hybridactiveandsemi-activecontrolforpantograph-catenary system of...

Hybrid active and semi-active control for pantograph-catenarysystem of high-speed train

I.U. Khan 1, D. Wagg 1, N.D. Sims 1

1 University of Sheffield, Department of Mechanical Engineering,S1 3JD, Sheffield, United Kingdome-mail: [email protected]

AbstractIn this paper a new hybrid control methodology using active actuator and semi-active device is proposedto minimize the oscillations between the pantograph and catenary by keeping the contact force betweenthem constant. One of the advantages of using the proposed hybrid controller is that a semi-active device caneasily be mounted on the pantograph upper arm without compromising the weight and the size. However, theperformance of a semi-active device is restricted because of the passivity constraint. To assist the semi-activedevice and to achieve the desired performance an active actuator is placed at the base of the pantograph. Theimmersion and invariance (I & I) methodology is used to design the controller for the active actuator, andsliding mode control (SMC) is used to design the controller for the semi-active device. Simulations showpromising results.

1 Introduction

The use of high speed trains not only improves the efficiency of transportation but also has a very positiveimpact on the environment in terms of controlling air pollution. High speed trains can achieve a very highspeed i.e around 350 km/h. Hence it is very important to have a permanent contact between pantograph andcatenary. The electric current from catenary to train transformer flows through the pantograph. In the idealscenario, the contact between pantograph and catenary should be permanent, but due to the flexibility in thestructure of catenary and pantograph, as the train speed increases the oscillations keep on increasing and thecontact is not guaranteed, which results in electric arcs and eventually deteriorating the current collectionfrom catenary. One solution to avoid loss of contact is to increase the contact force but this will result inwear and tear due to excessive contact force. It is very important to keep the contact force constant withoutcausing any damage to the pantograph-catenary system. Another solution is to increase the tension in thecontact wire which means increasing the equivalent stiffness. This solution is very expensive. The catenarypresents a time varying stiffness, which is depended on the train speed. To solve this problem differentcontrol strategies have been proposed using active actuators.

There are three types of pantograph-catenary models used in the literature for control purpose. In [1–4] thepantograph is modeled as a 2-DOF mass spring damper system and the catenary is designed as a springwith time varying stiffness. In [5–9] the pantograph is modeled as a 2-DOF mass spring damper systemand catenary is modeled as a spring with fixed stiffness. In addition to that a spring with fixed stiffnessis added to represent the pantograph shoe. In [10–12] the pantograph is modeled as a 3-DOF mass springdamper system and the catenary is designed a SDOF mass spring damper system with time varying mass,time varying stiffness and time varying damping.

In this paper to evaluate the performance of the proposed controller the first pantograph-catenary model isused. The hybrid controller is designed in a way that an active actuator is assisting a semi-active device toachieve a performance close to a fully active system. The semi-active device can only work in the energy

171

dissipative region. In the energy injection region, the semi-active controller has to be switched off and thesemi-active device behaves as a passive device. In the proposed control methodology, when the semi-activecontroller is about to switch off before going into the energy injection region, the active actuator injects therequired energy into the system and pushes the semi-active actuator back into the dissipative region.

Active controllers with active actuators are very effective for vibration control but sometimes it becomesvery difficult to place an active actuator at certain position in a structure because of the weight or size ofthe actuator, power consumption, mechanical design constraints etc. The idea presented in this paper isto overcome this difficulty by showing that an active actuator that is placed at a different location in thestructure can assist the semi-active actuator to achieve the performance close to a fully active actuator. Asper the authors’ knowledge, this idea has not been found in the literature. Immersion and invariance (I & I)methodology for the active actuator and sliding mode control (SMC) for the semi-active device is found to beparticularly suitable in this case. I & I was first introduced in [13], and it uses the concept of containing thesystem dynamics onto an invariant manifold. Further details of I & I controller and observer design can befound in [14]. Early studies on SMC are presented in [15, 16] and more recent surveys are given in [17–19].

In Section 2 the pantograph-catenary model is introduced. The hybrid controller design is presented with de-tail in Section 3. The simulation results are presented in Section 4 with detailed discussion, and conclusionsare given in Section 5.

2 Pantograph-catenary model

The pantograph-catenary system is represented as a 2-DOF mass spring damper system with a time varyingstiffness representing the catenary behavior as shown in Fig. 1b.

The system can be represented in state space form as

x1 = x2,

x2 =1

m1

(fa − fsa −K1x1 − C1x2 −K2(x1 − x3)− C2(x2 − x4)

),

x3 = x4,

x4 =1

m2

(fsa −K2(x3 − x1)− C2(x4 − x2)−K(t)x3

), (1)

where x1 and x2 are the position and velocity of mass m1 respectively, x3 and x4 are the position andvelocity of mass m2 respectively, fa represents the force of the active actuator, fsa represents the force ofthe semi-active device, m1, m2 represent the masses, K1, K2 are the linear spring stiffness, K(t) is the timevarying stiffness, C1 and C2 are the damping coefficients. K(t) is defined as

K(t) = K0

(1 + αcos(

2πV

Lt)), (2)

where V is the train speed, L is the span length, K0 is average equivalent stiffness, α is stiffness variationcoefficient in a span. K0 and α has been identified using (3).

K0 =Kmax +Kmin

2, α =

Kmax −Kmin

Kmax +Kmin. (3)

where Kmax and Kmin are the maximum and minimum values of the stiffness in a span respectively.

3 Hybrid controller design

To design the controller for an active actuator, the I & I methodology described in [14], is used. The relevanttheory is summarised in the appendix. The objective of the I & I methodology is to find a manifoldM =

172 PROCEEDINGS OF ISMA2016 INCLUDING USD2016

Messenger wireDroppers

TowerTower

Contact wire

Span

Pantograph

EngineBogie

Direction of travel

(a)

m2

m1

K2 X1

X2

C1

K(t)

C2

K1fa

fsa

(b)

Figure 1: Pantograph-catenary models, where fa represents the force of an active actuator and fsa representsthe force of a semi-active device. m1 & m2 represent the masses, K1 and K2 are the linear spring stiffness,K(t) is the time varying catenary stiffness, C1 and C2 are the damping coefficients (a) pantograph-catenarysystem, (b) 2-DOF pantograph-catenary model with time varying catenary stiffness.

{x ∈ Rn|x = π(ξ), ξ ∈ Rp} based on the actual system, target system and the mapping functions. Theorder of the target system is lower than the order of actual system and the mapping functions are defined asvirtual dynamics, to represent the actual system dynamics (off-the-manifold) that are not present in the targetsystem. The first step in the control design is to define a suitable target system. The target system shouldbe realizable and should also consider the physical constraints of the actual system. As a result the SDOFsystem shown in Fig. 2 is defined as the target system.

K(t)

mt

Kt

Ctf

1

Figure 2: Target system, where ξ1 and ξ2 represents the position and velocity of the mass mt, f will becomputed after defining the mapping functions,Kt is the linear spring stiffness,Ct is the damping coefficient.

The dynamics of the target system are given by

ξ1 = ξ2,

ξ2 =1

mt

(f −Ktξ1 − Ctξ2 −K(t)ξ1

), (4)

ACTIVE NOISE AND VIBRATION CONTROL 173

where ξ1 and ξ2 represent the position and velocity of the massmt respectively, and f =W+u, u representsthe controller signal and W is the function that needs to be chosen in a way that the target system shouldhave an asymptotically stable equilibrium at the origin. f is defined as

f =(− C2

K2

(K(t) +Kt

)+ Ct

)ξ2 + u. (5)

The next step is to design a controller for the target system. Any controller can be designed for the targetsystem as long as it can achieve the desired performance for the defined mapping functions. In this paper aproportional plus integral (PI) controller is designed in the same way as in [20]. The PI controller is given as

u = Kaep +Kb

∫epdt−KcK(t)ξ2 −KdK(t)ξ1. (6)

where Ka = KvKp, Kb = KiKp, Kc = Kv, Kd = Ki, Ki, Kv & Kp are control gains, and ep is the errorbetween the reference and the actual contact force. To check the asymptotic stability of the target system,the target system dynamics are compared with a single mass system dynamics (7). From the Lagrangianformulation the dynamics of a single mass are

ξ1 = ξ2,

ξ2 = −E′ − ξ2R, (7)

where E is the potential energy function and R is the damping function and a dash represents differentiationwith respect to the state vector.

Comparing (4) and (7) gives

E′

=1

mt

(mtK(t)

m2+Kt +K(t)

)ξ1, (8)

R =C2

K2

(K(t) +Kt

), (9)

and

E =1

2mt

(mtK(t)

m2+Kt +K(t)

)ξ1

2. (10)

A Lyapunov function is defined as a generalized energy function

Vi&i (ξ1, ξ2) =1

2ξ2

2 + E. (11)

The target system dynamics will have an asymptotically stable equilibrium at the origin if the followingconditions are satisfied by the Lyapunov function defined in (11)

V (0, 0) = 0, (12a)

V (ξ1, ξ2) > 0, in D − {0} . D → Rp (12b)

V (ξ1, ξ2) < 0, in D − {0} . (12c)

where V (ξ1, ξ2) is the energy function, andD is the subset of Rp in which the Lyapunov function is defined.


As a result

Vi&i (ξ1, ξ2) = −Rξ22. (13)

The first two conditions (12a) and (12b) are satisfied by the Lyapunov function defined in (11). The thirdcondition (12c) where Vi&i (ξ1, ξ2) should be negative definite, is satisfied whenR is positive. As can be seenfrom (9),R is always positive. Therefore, the selected target system has an asymptotically stable equilibriumat the origin.

The mapping functions that need to be defined are given by

π (ξ) =

π1(ξ1, ξ2)π2(ξ1, ξ2)π3(ξ1, ξ2)π4(ξ1, ξ2)

(14)

where π1(ξ1, ξ2), π2(ξ1, ξ2) need to be defined for off-the-manifold coordinates and π3(ξ1, ξ2) = x3(π1, π2),π4(ξ1, ξ2) = x4(π1, π2).

One of the requirements with I & I is to solve the partial differential equation (36). As the target systemdynamics (4) resembles the dynamics of the actual system (1), in which the vibration needs to be controlled,then π3(ξ1, ξ2) = ξ1. As x1 = x2, we can write π4(ξ1, ξ2) = ξ2. Based on (15), the mapping functionsπ1(ξ1, ξ2), π2(ξ1, ξ2) are derived from

ξ2 = π4, (15)

and

1

mt

(f −Ktξ1 − Ctξ2 −K(t)ξ1

)=

1

m2

(−K2(ξ1 − π1)− C2(ξ2 − π2)−K(t)ξ1

). (16)

The selection of the mapping functions is a non-trivial task and it is possible for more then one mappingfunction to exist. However, they should always satisfy (16) and by using these mapping functions, the targetsystem should have an asymptotically stable equilibrium at the origin. Therefore the mapping functionsselected are

π1 =(−m2

(K(t) +Kt

)K2mt

+ 1)ξ1 + α1ep + α2

∫epdt+ α3K(t)ξ2 + α4K(t)ξ1, (17)

π2 =(−m2

(K(t) +Kt

)K2mt

+ 1)ξ2 − α1K(t)ξ2 + α2ep + α3K(t)ξ2 + α4K(t)ξ2. (18)

The four unknowns α1, α2, α3, α4 are found by substituting π1, π2, f into (16). The error between theoff-the-manifold dynamics and the mapping functions is defined as

φ(x) = x1 − π1, (19)

and the manifold is defined as

M = −kaφ− kbφ. (20)

where φ(x) = x2 − π2. The gains ka and kb are chosen in such a way that (s2 + kbs+ ka) is Hurwitz.

The last step in the I&I methodology is to compute the control law, which is done using

φ = x2 − π2, (21)


and

φ =1

m1

(fa − fsa −K1x1 − C1x2 −K2(x1 − x3)− C2(x2 − x4)

)− ∂π2

∂x3x3 −

∂π2

∂x4x4. (22)

The control signal fa is given by

fa =

(− kaφ− kbφ+

∂π2

∂x3x3 +

∂π2

∂x4x4

)m1 +K1x1 + C1x2 +K2(x1 − x3) +K2(x1 − x3) (23)

+C2(x2 − x4),

where

∂π2

∂x3= −α2K(t)− α3

1

m2

(K2 +K(t)

), (24)

∂π2

∂x4= − m2

K2mt

(K(t) +Kt

)+ 1 + (α4 − α1)K(t)− α3

C2

m2. (25)

The next step is to design a controller for the semi-active device, and here we use a sliding mode controller.The error dynamics are defined as

e = x3 − ξ1. (26)

The sliding surface is defined in terms of the error dynamics as

S = λ1e+ λ2e. (27)

where λ1, λ2, are the design parameters, which will determine how fast the error dynamics will go to zeroand e = x4 − ξ2. In the next step the control signal is derived using (27).

fsa = fn −m2

λ2

(Ksmcsgn(S)

), (28)

where Ksmc is strictly positive and a design parameter and fn is given as

fn =m2

λ2

(− λ1(x4 − ξ2)

)+m2ξ2 +K2(x3 − x1) + C2(x4 − x2) +K(t)x3. (29)

The SMC control signal has two parts. One part represents the normalized control fn and the second partrepresents the discontinuous (signum function) control, which is responsible for the robustness. To makesure that the sliding surface has an asymptotically stable equilibrium at the origin towards which the systemwill slide, a Lyapunov function is defined as

Vsmc =1

2S2. (30)

The sliding surface will have an asymptotically stable equilibrium if (30) satisfies the conditions in (12).The first two conditions (12a) and (12b) are satisfied by the Lyapunov function defined in (30), for the thirdcondition (12c) to be satisfied, Vsmc needs to be analyzed, where

Vsmc = SS,

and

SS < 0,


To make sure that the system will reach the sliding surface in finite time, a more strict condition is imposedon SS

SS ≤ −η|S|, (31)

which leads to

S ≤ −ηsgn(S). (32)

where η is strictly positive. For the third condition to be satisfied for an asymptotically stable equilibrium,Ksmc should be greater than η. Of course, the semi-active device can only dissipate energy from the system.So, the controller will be switched-on, when the relative velocity vr across the semi-active device and thecontrol signal fsa have opposite signs and will be switched-off otherwise. This condition is imposed on fsa

in (33) and is called the passivity constraint.

fsa =

fn −m2

λ2

(Ksmcsgn(S)

)fsavr < 0

0 fsavr > 0(33)

where vr = x4 − x2.

4 Simulation results

The block diagram implementation of the hybrid controller is shown in Fig. 3. The SMC and I & I controller(as shown within the dotted lines) are forcing the pantograph-catenary system towards the manifold, so thatit starts behaving as the defined target system. When the pantograph-catenary system starts behaving as thetarget system, then the target system’s PI controller acts to maintain a constant contact force. Table 1 showsthe pantograph-catenary system parameters. The gains designed for the controllers are shown in Table 2.In order to introduce the actuator dynamics in the simulation; a second order low pass filter with the cutofffrequency of 50 Hz is incorporated in the simulation with a saturation limits of ± 250 N for both the activeactuator and the semi-active device.

Fig. 4a shows the contact force in actual and target system with a reference contact force of 100 N , undernormal conditions with a constant train speed of 300 km/h. It can be seen that the actual system is followingthe target system. The oscillations in the steady state shows a very small variation of ± 1 N in the contactforce. These oscillations are influenced by the speed of the train because the catenary is modeled as a timevarying stiffness, where one of the factor affecting the stiffness is the train speed. To check the robustness ofthe controller, Gaussian noise is added at mass m2, and the results are shown in Fig. 4b. The performance ofthe controller against external disturbance is good.

To check the performance of the controller against variable train speed, a speed profile is generated as shownin Fig. 5a. Fig. 5b shows the contact force in actual and target system for the variable train speed. Againthe performance of the controller is satisfactory. Fig. 6 shows the active and semi-active control signals inhybrid controller.

5 Conclusion

In this paper a new hybrid control methodology is presented to keep a constant contact force between thepantograph and catenary. I & I design method is used for the controller design for active actuator and forthe semi-active controller design SMC is used. The proposed controller has shown promising results bothunder normal conditions and in the presence of the Gaussian noise, which proves the robustness of thecontroller. Then the robustness is also checked against variable train speed with different slope variationsand the results are satisfactory. 2-DOF pantograph-catenary model with time varying stiffness representingthe catenary behavior is used for the validation of the controller.


Immersion & Invariance

control law

Mapping

and

Manifold

PI

controller

+_ fa

z1

z2

ue

x3

x=[x1,x2,x3,x4]

Figure 3: Block diagram implementation of Hybrid (Active and Semi-Active) control, where x1 and x2

are the position and velocity of mass m1 respectively, x3 and x4 are the position and velocity of mass m2

respectively, fa is the I & I control signal, fsa is the SMC control signal, z1 and z2 are the error dynamics inI & I controller, u and ut are the output of PI controllers in the actual and the target system, e and et are theerrors between reference and desired signal in actual and target system, ξ1 and ξ2 represent the position andvelocity of the mass mt respectively in the target system.

Parameters Notations Values

K0 3.6 kNm−1

Catenary α 0.5L 65 m

m2 8 kgPantograph head C2 120 Nsm−1

K2 10 kNm−1

m1 12 kgPantograph frame C1 30 Nsm−1

K1 100 Nm−1

Table 1: Pantograph-catenary system parameters

PI controller I&I controller SMC controller

Kp = 10 ka = 5000 Ksmc = 10

Kv = 1.1 kb = 450 λ1 = 1

Ki = 70 λ2 = 1

Table 2: Controller gains


0 2 4 6 8 10

time (sec)

0

20

40

60

80

100

120

contact force (N)

target systemactual system

105

951 2.5 4

(a)

0 2 4 6 8 10

time (sec)

0

20

40

60

80

100

120

contact force (N)

target systemactual system

105

951 2.5 4

(b)

Figure 4: Contact force in actual and target system, where solid line represents target system data, dotted linerepresents the actual system data (a) under normal conditions, (b) with Gaussian noise introduced at massm2.

0 10 20 30 40 50 60 70

time (sec)

0

50

100

150

200

250

300

train speed (km/h)

(a)

0 10 20 30 40 50 60 70

time(sec)

0

20

40

60

80

100

120

contact force (N) target system

actual system

20 25 40 50 6097

103

15 65

(b)

Figure 5: Contact force in actual and target system with train speed profile, where solid line represents targetsystem data, dotted line represents the actual system data (a) train speed profile, (b) contact force in actualand target system.

0.9 1 1.1 1.2 1.3 1.4 1.5

time(sec)

-200

-100

0

100

200

f a (N)

(a)

1 1.2 1.4 1.6 1.8 2 2.2

time(sec)

-50

0

50

100

150

200

250

f sa (N)

(b)

Figure 6: Control signals (a) active control signal, (b) semi-active control signal.


Acknowledgements

DJW would like to acknowledge the support of the EPSRC via grant EP/K003836/2.

References

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Appendix A : Immersion and Invariance Theorem

The immersion and invariance methodology defines a set of conditions for the existence of an invariantmanifold with an asymptotically stable target system within which the original system will be immersed. Weuse the standard I & I approach [14] for a nonlinear system

x = f (x) + g(x)u (34)

where x ∈ Rn is the system state, u ∈ Rm is the input signal, f(x) and g(x) are nonlinear functions of xand an over-dot represents the differentiation with respect to time. The equilibrium point to be stabilized isdenoted x? ∈ Rn.

The following properties should hold.(H1) The system

ξ = α(ξ) (35)

with transformed state vector ξ ∈ Rp has an asymptotically stable equilibrium at ξ? ∈ Rp, and

x? ∈ π(ξ?).

where α : Rp → Rp and π : Rp → Rn are smooth mapping functions with p < n.(H2) For all ξ ∈ Rp, substituting a smooth mapping x = π(ξ) in (34) leads to

f (π(ξ)) + g (π (ξ)) c (π (ξ)) =∂π

∂ξα(ξ). (36)

where c : Rp → Rm is the control signal that renders the manifold invariant.(H3) The set identity holds

{x ∈ Rn|φ (x) = 0} = {x ∈ Rn|x = π (ξ) , ξ ∈ Rp} . (37)

where φ : Rn → Rn−p represents the manifold. From (37), the manifold φ(x) = 0, when x = π(ξ), henceφ = x− π(ξ) and z = x− π(ξ), where z represents the distance between off-the-manifold coordinates andthe manifold.(H4) All trajectories of the system

z =∂φ

∂x[f (x) + g(x)ψ(x, z)], (38)

x = f (x) + g(x)ψ(x, z), (39)

are bounded and satisfy

limt→∞

z (t) = 0. (40)

where ψ : Rn×(n−p) → Rm is the equivalent control signal and right hand side of (38) is φ.Then x∗ is an asymptotically stable equilibrium of the closed loop system

x = f (x) + g(x)ψ(x, φ(x)). (41)

Once the close loop system (41), trajectories converges to the manifold and z = 0 then ψ(π(ξ), 0) = c(π(ξ)).


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