Neural compensation and modelling of a hot strip rolling mill using radial basis function

F.G.Rossomando^*, J. Denti F^† and A.Vigliocco^‡

^* Instituto de Automática (INAUT), Facultad de Ingeniería. Universidad Nacional de San Juan (UNSJ).
Av. San Martín 1109 Oeste - J5400ARL. San Juan - Argentina, frosoma@inaut.unsj.edu.ar
^† Laboratório de Controle e Instrumentação, Universidade Federal de Espirito Santo (UFES) CT2.
Av. Fernando Ferrari s/n CEP 29060-970 - Vitória-ES-Brasil, j.denti@ele.ufes.br
^‡ Gerencia de Laminación, TERNIUM SIDERAR-- San Nicolas-Pcia de Bs. As.- Argentina, avigliocco@ternium.com.ar

Abstract - In this paper a Neural Compensation Strategy for a hot rolling mill process is proposed. The target of this work is to built a RBF-NN compensation approximation for the classical force feed forward and speed controller. A strategy based on neural networks is proposed here, because they are capable of modelling many nonlinear systems and their neural control via RBF-NN approximation. Simulations demonstrate that the proposed solution deals with disturbances and modeling errors in a better way than classic solutions do. The analysis of the RBF-NN approximation error on the control errors is included, and control system performance is verified through simulations.

Keywords - Hot Strip Mill; Thickness Deviation; Neural Compensation; Radial Basis Functions.

I. INTRODUCTION

The efficiency of a hot strip mill can be increased if the amount of rejected material is reduced. A strip is considered as rejected material if it does not meet the requirements set by the customer and, therefore, has to be sold as lower quality product or has to be re-melted. This last option implies a tremendous amount of extra material handling and high energy costs

One way to improve the efficiency of the hot rolling mill is through a better finishing mill control.

In hot steel strip rolling processes, force feed forward (FFF) control is necessary to reduce the effects of rapid strip thickness variations due to temperature variations and other factors such as support and bearings deformation as well as long term or slow variations of thickness in the six- stand finishing mill (Huang et al. 2004;Pittner and Marwan 2006)

The aim of thickness control is to regulate the exit thickness from the roll gap and the speed output of the strip. Modern and advanced control methods can be employed to decrease the thickness standard deviation and decrease the thickness error by keeping the output speed of the strip constant.

In this approach, a complete model for the system is developed, which is identical to the process used in the facilities of TERNIUM SIDERAR-Argentina HSM, with real data and parameters provided by this company. This real-system model is used as to basis from which to build a simulator for validating new control strategies.

Some of the controllers designed so far are based only on linear models of the HSM. For instance, the controllers designed in Rigler et al. (1996) and Pedersen et al. (1995) compensate the roll gap dynamics, whereas in Galvez et al. (2003) the neural network based predictive control allows for overcoming the existing time delays in system dynamics. In Mahfouf et al. (2005) a neural compensation for roll gap and roll speed is proposed using MLP-NN, but the results shown are based only on off-line adjustment of the NN.

The work Kugi et al. (2001) presents a nonlinear thickness controller that is achieved using a dynamic feedback linearization technique, and implemented on HSM with a hydraulic adjustment system. Hearns et al. (2004) emphasized the importance of the interaction between loop length and exit thickness, and tried a performance trade off between mass flow control and gauge control by changing the output weightings, but the weights can be changed to ensure stable mill operation at the expense of exit gauge control.

Alvarez et al. (2002) describe the design and implementation of a supervisory system for real-time compensation of uneven thickness on both sides of a rolled strip, since it is based on visual inspection. And in Bouazza and Abbassi (2007)an off-line model-based controller and a process simulator are described.

In Rossomando and Denti (2006) a linear parameterization of two stands of a HSM is presented along with the design of a thickness and looper arm controller based on its entire model. One advantage of their controller is that its parameters are directly related to the mill stand parameters. However, if the parameters are not correctly identified or if the change with time because of, e.g. load variation, the performance of the controller can be severely affected.

In Lee et al. (2007) the proposed controller is implemented on the conventional control system, improving the performance of the system without any additional cost. The improvement was demonstrated by using the real data which are collected using the proposed control scheme in HSM, but it only adjusts the roll gap to reduce thickness variation in the HSM. In Ding et al. (2008), an adaptive threading technique was used to predict thickness and material hardness errors so as to modify the setup of the remaining stands without the metal strip rolling in.

In this work, a new strategy that shows a good performance under nonlinearities and disturbances is presented, such as the cold zones in the metal strip. The aim of this work is to built a neural compensation for thickness and speed variations in the roll system. Therefore, the dynamic controller is designed on the basis of a nominal model and a RBF-NN compensation controller capable of learning the dynamic difference between the desired and the actual models. Since the RBF-NN compensates only for a desired model error dynamics, its computational cost is significantly reduced as compared with a whole NN dynamics controller. An analysis is made to study the effects of the RBF-NN approximation error on the control error when the entire control system, the adaptive dynamics controller working, is applied to a roll gap and speed control task.

The paper is organized as follows: Section II presents the overview of the proposed system, and shows the mathematical representation of the complete screw and speed system model. The neural controller is discussed, in Section III, as well as the corresponding error analysis (Section III.A). Section IV presents some simulation results on the performance of the adaptive controller. Finally, conclusions are given in Section V.

II. PLANT MODELLIG

A. Process description

The rolling mill process consists of pressing a strip metal between a set of rotating rolls that reduces the thickness of the passing hot-metal strip. The mill housing, the back-up and work rolls and the roll gap adjustment system are the main machine sections that participate in the rolling mill process; a finishing mill section is a set of rolling mill stands; Fig. 2 shows a dynamic scheme of a simple rolling mill stand with four rolls each one, two work rolls and two back up rolls, with adjustment roll gap systems (screw). The rolls in contact with the strip are the work rolls, they are the main responsible for the rolling mill physical phenomena.

Figure 1: Finishing mill process

Figure 2: Mill stand dynamic model

Considering a given stand, the strip plate from the previous stands (flat product or a coil), is introduced in the gap between work rolls, which is smaller than the thickness of the strip, this gap is determined by the position of the screw.

The work rolls drag and press the strip into the gap to reduce its thickness. This strip has to leave and enter inthe next stand until the wished thickness is reached and coiled at the output of the last Stand. The finishing mill process in Ternium-Siderar consists of 6 rolling mill stands (F5 to F10). The complete finishing mill process is shown in Fig.1.

In this work, the mathematical model of the last Stand F10 of the rolling mill train to evaluate the proposed control technique is presented. The analytical modeling used is aimed at validating through simulations the obtained results.

The results are compared with the measured real values in the rolling mill stands controlled by the FFF technique (force feed forward; Bryant, 1973) .

To obtain an approximated rolling mill stand model, Alexander (1972) and Orowan (1944) theories were used since they are more reliable to obtain the rolling mill force, and can be used in a wide range of rolling mill conditions. The mathematical model was fit and calibrated with real values measured in the process, thus attaining a closer response to the real force value. This model is represented in a general form by Eq. (1).

(1)

and the rolling torque is expressed by:

(2)

where σ;₂ is the output strip tension for Stand F10, σ;₁ is the input strip tension for Stand F10, h₂ is the output strip thickness, h₁ is the input strip thickness, γ is the rolls gap for the Stand, μ is the friction Coefficient for the Stand, S is the Yield stress for the Stand and T is the Temperature of the strip.

B. Screw system modeling

The implemented mathematical model considers last stands process, In this case, the thickness adjustment by RBF-NN control technique is made, using the mathematical model expressed by the next equations. The output thickness of each stand is a function of the gap rolls and the stand stretching.

(3)

where h₂ is the output strip thickness of the stand, γ is the Rolls gap of the stand, Γ_s is the Rolling mill force of the stand and E(.) is the Elasticity function of the stand.

There is a slight difference between the roll gap (without mill force) and the thickness of the rolled product, caused by the elasticity effect of the mill structure.

The elasticity function can be expressed by:

(4)

The elasticity coefficients of Eq. (4) have a determined value depending on the location of the stand mill. The Table I shows the elasticity coefficients for each stand.

Table 1: Elasticity Coefficients

Table 2: Model parameters

The motor produces the following torque, which is proportional to the motor current :

(5)

where k_ms: is the motor torque constant, I_ms is the dc motor current and T_ms is the electric motor torque.

The electric counter force is proportional to the motor velocity

(6)

where θ_ms is the angular position of the screw motor, e_ms is the electromagnetic counter-force and k_es is the electrical constant of the motor.

The electric equation of DC motor is expressed by:

(7)

The mechanical equations of the DC motor, according to Newton's 2^nd Law, are

(8)

where J_ms is the mechanical inertia of the motor (constant), J_B is the inertia of the shaft and B_mr is the viscous friction of the motor.

The necessary torque to move the screw (T_p) is given by the following formula:

(9)

where F(t) is the resultant of the forces applied in the direction of the screw axis, expressed in tons; d_med is the Mean diameter of the displacement screw, given in meters (m); α is the angle of the screw's helix;μ is the friction coefficient between the displacement screw and the nut and H is the lead of the screw's helix.

Function f_s(θ₁) states whether the force F(t) is required to open or to close the roll gap (see Fig. 2).

The relation between torque and mill force is shown in Fig. 3. The translation movement of the system that involves the vertical movement of the backup rolls is ruled by the following equation.

Fig. 3: Mill force action on the screw

(10)

where M is the Mass of the backup rolls, work rolls and bearings (M₁ and M₂ in Table I), B_ca is the viscous friction coefficient between the Stands and bearings of the backup rolls; E^-1(.) is the elasticity inverse function of the stand. Γ_s can be derived from Eq. (3) and replacing in Eq. (10) and the resultant of the forces applied in the direction of the screw axis is:

(11)

The relation between the angular displacement of the shaft and the linear vertical displacement of the backup rolls is given by the following expression:

(12)

where p_p is the pitch of the screw, n_r is the Transmission ratio of the gear box.

The system responsible in charge of positioning the rolls in the desired gap is known as AGC (Automatic Gage Control) ( Dairiki et al ., 1989; and Guinzburg, 1989) .

The AGC control loop is closed by the feedback system of the gap, and the controller is a PI type (proportional integral), whose parameters are adjusted to obtain a fast under damped response of the gap (γ). (Guinzburg, 1989) shows the control system of the gap (AGC).

The gap controller is a PI controller (AGC control) and can be defined by the following equation:

(13)

where θ_u(t) is the input position of the screw system, sat is the saturation function, V_s(t) is the output voltage, K_Is is the integral gain and K_Ps is the proportional gain.

The total process, in which the proposed control should be applied, is constituted by the AGC system and by the load model of the rolling process.

The load model depends on the output thickness and at the same time the AGC system is affected by the rolling load (Γ_s).

The parameters to build the model were provided by TERNIUM SIDERAR and are shown in Table I

C. Main drive control system

In order to develop the roll speed response model, it is necessary to divide the mill into several mechanical components coupled by a flexible shaft link, and then develop the motion equations for the individual components using Newton's law. In this paper, a model of three masses (the motor inertia, the reduction gear and the roll mill) structure was initially adopted for the stand mill, whose simplified mechanical configuration is represented as:

(14)

where J_mr, J_gb, J_r are the mechanical inertia of the motor, gear box, and the flexible shaft, respectively; B_mr, B_gb, B_r are the viscous friction of the motor, gear box and the flexible shaft, respectively and K_mr, K_gb: are the elasticity of the flexible shaft link of the motor and the gear box, respectively.

The DC electric motor dynamics is the same as that represented by Eqs. (5),(6) and (7). Equations parameters are k_mr, k_er, L_r and R_r.

The motor speed controller is a PI type (proportional integral Speed control) and is determined by a similar Eq.(13) (with K_Irand K_Pr, the integral and proportional gains, respectively); but the input speed is ω_u(t).

D. Thickness Adjustment Dynamic

The whole system dynamic of the stand is represented in the state space as shown in Eq. (15) and eq. (16)

(15)

And the output vector is defined by:

(16)

where o_r(t) and o_s(t) are auxiliary variables to calculate the integral action, J_eq is the total inertia of the system (J_ms+J_B), v is the speed output of strip, r work roll radius and T₀ is the delay time (one step time)-

The torque of rolling τ(.) and the load force Γ(.) are delayed one sampling time (T₀) to avoid algebraic loops.

Expressing Eq. (15) and Eq. (16) in compact form.

(17)

where:

and

A basic problem with the thickness control is the difficulty to measure the thickness when the plate is being rolled. Instead, the thickness is estimated and the uncertainty introduced here is reflected in the stability analysis.

Some considerations on the model are:

1. The friction coefficient in the model, the yield stress tension and the stresses forward and backward from the strip are considered constants to this analysis.
2. The disturbances of the process cross through the mathematical model affect the output value of the plant.
3. The main consideration is that the control is applied when a strip exists into the stand. This means that the previous moment to the entrance of the strip in the last stand is not considered.
4. The sample time for purpose of control is T_o=0.04 sec. to design the delay time

Assumption: The rolling load is related to the strip's temperature, input and output thickness, the friction coefficient, the yield stress and stresses forward and backward from the strip. Any variation in said variables is considered a disturbance. These disturbances are bounded and affect the process outputs.

The whole process considered in this work is shown on Fig. 4.

Figure 4. Effect of water temperature on moisture content of samples in second month

III. PROCESS CONTROL

A. Error dynamics approximation

A simple control strategy, which is similar to the well-known model reference adaptive control (MRAC) scheme, is proposed in Fig. 4. This scheme is adopted in this study because it has the advantage of generating the desired input signal without requiring that the network be trained initially offline.

Moreover, the outputs of the neuro controller u_N which has fault tolerant ability through on-line learning only depend on the past and present value inputs, outputs and desired input reference of the system.

The selection of the model should not greatly exceed the dynamics of the system under control while the disturbances are considered constant, otherwise large control efforts might be required.

Assuming all the states of the system are accessible, the state error e is defined as e=x-x_d. The error dynamics for the overall system is,

(18)

Eq. (18) can be expanded in the desired reference using Taylor series.

(19)

where represents higher order terms. Substituting , by C, and neglecting all the higher order terms. The Eq. (19) can by expressed by:

(20)

Now applying the KYP (Kalman-Yacubovich-Popov) Lemma (Vidyasagar, 1993) in Eq (20).

Lemma 1: Let Z(s) = C(sI- A) ^-1B be a jxj transfer matrix such that Z (s)+ Z^T(-s) has normal rank j, where A is Hurwitz, (A,B) is stabilizable, and (C,A) is observable. Then Z(s) is Strictly Positive Real (SPR) if and only if there exist symmetric positive definite matrices P and Q such that:

(21)

(22)

In order to develop an algorithm to adjust u so that the system becomes asymptotically stable, in spite of parameter variations and modeling errors. Now considering

(23)

where u_ff is the correction signal (roll gap and speed) from the previous Stand and u_d is the set point of the Stand (constant).

Considering the correction signals and making,

(24)

Thus, the total control signal to the stand mill is the sum of the FFF adjustment, speed correction and the RBF-NN controller signals, from Eq. (24) and using Eq. (20), the error dynamics can be approximated as:

(25)

B. RBF-NN Parameterization

Setting the RBF-NN's inputs as ζ where , u_N can be approximated by a RBF-NN through on-line learning,

(26)

where w^* (m x j ) and ξ^*(m x 1) are optimal parameter vectors of weights w and radial basis functions ξ, respectively; c^* and η^* are optimal parameter vectors of centers c and widths η , respectively; and ε_n is the approximation error.

However, the optimal parameter vectors are unknown, so it is necessary to estimate the values. Define an estimative function.

(27)

where and are estimated parameter vectors of wand ξ, respectively; and and are estimated parameter vectors of c and η , respectively.

With the RBF-NN compensation, using Eq. (27) and replacing u_Nby , the control input vector u from (24) is,

(28)

Defining and , and replacing Eq. (23) and Eq. (28) in Eq. (25) the error dynamics may be written as,

(29)

where represents the learning error ψ_l and considering and . Therefore, a RBF-NN compensator is used for on-line learning of . The Eq. (29) can be expressed by

(30)

Using an approximation for the function . Ïn order to deal with , the Taylor's expansion of is taken about and .

(31)

where δ denotes the high-order arguments in a Taylor's series expansion, and and are derivatives of with respect to and at . They are expressed as¨:

(32)

Equation (31) can be expressed as

(33)

From (33) the high-order term δ is bounded by

(34)

where κ₁, κ₂, and κ₃ are some bounded constants due to the fact that RBF and its derivative are always bounded by constants (the proof is omitted here to save space).

Substitute (33) into (30), it can obtain that

(35)

where the uncertain , is reasonable to assume ε is bounded by a constant ε_H, and

(36)

C. Stability Analysis and neural parameters adjustment

To derive the stable tuning law, the following Lyapunov function is chosen,

(37)

where P is an j x j symmetric positive definite matrix, and Θ, Λ are m x m non-negative definite matrices. The derivative of the Lyapunov function is given by,

(38)

From eq. (21) b, since is Hurwitz, the Lyapunov function can always be found and has a unique solution and replacing Eq. (30) into Eq. (38),

(39)

Substituting Eq. (22) in Eq. (39)

(40)

Noting in eq. (39),

(41)

(42)

Eq. (40) becomes,

(43)

where is the i-th column of matrix and e_yi is the i-th row of vector e_y, If , and are selected as,

(44)

(45)

(46)

Then Eq. (43) becomes,

(47)

can be demonstrated negative according to Eq.(47),

(48)

Let , it can be shown directly that is negative when

Now considering, and , the tuning rules are,

(49)

(50)

(51)

(52)

The negativeness of the Lyapunov function is guaranteed, resulting in the overall system to be stable.

IV. SIMULATION RESULTS

In order to make the control technique evaluation collected real data of the rolling mill of the steel coil were used, with N^o 982 1612 laminated in the TERNIUM SIDERAR plant in Argentina.

The objective of the simulation shown in this section is to utilize the proposed control strategy to reduce variations of the parameters and modeling errors as well as decrease any possible disturbance.

In this simulation the number of RBF neurons is equal to five (m=5). The disturbance signals are shown in Fig. 7 and Fig. 8 shows the simulated output strip thickness with RBF-NN correction and the strip thickness with conventional control (FFF).

Figure 5: Input Disturbances

Figure 6: Gap and speed compensation from RBF-NN

Figure 7: Output speed of the strip for the RBF-NN compensation

Figure 8: Output thicknesses for the RBF-NN compensation and FFF.

The studies on control simulation involve parameter variations, modeling error and disturbance rejection cases (temperature and input thickness variations), where the output thickness and output speed are controlled using the RBF-NN roll gap and speed compensation as the manipulated variable.

The RBF-NN control variation error is lower than 60 μm and, in the case of conventional control (FFF), it reaches up to 100 μm.

Fig. 8 shows the output speed strip for stand F10, being the output expected reference speed equal to 5.45 m/s. The same results for different data collected in the rolling mill process were obtained and the thickness error in the case of the RBF-NN control is always smaller than the error of the control FFF, which is the actual control technique used in the plant.

Fig. 9 shows the load variation of the actual rolling load measured in the process, compared to the estimated load by the control model, in which the estimated load by the model has a similar behavior to the actual process.

Figure 9: Estimation of the load force from the rolling model and the measured load force.

The error between them could be caused by an error in the measurements, or because some of the involved variables were estimated.

V. CONCLUSIONS

The HSM model presented in this article is calibrated with real values measured in the rolling mill process and constructed with parameters provided by the firm TERNIUM SIDERAR.

In this way, a dynamic model for the last rolling mill stand is obtained with a behavior which is also similar to the real process with non-linearities that was considered.

Lyapunov's stability theory is used to derive a stable tuning rule to update all parameters in the RBF-NN, thus ensuring the local stability of the overall system.

The results show the strip thickness variation obtained with the two control techniques. The RBF-NN scheme of the controlled process evaluated by computational simulation shows a smaller thickness variation than the real system force feed forward FFF that was taken for comparison presents greater thickness dispersion than the RBF-NN compensation.

Simulation studies based on HSM model demonstrate that the proposed scheme can tolerate parameters variations, and this scheme has the advantage that it can be used in parallel, for system-performance verification under real operation conditions , allowing for a future probable complementing of the system's force feed forward (FFF) in HSM stands.

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Received: December 16, 2009
Accepted: September 27, 2010
Recommended by Subject Editor: José Guivant