robust pi--lpv tension control with elasticity … pi{lpv tension control with elasticity observer...

Robust PI–LPV Tension Control withElasticity Observer for Roll–to–Roll

Systems

Vincent Gassmann, Dominique Knittel 1

Design Engineering Laboratory, National Institute of Applies Sciences,Strasbourg,France

Web Handling Research Group, University of Strasbourg, France

Abstract: Flexible materials such as textiles, papers, polymers and metals are transportedon rollers during their processing. Maintaining web tension in the entire processing line underan expected web speed is a key factor in achieving good final product quality. It is commonpractice in industrial web transport systems to use decentralized PI–type controllers. Theperformances of such control strategies highly depend on web speed and elasticity because ofthe strong coupling between these two variables. Moreover web speed and elasticity are subjectto large variations during a same processing. The emphasis of this paper is on the design oflinear parameter varying PI controllers with quadratic H∞ performance to increase closed–loop system robustness regarding these parameters variations. First a polytopic model of theexperimental plant is derived from web speed and tension dynamics. A design methodology ofthe PI–LPV controllers, that guarantees closed–loop quadratic stability and H∞ performance, isthen presented. The method uses an optimization software with genetic algorithms to determinethe controllers parameters while minimizing the H∞ norm. Moreover, as it is difficult tomeasure web elasticity online, an observer is synthesized to estimate the Young’s modulus.The effectiveness of the proposed control strategy is illustrated with simulations.

Keywords: Roll–to–roll systems, PI–LPV controller, quadratic H∞ performance, observer,genetic algorithms.

1. INTRODUCTION

Roll-to-roll systems handling web material such as tex-tiles, papers, polymers or metals are very common inthe industry, because they represent a convenient wayof transporting and processing a product from one formto another. Printing, coating and drying are examples ofoperations that can be performed in different sections ofa web line. A web is usually described as any continuousand flexible material whose width is significantly less thanits length and whose thickness is less than its width. Webtension and speed are two key variables that need to bemonitored and controlled in order to achieve the expectedfinal product quality. One of the main objectives in webhandling machinery is to reach an expected web speedwhile maintaining the web tension within a close toleranceband in the entire processing line. This tolerance dependsobviously on the type of material that has to be processed.In the recent years, many works have focused on thetopic of tension control (Pagilla and Knittel (2005); Kocet al. (2002); Pagilla et al. (2007); Shin (2000); Gassmannet al. (2011)) and have proposed various ways to enhancethe performance: H∞ control, optimal control, etc. Butthe common practice in industrial web transport systemsremains the use of decentralized PI-type controllers. Animproved design methodology of these PI controllers withfixed–order and –structure H∞ techniques has been pre-

1 Corresponding author: [email protected].

sented in Knittel et al. (2007). Nevertheless, despite highperformances for a nominal working point, it has beennoticed two main drawbacks (Gassmann (2011)):

(1) the closed–loop system is unstable at low web speed;(2) the closed–loop system performances depend on web

elasticity since the dynamic behavior is strongly af-fected by the Young’s modulus.

Consequently, this paper investigates the design of PI–LPV controllers to figure out these issues. The conven-tional LPV controller synthesis is based on the existenceof a Lyapunov function with a convex characterizationand results in controllers that have the same order as theaugmented plant (Apkarian and Gahinet (1995)). Besides,the design of fixed–order PI controllers is formulated as anon–convex problem. Kwiatkowski et al. (2009) propose amethod to solve this problem: it is a non–convex optimiza-tion based on evolutionary algorithms. A similar methodis used in this contribution to take into consideration webspeed and elasticity variations in the decentralized controlstrategy.

The PI–LPV controllers require the knowledge of webelasticity at any time. An observer is therefore designed toestimate elasticity modulus over the entire processing line.Although the design of observers for roll-to–roll systemshas encountered some attention in the literature, theconcern is essentially on the estimation of web tension(Wolfermann (1997); Lynch et al. (2004); Gassmann and

Preprints of the 18th IFAC World CongressMilano (Italy) August 28 - September 2, 2011

Copyright by theInternational Federation of Automatic Control (IFAC)

8639

M1

M2

M3

M3’

M4

LC

LC

LC

LC

LCF1 F2

Unwinder Rewinder

Master-SpeedRoller

DrivenRoller

PDPD

Unwind Section Rewind SectionIntermediate Section

Fig. 1. Web Handling Platform

Knittel (2007)). Only few rare papers have focused onweb elasticity estimation (Boulter (1999); Angermann(2004)). In this paper, a PI–H∞ Luenberger state observeris introduced. The problem of the observer synthesis issimilar to the calculation of a static output feedbackcontroller. Consequently, the problem is non–convex andis solved by non–convex programming.

A sketch of the system that is used in this paper ispresented in figure 1. It mimics the classical structureof an industrial processing line. The system in Fig. 1is composed of an unwinder, two intermediate drivenrollers and a rewinder. For decentralized control, the plantis divided into several subsections that are controlledindependently either in tension or speed. Each subsectioncontains a driven roller, some idle rollers and web spans(a web span refers to the web between two consecutiverollers). In such a subsection, the goal is to regulate webtension at the desired value while transporting the webwith a prescribed speed. A master speed roller is usedto regulate web speed over the entire processing line.Generally, web tension measurement is obtained in a spanof the processing line and is used as a feedback signal toprovide a speed reference correction for a driven roller.The most commonly used measurement devices are loadcells and dancers. Tension control is performed using directmeasurement of web tension in the case of load cells,while dancer mechanism is an indirect method to ensuretension control. The variable that is regulated is not webtension but the position of a force loaded dancer whichprovides regulation of tension indirectly: a desired forceis applied to the dancer by a pneumatic or hydrauliccylinder that is compensated by web tension. In Fig. 1,web tension regulation in both unwind and rewind zonesis done by using a pendulum dancer (labeled as “PD”).Tension control is performed by using a load cell (LC)feedback in the intermediate section.

The paper is organized as follows. Section 2 recalls themain physical laws to model web handling systems. Sec-tions 3 and 4 are then respectively dedicated to the designsof PI–LPV controllers and elasticity observers.

2. SYSTEM MODELING

The nonlinear model of a web transport system is builtfrom the equations describing web tension behavior be-tween two consecutive rollers and the velocity of each roller(Brandenburg (1977); Koc et al. (2002)).

2.1 Web Speed Dynamics

Assuming the absence of slippage between the web and theroll, the velocity of the k th roll is given by torque balanceon it:

d(JkΩk)

dt= Rk (Tk − Tk−1) +KkUk − Crk , (1)

where Ωk = Vk/Rk is k th roller angular velocity (Vk isweb linear speed), Tk is web tension between the k th andthe (k+1)th rolls, Crk corresponds to friction torque, Jkis roller inertia and Rk is roller radius. If the roller isdriven, KkUk is motor torque (Uk is the torque referencevoltage sent to the drive calculator and Kk is the ratiofrom reference voltage to torque, i.e. the current loop isapproximated by the gain Kk), otherwise KkUk is zero.

2.2 Web Tension Dynamics

Web strain dynamics is derived from the equation ofcontinuity applied to the web transport system:

Lkdzkdt

= −Vk+1zk + Vkzk−1 (2)

with

zk =1

1 + εk(3)

where Lk is the web length between the k th and the(k+1)th rollers and εk is the strain in the correspondingweb span. Web tension is related to web strain by theHooke’s law:

Tk = ESεk = ES(z−1k − 1) (4)


8640

where E is the Young’s modulus and S is the cross–sectionarea. Assuming a nominal working point (V0, T0), (2) and(4) yield to a linear tension dynamics equation:

LkdTkdt

= (ES + T0) (Vk+1 − Vk) + V0 (Tk−1 − Tk) (5)

2.3 LPV Representation

The plant in figure 1 is described by the state representa-tion:

x(t) = A(θ(t))x(t) +Bu(t)y(t) = Cx(t)

(6)

where x(t) is the state vector composed of web tension ineach span and web speed of each roller, u(t) is the controlinputs and y(t) is the measurement outputs. For tensioncontrol, y(t) is composed of the variables that are usedto monitor web tension, i.e. direct measurement of webtension provided by a load cell or dancers position whentension control is performed indirectly. θ(t) is the varyingparameters vector.

From equations (1) and (5), A(θ(t)) can be expressed as alinear combination of the parameters V0 and E0 as follows:

A(θ(t)) = A0 + V0A1 + E0A2

E0 = ES + T0(7)

3. CONTROLLER DESIGN

The purpose of this paper is to determine PI–LPV con-trollers with quadratic H∞ performance to enhance ten-sion control of the roll–to–roll system in figure 1 regardingparameters variations. For each subsystem, the PI con-troller is defined as follows:

C(s, θ) = Kp(θ) +Ki(θ)

s, (8)

whereKp(θ) andKi(θ) are the parameters of the controllerto determine. The main difficulty comes from the fact thatthe synthesis of reduced–order controllers is typically non–convex and, subsequently, cannot be solved directly byLMIs or Riccati equations.

3.1 Optimization Issue

The synthesis of H∞ controllers for LPV systems is basedon two essential results: quadratic stability and quadraticH∞ performance (Apkarian et al. (1995)). These resultsare recalled in Theorems 1 and 2.

Theorem 1. (Quadratic stability) Consider a polytopicLPV plant described by the following state representation:[A(θ) B(θ)C(θ) D(θ)

]∈ P := Co

[Ai BiCi Di

], i = 1, . . . , N

. (9)

The LPV system is quadratically stable if there existsP > 0 satisfying the set of LMIs

ATi P + PAi < 0 , ∀i = 1, . . . , N. (10)

A preliminary requirement to guarantee the quadraticstability of the closed–loop LPV system is that the Aimatrices have to be locally Hurwitz, i.e. the largest realpart of the eigenvalues has to be negative.

Wp

Wu

WtGKr ue y

z1

z2

z3

zM0

Fig. 2. H∞ Synthesis Framework

Theorem 2. (Vertex property, Apkarian et al. (1995))Consider a polytopic LPV plant described by (9). Thefollowing statements are equivalent:

(1) the LPV system is stable with quadratic H∞ perfor-mance γ;

(2) there exists a single matrix P > 0 such that, for allthe admissible parameters trajectories,

M[A(θ),B(θ),C(θ),D(θ)](P, γ) < 0; (11)

(3) there exists P > 0 satisfying the set of LMIs

M[Ai,Bi,Ci,Di](P, γ) < 0, i = 1, . . . , N, (12)

with

M[A,B,C,D](P, γ) :=

ATP + PA PB CT

BTP −γI DT

C D −γI

. (13)

For the synthesis of the PI–LPV controllers, the objectiveof the H∞ synthesis is to minimize the γ–norm betweena set of inputs r and some performance outputs z whileensuring stability over the entire polytope. The S/KS/Tframework used in this work is presented in figure 2: thesystem is augmented with weighting functions to shapethe closed–loop system behavior. M0 denotes a transferfunction whose purpose is to give explicitly the expectedbehavior.

The main difficulty comes from the fact that the reduced–order controller problem is non–convex. To figure out thisissue, a method initiated in Farag and Werner (2004) andKwiatkowski et al. (2009) is used. The main idea is to sepa-rate the problem into a convex subproblem which providesstability and H∞ performance under LMIs constraints,and a non–convex subproblem with a limited numbers ofunknowns (controllers variables) which is performed bygenetic algorithms. It results in the following optimizationproblem.

At each vertex, the augmented plant with the controller isrepresented by:

Si =

[Ai BiCi Di

], i = 1, . . . , 4 (14)

For the evolutionary optimization, the cost function to beminimized is then defined by:

J =

λ+ p1 if Ai are not all Hurwitzλ+ p2 if not quadratically stablep3 if stable but (12) is infeasibleγ else

, (15)

where λ is the largest spectral abscissa of the Ai matrices.p1 > p2 > p3 are penalties that facilitate algorithmconvergence.


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0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5x 104

Time (s)

Coe

ffic

ient

E0 (

N)

E0min

E0max

(a) Parameter E0

0 10 20 30 40 50 60 700

100

200

300

400

500

Time (s)

Vel

ocity

(m/m

in)

V0min

V0max

(b) Web Speed V0

0 10 20 30 40 50 60 7030

40

50

60

Time (s)

Tens

ion

(N)

ReferenceIntermediate tension Ti

(c) Intermediate Tension Ti

0 10 20 30 40 50 60 70−1

−0.5

0

0.5

1

Time (s)

Posi

tion

(deg

)ReferenceDancer position α2

(d) Dancer Position α2

Fig. 3. System Time Response for the PI–LPV Controllers with 2 Varying Parameters – E0 and V0

LMIs (10) and (12) are solved by using the Matlab solverSeDuMi (Sturm (1999)) together with the YALMIP en-vironment (Lofberg (2004)). The minimization of the costfunction J is performed by the software ModeFRONTIER,tailored at solving optimisation problem with genetic al-gorithms (Rigoni and Poles (2005)).

By considering the two varying parameters E0 and V0,the controller C(θ) is then a linear interpolation of thecontrollers at the four vertices:

C(θ) =4∑i=1

αiCi =4∑i=1

αi

[Ai BiCi Di

],

4∑i=1

αi = 1, (16)

where Ai, Bi, Ci et Di are the controller state matricesat the ith vertex. Coefficients αi are then defined in thefollowing manner:

α1 = (1− κ1)(1− κ2)α2 = κ1(1− κ2)α3 = (1− κ1)κ2α4 = κ1κ2

, (17)

with

κ1 =E0(t)− E0

E0 − E0

, κ2 =V0(t)− V0V0 − V0

, (18)

where θ and θ represent respectively the lower and upperbounds of the parameter θ.

3.2 Results

The previous method is used to design the two positioncontrollers (tension control using dancers) and the tension

102 103 104 105 1060

1000

2000

3000

V 0 (m

/min

)

E0 (N)

Fig. 4. Stability Region Depending on E0 and V0

controller (use of a load cell in the intermediate section) forthe plant of Fig. 1. The results of the PI–LPV controllersare presented in figure 3. Despite large variations of E0

and V0 (figures 3(a) and 3(b)), both dancers positionsand web tension remain close to their reference values.In comparison, LTI PI–H∞ controllers, calculated asin Knittel et al. (2007), are unstable for this set ofparameters.

Furthermore, figure 4 gives the stability region of the pro-posed controller depending on E0 and V0. The controllerC(θ) is interpolated from the controller at each vertex ofthe red box. The proposed controller enables to cover awide range of web speed and elasticity.

4. OBSERVER DESIGN

The control strategy described previously requires theknowledge of web elasticity E at any time. In order to


8642

A

B C

∫Observer

u

x^+

+

-

B

Hd

∫

∫

A

Wp

Kp

Ki

+

+

+

++

+

+

x

x~z

y~

v

System

ζ

-

+

Cy

y^

Fig. 5. PI–H∞ Luenberger State Observer

get this information, a PI–H∞ Luenberger observer isdesigned.

4.1 PI–H∞ Luenberger Observer

The diagram of a system with its PI–observer is presentedin figure 5. By considering the LTI system (assumingD = 0):

x(t) = Ax(t) +Bu(t) +Hd(t)y(t) = Cx(t)

, (19)

the objective is to minimize the H∞–norm of the transferfunction between the disturbances d(t) and the perfor-mance outputs z(t). The system to optimize including thePI–observer is represented as follows:[

˙x(t)

ζ(t)

]=

[A−KpC −Ki

−C 0

] [x(t)ζ(t)

]+

[H0

]d

z(s) = Wp(s)x(s)

(20)

where x(t) = x(t)− x(t) is the state estimation error andWp is a weighting function used to shape the dynamicsof x(t). Kp and Ki are respectively the proportional andintegral gain of the observer to be determined.

The synthesis of a PI–H∞ observer is equivalent to thecalculation of a static (order 0) output feedback controller.The problem cannot be easily formulated under LMIScontraints as it turns out that it is non–smooth and non–convex. One way to solve this problem is the use of non–convex programming. The H∞ framework requires thesetwo points to be fulfilled for the system (20):

(1) Stabilization: the spectral abscissa, i.e. the largest realpart of the eigenvalues, has to be negative.

(2) Minimization: the H∞–norm of the transfer functionbetween a set of exogenous inputs and some perfor-mance outputs has to be minimized.

In this paper, this problem is solved with the non–convexoptimization algorithm HANSO (Burke et al. (2005)), partof the user–friendly Matlab toolbox HIFOO (Gumussoy

Decentralizedobserver

k-1

Decentralizedobserver

kεk-1 εk

TkTk-1

VkVk-1 Vk+1

ΔEk-1 ΔEk

εk-2^ ^^

Lk-1,εk-1,Ek-1 Lk,εk,Ek

Fig. 6. Decentralized Web Elasticity Observers

et al. (2009)) which is tailored at solving fixed–orderand –structure H∞ problem. The main asset of such anapproach is that there is no need to prove the existence ofa Lyapunov function: the only unknowns are the observerparameters.

4.2 Problem Formulation

In this paper, the PI–H∞ Luenberger observer is usedto estimate web elasticity over the entire processing line.The estimation structure that is presented in figure 6 isfully decentralized. Local observers are used to estimatean elasticity modulus in each section of the web transportsystem.

For each local observer, a linear model of the subsystem isbuilt as follows. First, a linear strain dynamics is derivedfrom equation (2):

Lkdεk(t)

dt= V0 (εk−1(t)− εk(t)) +

(1 + ε0) (Vk+1(t)− Vk(t)) .(21)

Then the Hooke’s law (4) can be rewritten in this manner:

Tk(t) = (E + ∆E(t))Sεk(t), (22)

where Tk(t) is a web tension measurement provided by aload cell in a web span, E is an arbitrary choice of theYoung’s modulus value. ∆E(t) represents the variationsof E around its chosen value. Relation (22) can also beexpressed as:

Tk(t) = ESεk(t) + Sb(t) (23)

with:

∆E(t) =b(t)

εk(t). (24)

Assuming b(t) = d(t), the state representation (19) for alocal subsystem is defined as follows:

xT (t) = [εk(t) b(t)],uT (t) = [εk−1(t) Vk(t) Vk+1(t)],

y(t) = Tk(t).(25)

4.3 Results

As many works concerning the design of observers forweb handling systems, the approach described above forthe estimation of web elasticity in each section of theprocessing line neglects idle rollers. Nevertheless, a closeattention is paid on the estimation accuracy. The systemin figure 1 requires the estimation of web elasticity inthe three sections that are concerned by tension control.


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20 30 40 50 60

100

150

200

Time (s)

Elas

ticity

Mod

ulus

(MPa

)

Actual modulusEstimated modulus without idlersEstimated modulus with idlers

(a) Estimated Web Elasticity

20 30 40 50 60−10

−5

0

5

10

Time (s)

Tens

ion

(N)

Without idlersWith idlers

(b) Estimated Measurement Error T = T − T

Fig. 7. Influence of Idle Rollers on Elasticity Estimation

Figure 7 gives the results of the proposed observer in theintermediate subsystem. Two cases are studied:

(1) the observers are simulated on the intermediate sub-system that does not contain the idle rollers (fig. 6);

(2) the observers are simulated on the real plant (fig. 1).

One can observe that a constant bias occurs on webelasticity estimate in the presence of idle rollers (fig. 7(a))even if the error between web tension measurement andthe estimated web tension is close to zero (fig. 7(b)). In thesame time, the case without idle rollers is very accurate.Nevertheless, the provided estimates remain suitable for ause in adaptive control strategies, e.g. PI–LPV controllers.

5. CONCLUSION

This contribution has presented a new strategy for tensioncontrol in roll–to–roll system by using PI–LPV controllerswith H∞ performance. The approach enables to reachhigh accuracy and effectiveness in the time response ofthe closed–loop system. Besides it has been proven thatsuch controllers ensure stability over a wide range of webelasticity and speed. In the second part, the design of aPI–H∞ observer is detailed to provide an estimation ofweb elasticity modulus.

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