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Disturbance Attenuation in a Hot Strip Rolling Mill via Feedforward Adaptive Control
6257
placed on a flat surface. Flatness is a function of the thickness profile across the strip width, and the manner in which this thickness profile, or 'crown'. evolves through the mill. Poor flatness arises when the reduction in this profile from st.'Uld to stand is not uniform, leading to uneven longitudinal extension across the strip width and thereby a buckling or warping effect. Flatness can be measured via a laser gauge at the mill exit, .md differential tension devices between st1mds. Such transducers can be expensive and hard to maintain; in their absence, skilled visual observation is necessary. With specialised flatness actuators such as roll bending jacks, the thickness profile is manipulated directly by altering the roll gap profile. Without independent actuators. the thickness profile must be influenced by suit.'lbly apportioning the total rolling force rP between the individual rolling stands, subject to the constraint that the resulting reduction in strip gauge across the mill must be such as to achieve final aim gauge. 2. MILL THREAD For the process of thread, the mill actuators (primarily roll gap and st.'Uld speeds, plus any flatness specific actuators) are set to calculated values. These initial setpoints must be set so that the strip will thread through successive stands smoothly and exit the mill with the important strip aims at, or close to, their target values. Such initial setpoints comprise the 'head end set-up'. responsible for the leading end of the strip. Following thread, the mill sensors st.'lft reading and dynamic controls begin operating in order to correct any errors resulting from the head end set-up, and thereafter to maintain the strip parmneters at their aim values despite the presence of disturbances. The main area of interest in this paper is the process of thread. and the adaptive me.ms which may be used to reduce its sensitivity to disturbance. Note that improving the mill thread will both improve head end performance and eLL',e the demands on the dynmnic control, given that less corrective action will be necessary. Calculation of the mill set-up requires accurate predictions of the way both the strip and the mill stands will behave as the strip threads the FM train. Key qmmtities to be modelled include: i) ii) iii) iv) v)
the evolution in strip temperature as the strip threads the mill the dependence of strip yield stress or hardness upon temperature the rolling force required to defonn the strip from one gauge to another the defonnation of the mill housing and roll gap when subjected to the rolling force the evolution in strip crown and flatness as the strip threads the mill
predictive models, which predict the strip and mill behaviour across a variety of conditions. The use of such models to generate a mill set-up has a number of subtleties, not least because of the requirement to achieve interacting, and sometimes conflicting, aims. State of the art approaches involve optimisation techniques, considered in (Bilkhu et at.. 1994). Inevitably modelling errors are observed, which can be addressed via adaptive techniques which take plant measurements collected following thread of a piece, using the errors observed to improve the set-up predictions for the next piece. Methods for this 'piece-to-piece' adaption are det.'liled in (Reeve .md MacAlister, 1987). By design. the model adaption cannot correct errors on the current piece; it will only have an effect on set-up for following pieces. Disturbances on the current piece are ultimately corrected by the mill dynmnic controls. However, such controls are feedback in nature and thus take a finite time to correct head end errors - particularly given that there are inherent transport delays in the mill environment. The remainder of this paper describes an extra level of control 'in piece adaption' - which recognises errors on the current piece and perfonns a feedforward correction of mill set-up so as to attenuate the effect of input disturbances and thereby improve head end performance. 3. IN PIECE ADAPTATION Prior to a piece threading the FM train, the mill set-up algorithm and its adapted models are used to generate the mill setpoints. The mill set-up is provided with strip aims, and the piece state at mill entry - primarily its dimensions, temperature profile and steel chemistry. These are fed into the adapted models, which essentially contain an embedded infonnation history of the rolling of similar products. The resulting mill set-up will be accurate, thereby threading the mill smoothly and achieving head end aims, under two basic assumptions: i) ii)
that the supplied piece entry state is accurate that the current product will exhibit behaviour similar to that of its observed predecessors
In practice however, these assumptions may fail to hold on a given piece, with the result that the mill set-up is not optimal for that piece. This may lead to poor performance on the head end (in terms of gauge, temperature or flatness) or a poor mill thread (e.g. mismatch of mass flow between stands. leading to possible instability). The principal disturbances for which the assumptions fail are: i) ii)
Such modelling requirements are implemented as a series of
6258
the supplied steel chemistry is inaccurate - import.'Ult since hardness is sensitive to variation in chemistry the estimate of mill entry temperature is poor possibly due to inaccurate or occasionally noisy pyrometer measurements, or where pieces are subject to unplanned delays. Temperature is an important factor since it has a large influence on strip hardness
6259
fN+IIN
+K[YN+I -g(fN+dN)]
(4)
q,N+dN - K /) KT
where K is the Kahnan gain, calculated via the linearisation (or Jacobian) C of the measurement function gO and the intermediate matrix D: D
Cq,N+dNCT+Rv
K
q,N+IIN
(5)
usually at exit of the most recently threaded stand. The infonnation contained in this state is used to perform a feedforward adjustment of mill set-up: this adjustment caters for deviations between the estimated state and the state assumed by set-up at the corresponding point. By this means the effect of certain of the input disturbances impinging on the process will be attenuated. In adjusting setpoints, FAT must: i)
C T D- I
ii) where the covm;ance matrix Rv describes the measurement noise on y. In the specific case of the Kalm,m filter update on stand thread, the function j(.) is constructed from: i)
ii)
iii)
the models governing evolution of gauge, crown and natness through the roll gap the temperature models governing both out of stand cooling (prior to stand thread) and through stand cooling (during the strip deformation) random walk behaviour of the model parmneters K, ex ,md S
It is a straightforward matter to linearise these models for the covari,mce extrapolation of equation (3), using model sensitivities calculated numerically. The key me,L,>urement from stand thread is the roll force, measured by load cells or hydraulic pressure sensors. A predicted value of roll force is produced as a part of the extrapolative step (2): this involves the solution of simultmeous equations in gauge ,md force. and in particular gives rise to a non diagonal co variance Rw due to the implicit correlations between elements of state. The predicted value of roll force is used in corrective step (4): the dependence of roll force on elements of state is linearised for use in the Kalm,m gain calculations (5). Improvements to mill instrumentation often involve interst.'llld measurements such as x-ray gauges or pyrometers. The Kahmm filter is particularly useful in such cases, updating the stlte estimate and covariance according to the accuracy of the additional measurements. The extrapolation stage is similar to that of stand thread, but without the need to consider strip deformation through the roll gap. The corrective stlge is straightforward, involving direct mea,>urements of an element of stlte so that the measurement function g(.) is linear.
calculate roll gap and (where available) t1atness actuator settings to improve or maintain strip natness, and to achieve aim mill exit gauge adjust mill speed setpoints to ensure mass now matching from stand to stand
Two different approaches are described in the following sub-sections. depending upon the availability of specialised natness actuators. In both cases it is assumed that the final rolling srnnd is N. ,md that stand M-I is the most recently threaded.
5.1
Control algorithm with specific flatness actuators
The availability of actuators such as stand based roll bending forces { J j } or roll side shifting distances { Dj }, in addition to the option of redistributing roll forces { P j } , mean that there are more control actuators than control aims. The problem of calculating the adjustment to such actuators is a non trivial one since the movement of an actuator on one stand affects downstream srnnds. The problem may be appropriately encapsulated in tenns of an incremental cost function
where successive elements of
(6)
represent deviations in mill exit gauge, crown, and stand exit natness from their target values. and deviations in jack force, side shift position and roll force from their current values. Q is a positive definite symmetric weighting matrix, which trades off the importance of achieving mill exit aims against the undesirability of large actuator movements as the strip is threading. Achieving mill exit gauge hN would typically be given highest priority. The mill exit aims are a function of the actuator values and current piece state, embodied in the constraints:
5. FEEDFORW ARD CONTROL ALGORITHM !lP,
As a result of the Kalman filter estimation, the scheme has ,m updated estimate of the strip state at the current point -
6260
i"'MtoN
(7)
6261
6262
placed on a flat surface. Flatness is a function of the thickness profile across the strip width, and the manner in which this thickness profile, or 'crown'. evolves through the mill. Poor flatness arises when the reduction in this profile from st.'Uld to stand is not uniform, leading to uneven longitudinal extension across the strip width and thereby a buckling or warping effect. Flatness can be measured via a laser gauge at the mill exit, .md differential tension devices between st1mds. Such transducers can be expensive and hard to maintain; in their absence, skilled visual observation is necessary. With specialised flatness actuators such as roll bending jacks, the thickness profile is manipulated directly by altering the roll gap profile. Without independent actuators. the thickness profile must be influenced by suit.'lbly apportioning the total rolling force rP between the individual rolling stands, subject to the constraint that the resulting reduction in strip gauge across the mill must be such as to achieve final aim gauge. 2. MILL THREAD For the process of thread, the mill actuators (primarily roll gap and st.'Uld speeds, plus any flatness specific actuators) are set to calculated values. These initial setpoints must be set so that the strip will thread through successive stands smoothly and exit the mill with the important strip aims at, or close to, their target values. Such initial setpoints comprise the 'head end set-up'. responsible for the leading end of the strip. Following thread, the mill sensors st.'lft reading and dynamic controls begin operating in order to correct any errors resulting from the head end set-up, and thereafter to maintain the strip parmneters at their aim values despite the presence of disturbances. The main area of interest in this paper is the process of thread. and the adaptive me.ms which may be used to reduce its sensitivity to disturbance. Note that improving the mill thread will both improve head end performance and eLL',e the demands on the dynmnic control, given that less corrective action will be necessary. Calculation of the mill set-up requires accurate predictions of the way both the strip and the mill stands will behave as the strip threads the FM train. Key qmmtities to be modelled include: i) ii) iii) iv) v)
the evolution in strip temperature as the strip threads the mill the dependence of strip yield stress or hardness upon temperature the rolling force required to defonn the strip from one gauge to another the defonnation of the mill housing and roll gap when subjected to the rolling force the evolution in strip crown and flatness as the strip threads the mill
predictive models, which predict the strip and mill behaviour across a variety of conditions. The use of such models to generate a mill set-up has a number of subtleties, not least because of the requirement to achieve interacting, and sometimes conflicting, aims. State of the art approaches involve optimisation techniques, considered in (Bilkhu et at.. 1994). Inevitably modelling errors are observed, which can be addressed via adaptive techniques which take plant measurements collected following thread of a piece, using the errors observed to improve the set-up predictions for the next piece. Methods for this 'piece-to-piece' adaption are det.'liled in (Reeve .md MacAlister, 1987). By design. the model adaption cannot correct errors on the current piece; it will only have an effect on set-up for following pieces. Disturbances on the current piece are ultimately corrected by the mill dynmnic controls. However, such controls are feedback in nature and thus take a finite time to correct head end errors - particularly given that there are inherent transport delays in the mill environment. The remainder of this paper describes an extra level of control 'in piece adaption' - which recognises errors on the current piece and perfonns a feedforward correction of mill set-up so as to attenuate the effect of input disturbances and thereby improve head end performance. 3. IN PIECE ADAPTATION Prior to a piece threading the FM train, the mill set-up algorithm and its adapted models are used to generate the mill setpoints. The mill set-up is provided with strip aims, and the piece state at mill entry - primarily its dimensions, temperature profile and steel chemistry. These are fed into the adapted models, which essentially contain an embedded infonnation history of the rolling of similar products. The resulting mill set-up will be accurate, thereby threading the mill smoothly and achieving head end aims, under two basic assumptions: i) ii)
that the supplied piece entry state is accurate that the current product will exhibit behaviour similar to that of its observed predecessors
In practice however, these assumptions may fail to hold on a given piece, with the result that the mill set-up is not optimal for that piece. This may lead to poor performance on the head end (in terms of gauge, temperature or flatness) or a poor mill thread (e.g. mismatch of mass flow between stands. leading to possible instability). The principal disturbances for which the assumptions fail are: i) ii)
Such modelling requirements are implemented as a series of
6258
the supplied steel chemistry is inaccurate - import.'Ult since hardness is sensitive to variation in chemistry the estimate of mill entry temperature is poor possibly due to inaccurate or occasionally noisy pyrometer measurements, or where pieces are subject to unplanned delays. Temperature is an important factor since it has a large influence on strip hardness
6259
fN+IIN
+K[YN+I -g(fN+dN)]
(4)
q,N+dN - K /) KT
where K is the Kahnan gain, calculated via the linearisation (or Jacobian) C of the measurement function gO and the intermediate matrix D: D
Cq,N+dNCT+Rv
K
q,N+IIN
(5)
usually at exit of the most recently threaded stand. The infonnation contained in this state is used to perform a feedforward adjustment of mill set-up: this adjustment caters for deviations between the estimated state and the state assumed by set-up at the corresponding point. By this means the effect of certain of the input disturbances impinging on the process will be attenuated. In adjusting setpoints, FAT must: i)
C T D- I
ii) where the covm;ance matrix Rv describes the measurement noise on y. In the specific case of the Kalm,m filter update on stand thread, the function j(.) is constructed from: i)
ii)
iii)
the models governing evolution of gauge, crown and natness through the roll gap the temperature models governing both out of stand cooling (prior to stand thread) and through stand cooling (during the strip deformation) random walk behaviour of the model parmneters K, ex ,md S
It is a straightforward matter to linearise these models for the covari,mce extrapolation of equation (3), using model sensitivities calculated numerically. The key me,L,>urement from stand thread is the roll force, measured by load cells or hydraulic pressure sensors. A predicted value of roll force is produced as a part of the extrapolative step (2): this involves the solution of simultmeous equations in gauge ,md force. and in particular gives rise to a non diagonal co variance Rw due to the implicit correlations between elements of state. The predicted value of roll force is used in corrective step (4): the dependence of roll force on elements of state is linearised for use in the Kalm,m gain calculations (5). Improvements to mill instrumentation often involve interst.'llld measurements such as x-ray gauges or pyrometers. The Kahmm filter is particularly useful in such cases, updating the stlte estimate and covariance according to the accuracy of the additional measurements. The extrapolation stage is similar to that of stand thread, but without the need to consider strip deformation through the roll gap. The corrective stlge is straightforward, involving direct mea,>urements of an element of stlte so that the measurement function g(.) is linear.
calculate roll gap and (where available) t1atness actuator settings to improve or maintain strip natness, and to achieve aim mill exit gauge adjust mill speed setpoints to ensure mass now matching from stand to stand
Two different approaches are described in the following sub-sections. depending upon the availability of specialised natness actuators. In both cases it is assumed that the final rolling srnnd is N. ,md that stand M-I is the most recently threaded.
5.1
Control algorithm with specific flatness actuators
The availability of actuators such as stand based roll bending forces { J j } or roll side shifting distances { Dj }, in addition to the option of redistributing roll forces { P j } , mean that there are more control actuators than control aims. The problem of calculating the adjustment to such actuators is a non trivial one since the movement of an actuator on one stand affects downstream srnnds. The problem may be appropriately encapsulated in tenns of an incremental cost function
where successive elements of
(6)
represent deviations in mill exit gauge, crown, and stand exit natness from their target values. and deviations in jack force, side shift position and roll force from their current values. Q is a positive definite symmetric weighting matrix, which trades off the importance of achieving mill exit aims against the undesirability of large actuator movements as the strip is threading. Achieving mill exit gauge hN would typically be given highest priority. The mill exit aims are a function of the actuator values and current piece state, embodied in the constraints:
5. FEEDFORW ARD CONTROL ALGORITHM !lP,
As a result of the Kalman filter estimation, the scheme has ,m updated estimate of the strip state at the current point -
6260
i"'MtoN
(7)
6261
6262