A Study on Control of Accumulators in Web Processing Lines

Transcription

1 Prabhakar R. Pagilla* Associate Professor Inderpal Singh Graduate Student Ramamurthy V. Dwivedula Graduate Student School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK A Study on Control of Accumulators in Web Processing Lines Design of a stable controller and observer for web tension regulation in an accumulator, in web processing lines, is considered in this paper. A simplified average dynamic model that includes accumulator carriage dynamics, average web tension dynamics in accumulator web spans, and the process and exit side driven roller dynamics is considered in the study. A feedback controller together with an observer for average web tension is proposed. It is shown that the proposed feedback controller and observer results in a stable closed-loop system. Simulation results on an industrial continuous web processing line are given and discussed for the proposed controller/observer and compared with a currently used industrial controller. Applications of this research can be found in web process lines where accumulators are used to store/release material to ensure continuous operation. DOI: / Introduction Any continuous material whose width is significantly less than its length and whose thickness is significantly less than its width can be described as a web. Plastic wrap, paper, film, and aluminum strip are examples of web. It is important that the tension in a web span be maintained within a close tolerance band of the desired tension during the processing of a web. For example, if the tension in the web changes during printing/perforating processes, the print perforation gets skewed. Further, excessive tension variations may cause wrinkles and may even tear the web. As the demand for higher productivity and better performance from the web processing industry increases, better models and more accurate control algorithms for the processes must be developed. Tension control plays a key role in improving the quality of the finished web. It is essential to keep the web in the process at a preset tension, which could change throughout the process by many conditions such as disturbances from uneven rollers and web speed variations. A continuous web processing line is a large-scale complex interconnected dynamic system with numerous control zones to transport the web while processing it. A continuous web processing line typically consists of an entry section, a process section, and an exit section. The entry section consists of an unwind stand, a tension leveller, and an entry accumulator. Operations such as wash, coat, and quench on the web are performed in the process section in the case of aluminum and steel webs, and printing, perforating, and laminating in the case of other consumer products. The exit section consists of an exit accumulator and a rewind stand. A typical process line layout of an aluminum strip processing line is shown in Fig. 1. Accumulators are primarily used to allow for rewind or unwind core change while the process continues at a constant velocity; they are also known by the name festoons in the non-metals industry. Dynamics of the accumulator directly affects the behavior of web tension in the entire process line. Tension disturbance propagation both upstream and downstream of the accumulator has been noticed due to motion of the accumulator carriage. Although sufficient amount of work has been done in Refs. *Corresponding author. Contributed by the Dynamic Systems, Measurement, and Control Division of THE AMERICA SOCIETY OF MECHAICAL EGIEERS for publication in the ASME OURAL OF DYAMIC SYSTEMS, MEASUREMET, AD COTROL. Manuscript received by the ASME Dynamic Systems and Control Division February 8, 2003; final revision, uly 8, Associate Editor: R. Mukherjee. 1 7, and is currently being done in tension control of a web, very little published research exists in modelling and control of accumulators in web processing lines. In Ref. 1, a mathematical model for longitudinal dynamics of a web span between two pairs of pinch rolls, which are driven by two motors, is given; this model does not predict tension transfer and does not consider tension in the entering span. A modified model that considers tension in the entering span was developed in Ref. 3. In Ref. 4, the moving web was considered as a moving continuum and general methods of continuum mechanics were used in the development of a mathematical model. An overview of the lateral and longitudinal behavior and control of moving webs was presented in Ref. 5. A review of the problems in tension control of webs can be found in Ref. 6. Discussions on tension control versus strain control and torque control versus velocity control were given in Ref. 7. Accumulators in web processing lines are important elements in all web handling machines. Functional importance of these in web processing lines is quite substantial as they are primarily responsible for continuous operation of web processing lines. A preliminary study on modelling and control of accumulators was given in Ref. 8 ; characteristics of the accumulator and its operation were explained; throughout the paper the dynamic behavior of the accumulator carriage, web spans, and the current methods used in controlling the carriage were discussed. A dynamic model for accumulator spans that consider the time-varying nature of the span length was developed in Ref. 9. In this paper, we consider control of the accumulator carriage in conjunction with control of the driven rollers both upstream and downstream of the accumulator. The average dynamic model developed in Ref. 8 is further simplified based on practical observations and is used for controller design. The design of the control algorithm is carried out based on Lyapunov s second method. An observer for estimating the average web tension in accumulator web spans is also developed during the design process. Stability of the closed-loop system with the proposed controller/observer is shown. Simulation results on an industrial continuous web process line for a typical operation of an accumulator are conducted using the proposed controller and observer; these results are compared with the simulation results of the current control techniques that are used in the industry. The remainder of the paper is organized as follows. Section 2 gives the dynamics of the accumulator carriage, tension in the web spans, and driven rollers upstream and downstream of the accumulator, together with a discussion on obtaining simplified ournal of Dynamic Systems, Measurement, and Control SEPTEMBER 2004, Vol. 126 Õ 453 Copyright 2004 by ASME

2 Fig. 1 Typical Web Process Line Layout and Terminology average tension model in the entire accumulator. Controller design and observer design are given in Section 3; stability of the proposed controller/observer is also shown. Simulation results using an industrial web process line as an example are given and discussed in Section 4. Conclusions and future research are given in Section 5. 2 Dynamics of Accumulator Carriage, Web Tension, and Driven Rollers A schematic of the carriage, web spans, and rollers within an accumulator is shown in Fig. 2; this figure shows an exit accumulator, that is, the web is fed into the accumulator from the process section. The accumulator carriage dynamics is given by M c ẍ c t u c t F d t M c g j 1 t j t, (1) where M c is the mass of the carriage, x c is the carriage position, v c is the carriage velocity ( ẋ c ), u c (t) is the controlled force, u c F r F p, F d is the disturbance force, g is the acceleration due to gravity, is the number of spans, and t j is the web tension in the j-th span. The controlled force, u c (t), is generated by a hydraulic system. The disturbance force, F d (t), includes friction in the hydraulic cylinder and the rod seals, friction in the carriage guides, and other external forces on the carriage. The torque shaft shown in Fig. 2 is included in the accumulator design to synchronize the side to side lifting action so that only vertical motion needs to be considered in the control system design. The number of rollers on the carriage is assumed to be /2. The number of rollers in the accumulator is 1. The strip tension and the roller dynamics in the jth accumulator span is given by 9, ṫ j t R x c t j t j 1 t R x c t t j 1 t j 1 t t j t j t x c t ẋ c t 1 x c t t j t ẋ c t, (2) j t B j j t R t j 1 t t j t, (3) where j 1,...,, A is the area of cross-section of the web, E is the modulus of elasticity of the web, R is the radius of each roller; it is assumed that all rollers are of same radius, which is typically the case in web processing lines, j is the angular velocity of the j-th roller, is the moment of inertia of the roller, and B j is viscous friction coefficient of the j-th roller. otice that there is a strong coupling between the carriage dynamics, Eq. 1, strip tension dynamics, Eq. 2, and the roller dynamics, Eq. 3. To obtain a simpler tension dynamic model for all the accumulator web spans in an average sense, an average tension is defined as: t c t 1 t j t. (4) j 1 This concept of average tension enables us to consider an idealized situation where the tension in each span in the accumulator is t c as shown in Fig. 3. Taking the sum from j 1 toj of both sides of Eq. 2 and dividing by results in ṫ c t R 1 x c t j t j 1 t j 1 1 j 1 R x c t t j 1 t j 1 t t j t j t 1 j 1 x c t ẋ c t 1 1 j 1 x c t t j t ẋ c t. (5) Fig. 2 Sketch of an Exit Accumulator 454 Õ Vol. 126, SEPTEMBER 2004 Transactions of the ASME

3 driven rollers which 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 process-side web velocities are: v 1 e t B fev e t R 2 t r t c t RK e u e t R 2 e t, (11) Fig. 3 Sketch of an Equivalent Exit Accumulator v 1 p t B fpv p t R 2 t c t t r RK p u p t R 2 p t. (12) Thus, the state-space form of the dynamics of the carriage, average tension and the entry/exit rollers is summarized in Eqs ṫ c t x c v t c t 1 v e t v p t, (13) ẋ c t v c t, (14) v c t 1 M c t c t F f v c t u c t g, (15) Evaluating the sum on the right-hand-side results in ṫ c t R 1 x c t t 0 t 1 R x c t t 0 t 0 t t t t x c t ẋ c t 1 x c t t c t ẋ c t. (6) We can further simplify the average tension dynamics, Eq. 6, under the assumption that the product is much larger than the tensions t 0, t, and t c. This is generally true for all web materials processed in web processing machines. Under this assumption the second and last term can be ignored from the average tension dynamic model, Eq. 6, resulting in the following simplified average tension dynamic model: ṫ c t R 1 x c t t 0 t x c t ẋ c t. (7) Assuming that there is no slip on the rollers 0 and shown in Fig. 2, the roller angular velocities and the web velocity are related as follows: v e R and v p R 0. Therefore, Eq. 7 can be written as: ṫ c t 1 x c t v e t v p t x c t ẋ c t. (8) The driven roller angular dynamics at the process side and the exit side of the accumulator are given by the following equations: e t B fe e t R t e t t c t K e u e t, (9) p t B fp p t R t c t t p t K p u p t, (10) where e (t) and p (t) are the exit side and process side driven roller angular velocities, respectively, B fe and B fp are the viscous friction coefficients in the exit side roller and process side roller, respectively, t e (t) and t p (t) are the web tension in the span downstream of the exit-side roller and in the span upstream of the process-side roller, K e and K p are positive gains, and u e (t) and u p (t) are exit side and process side driven roller control inputs. otice that t e (t) and t p (t) are given by the tension dynamics downstream of the exit-side driven roller and upstream of the process-side driven roller; since in this work we are interested in the web tension behavior within the accumulator, we assume that t e (t) and t p (t) are maintained close to the desired web tension, that is, t e (t) t r e (t) and t p (t) t r p (t), where t r is the desired web tension in the process line, and e (t) and p (t) are disturbances. Assuming that there is sufficient web wrap on the v 1 e t B fev e t R 2 t r t c t RK e u e t R 2 e t, (16) v 1 p t B fpv p t R 2 t c t t r RK p u p t R 2 p t. (17) 3 Controller Design The goal is to design control algorithms for the accumulator carriage, the exit-side and the process-side driven rollers such that the average web tension, the carriage position, the carriage velocity, the exit-side and process-side web velocities track their desired trajectories. Currently, the industrial controller uses only feed-forward of the desired profile for the accumulator carriage and feed-forward plus PI controllers for the exit and process-side driven rollers. The industrial controllers for the accumulator carriage, exit-side driven roller, and process-side driven roller are given by u ci t M c v cd t g v f v d M c t t c M c c d, (18) u ei t RK e B fe v e d t v ed t k pe e ve t k ie e ve t d, (19) u pi t RK p B fp v d p t v pd t k pp e vp t k ip e vp t d, (20) where k pe and k pp, and k ie and k ip are the proportional and integral gains, respectively. One can neither show closed-loop stability nor guarantee any performance in the presence of disturbances using the industrial controller. It was noticed on industrial process lines that periodic oscillation in the system variables due to disturbances in the process are sustained for a considerable time. Hence, there is a need to develop a stable controller that is capable of achieving stability as well as good performance in the presence of disturbances. Towards this end, we design a controller based on Lyapunov s second method. The approach for designing the proposed controller is to augment the industrial controller by auxiliary feedback terms that will ensure stability as well as result in improved performance. Further, the approach also allows us to construct an average tension observer to estimate the average tension in the accumulator spans based on the measurements. ournal of Dynamic Systems, Measurement, and Control SEPTEMBER 2004, Vol. 126 Õ 455

4 In the following, it is assumed that all the state variables are measurable except for the average web tension, t c (t). An observer will be designed to estimate the average web tension. Also, consider the following error variables: e tc (t) t c (t) t d c, e xc (t) x c (t) x d c (t), e vc (t) v c (t) v d c (t), e ve (t) v e (t) v d e (t), and e vp (t) v p (t) v d p (t), where t d c is the desired web tension, x d c (t) and v d c (t) are the desired accumulator position and velocity, respectively and v d e (t) and v d p (t) are desired exit and process-side velocities, respectively. Choose the following control inputs for the accumulator carriage, exit-side driven roller, and the process-side driven roller: u c t M c v cd t g v f M c v c d t M c t c d u ca t u e t RK e u p t RK p B fe v e d t v ed t u ea t B fp v p d t v pd t u pa t, (21), (22), (23) where u ca (t), u ea (t), and u pa (t) are auxiliary control inputs that will be designed later. Substituting the control input and using the error definitions, the error dynamics is given by ė tc t x c t e vc t x c t e ve t e vp t, (24) ė xc t e vc t, (25) ė vc t M c e tc t v f M c e vc t u ca t, (26) ė ve t 1 B fee ve t R 2 e tc t R 2 e t u ea t, (27) ė vp t 1 B fpe vp t R 2 e tc t R 2 p t u pa t. (28) Consider the following observer for estimating the average tension dynamics: tˆ c t x c t e vc t x c t e ve t e vp t fˆ t c, tˆc 0 tˆc0, (29) where tˆc(t) is the estimate of t c (t), and fˆ t c will be chosen later during the stability analysis. Define the observation error as ẽ tc (t) t c (t) tˆc(t). Also, define ê tc (t) tˆc(t) t c d. Using these definitions, the observer error dynamics is obtained as e8 tc t fˆ t c. (30) otice that we have used the fact that v c d (t) (v p d (t) v e d (t))/, that is, the difference between the desired process velocity and the desired exit velocity divided by the number of spans gives the desired carriage velocity. Consider the following Lyapunov function candidate for the accumulator carriage system: V c t 1 2 e tc 2 t 1 2 e 2 xc t 1 2 e 2 vc t 1 2 ẽ tc 2 t. (31) The derivative of the Lyapunov function candidate along the trajectories of the error dynamics is given by V c t e tc ė tc e xc ė xc e vc ė vc ẽ tc e8 tc e x tc e vc c x c t e ve e vp e tc e xc e vc e M tc e vc v f e 2 c M vc u ca t e vc c fˆ t c ẽ tc. (32) Choose the following auxiliary control input for the carriage: u ca t x c t ê tc t e xc t ê M tc t 3 e vc t, (33) c where 3 is a positive gain. Substituting Eq. 33 into Eq. 32, the derivative of the Lyapunov function candidate becomes V c t 3 v f M c e 2 vc x c t e ve e vp e tc M c ẽ tc e vc fˆ t c ẽ tc. (34) Consider the following Lyapunov function candidate for the exit and the process-side roller dynamics: V ep t 1 2 e 2 ve t 1 2 e vp 2 t. (35) The derivative of V ep (t) along the trajectories of the error dynamics, Eqs. 27 and 28, is V ep t 1 B fee 2 ve R 2 e tc e ve R 2 e t e ve e ve u ea 1 B fp e 2 vp R 2 e tc e vp R 2 p t e vp e vp u pa. (36) Consider the following Lyapunov function candidate for the combined system of the accumulator carriage, exit-side driven roller, and the process-side driven roller: The derivative of V(t) is V t V c t V ep t. (37) V t 3 v f M c e 2 vc x c t e ve e vp e tc x c M c ẽ tc e vc fˆ t c ẽ tc 1 B fee 2 ve R 2 e tc e ve R 2 e t e ve e ve u ea 1 B fpe 2 vp R 2 e tc e vp R 2 p t e vp e vp u pa. Assuming that e (t) and p (t) are bounded by some known constants, that is, e (t) e and p (t) p, we choose the following auxiliary control inputs: u ea t e e ve t x c ê tc t e sgn e ve, u pa t p e vp t x c ê tc t p sgn e vp, x c (38) where e and p are positive gains. Using these auxiliary control inputs and re-arranging terms, we obtain 456 Õ Vol. 126, SEPTEMBER 2004 Transactions of the ASME

5 Table 1 Parameters of the accumulator Description Symbol Value Mass of the carriage M c 7310 kg slugs Cross-section area of web A m in 2 Modulus of elasticity E /m 2 (10 7 psi) o of web spans 34 Viscous friction coefficient v f s/m lb-sec/in Radius of exit and process-side roller R m 6 in Moment of inertia kg-m slugs-in 2 Bearing friction coefficient B f m-s 0.02 lb-in-s V t 3 v f Choosing M c e 2 vc e B fe e 2 ve p B fp 2 e vp e ve e e t e vp p p t x c M c ẽ tc e vc x c ẽ tc e ve x c ẽ tc e vp fˆ t c ẽ tc. (39) fˆ tc x c M c e vc x c e ve e vp, (40) we get V t 3 v f M c e 2 vc e B fe e 2 ve p B fp e 2 vp. (41) Therefore, V(t) 0 is a non increasing function of time for all t 0. Hence, V(t) L and lim t V(t) V. Also, e tc (t), ẽ tc (t), e xc (t), e vc (t), e ve (t), e vp (t) L and e vc (t), e ve (t), e vp (t) L 2. From the error dynamics, Eqs , ė xc (t), ė vc (t), ė ve (t), ė vp (t) L. Therefore, using Barbalat s lemma 10, we have e vc (t), e ve (t), e vp (t) 0 ast. The following theorem summarizes the results of this section. Theorem 3.1 For the dynamics of the accumulator carriage and the driven rollers upstream and downstream of the accumulator given by Eqs. (13) through (17), the following control inputs u c t M c v cd t g v f v d M c t t d c M c c x c t ê tc t e xc t ê M tc t 3 e vc c t, (42) u e t R2 RK e e e ve t x c t ê tc t e sgn e ve B fe v e d t v t ed, (43) u p t R2 RK p p e vp t x c t ê tc t p sgn e vp B fp v d p t v t pd, (44) and the average tension observer tˆ c t 2 x c t M c e vc t 2 R2 x c t e ve t e vp t, tˆc 0 tˆc0, (45) will result in the signals e tc (t), ẽ tc (t), e xc (t), e vc (t), e ve (t), e vp (t) being bounded and further, the signals e vc (t), e ve (t), e vp (t) asymptotically converge to zero. Equations 42), (43, 44, and 45 give the accumulator carriage input, exit-side driven roller input, process-side driven roller input, and the observer dynamics, respectively. Remarks 1. The dynamics of the carriage, average tension, and the entry/ exit rollers, given by Eqs , is a strongly coupled system of equations. Thus, the input terms say u e ) in the dynamic equations affects the other equations. The effect of interconnections is ignored in the industrial controller given by Eqs The proposed controller incorporates the effects of interconnections and thus, improved performance is expected. otice that the proposed controller, Eqs. 42, 43, 44, is centralized, i.e., each control input requires the knowledge of the position of the carriage, and the estimate of the average tension, which in turn depends on velocity measurements of the carriage, process-side and exit-side driven rollers. 2. The industrial controller uses proportional and integral action for the exit-side and process-side driven rollers. It is difficult to show analytically that integral action ensures convergence of the states to their desired values. However, it is intuitive to expect, from linear control theory, that the integral action obviates steady state errors. The proposed controller does not have explicit integral action: the integral action is implemented indirectly via the observer dynamics. Each of the Eqs. 42, 43, and 44 contains the term, ê tc (t) tˆc(t) t d c and the term tˆc(t) is obtained by integrating the observer dynamics given by Eq The effect of the proportional and integral gains in the industrial controller on the stability is difficult to establish analytically since the system is nonlinear. Some values of these gains may result in instability. On the other hand, the effect of gains 3, e, and p on the stability is straight forward in the case of proposed controller: any positive gains ensure stability. 4. In the case of the proposed controller, the rate of convergence of the controlled variables to their reference values depends on the constants, 3, e, and p, in Eqs Rapid convergence requires large values of these constants. However, large values of these constants would mean higher magnitude of control efforts, u c, u e, and u p which translates to higher power consumption and/or the requirement of actuators with higher load capability. 4 Simulation Study In this section, the proposed control scheme is investigated by conducting simulations on an industrial continuous web process line. The simulations are performed using the parameters of an Alcoa continuous process line CPL and its exit accumulator. Different values of the parameters of the accumulator used in the simulations are given in Table 1. The desired tension in the web spans is lbs. The desired process speed is 3.3 m/s 650 fpm. A typical scenario of the exit speed and the carriage ournal of Dynamic Systems, Measurement, and Control SEPTEMBER 2004, Vol. 126 Õ 457

6 Fig. 4 Desired exit speed, carriage speed and carriage position during rewind roll-change Fig. 5 Three cases of sinusoidal disturbances speed during a rewind roll change is depicted in Fig. 4. The rewind roll change-over scenario, when the web velocity in the process section is maintained at a constant value, is described in the following steps by referring to Fig. 4: i AB velocity of the web in the rewind side is decelerated to zero from a value of 3.3 m/s 650 fpm, as a result of this the accumulator starts collecting the web and the carriage accelerates upwards; ii BC rewind stops and the carriage is moving up with constant velocity; iii CD after rewind roll change, exit side is accelerated up to the process speed, in this period the carriage is moving up while decelerating; iv DE exit side is accelerated up to a speed above the process speed, 4.8 m/s 950 fpm in this case; v EF exit speed is maintained at this constant speed; vi FG exit speed is reduced to the process speed. The desired profile for carriage velocity is given by: v d c t v d p t v d e t. From this velocity profile the desired carriage position profile is calculated. After each cycle of rewind roll-change operation the carriage will return to its original position; this means that the area under the carriage velocity curve is zero. The goal is to track the desired profiles of the carriage position and speed, exit velocity and process velocity while maintaining the desired level of the average web tension. The simulations are conducted using the system model given by Eqs and the control algorithms given by Eqs. 42, 43, 44, and the observer given by Eq. 29. Two types of controllers are considered for a comparative study via simulations: 1 Currently used industrial controller, and 2 the controller proposed in this paper. The control inputs for the exit and processor side driven rollers are torques generated by electric motors. The control input for the carriage is a force, which is either generated by an electro-hydraulic or electro-mechanical actuator. In the simulations, the values of the gains used in the industrial controller are k pe 100, k pp 100, k ie 0.1, and k ip 0.1. The gains used in the proposed controller are 3 100, e 100, and p 100. Observe that the magnitudes of these gains are comparable with the gains of the industrial controller thus, a comparison of the control inputs generated by the controllers is meaningful. The disturbances e and p considered in Eqs. 16 and 17 are of sinusoidal nature with a frequency of 0.2 Hz and amplitude lbs. For comparing the results of the two types of controllers, three types of sinusoidal disturbances are introduced into the accumulator carriage dynamics, that is, into Eq. 15. The amplitude of these disturbances is 0.25 m/s 2 10 in/s 2 and the frequency is 0.5 Hz. Low frequency disturbances are used because they are typical to accumulator carriages in industrial process lines; generally, the carriage of an accumulator does not have the ability to respond to high frequency disturbances due to its large mass. We consider the three time profiles shown in Fig. 5 for introducing sinusoidal disturbances into the accumulator carriage. In the first case the sinusoidal disturbance is introduced throughout the time duration of 400 s. In the second case the sinusoidal disturbance is introduced at specific time intervals of 20:30 s, 106:126 s, and then from 318:328 s. In the third case the disturbance is introduced at specific time intervals of 20:30 s, 40:60 s, 96:116 s, 136:156 s, and 308:328 s. The time durations in the second and third case are picked to reflect observations made on an industrial process line that periodic disturbances occur during the initiation of carriage motion from its stationary state and when the exit velocity starts to accelerate or decelerate. The simulations are performed for these three cases of disturbances and for both types of controllers. All the gains that are common to both controllers are chosen to be the same. The results are shown in Figs. 6 through 17. The three cases of disturbances shown in Fig. 5 are labeled as Disturbance 1, Disturbance 2, and Disturbance 3, respectively. For each disturbance the following errors are shown: web tension error (e tc ), carriage position error (e xc ), carriage velocity error (e vc ), exit velocity error (e ve ), and process velocity error (e vp ). The control signals of accumulator carriage, exit-side driven roller, and process-side driven roller for both controllers are also shown. Fig. 6 State errors for the industrial controller: Disturbance Õ Vol. 126, SEPTEMBER 2004 Transactions of the ASME

7 Fig. 7 State errors for the proposed controller: Disturbance 1. Fig State errors for the industrial controller: Disturbance Figures 6 and 7 show the errors for the industrial controller and the proposed controller, respectively, plotted to the same scale to facilitate comparison. The top plot in Fig. 7 shows that the error in the tension is negligible when the proposed controller is used whereas the top plot in Fig. 6 shows that the error in the tension is considerably large when the industrial controller is used. This clearly indicates that the proposed controller effectively rejects sinusoidal tension disturbances. Figures 8 and 9 show the plots for control effort required for the two controllers considered, again plotted to the same scale to facilitate comparison. The control effort required to move the accumulator carriage is of similar Fig Control inputs for the industrial controller: Disturbance Fig State errors for the proposed controller: Disturbance Fig Control inputs for the proposed controller: Disturbance Fig. 12 Control inputs for the industrial controller: Disturbance 2. ournal of Dynamic Systems, Measurement, and Control SEPTEMBER 2004, Vol. 126 Õ 459

8 Fig. 13 Control inputs for the proposed controller: Disturbance 2. Fig Control inputs for te industrial controller: Disturbance magnitude in both schemes, as indicated by the top plots in Figs. 8 and 9. However, the control effort required for both the driven rollers for the proposed controller is smaller than the industrial controller. Thus, the performance of the proposed controller is superior compared to the industrial controller with little or no additional control effort. Fig. 17 Control inputs for the proposed controller: Disturbance 3. Fig Fig State errors for the industrial controller: Disturbance State errors for the proposed controller: Disturbance Figures 10 and 11 show the state errors of the industrial controller and the proposed controller. The y-axis scale in Fig. 11 is chosen appropriately to show even very small error in the tension signal. The tension error is again seen to be much small in the case of the proposed controller. Such small error in the tension is quite often tolerable. Figures 12 and 13 show the control inputs for the industrial controller and the proposed controller. It can be seen that the control effort required for the process side and the exit side rollers is much smaller for the proposed controller than the industrial controller. Similar observations can be made from Figs. 14 through 17 which show the state errors and the control inputs for the case of disturbance 3. 5 Conclusions and Future Work In this paper, feedback control algorithms for the accumulator carriage, and for the upstream and downstream driven rollers to the accumulator, were designed for tracking the desired exit and process web velocities, and to maintain the web tension at the desired level. It is common in the web handling industry to just apply a desired force on the carriage using a hydraulic system in opposition to the carriage weight and the force required to produce desired tension in all the accumulator web spans; thus, ignoring the dynamics of the carriage motion. This strategy often leads to large tension variations not only in the accumulator web spans but also in web spans in the entire process line due to tension disturbance propagation both upstream and downstream of the accumulator. Utilizing the notion of the average tension dynamics in the web spans of the accumulator allows us to design 460 Õ Vol. 126, SEPTEMBER 2004 Transactions of the ASME

9 stable control algorithms for the accumulator system, that includes the carriage dynamics, process-side and exit-side driven roller dynamics and the tension dynamics of the web spans. The proposed design results in better disturbance rejection capability and improved performance. Simulation results comparing the proposed control algorithm with the currently used control scheme in industry show that the proposed control algorithm results in much less web tension variations with comparable magnitudes of control effort. In the current work, we have assumed knowledge of the friction coefficients; in practice, the value of friction coefficients is not known. Future work should include adaptively estimating the friction coefficients. Instead of assuming that the force on the carriage is directly accessible as an input, future work should also include the dynamics of the actuator, either hydraulic or electric, coupled with the dynamics of the carriage and the web spans. In the current work, we have neglected the effect of the weight of the web on the accumulator carriage; the weight of the web on the carriage can be substantial in metal process lines, where the thickness of the web can be as high as 0.2 inches. The weight of the web increases with the increase in the length of the accumulator spans as the accumulator moves up; the effect of this time-varying weight must also be investigated in the future. Lyapunov s second method was used in this paper to systematically facilitate the design of the control algorithms and the tension observer. In future, it may be a worthwhile effort to investigate other potential nonlinear control designs from literature and compare them with the proposed controller. Finally, comparative experiments on an industrial process line should be performed to highlight the advantages and disadvantages of the proposed controller over the existing industrial controller. Acknowledgments This work was supported by the ational Science Foundation under Grant o. CMS The authors would like to thank the anonymous reviewers for their constructive and helpful comments. omenclature A Area of cross section of the strip B j, B fe, B fp Coefficient of viscous friction E Young s modulus of the material of the strip F d Disturbing force on the accumulator carriage F p, F r Forces on either side of the carriage g Acceleration due to gravity Moment of inertia of the roller about axis of rotation K e, K p Gains of the actuators M c Mass of the carriage Total number of spans in the accumulator R Radius of the roller t c Average tension in the accumulator t j (t) Tension in the jth span in the accumulator t p, t e Strip tension at the process end and exit end of the accumulator t r Desired tension in the strip u c Control force on the carriage ( F r F p ) u p, u e Control inputs on the rollers at the process side and exit side of accumulator v p, v e Strip velocity at the process end and exit end of the accumulator x c, v c Displacement and velocity of the carriage j (t) Angular velocity of the jth roller in the accumulator References 1 Campbell, D. P., 1958, Dynamic Behavior of the Production Process, Process Dynamics, ohn Wiley and Sons, Inc., ew York, first edition. 2 Grenfell, K. P., 1963, Tension Control on Paper-Making and Converting Machinery, IEEE 9-th Annual Conference on Electrical Engineering in the Pulp and Paper Industry, Boston, MA, pp King, D. L., 1969, The Mathematical Model of a ews Paper Press, ewspaper Techniques, pp Brandenburg, G., 1977, ew Mathematical Models for Web Tension and Register Error, International IFAC Conference on Instrumentation and Automation in the Paper, Rubber and Plastics Industry, vol. 1, pp Young, G. E., and Reid, K.., 1993, Lateral and Longitudinal Dynamic Behavior and Control of Moving Webs, ASME. Dyn. Syst., Meas., Control, 115, pp Wolfermann, W., 1995, Tension Control of Webs, a Review of The Problems and Solutions in The Present and Future, Proceedings of the Third International Conference on Web Handling, pp Shelton,.., 1999, Limitations to Sensing Web Tension by Means of Roller Reaction Forces, Proceedings of the Fifth International Conference on Web Handling, Stillwater, Oklahoma. 8 Pagilla, P. R., Garimella, S. S., Dreinhoefer, L. H., and King, E. O., 2001, Dynamics and Control of Accumulators in Continuous Strip Processing Lines, IEEE Trans. Ind. Appl. 37, pp Pagilla, P. R., King, E. O., Dreinhoefer, L. H., and Garimella, S., 2001, Robust Observer-Based Control of an Aluminum Strip Processing Line, IEEE Trans. Ind. Appl. 36, pp Khalil, H. K., 1996, onlinear Systems, Prentice Hall, Upper Saddle River,. ournal of Dynamic Systems, Measurement, and Control SEPTEMBER 2004, Vol. 126 Õ 461