Characteristics of active and passive dancers: A comparative study

Transcription

1 Control Engineering Practice 14 (2006) Characteristics of active and passive dancers: A comparative study Ramamurthy V. Dwivedula, Yongliang Zhu, Prabhakar R. Pagilla School of Mechanical and Aerospace Engineering, Oklahoma State University, 218 Engineering North, Stillwater, OK 74078, USA Received 24 October 2003; accepted 13 February 2005 Available online 5 April 2005 Abstract A substantial number of web process lines use dancer mechanisms for attenuating web tension disturbances. Although passive dancer mechanisms are commonly used in web process lines, recently there has been a growing interest in active dancer mechanisms due to their ability to attenuate disturbances over a wide range of frequencies. The focus of the paper is on a comparative study of active and passive dancer mechanisms used in web processing machines for tension disturbance attenuation. Active and passive dancers are compared using simplified analytical models. To substantiate the analysis, results from experiments are shown and discussed. r 2005 Elsevier Ltd. All rights reserved. Keywords: Web processing; Web tension control; Active dancers; Passive dancers; Experiments 1. Introduction Any material in flexible, continuous strip form is called a web. Examples of webs include paper, plastic film, fabric, and sheet metals. Many consumer products are processed in the form of web at some stage of their production. The quality of the finished product is strongly affected by the web tension in the machine direction of the web transport system, winders, and processing sections. Fig. 1 shows an outline sketch of a typical web process line. Operations such as coating, printing, and drying are performed in the processing section. Attenuation of tension disturbances entering the process section is an important aspect in printing and converting industry. The ever increasing range of substrates used in today s packaging industry emphasizes the need for accurate tension control. Controlling tension within a tight tolerance zone is essential for precise registration between the tool and the product in Corresponding author. Tel.: ; fax: address: pagilla@ceat.okstate.edu (P.R. Pagilla). printing and converting machines. A commonly used tension control strategy is to measure the tension in a span upstream to the process section and use this signal as a feedback for the drive motor. Early development of mathematical models for longitudinal dynamics of a web can be found in Campbell (1958), Grenfell (1963), King (1969), Brandenburg (1977). An overview of lateral and longitudinal dynamic behavior and control of moving webs was presented in Young and Reid (1993). A review of the problems in tension control of webs can be found in Wolfermann (1995). Considerations in the selection of a dancer roller or load cell based tension control approach are discussed in Carlson (2001), Ebler, Arnason, Michaelis, and Sa (1993), Shelton (1999). The dancer subsystem shown in Fig. 1 may be used as a tension measurement device or as a device to attenuate tension disturbances as discussed in Reid and Lin (1993). When the dancer subsystem is used as a tension measurement device, the displacement of the dancer roller is measured and the tension in the span is inferred from the measured displacement. In contrast, when the dancer subsystem used as a device for attenuating tension disturbances, the dancer subsystem is designed to absorb tension /$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi: /j.conengprac

2 410 ARTICLE IN PRESS R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) Nomenclature A B f b E F J K i k L i M OPG cross-sectional area of web bearing friction viscous friction coefficient at the dancer roller modulus of elasticity force exerted by the actuator polar moment of inertia of roller web span spring constant ðea=l i Þ spring constant of the spring loading on the dancer mechanism length of the ith web span mass of the dancer roller offset-pivoted guide (displacement guide) R T i ðtþ T i ðsþ t i ðtþ t r UðtÞ UðsÞ V i ðtþ V i ðsþ v i ðtþ v r X ðtþ X i ðsþ radius of a roller change in tension from the reference Laplace transform of T i ðtþ tension in span reference web tension dancer translational velocity input Laplace transform of UðtÞ change in web velocity from the reference Laplace transform of V i ðtþ velocity of web on ith roller reference web velocity change in linear displacement of the dancer roller from the reference Laplace transform of XðtÞ variations. The focus in this paper is on the use of a dancer subsystem as a device to attenuate web tension disturbances. Fig. 2 shows a schematic of a typical dancer subsystem. The dancer subsystem can be classified into two types: (i) an active dancer, and (ii) a Unwind roll Dancer subsystem or load cell roller Process section Fig. 1. Outline of a typical web processing line. Rewind roll passive dancer. In an active dancer, an actuator is used to control the translational velocity of the dancer roller based on the tension measured by a load cell mounted on a roller downstream to the dancer roller. It is expected that an active dancer will offer a control engineer with an additional flexibility in controlling tension within tighter tolerance zones as discussed in Rajala (1995, 1997), Pagilla, Dwivedula, Zhu, and Perera (2001, 2004), Pagilla, Dwivedula, and Zhu (2002). Passive dancers are widely used for tension disturbance attenuation. Although almost all web process lines use some sort of a passive dancer mechanism, a clear understanding of how a passive dancer can or cannot reject tension disturbances is not fully investigated. In the case of a passive dancer mechanism, the dancer roller is free to move, generally about a pivot or From supply roller T 0 Web motion V 0 T 1 V 2 T 3 To takeup roller T 0 V 0 T 1 V 2 T 3 L 1 T 2 L 2 T 2 Mg T 1 Mg T 2 V 1 V 1 X b k bx. F kx (a) Schematic diagram (b) Freebody diagram Fig. 2. A typical dancer subsystem. Dancer subsystem is classified as an Active Dancer if the dancer roller is driven by an actuator (represented by a rectangle with an arrow inside it) otherwise it is a Passive Dancer.

3 R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) on linear slides. Instead of an actuator, a counter pressure from an air cylinder is used in Ebler et al. (1993) to avoid continuous motion of the dancer roller for small changes in the tension. This counter pressure acts as the tension set point. A variation in the tension causes movement of the dancer roller and this displacement, in turn, causes a change in the tension. Additional passive elements such as a spring and damper may be used to ensure contact between the dancer roller and the web and also to increase the speed of response of the dancer mechanism as discussed in Shin, Reid, and Kwon (1995). Extensive literature is available on control schemes (e.g., Reid & Lin, 1993; Shin et al., 1995; Lin & Lan, 1993; Shin, 1991) using both active dancer rollers and load cells. A generalized dynamic model for the dancer subsystem is presented in Reid and Lin (1993) wherein an example of the inertia compensated dancer subsystem is illustrated; it is demonstrated that an inertia compensated dancer mechanism acts as a lowpass filter and that a spring-loaded dancer mechanism normally responds to tension variations faster than a free roller. Effects of the PID gains for controller with a dancer mechanism are investigated in Lin and Lan (1993); a comparison of the effects of the individual gains in various controllers using P-control, PD-control, PI-control and PID-control is also presented. Modeling and control of active dancer mechanisms can be found in Pagilla et al. (2001, 2002, 2003), Rajala (1995, 1997), McDow and Rahn (1998). These and the other representative literature did not focus on the relative merits and demerits of active mechanism and passive dancer mechanisms. The aim of this work is to compare and contrast active and passive dancer mechanisms with respect to their tension disturbance attenuation capability. The remainder of the paper is organized as follows. Dynamic models of the active and passive dancers are given and discussed in Section 2. Benefits and limitations of the active and passive dancers are discussed in Section 3. Experimental study conducted on two different web process lines, one at the university and the other at a web handling industry (Fife Corporation, Oklahoma City), is described in Section 4. Section 5 gives conclusions and potential future research on the topic. 2. Dynamic models This section presents the dynamic models of the active and passive dancer subsystems. In Brandenburg (1972), Whitworth and Harrison (1983), Shelton (1986) beginning from the nonlinear model of span dynamics, linearized models are developed and from the linearized models, input output models are developed. Complete details about the development of the span tension dynamics, derived using the principle of conservation of mass, and the results of the system identification validating the linearized models may be found in Pagilla et al. (2004). The next two subsections present the derivation of input output models of active dancer subsystem and passive dancer subsystem Active dancer subsystem Schematic of a typical dancer subsystem is shown in Fig. 2. The design of this subsystem is generic in the sense that it can be included as a subsystem anywhere in the web process line where precise tension regulation is required. The nonlinear dynamics is linearized around a reference web tension ðt r Þ and a reference web velocity ðv r Þ: Defining the deviations T i ¼ t i t r and V i ¼ v i v r for i ¼ 0; 1; 2; 3; the linearized dynamics of the active dancer subsystem may be obtained as b _V 0 ¼ gv 0 þðt 1 T 0 Þ, (1) t 1 _T 1 ¼ T 1 þ T 0 þ aðv 1 V 0 Þþ a t 1 X þ au, (2) b _V 1 ¼ gv 1 þðt 2 T 1 Þ, (3) t 2 _T 2 ¼ T 2 þ T 1 þ aðv 2 V 1 Þþa 1 1 X þ au, t 2 t 1 (4) b _V 2 ¼ gv 2 þðt 3 T 2 Þ, (5) _X ¼ U, (6) where b ¼ J=R 2 ; g ¼ B f =R 2 ; a ¼ EA=v r ; t 1 ¼ L 1 =v r ; and t 2 ¼ L 2 =v r : Note that in deriving these equations, all the rollers in the dancer subsystem are assumed to be of equal radii and equal inertias. The input/output model for the active dancer subsystem can be derived by considering the translational velocity of the dancer roller (U) as the control input and tension ðt 2 Þ at the roller immediately downstream of the dancer roller as the measured output. Taking the Laplace transform of Eqs. (1) (6) and simplifying, the Laplace transforms of tensions, T 1 ðsþ and T 2 ðsþ; are obtained as T 1 ðsþ ¼G 1 ðsþt 0 ðsþþg 2 ðsþt 2 ðsþþg 3 ðsþxðsþ, (7) T 2 ðsþ ¼G 4 ðsþt 1 ðsþþg 5 ðsþt 3 ðsþþg 6 ðsþxðsþ, (8) where the transfer functions G 1 ðsþ through G 6 ðsþ are given by Eq. (A.1) in the Appendix. Substituting the expression for T 1 ðsþ from (7) into (8) and simplifying results in the following input/output model under the assumption that bearing friction is negligible ðg ¼ B f =R 2 0Þ: T 2 ðsþ ¼ D adðsþ C ad ðsþ UðsÞþA adðsþ C ad ðsþ T 0ðsÞþ B adðsþ C ad ðsþ T 3ðsÞ, (9)

4 412 ARTICLE IN PRESS R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) where LðT i ðtþþ9t i ðsþ; i ¼ 0; 1; 2; 3; LðUðtÞÞ9UðsÞ; Z9b=a; and A ad ðsþ ¼ðZs þ 1Þ 2, (10) B ad ðsþ ¼Zsðt 1 s þ 1Þþ2, (11) C ad ðsþ ¼Z 2 t 1 t 2 s 4 þ Z 2 ðt 1 þ t 2 Þs 3 þ ZðZ þ 2t 1 þ 2t 2 Þs 2 þ 3Zs þ 3, ð12þ D ad ðsþ ¼bZt 1 s 3 þ bz 1 þ t 1 s 2 t 2 þ b 3 þ Z s þ b 2 1. ð13þ t 2 t 2 t 1 Notice that, if t 2 42t 1 ; i.e., L 2 42L 1 ; then the constant term of the polynomial D ad ðsþ is negative, which results in a zero in the right-half of the complex plane. This fact forms an important constraint on the implementation of the active dancer, which will be discussed later Passive dancer subsystem In this section, a complete model of the passive dancer followed by a simplified model is presented; the simplified model will be used to analyze the action of the passive dancer in rejecting web tension disturbances. As a first step, notice that if the dancer roller is passive (that is, no external actuator is used to position the dancer roller, F ¼ 0; see Fig. 2), the equation governing the dynamics of the dancer roller is given by m X þ b _X þ kx ¼ T 1 ðtþþt 2 ðtþ. (14) Taking the Laplace transform of the above equation results in 1 XðsÞ ¼ ðms 2 þ bs þ kþ ðt 1ðsÞþT 2 ðsþþ. (15) It may be noticed here that Eqs. (7) and (8) are applicable for the case of passive dancer also; in the case of the active dancer XðsÞ in Eqs. (7) and (8) is controlled independently by an actuator, whereas, for the passive dancer XðsÞ is given by (15). Define b ¼ b M ; k ¼ k M b and ¼ b M : Substituting (15) into (7) and (8) and simplifying, T 1 ðsþ ¼H 1 ðsþt 0 ðsþþh 2 ðsþt 2 ðsþ, (16) T 2 ðsþ ¼H 3 ðsþt 1 ðsþþh 4 ðsþt 3 ðsþ, (17) where the transfer functions H 1 ðsþ through H 4 ðsþ are given in Eq. (A.2) in the Appendix. Substituting the expression for T 1 ðsþ given by (16) into (17), T 2 ðsþ is written as T 2 ðsþ ¼ A pdðsþ C pd ðsþ T 0ðsÞþ B pdðsþ C pd ðsþ ðsþt 3ðsÞ. (18) The polynomials A pd ðsþ; B pd ðsþ are of sixth order and the polynomial C pd ðsþ is of eighth order. These polynomials are listed in (A.3) in the Appendix. The coefficients in these polynomials depend on the mechanical features of the dancer and the properties of the web material. This, in turn, indicates that, in the case of a passive dancer, the behavior of the tension T 2 is solely determined by the mechanical features of the passive dancer (the mass of the dancer roller M, the spring constant k, and the viscous friction constant b), and the properties of the web material (Young s modulus E and the crosssectional area of the web A). Eqs. (9) and (18) are similar in structure except for the term involving the dancer velocity input, UðsÞ; in the active dancer input/output model. It is important to note that setting UðsÞ ¼0 in the active dancer input/ output model will not make the Eqs. (9) and (18) identical. Since UðsÞ is the dancer translational velocity, setting UðsÞ ¼0 would mean that the dancer roller is arrested at a particular position and thus acts as an idle roller. Though the linearized model of the passive dancer, presented in Eq. (18), is a fairly accurate model, it is not clear as to how one would conduct analysis to highlight the characteristics of the passive dancer using this model. In specific, the dynamics of the passive dancer is characterized by six independent parameters, viz., b; k; b; t 1 ; t 2 ; and Z: Thus, to analyze the system using classical techniques, such as frequency response technique, each combination of the six parameters need to be considered to arrive at meaningful conclusions about the performance of the dancer. Such an analysis is cumbersome and overwhelming. To bring out the inherent features of the passive dancer, a simple, intuitive explanation of the passive dancer is needed. Consequently, to illustrate the tension disturbance attenuation features of the passive dancer, a simplified model of the passive dancer is developed in the following. Fig. 3 shows a free-body diagram of the passive dancer roller. Assuming that the web is mostly elastic and obeys Hooke s law, the spans adjacent to the passive dancer can be represented as springs with stiffness K 1 and K 2 ; respectively; K 1 ¼ EA=L 1 and K 2 ¼ EA=L 2 : The net tension in span 1 is the sum of the d K 1 X K 2 X 1 d 2 Mg Span 1, t 1 Span 2,t 2 kx. bx X M Ẋ. Fig. 3. Free-body diagram of the passive dancer roller.

5 R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) Mg 1 X e K 1 + K + 2 Ms 2 + bs + k + - Fig. 4. An interpretation of the action of passive dancer tension control system. spring force due to dancer displacement, K 1 X; and any disturbance, d 1 ; entering the span, so t 1 ¼ K 1 X þ d 1 : Similarly, t 2 ¼ K 2 X þ d 2 : Notice that if the material of the web is visco-elastic, then the force in the span may be modeled as K 1 X þ B v _X where the damping B v reflects the viscosity of the web material. The effect of the air pressure is modeled as a spring force kx and a damping force b _X: From the free-body diagram, the equation of motion of the passive dancer is obtained by a force balance in the vertical direction as M X þ b _X þ kx ¼ Mg e, e9t 1 þ t 2 ¼ðK 1 þ K 2 ÞX þ w, w9d 1 þ d 2. ð19þ The reference value of e is obtained as the sum of tensions t 1 and t 2 at the reference point: e r ¼ t r þ t r ¼ 2t r : Eq. (19), is represented by a control block diagram shown in Fig. 4, where w denotes the tension disturbance and X denotes the displacement of the passive dancer roller around the equilibrium point. Equilibrium point here means that the position of the dancer roller when there are no disturbances, that is, when wðtþ ¼0: The tension variation around the equilibrium point is given by EðtÞ9e e r ¼ t 1 þ t 2 2t r ¼ðt 1 t r Þþðt 2 t r Þ¼T 1 ðtþþt 2 ðtþ: The passive dancer attempts to offer a reaction force to minimize EðtÞ whenever disturbance wðtþ appears. From the block diagram, and using Eqs. (19), it can be seen that Ms 2 þ bs þ k EðsÞ ¼ Ms 2 WðsÞ þ bs þ k þ K 1 þ K fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 G passive ðsþ K 1 þ K 2 þ Ms 2 Mg, þ bs þ k þ K 1 þ K fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 G mass ðsþ ð20þ where EðsÞ9LðeðtÞÞ and WðsÞ9LðwðtÞÞ are the Laplace transforms of eðtþ and wðtþ; respectively. Eq. (20) characterizes the control action of the passive dancer for tension disturbance attenuation and highlights many important features of the passive dancer. The next section compares passive and active dancers. w 3. Active and passive dancers: benefits and limitations From the simplified block diagram of the passive dancer, the following inferences may be made: At the equilibrium point, the sum of the tensions in the two adjacent spans of the dancer roller is given by K 1 þ K 2 Mg, (21) k þ K 1 þ K 2 which must be equal to two times the reference tension t r : Thus, this relationship shows that the mass of the dancer roller cannot be chosen arbitrarily small. Using Eq. (20), define the frequency response of the disturbance to output transfer function r as Ms 2 þ bs þ k rðoþ ¼ Ms 2 þ bs þ k þ K 1 þ K. () 2 s¼jo A small value of r means that the dancer subsystem has better tension disturbance attenuation properties. From Eq. (), it can be observed that, if the tension disturbance is a low frequency signal, the passive dancer can reduce the disturbance by a factor of r ¼ k=ðk þ K 1 þ K 2 Þ: Consider two extreme cases. (1) k!1; the passive dancer behaves like a fixed roller. Eq. () indicates that r ¼ 1 and so, no tension disturbance attenuation can be achieved. (2) k! 0 and b! 0; the passive dancer behaves like a floating mass; such passive dancers are termed inertia compensated dancers and are currently used in many existing web processing lines in industry. In this case, the transfer function between the disturbance ðwðsþþ and the tension ðeðsþþ is given by G passive ðsþ Ms 2 Ms 2 þ K 1 þ K 2. (23) It can be seen that (23) represents an undamped second-order system. For a low frequency tension disturbance, r ¼ lim o!0 jg passive ðjoþj ¼ 0; this implies that the disturbance is completely attenuated. However, the drawback of no damping is that the dancer will exhibit sustained oscillations even when subjected to a small tension disturbance. The value of r is small when K 1 þ K 2 is large; this can be seen from Eq. (). Thus, the passive dancer has better tension attenuation capability for high modulus web materials since both K 1 and K 2 are large. To make r small, that is to improve tension disturbance attenuation, k should be chosen small. But pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the resonant frequency of the dancer roller is ðk þ K 1 þ K 2 Þ=M: To avoid resonance, the resonant frequency should be high when compared to the frequency of potential tension disturbances. Consequently, the passive dancer should be designed with large k and/or small M. However, to achieve large

6 414 ARTICLE IN PRESS R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) disturbance rejection, small k is required. Hence, there is a trade-off between the ability to attenuate tension disturbances and resonance avoidance. The frequency response, rðoþ; is approximately one in the high-frequency region. Thus, the passive dancer does not have any tension disturbance attenuation capability at high frequencies. Contrary to a passive dancer, an active dancer can have different controller implementations with the same mechanical structure. Hence, it can be expected to provide tension disturbance attenuation over a wide range of frequencies. However, the following issues are critical in the design and implementation of the active dancer subsystem: From the dynamics given in Eq. (9), it may be noted that, if L 2 42L 1 ; the transfer function from UðsÞ to T 2 ðsþ has a non-minimum phase zero (zero in the right-half of the complex plane); if L 2 42L 1 ; t 2 42t 1 and the constant term in the polynomial D ad ðsþ; given by Eq. (13), becomes negative. It is well known that non-minimum phase zeros impose limitations on the performance of the system Morari and Zafiriou (1989). Thus, 2L 1 must be greater than L 2 for effective tension disturbance attenuation by the active dancer. The design constraint, which was derived from the analytical model, has an intuitive, physical interpretation. Assuming that the web is mostly elastic, the spans upstream and downstream to the dancer roller may be modeled as springs with spring constants, K 1 ¼ EA=L 1 and K 2 ¼ EA=L 2 ; respectively, as shown in Fig. 5. If L 1 4L 2 ; then K 1 ok 2 ; and any motion of the dancer roller results in larger tension variations in span 2 than in span 1. Thus, rejection of periodic disturbances from the spans upstream of the dancer roller into the spans downstream of the dancer roller is possible in this case. If L 1 ol 2 ; then K 1 4K 2 ; periodic dancer motion induces larger tension disturbances into span 1 than it rejects in span 2 due to feedback of tension T 2 : The tension disturbances injected into span 1 by the Loadcell Span 1 K 1 = EA/L 1 K 2 = EA/L 2 Span 2 Dancer Fig. 5. Interpretation of the effect of span lengths for an active dancer. dancer motion are propagated to span 2 by the transport of the web. Thus, when L 1 ol 2 ; the control action by the dancer results in creating more disturbances. Though the PID controller is versatile and simple to implement, it may not be adequate for control of tension while processing extremely thin materials at very high speeds. Advanced control approaches may have to be employed. Advanced control approaches often need more feedback signals. For the active dancer, the measurements of the dancer actuator velocity or the dancer roller position or both should be available for designing feedback loops in the event of implementation of more advanced controllers for the active dancer. The frequency range over which tension disturbances can be attenuated by using an active dancer mechanism depends on the bandwidth, o B ; of the actuator. An actuator with lower bandwidth may not respond to tension disturbances at frequencies higher than its bandwidth. In theory, periodic tension disturbances up to a frequency of o B may be attenuated by using the active dancer. However, practical considerations and accurate control of tension require that the bandwidth of the actuator be much above the frequency of potential disturbances. Actuators with higher bandwidth and higher load capability obviously cost more. In situations, such as processing at very high speeds, where accurate tension control is critical, the cost of using a high bandwidth actuator may be justified. An active dancer mechanism almost invariably involves a gearing and/or other mechanism to convert the rotary movement of the actuator to the translational movement of the dancer roller. This mechanism usually introduces nonlinearities such as backlash and friction. Such nonlinearities complicate the control algorithm. Tension disturbances in web process lines often occur due to the constructional and/or operational features of the web process line such as the misalignment of rollers and web velocity variations. Such imperfections result in periodic web tension disturbances. The frequency of the tension disturbances typically increases with increase in the web transport velocity. Thus, in high speed web lines, the passive dancer mechanism is not effective in attenuating tension disturbances. However, an active dancer mechanism, subject to the limitation of the bandwidth of the actuator, can attenuate these tension disturbances. Also, if a priori knowledge of the web process line and the nature of the tension disturbances is available, it is possible to design a control algorithm to attenuate the tension disturbances to a greater extent using the same mechanical structure of the dancer mechanism.

7 R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) Active dancer control This section discusses the three controllers implemented for active dancer control. The controllers are: (1) proportional-integral (PI) controller, (2) internal model controller (IMC), and (3) self-tuning controller (STC). Each one of these controllers have their own advantages. The PI controller is an industrial standard. Many industries use this controller because of its simplicity. Because the tension disturbances are often periodic, the IMC works well when the frequency of the disturbances is known. Self-tuning controllers find their place in situations where the process or its environment is changing continuously or at times. These systems are difficult to control if there are no mechanisms to detect these changes and adjust the controller parameters to achieve the control goal. The advantages of the STC are the automation of the tuning of the controller gains and its versatility. Fig. 6 shows the control block diagram for the hydraulic active dancer subsystem with an inner position loop and an outer tension loop and outer loop. The inner loop employs a PI controller while the controllers for the outer loop are the PI, IMC, and STC. Digital implementation of each controller is discussed in the following subsections PI controller One of the extensively used controllers in industry is the PI controller and variations of it. The discrete-time version of the PI controller is UðkÞ ¼K p EðkÞþK i T s X k j¼1 EðjÞ, (24) where T s is the sampling period, UðkÞ is the control input and EðkÞ is the error signal. The gains K p and K i can be tuned appropriately to obtain the desired performance IMC controller IMC is most effective when some properties about the disturbance are partially or completely known. Typically, this controller can be used to reject periodic disturbances whose frequency is known to the designer. The derivation of the internal model of the disturbance is given as follows. Full development of the IMC controller may be found in Morari and Zafiriou (1989). When the disturbance is in the form, dðtþ ¼A d sinðyt þ fþ, (25) where A d and f are unknown, but y is known approximately. The control objective is to select G c ðsþ (the controller) to eliminate the disturbances ðdðtþþ from the output. The following controller G c ðzþ ¼ K p þ z 1 K imc sinð$t s Þ 1 2z 1 cosð$t s Þþz 2 (26) with $ ¼ y; would attenuate the disturbance of the form given in (25). In Eq. (26) K p is the proportional gain and K imc is used to compensate for the amplitude of the periodic disturbance. With the choice of G c ðzþ as given in 26, G c ðzþ ¼1 when z ¼ cosðyt s Þþj sinðyt s Þ: The control algorithm for real-time implementation is UðkÞ ¼K p EðkÞþEðk 1Þ½K imc sinð$t s Þ 2K p cosð$t s ÞŠ þ K p Eðk 2Þ þ 2Uðk 1Þ cosð$t s Þ Uðk 2Þ. ð27þ 4.3. Self-tuning controller A brief description of the STC is given in this section. Full development of the direct self-tuning controller may be found in Astrom and Wittenmark (1995). Consider the process dynamics, the reference model and the controller given by (28) (30): Process Model: AðzÞyðkÞ ¼BðzÞuðkÞ, (28) Reference Model: A m ðzþyðkþ ¼B m ðzþu c ðkþ, (29) Controller: RðzÞuðkÞ ¼TðzÞu c ðkþ SðzÞyðkÞ, (30) Ram Position Feedforward Inner Loop Disturbance Reference Tension Tension + + Control Position Control Hydraulic Actuator + + Web Dynamics Actual Tension Position Sensor Load Cell Fig. 6. Active dancer control block diagram.

8 416 ARTICLE IN PRESS R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) where uðkþ is the output of the controller and yðkþ is the tension read by the load cell. It is possible to reparametrize the process dynamics in terms of the controller parameters using the reference model. Eqs. (28) (30) imply that yðkþ ¼ B mðzþ ½RðzÞuðkÞþSðzÞyðkÞŠ þ ðkþ, (31) A m ðzþtðzþ which can be written as yðkþ ¼j T ðkþy þ ðkþ, (32) where ðkþ ¼yðkÞ y m ðkþ; j T ðkþ is a vector of regression variables, and y is the vector of the parameters. The algorithm for a direct self-tuning controller can be written as Data: Specification in terms of A m ðzþ; B m ðzþ and d. Step 1: Estimate the coefficients of RðzÞ and SðzÞ using least squares methods on Eq. (32). Step 2: Compute the control using Eq. (30). Repeat steps 1 and 2 at each sampling period. 5. Experimental study To highlight the performance of the active dancer and the passive dancer, experiments were conducted on two web process lines: (i) an endless experimental platform, and (ii) an experimental platform with unwind/rewind stands. The experimental platforms are briefly described here followed by the results of the experiments Endless experimental platform A sketch of the open-architecture experimental web platform together with an active dancer system is shown in Fig. 7. The web platform consists of an endless web line with a number of rollers, a passive dancer, an active dancer system, and web guides for maintaining lateral position. The term endless web line refers to a web line without unwind and rewind rolls. This type of platform mimics most of the features of a process section of a web process line. The passive dancer system has a Bellofram super cylinder with a bore of 5.84 cm (2.3 in) and a stroke of 4.57 cm (1.8 in). This type of passive dancer system is commonly used in web process lines to attenuate tension disturbances. Air pressure in the pneumatic cylinder of the passive dancer system is used to set the desired reference web tension. The active dancer system consists of an electro-mechanical actuator and a guide way with the dancer roller mounted on it. This actuator has relatively low bandwidth (around 7 Hz). Sensors available on the experimental web platform (see sketch in Fig. 7) include load cells A and B to measure the web tension, tachometer to measure the velocity of the active dancer actuator, and an edge sensor to measure the lateral position of the web downstream of the guide system Fife experimental platform with unwind/rewind stands A picture of the Fife web process line is shown in Fig. 8. This machine has seventeen rollers including the guide rollers and the unwind/rewind rolls. Lateral sensors next to the unwind and rewind stands measure the lateral position of the web coming out of the unwind roll and going into the rewind roll. Based on these measurements, the unwind and rewind stands have the capability to displace laterally to provide lateral correction to the web. The lateral guiding motors are controlled by Fife CDP-01 controllers. Besides the lateral guiding at the unwind and rewind rolls, a displacement guide and a remotely pivoted guide are used to provide lateral correction at two intermediate locations as shown in Fig. 8. Reference tension in the web line is maintained by applying braking action on the unwind roll shaft. The braking torque is computed using the feedback from the loadcell located immediately downstream of it. Edge Sensor Passive Dancer Guide System Load Cell A Web Uneven Roll Surface Piston Air Piston Load Cell B Active Dancer Electric Motor Master Speed Roller Nip Roller Fig. 7. Sketch of the endless experimental web platform at OSU.

9 R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) Fig. 8. Picture of the unwind/rewind web process line at Fife Corporation. E M41 M4 OPG M3 OPG M5 D F H M1 I M6 G A M11 C Steering Guide M2 B OSU Addition FIFE Test Machine Sensors A Fife SE 31 edge sensor B Magpowr TS 150 FC load cell C Fife SE 17 infrared edge sensor D SE 34 infrared edge sensor E Fife SE 26 optical edge sensor F Magpowr US 2 radius sensor G Cleveland Kidder loadcell H Fife SE 31 edge sensor I MTS LDT sensor Motors/ Drives M1 Unwind roll motor M11 Lateral motor for unwind roll M2 Steering guide motor M3 OPG motor M4 Rewind roll Motor M41 Lateral Motor for rewind roll M5 Motor for OPG M6 Parker hydraulic cyllinder Fig. 9. Schematic of the unwind/rewind web process line at Fife Corporation. Similarly, the rewind roll is driven through a clutch to coordinate with the braking action of the unwind roll. The brake and the clutch are controlled by Magpowr Digitrac and Versatec controllers, respectively. A detailed layout of the web line and the details of the sensors and the actuators used in the set up are given in Fig. 9. To conduct experiments with the active dancer, the Fife web process line is augmented with an active dancer module as shown in Fig. 9. This schematic shows two

10 418 ARTICLE IN PRESS R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) Fig. 10. Picture of the active dancer module. main components in the web process line: (a) the original web process line (marked as Fife Test Machine ), and (b) an active dancer module incorporated as an extension (marked as OSU Addition ). A picture of the active dancer module is shown in Fig. 10. The active dancer module used in this process line consists of a displacement guide and an active dancer system. An electro-hydraulic actuator is used in the active dancer system. This actuator has larger bandwidth than that of the electro-mechanical actuator used in the endless experimental platform. This experimental platform does not have a passive dancer subsystem. Sensors available in this platform include edge sensors (A, C, D, E, H in Fig. 8) to measure the lateral position of the web, load cells (B, G) to measure web tension, a radius sensor (F), and a displacement sensor (I) to measure the displacement of the dancer roller Results from the endless experimental platform The performance of the passive and active dancers when they are subjected to tension disturbances at various frequencies is compared. The tension disturbances are generated by forming an uneven roll surface on a roller upstream to the active/passive dancer roller as shown in Fig. 7. The fundamental frequency of the tension disturbance generated is proportional to the web speed. Three sets of experiments were conducted at different web transport velocities: (a) Both passive dancer and active dancer rollers are fixed. (b) Active

11 R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) without dancer 13.3 without dancer (a) FFT (N) with passive dancer 13.3 with passive dancer (b) FFT (N) with active dancer 13.3 with active dancer (c) Time (sec.) FFT (N) Frequency (Hz) Fig. 11. Tension measured by load cell A (endless web platform). The fundamental frequency of the tension disturbance is around 0.74 Hz. dancer roller is fixed and only passive dancer is used for attenuating the tension disturbance, and (c) passive dancer roller is fixed and only active dancer is used for attenuating the tension disturbance. In each set of experiments, two load cells (A and B) measure the tension disturbance entering the process line. Reference tension in all experiments is N (8.2 lbf). Fig. 11 shows the tension measured by load cell A at the web speed of m/s (56 fpm). The top two plots show the measured tension and the FFT of the deviation of measured tension from the mean tension without attenuation by either passive or active dancer. The middle two plots show attenuation obtained by using passive dancer only. The bottom two plots show attenuation obtained by using active dancer only. From the figure, it can be seen that the fundamental frequency of the tension disturbance at m/s (56 fpm) is around 0.74 Hz. Passive dancer attenuated the tension disturbance at 0.74 Hz by around 28% and active dancer by around 75%. Notice that, since the tension disturbances are generated by an out-of-round roll surface, an increase in the web speed will result in an increase in the frequency of the tension disturbance. Fig. 12 shows the tension measured by load cell A when the web is running at m/s (200 fpm). The fundamental frequency of the tension disturbance is around 2.7 Hz. From this figure, it is seen that there is not much tension disturbance attenuation by passive dancer. This verifies the analysis of the passive dancer made earlier, that is, passive dancer can attenuate lowfrequency tension disturbances, but not high-frequency tension disturbances. Although active dancer exhibited better result at low frequency (0.74 Hz) than passive dancer, at high frequency, similar performance was not seen. This can be attributed to the fact that the electromechanical actuator used in the active dancer has a very low bandwidth (around 7 Hz). It is expected that an actuator with a higher bandwidth can reject tension disturbances to a greater extent. To verify whether actuator with higher bandwidth can attenuate tension disturbances occurring at higher frequencies, experiments were conducted using an electro-hydraulic actuator which has larger bandwidth. Further, the experiments are conducted on a web platform with unwind/rewind stands, which eliminates the possibility of back propagation of tension in the endless web line as the same web is transported repeatedly in the endless web line. The results of these experiments are shown in the next section Results from the experimental platform with unwind/ rewind stands Experiments were conducted at different line speeds and two different reference tensions. The tension disturbances were created by using an uneven surface roller in the process line. The fundamental frequency of the tension disturbance is determined by the web travel speed. The tension signals from load cell (G) located

12 420 ARTICLE IN PRESS R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) without dancer 13.3 without dancer (a) FFT (N) with passive dancer 13.3 with passive dancer (b) FFT (N) with active dancer 13.3 with active dancer FFT (N) (c) Time (sec.) Frequency (Hz) Fig. 12. Tension measured by load cell A (endless web platform). The fundamental frequency of the tension disturbance is around 2.7 Hz. immediately downstream of the active dancer roller were acquired using a data acquisition system. The three controllers, PI, IMC, STC, discussed in Section 4 were used for controlling the active dancer. Figs. 13 and 14 show a representative sample of the performance of the three controllers (PI, IMC and STC) at a reference tension of.72 N (15 lbf) and web speed of m/s (300 fpm) and 3.5 m/s (700 fpm), respectively. Fig. 13 shows the results of the experiments in which the active dancer is switched on at around 10 s. Results from all three controllers show good tension disturbance reductions. Similar results can be seen in Fig. 14 when the web travel speed was about 3.5 m/s (700 fpm). The results of the experiments indicate that the active dancer is able to reject tension disturbances by around 30 60%. The results of the controllers for the reference tension of.72 N (15 lbf) and N (20 lbf) at different web speeds are summarized in Fig. 15. More experimental results can be found in Pagilla et al. (2002). Experimental results from the endless web platform at the Web Handling Research Center at Oklahoma State University, and Fife web process line at Fife Corporation, Oklahoma City, indicate that the active dancer rejected tension disturbances effectively. The bandwidth of the actuator in the active dancer is a very important consideration in designing an active dancer. A higher bandwidth actuator can reject higher frequency tension disturbances. Whereas, the passive dancer is effective only in the region of low frequencies. 6. Conclusions In this paper, analytical and experimental study is undertaken to compare the performance of passive dancer and active dancer in attenuating tension disturbances. A simplified model of the passive dancer is developed (Eq. (19)). Analysis of the model predicts that a passive dancer is effective in lowfrequency region. The active dancer has the capability of tension regulation over wider frequency regions subject to the limitation of the bandwidth of the actuator. Also, the active dancer mechanism allows for implementation of advanced control approaches to achieve better tension disturbance attenuation with the same mechanical structure. However, there is a constraint on the design of the active dancer, namely, the length of the web span upstream to the dancer roller must be greater than half the length of the web span downstream to the dancer roller. Use of an actuator with higher bandwidth and higher load capability may be justified by the requirement for control of tension in a tight tolerance zone. This paper did not investigate the effect of the sensor (mainly load cell) characteristics on the tension control. Aspects of the sensor used as the feedback for active dancer controller, such as, response time, the location in the process line, and the bandwidth, are other important considerations that need to be taken into account.

13 R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) PI Controller Reference Tension =.72 N Internal Model Controller Self Tuning Controller Fig. 13. Tension disturbance attenuation (Fife web process line). The fundamental frequency of the tension disturbance is around 5.5 Hz. Time(s) PI Controller Reference Tension =.72 N Internal Model Controller Self Tuning Controller Time(s) Fig. 14. Tension disturbance attenuation (Fife web process line). The fundamental frequency of the tension disturbance is around 7.5 Hz.

14 4 ARTICLE IN PRESS R.V. Dwivedula et al. / Control Engineering Practice 14 (2006) Percentage Tension Reduction Reference Tension:.72 N IMC Controller Self Tuning Controller PI Controller H 1 ðsþ ¼ H 2 ðsþ ¼ H 3 ðsþ ¼ H 4 ðsþ ¼ ðzs þ 1Þðs 2 þ bs þ kþ ððzsðt 1 s þ 1Þþ2Þðs 2 þ bs þ kþ bsðs þ 1 ÞÞ, ðs 2 þ bs þ kþ bsðs þ 1 Þ ððzsðt 1 s þ 1Þþ2Þðs 2 þ bs þ kþ bsðs þ 1 ÞÞ, ðzs þ 1Þðs 2 þ bs þ kþþ bsðs þ Þ ððzsðt 2 s þ 1Þþ2Þðs 2 þ bs þ kþ bsðs þ ÞÞ, ðs 2 þ bs þ kþ ððzsðt 2 s þ 1Þþ2Þðs 2 þ bs þ kþ bsðs þ ÞÞ.ðA:2Þ Percentage Tension Reduction Hz 5.5 Hz 7.5 Hz 13.5 Hz 15.5 Hz Reference Tension: N IMC Controller PI Controller Self Tuning Controller A pd ðsþ ¼ððZsþ1Þðs 2 þ bs þ kþþ bsðs þ ÞÞ ðzs þ 1Þðs 2 þ bs þ kþ, B pd ðsþ ¼ððZsðt 1 s þ 1Þþ2Þðs 2 þ bs þ kþ bsðs þ 1 ÞÞ ðs 2 þ bs þ kþ, C pd ðsþ ¼C 1 pd ðsþc2 pd ðsþ C3 pd ðsþ, C 1 pd ðsþ ¼ððZsðt 1s þ 1Þþ2Þðs 2 þ bs þ kþ bsðs þ 1 ÞÞ, C 2 pd ðsþ ¼ððZsðt 2s þ 1Þþ2Þðs 2 þ bs þ kþ bsðs þ ÞÞ, C 3 pd ðsþ ¼ððs2 þ bs þ kþþ bsðs þ 1 ÞÞ ððzs þ 1Þðs 2 þ bs þ kþþ bsðs þ 1 ÞÞ. ða:3þ 3.5 Hz 5.5 Hz 7.5 Hz 13.5 Hz 15.5 Hz Fig. 15. Summary of tension disturbance attenuation (Fife web process line). Acknowledgements We thank Fife Corporation, Oklahoma City, for giving access to their unwind/rewind web line to conduct experiments for this study, and also for assisting us to run the experiments. We gratefully acknowledge the support of the US National Science Foundation under the CAREER Grant No. CMS and the Web Handling Research Center, Oklahoma State University. Appendix. Transfer functions Define 1 ¼ 1=t 1 ; and ¼ð1=t 2 1=t 1 Þ: ðbs þ g þ aþ G 1 ðsþ ¼ ððbs þ gþðt 1 s þ 1Þþ2aÞ, a G 2 ðsþ ¼ ððbs þ gþðt 1 s þ 1Þþ2aÞ, aðbs þ gþðs þ 1 Þ G 3 ðsþ ¼ ððbs þ gþðt 1 s þ 1Þþ2aÞ, ðbs þ g þ aþ G 4 ðsþ ¼ ððbs þ gþðt 2 s þ 1Þþ2aÞ, a G 5 ðsþ ¼ ððbs þ gþðt 2 s þ 1Þþ2aÞ, aðbs þ gþðs þ Þ G 6 ðsþ ¼ ððbs þ gþðt 2 s þ 1Þþ2aÞ. ða:1þ References Astrom, K. J., & Wittenmark, B. (1995). Adaptive control. Reading, MA: Addison-Wesley. Brandenburg, G. (1972). The dynamics of elastic webs threading a system of rollers. Newspaper Techniques, Brandenburg, G. (1977). New mathematical models for web tension and register error. In Proceedings of the third international IFAC conference on instrumentation and automation in the paper, rubber and plastics industry (vol. 1, pp ). Campbell, D. (1958). Dynamic behavior of the production process, process dynamics. New York: Wiley. Carlson, D. H. (2001). Considerations in the selection of a dancer or load cell based tension regulating strategy. In J. Good (Ed.), Proceedings of the sixth international conference on web handling (pp ). Ebler, N. A., Arnason, R., Michaelis, G., & Sa, N. D. (1993). Tension control: Dancer rolls or load cells. IEEE Transactions on Industry Applications, 29(4), Grenfell, K. (1963). Tension control on paper-making and converting machinery. In Proceedings of the ninth IEEE annual conference on electrical engineering in the pulp and paper industry (pp ). King, D. (1969). The mathematical model of a news paper press. Newspaper Techniques, 3 7(Interim Report). Lin, P., & Lan, M. (1993). Effects of PID gains for controller design with dancer mechanism on web tension. In J. Good (Ed.), Proceedings of the second international conference on web handling (pp. 76). McDow, B. C., & Rahn, C. D. (1998). Adaptive web-tension control using a dancer arm. Tappi Journal, 81(10), Morari, M., & Zafiriou, E. (1989). Robust process control. Englewood Cliffs: Prentice-Hall. Pagilla, P. R., Dwivedula, R. V., & Zhu, Y. -L. (2002). The role of active dancers in tension control of webs. Tech. rep., Oklahoma State University, Stillwater, WHRC Project

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