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

Transcription

1 Hybrid active and semi-active control for pantograph-catenary system 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 Kingdom iukhan1@sheffield.ac.uk Abstract In this paper a new hybrid control methodology using active actuator and semi-active device is proposed to minimize the oscillations between the pantograph and catenary by keeping the contact force between them constant. One of the advantages of using the proposed hybrid controller is that a semi-active device can easily be mounted on the pantograph upper arm without compromising the weight and the size. However, the performance of a semi-active device is restricted because of the passivity constraint. To assist the semi-active device and to achieve the desired performance an active actuator is placed at the base of the pantograph. The immersion and invariance I & I methodology is used to design the controller for the active actuator, and sliding mode control SMC is used to design the controller for the semi-active device. Simulations show promising results. 1 Introduction The use of high speed trains not only improves the efficiency of transportation but also has a very positive impact on the environment in terms of controlling air pollution. High speed trains can achieve a very high speed i.e around 350 km/h. Hence it is very important to have a permanent contact between pantograph and catenary. The electric current from catenary to train transformer flows through the pantograph. In the ideal scenario, the contact between pantograph and catenary should be permanent, but due to the flexibility in the structure of catenary and pantograph, as the train speed increases the oscillations keep on increasing and the contact is not guaranteed, which results in electric arcs and eventually deteriorating the current collection from catenary. One solution to avoid loss of contact is to increase the contact force but this will result in wear and tear due to excessive contact force. It is very important to keep the contact force constant without causing any damage to the pantograph-catenary system. Another solution is to increase the tension in the contact wire which means increasing the equivalent stiffness. This solution is very expensive. The catenary presents a time varying stiffness, which is depended on the train speed. To solve this problem different control 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] the pantograph is modeled as a 2-DOF mass spring damper system and the catenary is designed as a spring with time varying stiffness. In [5 9] the pantograph is modeled as a 2-DOF mass spring damper system and catenary is modeled as a spring with fixed stiffness. In addition to that a spring with fixed stiffness is added to represent the pantograph shoe. In [10 12] the pantograph is modeled as a 3-DOF mass spring damper 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 is used. The hybrid controller is designed in a way that an active actuator is assisting a semi-active device to achieve a performance close to a fully active system. The semi-active device can only work in the energy 171

2 172 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 dissipative region. In the energy injection region, the semi-active controller has to be switched off and the semi-active device behaves as a passive device. In the proposed control methodology, when the semi-active controller is about to switch off before going into the energy injection region, the active actuator injects the required 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 becomes very difficult to place an active actuator at certain position in a structure because of the weight or size of the actuator, power consumption, mechanical design constraints etc. The idea presented in this paper is to overcome this difficulty by showing that an active actuator that is placed at a different location in the structure can assist the semi-active actuator to achieve the performance close to a fully active actuator. As per 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 be particularly suitable in this case. I & I was first introduced in [13], and it uses the concept of containing the system dynamics onto an invariant manifold. Further details of I & I controller and observer design can be found 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 detail in Section 3. The simulation results are presented in Section 4 with detailed discussion, and conclusions are 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 varying stiffness representing the catenary behavior as shown in Fig. 1b. The system can be represented in state space form as ẋ 1 = x 2, ẋ 2 = 1 f a f sa K 1 x 1 C 1 x 2 K 2 x 1 x 3 C 2 x 2 x 4, m 1 ẋ 3 = x 4, ẋ 4 = 1 m 2 f sa K 2 x 3 x 1 C 2 x 4 x 2 Ktx 3, 1 where x 1 and x 2 are the position and velocity of mass m 1 respectively, x 3 and x 4 are the position and velocity of mass m 2 respectively, f a represents the force of the active actuator, f sa represents the force of the semi-active device, m 1, m 2 represent the masses, K 1, K 2 are the linear spring stiffness, Kt is the time varying stiffness, C 1 and C 2 are the damping coefficients. Kt is defined as Kt = K αcos 2πV L t, 2 where V is the train speed, L is the span length, K 0 is average equivalent stiffness, α is stiffness variation coefficient in a span. K 0 and α has been identified using 3. K 0 = K max + K min, α = K max K min. 3 2 K max + K min where K max and K min 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 relevant theory is summarised in the appendix. The objective of the I & I methodology is to find a manifold M =

3 ACTIVE NOISE AND VIBRATION CONTROL 173 Tower Tower Droppers Span Messenger wire Kt X 2 m 2 Contact wire Pantograph C 2 f sa K 2 X 1 Direction of travel m 1 Bogie Engine C 1 f a K 1 a b Figure 1: Pantograph-catenary models, where f a represents the force of an active actuator and f sa represents the force of a semi-active device. m 1 & m 2 represent the masses, K 1 and K 2 are the linear spring stiffness, Kt is the time varying catenary stiffness, C 1 and C 2 are the damping coefficients a pantograph-catenary system, b 2-DOF pantograph-catenary model with time varying catenary stiffness. {x R n x = πξ, ξ R p } based on the actual system, target system and the mapping functions. The order of the target system is lower than the order of actual system and the mapping functions are defined as virtual dynamics, to represent the actual system dynamics off-the-manifold that are not present in the target system. The first step in the control design is to define a suitable target system. The target system should be realizable and should also consider the physical constraints of the actual system. As a result the SDOF system shown in Fig. 2 is defined as the target system. Kt 1 m t f K t C t Figure 2: Target system, where ξ 1 and ξ 2 represents the position and velocity of the mass m t, f will be computed after defining the mapping functions, K t is the linear spring stiffness, C t is the damping coefficient. The dynamics of the target system are given by ξ 1 = ξ 2, ξ 2 = 1 m t f Kt ξ 1 C t ξ 2 Ktξ 1, 4

4 174 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 where ξ 1 and ξ 2 represent the position and velocity of the mass m t respectively, and f = W +u, u represents the controller signal and W is the function that needs to be chosen in a way that the target system should have an asymptotically stable equilibrium at the origin. f is defined as f = C 2 K 2 Kt + K t + C t ξ 2 + u. 5 The next step is to design a controller for the target system. Any controller can be designed for the target system as long as it can achieve the desired performance for the defined mapping functions. In this paper a proportional plus integral PI controller is designed in the same way as in [20]. The PI controller is given as u = K a e p + K b e p dt K c Ktξ 2 K d Ktξ 1. 6 where K a = K v K p, K b = K i K p, K c = K v, K d = K i, K i, K v & K p are control gains, and e p is the error between 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 Lagrangian formulation the dynamics of a single mass are ξ 1 = ξ 2, ξ 2 = E ξ 2 R, 7 where E is the potential energy function and R is the damping function and a dash represents differentiation with respect to the state vector. Comparing 4 and 7 gives E = 1 m t Kt + K t + Kt ξ 1, 8 m t m 2 R = C 2 K 2 Kt + K t, 9 and E = 1 m t Kt + K t + Kt ξ m t m 2 A Lyapunov function is defined as a generalized energy function V i&i ξ 1, ξ 2 = 1 2 ξ E. 11 The target system dynamics will have an asymptotically stable equilibrium at the origin if the following conditions are satisfied by the Lyapunov function defined in 11 V 0, 0 = 0, 12a V ξ 1, ξ 2 > 0, in D {0}. D R p 12b V ξ 1, ξ 2 < 0, in D {0}. 12c where V ξ 1, ξ 2 is the energy function, and D is the subset of R p in which the Lyapunov function is defined.

5 ACTIVE NOISE AND VIBRATION CONTROL 175 As a result V i&i ξ 1, ξ 2 = Rξ The first two conditions 12a and 12b are satisfied by the Lyapunov function defined in 11. The third condition 12c where V i&i ξ 1, ξ 2 should be negative definite, is satisfied when R is positive. As can be seen from 9, R is always positive. Therefore, the selected target system has an asymptotically stable equilibrium at the origin. The mapping functions that need to be defined are given by π 1 ξ 1, ξ 2 π ξ = π 2 ξ 1, ξ 2 π 3 ξ 1, ξ 2 π 4 ξ 1, ξ 2 14 where π 1 ξ 1, ξ 2, π 2 ξ 1, ξ 2 need to be defined for off-the-manifold coordinates and π 3 ξ 1, ξ 2 = x 3 π 1, π 2, π 4 ξ 1, ξ 2 = x 4 π 1, π 2. One of the requirements with I & I is to solve the partial differential equation 36. As the target system dynamics 4 resembles the dynamics of the actual system 1, in which the vibration needs to be controlled, then π 3 ξ 1, ξ 2 = ξ 1. As ẋ 1 = x 2, we can write π 4 ξ 1, ξ 2 = ξ 2. Based on 15, the mapping functions π 1 ξ 1, ξ 2, π 2 ξ 1, ξ 2 are derived from ξ 2 = π 4, 15 and 1 f Kt ξ 1 C t ξ 2 Ktξ 1 = 1 K 2 ξ 1 π 1 C 2 ξ 2 π 2 Ktξ m t m 2 The selection of the mapping functions is a non-trivial task and it is possible for more then one mapping function to exist. However, they should always satisfy 16 and by using these mapping functions, the target system should have an asymptotically stable equilibrium at the origin. Therefore the mapping functions selected are Kt + Kt π 1 = m ξ 1 + α 1 e p + α 2 e p dt + α 3 Ktξ 2 + α 4 Ktξ 1, 17 K 2 m t π 2 = Kt + Kt m ξ 2 α 1 Ktξ 2 + α 2 e p + α 3 Kt ξ K 2 m 2 + α 4 Ktξ t The four unknowns α 1, α 2, α 3, α 4 are found by substituting π 1, π 2, f into 16. The error between the off-the-manifold dynamics and the mapping functions is defined as and the manifold is defined as φx = x 1 π 1, 19 M = k a φ k b φ. 20 where φx = x 2 π 2. The gains k a and k b are chosen in such a way that s 2 + k b s + k a is Hurwitz. The last step in the I&I methodology is to compute the control law, which is done using φ = ẋ 2 π 2, 21

6 176 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 and φ = 1 f a f sa K 1 x 1 C 1 x 2 K 2 x 1 x 3 C 2 x 2 x 4 π 2 ẋ 3 π 2 ẋ m 1 x 3 x 4 The control signal f a is given by f a = k a φ k b φ + π 2 x 3 ẋ 3 + π 2 x 4 ẋ 4 m 1 + K 1 x 1 + C 1 x 2 + K 2 x 1 x 3 + K 2 x 1 x C 2 x 2 x 4, where π 2 1 = α 2 Kt α 3 K 2 + Kt, 24 x 3 m 2 π 2 = m 2 C 2 Kt + K t α 4 α 1 Kt α x 4 K 2 m t m 2 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 The sliding surface is defined in terms of the error dynamics as e = x 3 ξ S = λ 1 e + λ 2 ė. 27 where λ 1, λ 2, are the design parameters, which will determine how fast the error dynamics will go to zero and ė = x 4 ξ 2. In the next step the control signal is derived using 27. f sa = f n m 2 K smc sgns, 28 λ 2 where K smc is strictly positive and a design parameter and f n is given as f n = m 2 λ 2 λ 1 x 4 ξ 2 + m 2 ξ 2 + K 2 x 3 x 1 + C 2 x 4 x 2 + Ktx The SMC control signal has two parts. One part represents the normalized control f n and the second part represents the discontinuous signum function control, which is responsible for the robustness. To make sure that the sliding surface has an asymptotically stable equilibrium at the origin towards which the system will slide, a Lyapunov function is defined as V smc = 1 2 S2. 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 third condition 12c to be satisfied, Vsmc needs to be analyzed, where and V smc = SṠ, SṠ < 0,

7 ACTIVE NOISE AND VIBRATION CONTROL 177 To make sure that the system will reach the sliding surface in finite time, a more strict condition is imposed on SṠ which leads to SṠ η S, 31 Ṡ ηsgns. 32 where η is strictly positive. For the third condition to be satisfied for an asymptotically stable equilibrium, K smc 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 v r across the semi-active device and the control signal f sa have opposite signs and will be switched-off otherwise. This condition is imposed on f sa in 33 and is called the passivity constraint. f n m 2 K smc sgns f sa v r < 0 f sa = λ f sa v r > 0 where v r = x 4 x 2. 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 that it starts behaving as the defined target system. When the pantograph-catenary system starts behaving as the target system, then the target system s PI controller acts to maintain a constant contact force. Table 1 shows the 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 cutoff frequency of 50 Hz is incorporated in the simulation with a saturation limits of ± 250 N for both the active actuator and the semi-active device. Fig. 4a shows the contact force in actual and target system with a reference contact force of 100 N, under normal conditions with a constant train speed of 300 km/h. It can be seen that the actual system is following the target system. The oscillations in the steady state shows a very small variation of ± 1 N in the contact force. These oscillations are influenced by the speed of the train because the catenary is modeled as a time varying stiffness, where one of the factor affecting the stiffness is the train speed. To check the robustness of the controller, Gaussian noise is added at mass m 2, and the results are shown in Fig. 4b. The performance of the controller against external disturbance is good. To check the performance of the controller against variable train speed, a speed profile is generated as shown in Fig. 5a. Fig. 5b shows the contact force in actual and target system for the variable train speed. Again the performance of the controller is satisfactory. Fig. 6 shows the active and semi-active control signals in hybrid controller. 5 Conclusion In this paper a new hybrid control methodology is presented to keep a constant contact force between the pantograph and catenary. I & I design method is used for the controller design for active actuator and for the semi-active controller design SMC is used. The proposed controller has shown promising results both under normal conditions and in the presence of the Gaussian noise, which proves the robustness of the controller. Then the robustness is also checked against variable train speed with different slope variations and the results are satisfactory. 2-DOF pantograph-catenary model with time varying stiffness representing the catenary behavior is used for the validation of the controller.

8 178 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 x 3 e u + _ PI controller Immersion & Invariance f a control law z 1 z 2 Mapping and Manifold x=[x 1,x 2,x 3,x 4 ] Figure 3: Block diagram implementation of Hybrid Active and Semi-Active control, where x 1 and x 2 are the position and velocity of mass m 1 respectively, x 3 and x 4 are the position and velocity of mass m 2 respectively, f a is the I & I control signal, f sa is the SMC control signal, z 1 and z 2 are the error dynamics in I & I controller, u and u t are the output of PI controllers in the actual and the target system, e and e t are the errors between reference and desired signal in actual and target system, ξ 1 and ξ 2 represent the position and velocity of the mass m t respectively in the target system. Parameters Notations Values K knm 1 Catenary α 0.5 L 65 m m 2 8 kg Pantograph head C Nsm 1 K 2 10 knm 1 m 1 12 kg Pantograph frame C 1 30 Nsm 1 K Nm 1 Table 1: Pantograph-catenary system parameters PI controller I&I controller SMC controller K p = 10 k a = 5000 K smc = 10 K v = 1.1 k b = 450 λ 1 = 1 K i = 70 λ 2 = 1 Table 2: Controller gains

9 ACTIVE NOISE AND VIBRATION CONTROL 179 contact force N target system actual system contact force N target system actual system time sec a time sec b Figure 4: Contact force in actual and target system, where solid line represents target system data, dotted line represents the actual system data a under normal conditions, b with Gaussian noise introduced at mass m 2. train speed km/h time sec a contact force N target system actual system timesec b Figure 5: Contact force in actual and target system with train speed profile, where solid line represents target system data, dotted line represents the actual system data a train speed profile, b contact force in actual and target system. f a N f sa N timesec a timesec b Figure 6: Control signals a active control signal, b semi-active control signal.

10 180 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 Acknowledgements DJW would like to acknowledge the support of the EPSRC via grant EP/K003836/2. References [1] A. Levant, A. Pisano, and E. Usai, Output-feedback control of the contact-force in high-speed-train pantographs, in Proceedings of the 40th IEEE Conference on Decision and Control, vol. 2. IEEE, 2001, pp [2] A. Rachid, Pantograph catenary control and observation using the LMI approach, in 50th IEEE Conference on Decision and Control and European Control Conference CDC-ECC. IEEE, 2011, pp [3] S. Shin, K. Eum, and J. Um, Contact Force Control of Pantograph-Catenary System using Block Pulse Function, in Proceedings of the 7th World Congress on Railway Research WCRR, pp [4] C. K. Ide, S. Olaru, P. Rodriguez-Ayerbe, and A. Rachid, A nonlinear state feedback control approach for a Pantograph-Catenary system, in 17th International Conference System Theory, Control and Computing ICSTCC. IEEE, 2013, pp [5] R. Garg, P. Mahajan, P. Kumar, and V. Gupta, Design and study of active controllers for pantographcatenary system, in IEEE Conference and Expo Transportation Electrification Asia-Pacific ITEC Asia-Pacific, [6] P. Mahajan, R. Garg, V. Gupta, and P. Kumar, Design of controller for pantograph-catenary system using reduced order model, in 6th IEEE Power India International Conference PIICON. IEEE, 2014, pp [7] E. Karakose and M. T. Gencoglu, Adaptive fuzzy control approach for dynamic pantograph-catenary interaction, in 15th International Symposium MECHATRONIKA. IEEE, 2012, pp [8] Y. J. Huang and T. C. Kuo, Discrete pantograph position control for the high speed transportation systems, in IEEE International Conference on Networking, Sensing and Control, vol. 2. IEEE, 2004, pp [9] E. Karakose and M. T. Gencoglu, An investigation of pantograph parameter effects for pantographcatenary systems, in IEEE International Symposium on Innovations in Intelligent Systems and Applications INISTA. IEEE, 2014, pp [10] N. Mokrani and A. Rachid, A robust control of contact force of pantograph-catenary for the highspeed train, in 2013 European Control Conference ECC. IEEE, 2013, pp [11] A. Pisano and E. Usai, Contact force estimation and regulation in active pantographs: an algebraic observability approach, Asian Journal of Control, vol. 13, no. 6, pp , [12] N. Mokrani, A. Rachid, and M. A. Rami, A tracking control for pantograph-catenary system, in 54th IEEE Annual Conference on Decision and Control CDC. IEEE, 2015, pp [13] A. Astolfi and R. Ortega, Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems, IEEE Transaction on Automatic Control, vol. 48, no. 4, pp , [14] A. Astolfi and D. Karagiannis, Nonlinear and adaptive control with applications. Springer Science & Business Media, 2007.

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12 182 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 Appendix A : Immersion and Invariance Theorem The immersion and invariance methodology defines a set of conditions for the existence of an invariant manifold with an asymptotically stable target system within which the original system will be immersed. We use the standard I & I approach [14] for a nonlinear system ẋ = f x + gxu 34 where x R n is the system state, u R m is the input signal, fx and gx are nonlinear functions of x and an over-dot represents the differentiation with respect to time. The equilibrium point to be stabilized is denoted x R n. The following properties should hold. H1 The system ξ = αξ 35 with transformed state vector ξ R p has an asymptotically stable equilibrium at ξ R p, and x πξ. where α : R p R p and π : R p R n are smooth mapping functions with p < n. H2 For all ξ R p, substituting a smooth mapping x = πξ in 34 leads to f πξ + g π ξ c π ξ = π αξ. 36 ξ where c : R p R m is the control signal that renders the manifold invariant. H3 The set identity holds {x R n φ x = 0} = {x R n x = π ξ, ξ R p }. 37 where φ : R n R n p represents the manifold. From 37, the manifold φx = 0, when x = πξ, hence φ = x πξ and z = x πξ, where z represents the distance between off-the-manifold coordinates and the manifold. H4 All trajectories of the system ż = φ [f x + gxψx, z], 38 x are bounded and satisfy ẋ = f x + gxψx, z, 39 lim z t = t where ψ : R n n p R m 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 ẋ = f x + gxψx, φx. 41 Once the close loop system 41, trajectories converges to the manifold and z = 0 then ψπξ, 0 = cπξ.