Decoupling controller for use with a process having two input variables and two output variables

A decoupling controller for use with a process such as paper-making having two input variables such as stock flow and steam pressure and two output variables such as moisture and basis weight. Decoupling is accomplished by the use of linked internal model controllers where for each individual unit of the linked pairs, a P.I.D. (proportional, integral, derivative) unit includes all of the feedback loop gains and then the process itself is modeled by a first order transfer function and deadtime units with two cross-linked error signals fed back. A specific technique of cross-linking the internal model controllers eliminates cross-coupling between the input and output variables.

INTRODUCTION
 The present invention is directed to a decoupling controller for use with a
 process having two input variables and two output variables and, more
 specifically, to a paper-making process where the input variables are dry
 stock flow and steam pressure to a dryer section and the output variables
 are basis weight and moisture.
 BACKGROUND
 In the paper-making process, the process itself has long deadtimes relative
 to the process time constant. This makes control difficult. There are
 typically two unique deadtimes; one is for the time required for a change
 in basis weight when the input variable, stock flow, is changed and the
 other is the time for variation in steam to affect the final moisture
 carried by the paper sheet. A further difficulty in the paper-making
 process is the cross-coupling affect; that is, each input variable affects
 both output variables. Hence, a decoupling controller is desired to
 regulate the outputs independently; for example, the operator would like
 to change the setpoint of the basis weight controller without changing the
 value of the moisture.
 Two rather crude techniques have been utilized for decoupling. In a first,
 a setpoint is changed only once every five minutes, for example, for
 changes in stock flow and once every minute for changes in steam. This is
 a much longer period of time than the generation of output data by a
 sensor which scans across the width of the paper, for example, every 20
 seconds.
 A second proposed decoupling technique is to provide absolute decoupling
 constants between changes in stock flow and steam pressure. These might be
 termed "compensating changes". However, these are mere guesses and do not
 compensate for grade changes or speed changes and do not take into account
 that the coupling effect may be nonlinear. One other problem with a
 controller for a paper-making machine discussed above, is the fact that
 the measurements of outputs occur either at long intervals or can occur
 asynchronously due to sheet breaks or standardization. In any case, scan
 measurements (which may take up to 120 seconds) only occur every 20
 seconds at best.
 OBJECT AND SUMMARY OF INVENTION
 A general object of the present invention is to provide an improved
 decoupling controller for use with a process having two input variables
 and two output variables.
 In accordance with the above object, there is provided a decoupling
 controller for use with a process having two input variables U1, U2 and
 two output variables X1, X2 where in the process each input variable
 affects both output variables (that is they are coupled), such process
 having desired setpoints S1, S2 for the output variables. Such decoupling
 controller comprising two pairs of linked internal model controllers, each
 internal model controller (IMC) including a proportional, integral,
 derivative (P.I.D.) velocity unit C11, C21, C12, and C22 for respectively
 receiving from a first pair of difference junctions total process error,
 et1, et2, in a feedback loop for the process and producing said input
 variables U1, U2, which are control inputs to the process itself, such
 P.I.D. units taking into account loop, proportional, integral and
 derivative gains of the feedback loop for both direct and cross-coupling.
 Four first order transfer function units K11, K12, K21, and K22 receive as
 inputs U1, U2, the K11, K22 units providing predicted values of X1, X2,
 the K21, K12 units providing predicted outputs of X1, X2 due to
 cross-coupling.
 Means feed back to a pair of second summing junctions the outputs of K11,
 K12 and K22, K21 respectively.
 Means couple the outputs of the second summing junctions, which are total
 predicted values of X1 and X2 taking into account cross-coupling, to a
 pair of third summing junctions, which also receive modeling error signals
 representing the difference between the actual X1 and X2 values and
 estimated values Y1 and Y2, from a pair of fifth junctions.
 Means feed the summed output of the third pair of summing junctions to the
 first pair of difference junctions, which have as the other difference
 input the setpoints S1, S2 to provide the total process error inputs et1
 and et2 to C11, C21 and C12, C22;
 Means take the deadtime of the process into account (that is the lag time
 between the change of input variables and output variables), including
 four deadtime units, D11, D21, and D12, D22, having their inputs
 respectively connected to the outputs of K11, K21, K12, and K22, including
 a pair of fourth summing junctions having as outputs the current estimated
 values Y1, Y2 of the X1, X2 output variables, where one of the pair of
 fourth summing junctions, sums the outputs of D11, D12 and the other of
 the pair of summing junctions, sums the outputs of D22, D21.
 Means couple the outputs of the fourth pair of summing junctions, to the
 fifth pair of difference junctions to take the difference between the
 actual outputs X1, X2 and the estimated values Y1, Y2, such differences
 being the modeling error signals.
 Means feed back the modeling error signals to the third pair of summing
 junctions.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENT
 FIG. 1 illustrates a typical paper-making machine, which includes an
 associated control hardware and software configuration as used in the
 present invention. Raw paper stock is supplied to the machine via a stock
 valve 10 and a stock line 11 to a head box 12 by a fan pump 13. The pump
 and water mixture jets from head box 12 through a slice 14 on top of and
 parallel to wire 16. This forms a wet web 17, which on leaving wire 16
 passes through rollers 18, which remove much of the water from the web and
 essentially converts it to a sheet of wet paper. Thereafter, the paper
 sheet passes through a dryer section 19 consisting of several rollers to
 which steam is supplied by steam control valve 20. Steam heats the rollers
 and, consequently, evaporates much of the water in the paper sheet so that
 the paper emerging from the dryer section 19 has the desired moisture
 content (MOI). Thereafter, the paper passes through a calendar stack 22
 through scanning sensors 23 and is wound on the reel 24.
 Scanning sensors 23 scan across the width of the paper approximately every
 20 seconds and provide a measurement of the output variables of basis
 weight (B.W.) and moisture (MOI). In the present invention, either basis
 weight, dry weight, or conditioned weight may be used; the latter is
 preferred. Conditioned weight is from a practical standpoint, dry weight
 where a standard 8 percent moisture factor is added. The two output
 variables from the scanning sensors 23 are also designated X1 and X2. The
 scanned moisture and basis weight values are coupled to the digital
 computer 26 for processing along with an operator input 25. Typically, in
 the cross direction of the paper, 240 measurements are made and these are
 averaged to provide a single "end of scan" measurement. This is therefore
 one of the measurements made in the machine direction. Both basis weight
 and moisture (X1 and X2) are also connected to control units 30A and 30B
 in a manner which will be illustrated in FIG. 2. The two control units are
 actually part of a single interlinked decoupling controller embodying the
 present invention as indicated by the 6-wire interface 31. Controller
 portion 30A has as inputs a setpoint S1 from the digital computer 26 and
 X1 and as an output, the control value U1, which drives the stock valve 10
 to provide a dry stock (fiber) flow rate of U1. Similarly, controller
 portion 30B receives the setpoint S2 (actually, the steam pressure) from
 digital computer 26 along with X2 and by its output U2 drives the steam
 valve 20 to provide a U2 steam input.
 Digital computer 26 has as other feedbacks the steam pressure line 31 from
 dryer section 19 and also flow meter 27 and a consistency meter 32, which
 determine the pounds per minute of dry stock flow.
 Referring to FIG. 2, the decoupling controller portions 30A and 30B are
 shown in greater detail and especially how they are interlinked. The
 paper-making process, fully illustrated in FIG. 1 is shown twice, both at
 36A and 36B, as being driven by the input variables U1 (stock flow) and U2
 (steam) and having the outputs X1 (basis weight) and X2 (moisture).
 The real process, illustrated in FIG. 3, is represented by four unknown
 transfer functions P.sub.11, P.sub.12, P.sub.21, and P.sub.22. The P-type
 functions, with the 11 and 22 subscripts represent the direct transfer
 functions of stock to basis weight and steam to moisture; the other
 cross-coupling subscripts 21 and 12, relate to how moisture is affected by
 changing stock and how basis weight is affected by a change in steam. The
 P functions can be represented by the associated Laplace Transform where
 the theta superscript is a deadtime function or delay function and the
 remainder is a time constant and gain, or rather a first order transfer
 function K.
 FIG. 4 illustrates the delay or deadtime of the process. Assuming a time,
 t.sub.o, when a change is made for stock, there is a delay until basis
 weight reaches a constant value; and the same is true for the change in
 steam for moisture. In the background of the invention, prior attempts at
 decoupling were discussed.
 With regard to the present invention and referring to FIG. 5, it has been
 found that when a prior art type internal model controller (IMC), as shown
 in FIG. 5, for a single loop only and for synchronous measurements in
 time, is adapted to a multi-variable system, it will accommodate deadtime,
 asynchronous behavior, and when two pairs of IMC's are linked, as is
 illustrated in FIG. 2, provide very effective decoupling. However, first
 referring to FIG. 5, to serve as a background for the total decoupling
 controller of FIG. 2, the internal model controller has a setpoint input S
 and a process at 36 and an output variable X. A difference junction 37
 provides a total error signal, et, to a controller C, 40, which then
 provides an input control signal U to the process 36. This input control
 signal, however, also drives a simulated model of process 36, which is
 characterized by the Laplace Transform (see FIG. 3) of a K function 38 and
 a D function 39 as illustrated by the accompanying formulas. This process
 model provides an estimated X value at 41, which at the difference
 junction 42 is compared to the actual X value to generate a feedback error
 signal on line 43. However, this error signal is not coupled back to the
 input until it is summed at a junction 44 with the raw output of the K
 model 38 (that is before a deadtime is taken into account). Then on line
 46 and junction 37, a difference is taken with the setpoint, S, to provide
 the total error, et. The unit 40 is actually a proportional, integral,
 derivative (P.I.D.) type controller which is illustrated by the following
 equation which produces an output U in response to a total error input,
 et:
 ##EQU1##
 Its constants are:
 K.sub.L =loop gain
 K.sub.P =proportional gain (generally approx. less 1)
 K.sub.I /s=integral gain (this is an acceleration factor which is about
 0.5) and s is the Laplace operator.
 K.sub.d s=K.sub.d is derivative gain which is in the range of 0.3 to 0.8
 and s is the Laplace operator.
 Thus, in summary the internal model controller models the process 36 by the
 use of the first order transfer function K (unit 38) with a deadtime D
 (unit 39). In general, this internal model controller (IMC) is for a
 single loop only and not for asynchronous use. For use in the chemical
 industry see the book entitled, "Robust Control" by Manfred Morari and
 Evanghulos Zafiriou, Prentice-Hall, 1989. It is quite apparent, from
 examination of FIG. 5, if there is a deadtime equal to 0, that is D=0,
 then it becomes a standard proportional integral derivative (P.I.D.)
 controller. The loop gain, K.sub.L is not normally part of a standard IMC.
 In the present invention, it has been discovered that if K.sub.L is made
 the reciprocal of the model gain, K (unit 38), the above equation becomes
 non-dimensional to allow the loop to be easily pre-tuned. The value of the
 constants are believed ideal to pretune for a typical paper making
 machine. They were derived by trial and error.
 Referring back to FIG. 2, this illustrates the decoupling controller of the
 present invention, which in effect incorporates four internal model
 controllers which are linked together. The C-type unit 40 of FIG. 5,
 designated 40', has two pairs of linked P.I.D. units C11, C21, C12, and
 C22. The numerical designations, of course, conform to the transfer
 functions illustrated in FIG. 3. Thus, C11 and C22 are the direct model of
 the process change for driving U1 and U2, respectively, and then for the
 other two, C21 relates to S1, X2, and C12 to S2, X1. In other words, these
 P.I.D. units are cross-coupled to produce at the additive junctions 51 and
 52, the U1 and U2 control values to the process 36A, 36B. In order to
 provide fast response, the units 40' are of the P.I.D. velocity type to
 eliminate "reset windup".
 "Reset Windup" occurs in a feedback control system which integrates error.
 But when the system variable is constrained at a 100% value (and thus the
 setpoint cannot be achieved) an intolerable error is built up by
 integration. This cannot happen with the present P.I.D. unit since the
 term, "K.sub.I e.sub.t ", has no integral. The operation of units 40' is
 determined by the following equation which, of course is a form of its
 basic equation given above:
EQU sC(s)=sU(s)=K.sub.L (K.sub.P s+K.sub.I +Kds.sup.2)e.sub.t (s)
 or in time domain;
 ##EQU2##
 (where .delta. is the difference operator)
 As discussed in combination with FIG. 5, the units 40' take into account
 loop gain, proportional gain, integral gain, and derivative gains involved
 in the feedback loops illustrated in FIG. 2.
 Next, the K11, K21, and K12, K22 units, also designated 38', receive as
 respective inputs U1, U2. The K12 and K21 units provide predicted outputs
 of X1, X2 due to cross-coupling. These outputs, in manner somewhat similar
 to FIG. 5, are cross-coupled back to junctions 2a and 2b and summed with
 the Kl1 and K22 outputs. Thus, the output of the second summing injunction
 pair, 2a and 2b, is actually the total predicted value of X1 and X2
 (taking into account cross-coupling, but without deadtime). These are
 designated yp1 and yp2. These outputs of the 2a and 2b junctions are then
 connected to a third pair of junctions, 3a and 3b, which also receive on
 the lines 53 and 54 modeling error signals representing the difference
 between the actual X1 and X2 values and the estimated values, Y1 and Y2.
 Finally, the outputs of the junctions 3a and 3d are fed back to the first
 pair of difference junctions 1a and 1b to provide the total process error,
 et1 and et2.
 Deadtime is taken in account by the D units 39' which receive the outputs
 of the respective K units. The deadtime is, of course, the lag time
 between the change of input variables and output variables. The deadtime
 units D11, D21 and D12, D22 have their inputs respectively connected to
 the outputs of K11, K21 and K12, K22. A current estimate Y1, Y2 of the X1
 and X2 output variables is provided at the summing junctions 4a and 4b;
 junction 4a sums D11 and the cross-coupled D12 and junction 4b sums D22
 and the cross-coupled D21.
 And then lastly, the junctions 5a, 5b take the difference between the
 actual outputs X1 and X2 and the estimated values Y1 and Y2 to provide the
 modeling error signal on lines 53 and 54.
 Thus, in a robust and elegant manner, the internal model controller of FIG.
 5 has been converted by the specific interlinking as shown in FIG. 2 to
 compensate for deadtime and provide for instant response to setpoint
 changes and at the same time to effectively decouple weight from moisture.
 To estimate values for the K and D units, 38' and 39', bump tests are used
 along with the knowledge and experience of an operator of the process.
 The above process model with four transfer functions also accommodates use
 in a wide variety of processes in addition to paper machines, such as in
 the petrochemical, mining, waste water treatment and food processing
 industries. Typically, in these cases, the gas or liquid stream is sampled
 isokinetically with the sample drawn to a chemical analyzer, such as a gas
 or liquid chromatograph resulting in a time lag.
 In order to execute the above equations asynchronously, the differential
 equations are solved analytically. For example:
 ##EQU3##
 Solving for x, given u=constant over a short period.
 ##EQU4##
 So, given the measurement at t=t.sub.o ; x(t.sub.o); the future values of x
 can be computed using Eq.(1) as long as u is constant beginning at
 t.sub.o. To find yp(t), simply delay x(t) by the time delay. Thus, the
 foregoing demonstrates how to execute the equations asynchronously.
 Executing the controller at random intervals is ideal for paper machine
 control systems since the end of scan measurements are at random times.
 Also, since the P.I.D. unit 40' is of the velocity form, no movement of
 the inputs to the process are possible between scan intervals. The
 controller does not have to be programmed explicitly to handle scanner
 standardizations or sheet breaks.
 The P.I.D. units 40' are preferably implemented in the velocity mode to
 eliminate "reset wind-up". However, depending on the process other type
 modes might be used.
 FIG. 6 illustrates typical data received in the machine direction of either
 basis weight or moisture. Five data points are illustrated. The use of a
 sliding least squares method of polynomial filters provides a single best
 estimated value of X1 or X2, as well as dx/dt and dx.sup.2 /dt.sup.2
 which, of course, relate to velocity and acceleration.
 In order to provide a well-tuned feedback system, the loop gain, K.sub.L,
 (see FIG. 5) of the controller units 40' is set as the inverse of the
 related K functions of the K11, K12, K21, and K22 modeling units.
 In addition, the K11 first order transfer function is adjusted for an
 increase in the paper machines' speed (or trim). Here, with a speed
 increase the loop gain, K.sub.L, of the P.I.D. is decreased since it is
 the inverse of K11. This is computed by the following fiber balance
 equation:
EQU D.sub.1n *K.sub.11 =D.sub.w *S*t
 where
 D.sub.1n --input fiber rate, g/sec,
 D.sub.W --output dry weight scan average, g/m.sup.2,
 K.sub.11 --process gain,
 S--speed of paper, m/min,
 t--trim of paper, m
 FIG. 7 shows a method of displaying the two dependent variables such as
 basis weight and moisture. In the case of a paper machine, the display
 includes the last 20 values of weight and moisture displayed on an XY
 chart with connecting lines between the data points. The chart is known as
 a phase planes chart in control theory and is used to analyze regions of
 stability and performance under transient conditions. Thus, as shown by
 the straight line 61, this is a perfect transition between an original
 setpoint and a new setpoint. However, the dashed line 62 illustrates a
 somewhat imperfect transition which, however, presents to the operator a
 possibility of improving by tuning the feedback system, the transition.
 Thus, an improved decoupling controller for use with a process having two
 input variables and two output variables has been provided.