# LQR Active Control of Fractional-Order Pantograph-Catenary System Based on Feedback Linearization

Authors: Beining Wang; Shaofang Wen; Yongjun Shen
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Fractional-order calculus has exclusive advantages in modeling the viscoelastic components with obvious fractional-order characteristics such as air springs and metal rubbers in the pantograph structure. In this paper, the air spring is tested, and fractional-order calculus is applied to the modeling of pantograph-catenary system of the high-speed train. The parameter identification method of fractional-order derivative is analytically derived. The traditional lumped mass model is improved and a coupling two-degree-of-freedom model of the fractional-order pantograph-catenary system is established. The fractional-order derivative term in the pantograph-catenary model is approximately calculated by the Oustaloup filter algorithm. Taking the time-varying nature into consideration, the catenary is treated as an extended variable to obtain an augmented model. On this basis, the system is linearized based on differential geometry theory, and an LQR controller is designed to control the pantograph-catenary system. The feedback linearized LQR control and PID control are used to control the same type of traditional pantograph, and the results are compared. Meanwhile, the control effects of feedback linearized LQR control under different pantograph parameters and at different train speeds are analyzed. The results show that the feedback linearized LQR control can present a much better control performance than PID control, and the pantograph-catenary contact force and pantograph head vibration amplitude are both reduced obviously. Even at different train speeds or under different pantograph parameters, it can also effectively reduce these control indexes and provide more robust control performance. These results help to put forward new control ideas and theoretical basis for the vibration control of the pantograph-catenary or similar dynamical system.


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## Body

## 1. Introduction

With the constant improvement of train speeds, the fluctuation of the contact force between the pantograph and catenary increases, which will result in unstable current receiving quality. Therefore, higher requirements are put forward for the current receiving performance of high-speed trains. As the direct power supply equipment for electrified railway, the power supply performance of the pantograph-catenary system directly affects the safety and stability of the railway operation. Due to the nonlinear and nonsmooth characteristics of the pantograph structure, the vibration of the train body, and the special excitation of the catenary, the actual working conditions of the pantograph at the high speed are very complicated. How to accurately describe the dynamic behavior of the pantograph-catenary system of the high-speed train, how to effectively control it, and how to improve the current receiving quality and ensure the safe operation of the high-speed train have become a hot research topic.Since the 1950s, many studies on the dynamics of pantograph, catenary, and pantograph-catenary interaction have been carried out. A series of theoretical models of pantograph-catenary system have been established gradually, of which the pantograph model and catenary model were continuously improved and expanded [1–5]. Lee et al. [1] established a three-dimensional catenary model considering factors such as slackening of the dropper, pre-sag, and stagger, and the reliability of the model was verified. Massat et al. [2] coupled the finite element catenary model with the full flexible multibody pantograph model for co-simulation, which improved the simulation efficiency. Hu et al. [3] studied the transient dynamic response of the whole dropper under the interaction of pantograph and catenary based on the simple chain-shaped suspension catenary. Zhang et al. [4] established the models of train, line, and pantograph-catenary, respectively, and then coupled them into a unified system. The pantograph models mainly include lumped mass model, multirigid body model, rigid-flexible hybrid model, and fully flexible model [6]. However, most of the lumped mass models are integer-order ones, and the influences of nonlinear devices such as air springs in the pantograph structure are not considered enough. Vibration damping devices such as air springs, metal rubbers, and intelligent dampers all have memory characteristics. It is difficult for the traditional integer-order models to accurately describe these material characteristics. In contrast, fractional-order models can better reflect the memory characteristics of materials. Since the emergence of fractional calculus theory, scholars have carried out extensive research on it [7–12]. Lopes et al. [7] showed that fractional-order models could describe the memory characteristics of materials more accurately than integer-order models in electric engineering. Liu et al. [8] established a fractional-order Bingham model for magnetorheological fluid damper, and the experimental results showed that the fitting accuracy of the fractional-order Bingham model was significantly higher than that of the traditional Bingham model. Zhang et al. [9] proved that the fractional-order Maxwell model could effectively describe the whole creep process of asphalt mixture under different stress levels through experiments. Peng et al. [10] applied fractional calculus to the wellbore creep model. By comparing with the classical model, it was found that the fitting accuracy of the fractional-order model was higher for viscoelastic materials. Obviously, in the pantograph-catenary system with viscoelastic devices, the fractional-order model may achieve higher fitting accuracy with fewer parameters.In the research on vibration control of pantograph-catenary system, in addition to improving the dynamic characteristics of pantograph-catenary system by optimizing the performance parameters of pantograph and catenary structure, active control based on modern control theory is also applied to suppress the fluctuation of the contact force, and the current receiving quality of pantograph-catenary system will be improved. Optimal control [13–19], variable structure control [20–23], and fuzzy control [24–28] are commonly used in vibration control of pantograph-catenary system. LQR optimal control takes the minimum system performance index as the evaluation function [19], which can achieve the optimal control effect with small control energy. Guo et al. [13] applied the LQR control to the two-degree-of-freedom linear pantograph-catenary model, obtained the optimal control law by solving the Riccati equation, and compared the control effect of the active control force acting on different positions. Due to the time-varying stiffness of the catenary, there are limitations in directly applying the control strategy of the linear system to the pantograph-catenary system. If the algebraic Riccati equation is used to solve the problem, the influence of the time-varying stiffness of the catenary is not fully considered and then the calculation accuracy decreased. From the perspective of controlling the tracking error, Chen et al. [14] used the LQR control algorithm to design the output tracker to adapt to the time-varying stiffness of catenary and improve the calculation accuracy. For the time-varying stiffness of catenary, the linear pantograph-catenary model is transformed into a nonlinear model [15, 17–19], and the controller based on feedback linearization is designed. Ren et al. [29] designed a feedforward controller for wind turbine based on feedback linearization with sliding mode and fuzzy PID algorithm. Liu et al. [30] applies the feedback linearization method to the state observation of a nonlinear quarter car model with an air spring. Xia [31] designed controllers for two kinds of fractional-order T-S fuzzy systems based on state feedback and achieved good control effects. Accordingly, it is necessary and feasible to establish the fractional-order pantograph-catenary system model with an air spring and apply the feedback linear method to design its controller.In this paper, the viscoelastic characteristic of air springs is studied experimentally. The fractional-order model of the air spring is established by using fractional calculus theory, and then the traditional two-degree-of-freedom pantograph-catenary model is improved. Compared with the traditional integer-order pantograph-catenary model, a more accurate fractional-order pantograph-catenary model is established. The time-varying stiffness of catenary is treated as the extended state variable to obtain the augmented model. The inner and outer state equations of the augmented model are derived by using the theory of differential homeomorphic transformation. On this basis, the LQR controller is designed to actively control the pantograph-catenary system. Through numerical simulation, the effect of the LQR control designed on the vibration control of the pantograph-catenary system is analyzed under different pantograph parameters and at different speeds. At the same time, the PID controller is designed. The control effects and performance indicators of the two active controllers are compared and analyzed when the active control forces act on the frame.


## 2. Modeling of Fractional-Order Pantograph-Catenary System

### 2.1. Fractional-Order Modeling of Air Springs

The primary experimental equipment is the HT-9711 material fatigue testing machine produced by Taiwan Hongda Group Co., Ltd. The test equipment and the ContiTech air spring used are shown in Figure1.Figure 1
Air springs and experimental equipment.With different pressures, amplitudes, and frequencies for experiments, it is found that the viscoelastic characteristics of the air spring under dynamic load are more obvious. In order to correct the difference between the model and the experimental results, the fractional-order theory is introduced [32]. The fractional-order model of the air spring is established as(1)Fat=KDtαxt,where Fa represents the force of the air spring. K is the fractional-order differential coefficient, xt denotes the displacement, and Dtαxt represents the α -order derivative of x with respect to t (0< α <1).The forced excitation loaded in the experiment is a sinusoidal excitation, which is(2)xt=zsinωt+φ,where z is the excitation amplitude, ω is the angular velocity, and φ is the phase.We substitute (2) into (1), and through Laplace transform, it can be obtained(3)LFat=KLDtαxt=KzωcosφSαS2+ω2+sinφSα+1S2+ω2.According to the inverse Laplace transform,equation (3) becomes(4)Fat=KzωcosβL−1∇1+sinβL−1∇2,where ∇1=Sα/S2+ω2 , ∇2=Sα+1/S2+ω2. The singularity of ∇1 is S=±iω . According to the residue theorem, the inverse Laplace transform of ∇1 is(5)L−1∇1=iωαi2ωeiωt+−iωα−i2ωe−iωt.We use the equation(6)iα=eiπ/2α=eiαπ/2.We substitute equation (6) into (5), and simplify it(7)L−1∇1=ωαωsinαπ/2+ωt.According to the same process, it can be obtained(8)L−1∇2=ωαsinαπ/2+π/2+ωt.We substitute (7) and (8) into (4), and simplify it(9)Fat=Kzωαsinαπ/2+ωt+φ.Selecting the experimental parameters as the pressureP=0.1 MPa, the excitation amplitude z=6 mm, and the frequency as 1 Hz, the measured results of the displacement and force are shown in Figure 2, represented by “∗”. The fractional-order α and coefficient K of (1) are obtained by the least square method. The force-displacement fitting curve is also shown in Figure 2, represented by the solid red line. Through the data simulation and comparison, it is found that the data simulated by the fractional-order model agrees well with the experimental data. It shows that the fractional-order model proposed in this paper can well reflect the viscoelastic characteristics of the air spring.Figure 2
Comparison of experimental and fitting curves.


### 2.2. Pantograph-Catenary Model

To study the complex dynamics of pantograph-catenary system conveniently, the pantograph and catenary are simplified into mathematical models separately at first. Then, the pantograph model and catenary model are coupled to obtain the dynamic model of the pantograph-catenary system.The stiffness of the catenary varies continuously within and between spans due to the different contact locations. In the simplified analysis, the catenary is regarded as a variable stiffness elastic system. In ref. [13], Guo et al. used the least squares method to fit the actual stiffness curve calculated by the finite element method and obtained the simplified catenary stiffness formula. It can be described as(10)kt=k01+α1f1+α2f2+α3f12+α4f32+α5f42.Among them,f1=cos2πvt/L,f2=cos2πvt/L1,f3=cosπvt/L,f4=cosπvt/L1 , where v represents the train speed, m/s. L represents the catenary span, m. L1 represents the distance between the adjacent hanging strings of the catenary, m. k0 represents average stiffness, N/m. αii=1,2,3,4,5 represents stiffness difference coefficient.In this study, the simple chain suspension is used [13, 19]. For simple chain suspension catenary, there are k0=3684.5,α1=0.4665,α2=0.0832,α3=0.2603,α4=−0.2801,α5=−0.3364 .In the research of the vibration control of pantograph-catenary system, the pantograph is usually simplified to a lumped mass model composed of linear stiffness and linear damping, as is shown in Figure3. In this paper, a two-degree-of-freedom pantograph model is chosen as the basis. Take into consideration of the viscoelastic properties of vibration damping devices in pantograph-catenary systems, it is difficult to use the integer-order model to describe its performance well with fewer parameters. Therefore, the fractional-order model of the air spring is introduced into the model of the pantograph-catenary system, as is shown in Figure 4.Figure 3
Lumped mass model of pantograph.Figure 4
Two-degree-of-freedom model of pantograph-catenary.The system dynamical equation is(11)m1z¨1+k1z1−z2+c1z˙1−z˙2+ktz1=0,m2z¨2+k1z2−z1+c1z˙2−z˙1+c2z˙2−z˙r+kDtpz2−zr=F,where m1,m2 are the mass of the collector and frames of a pantograph, respectively. k1 represents the stiffness of the collector. k represents the fractional differential term coefficient. c1,c2 represent the damping of the collector and frame, respectively. z1,z2 denote the vertical displacement of each mass. zr is the locomotive body excitation. kt represents catenary stiffness. F is the active control force, which consists of the control force ut and the static lifting force f0.


### 2.3. Oustaloup Filter Approximation Algorithm for Fractional Calculus

At present, there are three main forms of definitions for fractional calculus [12], i.e., Riemann–Liouville definition, Grunwald–Letnikov definition, and Caputo definition. In this paper, the Caputo definition is used as(12)Dtpxt=1Γ1−p∫0tx˙τt−τpdτ,where Γy is the Gamma function and p is the fractional order.For simulation, the Oustaloup filter algorithm [12] is used in the approximate calculation of fractional calculus. In the selected frequency band ωb,ωh , the approximate substitution of fractional operators sp is obtained and the Oustaloup filter is constructed as(13)sp≈ωhp∏i=1Ms+ωi∗s+ωi,where M is the order of the filter, the zero point of the filter is ωi∗=ωbωh/ωb2i−1−p/M, and the pole point is ωi=ωbωh/ωb2i−1−p/M.When the fractional operator iss0.5 , M is selected as 5, 7, and 9, respectively, and the results are shown in Figure 5. In the frequency range (0.001 rad/s, 1000 rad/s), the higher the value of filter order M, the higher the approximate accuracy of Oustaloup filter algorithm. Considering the computation accuracy and complexity, the filter order M is selected as 9.Figure 5
Bode fitting curve of oustaloup filter.


## 2.1. Fractional-Order Modeling of Air Springs

The primary experimental equipment is the HT-9711 material fatigue testing machine produced by Taiwan Hongda Group Co., Ltd. The test equipment and the ContiTech air spring used are shown in Figure1.Figure 1
Air springs and experimental equipment.With different pressures, amplitudes, and frequencies for experiments, it is found that the viscoelastic characteristics of the air spring under dynamic load are more obvious. In order to correct the difference between the model and the experimental results, the fractional-order theory is introduced [32]. The fractional-order model of the air spring is established as(1)Fat=KDtαxt,where Fa represents the force of the air spring. K is the fractional-order differential coefficient, xt denotes the displacement, and Dtαxt represents the α -order derivative of x with respect to t (0< α <1).The forced excitation loaded in the experiment is a sinusoidal excitation, which is(2)xt=zsinωt+φ,where z is the excitation amplitude, ω is the angular velocity, and φ is the phase.We substitute (2) into (1), and through Laplace transform, it can be obtained(3)LFat=KLDtαxt=KzωcosφSαS2+ω2+sinφSα+1S2+ω2.According to the inverse Laplace transform,equation (3) becomes(4)Fat=KzωcosβL−1∇1+sinβL−1∇2,where ∇1=Sα/S2+ω2 , ∇2=Sα+1/S2+ω2. The singularity of ∇1 is S=±iω . According to the residue theorem, the inverse Laplace transform of ∇1 is(5)L−1∇1=iωαi2ωeiωt+−iωα−i2ωe−iωt.We use the equation(6)iα=eiπ/2α=eiαπ/2.We substitute equation (6) into (5), and simplify it(7)L−1∇1=ωαωsinαπ/2+ωt.According to the same process, it can be obtained(8)L−1∇2=ωαsinαπ/2+π/2+ωt.We substitute (7) and (8) into (4), and simplify it(9)Fat=Kzωαsinαπ/2+ωt+φ.Selecting the experimental parameters as the pressureP=0.1 MPa, the excitation amplitude z=6 mm, and the frequency as 1 Hz, the measured results of the displacement and force are shown in Figure 2, represented by “∗”. The fractional-order α and coefficient K of (1) are obtained by the least square method. The force-displacement fitting curve is also shown in Figure 2, represented by the solid red line. Through the data simulation and comparison, it is found that the data simulated by the fractional-order model agrees well with the experimental data. It shows that the fractional-order model proposed in this paper can well reflect the viscoelastic characteristics of the air spring.Figure 2
Comparison of experimental and fitting curves.


## 2.2. Pantograph-Catenary Model

To study the complex dynamics of pantograph-catenary system conveniently, the pantograph and catenary are simplified into mathematical models separately at first. Then, the pantograph model and catenary model are coupled to obtain the dynamic model of the pantograph-catenary system.The stiffness of the catenary varies continuously within and between spans due to the different contact locations. In the simplified analysis, the catenary is regarded as a variable stiffness elastic system. In ref. [13], Guo et al. used the least squares method to fit the actual stiffness curve calculated by the finite element method and obtained the simplified catenary stiffness formula. It can be described as(10)kt=k01+α1f1+α2f2+α3f12+α4f32+α5f42.Among them,f1=cos2πvt/L,f2=cos2πvt/L1,f3=cosπvt/L,f4=cosπvt/L1 , where v represents the train speed, m/s. L represents the catenary span, m. L1 represents the distance between the adjacent hanging strings of the catenary, m. k0 represents average stiffness, N/m. αii=1,2,3,4,5 represents stiffness difference coefficient.In this study, the simple chain suspension is used [13, 19]. For simple chain suspension catenary, there are k0=3684.5,α1=0.4665,α2=0.0832,α3=0.2603,α4=−0.2801,α5=−0.3364 .In the research of the vibration control of pantograph-catenary system, the pantograph is usually simplified to a lumped mass model composed of linear stiffness and linear damping, as is shown in Figure3. In this paper, a two-degree-of-freedom pantograph model is chosen as the basis. Take into consideration of the viscoelastic properties of vibration damping devices in pantograph-catenary systems, it is difficult to use the integer-order model to describe its performance well with fewer parameters. Therefore, the fractional-order model of the air spring is introduced into the model of the pantograph-catenary system, as is shown in Figure 4.Figure 3
Lumped mass model of pantograph.Figure 4
Two-degree-of-freedom model of pantograph-catenary.The system dynamical equation is(11)m1z¨1+k1z1−z2+c1z˙1−z˙2+ktz1=0,m2z¨2+k1z2−z1+c1z˙2−z˙1+c2z˙2−z˙r+kDtpz2−zr=F,where m1,m2 are the mass of the collector and frames of a pantograph, respectively. k1 represents the stiffness of the collector. k represents the fractional differential term coefficient. c1,c2 represent the damping of the collector and frame, respectively. z1,z2 denote the vertical displacement of each mass. zr is the locomotive body excitation. kt represents catenary stiffness. F is the active control force, which consists of the control force ut and the static lifting force f0.


## 2.3. Oustaloup Filter Approximation Algorithm for Fractional Calculus

At present, there are three main forms of definitions for fractional calculus [12], i.e., Riemann–Liouville definition, Grunwald–Letnikov definition, and Caputo definition. In this paper, the Caputo definition is used as(12)Dtpxt=1Γ1−p∫0tx˙τt−τpdτ,where Γy is the Gamma function and p is the fractional order.For simulation, the Oustaloup filter algorithm [12] is used in the approximate calculation of fractional calculus. In the selected frequency band ωb,ωh , the approximate substitution of fractional operators sp is obtained and the Oustaloup filter is constructed as(13)sp≈ωhp∏i=1Ms+ωi∗s+ωi,where M is the order of the filter, the zero point of the filter is ωi∗=ωbωh/ωb2i−1−p/M, and the pole point is ωi=ωbωh/ωb2i−1−p/M.When the fractional operator iss0.5 , M is selected as 5, 7, and 9, respectively, and the results are shown in Figure 5. In the frequency range (0.001 rad/s, 1000 rad/s), the higher the value of filter order M, the higher the approximate accuracy of Oustaloup filter algorithm. Considering the computation accuracy and complexity, the filter order M is selected as 9.Figure 5
Bode fitting curve of oustaloup filter.


## 3. LQR Active Control Based on Feedback Linearization

### 3.1. State Space Representation of Fractional Pantograph-Catenary System

In order to analyze the control strategy, the following state variables are defined as follows:(14)X=x1,x2,x3,x4T=z1,z˙1,z2,z˙2T.The input variable is the active control force:(15)F=ut+f0,where ut is the control force and f0 is the static lifting force.Letw=z˙r , the state space equation and output equation of (11) are(16)X˙=fx+gxF+Dw,Y=hx,,


### 3.2. Augmented Model of Fractional-Order Pantograph-Catenary System

The time-varying stiffnesskt of the catenary is treated as an extended variable to obtain an augmented model. Let k¯t=kt−k0 , and the extended state variables are x5=k¯t, x6=k¯˙t, x˙6=k¯¨t. Then, the expanded state variables are(17)X=x1,x2,x3,x4,x5,x6T=z1,z˙1,z2,z˙2,k¯t,k¯˙tT.The augmented nonlinear affine model is(18)X˙=fx+gxF+Dw,Y=hx,.


### 3.3. Feedback Linearization of Fractional Pantograph-Catenary System

For ann -order nonlinear system with r-order relative order, where r≤n , the system can be transformed into the form of internal and external state equations through differential homeomorphic transformation, so that the controller can be designed [33, 34].According to Lie derivative, whenk=0 , there is(19)LgLf0hx=∂Lf0hx∂XTgx=∂hx∂XTgx=0,when k=1, there is(20)LgLf1hx=Lf1hx∂XTgx=∂∂Lf0hx/∂XTfx∂XTgx=0,when k=2, there is(21)LgLf2hx=∂Lf2h1x∂XTgx=c1x5+k0m1m2≠0.From this, it can be found that the relative order of the system isr=3 , and the dimension of the system is n=6 , There is r≤n , so the system can be linearized by using homeomorphic differentiation [34].A differential homeomorphic transformation is performed on the system, and the transformed state isw, which consists of the external state ξ and the internal state η . It is(22)w=Wx=ξTηTT,where ξT=w1w2w3=hxLfhxLf2hx,ηT=w4w5w6=x4x5x6,w1=hx=k0x1+x5x1,w2=Lfhx=x2k0+x5+x1x6,w3=Lf2hx=2x2x6−k0+x5/m1k1x1−x3+c1x2−x4+k0x1+x5x1+x1x˙6.Take a derivative ofw, and we can get its Jacobi matrix(23)∂w∂xT=k0+x5000x10x6k0+x500x2x1①2x6−k0+x5c1m1k0+x5k1m1k0+x5c1m1①2x2000100000010000001,where ①=x˙6−k0+x5/m1k1+k0+x5,②=−1/m1k1x1−x3+c1x2−x4+2k0x1+2x5x1.It can be seen that this matrix is nonsingular, so it can be proved that the transformationw is a homeomorphic differential transformation that satisfies the requirements. After the transformation, the following state feedback control law is imposed on the system(24)ut=LgLf2hx−1−Lf3hx+v.After differential homeomorphism transformation, the dynamic equation of the system is obtained, which is composed of the internal state equation, the external state equation, and the output equation as(25)η˙=qw+pwu,ξ˙=Acξ+Bcv,y=Ccξ,whereqw=j2xw6w˙6T,pw=1/m200T,Ac=100010001,Bc=001T,Cc=100T.


### 3.4. Design of LQR Controller

In order to improve the working performance of pantograph, reduce the fluctuation of contact force, and improve the receiving quality, the vibration amplitude of pantograph head should be reduced as much as possible. The linear quadratic optimal control strategy is adopted to design the controller in the external state equation. The performance index function of the controller is determined as(26)J=limT⟶∞∫0TXTQX+vTRvdt,where R is the control energy weighting coefficient, and Q is the state weight matrix. Here, R=10−7 and Q=diag100000.The optimal control law that satisfies (26) is(27)v=−KX,K=R−1BTPT,where K=K1K2K3 is the solution of the Riccati equation(28)ATP+PA−PBR−1BTP+Q=0.Solving the Riccati equation and we can getK=100000431090 . Therefore, the optimal control force of LQR is(29)v=105w1+4310w2+90w3.Substitute (29) into (24), and the LQR optimal control force of the fractional-order pantograph-catenary system based on feedback linearization can be obtained(30)ut=LgLf2hx−1105w1+4310w2+90w3−Lf3hx.


## 3.1. State Space Representation of Fractional Pantograph-Catenary System

In order to analyze the control strategy, the following state variables are defined as follows:(14)X=x1,x2,x3,x4T=z1,z˙1,z2,z˙2T.The input variable is the active control force:(15)F=ut+f0,where ut is the control force and f0 is the static lifting force.Letw=z˙r , the state space equation and output equation of (11) are(16)X˙=fx+gxF+Dw,Y=hx,,


## 3.2. Augmented Model of Fractional-Order Pantograph-Catenary System

The time-varying stiffnesskt of the catenary is treated as an extended variable to obtain an augmented model. Let k¯t=kt−k0 , and the extended state variables are x5=k¯t, x6=k¯˙t, x˙6=k¯¨t. Then, the expanded state variables are(17)X=x1,x2,x3,x4,x5,x6T=z1,z˙1,z2,z˙2,k¯t,k¯˙tT.The augmented nonlinear affine model is(18)X˙=fx+gxF+Dw,Y=hx,.


## 3.3. Feedback Linearization of Fractional Pantograph-Catenary System

For ann -order nonlinear system with r-order relative order, where r≤n , the system can be transformed into the form of internal and external state equations through differential homeomorphic transformation, so that the controller can be designed [33, 34].According to Lie derivative, whenk=0 , there is(19)LgLf0hx=∂Lf0hx∂XTgx=∂hx∂XTgx=0,when k=1, there is(20)LgLf1hx=Lf1hx∂XTgx=∂∂Lf0hx/∂XTfx∂XTgx=0,when k=2, there is(21)LgLf2hx=∂Lf2h1x∂XTgx=c1x5+k0m1m2≠0.From this, it can be found that the relative order of the system isr=3 , and the dimension of the system is n=6 , There is r≤n , so the system can be linearized by using homeomorphic differentiation [34].A differential homeomorphic transformation is performed on the system, and the transformed state isw, which consists of the external state ξ and the internal state η . It is(22)w=Wx=ξTηTT,where ξT=w1w2w3=hxLfhxLf2hx,ηT=w4w5w6=x4x5x6,w1=hx=k0x1+x5x1,w2=Lfhx=x2k0+x5+x1x6,w3=Lf2hx=2x2x6−k0+x5/m1k1x1−x3+c1x2−x4+k0x1+x5x1+x1x˙6.Take a derivative ofw, and we can get its Jacobi matrix(23)∂w∂xT=k0+x5000x10x6k0+x500x2x1①2x6−k0+x5c1m1k0+x5k1m1k0+x5c1m1①2x2000100000010000001,where ①=x˙6−k0+x5/m1k1+k0+x5,②=−1/m1k1x1−x3+c1x2−x4+2k0x1+2x5x1.It can be seen that this matrix is nonsingular, so it can be proved that the transformationw is a homeomorphic differential transformation that satisfies the requirements. After the transformation, the following state feedback control law is imposed on the system(24)ut=LgLf2hx−1−Lf3hx+v.After differential homeomorphism transformation, the dynamic equation of the system is obtained, which is composed of the internal state equation, the external state equation, and the output equation as(25)η˙=qw+pwu,ξ˙=Acξ+Bcv,y=Ccξ,whereqw=j2xw6w˙6T,pw=1/m200T,Ac=100010001,Bc=001T,Cc=100T.


## 3.4. Design of LQR Controller

In order to improve the working performance of pantograph, reduce the fluctuation of contact force, and improve the receiving quality, the vibration amplitude of pantograph head should be reduced as much as possible. The linear quadratic optimal control strategy is adopted to design the controller in the external state equation. The performance index function of the controller is determined as(26)J=limT⟶∞∫0TXTQX+vTRvdt,where R is the control energy weighting coefficient, and Q is the state weight matrix. Here, R=10−7 and Q=diag100000.The optimal control law that satisfies (26) is(27)v=−KX,K=R−1BTPT,where K=K1K2K3 is the solution of the Riccati equation(28)ATP+PA−PBR−1BTP+Q=0.Solving the Riccati equation and we can getK=100000431090 . Therefore, the optimal control force of LQR is(29)v=105w1+4310w2+90w3.Substitute (29) into (24), and the LQR optimal control force of the fractional-order pantograph-catenary system based on feedback linearization can be obtained(30)ut=LgLf2hx−1105w1+4310w2+90w3−Lf3hx.


## 4. Simulation Study

The pantograph will be affected by external excitation from the locomotive body in train operation. In this paper, the locomotive body excitation is approximated as a Gaussian white noise signal. The passive control simulation model, PID control simulation model, and feedback linearized LQR active control simulation model are established in Simulink, respectively. The following simulation results are all in a steady state, and the blue line represents passive control, the red line represents the feedback linearized LQR active control, and the green line represents PID control.


### 4.1. Simulation of Different Pantograph Parameters

In order to prove that the LQR active control method in this paper can be applied to different types of pantographs, two types of traditional pantograph parameters [16, 23] are selected, and the specific parameters are shown in Table 1.Table 1
Pantograph parameters [16, 23].
ParametersType IType IIm1/kg812m2/kg1213k1/N·m100004740c1/N·s·m−112070c2/N·s·m−13070k/N·m100100f0/N100100When train speed is 250 km/h, the simulation results of Type I pantograph under passive control and active control are shown in Figure6, and the simulation results of Type II pantograph are shown in Figure 7. The comparison of pantograph-catenary contact force and pantograph head displacement in the steady state are shown in Tables 2 and 3, respectively. It can be seen that feedback linearized LQR active control can achieve good control effects for the two types of pantographs, and the active control force fluctuates are controlled within the range of (70–100) N and (50–90) N, respectively. It can be seen from Table 2 that for Type I pantograph, the feedback linearization LQR control reduces the mean and the standard deviation of the contact force by 14% and 23% respectively, and the mean and the standard deviation of pantograph head displacement are reduced by 17% and 31%, respectively. It can be seen from Table 3 that for Type II pantograph, the feedback linearization LQR control reduces respectively the mean and the standard deviation of the contact force 26% and 49%, respectively, and the mean and the standard deviation of pantograph head displacement are reduced by 28% and 50%, respectively. It can be concluded that the feedback linearized LQR active control can effectively reduce the pantograph-catenary contact force and the vibration amplitude of the pantograph head, and make more stable for different types of pantographs.Figure 6
The results of Type I pantograph. (a) Comparison of contact force. (b) Comparison of pantograph head displacement. (c)Active control force response.
(a)(b)(c)Figure 7
The results of Type II pantograph. (a) Comparison of contact force. (b) Comparison of pantograph head displacement. (c) Active control force response.
(a)(b)(c)Table 2
Performance index of type I pantograph.
(a) Statistical comparison of pantograph-catenary contact forceControl modesMax/NMin/NMean/NSTDPassive control167.1446.2098.4620.73Feedback linearized LQR active control131.6042.2985.1115.97(b) Statistical comparison of pantograph head displacementPassive control0.0640.0130.0360.013Feedback linearized LQR active control0.0510.0110.0300.009Table 3
Simulation performance index of type II pantograph parameters.
(a) Statistical comparison of pantograph-catenary contact force.Control modesMax/NMin/NMean/NSTDPassive control146.6045.3898.2520.72Feedback linearized LQR active control116.2743.2472.8410.67(b) Statistical comparison of pantograph head displacementPassive control0.0600.0140.0360.012Feedback linearized LQR active control0.0400.0130.0260.006


### 4.2. Compared with PID Control Simulation

In order to verify the control effect of feedback linearization LQR active control, PID active control and feedback linearization LQR active control are used for simulation and comparison.We take the performance index of the pantograph-catenary contact force as the control quantity, and a PID controller is designed to control the pantograph-catenary system. The expected value of the pantograph-catenary contact force is set to 100 N, and the parameter optimization objective function of the PID controller is constructed as follows:(31)J=limT⟶∞∫0TF2dt,where F is the actual pantograph-catenary contact force.The input signal of the PID controller is(32)et=FN−F,where FN is the expected value of pantograph-catenary contact force, and F is same as above.The output signal of the PID controller is the active control input force by the nonlinear fractional-order pantograph-catenary system(33)ut=Kpet+KI∫etdt+KDdetdt,where Kp,KI,KD are the proportional, integral, and differential coefficients of the PID active controller, respectively.For Type I pantograph, when train speed is 250 km/h, the response curve of the pantograph-catenary system is continuously analyzed through the trial and error method, and the optimal PID control parameters that meet the control requirements are selected after multiple comparisons. There areKp=4.5 , KI=0.5 , and KD=0.01 . The two control methods are simulated with Type I pantograph. When train speed is 250 km/h, the comparison of the results of the two control methods is shown in Figure 8, and the statistical comparison of the performance indicators is shown in Table 4.Figure 8
Comparison of control effects between the two control methods. (a) Comparison of contact force. (b) Comparison of pantograph head displacement. (c) Comparison of active control force.
(a)(b)(c)Table 4
Statistical comparison between the two control methods (v = 250 km/h).
(a) Statistical comparison of pantograph-catenary contact force.Control modesMax/NMin/NMean/NSTDPassive control167.1446.2198.4620.74Feedback linearized LQR active control131.5942.2985.1015.97PID control137.2463.6992.4014.25(b) Statistical comparison of pantograph head displacement.Passive control0.0640.0130.0350.013Feedback linearized LQR active control0.0510.0120.0300.009PID control0.0510.0160.0330.010(c) Statistical comparison of active control force.Feedback linearized LQR active control99.0266.4886.496.47PID active control202.98-92.9293.0764.27From Figure8 and Table 4, it can be seen that when train speed is 250 km/h, both PID control and feedback linearized LQR control can effectively improve the dynamic performance of the pantograph-catenary system, and the feedback linearized LQR control effect is slightly better than the PID control. Compared with the passive control, the feedback linearized LQR control reduces the mean and the standard deviation of contact force by 14% and 23%, respectively. Compared with the PID control, the active control force fluctuation range of the feedback linearized LQR control is smaller, the mean value is reduced by 8%, and the standard deviation is much smaller than the PID control force. It is shown that, compared with PID control, feedback linearized LQR control can achieve a better control effect with relatively stable and smaller control force, which is beneficial to reduce the energy consumed by the active control of pantograph. At the same time, the smaller fluctuation range can also reduce the adverse effects of time delay.


### 4.3. Simulation Analysis at Different Train Speeds

In reality, train speed is not a constant value, and it is always changing according to real-time acceleration and deceleration. Therefore, analyzing the response of the pantograph at different train speeds is helpful to better understand the control effect of the control method. Type I pantograph is adopted, and the train speed is increased from 200 km/h to 350 km/h and the step length is 50 km/h. Passive control simulation and feedback linearization LQR control simulation are carried out. The simulation results are shown in Figure9. Figures 9(a) and 9(b) show the comparison of the maximum, minimum, and average values of contact force and pantograph head displacement at different speeds, respectively. Figure 9(c) shows the maximum, minimum, and average value of the control force for active control. As can be seen from Figure 9, with the increase of train speed, the statistical values of pantograph-catenary contact force and pantograph head displacement under passive control gradually increase, while feedback linearization LQR active control can effectively reduce the pantograph-catenary contact force and pantograph head displacement at different train speeds. At the same time, the average value of the active control force also increases with the increase of speed, and the fluctuation remains within a reasonable range.Figure 9
Comparison of performance indexes of pantograph-catenary with increasing speed. (a) Contact force statistics. (b) Pantograph head displacement statistics. (c) Active control force statistics.
(a)(b)(c)


## 4.1. Simulation of Different Pantograph Parameters

In order to prove that the LQR active control method in this paper can be applied to different types of pantographs, two types of traditional pantograph parameters [16, 23] are selected, and the specific parameters are shown in Table 1.Table 1
Pantograph parameters [16, 23].
ParametersType IType IIm1/kg812m2/kg1213k1/N·m100004740c1/N·s·m−112070c2/N·s·m−13070k/N·m100100f0/N100100When train speed is 250 km/h, the simulation results of Type I pantograph under passive control and active control are shown in Figure6, and the simulation results of Type II pantograph are shown in Figure 7. The comparison of pantograph-catenary contact force and pantograph head displacement in the steady state are shown in Tables 2 and 3, respectively. It can be seen that feedback linearized LQR active control can achieve good control effects for the two types of pantographs, and the active control force fluctuates are controlled within the range of (70–100) N and (50–90) N, respectively. It can be seen from Table 2 that for Type I pantograph, the feedback linearization LQR control reduces the mean and the standard deviation of the contact force by 14% and 23% respectively, and the mean and the standard deviation of pantograph head displacement are reduced by 17% and 31%, respectively. It can be seen from Table 3 that for Type II pantograph, the feedback linearization LQR control reduces respectively the mean and the standard deviation of the contact force 26% and 49%, respectively, and the mean and the standard deviation of pantograph head displacement are reduced by 28% and 50%, respectively. It can be concluded that the feedback linearized LQR active control can effectively reduce the pantograph-catenary contact force and the vibration amplitude of the pantograph head, and make more stable for different types of pantographs.Figure 6
The results of Type I pantograph. (a) Comparison of contact force. (b) Comparison of pantograph head displacement. (c)Active control force response.
(a)(b)(c)Figure 7
The results of Type II pantograph. (a) Comparison of contact force. (b) Comparison of pantograph head displacement. (c) Active control force response.
(a)(b)(c)Table 2
Performance index of type I pantograph.
(a) Statistical comparison of pantograph-catenary contact forceControl modesMax/NMin/NMean/NSTDPassive control167.1446.2098.4620.73Feedback linearized LQR active control131.6042.2985.1115.97(b) Statistical comparison of pantograph head displacementPassive control0.0640.0130.0360.013Feedback linearized LQR active control0.0510.0110.0300.009Table 3
Simulation performance index of type II pantograph parameters.
(a) Statistical comparison of pantograph-catenary contact force.Control modesMax/NMin/NMean/NSTDPassive control146.6045.3898.2520.72Feedback linearized LQR active control116.2743.2472.8410.67(b) Statistical comparison of pantograph head displacementPassive control0.0600.0140.0360.012Feedback linearized LQR active control0.0400.0130.0260.006


## 4.2. Compared with PID Control Simulation

In order to verify the control effect of feedback linearization LQR active control, PID active control and feedback linearization LQR active control are used for simulation and comparison.We take the performance index of the pantograph-catenary contact force as the control quantity, and a PID controller is designed to control the pantograph-catenary system. The expected value of the pantograph-catenary contact force is set to 100 N, and the parameter optimization objective function of the PID controller is constructed as follows:(31)J=limT⟶∞∫0TF2dt,where F is the actual pantograph-catenary contact force.The input signal of the PID controller is(32)et=FN−F,where FN is the expected value of pantograph-catenary contact force, and F is same as above.The output signal of the PID controller is the active control input force by the nonlinear fractional-order pantograph-catenary system(33)ut=Kpet+KI∫etdt+KDdetdt,where Kp,KI,KD are the proportional, integral, and differential coefficients of the PID active controller, respectively.For Type I pantograph, when train speed is 250 km/h, the response curve of the pantograph-catenary system is continuously analyzed through the trial and error method, and the optimal PID control parameters that meet the control requirements are selected after multiple comparisons. There areKp=4.5 , KI=0.5 , and KD=0.01 . The two control methods are simulated with Type I pantograph. When train speed is 250 km/h, the comparison of the results of the two control methods is shown in Figure 8, and the statistical comparison of the performance indicators is shown in Table 4.Figure 8
Comparison of control effects between the two control methods. (a) Comparison of contact force. (b) Comparison of pantograph head displacement. (c) Comparison of active control force.
(a)(b)(c)Table 4
Statistical comparison between the two control methods (v = 250 km/h).
(a) Statistical comparison of pantograph-catenary contact force.Control modesMax/NMin/NMean/NSTDPassive control167.1446.2198.4620.74Feedback linearized LQR active control131.5942.2985.1015.97PID control137.2463.6992.4014.25(b) Statistical comparison of pantograph head displacement.Passive control0.0640.0130.0350.013Feedback linearized LQR active control0.0510.0120.0300.009PID control0.0510.0160.0330.010(c) Statistical comparison of active control force.Feedback linearized LQR active control99.0266.4886.496.47PID active control202.98-92.9293.0764.27From Figure8 and Table 4, it can be seen that when train speed is 250 km/h, both PID control and feedback linearized LQR control can effectively improve the dynamic performance of the pantograph-catenary system, and the feedback linearized LQR control effect is slightly better than the PID control. Compared with the passive control, the feedback linearized LQR control reduces the mean and the standard deviation of contact force by 14% and 23%, respectively. Compared with the PID control, the active control force fluctuation range of the feedback linearized LQR control is smaller, the mean value is reduced by 8%, and the standard deviation is much smaller than the PID control force. It is shown that, compared with PID control, feedback linearized LQR control can achieve a better control effect with relatively stable and smaller control force, which is beneficial to reduce the energy consumed by the active control of pantograph. At the same time, the smaller fluctuation range can also reduce the adverse effects of time delay.


## 4.3. Simulation Analysis at Different Train Speeds

In reality, train speed is not a constant value, and it is always changing according to real-time acceleration and deceleration. Therefore, analyzing the response of the pantograph at different train speeds is helpful to better understand the control effect of the control method. Type I pantograph is adopted, and the train speed is increased from 200 km/h to 350 km/h and the step length is 50 km/h. Passive control simulation and feedback linearization LQR control simulation are carried out. The simulation results are shown in Figure9. Figures 9(a) and 9(b) show the comparison of the maximum, minimum, and average values of contact force and pantograph head displacement at different speeds, respectively. Figure 9(c) shows the maximum, minimum, and average value of the control force for active control. As can be seen from Figure 9, with the increase of train speed, the statistical values of pantograph-catenary contact force and pantograph head displacement under passive control gradually increase, while feedback linearization LQR active control can effectively reduce the pantograph-catenary contact force and pantograph head displacement at different train speeds. At the same time, the average value of the active control force also increases with the increase of speed, and the fluctuation remains within a reasonable range.Figure 9
Comparison of performance indexes of pantograph-catenary with increasing speed. (a) Contact force statistics. (b) Pantograph head displacement statistics. (c) Active control force statistics.
(a)(b)(c)


## 5. Conclusion

In this paper, considering the influence of fractional-order devices such as air springs between the pantograph and the locomotive body, a two-degree-of-freedom fractional-order pantograph-catenary model is established. Its dynamic equation is given, and the fractional-order differential term is approximated by the Oustaloup filter algorithm. The extended variable method is used to deal with the time-varying stiffness of the catenary, the extended model of pantograph-catenary is linearized by the differential geometry method, and the LQR active controller was designed based on state feedback. Through simulation analysis, the control effect of feedback linearized LQR active control is analyzed and compared with the PID control. Furthermore, the control effect of feedback linearized LQR active control at different train speeds is analyzed. The main conclusions can be drawn as follows: (1) For different types of pantographs, although the feedback linearization LQR active control has different effects, but it can still effectively reduce the vibration amplitude of the pantograph head and the contact force between the pantograph and the catenary. (2) Compared with PID control, feedback linearized LQR active control has a similar effect in decreasing pantograph head vibration amplitude and pantograph-catenary contact force. However, considering the control force size, frequency, and peak value, feedback linearized LQR active control can achieve better control effects with relatively stable control force fluctuations. From the perspective of time delay and energy, it believes that feedback linearized LQR active control is better than PID control. (3) As train speed increases, the time-varying stiffness frequency of the catenary also increases. Feedback linearized LQR active control can play a better control effect at different train speeds.



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*Source: 2213697-2022-10-21.xml*