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- 1. Robust control strategies for an electric motor driven accumulator with elastic webs David Kuhm a,b,c , Dominique Knittel a,b,n , Marie-Ange Bueno c a Web handling research Center, UFR Physique et Inge´nierie, Campus Meinau, University of Strasbourg, 17, rue du Mare´chal Lefebvre, 67100 Strasbourg, France b Laboratoire de Ge´nie de la Conception INSA Strasbourg, 24, boulevard de la Victoire, 67084 Strasbourg, France c Laboratoire de Physique et Me´canique Textiles ENSISA, 11, rue Alfred Werner, 68093 Mulhouse, France a r t i c l e i n f o Article history: Received 12 July 2011 Received in revised form 26 January 2012 Accepted 19 May 2012 Available online 15 June 2012 Keywords: Industrial accumulator Roll-to-roll Dynamic modelling H1 controllers Multi-model control PI control Robustness a b s t r a c t This paper concerns the modelling of an accumulator used in industrial elastic web processing plant to allow changing material roll while the rest of the line remains at a constant web velocity. A nonlinear model of a motor actuated accumulator is ﬁrst summarized. This model is derived from the physical relationships describing web tension and velocity dynamics in each web span of this accumulator. A linear model is deduced from the nonlinear one around a working point for frequency domain analysis. Thus the effect of some mechanical accumulator parameter variations are analyzed. In a second part, multi-model industrial PI controllers, adjusted with evolutionary algorithm on our realistic nonlinear model are compared with multi-model H1 controllers. Both controllers allow good robustness against mechanical parameter variations. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Continuous process lines increase the productivity of the plant. Accumulators (Fig. 1) are used to permit rewind or unwind core changes while the process continues at a constant velocity. Accu- mulators are composed of many free rollers and of a carriage moved to store or release web during the production phases. Most of the time, the accumulator carriage is maintained at its nominal position and the web just follow the geometrical path made by the free rollers. When the unwinder wound roll is almost empty, the carriage is moved up in order to increase the web stored in the accumulator. During the unwinder roll change, the unwinder is stopped and the accumulator carriage is adequately moved down to restore the web. The objective is to maintain constant web line velocity. Finally, after the unwinder roll change, the carriage is moved back to its nominal position. The unwinder roll changing time has to be as short as possible being limited by the accumulated web length. Web tensions inside an accumulator are easy to control during the regular production phase (when the carriage is at its nominal position). However web tension variations often occur in the transition phases and therefore can generate web folds or breaks due to the inertia and friction of the free rollers. Different works in web tension control of roll-to-roll systems have been published in the past years. For example Zalhan et al. [1], Brandenburg [2,3], Benlatreche et al. [4], Knittel et al. [5,6], Gassmann et al. [7] , Giannoccaro et al. [8], Chen et al. [9] present the modelling and control of a continuous proces- sing line composed only with tractors and free rollers, whereas Wolfermann [10], Olsen [11], Knittel et al. [12] describe the unwind- ing and the winding process. Only few results present the modelling and the control of industrial accumulators. Important results can be found in Koc- et al. [13,14], Pagilla et al. [15,16], Kuhm et al. [17]. The last reference concerns the modelling and robust control of an industrial accumulator including an ideal carriage (the carriage actuator dynamics are neglected) and [18]compare the accumulator presented in this paper to one including a pneumatic actuated carriage. Knittel et al. [19] concerns the accumulator controllers optimization using genetic algorithm. This paper presents for the ﬁrst time robust control strategies for a complete modelled industrial accumulator including the dynamics of the motor actuated carriage. 2. Accumulator modelling 2.1. Nonlinear model of the accumulator web dynamics Equations describing web tension behavior between two consecutive rollers and the velocity of each roll enable to build the nonlinear model of a web transport system. Our accumulator is modelled by a succession of web span tensions and velocities between each roll and by the accumulator carriage motion. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.05.004 n Corresponding author at: Web handling research Center, UFR Physique et Inge´nierie, Campus Meinau, University of Strasbourg, 17, rue du Mare´chal Lefebvre, 67100 Strasbourg, France. Tel.: þ33 6 71 20 31 30; fax: þ33 3 68 85 49 72. E-mail address: knittel@unistra.fr (D. Knittel). ISA Transactions 51 (2012) 732–742
- 2. 2.1.1. Web tension calculation Assuming no sliding between web and rollers, the equation of continuity applied to a web span between two consecutive rollers (Fig. 2), is given by [20,21] d dt Lk 1þet k ! ¼ À Vk þ 1 1þet k þ Vk 1þet kÀ1 ð1Þ where Lk is the web length between the kth and the (kþ1)th rollers. This web length can vary due to the accumulator carriage motion. Vk and Vk þ 1 are respectively the velocity of the kth and the (kþ1)th rollers and et k the strain of the kth web span. Web tension Tk t is then deduced from general web tension relation (1) by applying Hooke’s law. The equivalent strain et k in the web (see relation (2)) is composed of the strain ew k induced by the web weight (relation (4)) and by the strain ee k induced by the web elongation without the web weight: et k ¼ ee k þew k ð2Þ The weight WLk of a web span is given by the following relation [22]: WLk ¼ rgSLk ¼ ESew k ð3Þ with r the web mass density, g the gravity and Lk the span length. The strain ew k in the web span resulting from the gravity is then obtained from (3): ew k ¼ rgLk E ð4Þ In our accumulator (web with a Young modulus of 3000 MPa, a mass density of 750 kg mÀ3 , a section of 9.35 Â 10À5 m2 and a nominal tension of 400 N, corresponding to a paper web) we have ee k bew k (ee k ﬃ387ew k ). 2.1.2. Web velocity calculation The velocity dynamic of the kth roller is calculated using the torque balance. Assuming there is no slippage between the web and the roll, the web velocity Vk is equal to the linear velocity of the roll. This velocity dynamic is given by d dt ðJkOkÞ ¼ CmkÀCrkÀCfk ð5Þ where Ok ¼ Vk=Rk is the angular velocity of the kth roll, Jk the roll inertia and Rk the roll radius. Cmk is the motor torque for a driven roller. Crk ¼ RkðTkÀTkÀ1Þ stands for the torque introduced by the web and Cfk is the friction torque between the roll and its shaft (composed of a term proportional to the roll velocity and a constant term). Eqs. (2) and (5) are applied to each web span and roller of the studied accumulator and enable the construction of a nonlinear longitudinal web dynamic simulator. 2.2. Nonlinear model of the accumulator carriage motion This part presents the nonlinear model of a motorized carriage, including static frictions representation. 2.2.1. Displacement of the accumulator carriage In the phase of wound roll change, the output speed Vout accu is maintained constant at nominal speed Vref and the input speed Vin accu decrease to zero. In order to maintain the output web tension and velocity constant during this change, it is necessary to move adequately the accumulator carriage. In our study, the accumulator is composed of nþ1 rollers and therefore has n web spans. The accumulator carriage displacement speed refer- ence during the wound roll change is given by the following Motor Unwinder Welding table 1 n+1 Accumulator ChainCarriage Vin accu Vout accu Tout L Fig. 1. Example of a motor actuated accumulator. Vk+1 Vk Lk k-1 Volume of control Roll k+1 Roll k k = k+ k e wt t Fig. 2. Indices of web span between two consecutive rollers. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 733
- 3. relation: Vaccu ¼ 1 n ðVin accuÀVout accuÞ ð6Þ This speed reference enables to ﬁnd the reference carriage position. 2.2.2. Unwinding speed during the accumulation phase During the web accumulation phase, the unwinder speed has to be increased to keep a constant web tension. This unwinding speed reference can be deduced from Eq. (6): Vin accu ¼ nVaccu þVout accu ð7Þ 2.2.3. Motor actuated carriage In order to move the accumulator carriage, in some industrial accumulator, a motor can be used. A representation of the motor actuator is shown in Fig. 3. The carriage is maintained at its nominal position thanks to a brake during the regular production phase. During the carriage motion phase, the brake is deactivated and the motor moves adequately the accumulator carriage. Due to the inclusion of the brake, there is no device to attenuate web tension variations into the motor actuated accumulator. The equation of the carriage motion is deduced by applying a torque balance on the carriage motor (8). The carriage position xc is related to wm (angular velocity) by the radius of the motor/chain gear Rm. In order to provide the carriage motion dynamic, a special attention has to be paid to static friction force Ffs. Static friction sign depends obviously on the direction of the carriage velocity. But the key point that has to be handled is when carriage velocity is zero. One has to make sure that the carriage remains immobile until the forces applied on the carriage are sufﬁcient to exceed the static friction force. These conditions are summarized in (9): Jm _wm ¼ udmkdmÀRmðFweb þFfvð _xc Þ þFfs þmtgÞÀFfvmwm if B1 or B2 satisfied Jm _wm ¼ udmkdmÀRmðFweb þFfvð _xc Þ ÀFfs þmtgÞÀFfvmwm if B3 or B4 satisfied Jm _wm ¼ 0 else 8 >>>>>>< >>>>>>: ð8Þ B1 : udmkdmÀRmðFweb þmtgÞ4FfsRm B2 : _wm 4e _wm B3 : udmkdmÀRmðFweb þmtgÞoÀFfsRm B4 : _wm oÀe _wm 8 >>>>< >>>>: ð9Þ with Jm is the motor inertia, udm is the reference voltage, kdm is the ratio from reference voltage to torque, Fweb is the force applied to the carriage by the web span and Ffvm is the motor dynamic friction force. The accumulator carriage motor is position con- trolled in order to move the carriage to the desired position. The control scheme is given in Fig. 3. 2.3. Linear model of the accumulator 2.3.1. Linearization The linear model is deduced from the nonlinear one and can be described by a state space representation. The web span dynamic linearization is detailed in Koc- [20] and Kuhm et al. [17]. The linear model is obtained by linearizing an accumulator, around a setting point, including an ‘‘ideal’’ carriage actuator (neglecting nonlinearities such as static friction, dead band, pressure depen- dant bulk modulus and saturations), according to the scheme given in Fig. 4. The linear model is useful for Bode diagrams calculation and for the frequency analysis of the transfer function. In steady state, ee k bew k in our application and therefore the web weight is neglected. However in other case, such as accumulators designed for metallic webs, the web weight is signiﬁcative and has to be taken into account by keeping the web weight terms during the linearization. By derivating relation (1) describing the tension and by keeping only the ﬁrst order terms, the relation after simpliﬁcation mc web Chain Motor Carriage Rm xc fMtg Cm - + Xref Controller Fig. 3. Generic motor representation including position control. Unwinder Welding table 1 n+1 Accumulator Carriage Actuator Vin accu Vout accu Tout L Fig. 4. Accumulator scheme representation used to build a linear model. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742734
- 4. can be written as follows: Lk dek dt ¼ Vk þ1ð1þekÞÀVkð1þ2ekÀekÀ1Þþ dLk dt ð1þekÞ ð10Þ With the notation ek ¼ ee k. An example of the developed relation, approximated before derivation (less accurate), including web weight, can be found in [22]. Hooke’s law gives the relation between the web tension and its elongation: Tk ¼ ESek ð11Þ Including the web tension (11) into relation (10) leads ﬁnally to the tension dynamic description: Lk dTk dt ¼ Vk þ 1ðESþTkÞÀVkðESþ2TkÀTkÀ1Þþ dLk dt ðESþTkÞ ð12Þ The relation (12) is then linearized around an operating point, while posing Tk ¼ T0 þtk, TðkÀ1Þ ¼ T0 þtðkÀ1Þ, Vk ¼ V0 þvk and Vðk þ1Þ ¼ V0 þvðk þ 1Þ. By preserving only the ﬁrst order terms Eq. (12) gives dtk dt ¼ E0 Lk vkþ 1À E0 Lk vkÀ V0 Lk tk þ V0 Lk tkÀ1 þ E0 Lk dLk dt ð13Þ whereas the web speed can be calculated from relation (5): dvk dt ¼ À R2 k Jk tkÀ1À Fk Jk vk þ R2 k Jk tk ð14Þ where respectively Rk is the roll radius of the kth roll, Fk the friction coefﬁcient between the kth roll and its shaft and Jk the inertia of the kth roll. 2.3.2. State space representation of the accumulator The linear model of the accumulator can be described by a state space representation given in (15). The states are the tension and velocity in each web span. The state space model has tree inputs uþv and one output Tnþ1, the accumulator output tension. The input u is the carriage displacement velocity, Vd is the unwinder speed, L is the web span length in the accumulator between two consecutive rollers. _X ¼ Aþ A1 L X þ B L uþB1v Y ¼ CX þDu 8 : ð15Þ where X ¼ Td Va Ta Va1 Ta1 ^ Vnþ 1 Tn þ1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 , u ¼ dL dt , v ¼ Vd Voutaccu ! ð16Þ C ¼ ð0 0 0 0 0 0 0 . . . 0 0 1Þ ð17Þ D ¼ 0 ð18Þ Simulation in Matlab/Simuling software environments shows that the linear model ﬁt on the nonlinear model, even if we neglect some nonlinearities (Fig. 5). The maximum relative error between the two model, during fast tension variations is 9.9%, therefore the linear model is used on the one hand to study the frequency response of the system and on the other hand to synthesize H1 controllers. 3. Accumulator performances analysis 3.1. Inﬂuence of the mechanical parameters Depending on the type of accumulator, mainly two control strategies are applied: one using Vin accu as control signal (con- troller output), the other using L. The synthesis of both control strategies need to calculate two transfer functions: a ﬁrst one 35 40 45 50 55 60 65 70 0 100 200 300 400 Time (s) N Web tension Reference tension Tension using non linear model Tension using linear model 35 40 45 50 55 60 65 70 0 5 10 Time (s) Error(%) Tension error between linear and non linear model Tension error Fig. 5. Linear and nonlinear model comparison during web span length variation. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 735
- 5. between the accumulator output tension Tout and the accumula- tor input velocity Vin accu named W1 and a second one between Tout and the web span length L named W2. In the following parts, the inﬂuences of some physical parameter variations around their nominal values are analyzed on simulated Bode diagrams. 3.1.1. Inﬂuence of the elasticity modulus and web span length The web elasticity inﬂuences the web dynamics (tension and velocity) in the transient phases. One can observe in Fig. 6 that the static gain and resonances (gains and frequencies) are depending on the Young modulus. The same observation have been made for the transfer functions W1 and W2. Very often in the industry, Young modulus changes occur during the manufacturing process and therefore the control performances decrease if the controllers are not robust enough for web elasticity variations. Consequently, the controllers have to be adjusted/calculated for each range of web elasticity, or robust for a given web elasticity range. Like the Young modulus, the web span length in the accumu- lator (between two consecutive rollers) has a signiﬁcant inﬂuence on the web dynamic (Fig. 7). A long web span will have a weaker resonance frequency and a lower gain. Such observations have been made for both transfer functions W1 and W2. Therefore the dynamic sensitivity to the web length variation has to be taken into account during controller synthesis because the web span length is varying each time when a wound roll change occurs. Moreover the nominal carriage position has to be ﬁxed ade- quately in order to increase the system bandwidth without resonances. The study of the inﬂuence of other parameters like roller inertias can be found in [17,18]. 3.2. Control strategy As mentioned previously, the accumulator can be controlled in two different ways: the controller can operate either on the web span length (by using the accumulator carriage as actuator) or the input velocity (by using the unwinder as actuator). More pre- cisely, the accumulator output web tension controller is synthe- sized using the input velocity of the accumulator as control signal, whereas the second control strategy uses the web length (by moving the accumulator carriage) as control signal. In industrial applications, both control schemes are implemented. Neverthe- less this paper studies a combination of the two strategies. During the regular production phase (when the accumulator carriage is located at its nominal position), we use the unwinder (accumulator input velocity) as actuator, whereas, during the wound roll change, the carriage is chosen as actuator. The on/off switching strategy of the two controllers is performed by weight- ing the controller output with a coefﬁcient continuously varying from 1 to 0. The control scheme is represented in Fig. 8, and the weighting coefﬁcient evolution is given in Fig. 9. 3.2.1. Linear time invariant (LTI) optimized PI controller In this part, industrial controllers (in our case a PI controller) are employed to control the output web tension. This controllers have the following form: CiðsÞ ¼ Kið1þtisÞ s , i ¼ 1 or 2 ðin Fig: 8Þ ð19Þ The PI parameters Ki and ti are determined with an optimization approach for our realistic nonlinear model using ITAE cost func- tion (20) and constraint (21) given as follows. The optimization was performed in modeFRONTIER software environment with MOGA-II genetic algorithm (population of 120 individuals, 100 generations, probability of directional cross-over equal to 0.5 and Frequency (rad/s) Gain(dB) E 2E 3E 4E 5E Young m odulus Fig. 6. Bode diagram of W2 for different Young modulii. Gain(dB) Frequency (rad/s) W eb span length (m ) Fig. 7. Bode diagram of W2 for different spans of web length. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742736
- 6. probability of mutation equal to 0.4). J ¼ Z tmax 0 t9XpÀXpref 9 dt ð20Þ Tmin oTi oTmax, 1rirn ð21Þ with Ti the web tension of the ith web span. Other cost function for PI optimization are presented in [23]. 3.2.2. Multi-model PI controller As shown previously, the web length has a signiﬁcant inﬂu- ence on the tension; static gain and bandwidth depend on the web span length. Therefore this LTI controller should not be optimal for full web length range and the controllers have to take into account the web length variation. To improve the accumu- lator output web tension controller, an another approach consists of synthesis a multi-model controller. A ﬁrst controller is opti- mized for a working point corresponding to the minimum web span length in the accumulator. Then, a second controller is optimized for the maximum web span length. The output of each controller is weighted by a coefﬁcient a (22) depending on the web length L stored in the accumulator. To avoid problem of integration during the controllers interpolation, the controllers integral terms are frozen when the controllers are not used. The evolution of a during web span length variation can be found in Fig. 10 The control scheme is given in Fig. 11 As observed in Fig. 11, controlling an accumulators using a multi-model struc- ture is more complex; in fact, we have to synthesis four con- trollers by optimization, and so the computational time will be longer. This control structure (Fig. 11) is similar to the LPV control structure, but the synthesis methods of the controllers is differ- ent. For LPV controllers [24,25], controllers are synthesized simultaneous to guaranty quadratic stability. LPV controllers a priori may be robust. With multi-model controllers, each con- troller is synthesized separately, so that the quadratic stability cannot be guaranty [26]. We have to take into account the controllers robustness during the optimization process (by pena- lizing unstable controller parameters set) and check the robust- ness range of our controllers, by simulation, after synthesis: a ¼ 1 L À 1 Lmin 1 Lmax À 1 Lmin ð22Þ Accumulator C1 Controller switching strategy C2 Vref Lref Vin accu Vout accu Tout accu L Tref + + + + - - Fig. 8. Control scheme for linear time invariant PI controllers. 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 Time (s) Velocity(m/s) Web velocity 0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 Actuator switching strategy Time (s) Fig. 9. Evolution of the controller on/off weighting coefﬁcient. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 737
- 7. 3.2.3. Linear time invariant H1 controller To improve the control performances and the robustness to the mechanical parameter variations shown in the previous part, a H1 controller has been synthesized. The H1 controller is calculated using the linear state space described in (15), according to the framework S/KS/T proposed in Fig. 12. The weighting functions Wp, Wu and Wt used for loop-shaping appear in the closed-loop transfer function between the exogenous input r, and the performances outputs z in the following manner: Tr-z ¼ WpS WuKS WtT 2 6 4 3 7 5 ð23Þ where S is the sensitivity function S ¼ ð1þKGu-yÞÀ1 and T ¼ IÀS. Exogenous input r is the web tension reference. The control scheme is the same as for PI controllers (see Fig. 8). The H1 problem consists to ﬁnd a stabilizing controller K which mini- mizes the H1-norm of the transfer function Tr-z between external input r and performance outputs z JTr-zJ1 ¼ sup o ðsðTr-zÞÞrg ð24Þ 0 10 20 30 40 50 60 70 80 90 100 1 1.5 2 2.5 3 3.5 Web length Time (s) Length(m) 0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 Alpha Time (s) Fig. 10. Evolution of the multi-model PI controllers weighting coefﬁcient a. Accumulator C1 Controller switching strategyC2 Vref Lref Tref + + -- + + L Vin accu Vout accu Tout accu 1- C3 C4 1- Lref Lref + + + + Fig. 11. Control scheme for multi-model PI controllers. K G Wp Wu Wt z1 z2 z3 y u e r + - Fig. 12. H1 framework. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742738
- 8. The weighting function Wp is usually taken with a high gain at low frequency in order to reject low frequency disturbances. The form of Wp is as follow [20]: WpðsÞ ¼ s M þo0 sþo0e0 ð25Þ where M is the maximum peak magnitude of the sensitivity S, o0 is the required frequency bandwidth, e0 is the steady state error allowed. The weighting function Wu is used to avoid large control signals. Wt is selected to attenuate high frequencies. As for PI controllers, the web length has a signiﬁcant inﬂuence on the web tension, therefore a synthesis of a multi-model H1 controller will be presented in the next part. 3.2.4. Multi-model H1 controller The synthesis of a multi-model H1 controller is based on the same approach as for a LTI H1 controller. For each actuator one H1 controller is ﬁrst synthesized for the minimum web length 0 10 20 30 40 50 60 70 80 90 100 0 100 200 300 400 500 600 700 Time (s) Tension(N) Web tension (N) Toutref ToutPILTI ToutPImulti_model 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 Time (s) Velocity(m/s) Web velocity (m/s) Vout VdPILTI VdPImulti_model 0 10 20 30 40 50 60 70 80 90 100 1 1.5 2 2.5 3 3.5 Web length (m) Time (s) Length(m) Lref LPILTI LPImulti_model a b c d e f g Fig. 13. Accumulator simulation results for industrial PI controllers. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 739
- 9. and second controllers for the maximum web length. The same control strategy as for a multi-model PI controller (Fig. 11) is used. 3.3. Nonlinear realistic simulation results 3.3.1. Industrial PI controllers simulation results The simulations represented in this part result from the realistic nonlinear model programmed in Matlab/Simulink software envir- onment. Our principal accumulator performance criteria concerns the maximum amplitude of tension variations in relation to the reference tension, during the wound roll change phase (when the unwinder is stopped). Accumulator simulation sequence: (a) from t¼25 to t¼35 s: the accumulator is at its nominal web length, velocity and tension. (b) from t¼35 to t¼41 s: the accumulator carriage is moved up to reach the maximum web length, the velocity is increased to maintain nominal web tension. (c) from t¼41 to t¼60 s: the accumulator is charged and main- tained at maximum web length. (d) from t¼60 to t¼64 s: the input velocity decreases and the accumulator carriage is moved down to restore web and maintain the nominal web tension. (e) from t¼64 to t¼66 s: the input velocity is equal to zero, the accumulator carriage is moved down to restore web and maintain the nominal web tension. (f) from t¼66 to t¼70 s: the input velocity increase to reach the nominal velocity, the accumulator carriage is moved down to restore web and maintain the nominal web tension. (g) from t¼70 to t¼100 s: the accumulator is at its nominal web velocity and tension, the accumulator is maintained at a constant position to maintain a constant web length in the accumulator. One can observe in Fig. 13 that optimized PI controller allows to ensure moderate tension peaks and variation, even during the wound roll change phase (d, e and f). As expected, multi-model PI controllers lead to better performances during the unwinder roll change phase (d, e and f). This performances improvement can be observed in Fig. 14 which represent the relative absolute tension error for both control strategies (LTI and multi-model) during the wound roll change phase. One can observe that for multi-model PI controllers, the maximum tension error is signiﬁcantly reduced (up to 57%). 3.3.2. H1 controllers simulation results Fig. 15 presents LTI H1 controllers simulation results. The simulation sequence is the same as indicated previously. Using H1 controllers leads to ensure moderate tension variation, even during the wound roll change phase. As expected, H1 controllers lead to better performances than PI controllers: LTI H1 controller reduces signiﬁcantly the maximum web tension peaks in relation to the reference tension, during all the simulation sequence, especially during the wound roll change phase (see phases d, e 60 62 64 66 68 70 72 0 20 40 60 80 100 Time (s) Error(%) Relative absolute tension error for PI control strategies PI LTI Fig. 14. Relative absolute tension error for LTI and multi-model PI control strategies during wound roll change phase. 0 10 20 30 40 50 60 70 80 90 100 0 100 200 300 400 500 600 700 Time (s) Tension(N) Web tension (N) Toutref ToutPILTI Tout a b c d e f g Fig. 15. Accumulator simulation results for linear time invariant H1 controllers. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742740
- 10. and f in Fig. 15). The little static error at the beginning of the simulation results from the terms neglected during the model linearization. Indeed, the H1 controller is calculated for the linear model and applied on the nonlinear one. Fig. 16 presents multi- model H1 controllers simulation results. Unlike PI controllers, multi-model H1 do not increases signiﬁcant performances, indeed, linear time invariant H1 performances are already improved compared to PI controllers. A minor performances improvement (tension peaks reduction) can be observed in Fig. 17 which represents the relative absolute tension error for both control strategies during the wound roll change phase. 4. Conclusion This paper presents the physical equations and the modelling of an industrial accumulator. The detailed nonlinear model, programmed in Matlab/Simulink software environment, enables 0 10 20 30 40 50 60 70 80 90 100 0 100 200 300 400 500 600 700 Time (s) Tension(N) Web tension (N) Toutref Tout Tout multi_model a b c d e f g 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 Time (s) Velocity(m/s) Web velocity (m/s) Vout Vd H Vd H 0 10 20 30 40 50 60 70 80 90 100 1 1.5 2 2.5 3 3.5 Time (s) ref H H a b c d e f g a b c d e f g Fig. 16. Accumulator simulation results for multi-model H1 controllers. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 741
- 11. a model based optimization of PI controllers. The linear model is used to study the effect of some mechanical parameters on the accumulator, with the help of Bode diagrams, and enables to synthesize H1 controllers. For both controllers, linear time invar- iant and multi-model control strategies are compared. Simulation results comparing both control strategy have also been presented. As expected, multi-model control strategy improves signiﬁcantly the performances. Acknowledgment The authors wish to thank Monomatic France Company for their helpful discussions. Also thanks to the Re´gion Alsace for having partially founded this work. References [1] Zahlan N, Jones DP. Modelling web traction on rollers. In: International conference on web handling IWEB’3, Stillwater, Oklahoma, USA, 1995. [2] Brandenburg G. ¨Uber das dynamische Verhalten durchlaufender elastischer Stoffbahnen bei Kraft¨ubertragung durch Coulomb’sche Reibung in einem System angetriebener, umschlungener Walzen. 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On the selection of tuning methodology of FOPID controllers for the control of higher order processes. ISA Transaction 2011;50:376–88. [24] Poussot-Vassal C. Commande Robuste LPV Multivariable de Chˆassis Auto- mobile. PhD thesis, Institut Polytechnique de Grenoble; 2008. [25] Gassmann V, Knittel D. Robust PI-LPV tension Control with elasticity observer for roll-to-roll systems. In: IFAC world congress, Milan, Italy, 2011. [26] Biannic J-M. Commande robuste des systemes a parametres variables. PhD thesis, SUPAERO; 1996. 60 62 64 66 68 70 72 0 10 20 30 40 50 Time (s) Error(%) Relative absolute tension error for H control strategies H H LTI Fig. 17. Relative absolute tension error for LTI and multi-model H1 control strategies during wound roll change phase. D. Kuhm et al. / ISA Transactions 51 (2012) 732–742742

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