Method and control system for tuning flatness control in a mill

Abstract

A method for tuning flatness control for rolling a strip in a mill including rolls controllable by means of a plurality of actuators, which mill is modeled by means of a mill matrix. The method includes: a) obtaining an equivalent movement range for each actuator, b) determining a scaled mill matrix by scaling the mill matrix based on the equivalent movement ranges, and c) obtaining a singular value decomposition of the scaled mill matrix for providing flatness control of the strip by means of the actuators. A computer program and a control system for carrying out the above method are also presented herein.

Claims

1. A method for tuning flatness control for rolling a strip in a mill comprising rolls controllable by means of a plurality of actuators, which mill is modeled by means of a mill matrix, wherein the method comprises: displaying an equivalent movement range window allowing a user input to change the equivalent movement ranges of the actuators, the equivalent movement ranges being user-selectable ranges of movement of the actuators within a full range of movement for each actuator, a) obtaining an equivalent movement range for each actuator, b) determining a scaled mill matrix by scaling the mill matrix based on the equivalent movement ranges, and c) obtaining a singular value decomposition of the scaled mill matrix for providing flatness control of the strip by means of the actuators.

2. The method as claimed in claim 1, wherein each equivalent movement range is an element of a vector.

3. The method as claimed in claim 1, comprising determining a scaling factor based on the equivalent movement ranges, wherein step b) comprises scaling the mill matrix with the scaling factor.

4. The method as claimed in claim 3, wherein the scaling factor is a diagonal matrix with its diagonal having as its diagonal elements the equivalent movement ranges.

5. The method as claimed in claim 1, wherein in step a) the equivalent movement range for each actuator is obtained via user input of each equivalent movement range.

6. The method as claimed in claim 1, comprising: d) determining a ratio of a largest singular value and a singular value that is larger than a predetermined flatness effect threshold value, of the scaled mill matrix and repeating steps a) to d) until a minimum ratio is obtained.

7. The method as claimed in claim 6, wherein the largest singular value is the numerator and the singular value that is larger than a predetermined flatness effect threshold value is the denominator of the ratio.

8. A computer program embodied in a non-transitory computer readable medium, which when loaded onto a processor of a control system are eecuted by the processor, performs the steps of: displaying an equivalent movement range window allowing a user input to change the equivalent movement ranges of the actuators, the equivalent movement ranges being user-selectable ranges of movement on the actuators within a full range of movement for each actuator, a) obtaining an equivalent movement range for each actuator, b) determining a scaled mill matrix by scaling the mill matrix based on the equivalent movement ranges, and c) obtaining a singular value decomposition of the scaled mill matrix for providing flatness control of the strip by means of the actuators.

9. A control system for providing flatness control for rolling a strip in a mill comprising rolls controllable by means of a plurality of actuators, which control system utilizes a mill matrix to model of the mill, wherein the control system comprises: a processing system arranged to: display an equivalent movement range window allowing a user input to change the equivalent movement ranges of the actuators, the equivalent movement ranges being user-selectable ranges of movement of the actuators within a full range of movement for each actuator, obtain an equivalent movement range for each actuator, determine a scaled mill matrix by scaling the mill matrix based on the equivalent movement ranges, and obtain a singular value decomposition of the scaled mill matrix for providing flatness control of the strip by means of the actuators.

10. The control system as claimed in claim 9, wherein each equivalent movement range is an element of a vector.

11. The control system as claimed in claim 9, wherein the processing system is arranged to determine a scaling factor based on the equivalent movement ranges, and to scale the mill matrix with the scaling factor.

12. The control system as claimed in claim 11, wherein the scaling factor is a diagonal matrix having as its diagonal elements the equivalent movement ranges.

13. The control system as claimed in any of claims 9, wherein the processing system is arranged to obtain each equivalent movement range from a user input.

14. The control system as claimed in claim 9, wherein the processing system is arranged to determine a ratio of the largest singular value and a singular value that is larger than a predetermined flatness effect threshold value, wherein the processing system is arranged to repeat: to obtain an equivalent movement range for each actuator, to determine a scaled mill matrix by scaling the mill matrix based on the equivalent movement ranges, to obtain a singular value decomposition of the scaled mill matrix for providing flatness control of the strip by means of the actuators, and to determine a ratio of a largest singular value and a singular value that is larger than a predetermined flatness effect threshold value until a minimum ratio is obtained.

15. The control system as claimed in claim 14, wherein the largest singular value is the numerator and the singular value that is larger than a predetermined flatness effect threshold value is the denominator of the ratio.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention and the advantages thereof will now be described by way of non-limiting examples, with reference to the accompanying drawings of which:

(2) FIG. 1 is a perspective view of an example of a cluster mill;

(3) FIG. 2 is a block diagram of a control system;

(4) FIG. 3a is an example of a user interface for tuning flatness control in a cluster mill;

(5) FIG. 3b is an example of an equivalent movement range window of the user interface in FIG. 3a for selecting actuator movement ranges; and

(6) FIG. 4 is a flow chart illustrating a method for tuning flatness control for rolling a strip in a mill comprising a plurality of rolls controllable by means of actuators.

DETAILED DESCRIPTION OF THE INVENTION

(7) FIG. 1 shows a perspective view of an example of a roll arrangement 1. The exemplified roll arrangement 1 comprises a cluster mill 2, an uncoiler 3 and a coiler 5. The cluster mill 2, hereafter referred to as mill 2, may be used for rolling hard materials, e.g. for cold rolling a metal strip.

(8) A strip 7 may be uncoiled from the uncoiler 3 and coiled onto the coiler 5. The strip 7 is subjected to a thickness reduction process by means of the mill 2 as the strip 7 moves from the uncoiler 3 to the coiler 5.

(9) The mill 2 comprises a plurality of rolls 9-1 and 9-2, including work rolls 19-1 and 19-2, respectively. The rolls 9-1 form a cluster of upper rolls above the strip 7. The rolls 9-2 form a cluster of lower rolls below the strip 7. The exemplified mill 2 is a 20-high mill with the rolls 9-1 and 9-2 arranged in a 1-2-3-4 formation above and below the strip 7, respectively. It is however to be noted that the present invention is likewise applicable to other types of mills such as 6-high and 4-high mills.

(10) Each roll may be actuated by means of actuators (not shown) in order to deform the work rolls 19-1 and 19-2 and thereby adjust a roll gap 21 which is formed between the work rolls 19-1 and 19-2. The process of thickness reduction the strip 7 is obtained when the strip passes the roll gap 21. The work rolls 19-1 and 19-2 are hence in contact with the strip 7 when the strip 7 moves through the mill 2.

(11) Each of the plurality of rolls 9-1 and 9-2 comprise backup rolls, such as backup rolls 11-1, 11-2, 11-3 and 11-4, forming an outer set of rolls of the mill 2. Each backup roll is segmented into a plurality of segments 13. Each of the segments 13 may be controlled by actuators. The segments 13 may by means of actuators be moved towards, or away from, the work rolls 19-1, 19-2. The movement of the rotating segments 13 permeates through the cluster of rolls toward the work roll 19-1 and/or work roll 19-2 for forming the strip 7 moving through the roll gap 21.

(12) In order to provide additional control of the thickness reduction process of the strip 7, the rolls 9-1 and 9-2 further comprise intermediate rolls 15 and 17 arranged between the work rolls 19-1, 19-2 and the backup rolls 11-1, 11-2, 11-3, 11-4. The intermediate rolls 15 and 17 may for instance have bending actuators and/or side-shift actuators, respectively.

(13) The roll arrangement 1 further comprises a measurement device 23, exemplified herein by a measurement roll. The measurement device 23 has an axial extension which is wider than the width of the strip 7 to enable force measurement along the width of the strip 7.

(14) The measurement device 23 comprises a plurality of sensors. The sensors may for instance be distributed in openings in the peripheral surface of the measurement device for sensing the forces applied by the strip to the measurement device. As the strip 7 moves over the measurement device 23, a strip tension profile may by means of the sensors be obtained. A strip tension profile having an even force distribution indicates that the strip has a uniform flatness along its width. A strip tension profile which is non-uniform indicates that the strip has a non-uniform flatness along its width at the associated measured position of the strip.

(15) The measured strip tension profile, translated into a deduced flatness profile, is provided by the measurement device 23 as measurement data to a control system 3.

(16) The measurement data is processed by the control system 3 for controlling the rolls 9-1 and 9-2 by means of the actuators of the mill 2 to thereby provide uniform flatness or a target flatness along the width of the strip 7.

(17) FIG. 2 depicts a schematic block diagram of control system 3. The control system 3 may for example be a multivariable model predictive controller, or it may comprise one control loop for each actuator realized by means of respective PI controllers.

(18) The control system 3 comprises an input/output unit (I/O) 3a, a processing system 3b and a memory 3c. The I/O unit 3a is arranged to be connected to the roll arrangement which it is to control. The control system 3 is arranged to receive measurement data from a measurement device via the I/O unit 3a, and to control the actuators via the I/O unit 3a. The memory 3c is arranged to store a model of the mill arrangement that the control system 3 is intended to control, and other computer-executable components for tuning flatness control. The model comprises a mill matrix G.sub.m. The I/O unit 3a may also be arranged to be connected to an input device such as a mouse or a keyboard, and to a display device adapted to display a user interface to users, such as commissioning engineers, such that tuning of the actuators may be performed by means of the control system 3.

(19) A method for tuning flatness control will now be described in more detail in the following with reference to FIGS. 3a-b and 4. FIG. 3a shows an example of a user interface 4 in which a first window 4a displays each pre-control flatness errors E1 as measured by the sensors of the measurement device, and each post-control flatness error E2 measured after actuator control has been initiated and the response has settled. According to the example, a second window 4b displays the actuator movements of crown actuators for obtaining the post-control flatness errors E2. A third window 4c displays the actuator movements of bend actuators for obtaining the post-control flatness errors E2. A fourth window 4d displays actuator movements of sideshift and skew actuators for obtaining the post-control flatness errors E2. Furthermore, an actuator tuning window 4e is displayed in the user interface 4. According to the example, a user may select the actuator tuning window 4e in order to open an equivalent movement range window 4f, as shown in FIG. 3b. The equivalent movement range window 4f allows a user to change the equivalent movement range of the actuators. A first column C1 indicates the actuators of the mill, which according to the present example has eleven actuators. A second column C2 indicates the equivalent movement ranges of the actuators. A value for each equivalent movement range may be selected by a user. The control system may thus receive user inputs of equivalent movement ranges via entry in the second column C2. A third column C3 may indicate the unit of each equivalent movement range, expressed in for example millimeter, or MPa in case of a hydraulic actuator. According to the example, a fourth column C4 indicates how large portion of the full range of movement each actuator is given as equivalent movement range. The equivalent movement range may for example correspond to 100% of the desired actuator movement span, i.e. the magnitude of a desired range of allowable actuator movement, or it may correspond to e.g. 2% or 1% of the desired actuator movement span.

(20) The equivalent movement range of each actuator in some sense characterizes how large movement of the actuators are considered to be equivalent, generally not in the sense that they provide the same flatness effect, but rather in that they are equally accepted by the mill. The equivalent movement ranges indicate roughly the ranges that the different actuators are expected to cover in their normal control actions, and they may thus also be viewed as preferred control ranges. But what matters in practice is only the relation between the equivalent movement ranges given to the different actuators. The equivalent movement range of an actuator may be a numeric value which is based on the actual physical range of allowed movement of that actuator. By means of the equivalent movement range window 4e, a user may select the equivalent movement ranges for the actuators. The user may observe simulations of flatness error control in windows 4a-4d based on the equivalent movement ranges selected, before deciding whether the selected equivalent movement ranges for the actuators is acceptable and is to be utilized for flatness control in the mill.

(21) FIG. 4 depicts a flow chart illustrating the flatness control tuning method in more detail. In a step a) an equivalent movement range for each actuator is obtained by the processing system 3b. The equivalent movement range for each actuator may for example be obtained by way of a user input via the user interface 4. Such a user input may for example be effected via the equivalent movement range window 4e.

(22) Each obtained equivalent movement range is an element of a vector p.sub.a. Each element of the vector p.sub.a is hence associated with a respective actuator and there is hence a one-to-one correspondence between the actuators and the coordinates of the vector.

(23) In a step b) a scaled mill matrix G.sub.s is determined by the processing system 2b of the control system 3 by scaling the mill matrix G.sub.m obtained from the memory 3c. The scaling is based on the equivalent movement ranges. The scaling of the mill matrix G.sub.m in step b) may be obtained by determining a scaling factor g.sup.1 based on the equivalent movement ranges p.sub.a and scaling the mill matrix G.sub.m with the scaling factor g.sup.1. Typically the scaling of the mill matrix G.sub.m is obtained by multiplying the scaling factor g.sup.1 with the mill matrix G.sub.m. According to one variation the scaling involves multiplying the mill matrix G.sub.m from the right with the scaling factor g.sup.1, i.e. G.sub.s=G.sub.m*g.sup.1. The scaling factor g.sup.1 may be a diagonal matrix with its diagonal having as its diagonal elements the equivalent movement range of each actuator, as shown in equation (1) below.
g.sup.1=diag(p.sub.a) (1)

(24) The scaling factor g.sup.1 is the inverse of g=(diag(p.sub.a)).sup.1 and can be derived as follows. Let u.sub.a denote the actuator positions expressed in original units. Then the actuators scaled by means of the equivalent movement ranges p.sub.a can be expressed u.sub.s=g*u.sub.a. Then the following relations hold.
G.sub.m*u.sub.a=G.sub.m*g.sup.1*g*u.sub.a=G.sub.m*g.sup.1*u.sub.s=G.sub.s*u.sub.s (2)
where G.sub.s=G.sub.m*g.sup.1, i.e. the mill matrix G.sub.m is scaled by means of g.sup.1.

(25) In a step c) a singular value decomposition of the scaled mill matrix G.sub.s is obtained by the processing system 3b. The scaled mill matrix G.sub.s may be utilized for providing flatness control of the strip by means of the actuators. In particular, the above-described tuning can be utilized in control systems comprising multivariable model predictive controllers or PI controllers.

(26) The singular value decomposition form of the scaled mill matrix G.sub.s may be expressed as follows.

(27) $\begin{array}{cc}{G}_s=U.Math.V^T=\left[\begin{array}{cc}{U}_1& U_2\end{array}\right]\left[\begin{array}{cc}{.Math.}_1& 0\\ 0& {.Math.}_2\end{array}\right]\left[\begin{array}{c}{V}_1^T\\ V_2^T\end{array}\right]U_1{.Math.}_1{V}_1^T& \left(3\right)\end{array}$

(28) The matrix is diagonal with the singular values of G.sub.s in its diagonal, with the largest singular value first, and arranged in decreasing order. The matrix U.sub.1 is associated with the flatness effects provided by specific actuator position combinations, i.e. actuator configurations, which do provide a flatness effect to the roll gap and which are defined by the row vectors of the matrix V.sub.1.sup.T. Each direction of the matrix V.sub.1.sup.T, i.e. each row vector, thus represents a specific actuator position combination. The singular values which form the diagonal of the matrix .sub.1 represent the magnitude of the flatness effect for the actuator position combinations of the matrix V.sub.1.sup.T.

(29) The matrix V.sub.2 is associated with those actuator position combinations which do not provide any flatness effect and the singular values which form the diagonal of the matrix .sub.2 are close to zero or zero. In particular, the column vectors of the matrix V.sub.2 span the null space of the mill matrix G.sub.s. In practice, the singular values which are seen to be zero for control purposes may be those singular values which are below a predetermined flatness effect threshold value. As an example, singular values which are a factor 10.sup.3 smaller than the largest singular value may be set to be zero. The column vectors of V which correspond to these singular values are hence defined to span the null space of the mill matrix G.sub.s.

(30) According to one variation of the tuning process, a ratio of a largest singular value and a singular value that is larger than a predetermined flatness effect threshold value, of the scaled mill matrix is determined in a step d) by means of the processing system 3b. Steps a) to d) may be repeated until the ratio is minimized. The largest singular value is hence the numerator and the singular value that has a predetermined flatness effect threshold value is the denominator of the ratio. This ratio determines the effective condition number which is the ratio between the largest singular value and a singular value which is not associated with a singular direction and which may be equal to or larger than the smallest such singular value. The singular value that is larger than a predetermined flatness effect threshold value may thus for example be the smallest singular value of the non-singular part of the matrix . However, often the condition number of the matrix .sub.1, taking the ratio between the largest singular value and the smallest singular value, is far too high. This means that one may have to settle for controlling fewer directions than a number corresponding to the rank of the scaled mill matrix. Thus, the singular value that is larger than a predetermined flatness effect value may be a singular value that is not the smallest singular value of the non-singular part of the matrix . The singular value that is larger than a predetermined flatness effect value may be selected by the user, for example the commissioning engineer.

(31) As an example, if the mill arrangement has eleven actuators, but a mill matrix of rank only eight, it is theoretically possibly to control eight directions. But the practical condition number, taking the ratio between the largest singular value and the eighth singular value, is probably far too high. This means that one may have to settle for controlling let us say just five directions instead. But the ratio between the first singular value and the fifth singular value will depend on the scaled mill matrix G.sub.s, i.e. on the actuator scaling. By minimizing the ratio, a minimum condition number for the non-singular part of the scaled mill matrix G.sub.s may be obtained, whereby more robust control may be provided. Thus, a scaled mill matrix G.sub.s based on equivalent movement ranges which minimizes the effective condition number may be used for flatness control. Alternatively, a scaled mill matrix G.sub.s based on a minimum condition number may be used as initial choice that may be adjusted according to the preferences for the particular case, for example via the equivalent movement range window 4e.

(32) As an alternative to step d), in a step d) a ratio of a largest singular value and a user-selected singular may be determined. Steps a) to d) may be repeated until the ratio is minimized. The user-selected singular value need not necessarily be larger than a predetermined flatness effect threshold value. The user-selected singular value may instead be that singular value in the number order of singular values, which corresponds to the number of singular value directions that the user, e.g. the commissioning engineer, would believe to be useful for efficient flatness control.

(33) The scaled mill matrix G.sub.s obtained via optimization by minimizing the ratio between the largest singular value and a singular value that is larger than a predetermined flatness effect threshold value or the ratio between the largest singular value and a user-selected singular value and/or by user selection of the scaling factor may be stored in the memory 3c for flatness control.

(34) As noted above, the herein presented tuning process may be utilised both for PI control systems and for multivariable model predictive control which may be implemented in software, in hardware or a combination thereof. In the former case a flatness error e can be determined by means of the processing system by the difference between the reference flatness of the strip and the measurement data. The flatness error e is adjusted to obtain an adjusted flatness error e.sub.p. The adjusted flatness error e.sub.p is to be construed as a parameterized flatness error, i.e. the adjusted flatness error e.sub.p is a parameterization of the flatness error e. The adjusted flatness error e.sub.p is determined based on the minimization of for example one of equations (4) and (5) herebelow. The determining of the adjusted flatness error e.sub.p is based on the difference between a mapping of the adjusted flatness error e.sub.p by means of the scaled mill matrix G.sub.s, and the flatness error e, while adding costs, i.e. weights, to the adjusted flatness error and the control unit outputs u and respecting constraints to the control unit outputs. Such constraints may for instance be end constraints, i.e. minimum and maximum allowed positions or possible positions of the actuators. Constraints can also relate to rate constraints, i.e. how fast the actuators are allowed to move, or can move. Furthermore, constraints may relate to differences between actuator positions.

(35) The error parameterization may be seen as a projection of the many original measurements onto exactly one measurement per actuator, which is normally a much lower number.

(36) $\begin{array}{cc}{e}_p\left(t\right)=\arg \left(\underset{u\left(t\right)\mathrm{allowed}}{\min }\left({.Math.G_m{e}_p\left(t\right)-e\left(t\right).Math.}^2+{e_p\left(t\right)}^T{\mathrm{VQ}}_e{V}^T{e}_p\left(t\right)+{u\left(t\right)}^T{\mathrm{VQ}}_u{V}^{T}u\left(t\right)\right)\right)& \left(4\right)\end{array}$

(37) The variable t in equation (4) indicates the time dependence of the flatness error e, the adjusted flatness error e.sub.p, and the control unit outputs u. The optimization is described in more detail in EP2505276.

(38) $\begin{array}{cc}{e}_p\left(t\right)=\arg \left(\underset{u\left(t\right)\mathrm{allowed}}{\min }\left(\begin{array}{c}{\left(G_m{e}_p\left(t\right)-e\left(t\right)\right)}^{T}Z\left(G_m{e}_p\left(t\right)-e\left(t\right)\right)+\\ {e_p\left(t\right)}^T{\mathrm{VQ}}_e{V}^T{e}_p\left(t\right)++{u\left(t\right)}^T{\mathrm{VQ}}_u{V}^{T}u\left(t\right)+\\ {u\left(t\right)}^T{Q}_{d}u\left(t\right)\end{array}\right)\right)& \left(5\right)\end{array}$

(39) If a multivariable model predictive controller (MPC) is used instead of PI controllers, the MPC controller also applies a criterion, but in that case for the direct determination in every sampling instant of the manipulated variable u(t) to be sent to the actuators. This criterion can be formulated as

(40) $\begin{array}{cc}u\left(t\right)=\arg \left(\underset{u\left(t\right)\mathrm{allowed}}{\min }\underset{k=t}{\overset{t+H}{.Math.}}\left[{\hat{e}\left(k\right)}^T{Q}_1\hat{e}\left(k\right)+{u\left(k\right)}^T{Q}_{2}u\left(k\right)\right]\right)& \left(6\right)\end{array}$
where H is the horizon and (k) is the predicted flatness error at sampling instant k. Also when an MPC solution is used, the singular value decomposition of the scaled mill matrix G.sub.s can be used in tuning of the control. Since actuator movement in directions coupled to small singular values are undesired, the weight matrix Q.sub.2 should be chosen with help of the singular vale decomposition, rather than the standard choice of a diagonal matrix. With the choice
Q.sub.2=VQ.sub.uV.sup.T (7)
and a diagonal matrix Q.sub.u, tuning parameters associated with the separate singular value directions are obtained. Beneficially large values in the elements of Q.sub.u are selected to be associated with small singular values. Similarly Q.sub.1 may be selected as
Q.sub.1=UQ.sub.yU.sup.T (8)
to be able to set weights on different shapes of the flatness error according to the singular values. In this case, with a diagonal matrix Q.sub.y large values for the elements associated with large singular values may beneficially be selected, since these are the error shapes that are generally desired to be eliminated, and low values for the elements associated with small singular values, as these are considered to be too hard to counteract.

(41) The skilled person in the art realizes that the present invention by no means is limited to the examples described hereabove. On the contrary, many modifications and variations are possible within the scope of the appended claims.