# Robust NonfragileH∞ Control of Lateral Semiactive Suspension of Rail Vehicles

Authors: Jianjun Meng; Ruxun Xu; Decang Li
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper focuses on the riding comfort of railway vehicle. A nonfragile control strategy is proposed based on robust theory for semiactive suspension system. First, the rail vehicle dynamic model was simplified reasonably, the control model of rail vehicle train lateral semiactive suspension was built, which involved lateral-moving, head-shaking, and side-rolling of the vehicle body and the lateral-moving of the two bogies, and the robust nonfragileH∞ control of head-shaking and side-rolling were designed, respectively. Then the H∞ norm was used to reflect riding comfort; the sufficient conditions for the existence of robust nonfragile H∞ control were developed based on linear matrix inequality (LMI) approach. The design of a robust nonfragile H∞ control with gain perturbation was changed into an optimization issue with linear inequality constraint and a linear objective function. And then the damping effect of the robust nonfragile H∞ controller was evaluated, the wavelet packet analysis theory was used to decompose and reconstruct the acceleration signal, the fragility of the controller was analyzed, and the theory of mathematical statistics was adopted to analyze the influence of robust nonfragile H∞ controller on the probability distribution of vibration acceleration. Finally, the simulation results show that the proposed control strategy can ensure the riding comfort of railway vehicle efficiently and has a comparatively strong nonfragility.


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## Body

## 1. Introduction

Remarkable achievements have been made in China’s high-speed train in recent years. In the course of high-speed operation, due to the unsmooth track, transverse vibration is inevitable. How to effectively suppress the transverse vibration of orbital vehicle is a hot research topic in the field of train control. Compared with the traditional passive suspension system, semiactive suspension system can repress vehicle’s vibration effectively and make the dynamic performance of vehicle better [1, 2]. In order to achieve these control objectives, it is necessary to design an efficient and reliable control algorithm for semiactive suspension system. Hsieh C. Y. raised a regenerative ceiling damping control strategy and got remarkable damping effect with the energy consumption and riding comfort of the semiactive suspension system considered [3]. Guo Jin raised a new semiactive control strategy based on the ceiling damping control through restraining the lateral vibration of the bogie and wheel set to reduce the derailment factor of railway vehicles [4]. Chen Shian and his team adopted LQG arithmetic and gave a systematic research to the optimization design of the suspension system, in order to raise vehicle’s riding comfort and reduce the manufacturing and operating costs [5]. Yang junghua realized the suspension system’s self-adaptive control by designing excitation current properly and proved feasibility of the control strategy by experiment [6]. Guo Konghui took overall consideration of the ceiling damping and floor damping and raised a mixed damping control that can improve trains’ vibration performance effectively [7]. Liu Hongyou made experiment and simulation on continuous adjustable damping semiactive suspension, with the damping effect of the suspension being analyzed in detail [8]. Li Guangjun set up the model of 17-DOF and introduced the variable universe fuzzy control theory and applied it to the design of semiactive suspension [9]. Li Zhongji and his team constructed the model of magnet or heliacal damper and designed a fuzzy controller for semiactive suspension system and proved the effectiveness of the semiactive suspension system by simulation [10].The rail vehicle under the actual running state is a very complex multicoupled system and is influenced by many uncertain factors; it is difficult to establish its exact mathematical model. RobustH∞ control can overcome the defects of the traditional control strategy effectively and keep system’s stability and restrain the vehicle’s vibration effectively when uncertain factors arise in vehicle’s suspension system. However, unignorably, the accuracy of robust controller execution has a great influence on its implementation. In the actual running state, frequent parameter perturbations are found in the controller. Therefore, it is necessary to comprehensively analyze the uncertainty of the controller when designing a semiactive suspension robust controller [11]. So that the stability of the controller can have a strong tolerance to the perturbation of the parameters that is to ensure that the controller is nonfragile [12–15].In this paper, the robust nonfragileH∞ control considering the perturbation of the controller is studied, which can simplify the lateral dynamic model of rail vehicle and establish the control model of lateral semiactive suspension of rail vehicle. The existence condition of the robust nonfragile H∞ controller for the movement of the moving body and the rolling motion of the vehicle is derived by using the linear matrix inequality. Meanwhile, rail vehicle dynamic model is set up by using ADAMS/Rail. Simulations to passive control and the method mentioned in this paper in time domain and frequency domain are conducted, respectively.


## 2. Dynamic Model of Semiactive Lateral Suspension of Railway Vehicles

In practical application, it is of paramount importance to set up a lower level controller to facilitate real-time control. The dynamics model used to design controller needs to adequately reflect the effects of the semiactive suspension on the train, but without regard to the details of the train. The rail vehicle’s lateral dynamic simulative model, which has taken into consideration the complicated interaction force among wheel sets and the characteristics of high DOF, is of great necessity to be simplified. The controller designed in this paper is mainly used to restrain vehicle body’s lateral vibration, focusing on the influence of the central suspension on the lateral vibration of the vehicle body but taking no consideration of the axle box suspension structure, etc. The simplified rail vehicle lateral semiactive suspension control model is shown as Figure1.Figure 1
The control model for lateral semiactive suspension of rail vehicles.Based on Newton's second law, the dynamic equation of transverse vibration of urban rail train body can be obtained from Figure1.Vehicle body lateral movement:(1)MCy¨c+K2yyc+lφc-h1θc-yt1+C2yy˙c+lφ˙c-h2θ˙c-y˙t1+K2yyc+lφc-h1θc-yt2+C2yy˙c-lφ˙c-h2θ˙c-y˙t2=u1+u2Vehicle body shaking movement:(2)Jczϕ¨c+K2ylyc+lϕc-h1θc-yt1+C2yly˙c+lϕ˙c-h2θ˙c-y˙t1-K2ylyc-lϕc-h1θc-yt2-C2yly˙c-lϕ˙c-h2θ˙c-y˙t2=u1l-u2lVehicle body side-rolling movement:(3)Jcxθ¨c-K2yh1yc+lϕc-h1θc-yt1-C2yh2y˙c+lϕ˙c-h2θ˙c-y˙t1-K2yh1yc-lϕc-h1θc-yt2-C2yh2y˙c-lϕ˙c-h2θ˙c-y˙t2+2K2zb22θc+2C2zb32θ˙c=-u1h2-u2h2In the equation,Mc is the quality of vehicle body,Jcz is the rotational inertia of vehicle body shaking movement,Jcx is rotational inertia of vehicle body side-rolling movement,yc is vehicle lateral motion, ϕc is the vehicle shaking angle motion, θc is vehicle sidewinder angle, yt1 and yt2 are lateral displacements of front and rear bogies, respectively,h1 is the vertical distance from vehicle core to the secondary level spring connection point,h2 is the vertical distance from vehicle core to the secondary level spring lateral damper,l is the half of distance,b2 is the half of lateral span of the secondary level vertical spring,b3 is the half of lateral span of the secondary level vertical shock absorber,K2y is the double lateral stiffness coefficient of the secondary level suspension,C2y is the double lateral damping coefficient of the secondary level suspension,K2z is the double vertical stiffness coefficient of the secondary level suspension,C2z is the double vertical damping coefficient of the secondary level suspension, andu1 andu2 are the controlling forces which the lateral actuator installed on the front and rear bogies exerts on vehicle body, respectively.They1 andy2 are defined as the lateral motion before and rear ends of vehicle body and the joint of semiactive suspension, the lateral motion (yc) and shaking angle motion (ϕc), respectively, are(4)yc=y1+y22(5)ϕc=y1-y22lSimplifying (1)~(3) we can get(6)Mcy¨c+2K2yyc-h1θc-K2yyt1+yt2+2C2yy˙c-h2θ˙c-C2yy˙t1+y˙t2=u1+u2(7)Jczϕ¨c+2K2yl2ϕc-K2ylyt1-yt2+2C2yl2ϕ˙c-C2yly˙t1-y˙t2=u1l-u2l(8)Jcxθ¨c-2K2yh1yc+2K2yh12+K2zb22θc+K2yh1yt1+yt2-2C2yh2y˙c+2C2yh22+C2zb32θ˙c+C2yh2y˙t1+y˙t2=-u1h2-u2h2In the above formula, it can be found that the shaking motion of the vehicle and the rolling motion are weakly coupled, so the controllers for these two movements can be designed separately to make the controller more simplified.


## 3. The Design of Robust NonfragileH∞ Controller

### 3.1. Shaking Motion Robust NonfragileH∞ Controller

For the vehicle body shaking movement, definingyYc=(y1-y2)/2, yYt=yt1-yt2/2, ΔyY=yYc-yYt, uY=u1-u2/2. Define xY=y˙Yc,ΔyYTas controller’s state vector. In order to restrain vehicle lateral vibration and adjust the size of control force in the suitable section effectively, selecting evaluating output as zY=y¨Yc,uYT; selecting the accelerated speed (yY=xY) sign that is easier to measure in the actual situation as measuring output, disturbance input is wY=y˙Yt. The state equation of vehicle shaking motion can be established by (5) and (7):(9)x˙Yt=AYxYt+BYuyt+BY1wYtzYt=CYxYt+DYuyt+DY1wYtyYt=xYtIn the equation,(10)AY=-2C2yl2Jcz-2K2yl2Jcz10BY=2C2yl2Jcz-1BY1=l2Jcz0CY=-2C2yl2Jcz-2K2yl2Jcz00DY=2C2yl2Jcz0DY1=l2Jcz1Consider the following fragile state feedback control:(11)uYt=KY+ΔKYyYt=KY+ΔKYxYtIn the equation, KY is the controller gain, ΔKY is the additive controller gain involved dynamic matrix, HY and EY are the constant matrixes with suitable dimension, and FY is the unknown matrix with Lebesgue that is measurable and satisfy additional gain perturbation:(12)ΔKY=HYFYEY,FYTFY≤IMaking AYcl=AY+BYKY+ΔKY,CYcl=CY+DYKY+ΔKY, then relevant closed-loop system is(13)x˙t=AYclxt+BY1wtzt=CYclxt+DY1wtAiming at vehicle shaking motion (9) and positive number given, γ>0 design form as (11) robust nonfragile H∞ controller, it makes closed-loop system (13) asymptotically stable, and the outside disturb an wY and the transmit function Twz of the evaluation output zY satisfy(14)Twzs∞=CYclsI-AYcl-1BY1+DY1∞<γFinally the rail vehicle lateral vibration is restrained effectively, and the safety and stability of train operation is improved.Theorem 1 (see [16]).
To system equation (13), assume γ>0 is a given constant, then the following conditions are equivalent.
( 1 ) System is asymptotically stable, and Tzws∞≤γ;
( 2 ) There is a symmetry enfilade matrix P>0 making(15)AYclTP+PAYclPBY1CYclTBY1TP-γ2IDY1TCYclDY1-I<0Lemma 2 (see [16]).
H, F, E are assumed as matrixes of random suitable dimension,FTF≥I, for random scalar invariant ε>0, and there is HFE+ETFTHT≤ε-1HHT+εETE.Theorem 3.
If there are constantsγ>0, ε>0, symmetry positive definite matrix X and matrix Y make linear matrix equation (16) established, so there is a robust nonfragile H∞ controller (17) which makes closed-loop system (13) internal asymptotically stable:(16)ΨBY1CYX+DYYTBYHYXEYT∗-γIDY1T00∗∗-γIDYHY0∗∗∗-εI0∗∗∗∗-εI<0(17)KY=YX-1In (16), Ψ=XAYT+AYX+BYY+YTBYT, “∗”expressed as symmetrical item of matrix.
Base on the above-mentioned existence conditions of state feedback controller, setting up the convex optimization problem with LMI in equation constraint and linear target function(18)min⁡γs,t.16X>0γ>0,ε>0By the LMI tool of MATLAB we can get the result of (18); then we can get robust nonfragile H∞ controller, and when H, F, E are all zero, (18) can be solved and robust regular H∞ controller can be obtained.


### 3.2. Side-Rolling Movement Robust NonfragileH∞ Controller

To vehicle side-rolling movement, definingyLc=y1+y2/2, yLt=yt1+yt2/2, ΔyL=yLc-yLt,uL=u1+u2/2. Choosing xL=y˙c,ΔyL,θ˙c,θcT as side-rolling movement controller’s state vector; choosing ZL=y¨Lc,uLT as evaluating output; choosing yL=y¨Lc as measuring output; choosing wL=y¨Lt as disturbance output. According to (4), (6), and (8) we can set up state function of vehicle side-rolling movement:(19)x˙Lt=ALxLt+BLuLt+BL1wLtzLt=CLxLt+DLuLt+DL1wLtyLt=xLtAmong them,(20)AL=-2C2yMc-2K2yMc2C2yh2Mc2K2yh1Mc10002C2yh2Jcx2K2yh1Jcx2C2yh22+C2zb32Jcx2K2yh12+K2zb22Jcx0010,BL=2C2yMc-1-2C2yh2Jcx0,BL1=2Mc-1-2h2Jcx0,CL=-2C2yMc-2K2yMc2C2yh2Mc2K2yh1Mc0000,DL=2C2yMc0,DL1=2Mc1Considering the following nonfragile feedback control,(21)uLt=KL+ΔKLyLt=KL+ΔKLxLtIn the equation, KL is controller gain, ΔKL is additional controller perturbation matrix, and the definition of ΔKL is given in (12).MakingALcl=AL+BLKL+ΔKL, CLcl=CL+DLKL+ΔKL, so the relevant closed-loop system is(22)x˙Lt=ALclxLt+BL1wLtzLt=CLclxLt+DL1wLtIf constant γ>0, ε>0 exist in (16), symmetry positive definite matrices X and Y make linear matrix in (23) established, so there is a robust nonfragile H∞ controller (24) making the inside of the closed-loop uncertain system (22) asymptotically stable.(23)ΨPBL1CLX+DLYTPBLHLELT∗-γIDL1T00∗∗-γIDLHL0∗∗∗-εI0∗∗∗∗-εI<0(24)KL=YX-1In (24) Ψ=XALT+ALX+BLY+YTBLT, “∗” represents the symmetry of the matrix.According to the above-mentioned existence conditions of state feedback controller, setting up the following convex optimization problem with LMI in equation constraint and linear target function(25)min⁡γs,t.25X>0γ>0,ε>0We can get the result of (25) by the LMI tool of MATLAB and then get robust nonfragile H∞ controller; meanwhile when H, F, E are all zero, we can solve (25) and get robust regular H∞ controller.


## 3.1. Shaking Motion Robust NonfragileH∞ Controller

For the vehicle body shaking movement, definingyYc=(y1-y2)/2, yYt=yt1-yt2/2, ΔyY=yYc-yYt, uY=u1-u2/2. Define xY=y˙Yc,ΔyYTas controller’s state vector. In order to restrain vehicle lateral vibration and adjust the size of control force in the suitable section effectively, selecting evaluating output as zY=y¨Yc,uYT; selecting the accelerated speed (yY=xY) sign that is easier to measure in the actual situation as measuring output, disturbance input is wY=y˙Yt. The state equation of vehicle shaking motion can be established by (5) and (7):(9)x˙Yt=AYxYt+BYuyt+BY1wYtzYt=CYxYt+DYuyt+DY1wYtyYt=xYtIn the equation,(10)AY=-2C2yl2Jcz-2K2yl2Jcz10BY=2C2yl2Jcz-1BY1=l2Jcz0CY=-2C2yl2Jcz-2K2yl2Jcz00DY=2C2yl2Jcz0DY1=l2Jcz1Consider the following fragile state feedback control:(11)uYt=KY+ΔKYyYt=KY+ΔKYxYtIn the equation, KY is the controller gain, ΔKY is the additive controller gain involved dynamic matrix, HY and EY are the constant matrixes with suitable dimension, and FY is the unknown matrix with Lebesgue that is measurable and satisfy additional gain perturbation:(12)ΔKY=HYFYEY,FYTFY≤IMaking AYcl=AY+BYKY+ΔKY,CYcl=CY+DYKY+ΔKY, then relevant closed-loop system is(13)x˙t=AYclxt+BY1wtzt=CYclxt+DY1wtAiming at vehicle shaking motion (9) and positive number given, γ>0 design form as (11) robust nonfragile H∞ controller, it makes closed-loop system (13) asymptotically stable, and the outside disturb an wY and the transmit function Twz of the evaluation output zY satisfy(14)Twzs∞=CYclsI-AYcl-1BY1+DY1∞<γFinally the rail vehicle lateral vibration is restrained effectively, and the safety and stability of train operation is improved.Theorem 1 (see [16]).
To system equation (13), assume γ>0 is a given constant, then the following conditions are equivalent.
( 1 ) System is asymptotically stable, and Tzws∞≤γ;
( 2 ) There is a symmetry enfilade matrix P>0 making(15)AYclTP+PAYclPBY1CYclTBY1TP-γ2IDY1TCYclDY1-I<0Lemma 2 (see [16]).
H, F, E are assumed as matrixes of random suitable dimension,FTF≥I, for random scalar invariant ε>0, and there is HFE+ETFTHT≤ε-1HHT+εETE.Theorem 3.
If there are constantsγ>0, ε>0, symmetry positive definite matrix X and matrix Y make linear matrix equation (16) established, so there is a robust nonfragile H∞ controller (17) which makes closed-loop system (13) internal asymptotically stable:(16)ΨBY1CYX+DYYTBYHYXEYT∗-γIDY1T00∗∗-γIDYHY0∗∗∗-εI0∗∗∗∗-εI<0(17)KY=YX-1In (16), Ψ=XAYT+AYX+BYY+YTBYT, “∗”expressed as symmetrical item of matrix.
Base on the above-mentioned existence conditions of state feedback controller, setting up the convex optimization problem with LMI in equation constraint and linear target function(18)min⁡γs,t.16X>0γ>0,ε>0By the LMI tool of MATLAB we can get the result of (18); then we can get robust nonfragile H∞ controller, and when H, F, E are all zero, (18) can be solved and robust regular H∞ controller can be obtained.


## 3.2. Side-Rolling Movement Robust NonfragileH∞ Controller

To vehicle side-rolling movement, definingyLc=y1+y2/2, yLt=yt1+yt2/2, ΔyL=yLc-yLt,uL=u1+u2/2. Choosing xL=y˙c,ΔyL,θ˙c,θcT as side-rolling movement controller’s state vector; choosing ZL=y¨Lc,uLT as evaluating output; choosing yL=y¨Lc as measuring output; choosing wL=y¨Lt as disturbance output. According to (4), (6), and (8) we can set up state function of vehicle side-rolling movement:(19)x˙Lt=ALxLt+BLuLt+BL1wLtzLt=CLxLt+DLuLt+DL1wLtyLt=xLtAmong them,(20)AL=-2C2yMc-2K2yMc2C2yh2Mc2K2yh1Mc10002C2yh2Jcx2K2yh1Jcx2C2yh22+C2zb32Jcx2K2yh12+K2zb22Jcx0010,BL=2C2yMc-1-2C2yh2Jcx0,BL1=2Mc-1-2h2Jcx0,CL=-2C2yMc-2K2yMc2C2yh2Mc2K2yh1Mc0000,DL=2C2yMc0,DL1=2Mc1Considering the following nonfragile feedback control,(21)uLt=KL+ΔKLyLt=KL+ΔKLxLtIn the equation, KL is controller gain, ΔKL is additional controller perturbation matrix, and the definition of ΔKL is given in (12).MakingALcl=AL+BLKL+ΔKL, CLcl=CL+DLKL+ΔKL, so the relevant closed-loop system is(22)x˙Lt=ALclxLt+BL1wLtzLt=CLclxLt+DL1wLtIf constant γ>0, ε>0 exist in (16), symmetry positive definite matrices X and Y make linear matrix in (23) established, so there is a robust nonfragile H∞ controller (24) making the inside of the closed-loop uncertain system (22) asymptotically stable.(23)ΨPBL1CLX+DLYTPBLHLELT∗-γIDL1T00∗∗-γIDLHL0∗∗∗-εI0∗∗∗∗-εI<0(24)KL=YX-1In (24) Ψ=XALT+ALX+BLY+YTBLT, “∗” represents the symmetry of the matrix.According to the above-mentioned existence conditions of state feedback controller, setting up the following convex optimization problem with LMI in equation constraint and linear target function(25)min⁡γs,t.25X>0γ>0,ε>0We can get the result of (25) by the LMI tool of MATLAB and then get robust nonfragile H∞ controller; meanwhile when H, F, E are all zero, we can solve (25) and get robust regular H∞ controller.


## 4. Simulation and Analysis

Setting up a semiactive suspension robust nonfragileH∞ controller in the MATLAB/SIMULINK and setting up a rail vehicle model by ADAMS/Rail, guiding the model into MATLAB/Simulink environment to conduct joint simulation via a control module, as is shown in Figure 2, “adams_sub” module is the model of rail vehicle, “controller_RP” module is the side-rolling motion robust nonfragile H∞ controller, “controller_yaw” is the head-shaking motion robust conventional H∞ controller.Figure 2
Robust nonfragileH∞ controller of lateral semiactive suspension system of rail vehicle.Writing German low-interference rail chart file in ADAMS/Rail, the circuit is a straight line rail, and the operating speed of vehicle is 300km/h. According to document [17], when adopting passive suspension, the lateral damping coefficient of vehicle secondary-level suspension is 26000N·s/m.


### 4.1. Time Domain Analysis

Figure3 is the lateral vibration accelerated speed time-domain plot in passive suspension and robust nonfragile H∞ controller situation. Table 1 lists the comparison of Sperling comfort index and the amplitude of the time-domain analysis vehicle lateral damping accelerated speed. We can see from Table 1 that after adopting robust nonfragile H∞ controller, compared with passive suspension, the maximums of accelerated speed of lateral-moving, head-shaking, and side-rolling vibration reduce by 51.22%, 35.30%, and 72.68%, respectively; and the Sperling value of accelerated speed of lateral-moving reduces by 21.48%.Table 1
Time domain analysis of the vehicle body lateral acceleration amplitude and Sperling value comparison.
Item Passive suspension Robust non-fragileH∞ control Improvement rate Lateral moving Maximum acceleration (m/s2) 0.2954 0.1441 51.22% Sperling value 1.7697 1.3896 21.48% Head shaking Maximum angle acceleration (rad/s2) 0.0677 0.0438 35.30% Side rolling Maximum angle acceleration (rad/s2) 0.2855 0.0780 72.68%Figure 3
Acceleration time domain diagram. (a) Lateral-moving vibration. (b) Head-shaking vibration. (c) Side-rolling vibration.


### 4.2. Frequency Domain Analysis

Figure4 shows the lateral vibration acceleration of the vehicle body in the two cases of passive suspension and robust nonfragile H∞ control, respectively. It can be seen from Figure 4 that the body lateral-moving, head-shaking, and side-rolling vibration acceleration which are mainly concentrated in the 0~5Hz, compared with the passive suspension, are significantly improved in the frequency range after the use of robust nonfragile H∞ control.Figure 4
Acceleration frequency domain map. (a) Lateral-moving vibration. (b) Head-shaking vibration. (c) Side-rolling vibration.The comparison of the frequency domain analysis acceleration maximum values is shown in Table2. It can be seen from Table 2 that the maximum nonfragile H∞ control lateral-moving, head-shaking, and side-rolling vibration acceleration are 0.0191m/s2, 0.0054rad/s2, and 0.0106rad/s2, respectively. Compared with the control effect of passive suspension, it can be seen that the maximum accelerations are reduced by 55.99%, 51.35%, and 65.92%, respectively.Table 2
Comparison of the maximum acceleration of vehicle body in frequency domain analysis.
Passive suspension Robust non-fragileH∞ control Improvement rate Lateral moving vibration acceleration (m/s2) 0.0434 0.0191 55.99% Head-shaking acceleration (rad/s2) 0.0111 0.0054 51.35% Side-rolling vibration acceleration (rad/s2) 0.0311 0.0106 65.92%The results of the analysis of the power spectral density (PSD) of the lateral-moving, head-shaking, and side-rolling of the rail vehicle are shown in Figure5. The results of the analysis of the maximum power spectral density are shown in Table 3.Table 3
Acceleration power spectral density maximum contrast.
Passive Robust non-fragileH∞ control Reduction rate Lateral-moving vibration PSD (m2/s3) 0.0044 0.0019 56.82% Head-shaking PSD (rad2/s3) 0.0011 0.0006 45.45% Side-rolling vibration PSD (rad2/s3) 0.0032 0.0011 65.63%Figure 5
Comparison result of power spectrum. (a) Lateral-moving vibration. (b) Head-shaking vibration. (c) Side-rolling vibration.It can be seen from Figure5 that the acceleration power spectral density function of the lateral-moving vibration, side-rolling vibration, and head-shaking vibration of the rail vehicle is mostly distributed in the low frequency range. According to the data of Table 3, it can be seen that for the passive suspension, lateral-moving, head-shaking, and side-rolling vibration acceleration reach the maximum values at 0.7532, 0.6433, and 0.7532Hz, respectively, and the maximum values are 0.0044m2/s3, 0.0011rad2/s, and 0.0032rad2/s3; for robust nonfragile H∞ control, the accelerations of the lateral-moving, head-shaking, and side-rolling vibration after control reach 0.6532, 0.1709, and 0.5311Hz, and the maximum values of acceleration are 0.0019m2/s3, 0.0006rad2/s3, and 0.0011rad2/s3. The maximum values of the acceleration power spectral density of the robust nonfragile H∞ control are 56.82%, 45.45%, and 65.63% lower than that of the passive suspension, respectively.


### 4.3. Controller Fragility Analysis

In order to verify the nonfragility of the semiactive control suspension system, the step responses of the simultaneously existing gain perturbations of the robust nonfragileH∞ controller and the conventional robust H∞ controller are studied by zero-pole analysis. Figures 6–11 are zero-pole distribution maps of transfer function of vehicle body head-shaking and side-rolling control system when gain perturbation arises or does not arise in controller. Tables 4–6 are zero-pore values of transfer function of vehicle body head-shaking and side-rolling control system, when gain perturbation arises or does not arise in controller.Table 4
Vehicle body head-shaking acceleration zero pole value.
Controller gain Conventional robustH∞ Control Robust non-fragileH∞ control Zero point z Pole p Zero point z Pole p Without perturbation -6.7296 -1.8805+4.6662i -3.7942 -13.7740 0.0000 -1.8805-4.6662i 0.0000 -1.0360  With perturbation -7.8835 0.2894 +5.4375i -4.9482 -8.5680 0.0000 0.2894 - 5.4375i 0.0000 -2.1720Table 5
Vehicle body vibration acceleration zero pole value.
Controller gain Conventional robustH∞ Control Robust non-fragileH∞ control Zero point z Pole p Zero point z Pole p Without perturbation -2.1322 + 8.9026i -32.5642 -1.1022 + 8.9834i -40.0488 -2.1322 - 8.9026i -2.1884 + 8.7767i -1.1022 - 8.9834i -0.6795 + 8.9088i -0.6741 -2.1884 -8.7767i -0.9034 -0.6795 - 8.9088i 0.0000 -0.6891 0.0000 -0.7523  With perturbation -6.3286 + 7.7512i -8.6896+22.3172i -5.3252 + 8.1739i -10.2541+18.6985i -6.3286 - 7.7512i -8.6896 -22.3172i -5.3252 - 8.1739i -10.2541 -18.6985i 0.1587 -1.9441 -0.0172 -2.8406 0.0000 0.4633 0.0000 -0.0412Table 6
Vehicle body vibration acceleration zero pole value.
Controller gain Conventional robustH∞ Control Robust non-fragileH∞ control Zero point z Pole p Zero point z Pole p Without perturbation -6.4136 -32.5642 52.6622 -40.0488 0.0004 -2.1884 + 8.7767i -0.0190 -0.6795 + 8.9088i 0.0001 -2.1884 - 8.7767i 0.0000 -0.6795 - 8.9088i -0.0001 -0.6891 0.0000 -0.7523  With perturbation -250.8154 -8.6896+22.3172i -191.76 -10.2541 +18.6985i -0.0164 -8.6896 -22.3172i -0.0200 -10.2541 -18.6985i -0.0000 -1.9441 0.0000 -2.8406 0.0000 0.4633 0.0000 -0.0412Figure 6
Conventional robustH∞ control for vehicle body head-shaking acceleration. (a) Without perturbation. (b) With perturbation.Figure 7
Robust nonfragileH∞ control for vehicle body head-shaking acceleration. (a) Without perturbation. (b) With perturbation.Figure 8
Conventional robustH∞ control for vehicle body lateral acceleration. (a) Without perturbation. (b) With perturbation.Figure 9
RobustH∞ nonfragile control for vehicle body lateral acceleration. (a) Without perturbation. (b) With perturbation.Figure 10
Conventional robustH∞ control for vehicle body side-rolling acceleration. (a) Without perturbation. (b) With perturbation.Figure 11
RobustH∞ nonfragile control for vehicle body side-rolling acceleration. (a) Without perturbation. (b) With perturbation.It can be seen from Figures6 and 7 and Table 4 that as for the vehicle body head-shaking controller when there is no controller gain perturbation the two poles of the robust nonfragile H∞ controller and the conventional robust H∞ controller are distributed in the left planes, and the controller is stable.When gain perturbation arises in controller, the two poles of the robust nonfragileH∞ controller are distributed in the left plane gain perturbation left half plane, while the two poles of the conventional robust H∞ controller are distributed in the right planes and the controller is unstable.From Figures10 and 11 and Tables 4–6, it can be seen that for the rolling motion control of the body roll movement controller, when no gain perturbation arises in controller, the total four poles of nonfragile H∞ controller and the conventional robust H∞ controller are distributed in the left half planes, and the controller is stable; when the controller gain perturbations occur, the four poles of robust and nonfragile H∞ controllers are located in the left planes, and the controller is stable, while when a pole of the conventional H∞ controller locates in the right planes, the controller is unstable.


## 4.1. Time Domain Analysis

Figure3 is the lateral vibration accelerated speed time-domain plot in passive suspension and robust nonfragile H∞ controller situation. Table 1 lists the comparison of Sperling comfort index and the amplitude of the time-domain analysis vehicle lateral damping accelerated speed. We can see from Table 1 that after adopting robust nonfragile H∞ controller, compared with passive suspension, the maximums of accelerated speed of lateral-moving, head-shaking, and side-rolling vibration reduce by 51.22%, 35.30%, and 72.68%, respectively; and the Sperling value of accelerated speed of lateral-moving reduces by 21.48%.Table 1
Time domain analysis of the vehicle body lateral acceleration amplitude and Sperling value comparison.
Item Passive suspension Robust non-fragileH∞ control Improvement rate Lateral moving Maximum acceleration (m/s2) 0.2954 0.1441 51.22% Sperling value 1.7697 1.3896 21.48% Head shaking Maximum angle acceleration (rad/s2) 0.0677 0.0438 35.30% Side rolling Maximum angle acceleration (rad/s2) 0.2855 0.0780 72.68%Figure 3
Acceleration time domain diagram. (a) Lateral-moving vibration. (b) Head-shaking vibration. (c) Side-rolling vibration.


## 4.2. Frequency Domain Analysis

Figure4 shows the lateral vibration acceleration of the vehicle body in the two cases of passive suspension and robust nonfragile H∞ control, respectively. It can be seen from Figure 4 that the body lateral-moving, head-shaking, and side-rolling vibration acceleration which are mainly concentrated in the 0~5Hz, compared with the passive suspension, are significantly improved in the frequency range after the use of robust nonfragile H∞ control.Figure 4
Acceleration frequency domain map. (a) Lateral-moving vibration. (b) Head-shaking vibration. (c) Side-rolling vibration.The comparison of the frequency domain analysis acceleration maximum values is shown in Table2. It can be seen from Table 2 that the maximum nonfragile H∞ control lateral-moving, head-shaking, and side-rolling vibration acceleration are 0.0191m/s2, 0.0054rad/s2, and 0.0106rad/s2, respectively. Compared with the control effect of passive suspension, it can be seen that the maximum accelerations are reduced by 55.99%, 51.35%, and 65.92%, respectively.Table 2
Comparison of the maximum acceleration of vehicle body in frequency domain analysis.
Passive suspension Robust non-fragileH∞ control Improvement rate Lateral moving vibration acceleration (m/s2) 0.0434 0.0191 55.99% Head-shaking acceleration (rad/s2) 0.0111 0.0054 51.35% Side-rolling vibration acceleration (rad/s2) 0.0311 0.0106 65.92%The results of the analysis of the power spectral density (PSD) of the lateral-moving, head-shaking, and side-rolling of the rail vehicle are shown in Figure5. The results of the analysis of the maximum power spectral density are shown in Table 3.Table 3
Acceleration power spectral density maximum contrast.
Passive Robust non-fragileH∞ control Reduction rate Lateral-moving vibration PSD (m2/s3) 0.0044 0.0019 56.82% Head-shaking PSD (rad2/s3) 0.0011 0.0006 45.45% Side-rolling vibration PSD (rad2/s3) 0.0032 0.0011 65.63%Figure 5
Comparison result of power spectrum. (a) Lateral-moving vibration. (b) Head-shaking vibration. (c) Side-rolling vibration.It can be seen from Figure5 that the acceleration power spectral density function of the lateral-moving vibration, side-rolling vibration, and head-shaking vibration of the rail vehicle is mostly distributed in the low frequency range. According to the data of Table 3, it can be seen that for the passive suspension, lateral-moving, head-shaking, and side-rolling vibration acceleration reach the maximum values at 0.7532, 0.6433, and 0.7532Hz, respectively, and the maximum values are 0.0044m2/s3, 0.0011rad2/s, and 0.0032rad2/s3; for robust nonfragile H∞ control, the accelerations of the lateral-moving, head-shaking, and side-rolling vibration after control reach 0.6532, 0.1709, and 0.5311Hz, and the maximum values of acceleration are 0.0019m2/s3, 0.0006rad2/s3, and 0.0011rad2/s3. The maximum values of the acceleration power spectral density of the robust nonfragile H∞ control are 56.82%, 45.45%, and 65.63% lower than that of the passive suspension, respectively.


## 4.3. Controller Fragility Analysis

In order to verify the nonfragility of the semiactive control suspension system, the step responses of the simultaneously existing gain perturbations of the robust nonfragileH∞ controller and the conventional robust H∞ controller are studied by zero-pole analysis. Figures 6–11 are zero-pole distribution maps of transfer function of vehicle body head-shaking and side-rolling control system when gain perturbation arises or does not arise in controller. Tables 4–6 are zero-pore values of transfer function of vehicle body head-shaking and side-rolling control system, when gain perturbation arises or does not arise in controller.Table 4
Vehicle body head-shaking acceleration zero pole value.
Controller gain Conventional robustH∞ Control Robust non-fragileH∞ control Zero point z Pole p Zero point z Pole p Without perturbation -6.7296 -1.8805+4.6662i -3.7942 -13.7740 0.0000 -1.8805-4.6662i 0.0000 -1.0360  With perturbation -7.8835 0.2894 +5.4375i -4.9482 -8.5680 0.0000 0.2894 - 5.4375i 0.0000 -2.1720Table 5
Vehicle body vibration acceleration zero pole value.
Controller gain Conventional robustH∞ Control Robust non-fragileH∞ control Zero point z Pole p Zero point z Pole p Without perturbation -2.1322 + 8.9026i -32.5642 -1.1022 + 8.9834i -40.0488 -2.1322 - 8.9026i -2.1884 + 8.7767i -1.1022 - 8.9834i -0.6795 + 8.9088i -0.6741 -2.1884 -8.7767i -0.9034 -0.6795 - 8.9088i 0.0000 -0.6891 0.0000 -0.7523  With perturbation -6.3286 + 7.7512i -8.6896+22.3172i -5.3252 + 8.1739i -10.2541+18.6985i -6.3286 - 7.7512i -8.6896 -22.3172i -5.3252 - 8.1739i -10.2541 -18.6985i 0.1587 -1.9441 -0.0172 -2.8406 0.0000 0.4633 0.0000 -0.0412Table 6
Vehicle body vibration acceleration zero pole value.
Controller gain Conventional robustH∞ Control Robust non-fragileH∞ control Zero point z Pole p Zero point z Pole p Without perturbation -6.4136 -32.5642 52.6622 -40.0488 0.0004 -2.1884 + 8.7767i -0.0190 -0.6795 + 8.9088i 0.0001 -2.1884 - 8.7767i 0.0000 -0.6795 - 8.9088i -0.0001 -0.6891 0.0000 -0.7523  With perturbation -250.8154 -8.6896+22.3172i -191.76 -10.2541 +18.6985i -0.0164 -8.6896 -22.3172i -0.0200 -10.2541 -18.6985i -0.0000 -1.9441 0.0000 -2.8406 0.0000 0.4633 0.0000 -0.0412Figure 6
Conventional robustH∞ control for vehicle body head-shaking acceleration. (a) Without perturbation. (b) With perturbation.Figure 7
Robust nonfragileH∞ control for vehicle body head-shaking acceleration. (a) Without perturbation. (b) With perturbation.Figure 8
Conventional robustH∞ control for vehicle body lateral acceleration. (a) Without perturbation. (b) With perturbation.Figure 9
RobustH∞ nonfragile control for vehicle body lateral acceleration. (a) Without perturbation. (b) With perturbation.Figure 10
Conventional robustH∞ control for vehicle body side-rolling acceleration. (a) Without perturbation. (b) With perturbation.Figure 11
RobustH∞ nonfragile control for vehicle body side-rolling acceleration. (a) Without perturbation. (b) With perturbation.It can be seen from Figures6 and 7 and Table 4 that as for the vehicle body head-shaking controller when there is no controller gain perturbation the two poles of the robust nonfragile H∞ controller and the conventional robust H∞ controller are distributed in the left planes, and the controller is stable.When gain perturbation arises in controller, the two poles of the robust nonfragileH∞ controller are distributed in the left plane gain perturbation left half plane, while the two poles of the conventional robust H∞ controller are distributed in the right planes and the controller is unstable.From Figures10 and 11 and Tables 4–6, it can be seen that for the rolling motion control of the body roll movement controller, when no gain perturbation arises in controller, the total four poles of nonfragile H∞ controller and the conventional robust H∞ controller are distributed in the left half planes, and the controller is stable; when the controller gain perturbations occur, the four poles of robust and nonfragile H∞ controllers are located in the left planes, and the controller is stable, while when a pole of the conventional H∞ controller locates in the right planes, the controller is unstable.


## 5. Conclusion

In this paper, the lateral semiactive suspension control model of the rail vehicle is established firstly, then the robust nonfragileH∞ controller of the vehicle body head-shaking vibration and the body rolling motion are established by taking full account of the gain perturbation of the controller, and the linear matrix inequality is taken to solve the problems. In this paper, the passive control, conventional robust H∞ control, and the robust nonfragile H∞ control established in this paper are compared and analyzed by using ADAMS and MATLAB joint simulation in German low-interference track spectrum. The results of simulation show that compared with the passive control, the robust nonfragile H∞ controller designed in this paper can effectively restrain the lateral-moving, head-shaking, and side-rolling of the vehicle body and effectively guarantee the riding comfort of the vehicle. When gains arise out of a controller, compared with the conventional H∞ robust controller, the robust nonfragile H∞ controller designed in this paper can still maintain its good performance and a comparatively strong nonvulnerability. The results of this paper are based on the linear matrix inequality (LMI) 2 technology, so nonconvex matrix inequality (nonlinear coupling) conditions will be encountered in further research [18].



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*Source: 1452380-2019-07-24.xml*