ADAPTIVE REJECTION OF ECCENTRICITY TENSION DISTURBANCES IN WEB TRANSPORT SYSTEMS. Yulin Xu, Michel de Mathelin, Dominique Knittel

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1 Copyright IFAC 5th Triennial Worl Congre, Barcelona, Spain ADAPTIVE REJECTION OF ECCENTRICITY TENSION DISTURBANCES IN WEB TRANSPORT SYSTEMS Yulin Xu, Michel e Mathelin, Dominique Knittel Strabourg I Univerity, ERT-Enroulement, LSIIT-UMR CNRS 75 ENSPS, B. Sébatien Brant, 674 Illkirch, FRANCE Tel: + () Fax: + () michel.emathelin@enp.u-trabg.fr Abtract: Thi paper preent aaptive algorithm to compenate roll eccentricity perturbation in web tranport ytem. Roll eccentricity create tenion iturbance whoe frequencie may be lowly varying ue to the change of raiu of the roll. The magnitue an phae of thee perturbation are etimate through an aaptive algorithm that cancel the effect of thee perturbation. The ability of the algorithm to reject eccentricity perturbation with lowly varying frequencie i hown through imulation on the phyical moel, a well a experimental reult. Keywor: Roll eccentricity perturbation, iturbance rejection, web tranport, aaptive algorithm.. INTRODUCTION We tuy a ytem that i quite common in inutry. The ytem ha at leat three motor (cf. Figure ): an unwiner, a traction motor an a winer, an it preent the inherent ifficultie of web tranport ytem. An important control problem in web tranport ytem i the effect of the perturbation ue to the eccentricity of the roll. In web tranport ytem, roll eccentricity create tenion iturbance that are perioic or quai-perioic ue to change of raiu of the roll when the roll i the winer or the unwiner. Thee perturbation can ometime inuce web break an fol or material amage. Thi paper focue on removing the effect of thee iturbance that are quai-perioic (perioic with a lowly varying frequency) an whoe amplitue an phae are unknown. There exit everal approache to tackle thi problem. One i bae on repetitive control principle (ee, e.g., (Garimella an Srinivaan, 994), (Hillertröm, 996), (Lee an Smith, 998), (Wei an Häfele, 999)). Another approach i bae on aaptive algorithm (ee, e.g., (Boon an Dougla, 997), (Boon et al., 998), an (Canua e Wit an Praly, 998)). In our application (high pee web tranport), it i important to eign an algorithm that can be ae to an exiting inutrial controller. Therefore, we propoe two aaptive algorithm bae on the aaptive algorithm for noie cancellation of (Boon an Dougla, 997). Thee algorithm lock on the amplitue an phae of the perturbation an cancel their effect. The paper i organize a follow, the firt ection preent the moel of the web tranport ytem bae on the law of phyic. The econ ection ecribe two aaptive algorithm cheme that reject the effect of quai-perioic perturbation. The lat ection how imulation an preliminary practical reult when thee algorithm are applie to web tranport ytem.

2 Fig.. Experimental etup T 5 T T 4 T 5 V V 5 V V V 4 T u = T+T T w = T4+T5 U u U t U w Fig.. Three motor web tranport ytem. MODEL OF THE WEB TRANSPORT SYSTEM Figure how a typical three motor ytem with winer, unwiner, tractor an two loa cell that i ue for our experiment. The complete moel of thi ytem i given by the following equation (ee, e.g., (Koç, ) an (Koç et al., )): Tenion between conecutive roll: T k L k = ES(V k V k )+T k V k T k (V k V k ) () k =,, 4, 5 (V k = R k Ω k ) where L k i the web length between roll k an roll k, T k i the tenion on the web between roll k anrollk, V k i the linear velocity of the webonrollk, Ω k i the rotational pee of roll k, R k i the raiu of roll k, E i the Young moulu an S i the web ection. Law of motion: Unwiner: (U = U u ) (J Ω ) = R T K U u C fu f vu Ω () Winer: (U 5 = U w ) (J 5 Ω 5 ) = R 5 T 4 + K 5 U w C fw f vw Ω 5 (6) where C fw, C ft, C fu are the ry friction torque of the three motor, repectively, f vu, f vt an f vw are the vicou friction coefficient, an K k U k are the motor torque. We can notice that the inertia J k an the raiu R k of the winer an the unwiner,are time epenent an may vary on a large cale uring the proce operation (about % for the raiu in our experiment). The effect of the eccentricity of roll k can be moele a a inuoial perturbation on the nominal raiu of roll k: R k (t) =R k (t)+a k in(θ k (t)) (7) where R k (t) i the nominal raiu of roll k an: θ k (t) = t t Ω k τ + δθ k (t) (8) Where δθ k (t) i a lowly varying phae-hift. In a contant pee web tranport ytem, if the eccentricity perturbation i on the winer or the unwiner, the perturbation frequency i hifting with time ue to the change of nominal raiu of the roll. Thi moel ha been valiate experimentally on the ytem hown in Figure. The Unwiner loa cell: J V = R (T T ) f V () Tractor: (U = U t ) J Ω = R (T 4 T )+K U t C ft f vt Ω (4) Winer loa cell: J 4 V 4 = R 4 (T 5 T 4 ) f 4 V 4 (5) Fig.. Moel matching

3 matching performe i excellent a it can be een in Figure, without perturbation.. ADAPTIVE PERTURBATION REJECTION We preent two aaptive algorithm for the rejection of eccentricity perturbation. Thee algorithm are bae on the work of Boon (ee, e.g., (Boon an Dougla, 997) an (Boon, 998)) on noie cancellation. The main ifference between thee two algorithm i the type of ientife noie parameter.. Aaptive algorithm I Thi algorithm i bae on a phae-locke loop tructure that etimate imutaneouly the phae an magnitue of the perturbation an then cancel it. Thi algorithm can be een a a moifie verion of a tructure ue in frequency emoulation. We aapt Boon approach to the cae of a feeback control ytem with output ignal y an reference input ignal r intea of the cae of a noie cancellation ytem. Another ifference with Boon approach i that the central frequency Ω c of the perturbation can be meaure an ae at the input of the frequency moulator in the loop. The aaptive algorithm cheme i hown r e + u K() y G() u witch A e Sat. D() δ ω e + α co(α) + Ω c -in(α) Fig. 4. Aaptive algorithm cheme I in Figure 4, where G() i the ytem, K() i the controller, i the unknown quai-perioic perturbation (aume to be equivalent to the eccentricity perturbation), A i the etimate of the magnitue of the perturbation, ω i the etimate of the intantaneou frequency of the perturbation an α i the etimate of it phae. Therefore, if u i the etimate of the perturbation, then: u = Aco(α) (9) It important that the perturbation cancellation algorithm oe not make matter wore even uring a tarting tranient. Therefore, we moify Boon algrithm by introucing a aturation block on the etimate magnitue A an a witching ytem that can lowly cloe or open the perturbation cancellation loop. The tranfer function matrix D() relate the error ignal e an e to the etimate magnitue A an etimate frequency hift δ, repectively. An approximate analyi of thi aaptive algorithm will allow u to euce a imple eign rule for D(). Thi harmonic analyi i bae on the following aumption: The value of A, δ an Ω c varie ufficiently lowly, o that the repone of the feeback ytem to the ignal u (t) can be approximate by the teay-tate output of the ytem for a inuoial input with the frequency ω. Thi i a tanar aumption in emoulation ytem. The intantaneou frequency ω = α i cloe to the known frequency Ω c o that G(jω ) can be replace by G(jΩ c ). We make the aumption that the perturbation to be cancelle i equivalent to an input pertubation, (t) =A (t)co(α (t)). Therefore, y = KG +KG [r] G +KG [u ] e = +KG [r]+ G +KG [u ] () Let efine: G G R = Re[ +KG (jω c)] G G I = Im[ +KG (jω c)] If we neglect the tranient ue to the variation of A, δ, Ω c,wehave: e = +KG [r]+ag Rco(α) AG I in(α) A G R co(α )+A G I in(α ) () e = AG R + AG Rco(α) AG Iin(α) +co(α) +KG [r] A G R co(α α ) A G I in(α α ) A G R co(α + α ) + A G I in(α + α ) () with α = ω = δ +Ω c () e = AG I AG Ico(α) AG Rin(α)

4 in(α) +KG [r] A G I co(α α ) + A G R in(α α )+ A G I co(α + α ) + A G R in(α + α ) (4) The filter D() i low-pa in nature, o the highfrequency component within the ytem are eliminate by proper low-pa filtering of the ignal e an e. Then, the ignal e an e are approximately given by: [ ] e e where P i efine a: P = [ ] A A co(α α ) A in(α α ) [ ] GR G I G I G R (5) (6) When the phae error α α i mall, thi equation can be linearize a: [ ] [ ] e A A (7) e A (α α ) Different metho can be ue to eign the filter D(). Let efine the two variable x an x a: [ ] [ ] x x e (8) e o that x = A A x = A [ t (δ(σ)+ω c )σ + α() α (t)] (9) Therefore, the ynamic of the ytem from the A + + e x D() A + P P α D () + e x δ ω α + Ω c Fig. 5. Linearize moel of algorithm I parameter A an δ to the variable x an x are approximatively ecouple from one another an can be repreente a in Figure 5. Then, D() may be eigne a the cacae of the tranformation (9) with the following filter: A = D () [x ] δ = D () [x ] () where the integrator are inclue to obtain a zero teay tate error on the etimate of A an α. The tranfer function D () and () are eigne to guarantee the cloe-loop tability of the two ub-ytem. Accoring to the frequency repone, D () and () can be electe a: c D ()=k Ω D ()= Ω c ( + Ω c /) () Ā ( + Ω c /) Where Ω c i an average value of the ytem rotation frequency Ω c an Ā i an average value for the amplitue A. Becaue the magnitue of the iturbance A act a a gain in the loop, the parameter in D () mut be eigne for a range of magnitue of the iturbance A.. Aaptive algorithm II Thi algorithm i hown in Figure 6. It i bae on the etimation of the amplitue of two component of the perturbation that are in phae quarature to each other. The etimate of the Ω c r e + u K() y G() u witch co(α) α e P e -in(α) -g -g Fig. 6. Aaptive algorithm cheme II perturbation, u,igivenby: δ δ q + α =Ω c () u = δ co(α) δ q in(α) The analyi of the aaptive cheme i bae on the aumption that the perturbation i equivalent to an input pertubation (t) =A (t)co(α (t)). Further, we aume that the value of Ω c varie ufficiently lowly, o that the hypothei of algorithm I are alo vali. Therefore, imilarly to algorithm I, if the high-frequency term are eliminate by proper low-pa filtering, the error ignal e an e are approximately given by: [ ] [ ] e δ A co(α α ) () e δ q + A in(α α ) where P i efine a in (6). Therefore, the ynamic of δ an δ q are ecouple a hown in the linearize moel of the iturbance cancellation loop in Figure 7. The parameter g et the etimation ynamic. Inee: [ ] [ ] δ δ A = g co(α α ) (4) δ q δ q + A in(α α )

5 + g A co(α α ) + e x + P P g A in(α α ) + e x + δ δ q v=45m/min v=5m/min v=6m/min v=68m/min v=77m/min Boe iagram: magnitue Fig. 7. Linearize moel of algorithm II Therefore, g mut be Ω c in orer to filter the high-pa frequency component, but large enough in orer to follow the phae variation of the perturbation,.inourexperimentg =. In thi cae, δ exponentially converge towar A co(α α )anδ q towar A in(α α ), o that u exponentially converge towar (with time contant g ). 4. APPLICATION TO THE WEB TRANSPORT SYSTEM In the web tranport ytem, if the eccentricity i ue to roll k, then the main tenion perturbation i on tenion T k+. Therefore, we a the aaptive algorithm on the controller of the motor of roll k with a feeback of the tenion T k+. For exemple, if the eccentricity i ue to the unwiner, we a the aaptive perturbation cancellation algorithm on the unwiner controller a hown in Figure 8, where Ω c i the rotation frequency of the. v=99m/min v=9m/min phae hift 4 v=45m/min 5 6 v=5m/min v=6m/min v=68m/min v=77m/min 7 v=99m/min 8 v=9m/min Frequency (ra/ec) Fig. 9. Boe plot where A max an A min can be electe accoring to the nominal output value of the controller. Web tenion (Kg).5.5 Web tenion output without algorithm at v=5 m/min With algorithm I With algorithm II.5 Ω c T T 6 T 4 T 5 V V5 Fig.. Tenion iturbance rejection at 5m/min V V V4 Web tenion output without algorithm at v= m/min u + aap + U u U v U w H multivariable controller With algorithm I e T + u = T+T V T w = T4+T With algorithm II T uref V uref T wref Web tenion (Kg) Fig. 8. Control cheme with aaptive algorithm unwining roll (Ω c = Ω ). The Boe plot of the tranport function between U u an T u i hown in Figure 9 for ifferent reference linear velocitie V uref. The controller in our experiment i eigne uing a H robut control approach (ee (Koç et al., ) an (Koç et al., )). Bae on the V uref value, an average value Ω c i compute for Ω c an the correponing matrix P i meaure from the Boe plot. In practice, Ā i compute from the etimate amplitue A uing a aturation function a in: A min if A A min Ā = A if A min A A max (5) A max if A A max Fig.. Tenion iturbance rejection at m/min Figure, an how imulation of both algorithm rejecting an input perturbation on the unwiner roll, at the web tranport pee V uref = 5 m/min, m/min an m/min, repectively, with a contant web tenion reference ignal. The cancellation algorithm i applie at time t = 5 ec. It can be een that the algorithm II i more robut to tenion et-point change than the algorithm I. Figure how the pectrogram of the web tenion at the unwiner uring a real experiment on

6 Web tenion output without algorithm at v= m/min Spectrogram of web tenion with algorithm at v=5m/min Web tenion (Kg).5.5 With algorithm I With algorithm II Fig.. Tenion iturbance rejection at m/min the ytem hown in Figure for the web tranport pee V uref = 5 m/min without aaptive iturbance cancellation algorithm. Then, Figure 4 how the pectrogram of the web tenion at the unwiner uring a real experiment at the web tranport pee V uref = 5 m/min with the aaptive iturbance cancellation algorithm II on the unwiner. One can ee that the lowly varying Web tenion (Kg) Spectrogram of web tenion without algorithm at v=5m/min.5 Frequency (Hz) Fig.. Spectrogram of the web tenion at 5m/min without algorithm firt harmonic i almot completely cancelle. The higher frequency perturbation term i ue to the traction motor. 5. REFERENCES Boon, M. (998). A icuion of Chaplin & Smith patent for the cancellation of repetitive vibration. Proceeing of the 7th IEEE Conference on Deciion an Control, Tampa, Fl. Boon, M. an S. C. Dougla (997). Aaptive algorithm for the rejection of inuoial iturbance with unknown frequency. Automatica. Boon, M., J. S. Jenen an S. C. Dougla (998). Active noie control for perioic iturbance. Proceeing of the American Control Conference, Philaelphia, Pa. 6 7 Web tenion (Kg) Frequency (Hz) Fig. 4. Spectrogram of the web tenion at 5m/min with algorithm II Canua e Wit, C. an L. Praly (998). Aaptive eccentricity compenation. Proceeing of the 7th IEEE Conference on Deciion an Control, Tampa, Fl. Garimella, S. S. an K. Srinivaan (994). Application of repetitive control to eccentricity compenation in rolling. Proceeing of the American Control Conference, Baltimore, Ma. Hillertröm, G. (996). Aaptive uppreion of vibration a repetitive control approach. IEEE Tranaction on control Sytem Technology. Koç, H. (). Moéliation et commane robute un ytème entrainement e bane flexible. PhD thei. Univerité Loui Pateur (Strabourg I univerity). Koç, H., D. Knittel, M. e Mathelin an G. Abba (). Robut gain-cheuling control in web wining ytem. Proceeing of the 9th IEEE Conference on Deciion an Control, Syney, Autralia. Koç, H., D. Knittel, M. e Mathelin an G. Abba (). Moeling an robut control of wining ytem for elatic web. IEEE Tranaction on Control Sytem Technology. Toappear. Lee, R. C. H. an M. C. Smith (998). Robutne an trae-off in repetitive control. Automatica 4, Wei, G. an M. Häfele (999). Repetitive control of MIMO ytem uing H eign. Automatica 5,