Sheetmaking system identification using synthetic measurement produced from redundant noisy measurements

In a continuous sheetmaking process, a synthetic measurement is constructed having a higher signal-to-noise ratio than that of each original measurement in situations where more than one (redundant) measurements are available. The input-output dynamic characteristics of each measurement are assumed to be identical except for a gain factor. Using the synthetic measurement, system parameters can be identified in less time with fewer input disturbances. The invention is especially advantageous for application to processes that are expensive to disturb and in which the noise amplitude is high compared to the permissible amplitude of the put disturbance. The signal-to-noise enhanced time-series response measurement may be produced by selecting a time series response measurement that exhibits a greatest magnitude. Alternatively, it may be produced by averaging in the cross-direction a plurality of time series response measurements. Preferably, the enhanced signal-to-noise ratio time-series response measurement is produced, in appropriate circumstances, by taking the root mean square of a plurality of time-series response measurements in the cross-direction. The root-mean-square or average calculation can be conducted at each sample interval such that only the synthetic measurement series needs to be collected in storage but not the matrix of all original measurements.

BACKGROUND OF THE INVENTION 
1. Related Applications 
The subject matter of the present application is related to U.S. patent 
application Ser. No. 07/901,044 filed Jun. 22, 1992 entitled Automatic 
Cross-Directional Controls Zone Alignment for Sheetmaking Systems now U.S. 
Pat. No. 5,400,258 and U.S. patent application Ser. No. 08/115,598 filed 
on even date herewith entitled Self-System Parameter Identification for 
Systems with Pure Time-Delay, both incorporated herein by reference. 
2. Field of the Invention 
The present invention relates to system identification of a 
cross-directional control system of a type used, for example, in 
sheetmaking process. In particular, the invention relates to techniques 
for increasing the signal-to-noise ratio of measurements used in 
identification of such systems. 
3. State of the Art 
In sheetmaking processes, on-line quality measurements and controls are 
used to control the quality of the product. In modem automated papermaking 
machines, for example, continuous paper webs, sometimes measuring as much 
as 400 inches across, can be produced at rate of up to 100 feet per 
second. To control the quality of the paper manufactured at such rates and 
to reduce the quantity of finished product that must be rejected if there 
are upsets in the manufacturing processes, properties of the paper web 
must be measured and adjusted while the machines are operating. 
Referring to FIGS. 1 and 2, in a papermaking process, a slurry of paper 
fibers and water mixture (or stock) is fed into a tank 10 called a 
"headbox", and the slurry then flows continuously through an opening 35 
defined by a "slice lip" 34. The slurry is deposited onto a continuous 
conveyor belt, or "wire" 13. The wire moves in a direction away from the 
headbox. The slurry thus forms a continuous mat 18 on the wire. The mat of 
paper slurry drains some of its water content as it is being transported 
by the wire and becomes a sheet that is then pressed by rollers 21 to 
remove additional moisture from the sheet. The "basis weight" (mass per 
unit area) or other property of the sheet is then measured using a sensor, 
typically a scanning sensor 30 as shown in FIG. 1. 
The vertical position of the slice lip is related to the size of the feed 
opening and hence to the amount of slurry deposited on the wire and 
ultimately to the basis weight of the sheet. The vertical position of the 
slice lip is controlled by a plurality of actuators 23 connected to the 
slice lip and to the headbox. Using information from a sensor, the 
actuators may be controlled to obtain the desired basis weight of the 
sheet. 
Machines which produce webs of sheet material such as paper, plastic and 
aluminum, face process control problems in producing webs which satisfy 
specifications for the given sheet material. Web specifications commonly 
include ranges for characteristics of the web including thickness, 
moisture content, weight per unit area, and the like. Quality control is 
complicated since the specified characteristics vary in both the machine 
direction (MD), or direction of movement of the web through the machine, 
and in the machine cross direction (CD), or laterally across the web. 
The MD variations are generally affected by factors that impact the entire 
width of the web, such as machine speed, the source of base material being 
formed into a web by the machine, supplies of working media like steam, 
and similar factors. CD variations, represented by profiles or profound 
signals, are normally controlled by arrays of actuation cells distributed 
across the width of the machine. On paper making machines, the CD 
actuation cells include basis weight actuators which control the slice of 
a headbox, steam shower nozzles, infrared heaters which control CD 
moisture variations, and other known devices. 
To maintain the CD quality profile of the sheet with CD actuators, it is 
important in know the effect of each actuator unit's adjustment. This 
effect has two aspects, namely spatial effect and time effect. The spatial 
effect is normally characterized by mapping and spatial response. Mapping 
describes the alignment of each actuator unit to its affected portion of 
measured profile. Spatial response describes the pattern of the profile 
change due to each actuator adjustment. The time effect refers to the 
relation between the adjustments of an actuator and the changes of its 
corresponding portion of the profile in terms of their dynamic evolution 
over time. It is characterized as dynamic response. This invention relates 
to a method of identifying the dynamic response of CD control actuators. 
As described in U.S. patent application Ser. No. 07/901,044, an automated 
tool may be used for identifying mapping and spatial response through a 
bump test. The same bump test result may be used to identify dynamic 
response of the CD control actuators. The dynamic response may be 
parameterizod as a time delay, a time constant, and a process gain. 
A typical response to a bump excitation is shown in FIG. 3. If the bump is 
applied at time t.sub.0, then some time later at time t.sub.1 the measured 
sheet property will begin to change at some rate and will continue to 
change at a rate that gradually decreases until the system reaches a 
steady state condition. The amount of change of the sheet property, i.e., 
the amplitude of the response, A.sub.R, divided by the amplitude of the 
bump excitation, A.sub.B, is defined as the process gain. The time t.sub.1 
-t.sub.0 is defined as the delay of the system, T.sub.D ; and the time at 
which the system would reach steady state if the measured sheet property 
changed at a non-decreasing rate is defined as the time constant of the 
system, T.sub.C. For a time-invariant, first-order linear system, the 
foregoing parameters completely describe the system's behavior. 
Thus, given a step response of a linear system, system parameters can be 
identified, for example, using a Least-Square (LS) algorithm, with little 
difficulty. However, noise in the response measurement, over a short 
transient response can be fatal to the identification result. Hence, 
multiple step disturbances, rather than one or two, and thus a longer 
response measurement time, may be required to obtain an acceptable 
identification result. 
The foregoing situation is illustrated in FIG. 4. A long sequence of 
disturbances x(t) is applied to a plant. The plant may be, for example, a 
sheetmaking system subject to considerable process noise. The output of 
the plant is measured, producing a long sequence of noisy measurements 
y(t). The disturbances x(t) are also input to a model identification 
processor, which estimates the plant output. The estimated plant output is 
compared to the measured plant output y(b) to produce an error signal 
e(t), which is used to adjust the model. As the signal-to-noise ratio of 
the measurements decreases the length of time required for successful 
system identification generally increases. 
In determining mapping and spatial response, cross-directional information 
is vitally important. In determining system parameters, on the other hand, 
such as time delay, time constant and process gain, cross-directional 
measurements through a transient response time include redundant 
information. Since all the actuators of the cross-direction are 
substantially identical and have substantially identical characteristics, 
each actuator is assumed to exhibit substantially the same time delay, 
time constant and gain. 
The present invention takes advantage of the redundancy of redundant noisy 
measurements to produce a synthetic measurement having a higher 
signal-to-noise ratio that may be used to perform system identification a 
relatively shorter period of time, within one or two step response times. 
SUMMARY OF THE INVENTION 
The present invention, generally speaking, provides for the construction, 
in a dynamic sheetmaking process, of a synthetic measurement having a 
higher signal-to-noise ratio than that of each original measurement in 
situations where more than one (redundant) measurements are available. The 
input-output dynamic properties of each measurement are assumed to be 
identical except for a gain factor. Using the synthetic measurement, 
system parameters can be identified in less time with fewer input 
disturbances. The invention is especially advantageous for application to 
processes that are expensive to disturb and in which the noise amplitude 
is considerably high compared to the permissible amplitude of the input 
disturbance. 
More particularly, in accordance with a preferred embodiment of the 
invention dynamic characteristics of a cross-directional control system 
having a plurality of substantially identical actuators arrayed in the 
cross-direction are determined by applying to the actuators respective 
excitation signals of equal magnitudes, collecting a profile made up of a 
multiplicity of measurements at different locations in the 
cross-direction, aggregating the multiplicity of measurements to produce a 
datum of a time-series response measurement; storing the datum, and 
repeating the collecting, aggregating and storing steps over a period of 
time of a response of the control system to the excitation signals. The 
resulting time-series response measurement has a signal-to-noise ratio 
substantially greater than would a time-series response measurement at any 
cross-direction location. Aggregating is preferably performed by forming 
the corrected root-mean-square of the multiplicity of measurements in a 
profile. Forming the corrected root-mean-square requires a noise estimate 
to be obtained. Therefore, prior to applying any excitation signal, a 
noise profile is collected and aggregated to produce a noise estimate. 
That estimate is refined by collecting and aggregating further noise 
profiles and updating the previous noise estimate.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
For purposes of the following description, passing familiarity with 
conventional systems identification techniques is assumed. Briefly, system 
identification is a process in which input and corresponding output 
signals of a system are used to estimate dynamic characteristics of the 
system. Further, details concerning system identification as applied to a 
sheetmaking system incorporating a pure time delay may be found in 
co-pending U.S. application Ser. No. 08/115,598. System identification is 
treated at length in Ljung and Soderstrom: Theory and Practice of 
Recursive Identification (MIT Press, Cambridge, Mass., 1983). 
The invention may be applied to advantage in papermaking systems, 
sheetmaking systems generally, and in other cross-directional control 
systems. For purposes of description, however, the invention will be 
described in the context of a sheetmaking system such as a papermaking 
system. 
In such a system, automatic tuning, whereby process dynamic parameters are 
identified, nay be performed using data obtained from a "bump test" in 
which a disturbance, is applied to the system and the system output is 
measured, for example, by performing repeated scans across the width of 
the sheet using a scanning head as in FIG. 1. As repeated scans are 
performed, there is accumulated a two-dimensional matrix of measurement 
data where one of the dimensions is time, t, and the other dimension is a 
cross-direction spatial index, i. In the presently-described method, this 
two-dimensional matrix of measurement data is used to produce a 
one-dimensional matrix (extending in the time direction only) of 
measurement data that has a greater signal-to-noise ratio than any 
one-dimensional sub-matrix of the two-dimensional matrix taken alone. As 
compared to the prior art (FIG. 4), the present method allows a shorter 
sequence of disturbances to be used in order to produce a shorter sequence 
of clearer measurements as shown in FIG. 5. A measurement synthesis unit, 
using different measurement synthesis techniques (such as averaging or 
RMS) produces from n cross-directional measurements a single synthetic 
output measurement y.sub.s (t) having an enhanced signal-to-noise ratio. 
In accordance with a first-order linear model, the time-varying output at 
different discrete measurement points i in the cross-direction output, 
y(t,i), of a sheetmaking (or other) system may be modelled in terms of its 
input signal, x(t), as: 
EQU y(t,i)=a.sub.1 y(t-1,i)+. . . +a.sub.n y(t-n,i) (1) 
EQU +p(i)[b.sub.1 x(t-d)++. . . +b.sub.m x(t-d-m+l)] (2) 
EQU +v(t,i),i+1, . . . N (3) (3) 
where a.sub.1,a.sub.2, . . . and b.sub.1, b.sub.2, . . . are process model 
parameters, d represents an unknown time-delay, p(i) n;presents the 
process gain at the different discrete measurement points in the 
cross-direction, and where v(t,i) is assumed to be Gausian and white noise 
with zero mean and variance .sigma..sub.v for all i. It is the gain p(i) 
that makes one measurement different from another. From a theoretical 
point of view, all of the measurements y(t,i) except one are redundant for 
the purpose of identifying the unknown parameters. 
The redundant measurements can be operated upon or combined in many 
different ways to construct a synthetic measurement that yields a higher 
signal-to-noise ratio. The simplest approach, if the process gain p(i) 
varies among the different discrete measurement points in the 
cross-direction, is to simply select as the output signal y(t) a 
measurement (or rather series of measurements) taken at a point in the 
cross-direction at which the gain is greatest: 
EQU y.sub.m (t)=y(t,k)=y(t) (4) 
where k is determined by p(k)=max.sub.1.gtoreq.i.gtoreq.N P(i). (Without 
losing generality, it is assumed that p(k)=1.) The maximum measurement is 
a single (non-synthetic) measurement that yields the maximum 
signal-to-noise ratio among all single measurements, y(t,i). This approach 
requires the knowledge of point in CD at which the gain is high. 
Otherwise, the two dimensional array of all point measurements needs to be 
stored. 
The signal-to-noise ratio may be increased beyond the maximum 
signal-to-noise ratio among all single measurements by producing a 
synthetic measurement. One method is to average together the measurements 
at all of the discrete measurement points in the cross-direction, as 
follows: 
##EQU1## 
where Y.sub.a is a normalization factor accounting for possible different 
gains p(i). 
By taking the average of all measurements in the cross-direction, the 
respective noise components, assumed to be uncorrelated, tend to cancel 
out. The average measurement may be shown to have the same mean value as 
that of the maximum measurement while yielding a signal-to-noise ratio 
that is 
##EQU2## 
times that of the maximum measurement. A larger number of measurements 
will lead to a higher signal-to-noise ratio, when the input to the system 
remains the same. Note that this approach requires only a one dimensional 
army to be stored during the dam collection. In other words, after one 
scan of a profile of measurements at different cross-directional 
locations, only the average of the profile is stored as y.sub.i (t) at the 
time t. 
An even more robust measurement may be constructed by taking Corrected 
Root-Mean-Square (CRMS) of measurement at all of the discrete measurement 
points, as follows: 
##EQU3## 
The CRMS measurement has the same mean value as the absolute value of the 
maximum measurement while yields a signal-to-noise ratio that is 
##EQU4## 
times that of the maximum measurement. 
Note that, for N.gtoreq.2, 
##EQU5## 
which indicates superiority of the CRMS measurement over the average 
measurement in terms of signal-to-noise ratio. The CRMS measurement yields 
the highest signal-to-noise ratio and should be used for applications 
where sign of the measurement is not varying or is known. The 
signal-to-noise ratio increases with the number of measurements available 
in power of 1/2. Unlike averaging, in the RMS technique, since sign 
information is obscured, separate noise components, instead of tending to 
cancel, accumulate. A correction must therefore be made, represented by 
the minuend term .sigma..sub.y.sup.2. In the CRMS measurement (9), 
.sigma..sub.y is the variance of each single measurement y(t,i). It is a 
function of the variance .sigma..sub.v, of each process noise measurement 
v,(t,i). For a time-invariant system and steady process noise, 
.sigma..sub.y can be approximated by a RMS value of the measurements 
during a period of time with zero input, i.e., 
##EQU6## 
In other words, in the CRMS measurement technique, prior to applying a 
disturbing input signal and collecting a series of measurement profiles, a 
series of noise profiles are collected without any disturbing input being 
applied. The mean square noise value is calculated across each profile. 
For each noise profile collected, a noise estimate is updated by computing 
the time average of the mean square noise values. Then, as the RMS 
measurement is computed, the RMS value is corrected to account for the 
process estimate. 
As in calculation of the average measurement, this approach only requires a 
one dimensional array of dam to be stored. 
In the average measurement and the corrected RMS measurement, the scaling 
factors Y.sub.a and Y.sub.r depends on p(i), which are usually unknown. 
With the exception of the overall input-output gain, these scaling factors 
will not affect the system parameter identification (a.sub.i, b.sub.i, and 
d). Therefore, these scaling factors p(i) can be estimate, d afterwards 
from the system response measurements and then used to scale the 
input-output gain identification. The estimation can be performed as: 
##EQU7## 
where T is a time when step responses reached or got close to steady 
state. 
In most common sheet production process Cross-Direction (CD) control 
systems, a greater number of measurements is performed during each scan 
than the number of corresponding actuators. Therefore, multiple 
measurements are affected and dominated by a single actuator, although the 
gain from actuator input to measurement output may not be the same for all 
of such measurements. The dynamic characteristics of each actuator's 
response can be approximated as a first order dynamic system. In such 
case, the system model, provided that process noise in each measurement 
zone is independent from the other (for a first order system), can be 
written as: 
EQU y(t,i)=ay(t-1,i)+bp(i)x(t-d)+v(t,i) (16) 
with i=1,2, . . . , N. In other words, a.sub.2, a.sub.3, . . . and b.sub.2, 
b.sub.3, . . . may be taken to be zero. Theoretically, the variance 
.sigma..sub.y of all the measurements is 
##EQU8## 
where .sigma..sub.v is the variance of the process noise v(t,i). In 
practice, .sigma..sub.y is estimated according to equation (13). Then the 
CRMS measurement can be constructed according to equations (8-10). 
A generalized version of the recursive Least Square algorithm can be used 
for parameter (a, b and d) identification as described in greater detail 
in U.S. application Ser. No. 08/155,598. 
The foregoing has described the principles, preferred embodiments and modes 
of operation of the present invention. However, the invention should not 
be construed as limited to the particular embodiments discussed. Instead, 
the above described embodiments should be regarded illustrative rather 
than restrictive, and it should be appreciated that variations may be made 
in those embodiments by workers skilled in the art without departing from 
the scope of the present invention as defined by the following claims.