EV model that reflects the influence of electromagnetic active suspension force and unbalanced vertical force on vehicle.
2.1. SRM Vertical Force Model
Because of the geometrically balanced motor structure, the SRM radial force is always considered zero. But the vehicle load and road excitation [17] will yield SRM airgap eccentricity, due to which, the radial force is always not zero. In this study, The primary objective is to reduce the vertical vibration of SRM and the negative influence on vehicle dynamic performance, the well-known 6/4 outside-rotor SRM, like [18, 19], is shown in Figure 1. The SRM parameter values are listed in Table 1. According to [20], the difference magnetic pull between each pair of poles yields the unbalanced radial force, and the radial force of opposite stator poles can be described as follows. The unbalanced radial force is follows.The vertical force that is the vertical component of the unbalanced radial force can be described aswhere is the overlap angle of stator and rotor, is tyre tangential force, is airgap length of SRM, Δg is the airgap eccentricity, and is the angle between stator and wheel longitudinal axis. Each pair of opposite stator with eccentricity will yield unbalanced radial force, for the convenience of presentation, only stators 1 and 4 are picked out to investigate the influence of SRM vertical force on SRM vibration and vehicle dynamic performance. When the is 90°, SRM unbalanced radial force is equal to SRM vertical force.
2.2. Vehicle Model
To simulate the effects of electromagnetic active suspension force on IWM-EV, the required vehicle dynamics model needs to reflect contributions of both electromagnetic active suspension force and SRM vertical force. A vehicle model which has been proved reasonably well in previous work [16, 17] is developed as shown in Figure 2, and the governing equations of the vehicle motions can be described aswhere , the suspension vertical forces at each corner, is as follows.Equations of the vertical motion of the aggregate mass of the tyres, the rims, and the SRM rotors at each corner are as follows. Equations of the vertical motion of the aggregate mass of the SRM stator and housing arewhere x, y, and z are the longitudinal displacement, the lateral displacement, and the vertical displacement, respectively, of the sprung mass of the vehicle; ρ, θ, and ψ are the vertical, pitch, and roll motion of the sprung mass of the vehicle; δ is the steering angle; Fc is the electromagnetic active suspension force; and are the longitudinal tyre force and the lateral tyre force at each wheel; is road displacement excitation at each wheel; is the unbalanced vertical force of each SRM; is the aggregate mass of tyre, hub, and SRM rotor; is the aggregate mass of SRM stator and housing; and is the bearing stiffness. The subscripts i in the equations refer to the front (f) or rear (r), and the subscripts j in the equations refers to the left (l) side or the right (r) side. The vehicle parameter values used for this study are listed in Table 2.
The Pacejka nonlinear tyre model [21] is adopted in this paper to simulate the nonlinearity tyre force. The equations of the Pacejka nonlinear tyre mode are where λ is the slip ratio and , , , , , , , and are empirical parameters. , are the longitudinal force and lateral force.
3. LQG Controller of Electromagnetic Active Suspension
In this paper, several performance indexes of IWM-EV are considered by linear quadratic Gaussian (LQG) controller, such us vehicle body acceleration, airgap eccentricity, SRM stator acceleration, suspension dynamic deflections, and tyre deformation. It is easy to use the methodology of Analytic Hierarchy Process (AHP) to select the weighted coefficients of performance indexes [22]. This LQG controller is based on optimal control theory and the controller can propose different objective functions according to design requirements and improve the vehicle performance by comprehensively considering various performance indexes.
3.1. State Equation
24 state variables and 14 output variables are chosen to constitute the vehicle system for controller, and the state equation can be described aswhere where X is the state variables, Y is the output variables, u is the control vector, and A is the system matrix. B is control matrix. G is disturbance matrix. C is the output matrix. D is transfer matrix.
3.2. Selection of Performance Indexes’ Weighted Coefficients for LQG Controller
AHP is a decision-making method. It decomposes elements that are always related to decision-making into objectives, criteria, and programs and then makes qualitative and quantitative analysis on this basis. It was used to select weighted coefficients of performance indexes for LQG controller [23].
3.2.1. Quantitating Scale Factor
The root mean square (RMS) of dynamic characteristics of the passive suspension vehicle can be produced by simulating the passive suspension vehicle model under the corresponding conditions. According to the literature, [22, 23], scale factors of vehicle dynamic characteristics can be described by the following equations:where are RMS of vehicle body vertical acceleration (BVA), pitch angle (PA), airgap eccentricity (AE), tyre deformation (TD), and suspension dynamic deflections (SDD). are pitch angle scale factor, airgap eccentricity scale factor, tyre deformation scale factor, and suspension dynamic deflections scale factor; vehicle body vertical acceleration (BVA) scale factor is 1.
3.2.2. Subjective Weighted Coefficient
(1) Making Judgment Matrix . (no unit) is the relative importance ratio of the index i and j. Table 3 is a comparison table of the relative importance of each index. According to Table 3, judgment matrix H shown in (23) can be constructed. Calculate multiplying vector of every row.Calculate and Wwhere W is the subjective weighted coefficient.
(2) Maximum Eigenvalue and Consistency Checking of Matrix Hwhere is the maximum eigenvalue of H.
If every element in matrix H satisfies the equations and , the matrix H is the consistency matrix. The method of consistency checking is comparing random consistency ratio CR. RI is the random consistency index. When n is 14, RI is 1. 59. If CR is less than 1, H passes the consistency checking. If CR is more than 1, H needs to be revised [24].
3.2.3. Weighted Coefficients
Subjective weighted coefficients of pitch angle, airgap eccentricity, tyre deformation, and suspension dynamic deflections can be calculated as follows.
3.3. LQG Controller Design
The control diagram for the electromagnetic active suspension is shown in Figure 3. Vehicle body acceleration and pitch angle are chosen to be the performance indexes for improving ride comfort and handling stability. Airgap eccentricity, tyre deformation, and suspension dynamic deflections of each wheel are chosen to be the performance indexes for reducing the vertical vibration of SRM.
The performance function is defined as follows:where q1 is the weighted coefficient of body vertical acceleration; q2 is the weighted coefficient of pitch angle; q3, q4, q5, and q6 are the weighted coefficients of airgap eccentricity; q7, q8, q9, and q10 are the weighted coefficients of tyre deformation; q11, q12, q13, and q14 are the weighted coefficients of suspension dynamic deflections; Q is the weight matrix of the state; u is the control vector of Fc; and R is the weighted coefficient of .
Rewrite (30) as a standard form:where
and the optimal control feedback matrix K can be given by the Riccati equation.
It is easy to calculate feedback matrix K by using Matlab function as follows.
4. Suppression of SRM Vertical Force and Promotion of Vehicle Dynamic Performance
SRM vibration and IWM-EV body movement influence each other, so it is difficult to improve the dynamic performance of the vehicle considering only the SRM structure and controller design. Therefore, this study used electromagnetic active suspension, connecting body and wheels, to reduce the SRM vibration and the negative influence on vehicle body.
4.1. Frequency Response on Stochastic Roads
For this simulation, the road excitations at four tyres are assumed to be the same, only the front left in-wheel SRM response is presented in this paper. In order to study the effects of electromagnetic active suspension under urban condition and high-speed condition, vehicle responses are simulated at 60 km/h vehicle speed on class C road and 100 km/h vehicle speed on class C road.
4.1.1. Response under Urban Condition
Vehicle responses are simulated at 60 km/h vehicle speed on class C road as shown in Figure 4.
Responses of vehicle body acceleration, airgap eccentricity, SRM stator acceleration, suspension dynamic deflections, and tyre deformation at 60 km/h are shown in Figures 5–9
It can be seen that the electromagnetic active suspension can reduce the power spectral density (PSD) values of vehicle body acceleration, airgap eccentricity, SRM stator acceleration, suspension dynamic deflections, and tyre deformation at their resonance frequency. Specifically, Figure 5 shows that the resonance frequency of vehicle body acceleration is 1.221 Hz, and the peak value of vehicle body acceleration decreases from 0.5395 (m/s2)2/Hz to 0.4343 (m/s2)2/Hz and reduced by 19.5%. This indicates that the electromagnetic active suspension can reduce the force acting between the wheel and the vehicle body, and ride comfort improved. Figure 6 shows that the resonance frequency of airgap eccentricity is 98.57 Hz, and the peak value of airgap eccentricity decreases from 1.307e-10 m2/Hz to 8.339e-11 m2/Hz and reduced by 36.2%. Figure 7 shows that the resonance frequency of stator acceleration is 98.57 Hz, and the peak value of stator acceleration decreases from 8.147 (m/s2)2/Hz to 5.520 (m/s2)2/Hz and reduced by 32.2%. Figures 6 and 7 mean that electromagnetic active suspension conspicuously suppressed SRM vertical vibration, and operating stability of SRM improved. Furthermore, SRM’s negative influence on the vehicle reduced. Figure 8 shows that the resonance frequency of suspension dynamic deflections is 1.221 Hz, and the peak value of suspension dynamic deflections decreases from 1.30e-4 m2/Hz to 1.06e-4 m2/Hz and reduced by 18.4%. Figure 9 shows that the resonance frequency of tyre deformation is 6.714 Hz, and the peak value of tyre deformation decreases from 1.154e-5 m2/Hz to 1.140e-5 m2/Hz and reduced by 1.21%. Because driving stability is related to suspension dynamic deflections and tyre deformation, decrease of them implies that electromagnetic active suspension can improve vehicle safety. Therefore, compared with passive suspension, electromagnetic active suspension can availably reduce values of body acceleration, airgap eccentricity, SRM stator acceleration, and so on. Ride comfort and safety under urban condition improved.
4.1.2. Response under High-Speed Condition
Vehicle responses are simulated at 100 km/h vehicle speed on class C road as shown in Figure 10. Vehicle body acceleration, airgap eccentricity, SRM stator acceleration, tyre deformation, and suspension dynamic deflections responses at 100 km/h are shown in Figures 11–15.
Under high-speed condition, the power spectral density (PSD) values of vehicle body acceleration, airgap eccentricity, SRM stator acceleration, suspension dynamic deflections, and tyre deformation can be reduced at their resonance frequency. Figure 11 shows that the peak value of vehicle body acceleration decreases from 0.9104 (m/s2)2/Hz to 0.8158 (m/s2)2/Hz and reduced by 10. 4%. This means that, even under high-speed conditions, the electromagnetic active suspension can still significantly reduce the vibration of the vehicle body, and the ride comfort improved obviously. As can be seen from Figures 12 and 13, the peak value of airgap eccentricity decreases from 1.867e-10 m2/Hz to 1.625e-10 m2/Hz and reduced by 13.0%; the peak value of stator acceleration decreases from 15.95 (m/s2)2/Hz to 13.91 (m/s2)2/Hz and reduced by 12.8%. This indicates that the vibration of SRM under high-speed condition was reduced effectively. Figure 14 shows that the peak value of suspension dynamic deflections decreases from 2.24e-4 m2/Hz to 2.02e-4 m2/Hz and reduced by 9.8%. Figure 15 shows that the peak value of tyre deformation decreases from 1.902e-5 m2/Hz to 1.894e-5 m2/Hz and reduced by 0.42%. Therefore, electromagnetic active suspension can improve ride comfort and reduce SRM vibration under high-speed condition. Because decreases of suspension dynamic deflections and tyre deformation can greatly enhance the driving stability under high-speed condition, the vehicle safety improved.
The effects of electromagnetic active suspension under high-speed condition are weaker than effects under urban condition, because the weighted coefficients of LQG controller are more suitable for urban condition. All the vehicle responses at both 60 km/h and 100 km/h were shown in Table 4.
4.2. Time Response on Representative Roads
4.2.1. Response on Bump Road
In order to simulate the speed bump, bump road model is shown in Figure 16. The simplified model can be described as (36), where L, h are the width and height of the bump and V is the vehicle speed. Setting different vehicle speed or changing the width of the bump can easily adjust road excitation frequency. To simulate the ordinary low frequency speed hump excitation, L=0.8m, h=0.07m, and V is 0.8 m/s. Vehicle body acceleration, pitch angle, airgap eccentricity, suspension dynamic deflections, and tyre deformation responses on class C road are shown in Figures 17–21.
Figure 17 shows the vehicle body acceleration responses. When t=1s, the vehicle body starts to vibrate under the excitation of bump road. When t=2.16s, the vehicle body acceleration reaches peak value, and the peak value decreases by 6.11%. Figure 18 shows the response of pitch angle. When t=1s, the bump road excitation starts to affect the vehicle. When t=3s, the vehicle system tends to be stable. The peak value of pitch angle decreases by 3.16%. Figures 17 and 18 mean that the vertical vibration of the body was reduced and the ride comfort improved, because the vertical motion of the unsprung mass and its impact on vehicle body were reduced. As can be seen from Figures 19 and 20, the trend of airgap eccentricity is similar to vehicle body acceleration, and the peak value of airgap eccentricity decreases by 8.89%. When t=1s, the suspension system starts to shrink. When t=3s, the electromagnetic active suspension system approaches steady state that is faster than passive suspension system at t=3.7s. The peak value of suspension dynamic deflection decreases by 3.64%. Figure 21 shows that the peak value of tyre deformation decreases by 7.44%. This implies that the electromagnetic active suspension can effectively reduce the airgap eccentricity, suspension dynamic deflection, and tyre deformation on bump road. The unsprung mass had less influence on the body and can provide higher stability for the vehicle.
4.2.2. Response on Single Lane Change
In order to study the effects of electromagnetic active suspension on handling stability, vehicle responses are simulated under single lane change condition. The steering angle model is shown in Figure 22. Airgap eccentricity, roll angle, sideslip angle, and yaw rate responses at 100 km/h on class C road are shown in Figures 23–26.
Figure 23 shows that the airgap eccentricity response does not have much relevance to steering angle, but the electromagnetic active suspension efficiently reduces the airgap eccentricity. Figure 24 shows that the vehicle starts to roll at 0.5s. The roll angle reaches peak value at 1.8s and decreases by 3.66%. Figures 25 and 26 show that the trends of sideslip angle and yaw rate are similar to steering angle. Although the contributions of the electromagnetic active suspension to sideslip angle and yaw rate under this condition are not very significant, the electromagnetic active suspension system does have positive influences on the lateral dynamic performance.
5. Conclusions
In order to reduce SRM vertical vibration and improve IWM-EV dynamic performance, this paper used electromagnetic active suspension and LQG controller to suppress the SRM unbalanced radial force. The controller based on optimal control theory can efficiently match electromagnetic active suspension. In theory, compared with the passive suspension, this electromagnetic active suspension can reduce SRM vertical vibration and provide better dynamic performance to the IWM-VE. Based on the analyzed simulation results, the following 3 conclusions can be summarize:
(1) This electromagnetic active suspension could effectively improve the ride comfort. Vehicle body acceleration was reduced on urban condition, high-speed condition, and representative roads. For urban condition, the vehicle body acceleration reduced by 19.5% near the vehicle body’s resonance frequency 1.22 Hz. For high-speed condition, the vehicle body acceleration reduced by 10.4% near the vehicle body’s resonance frequency 1.22 Hz.
(2) This electromagnetic active suspension could effectively reduce the SRM vertical vibration. SRM airgap eccentricity and stator acceleration were both reduced. Specifically, when the frequencies of the SRM airgap eccentricity and stator acceleration are near the stator’s resonance frequency of 107.4 Hz, the SRM vertical vibration shows a most obvious drop.
(3) This electromagnetic active suspension could improve the safety. Under both urban condition and high-speed condition, suspension dynamic deflection and tyre deformation were reduced. On representative roads, electromagnetic active suspension reduced roll angle, sideslip angle, and yaw rate. It indicated that the electromagnetic active suspension can improve lateral dynamic performance and reduce SRM vertical vibration at the same time.
In summary, this paper used electromagnetic active suspension and LQG controller to effectively improve the IWM-EV ride comfort and safety and also reduce the SRM vertical vibration. Due to limited research resource, this paper does not construct an experimental platform to verify the simulation results. But we are preparing an experimental platform and will give introduction in the next work.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (Grant No. 51605390), Foundation of Educational Commission of Sichuan Province (Grant No. 16ZA0161), the Open Research Subject of Key Laboratory of Sichuan Provincial Automotive Engineering (Grant No. szjj2015-048), Foundation of the Science and Technology Research and Development Program of Sichuan Province (Grant No. 2017GZ0103).