Fuzzy Control of the Lateral Position of a Moving Web in Roll-to-Roll Processes
Thanhtam Ho, Hyeunhun Shin, and Sangyoon LeeDepartment of Mechanical Design and Production Engineering
Konkuk University
1 Hwayang-dong, Gwangjin-gu, Seoul, South Korea
, {shinhh28 & slee}@konkuk.ac.kr
Abstract – This paper reports the development of simulation software and control logic for the lateral position control of a moving web in roll-to-roll (R2R) systems. A mathematical model is described first to explain the lateral dynamics of a moving web. Based on the model, simulation software named LACOSIM is designed and implemented to simulate the lateral dynamics and also to control the lateral position error of a web in R2R systems. As a lateral control method, fuzzy control logic is developed, embedded in LACOSIM, and verified by the simulations with sine and step input errors. In addition, a tabletop R2R hardware simulator is implemented as an experimental apparatus. Both the simulation software and the hardware simulator are considered as useful tools for the development of a web guide system for R2R systems.
Index Terms - lateral position control, roll-to-roll (R2R), moving web, fuzzy control.
I. Introduction
Roll-to-roll (R2R) systems employing a moving web made of plastic or metal have been an attractive choice for the mass production. In particular, R2R printing systems in the form of offset or gravure printing have been popularly used in the traditional packaging industry. Recently they are considered as an innovative solution for the printed electronics (also known as e-printing), which is a new field of electronic industry for the mass production of electronic devices by means of printing. A variety of devices including RFID, electronic paper, solar cells, displays are taken into consideration for the printed electronics [1].
The R2R method has advantages in the printed electronics because it can be highly effective in the cost and also applicable to the processing of flexible and thin materials. However, unlike traditional printing processes, the printed electronics requires a much higher accuracy in every process, and so the precision in the lateral position control of a moving web is also a significant requirement for the success of R2R printing systems in the printed electronics.
In general web guide systems are used to correct the lateral position error of a web in R2R systems. Typical web guide systems consist of a controller, an electromagnetic motor, and multiple sensors. When a web is thin and flexible like plastic and the width is small, the displacement guide system with a short span can be a suitable choice.
This paper reports the development of simulation software and a hardware simulator for the lateral position control of a moving web in R2R systems. The software named LACOSIM is designed for the simulation and control of the lateral position of a moving web in R2R processes. It employs a mathematical model and control logic in a 3-D graphical environment, and also it unifies the simulation and control of the lateral position of a web in a unique interface. In order to implement lateral control methods, mathematical models need to be set first to describe the lateral dynamics of a moving web.
In addition to the software system, a hardware web simulator employing two displacement guide systems is designed and built for verifying simulation results and for determining specific conditions of web-guiding. The simulator is composed of one DC motor, nine rollers including the drive roller and the tension roller, and two load cells.
II. Mathematical Model
In order to build a control system of the lateral position, we first introduce a mathematical model that describes the dynamics of a web. Shelton’s first order model [2, 3] presents the relation of the lateral velocity to the longitudinal velocity, the guide dynamics, and the input error that occurs at the previous roll in a single span guide.
The model is a simplified one because it is based on the fundamental law of static steering: the web in the entering span aligns itself perpendicularly to the roller. The model is built under the assumption that both the mass and the lateral stiffness of the web are negligible. As a result, the web is straight in each span and it also makes sharp angular breaks as it leaves each roller in a series of non-parallel rollers [2].
Fig. 1 shows the idealized web behaviour with a certain input error. The lateral displacement and the lateral velocity are expressed in (1) and (2), respectively. In the equations l is the distance between two rollers, V is the line speed, qr is the angle between the roller and y axis, and qL is the web angle measured with respect to the x axis [3].
(1)
(2)
If the roller moves laterally and has a position z, the movement should be included in the velocity of web edge. As a result a first order differential equation of the lateral velocity is obtained
. (3)
There is a specific case where two rollers are parallel and fixed as shown in Fig. 2. Since there is no roller movement component, the following equation is obtained by taking the Laplace transform with zero initial condition:
. (4)
If we let t = L/V be the time constant, the result is
. (5)
Fig. 1 Behavior of an idealized web.
Fig. 2 Model of fixed parallel guides.
III. Simulation Software
Simulink of Matlab provides a large advantage in the simulation work due to numerous built-in functions and toolboxes. However, the software lacks visualization ability in the simulation and is not easy to be connected to hardware systems. Other motion simulation softwares such as ADAMS, Nastran (MSC software corporation) or Recurdyn (FunctionBay, Inc.) can be useful for kinematics and dynamics analysis, but simulation work is usually complicated and computationally expensive. In addition, the dynamics of a moving web is relatively complex, and the simulation model for R2R system is rarely found in motion simulation softwares.
Therefore we developed our own simulation software named LACOSIM (LAteral motion and COntrol SIMulation). It has several useful features: first it allows one to implement the dynamics of a moving web and a web guide system composed of a DC motor and sensors; second control algorithms such as PID and fuzzy methods can be applied in the software to control a virtual web guide system in R2R systems; third LACOSIM itself can function as a control software for a real web guide system. Fig. 3 shows the model of web guide system in LACOSIM.
LACOSIM is constructed using C++ programming language with a graphics library OpenGL. The software interface has components including the main view, a graph window, control bars and dialog windows, as shown in Fig. 4. The main view window contains the 3-D model of web guide system. The user interface system of LACOSIM allows users to set and change the parameters of guide system, web properties, and controller gains. The simulation results are shown in graph windows, and can be saved to files.
All the ordinary differential equations (ODE) in LACOSIM are solved by the fourth order Runge-Kutta numerical method (RK4) [4, 5]. RK4 method is known to be very accurate and well-behaved for a wide range of problems. Since all the ODEs in the web simulator system are of higher order, RK4 could not be applied directly.
Therefore the mathematical models were converted to state variable forms and RK4 method was implemented. The results from solving ODE of mathematical models combined with the output of the controllers become the motions of guide and web in 3-D graphic environments. As a result, both the real motion of web and guide system and the response plot can be observed. In Fig. 4, the left window shows the plots of input error and sensor outputs.
Fig. 3 Web guide system model in LACOSIM.
Fig. 4 User interface of LACOSIM.
IV. Lateral Position Control
In R2R systems, web guide systems are usually used to correct the lateral position control of a web. When a web is thin and flexible like plastic and the width is small, the displacement guide (DG) with a short span can be useful. Fig. 5 illustrates the web simulator model, which is used for our lateral position control. The web simulator system includes two displacement guides where DG2 is the main guide that corrects the lateral position error as the web runs through, while DG1 serves as a disturbance generation source to generate sine or step input errors.
Fig. 5 The web simulator model
A. Fuzzy Control Logic
Fig. 6 shows the block diagram of the lateral control system where the guide system is the combination of a DC motor model, a guidance platform, and web dynamics model. Here fuzzy controller serves as a “black box” which maps the input space to the output space. The processing of mapping from the given input to the output is called the fuzzy inference.
Our fuzzy controller employs Mamdani inference type [6] considering that the output parameter membership function is a continuous function in the type. In Mamdani method, the output space is a fuzzy space determined by the continuous membership functions, and so the defuzzification process requires a complicated computation to find the centroid of the 2D-shape.
The fuzzy system has three input variables: error (e), rate of error () and sum of error (∑e). At the current time t that corresponds to a sampling point k, the input variables are calculated as
, (6)
, (7)
. (8)
Here yref(K) is the reference or desired lateral position at the downstream span, yn(K) is the actual position of web at the downstream, and yn(K) is the value measured by the edge detect sensor in LACOSIM. The variables are named Error, rate_Error, and sum_Error, respectively and they are linguistic variables of e, , and ∑e. We define four fuzzy variables for Error: so negative (SoNeg), Normal, Small, and so positive (SoPos).
Among a variety of shapes of fuzzy membership function, a triangle or a trapezoid can be a suitable choice because they often provide an adequate representation of the expert knowledge, and also simplifies the process of computation significantly. Hence in LACOSIM, only triangle and trapezoid are used. Fig.7 shows the membership functions of variable Error. Since the other two variables, sum_Error and rate_Error are supporting variables in our algorithm, only two fuzzy sets are used: negative (Neg) and Positive (Pos).
Since the Mamdani fuzzy inference is chosen as the inference machine, the output membership functions are also expected to be fuzzy sets. The output variable is named analog output(analog_out) which represents the DC voltage. It is applied to the DC motor model on web guidance platform. There are five fuzzy variables on the output (see Fig. 8): very negative(VeNeg), negative(Neg), little, positive(Pos) and very Positive(VePos). The triangle and trapezoid are used to describe these fuzzy variables too.
The fuzzy rules in LACOSIM are built such that the control mimics the principle of a conventional PID controller. As shown in Table 1, seven rules are developed in which rule No. 1, 2, and 7 are inspired from the proportion control. Rule No. 3, 4 are based on the derivative control and the other rules imitate the integration control.
Fig. 6 Fuzzy control model in LACOSIM
Fig. 7 Membership functions of Error variable.
Fig. 8 Membership functions of analog_out linguistic variable.
The implication operation in the fuzzy control is the method to apply the result of the antecedent evaluation to the membership function of the consequent. The most common implication method is cutting the consequent membership function at the level of the antecedent truth. Although the top of output membership function is cut and some information is lost, the clipping method has the advantage in the computation time and simplicity. In addition, the defuzzification process on the aggregated output surface is relatively easier.
Fig. 9 illustrates how the rule in our LACOSIM-Fuzzy works from the crisp input value to the result of the implication process. These final steps give the crisp output value for the next plant, but they often take much time for computation. The aggregation process combines all the cuts in 2-D shapes from the implication process into one in 2-D area. The defuzzification is the process which takes the crisp value from the area and produces the output value. One of most common methods for the defuzzification is the calculation of centroid or center of gravity (COG). In this method the crisp output value is considered as the horizontal coordinate of the centroid of the aggregated shape.
One large advantage of COG defuzzification method is that when the input values change continuously, the crisp output also change continuously. It is a desirable property in modeling and control applications. In most cases, the centroid calculation is performed by a numerical method, which can cause much computation time. In our LACOSIM-Fuzzy, since the membership functions have triangular or trapezoidal form only, the aggregated area can be divided into several rectangle or trapezoid subareas (ten subareas at most). Therefore the COG can be determined easily, and so the COG computation is quite efficient in the computation time. Fig. 10 shows the process.
TABLE I
Fuzzy Rule System
Rule No. / Antecedent / Consequent1 / IF Error IS so negative / THEN analog output SHOULD BE very positive
2 / IF Error IS so positive / THEN analog output SHOULD BE very negative
3 / IF Error IS normal AND rate of Error is negative / THEN analog output SHOULD BE positive
4 / IF Error IS normal AND rate of Error is positive / THEN analog output SHOULD BE negative
5 / IF Error IS small AND sum of Error is negative / THEN analog output SHOULD BE positive
6 / IF Error IS small AND sum of Error is positive / THEN analog output SHOULD BE negative
7 / IF Error IS small / THEN analog output SHOULD BE little