Improvement in the Tandem Rolling of Hot Metal ...

Feature Article

Improvement in the Tandem Rolling of Hot Metal Sheet Using a New Advanced Control Technique

A

n important step in the manufacturing and processing of metals is the tandem rolling of hot metal sheet. In the case of steel, almost half of the finished product made in the world is originally produced in a hot sheet rolling process, which in a great many cases is a tandem hot sheet finishing mill. In a typical tandem hot sheet finishing mill (Figure 1), large metal slabs which have been produced in a previous rolling or casting operation are placed in a reheating furnace and heated to temperatures suitable for intermediate processing and subsequent entry as transfer bars into the finishing mill. Or alternatively, in certain installations, transfer bars are entered into the tandem hot sheet finishing process directly from a continuous casting operation. Transfer bars of steel of typically about 30 mm in thickness enter the finishing mill at speeds of 1–2 m/second and at a temperature

of 850–1,100°C. In the mill, the sheet is passed through a set of five to seven pairs of independently driven work rolls, each supported by a backup roll of larger diameter. To keep the sheet at a reference tension, between each pair of work rolls there is a looper, which is a mechanism consisting of an arm and a roll driven by hydraulics, or by an electric motor in certain older mills. A schematic of a typical hot mill looper between two adjacent mill stands is shown in Figure 2. The sheet passes through individual pairs of work rolls in the mill, wherein its thickness is reduced to about 1.5–2 mm. This reduction in thickness is caused by very high compressive stress in a small region (denoted as the roll bite, or roll gap) between the work rolls. The metal is deformed in the roll bite, and there is slipping between the sheet and the surfaces of the work rolls. The necessary

The tandem rolling of hot metal sheet is a highly complex, nonlinear process that presents a difficult control challenge. The hostile hot metal rolling environment exacerbates this challenge, as it precludes the location of certain sensors to measure variables that are important for control. This paper presents a comprehensive model of this process plus the results of efforts toward the development of a novel process controller. The results of simulations with the new controller coupled to the model show the possibility of significant improvement in the quality of the final product.

Authors John Pittner research associate, University of Pittsburgh, Department of Electrical and Computer Engineering, Pittsburgh, Pa., USA [email protected]

Marwan A. Simaan

Figure 1

Florida 21st Century Chair and Distinguished Professor, University of Central Florida, Department of Electrical Engineering and Computer Science, Orlando, Fla., USA [email protected]

Seven-stand hot sheet finishing mill.

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December 2013 ✦ 1

Figure 2

Looper schematic.

compression force is applied by hydraulic rams, or by a screw arrangement driven by an electric motor in some older installations. The sheet temperature is monitored as it exits the last stand at a temperature in the range of 750–980°C, and in certain cases the mill speed may be increased (i.e., zoomed) to attempt to control the exit temperature. After it exits the last stand, the sheet travels toward a coiling mechanism over a runout table, where it is cooled at controlled rates by the application of water sprays to make it suitable for coiling and to achieve desired metallurgical properties. The reliable measurement of certain significant process variables, such as sheet speed and intermediate stand exit thickness, is precluded by an extremely hostile hot rolling environment. Certain variables which generally can be measured reliably include the roll force, position and angular velocity of the looper arm, roll gap actuator position, work roll speed, plus sheet thickness and temperature at the exit of the last stand. In addition, looper arm force and hydraulic cylinder pressure can be used to estimate interstand tension and looper torque. Due to a wide range of uncertainties, major disturbances and significant time delays, the development of a controller for this complex, non-linear process is quite difficult. Nonetheless, to date, many improvements have been realized in the conventional and advanced control concepts simulated in academia or those that have actually been implemented. In almost all of these control concepts, however, only two adjacent mill stands with their associated looper have been considered. Further, most concepts have failed to fully address the complex dynamic interactions between the other mill variables and the resulting effects of these interactions on the sheet thicknesses,

2 ✦ Iron & Steel Technology

sheet tensions and looper positions. There have been some attempts made at treating the mill as a single system using linearized models, but with resulting complexities that were found to be unacceptable in a practical setting, mostly due to difficulties in making the adjustments required during commissioning by personnel who have a limited background in advanced control theory. Thus, there is a need for a completely new approach that can fully address the complex interactions between variables, reduce the computational efforts associated with linearization, be user-friendly to design and commissioning personnel, and significantly improve the quality and yield of the mill output. In previous work for the development of a controller for the tandem cold rolling process, the authors have addressed certain requirements that are similar to those for tandem hot rolling.1–4 Therefore, it is considered that the methods that were highly successful in the case of tandem cold rolling could be expanded upon to develop a radically new technique for controlling the tandem hot rolling process. However, unlike the control of tandem cold rolling, the control of tandem hot rolling is more difficult, as it must address the additional issues of a mathematical model that is more complex due to the higher sheet temperatures, the control of the loopers which are not present in a tandem cold mill, and the lack of measurement of significant variables. This paper presents the results of an initial investigation into the development of a novel technique to control, as a single system, the thicknesses, tensions and looper positions of this complex process. It will also show by simulation that the newly developed method, when compared with conventional techniques, has the potential for significant improvement in performance.

Mathematical Model A mathematical model of the tandem hot sheet rolling process consists of a set of mathematical expressions which relate the rolling parameters to each other. As part of previous work, 5,6 a basic model that is suitable for the initial control research has been developed and verified, wherein simulation results were compared against data from actual installations and the simulation results of others. The following presents the salient features of this model which, along with the controller, accommodates a wide range of operating points. For this investigation, an operating point is selected based on a fully threaded condition at operating speed, a sheet tension of 0.01 kN/mm2 between adjacent stands, and each looper at an angle of 15°.

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 16(1 − u 2 )P  R p = R 1 + p E d  

Table 1 lists the operating point sheet thickness (hout), the average sheet temperature (T) at the mill entry and at the exit of each stand, the peripheral speed (V0) of the work rolls, and the undeformed work roll radius (R) of each stand. The prediction of the roll force in the roll bite area is based on Sims’ model.7 To better estimate the resistance to deformation of the material being rolled, Sims’ model has been enhanced by using the empirical results of Shida.8 The specific roll force as represented in Sims’ model is:

(Eq. 2) where R is the undeformed work roll radius,  is Poisson’s ratio and E is Young’s modulus which is taken as constant for this evaluation.

P = (kQ p − s ) R pd

A linear relationship estimates the output thickness hout as:

(Eq. 1)

hout = S + S 0 +

where

F M (Eq. 3)

k is the resistance to deformation of the material, Qp is a factor developed in Sims’ paper which compensates for friction and any inhomogeneities of deformation, s in + s out s =is the mean tension stress of the sheet 2

where S is the position of the roll gap position actuator, S0 is the intercept of the linearized approximation, F is the total rolling force (equal to PW, where W is the sheet width), and M is the mill modulus which represents the elastic stretch of the mill stand under the application of the roll force F.

s in + s out

(i.e., s = , where in and out are the 2 sheet tension stresses at the stand input and output), Rp is the deformed work roll radius which is estimated using the Hitchcock approximation9 and  is the stand draft (i.e.,  = hin – hout, where hin and hout are the sheet input and output thickness of the stand).

A measure of the sheet speed exiting the roll bite can be estimated using the forward slip f, where f is defined as the ratio of the relative velocity of the exiting sheet to the peripheral speed of the roll, i.e.:

The deformed work roll radius is estimated using the Hitchcock approximation as:

f =

Vout − V 0 V0 (Eq. 4)

Table 1 Mill Operating Point Stand

hout (mm)

T (°C)

V0 (m/second)

R (mm)

Entry

38.8

1058

—

—

1

21.6

988

1.188

360

2

14.4

973

1.823

336

3

8.6

957

2.957

353

4

6.1

938

4.294

343

5

4.7

922

5.665

388

6

3.9

904

6.946

348

7

3.5

894

7.880

369

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December 2013 ✦ 3

where Vout is the exit sheet speed and V0 is the peripheral speed of the roll. A more useful relationship for control development is that of Ford, Ellis and Bland,10 wherein the forward slip is expressed as:

 Rp  2 f = bn ) (   hout 

The Appendix gives the expression for (Eq. 5)

where

bn =

L((t)) is the sheet length between adjacent stands i and i+1 considering the motion of the looper,  is the looper angle and L0 is the distance between the centerlines of the adjacent stands, and Vout,i and Vin,i+1 are the output and input sheet speeds at the adjacent stands.

The controller of the position of the hydraulic cylinder that sets the work roll position at the roll bite, and the controller of the peripheral speed of the work rolls, are described as single, first-order lags,

j1 d k + s inhin − s out hout − 2 4k R p m

dS U S S = − , S (0 ) = S 0 dt t S t S (Eq. 6)

(Eq. 10)

with

 d  j1 =    Rp 

dL(q(t )) . dt

dV U V V = − , V (0 ) = V 0 dt tV tV

12

(Eq. 11) (Eq. 7) where

and

m is the friction coefficient, and other symbols are as defined previously. For hot rolling, m is taken as the coefficient for sticking friction, which is approximated by the empirical relationship given in Roberts as:

m = 0.00027TF – 0.08,

S is the cylinder position, US is the controller position reference, V is the roll peripheral speed, UV is the controller speed reference, and V and S are the time constants of the first-order lags. The interstand time delay is the time taken for an element of sheet to move between adjacent stands and is approximated as:

t d ,i ,i +1 =

(Eq. 8)

L Vout ,i

where TF is the temperature of the work piece in degrees F.11 Young’s modulus provides the basis for a relationship for sheet tension as:

 ds E  dL (q (t )) ≡ s = + Vin ,i +1 − Vout ,i  , s (0 ) = s 0  dt L0  dt  (Eq. 9) where E is Young’s modulus,

(Eq. 12) where L is the length of sheet between stands i and i+1 considering the looper movement and Vout,i is the sheet speed at the output of stand i. The angle of the looper position is determined as:

dq = w, q (0 ) = q 0 dt (Eq. 13)



x· = A(x)x + Bu, x(0) = x0

where  is the looper angular velocity which is derived from Newton’s second law of motion, and is described as:

(Eq. 18)

1 dw M lpr + M fct + M ld  , w (0 ) = 0 = dt J lpr  (Eq. 14) where Jlpr is the looper moment of inertia, Mlpr is the torque applied to the looper mechanism using a controlled hydraulic cylinder, Mfct is the friction torque of the looper mechanism and Mld is the load torque, which is: Mld = Mten + Mswt + Mlmas + Mbnd (Eq. 15) where Mten, Mswt, Mlmas and Mbnd are the torques resulting from the sheet tension force, the sheet weight, the mass of the looper mechanism, and the bending of the sheet at the looper roll, the values of which at the operating point, along with the value of Jlpr, are given in the Appendix. The looper hydraulic cylinder with its controller is approximated by the torque Mlpr, which is described as:

dM lpr dt

=

U Mlpr tM

−

M lpr tM

, M lpr (0 ) = M lpr ,0 (Eq. 16)

where UM is the torque controller reference and lpr M is the time constant of the first-order lag. The friction torque of the looper mechanism is approximated for this initial work as: Mfct = kvis (Eq. 17) where kvis is the viscous friction constant whose value is given in the Appendix. Using the above equations, a non-linear model of the process dynamics can be expressed in the following consolidated state-space form:

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y = C(x)x (Eq. 19) where x Rn is a vector whose elements represent the individual state variables, A(x) Rnxn is a state-dependent matrix, y Rp is a vector whose elements represent the individual output variables, C(x) Rpxn is a state-dependent output matrix, u Rm is a vector whose elements represent the individual control variables and B Rnxm is a constant matrix. Table 2 lists the variables represented by the elements of the state, control and output vectors, where U represents a control reference, with other symbols as noted previously. The variables represented by the elements of the state vector are derived from direct measurements so that all the states are available to the controller. The elements of the A(x), C(x) and B matrices are given in Pittner and Simaan.6

Process Controller The State-Dependent Riccati Equation Technique — The basis of the controller design is the state-dependent Riccati equation (SDRE) technique. Because of its simplicity and its capability for allowing the use of physical intuition in the design process, this technique is becoming increasingly recognized as a desirable and useful method for the control of many non-linear processes. The following provides, as background material, a brief general overview of this technique. More detailed treatments are provided in the applicable references.12–14 The SDRE method is similar to the well-established linear-quadratic-regulator (LQR) method, except that the coefficient matrices in the state and output equations, and the control and state weighting matrices, are state dependent. The non-linear plant dynamics are expressed in the form as noted previously in Equations 18 and 19. The optimal control problem then is defined in terms of minimizing the performance index:

J =

∞

1 2

∫ (x ′Q (x )x + u ′R(x )u )dt 0

(Eq. 20) December 2013 ✦ 5

Table 2 State, Control and Output Vector Variable Assignments State vector

Control vector

Output vector

x1 (12)

x21 (M12)

u1 (US1 )

y1 (hout1 )

y14 (P1 )

x2 (23 )

x22 (M23 )

u2 (US2 )

y2 (hout2 )

y15 (P2 )

x3 (34 )

x23 (M34 )

u3 (US3 )

y3 (hout3 )

y16 (P3 )

x4 (45 )

x24 (M45 )

u4 (US4 )

y4 (hout4 )

y17 (P4 )

x5 (56 )

x25 (M56 )

u5 (US5 )

y5 (hout5 )

y18 (P5 )

x6 (67 )

x26 (M67)

u6 (US6 )

y6 (hout6 )

y19 (P6 )

x7 (S1 )

x27 (12 )

u7 (US7 )

y7 (hout7 )

y20 (P7 )

x8 (S2 )

x28 (23 )

u8 (UV1 )

y8 (12 )

y21 (12 )

x9 (S3 )

x29 (34 )

u9 (UV2 )

y9 (23 )

y22 (23 )

x10 (S4 )

x30 (45 )

u10 (UV3 )

y10 (34 )

y23 (34 )

x11 (S5 )

x31 (56 )

u11 (UV4 )

y11 (45 )

y24 (45 )

y12 (56 )

y25 (56 )

y13 (67 )

y26 (67 )

x12 (S6 )

x32 (67)

u12 (UV5 )

x13 (S7)

x33 (12 )

u13 (UV6 )

x14 (V1)

x34 (23 )

u14 (UV7 )

x15 (V2)

x35 (34 )

u15 (UMlpr12 )

x16 (V3)

x36 (45 )

u16 (UMlpr23 )

x17 (V4 )

x37 (56 )

u17 (UMlpr34 )

x18 (V5 )

x38 (67 )

u18 (UMlpr45 )

x19 (V6 )

u19 (UMlpr56 )

x20 (V7)

u20 (UMlpr67)

with respect to the control vector u, subject to the constraint given in Equation 18, where Q(x)  0, R(x) > 0, Q(x) and R(x) C k for k ≥ 1 and where  indicates a positive semi-definite matrix, > a positive definite matrix, C k indicates that a matrix has continuous partial derivatives through order k and ' indicates the matrix or vector transpose.

A'(x)K(x) + K(x)A(x) – K(x)BR–1(x)B'K(x) + Q(x) = 0, (Eq. 21) is solved pointwise for K(x), where R–1 indicates the inverse of the matrix R. This results in the control law: u = –S(x)x, (Eq. 22) where, for the general case, S(x) = R –1(x)B'K(x). (Eq. 23)

The performance index in Equation 20 implies finding a control law which regulates the system to the origin. The state-dependent Algebraic Riccati Equation (ARE):


A solution to Equation 21 at each point is assured by requiring that the pair (A(x),B) be pointwise stabilizable (in a linear sense) for all x in the control space. Local asymptotic stability is assured under some A Publication of the Association for Iron & Steel Technology

Figure 3 fairly mild conditions;13 however, in general, stability over the entire control space must be confirmed by simulation. Definitions of pointwise stabilizability and asymptotic stability are provided in Pittner and Simaan.1 Control of Tandem Hot Sheet Rolling Using the SDRE Technique — Reductions in excursions in the interstand tensions and looper positions, as well as excursions in the sheet thicknesses, are essential to ensure the good quality of the mill output and the stability of rolling. This is done with the consideration that measurements of several of the more significant variables (such as sheet speeds, output thicknesses at the first and intermediate stands, sheet temperatures, and the material properties of the workpiece) are unavailable. To achieve this, the controller is configured such that the effects of excursions in the unmeasured variables are estimated by using measured variables, the model and data collected during norController structure. mal processing functions. As an example in Equation 3, except at the last stand where a thickness measurement is available, the exit thickness of the sheet is inferred by measurements of the roll force, the roll gap actuator position, and an estimation of the mill modulus, which uses collected data to refine the estimate. Additionally, the interactions between variables, both measured and unmeasured, throughout the entire mill are inherently handled by the non-linear multi-input/multi-output (MIMO) nature of the SDRE technique. As depicted in Figure 3, the SDRE technique is augmented by outer-loop single-input/single-output (SISO) trimming functions (trims). This reduces slight offsets in the looper position and provides zero steady-state error in the measured tensions and estimated sheet thicknesses. Moreover, the closed-loop control action of the trims also contributes heavily toward reducing the effects of uncertainties and of various unmodeled perturbations. This is because many or both of these occur inside the control loops of the trims, so that the trims significantly enhance the robustness of the controller to these types of effects. The trims also are effective in reducing the effects of the AIST.org

interstand time delays on the estimated sheet thicknesses at the stand exits, since they provide nearly immediate control action in response to excursions in the estimated thickness. Additionally, commissioning and operational functions are facilitated, as there is a one-to-one relationship between a trim adjustment and the variable being adjusted. As noted previously, since there is no supporting theory to assure stability, an SDRE-based system in general requires simulation to verify it. With the trims added, system performance, including stability, likewise must be verified by simulation. At the operating point the vectors x, u and y are represented as xop, uop and yop. The performance index (Equation 20) for the non-zero operating point is modified by the introduction of the vector z = x−xop to be:

J =

∞

1 2

∫(z ¢Qz + (u − u ) ¢R(u − u op

op

))dt

0

(Eq. 24) December 2013 ✦ 7

Figure 4 SDRE method, except that the on-line solution of the ARE is unnecessary.

Looper angle operating point trims.

where, for simplicity, Q and R are taken for this initial effort as diagonal matrices with adjustable constant elements. Each element of the state vector x is measurable, ym represents the measurable elements of the output vector y, ye is a vector whose elements are the measured (or estimated) elements of y, and y is an algorithm which generates ye . Inputs from measured variables are used to generate estimates of the stand output thickness and the tension stress. The uncertainties in the measured variables, in the estimated tensions and thicknesses, plus uncertainties and estimates of other variables, and the bases for these estimates as used in the simulation are as noted herein. The K P and K I blocks represent diagonal matrices whose elements are the proportional and integral gains for the thickness and tension trims. The looper operating point trims are implemented by the algorithm r , as shown in Figure 4. In Figure 4, xop,i (i = 27,…,32) is an element of the vector xop that represents the operating point for the looper between stands i,i+1; li,i+1,ref is the looper reference for stands i,i+1; xi is the element of the state vector which represents the measured looper position for stands i,i+1; and Ki,i+1 is a gain term for stands i,i+1. A direct feed-through is provided for the remaining elements of xop,i. At each mill stand, the output thickness is estimated using Equation 3, with an estimate of the mill modulus, M, that is updated as M changes with operational effects such as mechanical wear and roll temperature. A refined estimation of the mill modulus and Equation 3 are used to determine the sheet thickness at the output of the last stand.4 The structure of the controller does not require a lookup table for determining controller settings or an on-line solution of the ARE. This simplifies the controller structure, which includes simplification in the automatic setup for the next transfer bar to be processed and reduces the amount of off-line simulation needed. In the controller, there are three regimes of operation: (1) the basic regime, (2) the pre-roll regime and (3) the roll regime. The control law, S(x), of the inner loop of the controller (Figure 3) is determined on a pointwise basis, as in the general


The Basic Regime — The basic regime consists of an off-line simulation of a typical process operating at a typical operating point, x0. The point x0 is also the initial condition of the state vector x in the basic regime. The function of the basic regime is to establish a basis from which the initial settings of the controller for the full range of variations of the process can be determined. Accordingly, the simulations performed off-line in the basic regime would not need to be repeated at every installation, and that work done in the basic regime could be applied to the different mills in a larger group of mills without repetition for each mill. In the basic regime, experience is used to determine a typical operating point, x0 . Using the process model and the values of the elements of x0, the elements of A(x0) and C(x0) are computed. Using physical intuition and multiple simulations, a suitable controller is then obtained which includes determination of the elements of the diagonal Q and R weighting matrices and the settings of the PI gains of the trims to give a well-performing controller. The ARE is solved off-line to determine a gain SBA(x0) for the control law of the inner control loop of the controller, where:

S BA (x0 ) = R −1B ¢K (x0 ) (Eq. 25) and K(x0) is the solution of the ARE at x0. The inner controller gain SBA(x0) and the PI gains of the trims thus determine a basis to derive controllers for different mills wherein there are several ranges of variations of the process. The Pre-Roll Regime — The function of the pre-roll regime is to establish the settings of the controller just prior to when the transfer bar enters a specific mill. The model for use in this regime is updated by a separate system based on actual data collected during recent processing. Additionally, what was established in the basic regime to determine the initial inner loop controller gain and the PI settings of the outer loop trim functions are used. The initial setting of the controller gain of the inner loop is based on keeping the dynamic characteristics of the inner loop unchanged from those of the inner loop in the basic regime. For example, considering only the inner loop, the ordinary differential equation (ODE) of the closed loop can be written as: x· = (APR(x) – BSPR(x))x + Buop, x(0) = x0 (Eq. 26)


The ODE for the closed loop in the basic regime is: x· = (ABA(x) – BSBA(x))x + Buop, x(0) = x0 (Eq. 27) where ABA(x) and SBA(x) correspond to APR(x) and SPR(x) in the pre-roll regime and uop represents the vector u at the operating point. The dynamic characteristics of Equation 26 will be equivalent to those of Equation 27 if: APR(x) – BSPR(x) = ABA(x) – BSBA(x)

measurement of the variables represented by the elements of the state vector at a particular instance, j. For instances (j = 2,3,4,…), the value of the SR,j (x) matrix is computed using A R,j-1(x) and SR,j-1(x) matrices as determined in the previous update, and the value of the AR,j (x) matrix is computed using measurements of the elements of the vector x at instance j. For the first instance (j = 1), the values of the A PR (x0) and SPR (x0) matrices are used to determine the inner feedback gain as follows: SR,1(x) = B–L(AR,1(x) – APR(x0)) + SPR(x0) (Eq. 30)

(Eq. 28)

For subsequent instances (j = 2,3,4,…), the value of the inner control loop feedback gain is determined as:

from which SPR (x0) at the operating point of the preroll regime can be determined as:

SR,j(x) = B–L(AR,j(x) – AR,j–1(x)) + SR,j–1(x) (Eq. 31)

SPR(x0) = B –L(APR(x0) – ABA(x0)) + SBA(x0) (Eq. 29)

This is repeated in a pointwise manner during the processing of the remainder of the sheet without requiring a pointwise solution of an ARE. The settings of the gains of the PI trims are determined similarly using the pertinent elements of the C(x) matrix, such that the dynamic characteristics of the outer control loop area are kept very nearly invariant. However, in certain instances, automatic adjustment of the gains of the PI trims may be unnecessary, as would be determined on an individual case-by-case basis.

where APR(x0) is determined from the updated model at the operating point of the pre-roll regime, ABA(x0) and SBA(x0) are as previously determined for the basic regime and B –L is a left inverse of the B matrix, which inverse −1 exits and is computed as B − L = (B ¢B ) B ¢. Thus, the dynamics of the inner control loop remain essentially unchanged from those of the basic regime. The gains of the PI trims are then set to keep the overall outer loop characteristics very nearly unchanged from those as determined in the basic regime. The Roll Regime — The function of the roll regime is to adjust the settings of the pointwise controller at small successive instances of time as the sheet is processed through the mill, where each instance of time is understood to be a “point.” The time period between the successive points is set by the control designer to provide suitable control of the process, considering the process dynamics. The result is that the dynamic characteristics of the inner control loop and those of the overall outer control loop at any given instance of time (or point) remain very close to those determined in the previous instance of time and to the pre-roll regime at the first instance of time. The setting of the inner control loop feedback gain is therefore determined in a pointwise manner. The determination of the setting is based on the

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Disturbances and Uncertainties — The successful reduction in the effects of uncertainties in modeling and measurement and the effects of major disturbances as the sheet moves through the mill is a major issue in the control of this process. Typical uncertainties that are considered in modeling are the mill entry thickness (±5%), the workpiece temperatures (±10°C), the equivalent carbon content of the workpiece (±30%), plus the mill modulus (±10%). The uncertainties in measurement are taken to be sheet tension stress (±1%), roll gap actuator position, work roll peripheral speed, roll force, looper torque, angle and angular velocity, which are on the order of ±0.1%. The uncertainties in temperatures and equivalent carbon content are taken for this initial work as invariant for the entire length of the workpiece. Also, Young’s modulus, workpiece width and density are assumed to be constant. The major disturbances considered are changes in the incoming thickness and temperature due to skid chill in the reheating furnace, plus a rapid change in the equivalent carbon content of the workpiece of about 20% at the mill entry, as depicted in Figures

December 2013 ✦ 9

Figure 5 5 and 6. The disturbances and uncertainties are combined concurrently to simulate more severe scenarios. The data presented are based on manufacturer’s information, typical operational characteristics and experience. The percentages and values represent deviations from nominal operating point values.

Disturbances in thickness and temperature at the mill entry.

Figure 6

Disturbances in carbon content at the mill entry.

Figure 7

Increase in speed reference for zooming. 10 ✦ Iron & Steel Technology

Speed Increase of the Mill — In certain installations, to attempt to reduce excursions in the mill exit temperature, the mill speed is increased above its operating point value by a much slower separate controller. This is denoted as a zooming of the mill. The ability of the controller to handle this condition is included for completeness of the work. As shown in Figure 7, for this work, the mill speed reference for each stand is taken to be zoomed to about 20% above the operating point speed in approximately 10 seconds. Additional Advantages of the Controller — This controller considerably reduces the design effort over other controllers. This is because a lookup table, or other similar means, is not needed to determine the settings of the controller to accommodate changes in process characteristics or in the product. Commissioning efforts are reduced, as less tuning is required since changes in the A(x) and C(x) matrices during rolling are measured, with corresponding automatic compensation to maintain the changes in the dynamic characteristics of the inner and outer control loops essentially at a minimum between successive processing instances. In instances where the model changes very rapidly on the fly, such as in the case of “continuous” rolling, the changes to the controller for the next transfer bar are made rapidly and automatically without using a lookup table. This reduces undesirable A Publication of the Association for Iron & Steel Technology

excursions in tension, thickness and looper position, which improves the product quality in the neighborhood of the transition and promotes an improvement in rolling stability. Additionally, the design and commissioning of the controller is strongly based on physical intuition, while not requiring a background in advanced control theory. Thus the controller is user-friendly to most designers and commissioning personnel, with a corresponding reduction in costs and a rapid entry into productive operation.

Simulations The performance of the controller described in this work was assessed using simulations that were performed using MATLAB/Simulink software. Uncertainties and disturbances were introduced in the process described previously, and an unmodeled perturbation in the looper torque was also introduced according to:

Mlpr = a + b(sin(4t + p/6) + sin(7t)) (Eq. 32) where a, b are constants chosen so that the magnitude of the peak perturbation in torque is about 1.0% of the desired operating point value. The simulations were performed closed-loop with the controller coupled to the model. The mill and looper properties were taken to be as given in the Appendix, with the mill operating point as described by the data given in Table 1. The mill was assumed to be operating in a fully threaded condition and running initially at constant speed with the uncertainties applied, and then with a speed increase to simulate zooming. This was followed by the application of the disturbances and the unmodeled perturbation. A typical means

of compensation for mill modulus uncertainty was assumed to be operative so that the variations in the mill modulus M would be within ±2% of its nominal value (3,980 kN/mm). Additionally, a means of eccentricity compensation was assumed to be operational so that eccentricity effects were insignificant, and that control for setting the basic sheet profile and flatness was assumed to be operational and essentially non-interacting with this controller. Typically during controller development, these assumptions are made for the types of initial investigations that are described herein. As previously described in the basic regime, the settings of the elements of the diagonal Q and R matrices, with R being the identity matrix, and the settings of the trims, were established intuitively and confirmed by iterative simulations. Using these results, for the remainder of the simulations operation was shifted to the pre-roll regime and then to the roll regime. Three groups of cases were examined for various uncertainties in the mill modulus and in measured interstand tension stress. The disturbances were as depicted in Figures 5 and 6 for each group of cases. In the first group, the following were applicable: +2% uncertainty in the mill modulus M; uncertainties in the measured tension stresses of 0%, +1% and –1%; uncertainties in the mill entry thickness of +5%, in the equivalent carbon content and temperatures of the workpiece of +30% and –10°C, and in the looper positions of +0.1%. The next two groups of cases were the same as the first group except with uncertainties in M of –2% and 0%. Nine cases were examined, and in each case the same uncertainty was applied to each similar variable everywhere in the mill. Additionally, the mill speed was increased by zooming to about 20% above the operating point speed in 10 seconds. During zooming, each of the mill stand speed references was

Table 3 Maximum Magnitudes of Deviations From Desired Values in Variables With Typical Disturbances and Uncertainties Applied Max. of magnitudes of percent deviations in the sets of listed variables for the variations in certain uncertainties as shown below

Variable sets

Uncertainty in M of 2% with tension uncertainty of: –1% 1%

Uncertainty in M of –2% with tension uncertainty of: –1% 1%

12 to 67

1.00%

1.24

1.00

1.24

12 to 67

0.10

0.10

0.10

0.10

hout,1 to hout,6

0.66

0.66

0.69

0.69

hout,7

Neg.

Neg.

Neg.

Neg.

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December 2013 ✦ 11

Figure 8

(a) (b) Responses using SDRE-based controller (a); and responses using conventional controller (b).

Figure 9

Typical conventional controller (CS-tension controller, CM-torque controller, CG-thickness controller).

zoomed with the same percentage change in speed as in Figure 7.


Responses for interstand tension stress, looper position and sheet thickness at the area between stands 1 and 2 for the first group of cases, with an interstand tension uncertainty of +1%, are shown in Figure 8a. In all cases, a disturbance in a variable was added to its uncertainty. The controller performs well to mitigate the effects of disturbances in the presence of uncertainties, as is typified by the responses of Figure 8a. Responses for other interstand areas are similar except with slightly different maximums in the deviations from the operating points. Table 3 shows the maximum magnitudes of percent deviations from desired values of tensions, looper positions and sheet thicknesses for uncertainties in the mill modulus of ±2%, in the tensions of ±1%, with mill speed zooming, and with other uncertainties and disturbances applied as previously noted. Data for zero uncertainties in M and in tensions are not included, as they are equal to or less than the table entries. Also, a comparison with a well-performing conventional controller (Figure 9) was performed, with the results as depicted in Figure 8b. A typical conventional controller was assumed, with a measurement of sheet tension assumed to exist as feedback to provide


Table 4 Maximum Magnitudes of Deviations From Desired Values in Variables With More Severe Disturbances and Uncertainties Applied Max. of magnitudes of percent deviations in the sets of listed variables for the variations in certain uncertainties as shown below

Variable sets

Uncertainty in M of –10% with tension uncertainty of: –5% 5%

Uncertainty in M of 10% with tension uncertainty of: –5% 5%

12 to 67

5.00%

5.25

5.00

5.26

12 to 67

0.11

0.11

0.11

0.11

hout,1 to hout,6

3.24

3.24

3.92

3.93

hout,7

Neg.

Neg.

Neg.

Neg.

a better comparison with the SDRE-based controller. In the simulation, a similar conventional controller was taken to be applied to each adjacent looper-stand area. As seen in Figure 8, the improvement in performance of the new controller over the conventional controller is clearly significant in the interstand tension and looper position. While little improvement was observed in the sheet thickness variable, the deviations in thickness in both cases are quite small. Further, to evaluate the performance of the controller under conditions more severe than what is considered typical, certain uncertainties and disturbances were increased in magnitude. The choice of which variables to have their magnitudes increased was based on experience. For this set of simulations, the following disturbances were adjusted from what was noted in the aforementioned: specifically, the change in the entry thickness was adjusted from 5% to 10%, in the carbon disturbance from 20% to 40%, and in the entry sheet temperature from –10°C to –30°C. The uncertainties were adjusted as follows: uncertainty in the tension from ±1% to ±5%, in the mill modulus from ±2 to ±10%, in the entry thickness from +5% to +10%, and in the workpiece temperature from –10°C to –20°C. Other disturbances and uncertainties plus the mill zooming remained unchanged. Similar to Table 3, data for zero uncertainties in M and in tensions are not included, as they are equal to or less than the table entries. As can be seen from these results (Table 4), the new controller performed quite well in the presence of the more severe conditions.

a reduction in the overall cost of production to the mill owner and a savings in energy. The successful results as presented herein provide a sound basis for expanding the investigation beyond this initial work to consider control during threading, various product mixes and other regimes and conditions of operation.

Appendix Looper Characteristics6 — A detail of the looper is shown in Figure A-1. Dimensions: L0 = 5.478 m, y = 0.191 m, a = 1.943 m, r (radius) = 0.152 m, l = 0.762 m. Max. angle: 40°.

Figure A-1

Conclusion As shown by the results of the simulations, the new controller offers a major improvement in performance, which eventually results in an improved quality of the final product. Ultimately, this translates into

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Looper detail.


Pass line angle: 2.9°. Mass of looper arm: 300 kg. Mass of looper roll: 500 kg. Viscous friction constant: kvis = –2.0 kNm/rad/second. Moment of inertia: J lpr = 0.3542 (in kgm2/1,000). At the steady-state operating point:  = 15°,  = 0 rad/second, l1 = 2.684 m, l2 = 2.806 m, Mten = –4.289 kNm, Mswt = –1.515 kNm, Mlmas = –4.693 kNm, Mbnd = –0.132 kNm, Mfct = 0 kNm, Mld = –10.629 kNm and Mlpr = 10.629 kNm.

J lpr =

(A-7)

dL (q (t )) dt

dL dq dq dt

where L = l1 + l2 (A-9)

dL d (l 1) d (l 2) = + dq dq dq (A-10)

(

d (l 1) l (r − y ) cos q − a sinq = dq l1

)

1 2 2

l 1 = (r − y + A ) + (a + B ) 2

=

(A-8)

Calculations: With A = l sin ; B = l cos ; C = A + r; D = B + a, and as (for example) F67 = 67 hout6 W, where F67 is the tension force at the looper between stands 6 and 7, W is sheet width (1,290 mm), and where:

(

 r2 300l 2 + 500  l 2 +  3 2 

)

(A-1)

(

)

(

1 2 2

l 2 = (r − y + A ) + (L0 − a − B ) 2

(A-11)

d (l 2) l (r − y ) cos q − (L0 − a ) sinq = dq l2

)

(A-2)

M ten

  D (L − D )   1 1  = F67 C  − 0 − B (C − y )  +     l1 l 2  l2    l1

(A-12) and

dq =w dt

(A-3)

(A-13)

Mswt = (l1 + l2)W hout6 B(74.556·10–6) (A-4)

L((0)) = l1((0)) + l2((0)), (0) = 0

 300  M lmas = B (0.00981)  + 500  2 

(A-14) (A-5)

M bnd

 5 ⋅ 10 −5  1 1 2 = BWhout + 6   l 1 l 2   4  (A-6)

and


with

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4. J. Pittner, “Pointwise Linear Quadratic Optimal Control of a Tandem Cold Rolling Mill,” Ph.D. thesis, Department of Electrical and Computer Engineering, University of Pittsburgh, 2006. 5. J. Pittner and M.A. Simaan, “An Initial Model for Control of a Tandem Hot Metal Strip Rolling Process,” IEEE Transactions on Industry Applications, Vol. 46, Jan/Feb 2010, pp. 46–53. 6. J. Pittner and M.A. Simaan, “A Useful Control Model for Tandem Hot Metal Strip Rolling,” IEEE Transactions on Industry Applications, Vol. 46, Nov/Dec 2010, pp. 2251–2258. 7. R.B. Sims, “The Calculation of Roll Force and Torque in Hot Rolling Mills,” Proceedings of the Institution of Mechanical Engineers, Vol. 168, 1954, pp. 191–200. 8. M. Pietrzyk and J.G. Lenard, “Material Properties and Interfacial Friction,” Thermal-Mechanical Modeling of the Flat Rolling Process, Springer-Verlag, Berlin, 1991, pp. 12–13. 9. J.G. Lenard, M. Pietrzyk and L. Cser, “One-Dimensional Model of the Flat Rolling Process,” Mathematical and Physical Simulation of Hot Rolled Products, Elsevier, Amsterdam, 1999, p. 91.

10. H. Ford, E.F. Ellis and D.R. Bland, “Cold Rolling With Strip Tension, Part I — A New Approximate Method of Calculation and Comparison With Other Methods,” Journal of the Iron and Steel Institute, Vol. 168, May 1951, pp. 57–72. 11. W.L. Roberts, “Frictional Effects of Hot Rolling,” Hot Rolling of Steel, Marcel Dekker, New York, 1983, p. 782. 12. T. Cimen, “State-Dependent Riccati Equation (SDRE) Control: A Survey,” Milestone Reports and Selected Survey Papers, edited by M.J. Chung and P. Misra, 17th IFAC World Congress, Seoul, Korea, 11–13 July 2008. 13. J.R. Cloutier, N. D’Souza and C.P. Mracek, “Non-Linear Regulation and Non-Linear H∞ Control via the StateDependent Riccati Equation Technique: Part 1, Theory,” Proceedings of the International Conference on Non-Linear Problems in Aviation and Aerospace, Embry Riddle University, 1996, pp. 117–131. 14. J.R. Cloutier and D.T. Stansbery, “The Capabilities and Art of State-Dependent Riccati Equation-Based Design,” Proceedings of the American Control Conference, Anchorage, Ala., May 2002, pp. 86–91. F

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