EXTENDED SVD FLATNESS CONTROL Per Erik Modén and Markus Hol ABB AB, Västerås, Sweden ABSTRACT Cold rolling ills soeties do not see able to control flatness as well as exected, taking into account the nuber of actuators they are equied with. This aer, however, resents a systeatic aroach to the question of how to achieve the accurate flatness control that you would exect to get in a well-actuated cluster ill. The key to understanding the restrictions and difficulties is singular value decoosition (SVD) of the atrix that describes the flatness resonses for all actuators. This decoosition clarifies which flatness error shaes are easy to counteract, in the sense that they require little actuator oveent, and which error shaes are difficult or even iossible to reduce, since they would require very large cobined actuator oveents. Having sorted this out, a control strategy that uses this knowledge is resented. Exerience and results are rovided fro its successful use in ractice. KEYWORDS Flatness control; Singular value decoosition; Cold rolling; Cluster ill; Robustness; Perforance; INTRODUCTION Flatness control in cold-rolling is a ultivariable control roble. Abstractly seaking, you can view flatness errors and actuator oveents as occurring in saces of high diension. Different shaes of the flatness error corresond to different directions in the sace of flatness errors, and the diension of this sace equals the nuber of sensors. Different cobinations of actuator oveent corresond to different directions in the sace of actuator oveents, and the diension of this sace equals the nuber of actuators. A cobined oveent of actuators in the actuator sace will influence the errors in a certain direction in the error sace and this with a certain gain. This gain will vary a lot with the direction of the oveent. Singular value decoosition (SVD) is the systeatic way to get insight into this. It orders the influence into orthogonal directions in the saces according to the gains, fro highest gain to lowest, often ending with soe directions having zero gain. Use of SVD in flatness control has reviously been described in [], [2], and [3]. You soeties see flatness errors being described using a basis of orthogonal olynoials. The SVD rovides another orthogonal basis, with the additional advantage that for each error basis vector you also get the corresonding actuator basis vector and the gain. The largest singular value and the corresonding basis vectors tell which shae of flatness error is the easiest to eliinate, in the sense that it takes the least actuator oveents. It also tells how the required cobined actuator oveent looks. This is continued for gradually ore difficult-toreduce flatness error shaes, corresonding to directions in error sace that are orthogonal to the ones already treated. If the sallest singular values are zero, then the corresonding actuator directions for a null sace, eaning that oveents of actuators in this sace cancel out the flatness effect of each other. In other words, when there is a null sace the flatness effect obtained
with one actuator can alternatively be obtained by a certain cobined oveent of the other actuators. It often turns out that a ill with ten actuators or ore can still not control ore than around five directions in ractice, since the reaining ones would require too large actuator oveents. The roble is not only the restricted actuator ranges. For a direction related to a very sall singular value, the resonses to the large oveents are suosed to cancel in all directions excet the intended one. But to achieve this recise cancellation, the odel would have to be unrealistically erfect. This does not ean that the extra actuators are useless. It just exoses the control challenge. The control solution should use all actuators and all degrees of freedo, but not nervously send large oveents on chasing errors that are too hard to counteract in ractice. In contrast to the solutions resented in [] and [2], all degrees of freedo are still available for control with the solution resented in this aer. The aer resents the atheatical background in section. The control solution and soe of its roerties are resented in section 2. A discussion of the tuning of this solution is given in section 3. Practical exerience is exelified in section 4, and soe concluding rearks are rovided in section 5.. THE MILL MATRIX AND ITS SINGULAR VALUE DECOMPOSITION The influence of the actuators on the flatness can be described by a atrix which we call the ill atrix. Its coluns are fored by the steady state flatness resonses fro all actuators, one colun er actuator. Each flatness resonse is a colun vector with one eleent er easureent roll sensor osition. So the diension of the ill atrix is (nuber of sensor ositions) ties (nuber of actuators). We denote it by G and sell out the steady state relation as () Here is a vector describing a change in actuator ositions fro a revious steady state, and is the resulting vector of steady state change in flatness easureent. The inus sign is just a choice of convention. Flatness control is a ultivariable control roble, and in ultivariable rocess control directionality lays an iortant role. You can talk about high gain directions and low gain directions, where the forer are easy to control and the latter are difficult. It is often hard to get a good enough odel to be able to control the low gain directions, and you ight need to give u control of the. To find out what high and low gain directions we have in our flatness control case, we ake a singular value decoosition of the ill atrix. This is the atheatic exression for it: T U U U U V (2) T T V G UV 2 3 2 T V2 In this exression, the uer art of is a diagonal atrix holding the singular values {, 2, } in the diagonal, and they are ordered fro largest to sallest, and all are ositive or zero. The lower art of is fored by as any zero rows as there are ore sensor ositions than actuators. The atrix U is orthonoral and its coluns for a basis for the sace of flatness errors. In the sae way the atrix V is orthonoral and its coluns for a basis for the sace of actuators. We have assued that there are ore sensor ositions than actuators. The singular values are the gains fro
an actuator oveent along a colun of V to the resonse in flatness, which actually occurs along the corresonding colun of U. In the second equality in (2), we ake a artitioning of the atrices according to high gains and low gains (and those which are zero due to fewer actuators than sensor ositions). In the last equality in (2), it has been assued that all singular values in 2 are zero, or sall enough in coarison with the larger singular values to be aroxiated to zero. Now, we will try to exlain in sile words what the singular value decoosition (SVD in shor can tell about the roerties of the ill atrix and thus about the ultivariable flatness control roble. Suose we stand in the high diensional actuator sace and want to take a ste of size one, wondering what direction to choose in order to get the largest ossible flatness influence. The answer is: Choose the direction given by the first colun of V. The size of your flatness resonse to this articular cobined oveent of actuators will be, the largest singular value, and the shae of the steady state resonse will be given by the first colun of U. The singular value is the gain, and this was the high gain direction. If you then look for the highest gain aong reaining actuator directions orthogonal to that first one, you will find the second singular value. It is the gain fro actuator oveent along the second colun of V, which gives flatness resonse along the second colun of U. We can continue like this towards lower and lower singular values (gains), until we reach those which can be aroxiated to zero. We have then found also the low gain directions. What is it then that akes high gain directions easy to control and low gain directions hard? In the high gain direction, sall actuator oveents will be enough, and you ay disregard interference with other directions. Control in a low gain direction is troublesoe for at least two reasons: ) it will require large actuator oveents which ay cause rate saturation and even absolute saturation, which both have negative influence on erforance, 2) the effect of the large actuator oveents are suosed to cancel each other in the higher gain directions, and this uts hard deands on odel accuracy, often harder than ossible to fulfill. The nuber of non-zero singular values is what is called the rank of the ill atrix. We ay define the ractical rank as the nuber of singular values that we consider to be high enough gain to be addressed by the flatness control. The singular values that are saller define a (ractical) null sace of the ill atrix. Actuator oveents in the null sace have no (or insignifican influence on the flatness. The artitioning in (2) is such that the ractical rank is given by the size of, and 2 holds the singular values that are considered too sall. The null sace is sanned by the coluns of V 2. Another asect of this is that flatness errors that can be exressed as linear cobinations of the coluns in U 2 and U 3 cannot be counteracted at all. There are no cobinations of actuator oveents that influence errors of those shaes. All flatness errors that can be counteracted can be exressed as linear cobinations of the coluns of U. A recording fro a ill is shown in Figure, where the initial flatness and the final flatness are equal, but with quite different actuator ositions. It has been ascertained that this was not due to a changed disturbance situation. This illustrates the fact that the ill atrix for this ill has a null sace. The actuator oveent fro the initial ositions to the final ones did only cause a transient change of the flatness, but no change of steady state flatness, so the sae flatness effect fro the actuators was obtained with both final and initial ositions. Further, towards the end the actuators still ove without influencing the flatness, so this oveent also takes lace in the null sace. This ill has eleven actuators and the ill atrix has rank eight, so the null sace has diension three. The ractical rank is robably rather around five, leaving a ractical null sace of diension around six.
Figure. An actual recording fro a ill, illustrating that the sae flatness influence is obtained with quite different actuator ositions. The difference in initial and final ositions reresents a oveent in the null sace. In Figure 2 and Figure 3, an exale with seven actuators is used to illustrate the singular value decoosition of a ill atrix. This exale has five crown actuators and two side shifts. This exale does in fact not have an extreely high ratio between largest and sallest singular values. This ratio is called the condition nuber of the ill atrix. The largest singular value is in this case =3.6 and the sallest 7 =.86, giving a condition nuber of 42, which is not too bad. But if there is soe uncertainty in the resonse functions, one ight still want to treat actuator oveents along the last SVD direction as having no effect (i. e. as being in the null sace), since the effect of these oveents will be sall and uncertain. Total eliination of errors shaed according to the last SVD direction would in any case require very large actuator oveents, thereby easily causing the to reach their constraints. It is therefore wise to give u at least total eliination of such errors. 2.5 2.5 Flatness resonse functions Cr Cr2 Cr3 Cr4 Cr5 UerSS LowerSS.5 -.5 - - -.8 -.6 -.4 -.2.2.4.6.8 Figure 2. Steady state flatness resonse function for each of the actuators
5 crown actuators 2 side shifts 2 Flatness resonses in SVD directions SVD SVD 2 - - 2 3 4 5 6 7 2 3 4 5 6 7.5 SVD SVD 2 SVD 3 SVD 4 SVD 5 SVD 6 SVD 7 SVD 3.5-2 3 4 5 6 7 SVD 4 SVD 5 SVD 6 - - - 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 -.5 - -.5 SVD 7-2 3 4 5 6 7-2 - -.8 -.6 -.4 -.2.2.4.6.8 Figure 3. Flatness resonses along singular value directions. The left hand lot shows cobined actuator oveents that corresond to the resective singular value, and the right hand lot shows the flatness resonse for each of those cobined actuator oveents. Largest resonse is obtained when actuators are oved along the first SVD direction and sallest resonse when oved along the last. 2. CONTROL SOLUTION One coon control solution uses a araeterized flatness error obtained by iniizing a quadratic criterion while honoring actuator constraints. Based on SVD of the ill atrix, this solution can be extended to include weights on directions associated with sall singular values, thereby roviding the desired control erforance. This extended SVD control solution uses all actuators available. It can also ove the in the null sace, when needed due to constraints, but will not cause unnecessary oveent there. The standard control solution is illustrated in Figure 4. This standard control solution iniizes a criterion to find a lower diension araeterization e of the flatness error e. Each eleent of the araeterized error vector e is fed to one controller, for exale a PI controller, and the outut of each controller is fed to one of the actuators. In its basic original for the criterion to iniize could be exressed as W e 2 (, e( G e ( e( (3) The criterion (3) is to be iniized while taking constraints into account. Those constraints are the actuator constraints, including their rate and range constraints and for exale constraints on differences between adjacent crown actuators. Denoting the vector of actuator ositions at tie t by
Figure 4. Block schee showing the structure of a standard control solution u(, this u( which is subject to constraints will deend on the araeterized error e (, the revious ositions u(t-) and the controller states. In cases when the constraints do not becoe active, the solution to this sile iniization is given by the seudo-inverse G of the ill atrix G. Thus, for a case with non-singular ill atrix and no active constraints we get the araeterized error by the rojection T T G G G e( e ( G e( (4) This exression will however get very sensitive, if the ill atrix has any singular value that is sall in relation to the largest one. The inverse G T G will not even exist, if any singular value is zero. To ake this solution, relying on iniization of a criterion like (3), ractically usable for cases with a singular or near singular ill atrix, you need to either add soe kind of regularization to the criterion, or ake the ill atrix better conditioned for exale by gathering a nuber of actuators into fewer virtual actuators. The latter aroach will however reove soe degrees of freedo that the ill could have benefitted fro. Based on the singular value decoosition, we can introduce a systeatic regularization that enables the full use of all degrees of freedo while still avoiding the robles associated with a (near) singular ill atrix. A straightforward version of the extended criterion is 2 T T T T (, e(, u( G e ( e( e ( VQ V e ( u( VQ V u( W e (5) It is still to be iniized while taking constraints into account, just as with the original criterion (3). The idea here is that Q e and Q u are chosen as diagonal atrices. The entries in the diagonals ily certain weights in the criterion for araeterization of the error and the actuator ositions, resectively, along the directions sorted according to the singular values (or in other words sorted according to the gains). So, for exale, if we consider the gains to be too low fro the fifth singular value and on, then we should choose the fifth and further diagonal eleents of both Q e and Q u high enough. The first four diagonal eleents in these two atrices ay in this exale be set to zero, eaning that no extra weight is alied to araeterizations that give actuator oveents e u
in the first four directions. The choice of Q e will ainly influence the transient behavior, while the choice of Q u will ainly influence the steady state behavior. Assuing that, in the criterion (5), the non-zero weights in Q e and Q u corresond to the null sace, then oveent in the null sace will be avoided as long as no actuator gets saturated. But when any actuator hits a constraint, the control solution will use other actuators to accolish at least artially what the saturated one could not. This is ossible since oveent in the null sace is allowed. Such oveent is still avoided when not needed, since there is a enalty in the criterion for using it. 3. TUNING PARAMETERS AND TUNING TOOL It is aarent fro section 2 that the two atrices Q e and Q u are iortant for the control behavior in the extended SVD control solution. They are, however, not really the kind of intuitive tuning araeters that a coissioning engineer would require. What is needed in addition is a tuning tool that allows the tuning to be erfored on a higher level, roviding easily grased tuning knobs and clear resentations of what the exected results will be. In tuning, which is ade at coissioning, one consideration is the trade-off between robustness and noinal erforance. One easure of robustness is the eak of the sensitivity function. For rocess control it is often chosen to be between.2 and 2.. Lower values in this range ean higher robustness towards deviations between the odel used in tuning and the actual lant behavior, but also slower counteraction of disturbances and therefore lower noinal transient erforance. Only if you have a odel that you trust very well, you would tune the controller to get sensitivity eak values in the uer art of the suggested range. If you exect a need for very high robustness, due to an uncertain odel or varying actual behavior, you could very well tune for a sensitivity eak below.2 as well. The lower liit for what you can get at all is. With a tuning knob for the desired sensitivity eak value, the engineer doing the tuning will have good influence over the robustness to be obtained. This covers the ultivariable asect of robustness, as we consider the axiu singular value of the ultivariable sensitivity function. The sensitivity eak secification is an indirect secification of transient behavior, and it eventually leads to values of the diagonal of Q e, found to give the desired eak value. The user also has the freedo to select how any directions should be acted on with full force. Those will have zeros in the corresonding diagonal osition in Q e, and the reaining diagonal eleents will be used to get the secified sensitivity eak value. As a sanity check regarding ultivariable behavior, it is also ade sure that there is not too uch cross talk interference between different directions during transients. The user can secify an allowed ercentage for this cross talk. The tuning of the individual controllers (one er actuator) will actually also be art of the tuning of transient behavior. So beside soe eleents of Q e, also a araeter related to the settling tie for the individual control loos will be found autoatically to get the desired sensitivity eak, and cross talk below liits. An iortant art of any tuning tool is clear resentation of exected control erforance. The transient behavior can here be studied for a nuber of different disturbances, also with a validation odel that ay differ fro the noinal one used in the tuning. This way one ay check both the noinal erforance and the robustness. In addition to that, the actually obtained sensitivity eak value is resented, as well as the closed loo tie constant for the individual loos.
Figure 5. A tuning tool view. On to there are tuning inuts and soe resulting erforance easures. To the left there are siulation choices, in this case selecting a disturbance along the fourth SVD direction alied in closed loo with the noinal odel and ignoring actuator constraints. The grahs are divided as follows: The uer grahs show the flatness resonse and the lower grahs show the actuator ositions. The grahs to the left show the transient resonse while the grahs to the right shows the result at the end of the siulation. In the uer right grah only the initial errors are visible (white bars), since the final errors are zero in this siulation. An exale view fro the tuning tool can be seen in Figure 5. In this view, the evolution of the flatness error can be viewed in a 3D lot (like the figure), or exressed with either the SVD basis or a olynoial basis. The actuator oveents can be viewed as is (like the figure) or exressed with the SVD basis, and one can also choose to view their ossible oveent in the null sace. The steady state behavior with extended SVD control is ainly deterined by the choice of Q u. The concern of the tuning engineer is to avoid too high closed loo steady state gain fro disturbance to actuators, since that would too easily cause actuator saturation. For directions corresonding to large singular values of the ill atrix (high gain directions) this is no roble, but it ay be for low gain directions. So the Q u diagonal eleents corresonding to large singular values should be zero and the rest should be found based on the setting of a suitable intuitive tuning knob. This knob can be the highest allowed steady state gain fro disturbances to actuators. The tool can easily translate that to required values for the Q u diagonal eleents and calculate the resulting steady state attenuation of disturbances. In addition, it is ossible to let the engineer choose for how any SVD directions disturbances should be coletely eliinated in steady state, rovided the entioned gains do not exceed the secified liit. And it is ossible to state that if disturbances in any SVD direction cannot be attenuated ore than a certain ercentage, control of it should be given u totally. All these settings ay influence the finally obtained diagonal eleents of Q u.
To check the result of the steady state tuning a view is available in the tool, as exelified in Figure 6. In this exale, disturbances along the first six SVD directions are totally eliinated in steady state, but since a liit of 5 was secified for the steady state gain fro disturbance to actuators, disturbances along the last SVD direction will be attenuated only by 36%. The liit value is noralized with resect to the first SVD direction, which eans that a value of 5 allows 5 ties ore actuator oveents than the first SVD direction. This view will hel the tuning engineer check the exected behavior and select a suitable steady state tuning. Figure 6. A tuning tool view resenting the steady state resonses in closed loo. For each SVD direction, there is one lot showing an initial and final flatness error (white and blue bars resectively), when a disturbance of that shae occurs, and below that the final osition of the actuators (blue bars), assuing they were in zero osition before the disturbance occurred. Here, each disturbance has the sae nor, so the actuator oveents required to reach the resented final error can be directly coared. 4. PRACTICAL EXPERIENCE The resented control solution extended SVD has been coissioned in a cluster ill containing actuators with great results. The ill atrix in this ill has a theoretical rank of eight. The ractical rank, however, was considered to be four, leaving a ractical null sace of diension seven. The ratio between the largest (firs and sixth singular value was 3, which eans that it would require 3 ties larger actuator oveents to eliinate a disturbance according to the sixth SVD direction in coarison to a disturbance of the sae size for the first SVD direction. This is far too high to be ractically ossible. The corresonding ratios for the fourth and fifth singular values were 22 and 38. Both are lausible deending on the accuracy of the ill atrix. The better
the flatness resonse odels are, the ore SVD directions can be used in control. To deterine the odel accuracy, a syste identification exerient was erfored in the ill where the actuators were excited according to their SVD directions and the corresonding flatness resonses were recorded. Analysis showed that the recorded flatness resonses and the SVD directions fro the noinal ill atrix agreed very well u to the fourth direction. The fifth SVD direction, however, did not the atch the exected shae since the actuator oveents that were suosed to cancel each other out failed in doing so. Instead of a getting a low gain direction, the result was a quite high gain direction of a coletely different shae than exected. Of course, this direction could not be included in control, which resulted in a steady state tuning that included four SVD directions as can be seen in Figure 7. Figure 7. Steady state tuning used in control of the cluster ill. Four SVD directions were used in the control The forer flatness control solution ade the ill atrix better conditioned by aing actuators with siilar flatness resonses together, thereby reducing the nuber of control loos. In addition, the crown actuators were not used in the autoatic flatness control; they were only used for anual oeration by the ill oerators. Of course, this control solution has reoved valuable degrees of freedo. In Figure 8 below, a startu of a coil is lotted. The uer grah shows the actuators ositions, the iddle grah shows the flatness error (flatness target easured flatness) and the lower grah shows the ean flatness and the used control strategies. A thin blue line at the botto of the lower grah indicates the forer control solution and the thick blue line indicates the use of extended SVD control. As can be seen in both the 3D grah of the flatness error and the grah of the ean flatness, the forer control erfors oorly due to the fact that several actuators are saturated (as
can be seen in the uer grah as horizontal straight lines). When switching to extended SVD control, all actuators could be used in control and the full use of the available degrees of freedo resulted in a significant dro in flatness error and ean flatness. The average flatness was iroved fro a value of 7 to 4 I-units. Figure 8. A recording fro the startu of a coil, initially using the revious control solution and activating the extended SVD control after roughly two thirds of the recording (thick line in the botto grah). Both rocess engineers and oerators at the cluster ill were leased with the extended SVD control. First of all, all actuators were individually used in the autoatic control and thereby using all degrees of freedo. Secondly, the actuators were ore centered within their working ranges, which iniized saturation liit situations and thereby could reduce wear and aintenance of the actuators. 5. CONCLUSIONS We can conclude that the systeatic way of treating the ill atrix using singular value decoosition is an efficient way for understanding the ultivariable control roble inherent in flatness control for cases with any actuators. The extended SVD control solution that is based on it has also roved to give the desired control accuracy. It retains all degrees of freedo and uses all actuators available. In articular it handles actuator constraints efficiently, since it can use other actuators to rovide the sae effect, when one actuator has becoe saturated. REFERENCES [] J.V. Ringwood, Shae Control Systes for Sendziir Steel Mills, IEEE Trans. Control Syst. Technol. Vol. 8, No., (2), 7-86, [2] P. Bergsten, Method and device for otiization of flatness control in the rolling of a stri, EP89985, (28) [3] P.E. Modén and M. Hol, Method of flatness control for rolling a stri and control therefor, EP255276, (22)