Cross Directional Control - The University of Newcastle, Australia...In order to control the cross-directional proﬁle of the web, several actuators are evenly distributed along the

Cross Directional Control

Graham C. Goodwin

Day 4: Lecture 4

16th September 2004

International Summer SchoolGrenoble, France

Centre for Complex DynamicSystems and Control

1. Introduction

In this lecture we describe a practical application of recedinghorizon control to a common industrial problem, namelyweb-forming processes. Web-forming processes represent a wideclass of industrial processes with relevance in many different areassuch as paper making, plastic film extrusion, steel rolling, coatingand laminating.


In a general set up, web processes (also known as film and sheetforming processes) are characterised by raw material entering oneend of the process machine and a thin web or film being producedin (possibly) several stages at the other end of the machine. Theraw material is fed to the machine in a continuous orsemi-continuous fashion and its flow through the web-formingmachine is generally referred to as the machine direction [MD].


Sheet and film processes are effectively two-dimensional spatiallydistributed processes with several of the properties of the sheet ofmaterial varying in both the machine direction and in the directionacross the sheet known as the cross direction [CD].


cross directionactuators

sensors

machine direction

Figure: Generic web-forming process.


The main objective of the control applied to sheet and filmprocesses is to maintain both the MD and CD profiles of the sheetas flat as possible, in spite of disturbances such as variations inthe composition of the raw material fed to the machine, unevendistribution of the material in the cross direction, and deviations inthe cross-directional profile.


In order to control the cross-directional profile of the web, severalactuators are evenly distributed along the cross direction of thesheet. The number of actuators can vary from only 30 up to ashigh as 300. The film properties, on the other hand, are eithermeasured via an array of sensors placed in a downstream positionor via a scanning sensor that moves back and forth in the crossdirection. The number of measurements taken by a single scan ofthe sensor can be up to 1000.


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.


2. Problem Formulation

It is generally the case that web-forming processes can beeffectively modelled by assuming a decoupled spatial anddynamical response. This is equivalent to saying that the effect ofone single actuator movement is almost instantaneous in the crossdirection whilst its effect in the machine direction shows a certaindynamic behaviour.


These observations allow one to consider a general model for across-directional system of the form

yk = q−dh(q)Buk + dk , (1)

where q−1 is the unitary shift operator.


It is assumed that the system dynamics are the same across themachine and thus h(q) can be taken to be a scalar transferfunction. In addition, h(q) is typically taken to be a low order, stableand minimum-phase transfer function. A typical model is a simplefirst-order system with unit gain:

h(q) =(1 − α)q − α

. (2)

A transport delay q−d accounts for the physical separation thatexists between the actuators and the sensors in a typicalcross-directional process application.


The matrix B is the normalised steady state interaction matrix andrepresents the spatial influence of each actuator on the systemoutputs. In most applications it is reasonably assumed that thesteady state cross-directional profile generated by each actuator isidentical. As a result, the interaction matrix B usually has thestructure of a Toeplitz symmetric matrix.


The main difficulties in dealing with cross-directional controlproblems are related to the spatial interaction between actuatorsand not so much to the complexity of dynamics, which couldreasonably be regarded as benign.

A key feature is that a single actuator movement not only affects asingle sensor measurement in the downstream position but alsoinfluences sensors placed in nearby locations. Indeed, theinteraction matrix B is typically poorly conditioned in most cases ofpractical importance.


The poor conditioning of B can be quantified via a singular valuedecomposition

B = USVT (3)

where S,U,V ∈ Rm×m. S = diag{σ1, σ2, . . . , σm} is a diagonalmatrix with positive singular values arranged in decreasing order,and U and V are orthogonal matrices such that UU = UU = Imand VV = V V = Im, where Im is the m ×m identity matrix. If B issymmetric then U = V .


If B is poorly conditioned then the last singular values on thediagonal of S are very small compared to the singular values at thetop of the chain {σi}

mi=1. This characteristic implies that the control

directions associated with the smallest singular values are moredifficult to control than those associated with the biggest singularvalues, in the sense that a larger control effort is required tocompensate for disturbances acting in directions associated withsmall σi .


This constitutes not only a problem in terms of the limited controlauthority usually available in the array of actuators, but it is also anindication of the sensitivity of the closed loop to uncertainties in thespatial components of the model.


The control objective in cross-directional control systems is usuallystated as the requirement to minimise the variations of the outputprofile subject to input constraints. This can be stated in terms ofminimising the following objective function:

V∞ =∞∑

k=0

‖yk ‖22

subject to input constraints

‖uk ‖∞ ≤ umax. (4)


Another type of constraint typical of CD control systems is asecond-order bending constraint defined as1

‖∆ui+1k − ∆ui

k ‖∞ ≤ bmax for i = 1, . . . ,m, (5)

where ∆uik = ui

k − ui−1k is the deviation between adjacent actuators

in the input profile at a given time instant k .

1The superscript indicates the actuator number.Centre for Complex DynamicSystems and Control

3. Example 1

To illustrate the ideas involved in cross-directional control, weconsider a 21-by-21 interaction matrix B with a Toeplitz symmetricstructure and exponential profile:

bij = e−0.2|i−j| for i, j = 1, . . . , 21, (6)

where bij are the entries of the matrix B.


2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

Cross-directional index

Figure: Cross-directional profile for a unit step in actuator number 11.


We consider the transfer function

h(q) =1 − e−0.2

q − e−0.2, (7)

which is a discretised version of the first-order systemy(t) = −y(t) + u(t) with sampling period Ts = 0.2 sec.


The next figure shows the singular values of the interaction matrixB. We observe that there exists a significant difference betweenthe largest singular value σ1 and the smallest singular value σ21,indicating that the matrix is poorly conditioned. Dealing with thepoor conditioning of B is one of the main challenges in CD controlproblems as we will see later.


2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

Singular values index

Figure: Singular values of the interaction matrix B.


In order to estimate the states of the system and the outputdisturbance dk , a Kalman filter is implemented as described for anextended system that includes the dynamics of a constant outputdisturbance:

xk+1 = Axk + Buk ,

dk+1 = dk ,

yk = Cxk + dk ,


In our case

A = diag{e−0.2, . . . , e−0.2},

B = (1 − e−0.2)B,

C = Im.


The state noise covariance

Qn =

[

Im 00 100Im

]

,

and output noise covariance

Rn = Im,

were considered in the design of the Kalman filter.


We will consider the finite horizon quadratic objective function withboth prediction and control horizons set equal to one, that is

V1,1 =12

(y0Qy0 + u0Ru0 + x1Px1). (8)

Q = Im, R = 0.1Im. (9)


We assume the system is subject to physical constraints on theinputs of the form:

|uik | ≤ 1 for all k , i = 1, . . . , 21.


Design 1

The first control strategy that we try on the problem is a linearquadratic Gaussian [LQG] controller designed with the sameweighting matrices as above. This design clearly does not take intoconsideration the constraints imposed on the input profile. Asmight be expected, the application of such a blind (orserendipitous) approach to the problem would, in general, notachieve satisfactory performance.


2 4 6 8 10 12 14 16 18 20

−1

−0.5

0

0.5

1

1.5Input clipping


(a) Input profile.

2 4 6 8 10 12 14 16 18 20

−1.5

−1

−0.5

0

0.5

1

Input clippingdisturbance


(b) Output profile.

Figure: Input-output steady state profiles with input clipping.


The above results illustrates a phenomenon that is well known inthe area of cross-directional control, namely alternate inputsacross the strip converge to large alternate values, that is, “inputpicketing” occurs. We will see below, when we test alternativedesign methods, that this picketing effect can be avoided by carefuldesign leading to significantly improved disturbance compensation.


Design 2

We next try RHC considering initially only input constraints. Theachieved steady state input and output profiles are presentedbelow where we have also included, for comparison, the profilesobtained with the input clipping approach.


2 4 6 8 10 12 14 16 18 20

−1

−0.5

0

0.5

1

1.5Input clippingRHC


(a) Input profile.

2 4 6 8 10 12 14 16 18 20

−1.5

−1

−0.5

0

0.5

1

Input clippingRHCdisturbance


(b) Output profile.

Figure: Input-output steady state profiles using RHC (square-dashed line)and input clipping (circle-solid line).


We observe, perhaps surprisingly, that the steady state responseachieved with RHC does not seem to have improved significantlycompared with the result obtained by just clipping the control in theLQG controller. In addition, the input profile obtained with RHCseems to be dominated by the same high spatial frequency modesas those that resulted from the input clipping approach.


However, this is a reasonably well understood difficulty in CDcontrol systems: The “picket fence” profile in the input arises fromthe controller trying to compensate for the components of thedisturbance in the high spatial modes which, in turn, require biggercontrol effort, driving the inputs quickly into saturation.


Design 3 (Using the Singular Value Structure of theHessian

The commonly accepted solution to this inherent difficulty is to letthe controller seek disturbance compensation only in the lowspatial frequencies.


When the prediction horizon chosen is N = 1 then in the vectorformulation of the quadratic optimisation problem we can write

Γ = B

and the Hessian of the objective function is simply

H = BQB + R

= BB + R.

This implies that the singular values of the Hessian are simply thesingular values squared of the interaction matrix B shifted by theweighting in the input R.


We next limit the optimization to the first few singular vectors. Thishas two potential advantages:

(i) It avoids chasing hard to control high spatial frequencydistances.

(ii) It may improve robustness since the gain associated with highspatial frequencies may be poorly defined.


2 4 6 8 10 12 14 16 18 20

−1

−0.5

0

0.5

1

1.5SVD−RHC strategyRHC


(a) Input profile.

2 4 6 8 10 12 14 16 18 20

−1.5

−1

−0.5

0

0.5

1

SVD−RHC strategyRHCdisturbance


(b) Output profile.

Figure: Input and output steady state profiles using the SVD–RHC strat-egy (circle-solid line) and RHC (square-dashed line).


We observe that in this case the picket fence profile hasdisappeared from the input whilst the output profile has notchanged significantly. Clearly a slight degradation of the outputvariance is to be expected owing to the suboptimality of thestrategy. The steady state profiles obtained with RHC have beenrepeated in the above figure for comparison purposes.


4. Example 2

We conclude this lecture with a second example having 11actuators and 11 sensors.

Here we will run the simulations in real time so that you can seethe picketing develop when the disturbance is applied.






5. Conclusions

This lecture has presented a realistic application of constrained control to a difficult industrial control problem, namely cross directional control. This system has high complexity due to:

i. the large number of actuators (several hundred),

ii. large interactions, andiii. severe constraints on the actuator signals.

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Cross Directional Control - The University of Newcastle, Australia...In order to control the cross-directional proﬁle of the web, several actuators are evenly distributed along the