ACTUATOR SETTINGS OF A PAPER MAKING SYSTEM BACKGROUND FIG. 1 illustrates a system 100 that makes paper 102. A typical paper making system 100 may make a paper sheet that is thirty feet wide and produced at a rate of five- thousand feet per minute. The system 100 shown features a head box 104 that includes components that"spray"paper pulp onto a moving fabric. The system 100 also features a wide variety of actuators that control the paper making process based on their settings. For example, FIG. 1 a collection of actuators can control the thickness of a sheet of paper by exerting different amounts of force on a head box lip 108. Other actuators control other components in the paper making process such as actuators that control the amount of steam applied.

Paper quality can be measured in a variety of ways such as weight, moisture content, opacity, gloss, smoothness, and caliper/thickness. Additionally paper should generally be uniform along its length ("machine direction") and across its width ("cross-direction"). To measure these and other properties, the system 100 shown includes a scanning gauge 110 that moves across the paper and collects measurement data as the paper speeds past.

A gauge 110 may take as long as thirty seconds to move across a thirty-foot wide sheet. During this measurement interval, approximately fifteen-hundred feet of paper can tavel past the gauge 110 in the machine direction. Because both the paper 102 and gauge 110 move simultaneously, the gauge 110 traces a long diagonal across a given sheet.

The task of making paper involves setting the actuators such that gauge 110 measurements correspond to desired paper characteristics. Unfortunately, a variety of factors complicate the task of determining actuator settings. For instance, due to the diagonal path of the gauge 110 relative to the sheet, measurements may reflect both machine-direction (MD) and cross-direction (CD) variation in the sheet.

A wide variety of other factors complicate selection of actuator settings. For example, setting an actuator to a value that improves one characteristic may negatively effect another. For instance, changing the setting of an actuator to control paper weight can potentially produce undesired changes in moisture and caliper. Additionally, while the actuator may effect the paper weight in a small region near the actuator, the actuator may impact paper moisture in a much larger area. Further, there are limiting constraints on the actuators in terms of their absolute limits (e. g., their highest and lowest settings) and the

difference in settings between adjacent actuators. For example, actuator settings may permanently deform the head box lip if neighboring actuator settings vary too much. The effect of actuators can also vary with changes in the paper machine operating conditions (e. g., such as the type of fiber being used or workplace temperature).

SUMMARY In general, in one aspect, the disclosure describes a method of determining settings of actuators of a paper making system. The method includes generating a matrix representing the effect of the actuators on different values corresponding to process measurements, and at least one constraint upon the actuator settings. The method also includes determining a collection of actuator settings that satisfy the constraints represented by the matrix.

Embodiments may include one or more of the following. Determining actuator settings may include iterative application of a set of weights to the matrix. The method may include identify at least a portion (e. g., a sub-matrix) of the matrix as a sparse matrix. The method may further include adjusting the actuators based on the determined settings. The constraints may include an absolute limit constraint, a first order bending constraint, and/or a second order bending constraint.

BRIEF DESCRIPTION OF THE DRAWINGS Fig. 1 is a diagram of a paper making system.

Fig. 2 is a diagram illustrating the effect of different actuators on different measurements.

Fig. 3 is a flow-chart of a process for determining actuator settings.

DETAILED DESCRIPTION Referring to FIG. 1, the characteristics of paper 102 made by a paper making system 100 is controlled, at least in part, by the settings of different actuators. These actuators can control fabric speed, stock composition, steam pressure, Ti02 flow (whitener), clay additives, and so forth. The characteristics of the paper 102 (e. g., basis weight, reel moisture, ash, opacity, roll hardness, smoothness/gloss, and caliper) can be measured by sensors such sensors of scanning gauge 110. Often, different actuator settings involve a trade off between different paper characteristics. E. g., an improvement in weight

sometimes results in poorer moisture content. Due to the myriad of actuator interactions, determining actuator settings can be an extremely computationally intensive process.

Described herein are approaches that can quickly determine actuator settings that yield paper having desired characteristics despite the complex interactions between different actuators settings and the different constraints imposed on a given set of settings.

In greater detail, as described above, the scanning gauge 110 makes a very large number of measurements as the gauge 110 travels across the width of a paper sheet 102.

For example, the gauge 110 may make measurements at six-hundred different locations across the width of a sheet for each of a number of different paper characteristics. That is, the gauge 110 may make six-hundred measurements of weight, six-hundred measurements of moisture, six-hundred measurements of opacity, and so forth. Each measurement is referred to as a"databox". The databoxes may be combined (e. g., averaged) based on the number of actuators. For example, for a head box having fifty actuators, twelve adjacent databoxes may be merged into one to yield fifty databoxes. This process is known as "mapping".

Though mapping can establish a one-on-one ratio between actuators and databoxes, an actuator setting may effect more than one databox. For example, FIG. 2 shows a matrix 204 that illustrates the effect that a change of an actuator setting has on some measured variable (e. g., weight). In greater detail, each row of the matrix 204 shows the impact an actuator 202 has on different databoxes. For example, as shown, the setting of actuator 1 202a has a significant impact on the first two databoxes 206, represented by a plot of a hypothetical response, but has negligible effect on the remaining databoxes 208. As shown, the impact of a given actuator generally affects only a few neighboring databoxes. Thus, the resulting matrix 204 features a narrow diagonal band of significant matrix 204 values against a vast background of negligible values.

Matrix 204 is one of many different matrices representing variable responses to an actuator. For example, another matrix may feature the response of moisture measurements as the measured variable instead of weight. The different matrices may reflect different actuator effects on paper output. For example, as shown in FIG. 2, an actuator 202 setting generally effects two to three databoxes of weight measurements. However, the same actuator 202 may effect five databoxes of moisture measurements.

The matrix 204 may be used in a computations that determine actuator settings ("control variables"). Since the bulk of these values are negligible, a technique known as sparse matrix processing can greatly speed computations using the matrix.

In greater detail, FIG. 3 illustrates an approach that can determine actuator settings based on desired paper characteristics and actuator constraints. The approach generates a large matrix that includes different sub-matrices reflecting the effect of different actuators on measurement, desired actuator settings, and constraints upon a collection of actuator settings. The matrix represents a large collection of simultaneous equations that may be solved using a technique known as quadratic programming. The approach quickly converges on a solution to the equations that yields actuator settings that meet the constraints and yields a paper of a desired quality.

In greater detail, the error value of a given databox (or"datapoint") (i. e., the difference between the desired value and the measured value) can be expressed as: <BR> <BR> <BR> <BR> <BR> <BR> <BR> pi = a1,i##x1 + a2,i##x2 +###+aj,i##xj +###+aNaci-1,i##xNact-1 + aNact,i##xNact<BR> <BR> [Equation 1] where: p, error value at data point i i = 1, 2,3,... Nmeas N.. number of data points in the measurement profile dxj = change in position of actuator j au = effect or process gain of actuator j on data point i Nac, = number of actuators This general equation represents a set of linear algebraic equations that can be expressed in a matrix format as: Aerr [Equation 2] where: Axe"= actuator responses matrix g = change in actuator position (column vector) Pers = error profile (column vector) Aerr, #X, and Perr have the form : az,,... a, va" P a, 2 a2, 2... aNc,, 2 axi P2 i a13 aZ, 3 .. arlar3 aX p3 V 2 Aerr a1. 4 a2 4... aNacr4 (X Perr-p4 ... aNQ"s ax. Ps nazi al, Nmear a2, N" .. C1N IV pN, near [Equation 3]

Ae,, is usually a non-square matrix because the measurement profile usually has more elements than the number of actuators. Each actuator can have an effect on every data point of the error profile. However, in reality, an actuator has a limited effect. That is, an actuator typically provides no measurable effect for the data points that are further away from the central action point where the actuator has most of its effect. This can simplify the elements of the matrix A,,, considerably in that the non-zero values are located along the diagonal. a,,, 0 0... 0 al. 2 a2, 2 0... 0 a, 3 a23 a33 --O All"= a,, 4 a2, 4 . 4-* * 0 O a2p5 5 5--0 0 0 0 w aN a « [Equation 4] Because the matrix Aen has more rows than columns, the problem is over-determined and can be solved using a least squares method. The problem can therefore be stated as: minimize #Aerr##X - Perr# [Equation 5] In other words, given a matrix, A, representing the response of paper characteristics to a collection of actuators, and given data derived from sensor measurements, P, software can determine a collection of actuator settings, X, that minimize or reduce the error.

Representing other features of the paper making process in this matrix model can enable computation of actuator setting solutions that have desired properties. For example, often solutions tend toward either the high or the low absolute limits of actuator capacity.

To potentially reduce the likelihood of such a solution set, an optimization problem can be introduced into the general problem by defining an optimization function to be minimized: <BR> <BR> <BR> minimize #Aopt##X - Popt#<BR> [Equation 6] where where: COp, = [l 1 1-l] SaC = (see absolute limit definition) 0 = optimization target Thus, minimization of equation 6 can help ensure that a given set of actuator settings will, generally, not drive the actuators to their extreme settings.

Additionally, the matrix model can reflect constraints on a group of actuator settings. For example, a system may represent absolute limit constraints on the actuators (e. g., the limits of some actuators to some range of settings). The absolute limit constraints will be referred to by the subscript ac. In the same manner as for the general problem described in the previous section, a function to be minimized can be defined in terms of least squares. With this minimized function, the optimal solution can be found for Perr with the least movement of the actuators.

minimize#Aac##X-Pac# [Equation 7] where where: Cac = identity matrix sac = (hac - lac)/2 tac = (hac + lac)/2 Xfdbc = actuator feedback position vector lac = absolute low limit hac = absolute high limit The variable Aac is a diagonal matrix because the absolute limit constraint is independen of the adjacent actuators. The variable Pac is a column vector that reflects the actual feedback position of the actuators. Aac and Pac can advantageously be normalized to take into account the possible different ranges of operation of the actuators.

A first order bending limit may also be integrated into the approach. The limit represents the shear stress between two adjacent actuators. It can also be used to impose a local smoothing factor. This constraint will be referred to by the subscript fc. The function to be minimized can again be defined in terms of least squares: minimize #Afc##X - Pfc# [Equation 8] The variable Afc is a matrix that defines the way to calculate the first order bending value.

This value is calculated by using this general equation: Xi+1 - Xi = (Pfc)i [equation 9] From equations 8 and 9, we can define Afc and Pfc: #Cfc#Nact-1,Nact -#tfc#Nact-1,1<BR> Sfc Sfc [Equation 10]

sfc = bfc<BR> <BR> <BR> <BR> tfc = [Cfc]Nact-1,Nact#[Xfdbck]Nact,1 bfc = first order bending limit The computations may also involve a second order bending limit constraint associated with the bending moment around 3 adjacent actuators. The index sc will refer to the second order bending limit in the equations. Again we can define a function to be minimized in terms of least squares. <BR> <BR> <BR> <BR> <BR> minimize #Asc##X-Psc#<BR> [Equation 11] The matrix AS defines the calculation of the second order bending values. These values are calculated using this equation: 2Xi+1 - Xi - Xi+2 = (Psc)i <BR> <BR> <BR> <BR> [Equation 12]<BR> From Equations 11 and 12, we can define Asc and Psc: [Equation 13]

ssc = bfc<BR> tsc = [Cfc]Nact-2,Nact#[Xfdbck]Nact,1 bsc = second order bending limit Collecting the constraints and desired solution properties into a single expression yields, the goal of finding a set of actuators settings that minimize #[Atot]#[#X]-[Ptot]#<BR> [Equation 14]

The number of rows Nãct of matrix Axa, is equal to the number of actuators, with the number of columns Ntot being equal to the sum of Nmeas (= number of rows of matrix Art), 1 (= number of rows of matrix Aopt), Nmeans (= number of rows of matrix Arc), Nmeas -1 (= number of rows of matrix A, J, and Nmeas -2 (= number of rows of matrix Asc), for a total of Ntot = Nmeas + 1 + (3#Nact -3).

FIG. 3 depicts an illustrative technique of finding a solution for the change in actuator settings, #X, that meets the different constraints and desired paper characteristics

reflected in the model. The technique features a weight matrix (W) that iteratively adjusts the relative calculation emphasis of different constrained and unconstrained elements of the matrix. Initially, the weight matrix may be biased toward a weighting that emphasizes solution constraints. The values of the weight matrix are then iteratively adjusted until the solution set fails to improve by some amount, epsilon, in a subsequent iteration. More specifically, the approach repeatedly solves: minimize #([W]Ntot, Ntot#[Atot]Ntot, Nact #[#X]Nact,1)-([W]Ntot,Ntot#[Ptot]Ntot,1]# [Equation 15] where : [W]Ntot, Ntot = diagonal matrix for IL9x] until solution convergence. As also shown in FIG. 3, the iterative process can be described as follows: 1 Build [4j and [P,., 2 Calc [werr]next = [I]Nmeas+1,Nmeas+1 3 Calc [wtc]next = tc @next Ntc 4 [W] = #@[werr]next [@@]0 4 @@@ [w@]-#0 [wtc]next#Ntot,Ntot 5 Solve [WIAtot]#[#X]=[Ptot] # [#X]new 6 Calc [E]next = [Atc]Ntt,Nact#[#X]Nact,1|new-[Ptc]Ntc,1 7 Calc wnorm#next = #([wtc]Ntc,Ntc#prev##[E]Ntc,1#next) 8 Calc wnorm#prev = 0 9 While #wnorm#prev - wnorm#next# > epsilon 10 Update [werr]next 11 Update [wtc]next 12 Rebuild [W] 13 Solve again for [#X]new 14 Update [Elnert 15 Calc wnorm#prev = wnorm#next 16 Update wnorm#next ine ( 17 End While

With: fw 1 Iwerr JN", , +1, N"+I I pr Lwerr JN", , +I, N", , +I lnt Mormnoc ; [Equation 16] = weighting values for the unconstrained system Lwtc JN, , N, I pr yE, N, , l l tc N ; CsNre Inert w norm nert normett [Equation 17] = weighting values for the constrained system [Equation 18] where: Ntc = 3#Nact - 3 At the end of the iterative process, the vector [#X]new is added to the vector #Xfdbck# and becomes the new actuator set point profile. The number of iterations can be controlled by increasing or decreasing the constant epsilon.

As shown in FIG. 3, optimization process 700 starts 710 by building the matrix A, O, and computing Plot for specified changes in the measured variables. The initial weights W in the initial matrix [W] Ntot, Ntot are set equal to 1. The initial values for [We, linext for the unconstrained system are set 714 equal to the identity matrix and the initial values for [Wtc]#next for the constrained system are computed, and a combined matrix [W] is computed 716. Changes in the actuator column vector a Xlnew are computed 718 using the values for [W] and an error matrix [E] #next is computed 720 of the absolute limit constraint and the first and second order bending limit constraints of the actuators. Using this error matrix [E], new

normalized weights wnorm#next are computed 722 with the initial value wnom, Iprev set equal to 0 724. If (wnorm#next - wnorm#prev) is less than a predetermined value, epsilon, the solution is computed by adding the value for a Xlnew computed 718 to the original settings of the actuators.

If (wnorm#next - wnorm#prev) exceeds epsilon 726, [W,, Ilne,, and [WJLx. are updated 728, 730. Matrix [W] is rebuilt 732 using the new values for [Werr]#next and [Wtc]#next, enw values a Xlnew are computed 734 using the values of the rebuilt matrix [W]. Matrix [E] lneX, of the actuator profile is updated 736. wnorm#prev is set equal to Woom, Inext 738 Wnortnlnext is updated by matrix-multiplication of [w, J with [E]#next The process can repeat until (wnorm#next- Wnormlprev) is less than epsilon.

By determining actuator settings in using the approaches described herein, paper quality can be increased while paper variability decreases. Further, reducing variability allows economic target shifts. For example, reduced variability permits an increase in the moisture target which allows a paper machine to produce paper more quickly.

The process illustrated in FIG. 3 may be implemented as a MATLAB program.

Additionally, by identifying matrices or sub-matrices within the process as sparse (e. g., Atout), the process can enjoy substantial computation savings by speeding operations involving the multiplication of a matrix cell. For example, in some sparse matrix implementations, the location of all zeros within the matrix are identified and placed in a zero map file. Any multiplication functions that involve a parameter in the zero map file is immediately skipped and zero is returned as the answer.

The use of sparse matrix techniques is especially beneficial in this case since approximately 85% of all calculations involve multiplication of matrix elements that are zero. In tests, the system reduced computation time from 6.5 minutes to 6 seconds. As paper may be being fed during this computation, this time saving can save paper.

The techniques described above may form part of a paper making system that includes actuators and paper measurement sensors where actuator settings are determined, for example, using the method illustrated in FIG. 3 and then used to adjust the actuator setting. The computations may be implemented in hardware or software, or a combination of the two. Preferably, aspects of the techniques are implemented by computer programs executing on programmable computers that each include a processor, a storage medium readable by the processor (including volatile and non-volatile memory and/or storage elements), at least one input device, and one or more output devices. The computer

programs, for example, may be derived from MATLAB source code that use MATLAB's sparse matrix and quadratic programming features.

The computer program (s) are preferably stored on a storage medium or device (e. g., CD-ROM, hard disk, or magnetic disk) that is readable by a general or special purpose programmable computer for configuring and operating the computer when the storage medium or device is read by the computer to perform the procedures described herein. The system may also be considered to be implemented as a computer-readable storage medium, configured with a computer program, where the storage medium so configured causes a computer to operate in a specific and predefined manner.

Other embodiments are within the scope of the following claims.