observer-based tension feedback control with friction and inertia compensation

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003 109

Observer-Based Tension Feedback Control With Friction and InertiaCompensation

Ku Chin Lin

Abstract—Low cost and high productivity are two primarygoals in design of a web transport system. One approach to achievelow cost is through the implementation of observer techniques inreplace of tension transducers. To achieve high productivity, it isnormally required to increase the process speed. However, as theprocess speed or variation of the speed is high, system friction andinertia of rotation of rolls could cause problems in implementationof observer techniques for tension estimation and control. Fewof the previous studies have considered the problems of frictionand inertia in a single article. This paper proposes an observerwith friction and inertia compensation. The proposed observerhas a feedback configuration and it is able to estimate webtension precisely regardless of the effects of friction and inertia.Linearization and decentralization techniques are implemented.Design of the proposed observer-based tension feedback controlleris performed in the frequency domain. Eccentricity of unwindroll is considered as sinusoidal disturbances to the system. Aprocedure for design of the proposed controller and eccentricityof rolls are discussed. Simulation and experimental works havebeen performed and they show that the proposed observer-basedtension feedback controller performs as well as a classical tensionfeedback controller using a tension transducer.

Index Terms—Friction compensation, inertia compensation, ten-sion control, tension estimation, web transport systems.

I. INTRODUCTION

FEEDBACK tension control using tension transducers isubiquitous in the web industry. Now, low cost and high

productivity are two primary goals in design of a web transportsystem [1]–[9]. One approach to achieve low cost is throughthe implementation of observer techniques in replace of tensiontransducers. To achieve high productivity, it is normally requiredto increase the process speed. However, as the process speed orvariation of the speed is high, system friction and inertia of rota-tion of rolls could cause problems in implementation of observertechniques for tension estimation and control.

Implementation of an observer technique normally requiresmodeling the dominant dynamics of systems precisely. Previousstudies have demonstrated the effects due to system friction andinertia of rotation on web tension (e.g., [4], [6], [7], and [9]).But, few of the previous studies have considered the problemsof friction and inertia in a single article.

It is known that acceleration (deceleration) inertia of rota-tion can induce tension increase (decrease) in the process ofsystem start-up (shutdown). Friction arises as a shaft rotates.

Manuscript received June 20, 2001; revised April 9, 2002. Manuscript re-ceived in final form May 30, 2002. Recommended by Associate Editor K. Ko-zlowski. This work was supported in part by the National Science Council,Taiwan, R.O.C., under the Grant NSC 89-2212-E-168-015.

The author is with the Department of Mechanical Engineering,Kun Shan University of Technology, 71003 Taiwan, R.O.C. (e-mail:[email protected]).

Digital Object Identifier 10.1109/TCST.2002.806464

The Coulomb friction at a rotating shaft causes a change of ten-sion between two consecutive web spans. Viscous friction intro-duces a speed-dependent tension variation that can be substan-tial as the process speed is high. Tension estimation could beimproper if friction or inertia dynamics are ignored in the asso-ciated observer model.

Mathur et al. [6] addressed many practical issues on aprototype high-speed low-tension tape transport system.Multi-input–multi-output system identification, sequential loopclosing techniques, adaptive tension ripple cancellation, andfault compensation were incorporated into the control systemdesign. It was found that friction in a high-speed line is animportant issue in design of a control law. Trial-and-errorapproaches were used to estimate the Coulomb friction. Songet al. [4] have experimentally demonstrated a tension increaseinduced by inertia of rotation of unwind roll during systemstartup. Songet al.proposed a feedback control law to depressthe tension increase. In the control law, the acceleration inertiawas computed using the filtered derivative of angular velocityof unwind roll. Lin et al. [7] proposed a tension observer.The observer has a feedback configuration and it is able toestimate the tension increase induced by the accelerationinertia. However, both of Song’s and Lin’s observers neglectedthe friction effects.

In this paper, an observer-based tension feedback controllerwith friction and inertia compensation is proposed. The observermodels presented in [7] are extended to including the dynamicsof Coulomb and viscous friction. The adequacy of the proposedcontroller is justified on a direct unwind and rewind system byexperiment. This paper is organized as follows: mathematicalmodeling of the studied system and assumptions are given inSection II. In Section III, some observer structures includingcompensation of friction and inertia are presented along withsteady-state analysis of the observer outputs. A procedure fordesign of the proposed observer is summarized. Section IV illus-trates the proposed observer-based tension feedback controller.Linearization and decentralization techniques are implementedand design of the proposed controller is performed in the fre-quency domain. Results of simulation and experiment are givenand discussed in Section V. Finally, conclusions of this paperare given in Section VI.

II. SYSTEM MODELING AND ASSUMPTIONS

Fig. 1 shows a schematic layout of the direct unwind andrewind system studied in this paper. Web is unwound out of anunwind roll, moving through a load-cell subsystem consistingof two idle rollers, and then wound onto a rewind roll. The idlerollers guide the moving web around the load cell in a fixedangle. As shown in Fig. 1, the tension measured by the load cell

1063-6536/03$17.00 © 2003 IEEE

110 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003

Fig. 1. Schematic of the studied web transport system.

is denoted by . The tension in the upstream span of the loadcell is denoted by and in the downstream span is denoted by

. Driving the unwind roll is a motor which is under torquecontrol and acts as a brake against the moving web. A counter-clockwise torque is generated by the unwind motor againstthe tension of web. Another motor is used to drive the rewindroll and it is under speed control. The speed of the rewind motordetermines the overall process speed.

The dynamics of load cell and idle rollers are neglected. It isassumed that no web slippage occurs, web is perfectly elastic,and tension in all spans is the same. Based on first principles,the following equations describing the studied system dynamicscan be obtained [1], [2], [7]:

(1)

(2)

(3)

(4)

(5)

where the notations used are explained (hereafter, the suffixmeans unwind and suffix means rewind).

Total moment of inertia of the unwind roll and motor(kg/m/s ).Total moment of inertia of the rewind roll and motor(kg/m/s ).Web tension (kg).Angular velocity of the unwind roll (rad/s).Angular velocity of the rewind roll (rad/s).Radius of the unwind roll (m).Radius of the rewind roll (m).Torque due to friction at the unwind shaft (kg/m).Torque due to friction at the rewind shaft (kg/m).Torque generated by the unwind motor (kg/m).Torque generated by the rewind motor (kg/m).Tangential velocity at the periphery of the unwind roll(m/s).Tangential velocity at the periphery of the rewind roll(m/s).Total length of web (m).Wound-out tension of the unwind roll (assumed zeroin this paper).

Spring constant of web (kg/m).Input voltage to the unwind motor (V).Input voltage to the rewind motor (V).Torque constant of the unwind motor (kg/m/V).Torque constant of the rewind motor (kg/m/Vt).

Torque due to friction against the moving web acting on theunwind roll is modeled as

(6)

(7)

(8)

where denotes torque due to the bearing (viscous) frictionand denotes the coefficient of bearing friction; denotestorque due to the Coulomb friction and denotes the magni-tude of the torque.

III. T ENSION OBSERVERS

For observers with the input voltage and angular velocity ofunwind motor as the input signals, the observability of systemsdescribed by (1)–(5) but neglecting friction has been proven [8].In this section, some observers available in the literature are dis-cussed and a new observer with friction and inertia compensa-tion is proposed.

A. Tension Observers With Inertia Compensation

In the study of a web transport system similar to that shownin Fig. 1, Songet al. [4] used a formula to estimate web tensionas follows:

(9)

where denotes filtered angular acceleration of the unwindroll. The filtered angular acceleration is computed by takingderivative of the measured angular velocity of unwind roll, andthen passing through a second-order low-pass filter as follows:

(10)

By abuse of notation, denotes the Laplace transform of, while and denote the natural frequency and damping

ratio of the filter, respectively.An alternative approach to using the computational method of

(9) is utilization of an observer technique for tension estimation.Lin et al.[7] proposed a tension observer that has a feedback con-figurationand a filtered inertiablock asshownin Fig. 2. By the Mason’s rule, the output of the observer is

(11)

(12)

Hereafter, denotes the Laplace transform of , for ex-ample. If is a constant, then , and denotesthe constant. Proper values ofare 3 10 as described in [10].


Fig. 2. PI-type observer with a filtered inertia block [7].

Fig. 3. PI-type observer with inertia compensation [7].

The larger the value of is, the faster the observer responses canbe.ThestabilityofLin’sobservercanbeguaranteedbyproperde-sign of the PI gains, and . When is a constant, theestimated torque will converge to , even though theunwind roll has acceleration or deceleration inertia. The observeris good as a torque observer. However, it is not good as a tensionobserver if acceleration or deceleration inertia of the roll arises.Lin et al.[7] continued to propose another observer. The outputsof the filtered inertia block were used as feedforward signals andadded into theestimatedtorque.Thesumof thefiltered inertiaandestimated torque provides good estimates of web tension in spiteofaccelerationordeceleration inertiaof the roll.Ablockdiagramof the observer with inertia compensation is shown in Fig. 3. Theoutput of the observer is

(13)

In case of system startup, the unwind roll rotates in constantangular acceleration (i.e., ). Duringsuch system startup, a steady state of web tension could exist ifthe input voltage to the unwind motor maintains constant [i.e.,

]. By the final-value theorem, the steady state ofweb tension will converge to

(14)

Fig. 4. Proposed observer with compensation of inertia, bearing, and Coulombfriction.

According to (9) and (14), Song’s and Lin’s observers havethe same steady-state estimate of tension during the specificsystem startup. However, the latter is more advantageous thanthe former since better transient responses and disturbance re-jection can be achieved through proper design of PI gains. Theabove observers were developed without taking friction effectsinto account. Discrepancy between real and estimated tensioncould be significant if dominant friction effects arise.

B. Tension Observers With Friction and Inertia Compensation

Fig. 4 shows an extension of the observer in Fig. 3 to in-cluding bearing and Coulomb friction into the observer model.The output of the observer in Fig. 4 is

(15)

where

(16)

Note that the observers in Figs. 2 and 3 are special cases of thatin Fig. 4. For the observer in Fig. 2, and ,. For the observer in Fig. 3, and , .

In case the unwind roll rotates in constant angular accelerationand the input voltage to the unwind motor maintains constant,we have

(17)


The speed of observer responses is normally much faster thanthe speed of changes in the angular velocity of the unwind roll.Suppose that the time required for the estimated tension ap-proaching to a steady state is denoted by. As the time ap-proaches to zero, the speed of changes in the angular velocity ofthe unwind roll can be neglected, i.e.,

(18)

Substituting (18) into (17) leads to

(19)

Equation (19) shows that the proposed observer is able to esti-mate web tension precisely even though the system is under theeffects of Coulomb and bearing friction.

C. Design of Observer Gains

In Fig. 4, the proposed tension observer with friction and in-ertia compensation has two input signals (i.e., the input voltageto the unwind motor and the angular velocity of the unwind roll).The output of the proposed observer can be written as follows:

(20)

where

(21)

(22)

where is given in (16).In general, the proposed observer has one real pole and two

complex conjugated poles. The real pole will become less dom-inant by selecting a larger value of. The observer time con-stant is . The larger the proportional gainis selected, the more damped the observer responses are. As arule of thumb, observer responses are designed to be five–tentimes faster than those of open-loop systems [10]. A procedurefor calculating the PI gains of the proposed observer is summa-rized as follows.

1) Assign a larger value of (e.g., ).2) Let the observer time constant be five–ten times less than

that of the open-loop system.3) Select a proper observer damping ratio (e.g., ).4) Compute the observer PI gains, and .

An example study will be given later to demonstrate the use ofthe above procedure for design of the proposed observer.

IV. OBSERVER-BASED TENSIONFEEDBACK CONTROLLERS

Simultaneous control of the process speed and web tensionis required in the studied system. A common strategy involvesin control of the speed of the rewind motor and control of webtension through regulating the torque generated by the unwind

Fig. 5. Direct unwind and rewind system with measured tension feedbackcontrol.

Fig. 6. Direct unwind and rewind system with estimated tension feedbackcontrol.

motor. Fig. 5 shows a classical approach of PI control of thespeed and the tension. The speed loop is closed with measuredangular velocity of the rewind roll and the tension loop is closedwith measured tension as the feedback signals.

This study is primarily concerned with observer-based ten-sion feedback control without using a tension transducer. Thetension observers depicted in the previous section are to be em-ployed to estimate web tension. The estimated tension will beused as feedback signals to form the tension loop. Fig. 6 showsa block diagram of the studied system under the proposed ob-server-based tension feedback control and speed control. In thefigure, the block named tension observer can be one of the ob-servers shown in Figs. 2, 3, or 4.

For the studied system, the dynamics of web tension are non-linear. System stability analysis and analytic design of the con-trol gains based on (1)–(5) are difficult. Therefore, we performlinearization of the system equations around an operating condi-tion, and write the linearized equations in the following matrix


form:

(23)

where the following notations are used.: Changes in web tension from an operating value (kg).: Changes in angular velocity of the unwind roll from

an operating value (rad/s).: Changes in angular velocity of the rewind roll from

an operating value (rad/s).: Changes in input voltage to the unwind motor from

an operating value (V).: Changes in input voltage to the rewind motor from an

operating value (V).: Operating value of tangential velocity of the rewind

roll (m/s)For simplicity, measured tension is used in design of the ten-

sion control law. The control law is

(24)

and the speed control law is

(25)

By taking the Laplace transform of (23)–(25) and after manip-ulation, we can have

(26)

where

Fig. 7. Block diagram of the tension control subsystem to which changes intangential velocity of the rewind roll are considered as disturbances.

Fig. 8. Block diagram of the speed control subsystem to which changes intangential velocity of the unwind roll are considered as disturbances.

System stability analysis and analytic design of the controlgains based on (26) are still difficult. The decentralizationtechnique is adopted by reducing the studied system into twoseparated subsystems. Fig. 7 shows a block diagram of one ofthe subsystems to which changes in tangential velocity of therewind roll are considered as disturbances. Fig. 8 shows a blockdiagram of the other subsystem to which changes in tangentialvelocity of the unwind roll are considered as disturbances.

A. Design of Control Gains

The block diagrams in Figs. 7 and 8 can be further reducedto those shown in Fig. 9(a) and (b), respectively. The open-looptransfer functions of the subsystems are

(27)

(28)

The reduced block diagrams are in a standard form forfrequency-response analysis. A procedure for designing thePI gains in the speed and tension control loops is available[11]. For instance, a design procedure for and withspecified phase margin is summarized as follows.

1) Determine the frequency at which the angle ofequals to .

2) The PI gains are then given by and

B. Computation of Roll Radius and Inertia

The radius of the unwind roll is required in estimation of webtension. The radius of the rewind roll is required in control ofthe angular speed of the rewind roll. As the web transport is inprogress, the radii of both rolls vary with time. Two methodsare typically employed to achieve the time-varying radii of rollson line. One method is to directly measure the roll radius usingan ultrasonic or photo detector. The other method is based ononline computation of the roll radius by integrating the angulardisplacement of the roll. The angular displacement of roll is


Fig. 9. Reduced block diagrams of (a) the tension control subsystem (b) thespeed control subsystem.

measured using an encoder. Since each of the motor drive unitsused in this study has a built-in encoder, the online computationmethod is adopted for the reason of cost. The radii of rolls arecomputed according to the following equations:

(29)

(30)

where and denote the initial radii of the unwind andrewind rolls, respectively, and denote the integrated an-gular displacements of the rolls, anddenotes the web thick-ness.

C. Measurement of Coefficients of and

By hardware settings, the unwind motor drive unit can beset under torque control mode. Different input voltages wereapplied to the motor drive unit without load. The steady statesof angular velocity of the motor were measured. A typicaltorque-speed plot of the motor is shown in Fig. 10. An amountof motor torque is required to overcome the Coulomb frictionbefore the motor starts up. After that, linear relationshipbetween the motor torque and speed is assumed, i.e.,

(31)

From Fig. 10, the bearing friction coefficient and the magni-tude of the torque due to Coulomb friction are estimated ap-proximately, and they are e-5 kg-m-s/rad and

e-3 kg-m.

D. Eccentricity of Unwind Roll

Due to the weight of roll, eccentricity of the roll will induce anadditional torque on the rotational shaft of the roll. For instance,the torque induced by the eccentricity of unwind roll can beincorporated into (1) as follows [1]:

(32)

Fig. 10. Typical torque/speed plot of the unwind motor used in this study.

Fig. 11. Experimental setup for this study.

where denotes the weight of the unwind roll, denotesthe eccentricity of the roll, and denotes the initial angulardisplacement of the roll.

The torque induced by the roll eccentricity acts as system dis-turbance. The disturbance is sinusoidal and its frequency is re-lated to the angular speed of the roll. As the process speed in-creases, the disturbance frequency is getting close to the systemresonant frequency. Hence, the disturbance could induce largetension fluctuation and deteriorate the performance of tensioncontrol significantly.

V. SIMULATION AND EXPERIMENTAL RESULTS

Simultaneous control of the process speed and ob-server-based tension feedback control for the studied systemduring an acceleration start-up has been investigated. Simu-lation and experimental results are presented in this section.Fig. 11 shows a photo of the experimental setup for this study.Two dc servomotors (not shown in the figure) are used todrive the unwind roll and rewind roll, respectively. The unwindmotor is under torque control and the rewind motor is underspeed control. Table I summarizes the parameter values of themotors and web used. A motion control card with an onboardTMS320C32 DSP is employed to execute the control tasks inreal time.

Four cases of simulation and experimental results are illus-trated below to demonstrate the adequacy of the proposed ob-server-based tension feedback control. The results are given to


TABLE IPARAMETER VALUES OF THEMOTORS ANDWEB USED IN THEEXPERIMENT

emphasize the individual influences of compensation for accel-eration inertia, bearing, and Coulomb friction on tension con-trol.

Case 1) Observer-based tension feedback control withoutcompensation for acceleration inertia and friction (i.e.,

, , ).Case 2) Observer-based tension feedback control with com-

pensation for Coulomb friction only (i.e., ,, e-3 kg-m).

Case 3) Observer-based tension feedback control with com-pensation for bearing and Coulomb friction (i.e.,

, e-5 kg-m-s/rad, e-3kg-m).

Case 4) Observer-based tension feedback control with com-pensation for acceleration inertia, bearing, andCoulomb friction (i.e., , e-5kg-m-s/rad, e-3 kg-m).

The initial radii of rolls were m andm for all of the case studies. The total length of web was 0.3m. The reference speed commandwas first set at 0 m/s for2 s, increased in a rate of 0.7 m/sin 5 s, and then maintainedat 3.5 m/s for 3 s. The operating value of tangential velocityof rewind roll, , was chosen to be 3.5 m/s. The disturbancefrequencies induced by the eccentricity of unwind roll are about0 110 rad/s. The time constant of the open-loop system is about1.5 s. The observer responses were designed to be ten timesfaster than those of the open-loop system. To have the observerresponses critically damped, we chose that and

.In design of the tension controller, we specified the phase

margin to be 50. The frequency at which the angle ofequals to 125 is 130.0 rad/s. Hence, we had

and . In design of the speedcontroller, we specified the phase margin to be 85. Thefrequency at which the angle of equals to 90 is80.0 rad/s. Hence, we had , and .

Figs. 12 and 13 show the Bode plots of and, respectively. It is shown in the plots that using

the decentralization technique in design of system controllersis adequate. The closed-loop system is stable and it has good

Fig. 12. Bode plots ofT (j!)=T (j!).

Fig. 13. Bode plots of (j!)= (j!).

gain and phase margins. Hence, we believe that the closed-loopsystem is robust against system parameter variations. In addi-tion, good steady-state accuracy of tension and speed at lowfrequencies can be achieved. Resonant frequency of 150 rad/sexists for the controlled system. The resonant frequency is rea-sonably away from the disturbance frequencies induced by theeccentricity of the unwind roll.

Simulation and experimental results of the speed control areshown in Fig. 14. The results show that the tangential velocityof the rewind roll follows the reference speed command well.Figs. 15 and 16 show simulation and experimental results of ten-sion control where the reference tensionwas set to be 0.5 kg.As shown in Figs. 15(a) and 16(a) (i.e., Case 1), the Coulombfriction without being compensated induces a tension increaseof the magnitude of (about 0.1 kg) above the referencetension level during the initial period of 2 s.

During the next 5 s of startup period, the bearing frictionand acceleration inertia without being compensated inducemore tension increases. The acceleration inertia induces a stepchange of web tension, while the bearing friction induces aramp change of it. Tension increases due to the bearing and


Fig. 14. Simulation and experimental results of the speed control.

Fig. 15. Simulation results of proposed observer-based tension feedbackcontrol,t = 0:5 kg. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4.

Coulomb friction can be compensated by using the proposedcontroller with friction compensation as shown in Figs. 15(b),15(c), 16(b), and 16(c) (i.e., Case 2 and Case 3). The tensionincrease due to acceleration inertia of the unwind roll canbe also reduced if compensation of the acceleration inertia isconsidered. As shown in Figs. 15(d) and 16(d) (i.e., Case 4),with friction and acceleration inertia being compensated, webtension under control maintains at the desired level for mostof the time concerned. At the beginning of the startup period,tension overshoot occurs due to the need to overcome the static

Fig. 16. Experimental results of proposed observer-based tension feedbackcontrol,t = 0:5 kg. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4.

friction. At the end of the startup period, the occurrence of ten-sion undershoot is attributed to the abrupt loss of accelerationinertia of the unwind roll.

Figs. 17 and 18 show the experimental results as the refer-ence tension was set to be 0.3 kg and 0.8 kg, respectively. Ten-sion dynamic responses similar to those shown in Fig. 16 wereachieved. Note that the amount of tension increase induced bythe acceleration inertia of the unwind roll is independent on thereference tension level being set.

For the comparison purpose, the above case studies werereinvestigated with the classical tension feedback control usinga load-cell transducer. The experimental results achieved areshown in Fig. 19(a)–(c). By comparing the results shown inFig. 19 with those shown in Figs. 16(d), 17(d), and 18(d), weare convinced that the proposed observer-based tension feed-back controller with compensation of friction and accelerationinertia performs as well as the classical feedback controllerusing a load-cell transducer.

VI. CONCLUSION

Friction and inertia compensation are necessary in implemen-tation of observer techniques for tension estimation and control.This paper presents a PI-type observer that is able to estimateweb tension under dominant influences of system friction and



rotational inertia. The proposed observer has a feedback config-uration with friction and inertia being compensated.

For ease in design of control gains, linearization and decen-tralization techniques are implemented to reduce the studiedsystem into two subsystems. For each of the subsystems, thefrequency-response analysis is performed for control design. Aprocedure for determining the control gains is summarized. Theadequacy of using the decentralization technique in analysis ofthe studied system and control design is verified from the Bodeplots of and . The proposedobserver-based tension feedback controller is robust against thevariation of system parameters. Eccentricity of the unwind rollis modeled as sinusoidal disturbance to the system. The fre-quency of the sinusoidal disturbance is related to the angularspeed of the unwind roll and it must be reasonably away fromthe system resonant frequency.

Illustrative case studies are included in this paper to demon-strate the influences of acceleration inertia and friction duringsystem startup. Simulation and experimental results have shownthat the proposed observer-based tension feedback controllerwith compensation of friction and acceleration inertia performsas well as the classical feedback controller using a load-celltransducer. Good experimental results are achieved. This is pri-marily attributed to the adequacy in the system modeling, com-putation of roll radii and inertia, and measurement of and

, so that good estimation of web tension can be achieved.


Fig. 19. Experimental results of classical feedback tension control using aload-cell transducer.


ACKNOWLEDGMENT

The author would like to thank the reviewers and the Asso-ciate Editor, Prof. K. Kozłowski, for their comments that havehelped to improve the manuscript.

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observer-based tension feedback control with friction and inertia compensation

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