International Journal of Control, Automation, and Systems (2015) 13(3):1-8 DOI 10.1007/s12555-014-0097-1 ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555 Transverse Vibration Control of Axially Moving Web Systems by Regulation of Axial Tension Quoc Chi Nguyen*, Thanh Hai Le, and Keum-Shik Hong Abstract: In this paper, an active control scheme to suppress the transverse vibrations of an axially moving web system by regulating its transport velocity to track a desired profile is investigated The spatially varying tension and the time-varying transport velocity of the moving web are inter-related The system dynamics includes the equations of motion of the moving web and the dynamics of the drive rollers at boundaries of the web span The two roller motors provide control torque inputs to the web system The strategy for vibration control is the regulation of the axial tension in reference to a designed profile, so that the vibration energy of the moving web system decays The designed profile for the axial tension is designed via the total mechanical energy of the axially moving web system The Lyapunov method is employed to derive the model-based torque control laws ensuring that the transverse vibration and the velocity tracking error converge to zero exponentially The effectiveness of the proposed control scheme is demonstrated via numerical simulations Keywords: Axially moving string, boundary control, exponential stability, Lyapunov method, tension regulation INTRODUCTION There are various industries that use web-material transport systems such as papers, textiles, metals, polymers, and composites In these systems, the application of roll-to-roll (R2R) processing yields better performance and allows a mass production with highspeed automation However, the mechanical vibrations (particularly in the transverse direction) of the web material have been the main quality- and productivitylimiting factor, especially in high-speed precision R2R systems Therefore, reduction of the transverse vibrations in R2R systems has become an important research area To solve the vibration problems in R2R systems, boundary control (the application of control actions at the left or right boundary) has been developed [1-16] Since the provision of control inputs through a supporting roller is more cost effective than the addition of an extra actuator at a middle point in the system, boundary control is an efficient method to control the Manuscript received February 24, 2014; revised July 22, 2014; accepted August 26, 2014 Recommended by Associate Editor Won-jong Kim under the direction of Editor Fuchun Sun This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.04-2012.37 and by Vietnam National University HoChiMinh City (VNU-HCM) under grant number C2013-20-01 Quoc Chi Nguyen and Thanh Hai Le are with the Department of Mechatronics, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet, District 10, Ho Chi Minh City, Vietnam (emails: {nqchi, lthai}@hcmut.edu.vn) Keum-Shik Hong is with the School of Mechanical Engineering, Pusan National University, Busandaehak-ro, Geumjeonggu, Busan 609-735, Korea (e-mail: kshong@pusan.ac.kr) * Corresponding author © ICROS, KIEE and Springer 2015 R2R systems The tension and speed control problems of web handling systems have been an important research subfield [17-25] An integrated boundary control scheme that stabilizes the axial tension and the transport speed at desired set-points for axially moving materials was introduced in [17] Koc et al [18] developed an Hinfinity robust control strategy including varying control gains for unwind/rewind sections to improve the robust control property under the effects of variations on the radii of unwind/rewind rollers Meanwhile, Pagilla et al [19] proposed a decentralized control scheme for a web processing line based on a dynamic model taking variations on the radii and inertia of unwind/rewind rollers into account In [21], a multivariable H-infinity control scheme is used to independently regulate the speed and tension and to reject the disturbance due to radius variations of unwind/rewind rollers for a web span Using an iterative learning control (ILC) algorithm, Zhao and Rahn [22] presented a control scheme for an axially moving system that enables a precise regulation of the tension and transport velocity It is noted that the majority of aforementioned results were obtained under the assumption that the web tension and the transport velocity were constant In practice, when the set-points of tension and speed are changed during operation, the variations of the transport velocity and the tension can result in the transverse vibrations of the moving material [2,12] and consequently the stability of the R2R systems Transverse vibration control for axially moving systems was studied in [1-16]: Vibration control schemes for axially moving strings include [1-11] Those for axially moving beams include [12-16] Fung et al [1] developed a boundary control scheme for an axially moving string system in which adaptive boundary Quoc Chi Nguyen, Thanh Hai Le, and Keum-Shik Hong control laws were employed on a mass-damper-spring mechanism to suppress the vibrations Nguyen and Hong [7] proposed a robust adaptive boundary control scheme for an axially moving string of unknown system parameters under spatially varying tension and unknown boundary disturbance Li and Rahn [11] introduced an adaptive isolation scheme for an axially moving system, which were divided into two spans by a transverse force actuator, to reduce the transverse vibration of the controlled span to zero under bounded disturbances in the uncontrolled span These achievements show that the vibration control problem has been intensively investigated by researchers However, there are few researches that investigate vibration control problem together with other control problems, e.g., tension control and speed control, etc The first work that studied a combination of transverse vibration control and speed control for an axially moving web was presented in [2], where the transport speed was regulated by two control torques applied to the drive rollers at boundaries, and the transverse vibration was suppressed via an actuator in the middle of the web span Recently, Nguyen and Hong [8] developed an active control scheme that can suppress both longitudinal and transverse vibrations and to drive the transport velocity of the string simultaneously In this paper, we develop an active control scheme that suppresses the transverse vibrations while regulating the transport speed of an axially moving web system It is noted that the investigated control problem differs from the work [2] In [2], the transverse vibration was suppressed by control forces exerted from a mechanical guide located at the middle of the web span In this paper, the vibration control strategy is to regulate the axial tension to track a designed profile according to which the vibration energy is dissipated This control strategy is implemented using two control torques applied to the rollers at boundaries These control torques also play the role of control inputs in making the transport velocity In this formulation, the web dynamics and the roller dynamics are coupled The Lyapunov energy-based method is employed to derive the torque control laws ensuring that the transverse vibration and the speed tracking error converge to zero exponentially Finally, the numerical simulations are provided to confirm the effectiveness of the proposed control method Contributions of this paper are: First, a novel active control scheme for suppressing the transverse vibration of an axially moving web is presented: Contrary to the conventional methods (boundary or distributed controls) that use external forces, the proposed control method regulates the axial tension of the web Henceforth, an unexpected damage on the web surface due to external forces can be prevented Second, the proposed control method is able to provide vibration suppression and transport velocity tracking simultaneously Third, the exponential stability of the proposed control system is achieved The remainder of this paper is organized as follows Section introduces a dynamic model of the considered system (an axially moving viscoelastic string), in which a nonlinear partial differential equation (PDE) governs the transverse displacement, and the boundary conditions determine the dynamics of the rollers Section presents the proposed control scheme design A Lyapunov function-based stability analysis of the closed-loop control system and the proof of exponential stability also are discussed Section includes numerical simulation results that illustrate the effectiveness of the proposed control scheme Finally, Section draws conclusions PROBLEM FORMULATION Fig shows the schematic of the axially moving web driven by two rollers at boundaries The left boundary is fixed in the sense that the movement of the web in the vertical direction is restricted Conversely, at the right boundary, a damper mechanism allows its transverse (vertical) movement In this paper, the moving web is modeled as an axially moving string Let t be the time, x the spatial coordinate along the longitude of motion, w(x,t) the transverse displacement at the spatial coordinate x and time t, l the distance between two fixed rollers, ρ the mass per unit length of the web, cv the viscous damping coefficient of the web, ca the damping coefficient of the damper, and v(t) the timevarying transport velocity T(x,t) denotes the spatially varying tension of the web Two parameters of the rollers include the moment of inertia J and the radius r In this paper, the partial derivatives are denoted as follows: ()t = ∂ / ∂t and () x = ∂ / ∂x For notational convenience, instead of wx (x,t) and wt (x,t), wx and wt will be used; other similar abbreviations will be employed subsequently The equations of motion of the axially moving string are given as follows, see [2] ρ ( wtt + vwx + 2v(t ) wxt + v wxx ) (1) − (Twx ) x + cv ( wt + vwx ) = with boundary conditions wt (0, t ) = 0, (2) (ca − ρ v) wt (l , t ) + (T (l , t ) − ρ v ) wx (l , t ) = (3) The initial transverse displacement and velocity, respectively, are given as y ca w(x,t) x J T1 r τ1 τ2 J r l Fig The schematic of the axially moving web T2 Transverse Vibration Control of Axially Moving Web Systems by Regulation of Axial Tension w( x, 0) = h( x), wx ( x, 0) = (4) The velocity dynamics is obtained as 2J ⎛ ⎜ ρ lr + r ⎝ ⎞ ⎟ v(t ) = τ1 (t ) + τ (t ) − (T1 (t ) − T2 (t ))r , ⎠ (5) where T1(t) and T2(t) are the time varying tensions from the respective adjacent spans The distributed web tension is defined as τ1 (t ) Jv + T1 + r r (6) x ⎛ τ1 (t ) + τ (t ) Jv(t ) ⎞ + ⎜ − − T1 (t ) + T2 (t ) ⎟ , l⎝ r r2 ⎠ T ( x, t ) = T0 − where T0 is the axial tension of the undisturbed web Subsequently, the following equations are obtained a1 (τ1 + τ ) + (T1 (t ) − T2 (t ))(a2 − a3 ) Tx ( x, t ) = la3 Tt ( x, t ) = − , τ1 (t ) ( Jl + xa1 )(τ1 + τ ) + + T1 (t ) r a3 xa (T (t ) − T1 (t )) + , la3 (7) (8) where kv is a positive control gain Utilizing (14), we obtain 2J ⎛ ⎜ ρ lr + r ⎝ ⎞ ⎟ e(t ) = −kv e(t ) ⎠ (15) Assumption 1: It is assumed that the desired speed, the time derivatives of the desired speed, and the initial speed tracking error are bounded as follows: vd (t ) ≤ ξ1 , vd (t ) ≤ ξ , e(0) ≤ ξ3 , T1 (t ) ≤ ξT1 , T2 (t ) ≤ ξT2 (17) Assumption 3: The axial tension of the undisturbed web T0, the material tension in the adjacent span T2, and the damping coefficient of the damper ca are assumed to be sufficiently large such that the following inequalities hold T0 > kvξ3 (3 J + a1 )ξ + r r2 + J + r a1 + ra2 + ρ rl (ξT1 + ξT2 ) ra3 a2 = Jr , (9) (10) a3 = ( ρ lr + J )r ⎛ J + ρ 2lr ρξ1 Ja3 +⎜ + + ⎜ a r r kv ⎝ (11) + (ca + ρ r 2ξ1 + kv )(ξ1 + ξ3 ), The control objective is to suppress the transverse vibration of the moving web while regulating the transport velocity so that the desired profile is tracked Based on the Lyapunov energy-based method, a control scheme employing the control torques τ1(t) and τ2(t) is derived to achieve the exponential convergence of the transverse vibration and the speed tracking error to zero The speed tracking error is defined as follows: e(t ) = v(t ) − vd (t ), (12) where vd (t) is the desired speed of the moving web Differentiating (12) with respect to time and substituting (5) into (12), we arrive at 2J ⎛ ⎜ ρ lr + r ⎝ ⎞ ⎟ e(t ) = τ1 (t ) + τ (t ) − (T1 (t ) − T2 (t ))r ⎠ 2J ⎞ ⎛ − vd (t ) ⎜ ρ lr + ⎟ r ⎠ ⎝ (18) ⎞ ⎟⎟ e(0) ⎠ ca > ρ (ξ1 + ξ3 ) CONTROL DESIGN (16) where ξi (i = 1, 2,3) are positive constants Assumption 2: The time varying tensions T1(t) and T2(t) are assumed to be bounded as follows: where a1, a2, and a3 are defined as a1 = ρ lr , (19) The vibration energy of the moving web [1,12] is given as E (t ) = l l ρ ( wt + v(t ) wx )2 dx + ∫ Twx2 dx ∫ 2 (20) Applying the material derivative operator d / dt = ∂ / ∂t + v∂ / ∂x, the time derivative of E(t) is obtained as follows l E (t ) = ∫ ( wt + vwx )( wtt + vwx + 2vwxt + v wxx )dx l + ∫ Twx ( wxt + vwxx )dx + l (Tt + vTx )wx2 dx ∫0 l = ∫ ( wt + vwx )((Twx ) x − cv ( wt + vwx ))dx (13) Based on the structure of (13), the input control torques are designed to satisfy the following equation 2J ⎞ ⎛ τ1 (t ) + τ (t ) = ⎜ ρ lr + ⎟ vd (t ) + (T1 − T2 ) r − kv e(t ), r ⎠ ⎝ (14) l (Tt + vTx )wx2 dx, ∫0 (21) where (1) has been utilized Integrating by parts, the terms in (21) are further simplified as follows: l + ∫ Twx ( wxt + vwxx )dx + l l ∫ (wt (Twx ) x + Twx wxt )dx = [wt Twx ]0 , l l l ∫ wx (Twx ) x dx = [Twx ]0 − ∫ Twx wxx dx (22) (23) Quoc Chi Nguyen, Thanh Hai Le, and Keum-Shik Hong where λ is given as Utilizing (14)-(15), we obtain l E (t ) = −cv ∫ ( wt + vwx )2 dx − vT (0, t ) wx2 (0, t ) + wt (l , t )T (l , t ) wx (l , t ) + l (Tt + vTx )wx2 dx (24) ∫0 + vT (l , t ) wx2 (l , t ) Remark 1: From (24), the following notes are made (i) The viscous damping force reduces the mechanical vibration energy This can be seen from the presence of l −cv ∫ ( wt + vwx ) dx (ii) The varying tension can add energy to the web span This is justified by the last two terms of (24) On this technical basis, a control scheme that uses the two control torques at boundaries to regulate the spatially varying tension to make the time derivative of T(x,t) (i.e., Tt + vTx ) negative is now developed To stabilize the system given by the governing equation (1) and the boundary conditions (2) and (3), the following control torque laws are proposed τ1 (t ) = a3 r2 vd + J + ρ rl T1 (t ) − T2 (t ) a3 ⎛ kv r ⎛ Jr + ρ r 2l ρξ1 ⎞ + ⎜⎜ + ⎟⎟ e(0) exp ⎜⎜ − a3 r ⎠ ⎝ ⎝ a3 ⎞ t ⎟ (25) ⎟ ⎠ J + ρ rl T1 (t ) − T2 (t ) a3 ⎛ kv r ⎞ ⎛ Jr + ρ r 2l ρξ1 ⎞ t⎟ − ⎜⎜ + ⎟⎟ e(0) exp ⎜⎜ − ⎟ a3 r ⎠ ⎝ ⎝ a3 ⎠ − ρ r ξ1 (sgn(v) + + kv )v (26) The axially moving web system under the proposed control laws is a closed-loop system in that the control signals use the transport speed information Prior to the analysis of the stability of the closed-loop system, the following lemma is introduced Lemma [7]: Given u ( x, t ) : [0, l ] × ℜ+ → ℜ, if u (0, t ) = 0, then l u ( x, t ) ≤ l ∫ u x2 ( x, t )dx, l ∫0 l u ( x, t )dx ≤ l ∫ u x2 ( x, t )dx (27) (28) Theorem 1: Consider system (1) with boundary conditions (2) and (3), and speed tracking error dynamics (13) The control gain kv in control laws (25) and (26) is chosen to satisfy inequality (18) Then, (i) control laws (25) and (26) ensure that the transverse displacement of the web converges to zero exponentially in the following sense w( x, t ) ≤ 2l E (0) exp(−λ t ), Tmin (30) (ii) The transport speed tracking error converges to zero exponentially in the following sense ⎛ ⎞ kv r e(t ) = e(0) exp ⎜ − ⎟ ⎝ ρ lr + J ⎠ (31) Proof: (i) Substituting (3), (7), and (8) into (24), we obtain l E (t ) = −cv ∫ ( wt + vwx ) dx − vT (0, t ) wx2 (0, t ) ⎡ 2J ⎞ ⎛ ρ lr + ⎟ Jl + ρ lr x l ⎢ τ1 ⎜⎝ r ⎠ + ∫ ⎢− − vd 0⎢ r lr ( ρ lr + J ) ⎢ ⎣ ( + (29) ) k ( Jl + ρ 2lrx) ( Jl + ρ lrx) (T1 − T2 ) + l (vρ lr + J )2 e(t ) l ( ρ lr + J ) + ρ r ( vvd + vvd ) − ρ r vvd − + ρ r 2ξ1 (sgn(v) + + kv )v, τ (t ) = (T1 (t ) − T2 (t ))r − kv e(t ) − ⎪⎧ c ρ r ξ1kv v ⎪⎫ λ = ⎨ v , ⎬ Tmax ⎪⎭ ⎪⎩ ρ ρ rkv ρ lr + J e (t ) T1 − T2 ⎤ ⎥ wx dx ρ lr + J ρ lr + J ⎦ ⎡ ⎤ ( ρ v − T (l , t )) + ⎢vT (l , t ) + T (l , t ) ⎥ wx2 (l , t ) (32) ca − ρ v ⎣ ⎦ ρ rkv − vd e(t ) − It should be noted that the following equation is employed to obtain (32) 2J ⎛ τ1 (t ) + τ (t ) = ⎜ ρ lr + r ⎝ kv r ⎞ e(t ) ⎟ vd (t ) − ρ lr + J ⎠ (33) Substitution of (25) into (32) yields: l E (t ) ≤ −cv ∫ ( wt + vwx )2 dx − vT (0, t ) wx2 (0, t ) − ρ r 2ξ1kv v l T ( x, t ) wx2 dx Tmin ∫ − (T (l , t ) − ca v)T (l , t ) wx (l , t ) ca − ρ v (34) Since Assumptions 1~3 hold, the following inequalities are obtained E (t ) ≤ −λ E (t ), (35) which yields E (t ) ≤ E (0) exp(−λ t ) (36) Using (27) in Lemma 1, we obtain Tmin l w ≤ ∫ Twx dx ≤ E (t ) ≤ E (0) exp(−λ t ) 2l (37) From (37), (29) is proved (ii) The solution of (15) gives (31) Transverse Vibration Control of Axially Moving Web Systems by Regulation of Axial Tension 2.5 v(t) [m/s] Remark 2: The implementation of torque control laws (25) and (26) requires the measurements of v(t), T1(t), and T2(t) and exact knowledge of r, l, J, vd(t) The axial speed can be achieved with a tachometer or an encoder at the right roller The tensions T1(t) and T2(t) in the respective adjacent spans can be measured by adding tension sensors near the left and right rollers Remark 3: From (29), the boundedness of w(x,t) is obtained, and it is concluded that w(x,t) converges to zero exponentially From (31), the exponential convergence of the axial velocity v(t) to the desired speed vd (t) is achieved It is shown that the larger value of kv results in the faster convergence of v(t) to vd (t) However, the limit of kv is given by (18) Remark 4: From (34), it can be concluded that, with the proposed control laws (25) and (26), the stability of the axially moving system (1)-(3) is independent to the viscous damping and the mass per unit length desired velocity velocity in vibration control case velocity in no vibration control case 0 10 12 Time [s] 14 16 18 20 Fig Comparison of transport velocity tracking in the cases of no vibration control (red) and vibration control (blue) 0.6 (38) (39) In the second case, the torque control laws (25) and (26) are applied to suppress the transverse vibration and to drive the axial speed The control gain kv selected for the case of no vibration control is 40 Meanwhile, the control gain kv in (25) and (26) is set to satisfy the inequality (17), that is, 1.6 As shown in Fig 2, the performance of the transport speed tracking control in the case of no vibration control (red curve) has better quality than the one in the case of vibration control (blue curve): the settling time in the case of no vibration control is 1.5 s, whereas it takes s in the case of vibration control This can be explained by referring to (15) again, where the convergence speed of the speed tracking error e(t) depends on the value of kv 0.4 0.2 w(0.5l,t) [m] The finite difference method is employed to find an approximate solution for the PDE with the initial and boundary conditions given by (1)-(4) The convergence scheme is based on the central (for the string span) and forward/backward (for the left/right boundary) difference methods The system parameters used in simulation are the following: ρ = 0.7, l = 4, cv = 0.001, ca = 0.25, r = 0.2, J = 2.2, T0 = 200, T1 = 100 + 10sin(20t), and T2 = 150 + 10sin(20t) Let the initial conditions of the string be w(x,0) = 0.5sin(πx/l) and wt (x,0) = The dynamical responses of the axially moving web were simulated in two cases In the first case, no vibration control was considered, and only the damping force exerted from the damper and the viscous damping force of the web reduced the vibration energy The transport velocity was driven to track a typical velocity profile (dotted line) widely used in practice as shown in Fig In this case, the following simple control laws are used ⎞ ⎟ vd (t ) + T1 − T2 − kv e(t ), ⎠ 1.5 0.5 SIMULATION RESUTLS 2J ⎛ τ1 (t ) = ⎜ ρ lr + r ⎝ τ (t ) = -0.2 -0.4 10 12 Time [s] 14 16 18 20 Fig Transverse displacement of the web at x = l/2 in the case of no vibration control Fig shows that the transverse vibration can be suppressed if the viscous damping coefficient and the damping coefficient of the damper are sufficiently large But this type of suppression requires a great amount time: in our case, it took almost 14 s Meanwhile, the active vibration control scheme using control laws (24) and (25) obtains a good performance as shown in Fig In the acceleration period, the transverse displacement converges to zero within s When the acceleration has ended, and the constant axial speed is maintained, there is residual vibration that is also suppressed within s This also happens with the deceleration period To illustrate the robustness of the proposed control laws to the unknown system parameters, the dynamic response of the web-handling system was simulated with the assumption: The proposed control laws with the control gain kv = 1.6 is obtained by using the nominal value of the viscous damping cv = 0.001 while the actual viscous damping in the web-handling system is cv = 0.0005 As shown in Fig 5, the control laws stabilized the web-handling system The vibration suppression was achieved within s The similar simulation was carried Quoc Chi Nguyen, Thanh Hai Le, and Keum-Shik Hong 2.5 0.6 0.4 v(t) [m/s] w(0.5l,t) [m] 0.2 1.5 -0.2 0.5 -0.4 -0.6 10 12 Time [s] 14 16 18 Fig Transverse displacement of the web at x = l/2 in the case of vibration control 0.4 0.2 2.5 -0.2 -0.4 -0.6 10 12 Time [s] 14 16 18 20 1.5 10 12 Time [s] 14 16 18 20 Fig Transverse displacement at x = l/2: The difference between the nominal value (used in the control laws) and the actual value of the viscous damping is 0.0005 0.6 0.4 0.2 -0.2 -0.4 -0.6 Fig Comparison of transport velocity tracking with the system parameter uncertainties of the viscous damping (red) and the mas per unit length (blue) v(t) [m/s] w(0.5l,t) [m] 0.6 w(0.5l,t) [m] 0 20 10 12 Time [s] 14 16 18 20 Fig Transverse displacement of the web at x = l/2: The difference between the nominal value (used in the control laws) and the actual value of the mass per unit length is 0.3 out, where the unknown mass per unit length was assumed The nominal value of mass per unit length is 0.7 while the actual value is With the control law 0.5 0 10 12 Time [s] 14 16 18 20 Fig The transport velocity tracks the desired velocity profile (three levels of velocity) using the nominal value of the mass per unit length, it took s for the vibration suppression, see Fig The two simulation results shown in Figs 5-6 illustrate the robustness of the proposed control laws to the variations of the viscous damping and the mass per unit length This is consistent with the theoretical point inferred in Remark Since the convergence of the speed tracking error depends on the control gain kv and the mass per unit length ρ (see (31)), the difference between the nominal value and the actual value of the viscous damping does not affect to the performance of the speed tracking control (red curve in Fig 7) The larger mass per unit length made slower convergence of the speed tracking error (blue curve in Fig 7) The velocity profile (dotted line) including three levels (as shown in Fig 8) was used to verify the effectiveness of the proposed control laws As shown in Fig 8, the transport velocity (solid line) tracks the desired velocity profile Fig shows the convergence of the transverse vibration Through these simulation results, the effectiveness of the proposed control scheme was verified Transverse Vibration Control of Axially Moving Web Systems by Regulation of Axial Tension 0.6 w(0.5l,t) [m] 0.4 [7] 0.2 [8] -0.2 -0.4 -0.6 10 12 Time [s] 14 16 18 20 [9] Fig Transverse displacement at x = l/2 in the case of the desired velocity profile (dotted line) in Fig CONCLUSION [10] In this paper, an active control scheme was developed for the transverse vibration suppression and the transport speed tracking of an axially moving web system The control scheme employed two control torques applied to the rollers at boundaries The basis of the proposed control strategy for vibration suppression was the regulation of the axial tension to dissipate the vibration energy The torque control laws were designed based on the original PDE model and Lyapunov energy-based method The exponential convergence of the transverse vibration and the speed tracking error to zero were achieved It is concluded that the proposed control scheme can provide a viable solution for web-handling systems, in which the vibration control is required to be considered together with speed 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Systems, vol 11, no 6, pp 1138-1148, 2013 Quoc Chi Nguyen received his B.S degree in Mechanical Engineering from Ho Chi Minh City University of Technology, Vietnam, in 2002, an M.S degree in Cybernetics from Ho Chi Minh City University of Technology, Vietnam, in 2006, and a Ph.D degree in Mechanical Engineering from the Pusan National University, Korea, in 2012 Dr Nguyen was a Marie Curie FP7 postdoctoral fellow at the School of Mechanical Engineering, Tel Aviv University, from 2013 to 2014 He has been a faculty member in the Department of Mechatronics Engineering, Ho Chi Minh University of Technology since 2002 Dr Nguyen’s current research interests include MEMS control, nonlinear systems theory, adaptive control, and distributed parameter systems Thanh Hai Le received his B.S degree in Mechatronics Engineering from Ho Chi Minh City University of Technology, Vietnam, in 2003, an M.S degree in Biomechatronic Engineering from SungKyunKwan University, Korea, in 2007, and a Ph.D degree in Bio-mechatronic Engineering from the SungKyunKwan University, Korea, in 2011 He has been a faculty member in the Department of Mechatronics Engineering, Ho Chi Minh University of Technology since 2011 Dr Le’s current research interests include nonlinear systems theory, robotics, and image processing Keum-Shik Hong received his B.S degree in Mechanical Design and Production Engineering from Seoul National University in 1979, an M.S degree in Mechanical Engineering from Columbia University, New York, in 1987, and both an M.S degree in Applied Mathematics and a Ph.D degree in Mechanical Engineering from the University of Illinois at Urbana-Champaign (UIUC) in 1991 Dr Hong served as Editor-in-Chief of the Journal of Mechanical Science and Technology (2008-2011), and served as an Associate Editor for Automatica (2000-2006), and as Deputy Editor-in-Chief for the International Journal of Control, Automation, and Systems (2003-2005) He also served as General Secretary of the Asian Control Association (2006-2008) Dr Hong was Organizing Chair of the ICROS-SICE International Joint Conference 2009, Fukuoka, Japan His laboratory, Integrated Dynamics and Control Engineering Laboratory, was designated as a National Research Laboratory by the MEST of Korea in 2003 Dr Hong received various awards including the Presidential Award of Korea (2007) for his contributions in academia Dr Hong’s current research interests include brain-computer interface, nonlinear systems theory, adaptive control, distributed parameter systems, autonomous systems, and innovative control applications in brain engineering ... The solution of (15) gives (31) Transverse Vibration Control of Axially Moving Web Systems by Regulation of Axial Tension 2.5 v(t) [m/s] Remark 2: The implementation of torque control laws (25)... Through these simulation results, the effectiveness of the proposed control scheme was verified Transverse Vibration Control of Axially Moving Web Systems by Regulation of Axial Tension 0.6 w(0.5l,t)... Vibration, vol 330, no 20, pp 4676-4688, 2011 Q C Nguyen and K.-S Hong, Transverse vibration control of axially moving membranes by regulation of axial velocity,” IEEE Trans on Control Systems