Cross Directional Control Graham C. Goodwin Day 4: Lecture 4 16th September 2004 International Summer School Grenoble, France Centre for Complex Dynamic Systems and Control
1. Introduction In this lecture we describe a practical application of receding horizon control to a common industrial problem, namely web-forming processes . Web-forming processes represent a wide class of industrial processes with relevance in many different areas such as paper making, plastic film extrusion, steel rolling, coating and laminating. Centre for Complex Dynamic Systems and Control
In a general set up, web processes (also known as film and sheet forming processes ) are characterised by raw material entering one end of the process machine and a thin web or film being produced in (possibly) several stages at the other end of the machine. The raw material is fed to the machine in a continuous or semi-continuous fashion and its flow through the web-forming machine is generally referred to as the machine direction [MD]. Centre for Complex Dynamic Systems and Control
Sheet and film processes are effectively two-dimensional spatially distributed processes with several of the properties of the sheet of material varying in both the machine direction and in the direction across the sheet known as the cross direction [CD]. Centre for Complex Dynamic Systems and Control
actuators cross direction sensors machine direction Figure: Generic web-forming process. Centre for Complex Dynamic Systems and Control
The main objective of the control applied to sheet and film processes is to maintain both the MD and CD profiles of the sheet as flat as possible, in spite of disturbances such as variations in the composition of the raw material fed to the machine, uneven distribution of the material in the cross direction, and deviations in the cross-directional profile. Centre for Complex Dynamic Systems and Control
In order to control the cross-directional profile of the web, several actuators are evenly distributed along the cross direction of the sheet. The number of actuators can vary from only 30 up to as high as 300. The film properties, on the other hand, are either measured via an array of sensors placed in a downstream position or via a scanning sensor that moves back and forth in the cross direction. The number of measurements taken by a single scan of the sensor can be up to 1000. Centre for Complex Dynamic Systems and Control
Difficulties the high dimensionality of the cross-directional system; the high cross-direction spatial interaction between actuators; the uncertainty in the model; the limited control authority of the actuators. Centre for Complex Dynamic Systems and Control
2. Problem Formulation It is generally the case that web-forming processes can be effectively modelled by assuming a decoupled spatial and dynamical response. This is equivalent to saying that the effect of one single actuator movement is almost instantaneous in the cross direction whilst its effect in the machine direction shows a certain dynamic behaviour. Centre for Complex Dynamic Systems and Control
These observations allow one to consider a general model for a cross-directional system of the form y k = q − d h ( q ) ¯ Bu k + d k , (1) where q − 1 is the unitary shift operator. Centre for Complex Dynamic Systems and Control
It is assumed that the system dynamics are the same across the machine and thus h ( q ) can be taken to be a scalar transfer function. In addition, h ( q ) is typically taken to be a low order, stable and minimum-phase transfer function. A typical model is a simple first-order system with unit gain: h ( q ) = ( 1 − α ) (2) q − α . A transport delay q − d accounts for the physical separation that exists between the actuators and the sensors in a typical cross-directional process application. Centre for Complex Dynamic Systems and Control
The matrix ¯ B is the normalised steady state interaction matrix and represents the spatial influence of each actuator on the system outputs. In most applications it is reasonably assumed that the steady state cross-directional profile generated by each actuator is identical. As a result, the interaction matrix ¯ B usually has the structure of a Toeplitz symmetric matrix. Centre for Complex Dynamic Systems and Control
The main difficulties in dealing with cross-directional control problems are related to the spatial interaction between actuators and not so much to the complexity of dynamics, which could reasonably be regarded as benign. A key feature is that a single actuator movement not only affects a single sensor measurement in the downstream position but also influences sensors placed in nearby locations. Indeed, the interaction matrix ¯ B is typically poorly conditioned in most cases of practical importance. Centre for Complex Dynamic Systems and Control
The poor conditioning of ¯ B can be quantified via a singular value decomposition B = USV T ¯ (3) where S , U , V ∈ R m × m . S = diag { σ 1 , σ 2 , . . . , σ m } is a diagonal matrix with positive singular values arranged in decreasing order, and U and V are orthogonal matrices such that UU = U U = I m and VV = V V = I m , where I m is the m × m identity matrix. If ¯ B is symmetric then U = V . Centre for Complex Dynamic Systems and Control
If ¯ B is poorly conditioned then the last singular values on the diagonal of S are very small compared to the singular values at the top of the chain { σ i } m i = 1 . This characteristic implies that the control directions associated with the smallest singular values are more difficult to control than those associated with the biggest singular values, in the sense that a larger control effort is required to compensate for disturbances acting in directions associated with small σ i . Centre for Complex Dynamic Systems and Control
This constitutes not only a problem in terms of the limited control authority usually available in the array of actuators, but it is also an indication of the sensitivity of the closed loop to uncertainties in the spatial components of the model. Centre for Complex Dynamic Systems and Control
The control objective in cross-directional control systems is usually stated as the requirement to minimise the variations of the output profile subject to input constraints. This can be stated in terms of minimising the following objective function: ∞ � � y k � 2 V ∞ = 2 k = 0 subject to input constraints � u k � ∞ ≤ u max . (4) Centre for Complex Dynamic Systems and Control
Another type of constraint typical of CD control systems is a second-order bending constraint defined as 1 � ∆ u i + 1 − ∆ u i k � ∞ ≤ b max for i = 1 , . . . , m , (5) k where ∆ u i k = u i k − u i − 1 is the deviation between adjacent actuators k in the input profile at a given time instant k . 1 The superscript indicates the actuator number. Centre for Complex Dynamic Systems and Control
3. Example 1 To illustrate the ideas involved in cross-directional control, we consider a 21-by-21 interaction matrix ¯ B with a Toeplitz symmetric structure and exponential profile: b ij = e − 0 . 2 | i − j | for i , j = 1 , . . . , 21 , (6) where b ij are the entries of the matrix ¯ B . Centre for Complex Dynamic Systems and Control
1 0.8 0.6 0.4 0.2 0 2 4 6 8 10 12 14 16 18 20 Cross-directional index Figure: Cross-directional profile for a unit step in actuator number 11. Centre for Complex Dynamic Systems and Control
We consider the transfer function h ( q ) = 1 − e − 0 . 2 (7) q − e − 0 . 2 , which is a discretised version of the first-order system y ( t ) = − y ( t ) + u ( t ) with sampling period T s = 0 . 2 sec . ˙ Centre for Complex Dynamic Systems and Control
The next figure shows the singular values of the interaction matrix ¯ B . We observe that there exists a significant difference between the largest singular value σ 1 and the smallest singular value σ 21 , indicating that the matrix is poorly conditioned. Dealing with the poor conditioning of ¯ B is one of the main challenges in CD control problems as we will see later. Centre for Complex Dynamic Systems and Control
8 7 6 5 4 3 2 1 0 2 4 6 8 10 12 14 16 18 20 Singular values index Figure: Singular values of the interaction matrix ¯ B . Centre for Complex Dynamic Systems and Control
In order to estimate the states of the system and the output disturbance d k , a Kalman filter is implemented as described for an extended system that includes the dynamics of a constant output disturbance: x k + 1 = Ax k + Bu k , d k + 1 = d k , y k = Cx k + d k , Centre for Complex Dynamic Systems and Control
In our case A = diag { e − 0 . 2 , . . . , e − 0 . 2 } , B = ( 1 − e − 0 . 2 ) ¯ B , C = I m . Centre for Complex Dynamic Systems and Control
The state noise covariance � � I m 0 Q n = , 0 100 I m and output noise covariance R n = I m , were considered in the design of the Kalman filter. Centre for Complex Dynamic Systems and Control
We will consider the finite horizon quadratic objective function with both prediction and control horizons set equal to one, that is V 1 , 1 = 1 2 ( y 0 Qy 0 + u 0 Ru 0 + x 1 Px 1 ) . (8) Q = I m , R = 0 . 1 I m . (9) Centre for Complex Dynamic Systems and Control
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