WO2007041179A2 - System for controlling a rolling mill and method of controlling a rolling mill - Google Patents

Abstract

The invention generally relates to a method of controlled rolling of a metal strip moving through a rolling mill. The method comprises: monitoring a plurality of input control parameters; and generating a plurality of responsive operational control parameters in the rolling mill to selectively control the rolling mill and selectively control a plurality of output parameters of the rolling mill. A control system for controlled rolling of a metal strip moving through a rolling mill is also provided.

Description

SYSTEMFORCONTROLLINGAROLLINGMILLAND METHOD OF CONTROLLINGAROLLING MILL

Parent Case Text

[0001] This patent application claims priority under 35 USC § 119(e)(l) to provisional patent application No. 60/721,736, filed September 29, 2005, the contents of which is hereby incorporated by reference into this patent application in its entirety as if fully set forth herein.

Field Of The Invention

[0002] The invention generally relates to a system for controlling a rolling mill and a method of controlling a rolling mill.

Background Of The Invention

[0003] The cold rolling of metal strip is a complex nonlinear multivariable process whose optimization presents significant challenges to developing control systems for rolling mills. In general, the current technology relies on a structure wherein the effects of interaction between process variables are partially mitigated by single-input-single-output ("SISO") and single-input- multi-output ("SIMO") control loops each operating on a selected measured variable of the rolling mill. As such, the overall control system is based on several separate single input variable problems which have the objective of independent adjustment of strip tension and thickness anywhere in the rolling mill. While such a control system and variations of it have been effective in producing an acceptable metal strip, other control system design techniques for rolling mills may result in improvements in performance and in robustness to uncertainties, disturbances and the like found in a rolling mill.

[0004] Accordingly, a need exists in the art for an improved system for controlling a rolling mill and a method of controlling a rolling mill.

Summary Of The Invention

[0005] An object of the invention is to provide a method of controlled rolling of a metal strip moving through a rolling mill. [0006] Another obj ect of the invention is to provide a method of controlled rolling of a metal strip moving through a rolling mill that monitors a plurality of input control parameters and generates a plurality of responsive operational control parameters in the rolling mill to selectively control the rolling mill and selectively control a plurality of output parameters of the rolling mill.

[0007] Certain objects of the invention are achieved by providing a control system for controlled rolling of a metal strip moving through a rolling mill. The control system has a plurality of stands each of which includes a plurality of work rolls and a plurality of backup rolls associated with the plurality of work rolls. A plurality of work roll devices are provided that monitor speed of the work rolls and may actuate speed of the work rolls which work roll devices are associated with at least some of the plurality of work rolls. The control system also has a plurality of loading devices and a plurality of loading device position monitors associated with at least some of the plurality of stands. A plurality of load cells are associated with at least some of the plurality of stands, a plurality of tensiometers are located proximate to the metal strip moving through the work rolls, a plurality of thickness gauges are located proximate to the metal strip moving through the work rolls and a plurality of metal strip speed monitors are located proximate to the metal strip moving through the work rolls. The work roll devices, the loading devices, the loading device position monitors, the load cells, the tensiometers, the thickness gauges and the metal strip speed monitors monitor a plurality of input control parameters to the control system which generates a plurality of responsive operational control parameters in the rolling mill to selectively control the rolling mill and selectively control a plurality of output parameters in the rolling mill.

[0008] Other objects of the invention are achieved by providing a method of controlled rolling of a metal strip moving through a rolling mill. The method comprises: monitoring a plurality of input control parameters; and generating a plurality of responsive operational control parameters in the rolling mill to selectively control the rolling mill and selectively control a plurality of output parameters of the rolling mill.

Brief Description Of The Drawings

[0009] FIG. 1 is a schematic drawing of a rolling mill of the invention;

[0010] FIG. 2A is a graph of rolling mill typical entry disturbances at one hundred percent speed of the rolling mill; [0011] FIG. 2B is a graph of rolling mill typical entry disturbances at five percent speed of the rolling mill;

[0012] FIG. 3 is a schematic diagram of a system configuration;

[0013] FIG. 3 A is a schematic diagram of an interstand tension trim function;

[0014] FIG. 4 is a schematic diagram of a first stand output thickness estimation of the metal strip;

[0015] FIG. 5 is a schematic diagram of a fifth stand output thickness estimation of the . metal strip;

[0016] FIG. 6A is a graph of response in mill exit thickness to mill entry disturbances at one hundred percent speed of the rolling mill without uncertainties;

[0017] FIG. 6B is a graph of response in mill exit thickness to mill entry disturbances at five percent speed of the rolling mill without uncertainties;

[0018] FIG. 7 is a graph of response in mill exit thickness during deceleration from one hundred percent speed to five percent speed with mill entry disturbance applied without uncertainties; and

[0019] FIG. 8 is a graph of response in mill exit thickness during acceleration from five percent speed to one hundred percent speed with mill entry disturbance applied without uncertainties.

Detailed Description Of The Preferred Embodiments

[0020] For purposes of the description hereinafter, the terms "upper", "lower", "vertical",

"horizontal", "axial", "top", "bottom", "aft", "behind", and derivatives thereof shall relate to the invention, as it is oriented in the drawing FIGS. However, it is to be understood that the invention may assume various alternative configurations except where expressly specified to the contrary. It is also to be understood that the specific elements illustrated in the FIGS, and described in the following specification are simply exemplary embodiments of the invention. Therefore, specific dimensions, orientations and other physical characteristics related to the embodiments disclosed herein are not to be considered limiting.

[0021] As used herein, the symbols listed below have the meanings provided below.

SYMBOLS

i Subscript, stand i in Subscript, stand input parameter

[0022] Turning to FIG. 1 , an exemplary rolling mill 10 is shown in which aspects of the invention may be used. The displayed rolling mill 10 is a tandem cold rolling mill, but other rolling mills and alternate configurations of the rolling mill may also fall within the scope of the invention. Tandem cold rolling is typically performed after a hot rolling process in which metal slabs such as steel, for example, are heated in a furnace and then rolled into coils of reduced thickness suitable for further processing. After hot rolling and prior to cold rolling, the hot rolled material typically undergoes a pickling process wherein coiled strip is unwound and passed through an acid bath to remove oxides formed during hot rolling. Prior to recoiling, oil is applied to the strip to prevent rusting, eliminate damage due to scraping of adjacent coils and often to act as a lubricant for the first stand of a rolling mill.

[0023] The rolling mill 10 has a coil 12 which is unwound and feeds metal strip 14 into the rolling mill 10. The rolling mill 10 includes a plurality of stands or stations 16 through which the metal strip 14 passes. While the exemplary rolling mill 10 shows the use of five stands 16, the teachings of the invention are believed applicable to rolling mills having two or more stands and the depiction of five stands 16 in the patent application should not be considered a limitation of the invention.

[0024] Each of the stands 16 may have a plurality of rotating rolls or work rolls 18 and a plurality of backup rolls 20 associated with the plurality of work rolls 18. Coupled to some or all of the plurality of work rolls 18 is a work roll device 21 for selectively monitoring the speed of the work rolls 18 and/or actuating the speed of the work rolls 18. As depicted in FIG. 1, there is shown a first set of work rolls 22 and a first set of backup rolls 24 at a first stand 25, a second set of work rolls 26 and a second set of backup rolls 28 at a second stand 29, a third set of work rolls 30 and a third set of backup rolls 32 at a third stand 33, a fourth set of work rolls 34 and a fourth set of backup rolls 36 at a fourth stand 37, a fifth set of work rolls 38 and a fifth set of backup rolls 40 at a fifth stand 41. As the metal strip 14 passes through each of the work rolls 18, the metal strip 14 is reduced in thickness. The reduction in thickness is caused by high compression stress in the roll gap which is defined as a small region between the work rolls 18. In the roll gap, the metal is plastically deformed and there is occasional slipping between the metal strip 14 and the work rolls 18. The energy required to achieve reduction in the metal strip 14 thickness causes a temperature rise at the roll gap, which is reduced considerably by the cooling effects of air and/or use of a lubricant on the metal strip 14. Cold rolling of metal strip 14 is typically done to reduce metal thickness, improve the surface finish or to produce desired mechanical properties in the metal strip 14 by cold working the metal strip 14 to make it suitable for the manufacture of various products.

[0025] As shown, the backup rolls 20 may have a plurality of hydraulic devices, pneumatic devices, loading devices, screwdowns or the like 42 to apply a preselected load to the backup rolls 20 against the rolls work 18. The loading devices 42 may selectively actuate the force applied to the backup rolls 20. As depicted in FIG. 1, there is shown a first loading device 44, a second loading device 46, a third loading device 48, a fourth loading device 50 and a fifth loading device 52.

[0026] As shown, the stands 16 may also have a plurality of loading device position monitors used to monitor the position of the loading devices 42. The load applied to the work rolls 18 by some or all of the loading devices 42 via the backup rolls 20 is an operational control parameter that acts, along with some or all of the work roll devices 21 to selectively control output parameters of the rolling mill 10 such as tension in the metal strip 14 between some or all of the stands 16, the load applied by some or all of the rolls 18 to the metal strip 14, the thickness of the metal strip 14 as it exits some or all of the stands 16 and the speed in which the metal strip 14 exits some or all of the stands 16.

[0027] As shown, the rolling mill 10 may have monitoring equipment such as load cells

54 that monitor the load at some or all of the plurality of stands 16. As depicted in FIG. 1, there is shown a first load cell 56, a second load cell 58, a third load cell 60, a fourth load cell 62 and a fifth load cell 64 associated with each of the plurality of stands 16. Also, the rolling mill 10 may have a plurality of tensiometers 66 located proximate to the metal strip 14 moving through the work rolls 18 that monitor the tension between a pair of some or all of the plurality of stands 16. As depicted in FIG. 1, there is shown a first tensiometer 68, a second tensiometer 70, a third tensiometer 72 and a fourth tensiometer 74 associated between a pair of each of the plurality of stands 16. Additionally, the rolling mill 10 may have a plurality of thickness gauges 76 located proximate to the metal strip 14 moving through the work rolls 18 that monitor the thickness of the metal strip 14 as it exits some or all of the plurality of stands 16. As depicted in FIG. 1, there is shown a first thickness gauge 78 and a second thickness gauge 80 that are respectively placed beyond the exit of first stand 25 and a fifth stand 41. Additional thickness gauges 76 could be placed after some or all of the other stands 16 in order to monitor thickness of the metal strip 14 throughout the rolling mill 10. Also, the rolling mill may have a plurality of strip speed monitors 81 located proximate to the metal strip 14 moving through some or all of the work rolls 18 that monitor the speed of the metal strip 14 as it moves in and out of some of the work rolls 18. As depicted in FIG. 1, there are shown strip speed monitors 81 before the second stand 29, before the third stand 33, before the fourth stand 37, before the fifth stand 41 and after the fifth stand 41. [0028] Some or all of the work roll devices 21, some or all of the loading devices 42, some or all of the loading device position monitors 43, some or all of the load cells 54, some or all of the tensiometers 66, some or all of the thickness gauges 16 and some or all of the metal strip speed monitors 81 may act as input control parameters that selectively control the position of some or all of the loading devices 42, the speed of some or all of the work rolls 18, the speed of the work rolls 18, the speed of the metal strip 14, the load applied to the metal strip 14 between some or all of the rolls 18, the tension in the metal strip 14 between some or all of the stands 16, and the thickness of the metal strip 14 as it exits some or all of the work rolls 18. As the metal strip 14 exits the rolling mill 10, the metal strip 14 is collected and coiled on a rewinder 82. After cold rolling, the metal strip 14 may be cleaned and annealed to restore ductility, which was reduced by an increase in hardness and a decrease in formability caused by the strain hardening of the cold rolling process.

[0029J The rolling mill 10 instrumentation generally consists of various monitoring instruments such as work roll devices 21 that measure work roll 18 rotational speeds and/or actuate work roll 18 rotational speed, position monitors 43 that measure the positions of the loading devices 42, load cells 54 that measure roll force at each stand 16, tensiometers 66 that measure interstand metal strip 14 tension force, thickness gauges 78 that measure strip thickness at the exit of some or all of the stands 16 and strip speed monitors that measure metal strip speed 14 as it moves in and out of some of the work rolls 18.

[0030] A mathematical model of a cold rolling process was developed to take account of certain input control parameters and certain operational control parameters of a rolling mill 10. The mathematical model is a group of expressions which relate the rolling parameters of the rolling mill 10 to each other. The type of model used for the invention described herein is one which relates a plurality of input control parameters of the cold rolling process that are significant in the development of a process control strategy, and are capable of providing a plurality of operational control parameters for dynamic adjustment of the operational control parameters on a continuous basis of the rolling mill 10 operation in a straightforward manner without being computationally demanding. Accordingly, the relationships which comprise the mathematical model are based on a series of algebraic equations developed for control purposes by Bryant, C.F., Automation of Tandem Mills. British Iron and Steel Institute, London 1973 as a simplification of more complex classical models, and on empirical equations given in Roberts, W.L., Cold Rolling of Steel. Marcel Dekker, New York, 1978. [0031] The expressions which have been derived are given below for specific roll force

(1), forward slip (3), interstand tension stress (6), output thickness (7), work roll actuator position (i.e. loading device position) (8), work roll speed (9), and interstand time delay approximations (10), with symbols as previously defined, and with stand / understood where no subscript is given. The expressions for input thickness, mean yield stress, mean tension stress, mean thickness, friction coefficient, deformed work roll radius, draft, stand output strip speed, and stand input strip speed are given in Table 6 provided below.

[0032] The specific roll force is approximated in the zone of plastic deformation in the roll gap area of a mill stand as

where

[0033] The term (k - σ) in (1) represents the contribution of the product hardness as reduced by the effects of an average of the entry and exit tension stresses, while the term JWδ is the length of the arc of contact of the work rolls with the product being rolled, and the term (l+.4a) is an amplification effect mostly due to friction. The elastic recovery contribution is omitted from (1) since it is small and therefore can be neglected.

[0034] The forward slip is a measure of the increase in the speed of the strip exiting the roll gap area, with respect to the work roll peripheral speed which is taken as the strip speed at the neutral plane. The forward slip is approximated by

where the angle at the neutral plane and the contact angle respectively are

and

[0035] An expression for the interstand tension stress is obtained by applying Hooke's law to a length of strip between successive stands, assuming some average thickness and neglecting any stretching of the strip, as

[0036] A linear approximation for the mill stretch characteristic is used to estimate the thickness at the exit of a stand as

where F = PW is the total roll force and SO is the intercept of the linearized approximation.

[0037] The work roll actuator position controller (i.e. the position controller for the loading device) and the work roll speed controller are modeled respectively as single first order lags based on typical mill data

and

[0038] The interstand time delay is the time taken for a small element of strip to travel a distance Lo between successive stands and is approximated at any instant of time as

[0039] The theoretical system equations (1) through (9) and the approximations (10) for the interstand time delays are put into the form of a state equation (11) and an output equation (12),

y = g(x) (12) where x e R" is a vector whose elements represent the individual state variables, a(x) e R" is a

state-dependent vector, j/ e ^is a vector whose elements represent the individual output

variables, g(x) e R^p is a state-dependent vector, w e /?'" is a vector whose elements represent the

individual control variables, and B e R^mm is a constant matrix.

[0040] The individual state variables, control variables, and output variables represented by the elements of the vectors x, u, and y respectively in (11) and (12) are as shown in Table 1.

00 (b) (C)

Table 1. (a) State Vector, (b) Control Vector, and (c) Output Vector Elements,

Variable Assignments

[0041J Relationships which express P, h_out , and (V_in,i₊ι - V_out,i) as functions of the state variables are derived in Table 7 provided below.

[0042] The resulting state space model was verified by open loop simulations using three operating points similar to the typical production schedules given by Bryant. Open loop simulations refer to simulating the rolling mill 10 without feedback from the monitored or estimated input control parameters of the rolling mill 10. The results were compared to Bryant's results provided in Bryant, C. F., Automation of Tandem Mills, British Iron and Steel Institute, London 1973. The results were also compared to Geddes' results provided in Geddes, E. J. M., Tandem Cold Rolling and Robust Multivariable Control. PhD thesis, University of Leicester UK 1998. Geddes' results were based on reduction patterns similar to Bryant's. Simulations were performed at mill exit speeds of about 4000 feet per minute and at thread speeds of about 200 feet per minute. The results showed good consistency with the results of both Bryant and

Geddes.

[0043] The pointwise linear quadratic optimal control strategy evaluated for this invention is a pointwise application of the state-dependent Riccati equation method which has seen several recent successful applications in aerospace technology and other areas for control of nonlinear dynamical systems, hi the pointwise linear quadratic method, the nonlinear plant dynamics are expressed in

y = s(^χ)- (14)

[0044] By factorizing the state-dependent vectors a(x) into A(x)x , g(x) into C(x)x , and with b(x) = B , the above becomes a form resembling linear state space equations

y = C(X)X, (16)

where A(x) e R^m" is a state-dependent matrix, C(x) e R^px" is a state-dependent matrix, and

x, u, y, B are as noted previously.

[0045] The optimal control problem is to minimize the performance index

with respect to the control vector u, subject to the constraint (15), where Q(x) > 0 , R(x) > 0 ,

[0046] Under the assumptions a(0) = 0 and B ≠ 0 , the objective (17) is to find a control law which regulates the system to the origin. [0047] The method of solution is first to find a factorization of a(x) such that (13) can be expressed in the form of (15). Then the state-dependent algebraic Riccati equation

A \x)K(x) + K(x)A '(x) - K(X)BR-' (x)B'K(x) + Q(x) = 0 (18)

is solved pointwise for K(x) , resulting in the control law used in the control system of rolling mill 10

u = - R-¹ (x)B'K(x)x. (19)

[0048] As can be seen from the control law and Table 1 , a plurality of input control parameters of the rolling mill 10 are monitored and/or actuated by using work roll devices 21, loading devices 42, loading device position monitors 43, load cells 54, tensiometers 66, thickness gauges 76 and/or strip speed monitors 81 to selectively control the position of the loading devices 42 of some or all of the stands 16, the speed of some or all of the rolls 18, the speed of the metal strip 14, the load applied to the metal strip 14 between some or all of the rolls 18, the tension in the metal strip 14 between some or all of the stands 16, and the thickness of the metal strip 14 as it exits some or all of. the stands 16. Such input control parameters generate a plurality of responsive operational control parameters in the rolling mill 10 for adjusting a position of the loading devices 42 and adjusting the rotational speed of the work rolls 18 to selectively control the rolling mill 10 and selectively control a plurality of output parameters of the rolling mill 10 such as tension in the metal strip 14 between some or all of the stands 16, the load applied by some or all of the rolls 18 to the metal strip 14, the thickness of the metal strip 14 as it exits some or all of the stands 16 and the speed in which the metal strip 14 exits some or all of the stands 16. The control law is continuously calculated every 2 to 5 milliseconds in order to continuously monitor the rolling mill 10. If a deviation in the responsive operational control parameter is determined by the control law, the responsive operational control parameters are adjusted and updated such as position of the loading device 42 and adjusting the speed of the work rolls 18. [0049] In order to ensure a solution to (18) at each point, the method requires that the pair ( A(x), B) be pointwise stabilizable (in a linear sense) for all x in the control space, assuming the availability of full state measurement.

[0050] Local asymptotic stability is assured if (Λ(x), B) is pointwise stabilizable, if there exists a matrix Ci (x) such that Q(x) = C'i(x)Ci(x), and if (Λ(x),Ci(x)) is pointwise detectable, assuming that A(x) e C^k- Global asymptotic stability must be confirmed by simulation since, except for certain special cases, at present there is no useful theory which assures it.

[0051] In general, the necessary condition for the optimal control problem is not satisfied in the case of pointwise linear quadratic optimal control. However, if each element of A(x) ,

K(x) , Q(x) , R(x) , and each element of their partial derivatives A_x(x) , K_x(x) , Q_x(^χ) _> Rx(x) is bounded for all x in the control space, and under global asymptotic stability, then the state trajectories converge to the optimal state trajectories as the states are driven to zero. This is taken to be a near optimal (i.e. suboptimal) condition.

[0052] The application of the pointwise linear quadratic control technique to the tandem cold rolling process relies heavily on physical intuition and simulation to develop and confirm a controller design. This is mostly because no useful theory presently exists which assures global asymptotic stability or robustness. In addition, the process is large, is highly nonlinear with complex interactions between variables, and has significant time delays, which make estimations of performance and robustness to disturbances and uncertainties difficult using analytical methods.

[0053] As an example, an operating point using a typical production schedule, plus the mill and strip parameters, are provided in Table 2 and in Table 3 below.

Table 2. Operating Point

Table 3. Mill and Strip Properties

[0054] The initial state XQ at the operating point is an open loop equilibrium point established by the control vector uo whose elements are given in Table l(b). The operating point is shifted to the origin by introducing the variable z = x-xo. Minimization of the performance index J is then with respect to the vector u-uo,

where Q and R initially are taken as diagonal matrices with tunable constant elements.

[0055] The most significant external disturbances are deviations in mill entry thickness and mill entry hardness over a factor of time as depicted in FIG. 2 A at one hundred percent speed of the rolling mill 10 and FIG. 2B at five percent speed of the rolling mill 10. These disturbances result from the contact of hot metal slabs with colder support skids in the reheat furnace and from roll eccentricities in the hot rolling process. [0056] The most significant internal disturbances resulting from cold mill roll eccentricities, for example, are assumed to be mitigated by an active roll eccentricity compensation scheme.

[0057] A disturbance changes the matrix A(x) by δA(x) which results in a disturbance effect SA(x)x as shown in FIG. 3.

[0058] A control objective of the invention is to keep deviations in individual stand 16 output thicknesses and interstand tensions are as low as reasonably achievable in the presence of external and internal disturbances applied during steady speed and during speed changes. In addition, the stand exit thicknesses and the interstand tensions must be independently adjustable. [0059] In the pointwise linear quadratic technique, the algebraic Riccati equation is solved on a pointwise basis. Disturbance rejection is improved by adding an integrator function and a proportional function to trim the control reference for the position of the position actuator (i.e. the position of the loading device) of each stand. These added trim functions produce zero steady-state error in the control of the estimated individual stand output thickness and reduce the effect of the interstand time delay. In addition, a function was added to estimate the unmeasured elements of the output vector;;. Monitoring instruments to measure the strip speed at the input of the second stand 29, the third stand 33, the fourth stand 37 and the fifth stand 41, and at the output of the fifth stand 41 were added to provide speed signals for the estimation of strip thicknesses at the outputs of stand 29 through stand 41 using mass flow techniques, and for tracking of strip thickness. Elements Q(IJ), Q(2,2), Q(3,3), and Q(4,4) of weighting matrix Q were set during initial simulation to reduce deviations in the interstand tension stresses. In addition, a trim function φ_r for each interstand tension was added to correct for slight offsets from the operating point. The control law computed by the pointwise linear quadratic controller provides signals to the loading device position controllers and the work roll speed controllers for the final control of the tensions, so that excursions in the tensions are significantly reduced which is essential for the stability of rolling.

[0060] The system configuration is depicted in FIG. 3, where each element of the state vector jc is measurable, y_m represents the measurable elements of the output vector y, y_e e R^p(p = 14) is a vector whose elements are the measured (or estimated) elements of y, φ_y is an algorithm which generates y_e , F,,,,(z = 2,3,4,5) are the measured strip speeds at the inputs of stands 29, 33, 37, 41, V_outs is the measured strip speed at the output of stand 41, h_outi_m and houtsm are the measured strip thicknesses at stand 25 and stand 41, Ki e R^mxp(m = 10,p = 14) and Kp e R^mxp are matrices whose elements are zero except for elements (j,j), (j = 1,2,3,4,5), which are the gains for the integral and proportional trim functions for each stand 16. φ_r is an algorithm which implements the interstand tension operating point trims as shown in Fig. 3 A, where x_0Pιi (i=l,2,3,4) is an element of the vector x_op which represents the operating point for the interstand tension for stands i,i+l, σ,,,+i,_re/ is the interstand tension reference for stands i,i+l, Xi is the element of the state vector which represents the measured interstand tension for stands i,i+l, and Kμ₊i is a gain term for stands i,i+l. A direct feed-through is provided for elements X_op,i (i=5, ...,14).

[0061] The algorithm φ_y computes h_outie(y_le) as an estimate of h_Ouu(y₁) using a

British Iron and Steel Research Association measurement houtib (21) and as depicted in FIG. 4,

where the notation h_outie(y_le) indicates that variable homie is represented by element 1 of vector

y_e , and similarly for other variables represented by the elements of y and y_e . [0062] The effects on h_outib of roll eccentricity and the uncertainty in Mai are addressed in the sequel. The time delay from stand 25 to the thickness gauge is approximated as a_e using Vj,,₂ aadLmi . [0063] Thickness h_mt2e(y2e) is computed using V_in2, V_in3, and hue as

where h_n2e is h_outi_m delayed by the transport lag from the thickness gauge to second stand 29,

and 1i_2e is a correction factor for small errors such as changes in thickness caused by spreading, reductions in width, or other effects, which is computed by a separate mill adaptation system which is not a part of the controller.

[0064] Thicknesses h_out3e(y3e) and h_Out4e(y4e) are determined similarly, except that tracking is from the previous stand 16. Thickness h_out5e(yse) is obtained as depicted in FIG. 5, where houtsb is computed as

where hmse is h_out4e(y4e) delayed by the time delay from the fourth stand 37 to the fifth stand

41, and the time delay from the fifth stand 41 to the thickness gauge is approximated as b_e using

Vouts and LmS .

[0065] Interstand tension stresses σi2(y_6e) through σ«0W are computed using strip thicknesses and direct measurement of tension forces. Specific roll forces Pι(ym) through

Ps(yue) are computed using strip width and direct measurement of roll forces. [0066] Adjustments of the individual stand output thicknesses and the individual interstand tension stresses are made simply by changing the variables represented by the elements of the vector y_0Pιi (i=l, ...,5) and the elements of the vector x_op,j (i=l, ...,4) , respectively (Table 1). The independence of adjustment was confirmed by simulation which showed that an adjustment of 2% in a stand output thickness, or an adjustment of 5% in an interstand tension, resulted in a negligible effect on the unadjusted interstand tensions and on the unadjusted steady- state stand output thicknesses.

[0067] Generally speaking, roll eccentricity is an axial deviation between the roll barrel and the roll neck caused by irregularities in the work rolls 18, in the roll bearings, or in both, which results in cyclic variations in the metal strip 14 thickness. In the model, roll eccentricity modifies (7) as

where e is the roll eccentricity.

[0068] While compensation for roll eccentricity is not part of the pointwise linear quadratic controller, it must be considered for consistency with data reported from operating mills which usually includes the effects of roll eccentricity. Numerous methods of eccentricity compensation are described in the literature and many have been successfully implemented. A method of active compensation that fits nicely into the framework of the pointwise linear quadratic controller is a form of adaptive noise cancellation, similar (but not identical) to that described in Kugi, A., et al., 2000, "Active Compensation of Roll Eccentricity in Rolling Mills," IEEE Transactions on Industry Applications, Vol. 36, No. 2, pp. 625-632, which relies on the eccentricity as being always periodic with a frequency proportional to the measured angular velocity of the rolls, so that after compensation the stand exit thickness h_out , the measured roll force F , and the measured position S of the loading device are nearly eccentricity free. The eccentricity components remaining in the mill exit thickness after compensation have been estimated by simulation and combined with the deviations in output thickness as noted later herein.

[0069] The reduction of errors caused by modeling uncertainties and by measurement uncertainties is significant to attaining strong robustness. Estimates of these uncertainties and the sources for each estimate are listed in Table 4 and in Table 5. In these tables, the listed estimated uncertainties are percentages of the measured values except for F, σ, and S (Table 5) which are percentages of full scale values, and for purposes of comparison with other controllers (Table 11) the estimated uncertainty for h_outim(5m) (Table 5) is taken to be zero. Assuming stability, the steady-state errors in stand output thicknesses resulting from these uncertainties are attenuated since they occur inside the closed loops of the trims (FIG. 3). An exception is the errors caused by the measurements of Pj_w- (i=2,3,4) which are used in the computation of houtie for the second, third, fourth stands 16, e.g. (22). These errors are small since the uncertainties in the measurement of V_M are small.

Table 4. Modeling Uncertainties

Table 5. Measurement Uncertainties

[0070] Transient errors in thicknesses due to uncertainties also are small because any changes in the uncertainties are slow compared to the responses of the trim control loops, or the errors are small. In the case of the first stand 25, where a BISRA measurement (21) is used, the estimate of homib is very sensitive to the uncertainty inM/_e . To reduce the transient effects of this uncertainty, Mu is determined using (7) and measurements of h_outim , Fi , and Si , where the measurements of Fi and Si are delayed by the transport lag from the first stand 25 to the thickness gauge. Mu is taken to be equal to Mi since changes in Mi are slow compared to the transport lag. where

and λ₁, a,b, c,γ are constants.

where λ₂ is a constant.

where λ₃ is a constant.

where Ki,K2 are constants.

Table 6. Expressions for Various Parameters

A. Relationships for h_out ,Pas Functions of the State Variables

Using the expressions of Table 6, ξ and α are computed (during each scan of the controller)

at a number of equally spaced points in a predetermined neighborhood of h_oul0 as

and

Using (1) and noting that F = PW, the total roll force is then computed (at each point) as

In the neighborhood of h_out0, F is approximated by a linear fit, which is reasonable because

the neighborhood is not large.

where C₁ and c₂ are constants.

Using (7) and (7-4) h_out is then

and the specific roll force is

Thus hut and P become functions of the state variables which are represented by the state

vector elements (Table 1 (a)).

B. Relationship for ψi,,,i₊i -Voutj) as a Function of the State Variables

From (6-13) the strip speed at the exit of the roll bite is

where calculation of the forward slip/is as given in (3) and calculation of h_out is as given in (7-

5). The variables used in calculation of/(and V_oM ) thus become functions of the state variables.

By conservation of volume through the roll bite,

and ψm,M — Vout.i) becomes a function of the state variables.

Table 7. Derivations of relationships which express

P, hut _> and (Vm₁M -V_out,i) as functions of state variables.

[0071] Open loop simulations and closed loop simulations were performed on the invention using Matlab and Simulink. Matlab and Simulink are products of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098. Open loop simulations or open loop systems refer to simulating the rolling mill 10 without feedback from the monitored or estimated input control parameters of the rolling mill 10. Closed loop simulations or closed loop systems refer to operating the rolling mill by monitoring the input control parameters, and generating (by feedback control) operational control parameters to consistently produce metal strip 14 with certain output control parameters. The open loop simulations confirmed the validity of the model by comparing the simulation results with the results of others, as noted previously. Closed loop simulations, performed to verify control performance and robustness, were done with the controller coupled to the model. For these simulations, Q was set to Iu except for Q(I₁I), Q(2,2), Q(3,3) and Q(4,4), which were set to 10⁸, and R was set to I₁₀. Other parameters were set as noted in Table 8.

Table 8. Parameter Settings

[0072] To highlight the performance and robustness of the controller, total compensation of eccentricity was assumed for the simulations. An estimated eccentricity component was then added to the mill exit thickness for comparison to other systems. See, e.g. Table 11. [0073] The mill entry disturbances of FIG. 2A and FIG. 2B were applied at 100% speed of the rolling mill 10 and at 5% speed of the rolling mill 10, with zero uncertainties. The resulting percentage changes in mill exit thicknesses at 100% speed and at 5% speed are displayed in FIGS. 6 A and 6B. The mill then was decelerated from 100% speed to 5% speed by changing Uvi, Uv2, Uv3, Uv4, and Uvs proportionally to the master speed reference and similarly changing the corresponding elements of xo, with the mill entry disturbances applied and with zero uncertainties. The master speed reference and the response in mill exit thickness are shown in FIG. 7. The mill next was accelerated from 5% speed to 100% speed similarly to deceleration. The results are shown in FIG. 8. Table 9 summarizes the magnitude of the maximum percent deviations of stand exit thicknesses and interstand tension stresses from their operating point values during steady speed and during speed changes, assuming no uncertainties.

Table 9. Maximum Percent Deviation of Stand Exit Thicknesses and Interstand Tension Stresses, with Mill Entry Disturbances Applied, without Uncertainties

[0074] The previous simulations were repeated except with the uncertainties of Table 4 and Table 5 applied simultaneously, with magnitudes and directions such that the worst deviation in mill exit thickness was realized for each case simulated. The results are summarized in Table 10, which shows no significant deviations from the results of Table 9, implying good robustness to external disturbances and to modeling and measurement uncertainties. [0075] For a change in product which changes the operating point, the material properties, or both, the simulations are repeated to verify stability, performance, and robustness, and to establish the settings of weighting matrices Q and R, and of parameters KJJ, Kpj, K_gjntιi(s), and K_υ+i.

Table 10. Maximum Percent Deviation of Stand Exit Thicknesses and Interstand Tension Stresses, with Mill Entry Disturbances Applied, with Uncertainties

[0076] The previous results were compared with data from two operating industrial controllers of Tezuka, T., et al. and Sekiguchi, K., et al. described in Tezuka, T., et. al., "Application of a New Automatic Gauge Control System for the Tandem Cold Mill," in IEEE IAS 2001 Conference Record of the 36^th IAS Annual Meeting. Vol. 2, Sept/Oct, 2001 and Sekiguchi, K., et. al., "The Advanced Set-Up and Control System for Dofasco's Tandem Cold Mill," in IEEE Transactions on Industry Applications. Vol. 32, No. 3, May/June 1996. While differences in mill properties, in operating points, and in material properties, and an absence of disturbance data in the case of the industrial controllers precluded specific comparisons, some general comparisons using mill exit thickness measurements could be made. As noted, mill exit thickness of metal strip 14 is expected by operators to be within 0.8% of the operating point value, which is met by the pointwise linear quadratic controller. The two industrial controllers used for comparison purposes generally conformed to this guideline. Table 11 summarizes the results of the comparisons, assuming that the mill exit thickness measurements have zero uncertainties. For comparison purposes, a maximum eccentricity component of 0.05% (after compensation) was assumed for the pointwise linear quadratic controller, which was confirmed by initial simulation of an active compensation method, using an eccentricity of 0.0012 inches and considering changes in the roll diameter due to mechanical wear and heating. The 0.2% of Table 11 was obtained by adding the 0.05% plus .08% for the fifth stand 16 maximum deviation in output thickness from Table 10, plus .07% for conservatism.

Table 11. Comparison of Magnitudes of

Maximum Percent Deviation of Mill Exit Thickness with Other Controllers

[0077] As shown in Table 11, the pointwise linear quadratic control method with appropriate trimming functions provides the potential for significant improvement over exiting controllers in maintaining the tolerance in mill exit thickness during steady speed and during speed change, in the presence of disturbances and uncertainties. In addition, this method offers the following features, and advantages over existing control strategies and over proposed control methods:

1. The structure of the controller allows for the use of physical intuition in the design process, and does not require a linearized model or gain scheduling as in the case of existing control methods and certain proposed strategies (e.g. linear H⁰⁰). This simplifies the design and thus reduces the design effort and cost.

2. Design effort and cost are further reduced since the pointwise linear quadratic MIMIO control strategy requires no feedforward control, as in the case of most existing industrial control schemes.

3. The configuration of the controller is very simple as compared to existing controllers and to proposed control strategies. This results in a user-friendly environment for commissioning and maintenance personnel which reduces efforts and costs in these areas. 4. The strip thicknesses and interstand tensions are independently adjustable by the operator which is essential during mill operation.

5. The tight control of thickness and interstand tensions strongly contributes to the stability of rolling.

6. The capability of the controller to maintain tight control of thickness and interstand tensions during speed change is especially useful for continuous rolling applications where the speed is changed during weld seam passage.

7. The controller is easily configured to accommodate new applications or revamps.

[0078] While specific embodiments of the invention have been described in detail, it will be appreciated by those skilled in the art that various modifications and alternatives to those details could be developed in light of the overall teachings of the disclosure. Accordingly, the particular arrangements disclosed are meant to be illustrative only and not limiting as to the scope of the invention which is to be given the full breadth of the claims appended hereto and any and all equivalents thereto.

Claims

What is claimed is:

1. A control system for controlled rolling of a metal strip moving through a rolling mill comprising: a plurality of stands each of which includes a plurality of work rolls and a plurality of backup rolls associated with the plurality of work rolls; a plurality of work roll devices that monitor speed of the work rolls and may actuate speed of the work rolls which work roll devices are associated with at least some of the plurality of work rolls; a plurality of loading devices and a plurality of loading device position monitors associated with at least some of the plurality of stands; a plurality of load cells associated with at least some of the plurality of stands; a plurality of tensiometers located proximate to the metal strip moving through the work rolls; a plurality .of thickness gauges located proximate to the metal strip moving through the work rolls, a plurality of metal strip speed monitors located proximate to the metal strip moving through the work rolls, wherein the work roll devices, the loading devices, the loading device position monitors, the load cells, the tensiometers, the thickness gauges and the metal strip speed monitors monitor a plurality of input control parameters to the control system which generates a plurality of responsive operational control parameters in the rolling mill to selectively control the rolling mill and selectively control a plurality of output parameters in the rolling mill.

2. The system of claim 1 wherein the work rolls are rotating at a certain speed, wherein the metal strip is moving at a certain speed, wherein a load is applied to the metal strip, wherein tension is applied to the metal strip between the stands, wherein the metal strip has a certain thickness, and wherein the input control parameters are selected from the group consisting of monitoring a position of the loading devices, the speed of the work rolls, the speed of the metal strip, the load applied to the metal strip, the tension applied to the metal strip between the stands, and the thickness of the metal strip as it exits some of the work rolls.

3. The system of claim 1 wherein the work rolls are rotating at a selected speed, and wherein the plurality of responsive operational control parameters are selected from the group consisting of adjusting a position of the loading device and adjusting the speed of the work rolls.

4. The system of claim 1 wherein the metal strip has tension between the stands, wherein a load is applied by the work rolls to the metal strip, wherein the metal strip has a thickness, wherein the metal strip is moving at a certain speed, and wherein the output control parameters are selected from the group consisting of adjusting the tension in the metal strip between the stands, the load applied by the rolls to the metal strip, the thickness of the metal strip and the speed of the metal strip.

5. The system of claim 1 wherein the responsive operational control parameters are continuously calculated.

6. The system of claim 5 wherein a deviation in the responsive operational control parameters is determined and the responsive operational control parameters are adjusted and updated.

7. The system of claim 1 wherein the responsive operational control parameters are calculated about every 2 to 5 milliseconds.

8. The system of claim 1 wherein tension is applied to the metal strip between the stands, wherein tension in the metal strip is partially controlled by the equation -R^{" 1}Cx)B¹K(X)_x, and wherein -R^"1 is a matrix inverse of a control weighting matrix, B' is a matrix transposition of a control matrix, K(x) is a state dependent solution of an algebraic Riccati equation and x is a state vector.

9. The system of claim 1 wherein tension is applied to the metal strip between the stands, and wherein tension in the metal strip is partially controlled by solving matrix equations.

10. The system of claim 1 wherein the metal strip has a certain thickness, wherein the metal strip is moving at a certain speed, and wherein the thickness of the metal strip is partially controlled by taking a selected metal strip speed and dividing that metal strip speed with another selected metal strip speed and multiplying that product by the metal strip thickness at a selected point which is then multiplied by a correction factor.

11. A method of controlled rolling of a metal strip moving through a rolling mill comprising: monitoring a plurality of input control parameters; and generating a plurality of responsive operational control parameters in the rolling mill to selectively control the rolling mill and selectively control a plurality of output parameters of the rolling mill.

12. The method of claim 11 wherein the work rolls are moving at a selected speed, wherein tension is applied to the metal strip between a plurality of stands, and wherein the monitoring of the input control parameters includes monitoring a position of the loading device, the speed of the work rolls and the tension applied to the metal strip between the stands.

13. The method of claim 12 wherein a load is applied to the metal strip, wherein the metal strip has a certain thickness, and wherein the monitoring of the input control parameters further comprises monitoring the load applied to the metal strip and the thickness of the metal strip as it exits some of the work rolls.

14. The method of claim 11 wherein the rolling mill has work rolls moving at a selected speed, and wherein the selective control of the rolling mill includes adjusting a position of a loading device associated with a stand and adjusting the speed of the work rolls.

15. The method of claim 11 wherein the metal strip has tension between a plurality of stands, wherein a load is applied by the work rolls to the metal strip, wherein the metal strip has a thickness, and wherein the output parameters include adjusting the tension in the metal strip between the stands, the load applied by the work rolls to the metal strip and the thickness of the metal strip.

16. The method of claim 15 wherein the metal strip is moving at a selected speed, and wherein the output parameters further comprises adjusting the speed of the metal strip.

17. The method of claim 11 wherein the responsive operational control parameters are continuously calculated.

18. The method of claim 17 wherein a deviation in the responsive operational control parameters is determined and the responsive operational control parameters are adjusted and updated.

19. The method of claim 11 wherein the responsive operational control parameters are calculated about every 2 to 5 milliseconds.

20. The method of claim 11 wherein tension is applied to the metal strip between a plurality of stands, wherein tension in the metal strip is partially controlled by the equation -R^{" 1}Cx)B¹K(X)X, and wherein -R^"1 is a matrix inverse of a control weighting matrix, B' is a matrix transposition of a control matrix, K(x) is a state dependent solution of an algebraic Riccati equation and x is a state vector.

21. The method of claim 11 wherein tension is applied to the metal strip between a plurality of stands, and wherein tension in the metal strip is partially controlled by solving matrix equations.

22. The method of claim 11 wherein the metal strip has a certain thickness, wherein the metal strip is moving at a certain speed, and wherein the thickness of the metal strip is partially controlled by taking a selected metal strip speed and dividing that metal strip speed with another selected metal strip speed and multiplying that product by the metal strip thickness at a selected point which is then multiplied by a correction factor.