A study on control of accumulators in web processing lines 1

Transcription

1 A study on ontrol of aumulators in web proessing lines Prabhakar R. Pagilla, Inderpal Singh, and Ramamurthy V. Dwivedula Abstrat: Design of a ontrol algorithm for web tension regulation in an aumulator, in web proessing lines, is onsidered in this paper. A simplified average dynami model that inludes aumulator arriage dynamis, average web tension dynamis in aumulator web spans, and the proess and exit side driven roller dynamis is onsidered in the study. A feedbak ontroller together with an observer for web tension is proposed. It is shown that the proposed feedbak ontroller results in a stable losed-loop system. Simulation results on an industrial ontinuous web proessing line are given and disussed for the proposed ontroller/observer and ompared with a urrently used industrial ontroller. Appliations of this researh an be found in all ontinuous web proess lines. Introdution Any ontinuous material whose wih is signifiantly less than its length and whose thikness is signifiantly less than its wih an be desribed as a web. Plasti wrap, paper, film, and aluminum strip are examples of web. It is important that the tension in a web span be maintained within a lose tolerane band during the proessing of a web. For example, if the tension in the web hanges during printing/perforating proesses, the print (perforation) gets skewed. Further, exessive tension variations may ause wrinkles and may even tear the web. As the demand for higher produtivity and better performane from the web proessing industry inreases, better models and more aurate ontrol algorithms for the proesses must be developed. Tension ontrol plays a key role in improving the quality of the finished web. It is essential to keep the web in the proess at a preset tension, whih ould hange throughout the proess by many onditions suh as disturbanes from uneven rollers and web speed variations. A ontinuous web proessing line is a large-sale omplex interonneted dynami system with numerous ontrol zones to transport the web while proessing it. A ontinuous web proessing line typially onsists of an entry setion, a proess setion, and an exit setion. The entry setion onsists of an unwind stand, a tension leveller, and an entry aumulator. Operations suh as wash, oat, and quenh on the web are performed in the proess setion in the ase of aluminum and steel webs, and printing, perforating, and laminating in the ase of other onsumer produts. The exit setion onsists of an exit aumulator and a rewind stand. A typial proess line layout of an aluminum strip proessing This work was supported by the ational Siene Foundation under Grant CMS The authors are with the Shool of Mehanial and Aerospae Engineering, Oklahoma State University, Stillwater, OK. Corresponding author pagilla@eat.okstate.edu. line is shown in Figure. Aumulators are primarily used to allow for rewind or unwind ore hange while the proess ontinues at a onstant veloity. Dynamis of the aumulator diretly affets the behavior of web tension in the entire proess line. Tension disturbane propagation both upstream and downstream of the aumulator has been notied due to motion of the aumulator arriage. Entry Setion Tension Leveler A Bridle B Bridle Unwind Entry Aumulator Carriage 2Bridle Proess Exit Aumulator Exit Setion Setion Carriage Wash,Coat, Quenh, et., 3Bridle 4Bridle Rewind Figure : Typial web proess line layout and terminology. Although suffiient amount of work has been done [, 2, 3, 4,, 6, 7] and is urrently being done in tension ontrol of a web but very little published researh exists in modelling and ontrol of aumulators in web proessing lines. In [], a mathematial model for longitudinal dynamis of a web span between two pairs of pinh rolls, whih are driven by two motors, is given; this model does not predit tension transfer and does not onsider tension in the entering span. A modified model that onsiders tension in the entering span was developed in [3]. In [4], the moving web was onsidered as a moving ontinuum and general methods of ontinuum mehanis were used in the development of a mathematial model. An overview of the lateral and longitudinal behavior and ontrol of moving webs was presented in []. A review of the problems in tension ontrol of webs an be found in [6]. Disussions on tension ontrol versus strain ontrol and torque ontrol versus veloity ontrol were given in [7]. Aumulators in web proessing lines onstitute an important element in all of the web handling mahines. Funtional importane of these in web proessing lines is quite substantial as they are primarily responsible for ontinuous operation of web proessing lines. A preliminary study on modelling and ontrol of aumulators is given in [8]; harateristis of an aumulator and its operation are explained; throughout the paper disussions are arried out to gain more insight into the dynami behavior of the aumulator arriage, web spans, and the urrent methods used in ontrolling the arriage. A dynami model for aumulator spans that onsider the time-varying nature of the span length was developed in [9]. In this paper, we onsider ontrol of the aumulator arriage in onjuntion with ontrol of the driven rollers both upstream and downstream of the aumulator. The average /3/$7. 23 IEEE 3684 Denver, Colorado une 4-6, 23

2 dynami model developed in [8] is further simplified based on pratial observations and is used for ontroller design. The design of the ontrol algorithm is arried out based on Lyapunov s seond method. An observer for estimating the average web tension in aumulator web spans is also developed in the proess. Simulation results on an industrial ontinuous web proess line for a typial operation of an aumulator are onduted using the proposed ontroller and observer; these results are ompared with the simulation results of the urrent ontrol tehniques that are used in the industry. The remainder of the paper is organized as follows. Setion 2 gives the dynamis of the aumulator arriage, tension in the web spans, and driven rollers upstream and downstream of the aumulator, together with a disussion on obtaining simplified average tension model in the entire aumulator. Controller design and observer design are given in Setion 3; stability of the proposed ontroller/observer is also shown. Simulation results on an industrial web proess line are given and disussed in Setion 4. Conlusions and future researh are given in Setion. omenlature A Area of ross setion of the strip B j ;B fe ;B fp Coeffiient of visous frition E Young s modulus of the material F d Disturbing fore on the arriage F p ;F r Fores on either side of the arriage g Aeleration due to gravity Moment of inertia of the roller K e ;K p Gains of the atuators Mass of the arriage umber of spans in the aumulator R Radius of the roller t Average tension in the aumulator t j (t) Tension in j th span in the aumulator t p ;t e Strip tension at the proess/exit end t r Desired tension in the strip u Control fore on the arriage (=F r + F p ) u e ;u p Control inputs on the proess/exit rollers v p ;v e Strip veloity at the proess and exit end v ;x Displaement/veloity of the arriage! j (t) Angular veloity of the j th roller 2 Dynamis of aumulator arriage, web tension, and driven rollers A shemati of the arriage, web spans, and rollers within an aumulator is shown in Figure 2; this figure shows an exit aumulator, that is, the web is fed into the aumulator from the proess setion. The aumulator arriage dynamis is given by d 2 x (t) 2 = u (t) F d (t) g X j= t j (t) () F r Carriage t Torque Shaft t F d j Datum t e 2 t p v e v p Strip to Rewind Strip from Proess t 2 t F p The ontrolled fore, u (t), is generated by a hydrauli sysx Figure 2: Sketh of an exit aumulator. tem. The disturbane fore, F d (t), inludes frition in the hydrauli ylinder and the rod seals, frition in the arriage guides, and other external fores on the arriage. The torque shaft shown in Figure 2 is inluded in the aumulator design to synhronize the side to side lifting ation so that only vertial motion needs to be onsidered in the ontrol system design. The number of rollers on the arriage is assumed to be =2. The number of rollers in the aumulator is +. The strip tension and the roller dynamis in the j-th aumulator span is given by [9], j (t) = R x (t) (! j(t)! j (t)) + R x (t) [t j (t)! j (t) t j (t)! j (t)] + x (t) _x (t) x (t) t j(t)_x (t) (2) d! j(t) = B j! j (t) +R(t j+ (t) t j (t)) (3) where j =:. It is assumed that all the rollers are of the same radius (R), whih is typially the ase in web proess lines. otie that there is a strong oupling between the arriage dynamis (), strip dynamis (2), and roller dynamis (3). To obtain a simpler tension dynami model for all the aumulator web spans in an average sense, an average tension is defined as: X t (t) = t j(t): (4) j= This onept of average tension enables us to onsider an idealized situation where the tension in eah span in the aumulator is t. Taking the sum from j = to j = of both sides of (2) and dividing by results in (t) = R (! (t)!(t)) x (t) + R (t(t)!(t) t (t)! (t)) x (t) + x _x(t) t(t)_x(t): () (t) x (t) We an further simplify the average tension dynamis, (), under the assumption that the produt is muh larger than the tensions 368 Denver, Colorado une 4-6, 23

3 t ;t, and t. This is generally true for all web materials proessed in web proessing mahines. Under this assumption the seond and last term an be ignored from the average tension dynami model, (), resulting in the following simplified average tension dynami model: (t) = R x (t) (! (t)!(t)) + _x(t): (6) x (t) Assuming that there is no slip on the rollers and shown in Figure 2, the roller angular veloities and the web veloity are related as follows: v e = R! and v p = R!. Therefore, (6) an be written as: (t) = (ve(t) vp(t)) + x (t) _x(t): (7) x (t) The driven roller angular dynamis at the proess side and the exit side of the aumulator are given by the following equations: _! e(t) = B fe! e(t) +R(t e(t) t (t)) + K eu e(t) (8) _! p(t) = B fp! p(t)+r(t (t) t p(t)) + K pu p(t) (9) where! e(t) and! p(t) are the exit side and proess side driven roller angular veloities, respetively. otie that t e(t) and t p(t) are given by the tension dynamis downstream of the exit-side driven roller and upstream of the proess-side driven roller; sine in this work we are interested in the web tension behavior within the aumulator, we assume that t e(t) and t p(t) are maintained lose to the desired web tension, that is, t e(t) = t r + ffi e(t) and t p(t) = t r + ffi p(t), where t r is the desired web tension in the proess line, and ffi e(t) and ffi p(t) are disturbanes. Assuming that there is suffiient web wrap on the driven rollers whih will ensure that there is no web slip, we have v e(t) =R! e(t) and v p(t) =R! p(t), the equations for exit-side and proess-side web veloities are: _v e(t) = B fe v e(t) +R 2 (t r t (t)+ffi e(t)) + RK eu e(t) () _v p(t) = B fp v p(t)+r 2 (t (t) t r ffi p(t)) + RK pu p(t): () 3 Controller design The goal is to design ontrol algorithms for the aumulator arriage, the exit-side and the proess-side driven rollers suh that the average web tension, the arriage position, the arriage veloity, the exit-side and proess-side web veloities trak their desired trajetories. It is assumed that all the state variables are measurable exept for the average web tension, t (t). An observer will be designed to estimate the average web tension. Consider the following error variables: e t(t) = t (t) t d, e x(t) = x (t) x d (t), e v(t) = v (t) v d (t), e ve(t) = v e(t) ve d (t),and e vp(t) =v p(t) vp(t), d where t d is the desired web tension, x d (t) and v d (t) are the desired aumulator position and veloity, respetively and ve d (t) and vp(t) d are desired exit and proess-side veloities, respetively. Choose the following ontrol inputs for the aumulator arriage, exit-side driven roller, and the proess-side driven roller: u (t) =(_v d (t) +g + v f v d (t)+ t d + u a(t)) (2) M u e(t) = u B fe ea(t)+ RK e vd e (t) + _v d e (t) (3) u p(t) = u B fp pa(t)+ RK p vd p(t) + _v p(t) d (4) where u a(t), u ea(t), and u pa(t) are auxiliary ontrol inputs that will be designed later. Substituting the ontrol input and using the error definitions, the error dynamis is given by: _e t(t) = x ev(t)+ (eve(t) evp(t)) () (t) x (t) _e x(t) =e v(t) (6) _e v(t) = e t(t) v f e v(t)+u a(t): (7) _e ve(t) = ( B fee ve(t) R 2 e t(t)+r 2 ffi e(t)) + u ea(t) (8) _e vp(t) = ( B fpe vp(t)+r 2 e t(t) R 2 ffi p(t)) + u pa(t): (9) Consider the following observer for estimating the average tension dynamis: _^t (t) = x ev(t) + (t) x (eve(t) evp(t)) + ^f t ; (2) (t) where ^t (t) is the estimate of t (t), ^t () = ^t and ^f t will be hosen later during the stability analysis. Define the observation error as ~e t(t) =t (t) ^t (t). Also, define ^e t(t) =^t (t) t d. Therefore, the observer error dynamis is _~e t(t) = ^f t : (2) otie that we have used the fat that v d (t) =(v d p(t) v d e (t))=, that is, the differene between the desired proess veloity and the the desired exit veloity divided by the number of spans gives the desired arriage veloity. Consider the following Lyapunov funtion andidate for the aumulator arriage system: V (t) = 2 e2 t(t) + 2 e2 x(t) + 2 e2 v(t)+ 2 ~e2 t(t): (22) The derivative of the Lyapunov funtion andidate along the trajetories of the error dynamis is given by _V (t) =e t _e t + e x _e x + e v _e v +~e t ~et _ = e te v + (eve evp) et + exev x x (t) e te v v f e 2 v + u a(t)e v ^f t ~e t: (23) Choose the following auxiliary ontrol input for the arriage: u a(t) = x (t) ^et(t) ex(t)+ ^e t(t) fl 3e v(t) (24) where fl 3 is a positive gain. Substituting (24) into (23), the derivative of the Lyapunov funtion andidate beomes _V (t) = fl 3 + v f e 2 v + (eve evp) et x (t) + x ~e te v ^f t ~e t: (2) Consider the following Lyapunov funtion andidate for the exit and the proess-side roller dynamis: V ep(t) = 2 e2 ve(t) + 2 e2 vp(t): (26) The derivative of V ep(t) along the trajetories of the error dynamis, (8) and (9), is _V ep(t) = Bfe e 2 ve R 2 e te ve + R 2 ffi e(t)e ve + eveu ea + Bfp e 2 vp + R 2 e te vp R 2 ffi p(t)e vp + evpu pa: 3686 Denver, Colorado une 4-6, 23

4 Consider the following Lyapunov funtion andidate for the ombined system of the aumulator arriage, exit-side driven roller, and the proess-side driven roller: V (t) =V (t)+v ep(t): (27) The derivative of V (t) is _V (t) = fl 3 + v f e 2 v + (eve evp) et x (t) + ~e te v x M ^f t ~e t + Bfe e 2 ve R 2 e te ve + R 2 ffi e(t)e ve + eveu ea + Bfp e 2 vp + R 2 e te vp R 2 ffi p(t)e vp + evpu pa: 4 Simulation study In this setion, the proposed ontrol sheme is investigated by onduting simulations on an industrial ontinuous web proess line. The simulations are performed using the parameters of an ALCOA ontinuous proess line (CPL) and its exit aumulator. Values of the parameters of the aumulator used in the simulations are: = 73 kg (.6 slugs), A = m 2 (. in 2 ), E = 6.9 =m 2 ( 7 psi), = 34, v f = s/m (2lb-se/in), R =.24 m(6 in), = 2.42 kg-m 2 ( slugs-in 2 ), and B f =.2. The desired tension in the web spans is 8 (6 lbs). The desired proess speed is 3.3 m/s(6 fpm). A typial senario of the exit speed and the arriage speed during a rewind roll hange is depited in Figure 3. The rewind roll hange- 6 Assuming that ffi e(t) and ffi p(t) are bounded by some known onstants, that is, jffi e(t)j» ffi μ e and jffi p(t)j» ffi μ p, we hoose the following auxiliary ontrol inputs: bet u ea = fl ee ve u pa = fl pe vp + x x μ ffi esgn(e ve) (28) bet μ ffi psgn(e vp) (29) where fl e and fl p are positive gains. Using these auxiliary ontrol inputs and re-arranging terms, we obtain _V (t)» fl 3 + v f e 2 v fl e + B fe e 2 ve fl p + B fp e 2vp jevej(μ ffi e jffi e(t)j) Choosing + jevpj(μ ffi p jffi p(t)j) + x ^f t = x e v + we get _V (t)» fl 3 + v f e 2 M v x ~e te v (~e te ve ~e te vp) ^f t ~e t: (e ve e vp); (3) x fl e + B fe e 2 ve fl p + B fp e 2 vp: (3) Therefore, V (t) is a non inreasing funtion of time for all t. Hene, V (t) 2 L and lim t! V (t) = V <. Also, e t(t); ~e t(t);e x(t);e v(t), e ve(t);e vp(t) 2 L and e v(t), e ve(t);e vp(t) 2 L 2. From the error dynamis, (6) (9), _e x(t); _e v(t); _e ve(t); _e vp(t) 2 L. Therefore, using Barbalat s lemma, we have e v(t);e ve(t);e vp(t)! as t!. The following theorem summarizes the results of this setion. Theorem 3. For the dynamis of the aumulator arriage and the driven rollers upstream and downstream of the aumulator given by equations (7), (), and (), the ontrol inputs given by equations (2),(3), (4), the auxiliary inputs given by equations (24), (28), (29), the observer dynamis given by equation (2), and the observer auxiliary term given by equation (3) will result in the signals e t(t); ~e t(t);e x(t);e v(t);e ve(t);e vp(t) being bounded and further, the signals e v(t);e ve(t);e vp(t) asymptotially onverge to zero Carriage Vel. m/se Exit Vel. m/se Carriage Position m time se Figure 3: Desired exit speed, arriage speed and arriage position during rewind roll-hange. over senario, when the web veloity in the proess setion is maintained at a onstant value, is desribed in the following steps by referring to Figure 3: (i) AB veloity of the web in the rewind side is deelerated to zero from a value of 3.3m/s(6 fpm), as a result of this the aumulator starts olleting the web and the arriage aelerates upwards; (ii) BC rewind stops and the arriage is moving up with onstant veloity; (iii) CD after rewind roll hange, exit side is aelerated up to the proess speed, in this period the arriage is moving up while deelerating; (iv) DE exit side is aelerated up to a speed above the proess speed, 4.8 m/s(9 fpm) in this ase; (v) EF exit speed is maintained at this onstant speed; (vi) FG exit speed is redued to the proess speed. The desired profile for arriage veloity is given by: v d (t) = vd p(t) ve d (t) : From this veloity profile the desired arriage position profile is alulated. After eah yle of rewind roll-hange operation the arriage will return to its original position; this means that the area under the arriage veloity urve is zero. The goal is to trak the desired profiles of the arriage position and speed, exit veloity and proess veloity while maintaining the desired level of the average web tension. The simulations are onduted using the system model given by equations (7), (), and () and the ontrol algorithms given by (2), (24), (4), (29),(3), and (28) and the observer given by equations (2) and (3). Two types of ontrollers are onsidered for omparative study via simulations: () Currently used industrial ontroller, and (2) the ontroller proposed in this paper. Currently, the industrial ontroller uses only feedforward of the desired profile for the aumulator arriage and PI ontrollers for the exit and proess-side driven Denver, Colorado une 4-6, 23

5 rollers. The industrial ontrollers for the aumulator arriage, exitside driven roller, and proess-side driven roller are given by: u I(t) =(_v d (t)+g + v f v d (t) + t d ) (32) u ei(t) = RK e Bfe vd e (t) + _v d e (t) k pee ve(t) k ie Z e ve(t)dfi (33) u pi(t) = Bfp Z RK p vd p(t) + _v p(t) d k ppe vp(t) k ip e vp(t)dfi (34) where k pe and k pp, and k ie and k ip are the proportional and integral gains, respetively. The disturbanes ffi e and ffi p onsidered in equations () and () are of sinusoidal nature with a frequeny of.2 Hz and amplitude 44 ( lbs). For omparing the results of the two types of ontrollers, sinusoidal disturbane as shown in Figure 4 is introdued into the aumulator arriage dynamis, that is, into equation (). The amplitude of this disturbane is.2 m/s 2 ( in/se 2 ) and the frequeny is. Hz. Low frequeny disturbanes are used beause they are typial disturbanes on the aumulator arriage; the aumulator arriage does not have the ability to respond to high frequeny disturbanes due to its large mass. The 4 3 aumulator, are designed for traking the desired exit and proess web veloities, and to maintain the web tension at the desired level. It is ommon in the web handling industry to just apply a desired fore on the arriage using a hydrauli system in opposition to the arriage weight and the fore required to produe desired tension in all the aumulator web spans; thus, ignoring the dynamis of the arriage motion. This strategy often leads to large tension variations not only in the aumulator web spans but also in web spans in the entire proess line due to tension disturbane propagation both upstream and downstream of the aumulator. Simulation results omparing the proposed ontrol algorithm with the urrently used ontrol sheme in industry show that the proposed ontrol algorithm results in muh less web tension variations. In the urrent work, we have assumed knowledge of the frition oeffiients, whih may not be pratially feasible. Future work must inlude adaptively estimating the frition oeffiients. Instead of assuming that the fore on the arriage is diretly aessible as an input, future work should also inlude the dynamis of the atuator, either hydrauli or eletri, oupled with the dynamis of the arriage and the web spans. In the urrent work, we have negleted the effet of the weight of the web on the aumulator arriage; the weight of the web on the arriage an be substantial in metal proess lines where the thikness of the web an be as high as.2 inhes. The weight of the web inreases with the inrease in the aumulator spans length as the aumulator moves up; the effet of this time-varying weight must also be investigated in the future. Disturbane time se Figure 4: Sinusoidal disturbane. time duration of the disturbanes is hosen to simulate periodi disturbanes that our during the initiation of arriage motion from its stationary state and when the exit veloity starts to aelerate or deelerate. In the simulations, all the gains that are ommon to both ontrollers are hosen to be the same. Figures and 6 show the state errors for the proposed ontroller and Figures 8 and 9 show the state errors for the industrial ontroller. From the error in web tension, e t plotted in Figures 6 and 9, it an be learly seen that the proposed ontroller does a muh better job at rejeting the sinusoidal disturbanes. Also, Figures and 8 learly indiate that the proposed ontroller exhibits muh lesser errors in the exit and proess side roller veloities. Figures 7 and show the ontrol inputs for the proposed ontroller and the industrial ontroller. The ontrol effort required for aumulator arriage (u ) is similar in both shemes but the ontrol effort required for driven rollers (u p and u e) for the proposed ontroller is smaller than the industrial ontroller. Overall, the performane of the proposed ontroller is muh improved as ompared to the industrial ontroller with no additional ontrol effort. Conlusions and future work In this paper, feedbak ontrol algorithms for the aumulator arriage, and for the upstream and downstream driven rollers to the Referenes [] D.P. Campbell, Dynami Behavior of the Prodution Proess, Proess Dynamis. ohn Wiley and Sons, In., ew York, 98. [2] K.P. Grenfell, Tension ontrol on paper-making and onverting mahinery, Pro. of the 9th IEEE Annual Conf. on Eletrial Engineering in the Pulp and Paper Industry, Boston, MA, une, 963. [3] D. King, The mathematial model of a newspaper press, ewspaper Tehniques, Deember, 969. [4] G. Brandenburg, ew mathematial models for web tension and register error, Pro. of the 3rd Intl. IFAC Conf. on Instrumentation and Automation in the Paper, Rubber, and Plastis Industry, vol., 977. [] G.E. Young and K.. Reid, Lateral and longitudinal dynami behavior and ontrol of moving webs, ASME ournal of Dynami Systems, Measurement, and Control, vol., pp , une, 993. [6] W. Wolfermann, Tension ontrol of webs A review the problems and solutions in the present and future, Pro. of the Third Intl. Conf. on Web Handling, Stillwater, OK, une, 99. [7].. Shelton, Limitations to sensing of web tension by means of roller reation fores, Pro. of the Fifth Intl. Conf. on Web Handling, Stillwater, OK, une, 999. [8] P.R. Pagilla, S.S. Garimella, L.H. Dreinhoefer, and E.O. King, Dynamis and ontrol of aumulators in ontinuous strip proessing lines, IEEE Transations on Industry Appliations, vol. 37, no. 3, pp , 2. [9] P.R. Pagilla, E.O. King, L.H. Dreinhoefer, and S.S. Garimella, Robust observer-based ontrol of an aluminium strip proessing line, IEEE Transations on Industry Appliations, vol. 36, no. 3, pp , Denver, Colorado une 4-6, 23

6 e ve x 3 e ve... e vp 4 2 x Figure : State errors of the proposed ontroller. e vp Figure 8: State errors of the industrial ontroller. e t e x (m) e v x x Figure 6: State errors of the proposed ontroller. e t 4 2 x 3 e x (m) e v 4 x Figure 9: State errors of the industrial ontroller. x x u u u e ( m) u p ( m) Figure 7: Control inputs for the proposed ontroller. u e ( m) u p ( m) Figure : Control inputs for the industrial ontroller Denver, Colorado une 4-6, 23