Decentralized Control of Web Processing Lines - IEEE Xplore

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 1, JANUARY 2007

Decentralized Control of Web Processing Lines Prabhakar R. Pagilla, Member, IEEE, Nilesh B. Siraskar, and Ramamurthy V. Dwivedula

Abstract—The focus of this research is on modeling and design of a decentralized controller for web processing lines. First, an accurate dynamic model is developed for the unwind (rewind) roll in a web processing line by explicitly taking into account the timevarying nature of the roll inertia and radius. The unwind roll in a web processing line releases unfinished web to the process section; the rewind roll accumulates the finished web. Second, a strategy for computing the equilibrium inputs and reference velocities for each driven roll/roller is given; this strategy is based on dividing the web processing line into tension zones and using the reference web tension of each zone and the reference velocity of the master speed roller, which sets the desired web transport speed for the process line. Based on the new model developed, a decentralized controller is proposed. Variations in web tension and transport velocity in each tension zone are shown to exponentially converge to zero. A large experimental web platform, which mimics most of the features of an industrial process line, is used for experimentation. Extensive comparative experiments were conducted with the proposed decentralized controller and an often used decentralized industrial proportional-integral (PI) controller. A representative sample of the experimental results is shown and discussed. Index Terms—Decentralized control, distance to stability, inertia compensation, large-scale systems, Riccati equation, tension control, velocity control, web handling.

I. INTRODUCTION

A

WEB is any material which is manufactured and processed in a continuous, flexible strip form. Examples include paper, plastics, textiles, strip metals, and composites. Web processing allows us to mass produce a rich variety of products from a continuous strip material. Products that include web processing somewhere in their manufacturing include aircraft, appliances, automobiles, bags, books, diapers, boxes, newspapers, and many more. Web tension and velocity are two key variables that influence the quality of the finished web, and hence, the products manufactured from it. Web handling refers to the physical mechanics related to the transport and control of web materials through processing machinery. The primary goal of research in web handling is to define and analyze underlying sciences which govern unwinding, web guiding, web transport, and rewinding in an effort to minimize the defects and losses which may be associated with han-

Manuscript received September 8, 2005; revised April 5, 2006. Manuscript received in final form July 20, 2006. Recommended by Associate Editor D. Gorinevsky. This work was supported by the National Science Foundation under Grant CMS 9982071. P. R. Pagilla is with the School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078-5016 USA (e-mail: [email protected]). N. B. Siraskar is with Dexterous Technologies, Nashik 422001, India. R. V. Dwivedula is with Research and Development Department, Fife Corporation, Oklahoma City, OK 73114 USA (e-mail: [email protected]). Color versions of Figs. 1, 5, and 8–14 are available online at http://ieeexplore. ieee.org. Digital Object Identifier 10.1109/TCST.2006.883345

dling of the web. Web handling systems facilitate transport of the web during its processing which is typically an operation specific to a product; for example, in the case of an aluminum web, the web is brought to a required thickness, cleaned, heattreated, and coated; and in the case of some consumer products, lamination and/or printing on the web may be performed. Early development of mathematical models for longitudinal dynamics of a web can be found in [1]–[6]. In [1], a mathematical model for longitudinal dynamics of a web span between two pairs of pinch rolls, which are driven by two motors, was developed. 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 [3]. In [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. The study in [4] included the steady state and transient behavior of tensile force, stress, and strain in a web as functions of variables such as wrap angle, position and speed of the driven rollers, density, cross-sectional area, modulus of elasticity, and temperature. A dynamic model for the behavior of span tension in a web process line was developed in [5] and [6]. Based on the work in [4] and using conservation of mass principle, Whitworth derived the span tension dynamics which formed the basis for almost all the subsequent work on span tension dynamics. In [7], equations describing web tension dynamics were derived based on the fundamentals of web behavior and the dynamics of the drives used for web transport; an example system was considered to compare torque control versus velocity control of a roll for regulation of web tension. Nonideal effects such as temperature and moisture change on web tension were studied in [8]; based on the models developed, methods for distributed control of tension in multispan web transport systems were studied. An overview of lateral and longitudinal dynamic behavior and control of moving webs was presented in [9]. A review of the problems in tension control of webs can be found in [10]. Using Whitworth’s model for span tension dynamics, the dynamics of active/passive dancer mechanisms and an investigation of advanced control strategies for tension control were discussed in [11] and [12]. A decentralized adaptive control scheme for a class of large scale systems with application to web processing lines is discussed in [13]. controller for a web winding system A robust centralized consisting of an intermediate driven roller and unwind/rewind controller with rolls was proposed in [14]. A multivariable one or two degrees of freedom is proposed in [15]; two different controller structures, centralized and semi-decentralized, with and without overlapping were studied. Two schemes for decomposing a web transfer system into subsystems were presented in [16]: 1) disjoint decomposition and 2) overlapping decomposition. For each decomposition scheme, control design based on a linear, time-invariant dynamic model was considered. Dy-

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PAGILLA et al.: DECENTRALIZED CONTROL OF WEB PROCESSING LINES

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Fig. 2. Sketch of the experimental web line showing tension zones.

Fig. 1. Experimental web process line.

namics and control of accumulators in continuous strip processing lines was considered in [17]. The role of active dancers in attenuation of periodic tension disturbances was studied in [18] and [19]. A large amount of results and literature on decentralized control of large-scale systems can be found in [20]; although web processing application was not considered, the material is relevant to web processing systems. The work in this paper is aimed at developing an accurate dynamic model of the web handling machines and a stable controller that can accurately regulate web tension and transport velocity. A large experimental web line (Fig. 1), which mimics most of the features of an industrial web processing line, is considered as an example for model development. The experimental web line consists of unwind/rewind stations together with a number of intermediate rollers, some of which are driven to provide transport of the web from the unwind to rewind roll; details of the web line are given in Section IV-A. First, a model for the unwind (rewind) roll is developed by explicitly considering the variation of radius and inertia resulting from release (accumulation) of material to (from) the process. A dynamic model of the entire experimental web line is presented by considering each section of the web between two driven rollers as a tension zone; a natural subsystem formulation is proposed based on this strategy; each subsystem consists of the driven roller/roll dynamics and web tension dynamics, except for the master speed roller. The master speed roller is the first driven roller upstream of the unwind roll, whose sole purpose is to set and maintain the web transport speed; the subsystem corresponding to the master speed roller contains just the roller velocity dynamics. A strategy for computing the equilibrium inputs and reference velocities based on the reference of the master speed roller and reference tensions is given. Decentralized controllers are often preferred, and mostly used, by the web handling industry due to the ease of tuning individual stations; decentralized controllers also provide reliable operation of the process line in the event of occasional actuator and sensor malfunctions. A decentralized controller is proposed for the system, and a sufficient condition for exponential stability of the closed-loop system is developed. Extensive experiments were conducted with the proposed decentralized controller and results are compared with the implementation of a classical decentralized PI controller used in the web handling industry.

The contributions of this paper can be summarized as follows: 1) improved model for unwind/rewind roll dynamics; 2) systematic representation of subsystems and computation of reference velocities and equilibrium inputs; 3) design of a stable decentralized controller; and 4) comparative experimental evaluation of the proposed decentralized controller with an industrial controller on a large web processing line. The rest of this paper is organized as follows. Section II develops the dynamic model. The equilibrium control inputs and the decentralized feedback controller design are described in Section III. Description of the experimental platform and a representative sample of the comparative experimental results are shown and discussed in Section IV. Section V gives conclusions of the present work and some topics for future study. II. DYNAMIC MODELS It is common in the web handling industry to divide a process line into several tension zones by denoting the span between two successive driven rollers as a tension zone. Since the free roller dynamics has an effect on the web tension only during the transients due to acceleration/deceleration of the web line and negligible effect during steady-state operation, the assumption that the free rollers do not contribute to web dynamics during steady-state operation is reasonable and is often of value in practice, and is extensively used in the industry. This assumption will be used in developing the dynamic model in this section. Further, it will also be assumed that the web is elastic and there is no web slip on the rollers. Fig. 2 shows a web line with three tension zones; the line consists of the unwind/rewind rolls and two intermediate driven rollers. In the figure, LC denotes the load cell roller, which is mounted on a pair of load cells on either side for measuring for web tension. The driving motors are represented by represents input torque from the th motor, represents the transport velocity of the web on the th roller, and represents web tension in the span between th and th driven rollers. There are four sections in the web line shown in Fig. 2, which are the unwind section, master speed roller, process section, and rewind section. The name master speed roller is given to a driven roller which sets the reference web transport speed for the entire web line, and is, generally, the first driven roller upstream of the unwind roll in almost all web process lines; the purpose of the master speed roller is to regulate web line speed and is not used to regulate tension in the spans adjacent to it. The unwind/rewind rolls release/accumulate material to/from the processing section of the web line. Thus, their radii and inertia are time-varying. The dynamics of each of the four sections is presented in the following.

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Substitution of (4) and (5) into the velocity dynamics given by (3), and simplifying results in

(6) But the rate of change of radius , is a function of the velocity and the web thickness , and is approximately given by Fig. 3. Cross-sectional view of the unwind roll.

(7) Unwind section: A cross-sectional view of the unwind roll is shown in Fig. 3. The associated local state variables for the and tension . At any inunwind section are web velocity stant of time , the effective inertia of the unwind section is given by (1) where is the gearing ratio between the motor shaft and unis the inertia of all the rotating elements wind roll shaft, on the motor side, which includes inertia of motor armature, is the inertia of driving pulley (or gear), driving shaft, etc., is the inthe driven shaft and the core mounted on it, and ertia of the cylindrically wound web material on the core. Both and are constants, but the inertia due to cylindrically wound web material , is not constant because the web is , is continuously released into the process. The inertia, given by (2) is the density of the web material, where is the web width, is the radius of the empty core mounted on the unwind is the radius of the material roll. roll-shaft, and From Fig. 3, the velocity dynamics of the unwind roll can be written as

(3) where is the angular velocity of the unwind roll and is the coefficient of friction in the unwind roll shaft. The rate of is only because of the change in , and change in is given by from (2), the rate of change of (4) The velocity of the web coming off the unwind roll is related to the angular velocity of the unwind roll by . Hence, in terms of as one can obtain (5)

Notice that (7) is approximate because the thickness affects the rate of change of the radius of the roll only after each revolution of the roll; the continuous approximation is valid since the thickness is generally very small. Also, notice that the last term in the velocity dynamics (6) is often ignored in the literature under the assumption that the roll radius is slowly time-varying. But in practice the last two terms in (6) are significant for large transport velocities. Hence, equation (6) can be simplified to

(8) Dynamic behavior of the web tension , in the span immediately downstream of the unwind roll is given by (9) where is the length of the web span between unwind roller (M0) and master speed roller (M1), is the area of cross section of the web, is the modulus of elasticity of the web material, and represents the wound-in tension of the web in the unwind roll. Equation (9) is derived using the principle of conservation of mass applied to the control volume defined by the web span between two rollers. Complete details of this derivation and various other aspects such as effect of slip and span tension dynamics may be found in [5]. Slightly different versions of (9), obtained using the same principle but different approximation schemes, were discussed in [12] and [14]. Master Speed Roller: The dynamics of the master speed roller are given by (10) Process Section: The web tension and web velocity dynamics in the process section are given by (11) (12) Rewind Section: The web velocity dynamics entering the rewind roll can be determined along similar lines as those


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presented for the unwind roll. The web tension and velocity dynamics in the rewind section are

The web tension dynamics in the unwind section can be written as

(13) (17) (14) Equations (8)–(14) represent the dynamics of the web and rollers for the web line configuration shown in Fig. 2. Extension to other web lines can be easily made based on this model. For web process lines that have a series of process sections between the master speed roller and the rewind roll, (11) and (12) can be used for each process section. III. CONTROL DESIGN The control goal is to regulate web tension in each of the tension zones while maintaining the prescribed web transport velocity. To achieve this, first, one has to systematically calculate the control input required to keep the web line at the forced equilibrium of the reference web tension and web velocity in each of the zones. Then, some additional compensation must be included to provide error convergence in the presence of uncertainties and disturbances. We give a simple procedure for the calculation of equilibrium control inputs that is easy to understand and implement by practicing engineers. and Define the following variables: , where and are tension and velocity references, and are the variations in tension and verespectively, is the locity, respectively, around their reference values, control input that maintains the forced equilibrium at the refis the variation of the control erence values, and input. Define the state vector for the unwind section as and the state for the master speed roller as . After master speed section, define the state vector for the th for 2, 3. In the following, equisubsystem as librium control inputs and reference velocities are determined for each driven roll/roller based on the reference velocity of the master speed roller and reference tension in each tension zone.

From (17), assuming , , and as zero at the forced equilibrium, the relationship between the reference velocities and is given by (18) By choosing the reference velocity of the unwind roll as a function of the master speed roller as given by (18), the variational dynamics of the unwind section can be written as (19)

(20) Assuming the product of variations ational dynamics can be written as

, is negligible, the vari-

(21)

where

A. Equilibrium Control and Reference Velocities The velocity dynamics in the unwind section can be written as

(15) and At the forced equilibrium, assuming the variations and their derivatives as zero, the input that maintains this equilibrium is given by

(16)

where and are null matrices. For the master speed roller, similar equilibrium analysis gives the following velocity variation dynamics: (22) where

and

(23)

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Since arranged as


, dynamics for the master speed roller can be

(24) where

(30) B. Feedback Control Design

For the process section driven roller, the reference velocity and the equilibrium input are given by

Choose the decentralized control input for each section of the web line as follows:

(26)

(31) (32) (33) (34)

Therefore, the variational dynamics in the process section is given by

0, 1, 2, 3 are feedback gain vectors. The dynamics where , of the system under these decentralized control inputs becomes

(27)

(35)

(25)

. To generalize the problem to any web where processing line, we consider the general case of the previous . The following dynamics, that is, theorem gives the stability result. , of the dynamics given Theorem 1: The equilibrium, by (35) is exponentially stable, if the feedback gain vectors are chosen such that

where

(36) where gular value of

denote the smallest (largest) sin, and

The dynamics of the rewind section is given by

(28) where

and (29)

The quantity on the left-hand side of the inequality in (36) is [21]. The definition of disknown as distance to stability tance to stability and the proof of Theorem 1 are given in the appendix. Note that stability is guaranteed only when the mathematical model accurately approximates the tension behavior in the process. Further work is needed to study the robustness of the algorithm to modeling uncertainties such as unmodeled dynamics.


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Fig. 4. Typical web processing line.

To summarize, the control input is split into two components: one component maintains the forced equilibrium of the system and the other component compensates for perturbations from the equilibrium position. The design procedure previously discussed is listed as follows. 1) Decompose the given web processing plant into tension zones such as unwind section, master speed section, process section(s), and rewind section. The decomposition procedure followed in the paper is based on the physical arrangement of the system, i.e., a subsystem is considered by grouping variables near each drive motor as shown and as in Fig. 4. For example, subsystem 0 has elements in its state vector, subsystem 1 has as its state variable, and subsystem has and as elements in its state vector, and so on. Note that in an industrial process line, there could be more than one process section. The numbering scheme for denoting tensions and velocities in a general web processing line is shown in Fig. 4. 2) Obtain the dynamic model for each of the sections as discussed in Section II. If there are many process sections, (11), (12), and (27) are written for each of the process sections. 3) Compute the reference velocities of each of the sections using (18), (25), (29), and the given reference tensions. that satisfy condition (36); 4) Compute the gain matrices the gains may be found either by a pole placement technique or by posing it as an LQ problem (see Section IV-B). and state-feedback 5) Compute the equilibrium inputs . inputs Implementation procedure based on this proposed control design is explained in Section IV-B. IV. EXPERIMENTS A. Experimental Platform Fig. 5 shows a sketch of the experimental platform and the web path for conducting experiments using the proposed controller. The line mimics most of the features of an industrial web process line. It is developed with the aim of creating an openarchitecture design that allows for modifying the line to conform to research experimentation. The line contains a number of different stations, as shown in Fig. 5, and a number of driven rollers. For conducting the experiments for this work, the web is threaded through four driven rollers to as shown, and through many other idle rollers throughout the line to facilitate transport of the web from the unwind to rewind. The nip rollers (denoted by NR), which are pneumatically driven, are used to

Fig. 5. Sketch of the experimental platform.

Fig. 6. Decentralized control strategy with two PI controllers.

maintain contact of the web with the driven rollers. The two controlled lateral guides (guides are denoted by DG and the web edge sensors by E), near unwind and rewind sections, respectively, are used to maintain the lateral position of the web on the rollers during web transport. Three-phase induction motors, with 30-HP capacity, from Rockwell Automation are used to drive the unwind and rewind rolls, whereas master speed and process section rollers are driven by 15-HP induction motors. The motor drive system and the realtime architecture, which includes microprocessors, input/output (I/O) cards, and realtime software (AUTOMAX), are industrial components manufactured by Rockwell Automation. In the experimental platform, each motor is driven by a dedicated vector controller. Reference torque signals for each of the vector controllers are generated by control algorithm software in microprocessors, which are a part of the AUTOMAX distributed control system. To implement the desired control algorithms, programs in AUTOMAX can be modified offline using personal computer, and then uploaded to the dedicated microprocessors. Similar to a typical industrial web line control system, microprocessors used in the experimental platform are located in two racks, namely A00 and A01. Rack A00 has microprocessors and vector control drives for the rewind and process section roller . Rack A01 contains roll microprocessors and vector control drives for the unwind roll and master speed roller . Depending upon the number of process sections, actual industrial setup may have a large number of such racks. In most industrial web process lines, the decentralized control scheme for each section has two cascaded PI control loops, as shown in Fig. 6. Notice that the PI action is not acting on tension and velocity errors individually; the output of the tension loop becomes reference velocity error correction for the velocity loop. The implementation strategy for the proposed decentralized controller is shown in Fig. 7. The web material used in the experiments is Tyvek, which is a product made by Dupont. The product of the elasticity of the

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Fig. 7. Decentralized control strategy with proposed controller.

web material and its cross-sectional area , is equal to 8896 N. For both the master speed and process section driven rollers, the values of the radii and inertia are 0.0914 m and 0.0842 kg m , respectively. For inertia compensation and equilibrium control of unwind and rewind rolls, instantaneous radius of each roll is calculated using the encoder signal. Effective coefficient of is taken as 0.685 N ms/rad. The friction for all m, lengths of the three different web spans are m, and m. The Rockwell hardware uses a first-order tension filter with a cutoff frequency at 3 cycles/s, and encoders mounted on the motors are used to compute the motor speed signals. B. Implementation Procedure

US: MSS: PS:

To provide comparative results, two control schemes were implemented on the experimental platform shown in Fig. 5. These two schemes are shown in Figs. 6 and 7. In the following, we give an implementation procedure for each of these. The control scheme shown in Fig. 6 has two loops: an inner web transport velocity loop and an outer tension loop. To tune these control loops, first, the proportional and the integral gains of the inner velocity loops of individual motors are tuned to obtain best motor performance, i.e., to obtain smallest peak overshoot, and fastest settling time. Then, the gains of the outer tension correction loops are tuned to result in stable tension behavior. This tuning resulted in a lot of experimentation to arrive at the best possible combination of the P and I gains of both the loops on all the motors over the entire range of the radii of unwind/rewind rolls. Design procedure for the proposed controller is given in Section III. Implementation details related to computing the state feedback gains are given here. Since the pair is controllable, it is possible to adjust eigenvalues of arbifor the th subsystem trarily. We first choose a stable matrix that satisfies the sufficient condition (36), and then compute the feedback controller gain vector . The following nomenclature is used in the following: unwind section (US), master speed section (MSS), process section (PS), and rewind section (RS). Choose the following matrices for each subsystem:

RS: (38) It is also possible to cast the problem of finding as an LQ-problem, thus, ensuring that the feedback gains computed not only satisfy the sufficient condition (36) but also inherit the properties of LQ-control strategy. The idea of posing an LQ-problem is presented for the unwind section in the following. 1) LQ Approach for Finding the Closed-Loop State Matrix: Defining (39) we see that if were to be the LQ-optimal closed-loop state , its second row is the LQ-opmatrix for the pair timal state feedback gain for the system . To show this more clearly, consider the dynamics of the unwind section given in (21) along with the feedback law given in (31); the closed-loop dynamics of the unwind section is (40) , To cancel the time-varying terms in the second row of , where consider and is an intermediate gain variable that will be obtained from the LQ solution. Defining , (40) simplifies to

US: MSS: PS: RS:

and are positive constants chosen such that the where are asymptotically stable and satisfy the sufficient matrices is not time-varying becondition given by (36). Note that cause it is not a function of the time-varying parameters and . Hence, the choice of is fixed and does not change with the change in the radius and inertia of the unwind or rewind roll. contain reference values and . Since But the matrices the web processing line runs at a predetermined set of reference , one values, for each pair of operating reference values can obtain the corresponding . is controllable, one may choose pole Since the pair placement to satisfy condition (36). The process of selecting gains so that the closed-loop matrices satisfy the constraint (36) is iterative; it consists of reducing the eigenvalues of the closedin increments and checking whether the conloop matrix straint (36) is satisfied. Once an that satisfies (36) is found, may be found from the feedback gain vectors, . The gain vectors, thus, obtained are given by

(37)

(41)


Fig. 8. Sufficient condition check for different reference velocities ( = N ( + ) and = min (A j!I )).

0

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3) Online Computations: 1) Assign the values of machine parameters such as radii of rollers and their inertias, initial radii of unwind/rewind rolls, gearing ratios, span lengths, and web material properties. 2) At each sampling instant, perform the following computations: a) acquire the speed signals from encoders and the tension signals from loadcells; b) update the radii of unwind and rewind rolls using (7); c) compute the inertias of the unwind and rewind rolls using (1) and (2); d) compute the equilibrium inputs given by (16), (23), (26), and (30); from (38) and the e) compute the feedback gains decentralized control inputs from (31)–(34); , to the motor f) send the control input drivers. C. Experimental Results

Notice that the intermediate variable , is now absorbed into . We now find the state feedback gain to minimize the quadratic performance index and use the feedback law ; the term with underbrace in (41) simplifies to . Thus, if is the LQ-optimal closed-loop state matrix , then the second row of is obtained for the pair . It is emphasized as the LQ-optimal gain for the pair that the sufficient condition (36) still needs to be satisfied for the optimal closed-loop matrix . Thus, computation of gains using the LQ-formulation is also iterative. The proposed control algorithm may be considered to have a set of offline computations and a set of online computations, which are discussed in the following. 2) Offline Computations: 1) Assign the values of radii of rollers, initial radii of unwind and rewind rolls, and web material properties. for various zones are 2) Assign the reference tensions required by the process specifications. 3) Compute the reference velocities as given by (18), (25), and (29). , such that the matrices given in 4) Find (37) satisfy the sufficient condition (36). These may be found either by pole placement or the LQ-approach as discussed earlier. is much larger than for most Since the quantity is of webs, the sufficient condition as a function of value. Fig. 8 shows the inequality (36) as a plot from which we can see that at each of the line speeds within the operating range. The reference tensions used to gen63.83 N (14.35 lbf) and erate the plot in Fig. 8 are for all . The following and values are , , used to obtain these plots: , , , , and . A similar check is performed for each set of operating conditions chosen for experimentation.

Extensive experiments at different web transport speeds were conducted with the currently used industrial decentralized PI controller and the proposed decentralized controller. Sampling time for control scanning loop and data acquisition was chosen to be 5 ms. Experimental results for the following two cases are shown in this paper as follows. Case 1) Reference velocity 5.08 m/s (1000 ft/min); 109.42 N (24.6 lbf), 91.18 N (20.5 lbf), and 72.95 N (16.4 lbf); the roll diameter varies from 0.46 m (18 in) to 0.33 m (13 in). 7.62 m/s (1500 ft/min); Case 2) Reference velocity the reference tension is the same in all zones and is chosen 63.83 N (14.35 lbf); the roll diameter varies from as 0.36 m (14 in) to 0.20 m (8 in). Notice that to evaluate the proposed controller, differential tension zones were created throughout the web line with different tension references in different sections in case 1, whereas the same reference tension in all tension zones was used in case 2). In each case, variation of the web line speed at the master speed roller and tension variations in each tension zone are shown. In each figure shown subsequently, the top plot shows the variaand the tion in master speed velocity from its reference remaining three plots show the tension variations in the three . tension zones Results of the experiments for case 1) are shown in Figs. 9 and 10. Fig. 9 shows the results using a well tuned decentralized PI controller (block diagram shown in Fig. 6). Fig. 10 shows the results using the proposed decentralized controller. As compared to the existing decentralized PI controller, results using the proposed decentralized controller show much improved regulation of web line speed and tension in each of the zones. Similar comparative results can be obtained for the higher reference velocity of 7.62 m/s case, as shown in Figs. 11 and 12. Figs. 13 and 14 show the four control inputs corresponding to case 1). The magnitude of control inputs for the proposed controller is larger than that of the PI controller. The increase in magnitude of the control inputs for the proposed controller is due to its ability to

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Fig. 9. Decentralized PI controller: Reference velocity 5.08 m/s.

Fig. 12. Proposed decentralized controller: Reference velocity 7.62 m/s.

Fig. 10. Proposed decentralized controller: Reference velocity 5.08 m/s.

Fig. 13. Control inputs with PI controller at 5.08 m/s.

Fig. 14. Control inputs with proposed controller at 5.08 m/s. Fig. 11. Decentralized PI controller: Reference velocity 7.62 m/s.

use higher tension gains to reduce large tension variations while keeping the entire system stable. Notice that the industrial PI

controller has a cascaded structure (see Fig. 6); the output of the tension PI controller provides reference velocity corrections for the inner velocity loop; this places constraint on choosing


large gains in the tension loop to compensate for tension errors. Whereas, the proposed controller structure (see Fig. 7) is not cascaded as in the PI strategy, which allows choice of tension gains independent of the velocity loop. Although the proposed controller requires larger control inputs, it offers much better performance in terms of tension regulation. Further, the control torques for both the controllers are well within the torque capacities of the motors.

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has no eigenvalue Definition 1: [21] Suppose be the set of matrices on the imaginary axis. Let with at least one eigenvalue on the imaginary axis. The distance from to is defined by (42) It can be shown that [21]

V. CONCLUSION An accurate dynamic model for web process lines that explicitly takes into account the time-varying nature of the unwind/rewind roll inertia was developed. A subsystem formulation of the web process line is given based on the creation of a number of tension zones. This formulation facilitates efficient design of decentralized controllers. A strategy for computing the equilibrium inputs based on the reference tensions in each zone and reference velocity of the web line was developed. A fully decentralized state feedback controller was proposed. Exponential regulation of tension and velocity in each zone was shown using the proposed controller. Experimental results on a large web platform show that the proposed decentralized strategy with the new dynamic model gives much improved tension regulation over the existing industrial decentralized PI controller. Although the exhaustive experimental analysis on a large platform with commercial hardware/software lends much credibility to the proposed scheme, the ultimate test of its performance can be realized only upon successful implementation in an industrial environment. Future work should focus on the robustness properties of the proposed controller to unmodeled dynamics; uncertainties in web material properties and machine parameters must be explored; relevant experimentation with various disturbances under different dynamic conditions must be conducted to ascertain the robustness of the algorithm. Since the stability of the current algorithm is based on the mathematical model of the process, stability of the algorithm when the model does not accurately reflect the process must be investigated. The selection of controller gains is based on closed-loop system matrices satisfying a sufficient condition. The choice of the gains to satisfy this sufficient condition is iterative; future work should include how the gains that satify the sufficient condition can be determined without going through the iterative process. The proposed decentralized controller is a static state feedback controller and as such does not contain integral action, which can be introduced in a similar fashion as given in [15], [22], and [23]; we also plan to experimentally investigate this in the future. It is possible that load cells may not be available in each tension zone. In such cases, design of a decentralized tension observer and a decentralized output feedback controller is desired, which could be a topic for future research. Our future work will include investigating the above issues in an effort to obtain an algorithm that can be implemented with ease on industrial web process lines. APPENDIX Proof of Theorem 1: The proof of the theorem involves the following definitions and results.

(43) Lemma 1: [24]–[26] Consider the algebraic Ricatti equation (ARE) (44) If

,

,

is Hurwitz, and the associated is hyperbolic, i.e.,

Hamiltonian matrix

has no eigenvalues on the imaginary axis, then there exists a , which is the solution of the ARE (44). unique Proof of Theorem 1: Consider the following Lyapunov function candidate:

(45)

The derivative of

along the trajectories of (35) is

(46)

Using the following inequality

(47) for

in the last two terms of (46), we obtain

(48)

Also, one can obtain the following bound for the last term of

:

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As a result,


satisfies

ACKNOWLEDGMENT (49)

Therefore, we have the following. If there exist positive definite solutions to the AREs

The authors would like to acknowledge the equipment support from the Web Handling Research Center at Oklahoma State University. One of the authors, P. R. Pagilla, would also like to thank D. Knittel for useful discussions on modeling and control of web processing lines. REFERENCES

(50) then (51) Hence,

is a Lyapunov function. Defining , , and , where denotes the maximum eigenvalue, we have and . Thus, we have which gives us . Hence, globally exponentially converges to zero. Proof of Theorem 1 now rests on the existence of symmetric to the ARE (50). To this end, we positive definite matrices invoke Lemma 1 and write the Hamiltonian for the ARE (49) as (52) The eigenvalues of the Hamiltonian writing

may be obtained by

(53) From (53) we see that Notice that

is hyperbolic if

is nonsingular.

(54) From (43), we see that the term in braces in (54) is always . Thus, if greater than (55) we can always choose a value for as for any in the range to make in (54) positive definite, thus, ensuring the existence of a symmetric positive definite to satisfy the ARE (50).

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Prabhakar R. Pagilla (SM’91–M’96) received the B.Eng. degree from Osmania University, Osmania, India, and the M.S. and Ph.D. degrees from the University of California, Berkeley, in 1990, 1994, and 1996, respectively, all in mechanical engineering. He is currently a Professor in the School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater. His research activities are mainly in the areas of large-scale systems, adaptive control, mechatronics, disc drives, biomedical systems, and web handling systems. He is an Associate Editor of the ASME Journal of Dynamic Systems, Measurement, and Control and a Technical Editor of the IEEE/ASME TRANSACTIONS ON MECHATRONICS. Dr. Pagilla was a recipient of the National Science Foundation CAREER Award in 2000.

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Nilesh B. Siraskar received the B.Eng. degree from the Government College of Engineering, Pune, India, and the M.S. degree from the Oklahoma State University, Stillwater, in 2002 and 2004, respectively, all in mechanical engineering. He worked as a Control Systems Engineer in Metro Automation, Dallas, TX. He is currently a CEO of Dexterous Technologies, Nashik, India, producing servo control systems for ac and dc motors. His research focus is mainly on the areas of large-scale systems, adaptive control, mechatronics, and web handling systems. Dr. Siraskar was a recipient of the Graduate Research Excellence Award from Oklahoma State University in 2005.

Ramamurthy V. Dwivedula received the B.Eng. degree from Andhra University, Andhra, India, the M.Tech. degree from Indian Institute of Technology, Delhi, India, and the Ph.D. degree from Oklahoma State University, Stillwater, in 1987, 1992, and 2006, respectively, all in mechanical engineering. He is currently a Research and Development Mechanical Engineer at Fife Corporation, Oklahoma City, OK. His research interests include adaptive control of web guides, web tension/speed control, and dancer systems to reject tension disturbances.