Robust PI LPV Tension Control with Elasticity Observer for Roll to Roll Systems

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1 Milano (Italy) August 28 - September 2, 211 Robust PI LPV Tension Control with Elasticity 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 transported on rollers during their processing. Maintaining web tension in the entire processing line under an expected web speed is a key factor in achieving good final product quality. It is common practice in industrial web transport systems to use decentralized PI type controllers. The performances of such control strategies highly depend on web speed and elasticity because of the strong coupling between these two variables. Moreover web speed and elasticity are subject to large variations during a same processing. The emphasis of this paper is on the design of linear parameter varying PI controllers with quadratic H performance to increase closed loop system robustness regarding these parameters variations. First a polytopic model of the experimental plant is derived from web speed and tension dynamics. A design methodology of the PI LPV controllers, that guarantees closed loop quadratic stability and H performance, is then presented. The method uses an optimization software with genetic algorithms to determine the controllers parameters while minimizing the H norm. Moreover, as it is difficult to measure 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 textiles, papers, polymers or metals are very common in the industry, because they represent a convenient way of transporting and processing a product from one form to another. Printing, coating and drying are examples of operations that can be performed in different sections of a web line. A web is usually described as any continuous and flexible material whose width is significantly less than its length and whose thickness is less than its width. Web tension and speed are two key variables that need to be monitored and controlled in order to achieve the expected final product quality. One of the main objectives in web handling machinery is to reach an expected web speed while maintaining the web tension within a close tolerance band in the entire processing line. This tolerance depends obviously on the type of material that has to be processed. In the recent years, many works have focused on the topic of tension control (Pagilla and Knittel (25); Koc et al. (22); Pagilla et al. (27); Shin (2); Gassmann et al. (211)) and have proposed various ways to enhance the performance: H control, optimal control, etc. But the common practice in industrial web transport systems remains the use of decentralized PI-type controllers. An improved design methodology of these PI controllers with fixed order and structure H techniques has been pre- 1 Corresponding author: knittel@unistra.fr. sented in Knittel et al. (27). Nevertheless, despite high performances for a nominal working point, it has been noticed two main drawbacks (Gassmann (211)): (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 affected by the Young s modulus. Consequently, this paper investigates the design of PI LPV controllers to figure out these issues. The conventional LPV controller synthesis is based on the existence of a Lyapunov function with a convex characterization and results in controllers that have the same order as the augmented plant (Apkarian and Gahinet (1995)). Besides, the design of fixed order PI controllers is formulated as a non convex problem. Kwiatkowski et al. (29) propose a method to solve this problem: it is a non convex optimization based on evolutionary algorithms. A similar method is used in this contribution to take into consideration web speed and elasticity variations in the decentralized control strategy. The PI LPV controllers require the knowledge of web elasticity at any time. An observer is therefore designed to estimate elasticity modulus over the entire processing line. Although the design of observers for roll-to roll systems has encountered some attention in the literature, the concern is essentially on the estimation of web tension (Wolfermann (1997); Lynch et al. (24); Gassmann and Copyright by the International Federation of Automatic Control (IFAC) 8639

2 Milano (Italy) August 28 - September 2, 211 Driven Roller Master-Speed Roller Unwinder M 2 Rewinder M 3 M 3 M 1 F 1 F 2 PD PD M 4 Unwind Section Intermediate Section Rewind Section Fig. 1. Web Handling Platform Knittel (27)). Only few rare papers have focused on web elasticity estimation (Boulter (1999); Angermann (24)). In this paper, a PI H Luenberger state observer is introduced. The problem of the observer synthesis is similar to the calculation of a static output feedback controller. Consequently, the problem is non convex and is solved by non convex programming. A sketch of the system that is used in this paper is presented in figure 1. It mimics the classical structure of an industrial processing line. The system in Fig. 1 is composed of an unwinder, two intermediate driven rollers and a rewinder. For decentralized control, the plant is divided into several subsections that are controlled independently either in tension or speed. Each subsection contains a driven roller, some idle rollers and web spans (a web span refers to the web between two consecutive rollers). In such a subsection, the goal is to regulate web tension at the desired value while transporting the web with a prescribed speed. A master speed roller is used to regulate web speed over the entire processing line. Generally, web tension measurement is obtained in a span of the processing line and is used as a feedback signal to provide a speed reference correction for a driven roller. The most commonly used measurement devices are load cells and dancers. Tension control is performed using direct measurement of web tension in the case of load cells, while dancer mechanism is an indirect method to ensure tension control. The variable that is regulated is not web tension but the position of a force loaded dancer which provides regulation of tension indirectly: a desired force is applied to the dancer by a pneumatic or hydraulic cylinder that is compensated by web tension. In Fig. 1, web tension regulation in both unwind and rewind zones is done by using a pendulum dancer (labeled as PD ). Tension control is performed by using a load cell () feedback in the intermediate section. The paper is organized as follows. Section 2 recalls the main physical laws to model web handling systems. Sections 3 and 4 are then respectively dedicated to the designs of PI LPV controllers and elasticity observers. 2. SYSTEM MODELING The nonlinear model of a web transport system is built from the equations describing web tension behavior between two consecutive rollers and the velocity of each roller (Brandenburg (1977); Koc et al. (22)). 2.1 Web Speed Dynamics Assuming the absence of slippage between the web and the roll, the velocity of the k th roll is given by torque balance on it: d(j k Ω k ) = R k (T k T k 1 ) K k U k C rk, (1) dt where Ω k = V k /R k is k th roller angular velocity (V k is web linear speed), T k is web tension between the k th and the (k1) th rolls, C rk corresponds to friction torque, J k is roller inertia and R k is roller radius. If the roller is driven, K k U k is motor torque (U k is the torque reference voltage sent to the drive calculator and K k is the ratio from reference voltage to torque, i.e. the current loop is approximated by the gain K k ), otherwise K k U k is zero. 2.2 Web Tension Dynamics Web strain dynamics is derived from the equation of continuity applied to the web transport system: with L k dz k dt = V k1z k V k z k 1 (2) z k = 1 1 ɛ k (3) where L k is the web length between the k th and the (k1) th rollers and ɛ k is the strain in the corresponding web span. Web tension is related to web strain by the Hooke s law: T k = ESɛ k = ES(z 1 k 1) (4) 864

3 Milano (Italy) August 28 - September 2, 211 where E is the Young s modulus and S is the cross section area. Assuming a nominal working point (V, T ), (2) and (4) yield to a linear tension dynamics equation: L k dt k dt = (ES T ) (V k1 V k ) V (T k 1 T k ) (5) 2.3 LPV Representation r e K M u G y W p W u W t z 1 z 2 z 3 } z The plant in figure 1 is described by the state representation: { ẋ(t) = A(θ(t))x(t) Bu(t) (6) y(t) = Cx(t) where x(t) is the state vector composed of web tension in each span and web speed of each roller, u(t) is the control inputs and y(t) is the measurement outputs. For tension control, y(t) is composed of the variables that are used to monitor web tension, i.e. direct measurement of web tension provided by a load cell or dancers position when tension control is performed indirectly. θ(t) is the varying parameters vector. From equations (1) and (5), A(θ(t)) can be expressed as a linear combination of the parameters V and E as follows: A(θ(t)) = A V A 1 E A 2 E = ES T (7) 3. CONTROLLER DESIGN The purpose of this paper is to determine PI LPV controllers with quadratic H performance to enhance tension control of the roll to roll system in figure 1 regarding parameters variations. For each subsystem, the PI controller is defined as follows: C(s, θ) = K p (θ) K i(θ), (8) s where K p (θ) and K i (θ) are the parameters of the controller to determine. The main difficulty comes from the fact that the synthesis of reduced order controllers is typically non convex and, subsequently, cannot be solved directly by LMIs or Riccati equations. 3.1 Optimization Issue The synthesis of H controllers for LPV systems is based on two essential results: quadratic stability and quadratic H performance (Apkarian et al. (1995)). These results are recalled in Theorems 1 and 2. Theorem 1. (Quadratic stability) Consider a polytopic LPV plant described by the following state representation: [ ] {[ ] } A(θ) B(θ) Ai B P := Co i, i = 1,..., N. (9) C(θ) D(θ) C i D i The LPV system is quadratically stable if there exists P > satisfying the set of LMIs A T i P P A i <, i = 1,..., N. (1) A preliminary requirement to guarantee the quadratic stability of the closed loop LPV system is that the A i matrices have to be locally Hurwitz, i.e. the largest real part of the eigenvalues has to be negative. Fig. 2. H Synthesis Framework Theorem 2. (Vertex property, Apkarian et al. (1995)) Consider a polytopic LPV plant described by (9). The following statements are equivalent: (1) the LPV system is stable with quadratic H performance γ; (2) there exists a single matrix P > such that, for all the admissible parameters trajectories, M [A(θ),B(θ),C(θ),D(θ)] (P, γ) < ; (11) (3) there exists P > satisfying the set of LMIs M [Ai,B i,c i,d i](p, γ) <, i = 1,..., N, (12) with M [A,B,C,D] (P, γ) := AT P P A P B C T B T P γi D T C D γi. (13) For the synthesis of the PI LPV controllers, the objective of the H synthesis is to minimize the γ norm between a set of inputs r and some performance outputs z while ensuring stability over the entire polytope. The S/KS/T framework used in this work is presented in figure 2: the system is augmented with weighting functions to shape the closed loop system behavior. M denotes a transfer function whose purpose is to give explicitly the expected behavior. The main difficulty comes from the fact that the reduced order controller problem is non convex. To figure out this issue, a method initiated in Farag and Werner (24) and Kwiatkowski et al. (29) is used. The main idea is to separate the problem into a convex subproblem which provides stability and H performance under LMIs constraints, and a non convex subproblem with a limited numbers of unknowns (controllers variables) which is performed by genetic algorithms. It results in the following optimization problem. At each vertex, the augmented plant with the controller is represented by: [ ] Ãi Bi S i =, i = 1,..., 4 (14) C i Di For the evolutionary optimization, the cost function to be minimized is then defined by: λ p 1 if Ãi are not all Hurwitz λ p J = 2 if not quadratically stable, (15) p 3 if stable but (12) is infeasible γ else where λ is the largest spectral abscissa of the Ãi matrices. p 1 > p 2 > p 3 are penalties that facilitate algorithm convergence. 8641

4 Milano (Italy) August 28 - September 2, 211 x Coefficient E (N) E max E min Velocity (m/min) V min V max (a) Parameter E (b) Web Speed V 6 1 Tension (N) 5 4 Reference Intermediate tension T i (c) Intermediate Tension T i Position (deg).5.5 Reference Dancer position α (d) Dancer Position α 2 Fig. 3. System Time Response for the PI LPV Controllers with 2 Varying Parameters E and V LMIs (1) and (12) are solved by using the Matlab solver SeDuMi (Sturm (1999)) together with the YALMIP environment (Löfberg (24)). The minimization of the cost function J is performed by the software ModeFRONTIER, tailored at solving optimisation problem with genetic algorithms (Rigoni and Poles (25)). By considering the two varying parameters E and V, the controller C(θ) is then a linear interpolation of the controllers at the four vertices: 4 4 [ ] Âi 4 ˆBi C(θ) = α i C i = α i, α i = 1, (16) ˆDi i=1 i=1 Ĉ i i=1 where Âi, ˆBi, Ĉ i et ˆD i are the controller state matrices at the i th vertex. Coefficients α i are then defined in the following manner: with α 1 = (1 κ 1 )(1 κ 2 ) α 2 = κ 1 (1 κ 2 ) α 3 = (1 κ 1 )κ 2, (17) α 4 = κ 1 κ 2 κ 1 = E (t) E E E, κ 2 = V (t) V V V, (18) where θ and θ represent respectively the lower and upper bounds of the parameter θ. 3.2 Results The previous method is used to design the two position controllers (tension control using dancers) and the tension V (m/min) E (N) Fig. 4. Stability Region Depending on E and V controller (use of a load cell in the intermediate section) for the plant of Fig. 1. The results of the PI LPV controllers are presented in figure 3. Despite large variations of E and V (figures 3(a) and 3(b)), both dancers positions and web tension remain close to their reference values. In comparison, LTI PI H controllers, calculated as in Knittel et al. (27), are unstable for this set of parameters. Furthermore, figure 4 gives the stability region of the proposed controller depending on E and V. The controller C(θ) is interpolated from the controller at each vertex of the red box. The proposed controller enables to cover a wide range of web speed and elasticity. 4. OBSERVER DESIGN The control strategy described previously requires the knowledge of web elasticity E at any time. In order to 8642

5 Milano (Italy) August 28 - September 2, 211 d u H B B Observer Fig. 5. PI H Luenberger State Observer z v A W p A ~ x K p K i x - x^ ζ C C y^ System get this information, a PI H Luenberger observer is designed. 4.1 PI H Luenberger Observer The diagram of a system with its PI observer is presented in figure 5. By considering the LTI system (assuming D = ): { ẋ(t) = Ax(t) Bu(t) Hd(t), (19) y(t) = Cx(t) the objective is to minimize the H norm of the transfer function between the disturbances d(t) and the performance outputs z(t). The system to optimize including the PI observer is represented as follows: [ x(t) ζ(t) ] = [ A Kp C K i C z(s) = W p (s) x(s) ] [ x(t) ζ(t) - y ~ y ] [ ] H d (2) where x(t) = x(t) ˆx(t) is the state estimation error and W p is a weighting function used to shape the dynamics of x(t). K p and K i are respectively the proportional and integral gain of the observer to be determined. The synthesis of a PI H observer is equivalent to the calculation of a static (order ) output feedback controller. The problem cannot be easily formulated under LMIS contraints 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 these two points to be fulfilled for the system (2): (1) Stabilization: the spectral abscissa, i.e. the largest real part of the eigenvalues, has to be negative. (2) Minimization: the H norm of the transfer function between a set of exogenous inputs and some performance outputs has to be minimized. In this paper, this problem is solved with the non convex optimization algorithm HANSO (Burke et al. (25)), part of the user friendly Matlab toolbox HIFOO (Gumussoy L k-1,ε k-1,e k-1 T k-1 V k-1 V k V k1 Decentralized observer k-1 L k,ε k,e k Decentralized observer k ^ ^ ^ ε k-2 ε k-1 ε k ΔE k-1 ΔE k Fig. 6. Decentralized Web Elasticity Observers et al. (29)) which is tailored at solving fixed order and structure H problem. The main asset of such an approach is that there is no need to prove the existence of a Lyapunov function: the only unknowns are the observer parameters. 4.2 Problem Formulation In this paper, the PI H Luenberger observer is used to estimate web elasticity over the entire processing line. The estimation structure that is presented in figure 6 is fully decentralized. Local observers are used to estimate an elasticity modulus in each section of the web transport system. For each local observer, a linear model of the subsystem is built as follows. First, a linear strain dynamics is derived from equation (2): L k dɛ k (t) dt T k = V (ɛ k 1 (t) ɛ k (t)) (1 ɛ ) (V k1 (t) V k (t)). (21) Then the Hooke s law (4) can be rewritten in this manner: T k (t) = (E E(t))Sɛ k (t), (22) where T k (t) is a web tension measurement provided by a load cell in a web span, E is an arbitrary choice of the Young s modulus value. E(t) represents the variations of E around its chosen value. Relation (22) can also be expressed as: with: T k (t) = ESɛ k (t) Sb(t) (23) E(t) = b(t) ɛ k (t). (24) Assuming ḃ(t) = d(t), the state representation (19) for a local subsystem is defined as follows: 4.3 Results x T (t) = [ɛ k (t) b(t)], u T (t) = [ˆɛ k 1 (t) V k (t) V k1 (t)], y(t) = T k (t). (25) As many works concerning the design of observers for web handling systems, the approach described above for the estimation of web elasticity in each section of the processing line neglects idle rollers. Nevertheless, a close attention is paid on the estimation accuracy. The system in figure 1 requires the estimation of web elasticity in the three sections that are concerned by tension control. 8643

6 Milano (Italy) August 28 - September 2, 211 Elasticity Modulus (MPa) Tension (N) Actual modulus Estimated modulus without idlers Estimated modulus with idlers (a) Estimated Web Elasticity Without idlers With idlers (b) Estimated Measurement Error T = T ˆT Fig. 7. Influence of Idle Rollers on Elasticity Estimation Figure 7 gives the results of the proposed observer in the intermediate subsystem. Two cases are studied: (1) the observers are simulated on the intermediate subsystem 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 web elasticity estimate in the presence of idle rollers (fig. 7(a)) even if the error between web tension measurement and the estimated web tension is close to zero (fig. 7(b)). In the same time, the case without idle rollers is very accurate. Nevertheless, the provided estimates remain suitable for a use in adaptive control strategies, e.g. PI LPV controllers. 5. CONCLUSION This contribution has presented a new strategy for tension control in roll to roll system by using PI LPV controllers with H performance. The approach enables to reach high accuracy and effectiveness in the time response of the closed loop system. Besides it has been proven that such controllers ensure stability over a wide range of web elasticity and speed. In the second part, the design of a PI H observer is detailed to provide an estimation of web elasticity modulus. REFERENCES Angermann, A. (24). Entkopplung von Mehrgrössensystemen durch Vorsteurerung am Beispiel von kontinuierlichen Fertigungsanlagen. Ph.D. thesis, Technische Universität München, Germany. Apkarian, P. and Gahinet, P. (1995). A convex characterization of gain-scheduled H controllers. IEEE Transactions on Automatic Control, 4(5), Apkarian, P., Gahinet, P., and Becker, G. (1995). Selfscheduled H control of linear parameter-varying systems: a design example. Automatica, 31(9), Boulter, B.T. (1999). Estimating modulus of elasticity, torque loss, and tension using an extended Kalman filter. In Int. Conf. on Web Handling. Stillwater, USA. Brandenburg, G. (1977). New mathematical models for web tension and register error. In 3 rd Int. IFAC Conf. Instrum. Autom. Paper, Rubber Plastics Ind., Burke, J.V., Lewis, A.S., and Overton, M.L. (25). A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim., 15, Farag, A. and Werner, H. (24). A Riccati-genetic algorithms approach to fixed-structure controller synthesis. In American Control Conference, Gassmann, V. (211). Commande décentralisée robuste de systèmes d entraînement de bandes à élasticité variable. Ph.D. thesis, University of Strasbourg, France. Gassmann, V. and Knittel, D. (27). Tension observers in elastic web unwinder-winder systems. In ASME IMECE. Seattle, WA, USA. Gassmann, V., Knittel, D., Pagilla, P.R., and Bueno, M.A. (211). Fixed-order H tension control in the unwinding section of a web handling system using a pendulum dancer. IEEE Trans. on Cont. Sys. Tech. Gumussoy, S., Henrion, D., Millstone, M., and Overton, M.L. (29). Multiobjective robust control with HIFOO 2.. In IFAC Symp. on ROCOND. Haifa, Israel. Knittel, D., Henrion, D., Millstone, M., and Vedrines, M. (27). Fixed-order and structure H control with model based feedforward for elastic web winding systems. In IFAC Symp. on Large Scale Systems. Gdansk, Poland. Koc, H., Knittel, D., De Mathelin, M., and Abba, G. (22). Modeling and robust control of winding systems for elastic webs. IEEE Trans. on Cont. Sys. Tech., 1(2), Kwiatkowski, A., Werner, H., Blath, J., Ali, A., and Schultalbers, M. (29). Linear parameter varying PID controller design for charge control of a spark-ignited engine. Cont. Eng. Pract., 17(11), Löfberg, J. (24). YALMIP : A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference. Taipei, Taiwan. Lynch, A., Bortoff, S., and Röbenack, K. (24). Nonlinear tension observers for web machines. Automatica, 4, Pagilla, P. and Knittel, D. (25). Recent advances in web longitudinal control. In Int. Conf. on Web Handling. Stillwater, USA. Pagilla, P., Siraskar, N., and Dwivedula, R. (27). Decentralized control of web processing line. IEEE Transactions on Control Sytems Technology, 15(1), Rigoni, E. and Poles, S. (25). NBI and MOGA-II, two complementary algorithms for multi-objective optimizations. In Pract. Appr. to Multi-Objective Optimization. Shin, K. (2). Tension control. Tapi press edition. Sturm, J.F. (1999). Using SeDuMi 1.2, a Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software, 11(1), Wolfermann, W. (1997). Sensorless tension control of webs. In Int. Conf. on Web Handling. Stillwater, USA. 8644