Contact force estimation and regulation in active pantographs: an algebraic observability approach

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1 Asian Journal of Control, Vol., No., pp. 1 9, Month 28 Published online in Wiley InterScience ( DOI: 1.12/asjc. Contact force estimation and regulation in active pantographs: an algebraic observability approach Alessandro PISANO and Elio USAI ABSTRACT In this paper we extend our previous work concerning the regulation of the contact force in active train pantographs with wire-actuators. In particular, we suggest a second-order sliding mode based control scheme that avoids the direct measurement of the contact force, which is quite unpractical in real applications. The idea is to estimate the contact force using the measured displacements of the upper and lower pantograph frames. Unlike the contact force, the frames displacements can be measured accurately and reliably in practice. The estimation method, based on the algebraic observability theory, entails the use of real-time sliding-mode differentiators. It makes use explicitly of the mechanical characteristics of the upper frame only. The considered secondorder sliding mode control scheme also includes proper linear compensating cascade filters devoted to cope with the de-stabilizing effects of the resonant wire actuator. We show that such a control scheme can be effectively combined with the given contact force estimation method. Key Words: Active Pantographs, Contact force estimation, Algebraic observability, Sliding Mode Control. I. INTRODUCTION The use of active pantographs in high speed railways has been investigated during the last decades [4, 14, 16, 18, 19, 22, 25, 29, 31, 34] and most of the proposed control schemes assume that the contact force is measured with sufficient accuracy. Usually the contact force is evaluated by means of load cells whose measurement are compensated by accelerometers [1, 21, 13]. Nevertheless, measuring the contact force is considered as a very challenging problem due to the adverse environmental conditions (e.g., wire vibrations, electromagnetic interferences, parasitic dynamics). In order to avoid the direct measurement of the contact force, Diana et al. [14, 15] proposed the use of an Extended Kalman Filter. In this case the estimation Corr. author: Alessandro Pisano. The authors are with the Department of Electrical and Electronic Engineering, University of Cagliari, Piazza d armi, 91 Cagliari, Italy. E- mails {pisano,eusai}@diee.unica.it accuracy depends on the accuracy of the mathematical model and on the matching between the nominal and actual statistical properties of the process and output noise signals. We consider a linear model of the catenary/pantograph system, where the lower and upper pantograph frames, as well as the catenary, are modeled in terms of lumped masses, springs and dampers [25, 29, 34]. A schematic representation is depicted in the Figure 2. The mechanical parameters of the upper and lower frame are constant and can be considered known with good accuracy. The catenary parameters, on the contrary, will be space-dependent functions of the current distance of the contact point from the closest adjacent towers and droppers (see Fig. 1) In order to reduce the information needed, we consider the Algebraic Observability approach [17, 2, 11, 12] and our observation scheme requires the knowledge of: i. the mechanical parameters of the upper frame. c 28 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [Version: 28/7/7 v1.]

2 2 Asian Journal of Control, Vol., No., pp. 1 9, Month 28 IV describes the contact force estimation method, and Section V investigates the closed-loop control performance with the estimates contact force used for feedback. The final Section VI gives some concluding remarks. Fig. 1. The system train pantograph catenary ii. the value of any control force applied to the upper frame. iii. position, velocity and acceleration of the upper frame. iv. position and velocity of the lower frame. The velocities and acceleration are estimated by differentiating in real time the measured displacements of the pantograph frames, and to this end we make use of the Levant s arbitrary-order robust differentiator [24]. The use of wire actuators pulling down the actuated frame by a strengthened iron cable was considered in [28, 4, 1] for the practical realization of the control. Wire actuators offer some practical advantages since the motor be safely located on the train roof. Directly coupled motors, on the contrary, are typically mounted on the actuated frame, increasing the weight and inertia of the suspended masses with negative impact on the system performance. It was shown in [2] that the resonance of the wire actuators may cause unacceptable chattering effect if not properly taken into account. In [2] it was shown that a remarkable improvement of the system performance can be achieved by cascading the Suboptimal controller and the integrator at its output with a further, linear, low-pass, compensating filter. This method can be theoretically justified by performing a frequency-based analysis of the steady-state motion, following the approach detailed in [1, 9] and using a nominal LTI model for the pantograph/catenary dynamics. A similar technique has been also studied in [32]. The actuator dynamics effect in variable structure control systems for linear systems has been the subject of several research efforts (see e.g. [33, 8]) We show that such a control scheme can be effectively combined with the given contact force estimation method. We compare the obtained accuracy in the two cases in which the contact force is measured or estimated. Both cases are dealt with by including appropriate error sources in the measured signals. The plan of the paper is as follows: Section II presents the adopted mathematical model and Section III outlines the control scheme suggested. Section II. SYSTEM MODEL The rigorous mathematical modeling of the overhead suspended system, the catenary, would give rise to a distributed-parameter system [31, 3] Nevertheless, simplified models with lumped, possibly time-varying, parameters have been shown to be sufficiently accurate for control system analysis and design purposes [34, 4, 1]. It was shown, in particular, that a linear system can approximate the pantograph dynamics with sufficient accuracy in a vicinity of the working configuration [1, 25, 34]. Let us consider a 3DOF representation of the pantograph and catenary dynamics in terms of lumped masses, springs and dampers (Figure 2). The block scheme in Figure 2 highlights the separate, interacting, dynamics of the catenary and of the pantograph. k c (t) k 3 k 2 k 1 m c (t) t λ ( ) m 2 m 3 b 2 b 3 b c (t) b 1 f c f c Fig. 2. A lumped-parameters model of the pantograph-catenary system. The mechanical parameters of the catenary present a periodic behaviour along each span [4], then it makes sense to consider the following Fourier series F f q x c x 1 x 2 x 3 c 28 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

3 A. Pisano and E. Usai: Contact force estimation and regulation in active pantographs 3 approximations: m c (t) = m c + 3 i=1 m cicos( 2iπ L x) + m c7cos( 14π L x) b c (t) = b c + 3 i=1 b cicos( 2iπ L x) + b c7cos( 14π L x) k c (t) = k c + 3 i=1 k cicos( 2iπ L x) + k c7cos( 14π L x) (1) where x = x(t) the actual distance of the train from a tower, L is the span length, and the 7-th harmonic is considered since it takes into account the periodicity due to the droppers (there are, indeed, six equally spaced droppers along each span [4] that divide it in seven sectors). By considering the system in Figure 2 the contact force takes the following expression λ = max{k 1 (x 2 x 1 ) + b 1 (ẋ 2 ẋ 1 ), } (2) Relationship (2) also include the possible loss of contact. However, during normal operation the pantograph will be always in contact with the catenary. This implies that x 1 x c (see Fig. 2), and that the contact force λ is positive. Therefore, the following sixth-order system derives: m c(t)ẍ 1 + b c(t)ẋ 1 + k c(t)x 1 = k 1(x 2 x 1) + b 1(ẋ 2 ẋ 1) m 2ẍ 2 + b 2ẋ 2 + k 2x 2 = k 2x 3 + b 2ẋ 3 + f c(t) k 1(x 2 x 1) b 1(ẋ 2 ẋ 1) m 3ẍ 3 + (b 2 + b 3)ẋ 3 + (k 2 + k 3)x 3 = k 2x 2 + b 2ẋ 2 + f q(t) (3) where f q (t) is a control force acting on the lower frame and f c (t) is a control force acting on the upper frame. Let z(t) = [x 1, ẋ 1, x 2, ẋ 2, x 3, ẋ 3, ] T be the system state vector, then the linear time-varying state-space model (3) can be compactly rewritten as follows: ż(t) = A(t)z(t) + B 1 f c (t) + B 2 f q (t) λ(t) = Cz(t) (4) where y(t) = λ(t) is the contact force, the matrix terms B 1, B 2 and C take the following simple form B 1 = [ ] T [ 1 1, B 2 = m 2 m 3 (5) C = [ k 1 b 1 k 1 b 1 ] (6) and matrix A(t), depending on the time-varying catenary parameters, is implicitly defined. In practice, only one adjustable control action is employed among the two alternatives of actuating the upper frame or the lower pantograph frame. This choice strongly impacts the output regulation control problem since the input-output relative degree is different for the two cases. ] T III. CONTACT FORCE ESTIMATION We exploit the Algebraic Observability Approach [17, 2] in order to estimate the contact force considered as an unknown system input. Let the displacements of the lower and upper frame, x 2 and x 3, be measured. The second equation of system (3), together with the contact force definition (2), lead to the following relationship involving the variables x 3 and x 2, some of their time derivatives, the contact force λ and the control force f c (possibly zero) exerted on the upper frame: λ = m 2 ẍ 2 + b 2 (ẋ 3 ẋ 2 ) + k 2 (x 3 x 2 ) + f c (t) (7) According to previously given considerations, the upper frame mechanical parameters m 2, b 2 and k 2 can be considered known with a good accuracy and the force f c applied to the upper frame can be also considered accessible for measurements. In order to estimate the required derivatives of x 2 and x 3, a real-time differentiation algorithm based on the higher-order sliding mode control theory is considered [24]. Here we need two instances of such an algorithm, which can provide the estimate of an arbitrary number of derivatives. The first instance, which is used for estimating ẋ 2 and ẍ 2, consists of the following third-order system ż = λ z x 2 2/3 sign(z x 2 ) + z 1 ż 1 = λ 1 z x 2 1/2 sign(z x 2 ) + z 2 ż 2 = λ 2 z x 2 sign(z x 2 ) ˆẋ 2 = z 1 ˆẍ 2 = z 2 (8) where z 1 and z 2 define the finite-time converging estimates of ẋ 2 and ẍ 2, respectively. The tuning of the three coefficients λ, λ 1, λ 2 can be carried out by means of simple formulas that involve a Lipschitz constant L 2 for ẍ 2, i.e., any constant value L 2 such that x (3) 2 L 2 (9) Among many possible feasible alternatives, the following formulas were suggested in [24]: λ = 3L 1/3 2, λ 1 = 1.5L 1/2 2 λ 2 = 1.1L 2 (1) The second differentiator instance, used to estimate ẋ 3, is the classical Super-Twisting differentiator [23, 24] described by the following second-order system ẇ = β w x 3 1/2 sign(w x 3 ) + w 1 ẇ 1 = β 1 sign(w x 3 ) ˆẋ 3 = w 1 (11) c 28 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

4 4 Asian Journal of Control, Vol., No., pp. 1 9, Month 28 The tuning of the two coefficients β, β 1 is carried out by means of simple formulas that involve the Lipschitz constant L 3 of ẋ 3, i.e., the constant value L 3 such that ẍ 3 L 3 (12) The following formulas were suggested in [23, 24]: β = 1.5L 1/2 3, β 1 = 1.1L 3 (13) The contact force is then estimated as follows ˆλ = m 2 z 2 + b 2 (w 1 z 1 ) + k 2 (x 3 x 2 ) + f c (t) (14) where the signals z 2, z 1 and w 1 can be shown to converge in finite time towards the corresponding estimated derivatives. IV. SYSTEM ANALYSIS AND CONTROLLER DESIGN By analyzing the input-output dynamics it follows that the output variable y has a globally defined relative degree one with respect to the control force f c, while it has globally defined relative degree two with respect to the control force f q. The sliding variable is defined by the output regulation error σ(t) = y(t) λ (15) where λ is a constant set-point usually taken as 1N. Obviously, the relative degree of σ and y are the same. We consider control laws having two components, a constant feed-forward aiding term and a timevarying feedback term. Those components are denoted with the subscripts FF and FB, respectively. Since it is easier to locate an actuator for the lower frame control, the feed-forward term f qff is always a component of f q (t) such that the control forces are f c (t) = f cfb (t) (16) f q (t) = f qff + f qfb (t) (17) with f cfb (t) and f qfb (t) such that one of them is set to zero. The feed-forward term f qff should be also large enough such that the wire of the actuator is always strengthened. This force also acts as a lift force before the contact of the pantograph shoe with the catenary is established. The above consideration further supports the choice of locating the main actuator on the train roof. To cope with the uncertainties affecting the system dynamics, the feedback control terms are suitably defined according to the second-order sliding-mode control (2-SMC) approach [5], and, in particular, the Generalized Sub-optimal (G-SO) controller [?] is considered u = sub(s; U, β) = Usign (s βs Mi ) (18) where U is a sufficiently large constant control gain, β is a constant belonging to the interval [.5, 1), and s Mi is the latest singular point of the input signal s, i.e., the value of s at the most recent time instant t Mi (i = 1, 2,...) such that ṡ(t Mi ) =. In [2], we proposed a control algorithm for the case of upper frame control (UFC). The control laws (16) and (17) are considered along with the following expression for the feedback control terms f cfb : UFC : f qfb (t) = f cfb () = f cfb (t) = sub(σ; U c, β c ) (19) By following analogous design considerations as those given in [2], one can also define a LFC version of the control law as follows: { fqfb (t) = sub(σ; U LFC : q, β q ) (2) f cfb (t) = In both cases the same control algorithm is considered. In the upper frame control the time derivative of the feedback control component f cfb is considered as the auxiliary control variable. This implies that f cfb (t), obtained according to (19) by time-integration of a discontinuous signal, will be a continuous function of time, while f qfb (t) in (2) will be a discontinuous switching signal. Clearly, the reason motivating this difference is the different relative degree of the sliding variable with respect to the two considered control forces. It was proven in [2] that with sufficiently large U c and properly chosen β c the UFC implementation leads to the finite time vanishing of the output error. The lower frame control case was not explicitly considered in [2], but, since the sliding variable dynamics are formally equivalent, it can be derived by analogous considerations. Remark 1. Typical SMC laws, including the considered sub-optimal 2-SMC algorithm, depend on a constant gain parameter to be tuned sufficiently large to ensure the closed-loop stability. By a conservative worst-case analysis the gain parameter can be computed off-line solving proper inequalities that involve the gain parameter and the worst-case uncertainty bounds. c 28 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

5 A. Pisano and E. Usai: Contact force estimation and regulation in active pantographs 5 In most cases such a value, computed off-line, is redundantly large and practically useless, since it leads to an overall degradation of the control performance in actual implementation. In practice, the controller is better calibrated by progressively increasing the gain until stable closed-loop operation is observed. V. CONTROL IMPLEMENTATION BY WIRE-ACTUATORS In [28, 1, 4], the implementation of the control forces (either f c or f q ) by means of a wire pulling down the corresponding frame was suggested. The use of such a type of actuator constrains the control action to be negative (i.e., f c, f q ). Nevertheless this can be accomplished by increasing the value of the feedforward term f qff. The wire actuator can be well modeled by a linear second-order dynamics with a small damping and a resonance peak [1, 2] according to f c + 2ξ c ω cn f c + ωcn 2 f c = ωcn 2 v c(t) f q + 2ξ q ω qn f q + ωqnf 2 q = ωqnv 2 q (t) (21) where v c and v q are the actuator commands. Depending on the particular frame to which the wire-actuator is connected, the length of the wire is different (obviously, a longer wire is required when the upper frame is actuated). Clearly, this difference in length affects both the the resonance frequency ω n and the damping coefficient ξ, therefore the following pairs of actuator parameters are considered in the Upper Frame Control (UFC) and Lower Frame Control LFC cases [28, 4]: UFC : ξ c =.24, ωcn 2 = 45.3 Hz. (22) LFC : ξ q =.17, ωqn 2 = 8.89 Hz. (23) VI. CHATTERING REDUCTION VIA COMPENSATING FILTERS The direct introduction of the wire actuators (21)- (23) in the control system previously described causes a severe deterioration of the control performance even at very low train velocity, because of the neglected actuator dynamics [3]. The presence in the control loop of parasitic dynamics was recently shown to be one of the main causes of chattering in 2-SMC systems [7]. It was shown in [9, 2, 3] that chattering can be attenuated by shaping the open-loop frequency response W(jω) of the closed loop system with the suboptimal algorithm as the nonlinear relay element. The describing function (DF) method was the main underlying tool of analysis in the above works. The frequency-based analysis which motivates the W ( jω ) 2-SMC algorithm Fig. 3. The feedback control system compensator design requires a LTI plant dynamics. That is why the LTV pantograph/catenary dynamics is approximated by means of the nominal (or averaged) model, i.e. the catenary parameters m c (t), b c (t), k c (t) in the characteristic matrix A(t) are replaced by their nominal mean values m c, b c and k c. The analysis of the resulting transfer function was made in [3]. It was shown there that a linear low-pass filter located at the input of the wire-actuator can be an effective precompensation shaping filter. Considering the parameter values reported in the Appendix a firstorder compensator H c (s) with time constant.1s was derived, together with an additional measurement filter H m (s), with time constant.1s, located at the controller input in order to filter out the high-frequency measurement noise: H c (s) = H m (s) = s s (24) (25) The overall proposed control system, including the subsystem for the estimation of the contact force, can be therefore summarized by the block-scheme in Fig. 4. The main aspects underlying the separation between the suboptimal 2-SMC controller and the estimator are now investigated. Let η λ = λ ˆλ be the contact force estimation error. The main point is that the estimation error tends to zero in finite time [23, 24]. The finite time vanishing of the estimation error is guaranteed as long as the Lipschitz conditions (9), (12) relevant to the differentiators tuning are not violated. During the transient of the differentiators, the contact force differs from the actual value by a uniformly bounded error η λ that can be considered as a bounded and vanishing measurement noise (see [6] c 28 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

6 6 Asian Journal of Control, Vol., No., pp. 1 9, Month 28 f qff FEED-FORWARD PRE-LOADING FORCE H c ( s) v c WIRE-ACTUATOR f c CATENARY-PANTOGRAPH SYSTEM x 2, x 3 COMPENSATOR SUB-OPTIMAL 2-SMC + 1N CONTACT FORCE SET-POINT H m ( s) MEASUREMENT FILTER λˆ f c CONTACT FORCE ESTIMATION ALGORITHM Fig. 4. Control system block scheme in the UFC case for the analysis of the robustness properties in the nonvanishing case). Due to the finite time convergence of the estimate, the performance of the scheme that directly measures the contact force is restored in a finite time. A residual permanent estimation error will always exist in practice (e.g., due to errors in the upper frame parameters) that causes a bounded residual error for the regulation mismatch σ = λ λ [6]. VII. SiMULATION RESULTS The overall UFC and LFC systems, including the compensators and the measurement filters, have been simulated in the Matlab-Simulink environment by considering the mechanical parameters reported in the Appendix and the wire-actuator parameters given in (21)-(23). The train is kept at rest for 2 seconds, during which the nominal contact force is attained, and then a constant acceleration, according to the velocity profile reported in the Figure 5-left, is applied. Such a velocity profile is far from a real situation but it allows for verifying the system behaviour at all speeds. The corresponding position is shown in the Figure 5-right. measurements, as suggested in the present paper, or by measuring it directly. The upper and lower frame positions, to be differentiated in real time in order to derive the contact force estimate, have been corrupted by random measurement noises of magnitude less than.5mm. The measured contact and control forces (the latter is needed for estimation purposes) are corrupted by measurement errors less than 2N. In TEST 1 and TEST 2, the UFC case was considered. TEST 1 refer to the block scheme in Fig. 4, i.e. the contact force is estimated by the suggested method (14), (8), (11) with the differentiator parameters (1), (13) and the Lipschitz constants L 2 = L 3 = 1. In TEST, 2 the noisy measurement of contact force is taken for grant. The suboptimal controller parameters are set as β c =.9, U c = 3. The comparison between the plots in Figure 6 points out that the estimation of the contact force causes only a slight detriment in the regulation accuracy as compared to the case in which the contact force is measured directly. Figure 7-left shows the contact force estimation error for TEST 1, while Figure 7-right show the control force f c generated by the wire-actuator. The train velocity profile [km h 1 ] Train position [m] The contact force [N] The contact force [N] Fig. 5. The velocity and position profiles used in all simulations. Comparative tests have been made both by estimating the contact force via frame position Fig. 6. The actual contact force in TEST 1 (left plot) and TEST 2 (right plot) The comparative analysis between the closed-loop control systems with the estimated or measured contact c 28 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

7 A. Pisano and E. Usai: Contact force estimation and regulation in active pantographs 7 1 Contact force estimation error [N] 5 The feedback control force f cfb (t) [N] Contact force estimation error [N] 3 2 The feedback control force f qfb Fig. 7. TEST 1. The contact force estimation error (left plot) and the feedback component of the control force (right plot). Fig. 9. TEST 3. The contact force estimation error (left plot) and the feedback component of the control force (right plot). force has been performed also the LFC case, with the suboptimal controller parameters set as β q =.99 and U q = 3. This analysis is made in TEST 3 (estimated contact force) and TEST 4 (measured contact force). In Fig. 8 the actual contact force profiles in TEST 3 and TEST 4 are compared. It can be seen that the performance deterioration is much more significant as compared with the UFC case. Indeed, the differentiation of the frame positions (especially the lower one) is much more problematic in the LFC case. This is due to the fact that the control acting on the lower frame has more effort and less smoothness degree than the control acting on the upper frame, which acts closer to the pantograph/caterary contact point. This statement can be checked by comparing the right plots of Figure 7 and Figure 9. As expected, the force estimation error in the LFC case is larger, as it clearly follows by comparing Fig.7-left and Fig.9-left. controller parameters β q =.99, U q = 4 and ε = 1.5. A steady-state error component is present on the contact force, as it can be observed in Figure 1-left. This is reasonable since the adopted sat-based control is a simple proportional one in a neighborhood of the sliding manifold. In the right plot a zoom of the contact force versus the train position is shown, while the train is moving across three spans. The contact-force oscillation has a lower-frequency component, due to the towers, and an higher-frequency one due to the droppers encountered. Furthermore, the sat-based implementation of the control law leads to significant improvements in the smoothness of the control force (Figure 11, left) and in the accuracy of the contact force estimation (Figure 11, right). 2 The contact force [N] Contact force [N] vs. train position [m]. A zoom The contact force [N] 2 The contact force [N] Train position [m] 5 Fig. 1. TEST 4. Left plot: the contact force in the whole control time interval. Right plot: a zoom in the time interval [7, 8 ] s. Fig. 8. The actual contact force in the TEST 3 (left plot) and TEST 4 (right plot). In order to reduce the significant chattering which is seen in Figure 8-left, a modification is made to the 2-SMC controller. The sign function appearing in (18) has been replaced with the sat-approximation which smooths out the discontinuity, i.e. sign(σ) sat(σ; ε) = σ/( σ + ε), with ε being a positive coefficient. After some trial-and-error, satisfactory performance has been found by means of the triple of Finally, the plot of the frames and catenary height positions are shown VIII. CONCLUSIONS The problem of contact force estimation and regulation in active train pantographs with wire-type actuators has been addressed and solved by combining second-order sliding-mode control techniques, used to design the controller, and an algebraic observability c 28 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

8 8 Asian Journal of Control, Vol., No., pp. 1 9, Month Contact force estimation error [N] The feedback control force f qfb (t) [N] 3 Fig. 11. TEST 4. The contact force estimation error (left plot) and the feedback component of the control force (right plot) Catenary vertical displacement [m] Upper frame vertical displacement [m] Lower frame vertical displacement [m].5 Fig. 12. TEST 4. The catenary and pantograph frames vertical displacement approach to estimate the contact force. The algebraic observability approach entails the estimation of the lower and upper frame velocity and acceleration, which has been provided by using sliding-mode differentiators. The presented simulation results highlighted that the proposed estimator can be combined very effectively with the 2-SM controller in the UFC case, while it suffers of some problem in the LFC case. Nevertheless, also in the LFC case the obtained accuracy of the closed-loop system can be considered acceptable for the considered application. The possibility of measuring directly the vertical acceleration ẍ 2, instead of estimating it by double numerical differentiation, will be investigated in next works since accelerometers are cheap and accurate sensors that could be sensibly used in the present context. IX. ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the FP7 European Research Projects PRODI - Power plants Robustification by fault Diagnosis and Isolation techniques, grant no REFERENCES 1. Allotta B., Papi M., Pugi L., Toni P., Violi A.G., Experimental campaign on a Servo-Actuated Pantograph, Proc. 21 IEEE/ASME Int.l Conf. on Advanced Intelligent Mechatronics, vol.1, pp , B. Allotta B., A. Pisano, L. Pugi, E. Usai, VSC of a servo-actuated ATR9-type pantograph, Proc. of the 44th Conference on Decision and Control CDC 25, Siviglia, December M. Arnold, B. Simenon Pantograph and catenary dynamics: a benchmark problem and its numerical solution, Appl. Num. Math., 34, pp , Balestrino A., Bruno O., Landi A., Sani L., Innovative Solutions for Overhead Catenary-Pantograph Systems: Wire Actuted Control and Observed Contact Force, I Vehicle System Dynamics, 33, pp , G. Bartolini, A. Ferrara, A. Levant, and E. Usai, On Second Order Sliding Mode Controllers, in Variable Structure Systems, Sliding Mode and Nonlinear Control, K.D. Young and U. Ozguner Eds. Berlin: Springer-Verlag, Lecture Notes in Control and Information Sciences,vol. 247, 1999,pp bibitembfpuijc G. Bartolini, A. Pisano, E. Punta and E. Usai, A survey of applications of second order sliding mode control to mechanical systems, Int. J. Control, vol. 76, 9/1, pp , G. Bartolini, A. Pisano, E. Usai An improved second-order sliding mode control scheme robust against the measurement noise, IEEE Trans. on Automatic Controll, vol. 49, n. 1, pp , Boiko, I., Fridman, L., & Castellanos, M.I. (24). Analysis of second order sliding mode algorithms in the frequency domain. IEEE Transaction on Automatic Control, 49 (6), Boiko, I., (23). Frequency Domain Analysis Of Fast And Slow Motions In Sliding Modes. Asian Journal of Control, 5 (4), I. Boiko, Frequency-domain design of compensating filters for sliding mode control systems, Proc. 9-th Int. Workshop on Variable Structure Systems VSS 6, Alghero, Italy, 5-7 June Boiko I., Fridman L., Iriarte R., Pisano A. Usai E. Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators, Automatica, vol. 42, pp , Cannas B., Cincotti S., Usai E., An algebraic observability approach to chaos synchronisation by sliding differentiators, IEEE Trans. Circuit and Systems-I: Fundamental Theory and Applications, vol. 49, no. 7, pp. 1-16, July Cannas B., Cincotti S., Usai E., A chaotic modulation scheme based on algebraic observabilty c 28 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

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