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Looper and tension control in hot strip finishing mills based on different control approaches
Journal of Engineering and Applied Science volume 69, Article number: 100 (2022)
Abstract
The looper control of hot strip finishing mill is one of the most critical control items in hot strip rolling mill process. It is a highly complex nonlinear system, with strong states coupling and uncertainty that present a difficult control challenge. Loopers are placed between finishing mill stands not only to control the mass flow of the two stands but also to generate a constant specific strip tension during rolling, independent of the actual strip stock which influences the width of the strip. Some of the applicable control architectures for the finishing mill are reviewed. Two general approaches are considered: the first is based on a loop or SISO (single input single output) strategy, while the second is based on a MIMO strategy:
In the loop approach, two different schemes are considered with distinct loop configuration. For both schemes, PID and fuzzyPID are considered for comparison purposes. Since in the second scheme the most important loop is the looper control loop, it is investigated whether a nonlinear sliding mode control can significantly improve the response.
In the multivariable or MIMO approach, standard LQR (linear quadratic regulator) is investigated after linearization of the system.
Introduction
A hot rolling of steel strip is a sequential process performed to convert the semifinished slab which is reheated to a predetermined temperature and then rolled to make the specified hot strip in a coiled form. A hot slab mill primarily consists of the following units or areas (illustrated in Fig. 1): reheat furnace, roughing mill, transfer table, coil box, crop shear, finishing mill, runout table, and coiler. Slabs of around 250mm thickness are reheated to a temperature of approximately 1200°C in the reheat furnace. In the roughing mill, the reheated slabs are reduced to a thickness of 25–50 mm. The resulting transfer bar is then transported to the finishing mill, where it is further reduced to the final thickness 0.8–20 mm.
The finishing mill in hot strip mill has typically five to seven stands to reduce the transfer bar thickness down to the appropriate thickness. The speed of the rolling is set to perform the final reduction between 820 and 900 °C at the finishing temperature to achieve particular mechanical properties. The finishing mills roll the transfer bar in tandem, which mean each bar is rolled at once through all the finishing stands. The hot steel is quite fragile as it is rolled and tension between the stands of finishing mill must be tightly controlled at very low levels to prevent stretching or tearing the strip. The position of each roll is fed back to the finishing mill’s automation system which, along with information from the load cells that monitor rolling force and from the gauge measuring final strip thickness, work to smoothly adjust the roll gaps and speeds to maintain stable rolling of strip to the necessary thickness despite the temperature variations present in every bar.
For the last three decades, many advanced looper control techniques have been proposed, such as PI controller, inverse linear quadratic ILQ [1], neurofuzzy methodologies [2, 3], adaptive control [4], and H_{∞} [5, 6]. However, sufficient performance has not been achieved due to the following: the model is approximately linearized around the reference point; the controller ignores the interactions between the strip gauge, tension, and angle; or the external unknown nonlinear disturbances are not considered. Recently, the nonlinear sliding mode techniques have been proposed due to robustness performance and the ability to withstand external disturbance and parameter uncertainties [7, 8].
This paper’s objective is to illustrate and compare the performance of different control techniques in the presence of disturbances, uncertainty, and unmodeled dynamics. The paper’s goal is achieved by applying the traditional PID controller as the first control technique via two control schemes. The second controller adopted is achieved by replacing the PID controller by fuzzylike PID in both control schemes. The super twisting sliding mode control is considered the third adopted controller technique, in which the control input is the reference of the looper torque and the controlled variable is the looper angle. The last control technique proposed in this paper is the linear quadratic regulator (LQR). All the proposed controllers have been implemented individually for a complete nonlinear six stands model.
Besides this introductory section which includes a brief introduction and the objective of this research, the paper includes four sections. The process model is introduced in the “Process model” section, the proposed controller is explained in the “Control approaches” section, simulation results are illustrated in the “Simulation model” section, and conclusions are drawn in the “Conclusions” section.
Process model
Between all the stands of the finishing mill, a looper is installed for the purpose of ensuring that there is a particular strip tension between any two stands. The reason is to prevent the necking of the strip as a result of excessive tension. Too strong tension will finally lead to a strip break. This can occur when the mass flow through the following stand is higher than the flow through the previous stand as a result of a too high speed in the following stand. When the mass flow through the following stand is smaller, a loop is built up. A steadily increasing loop leads to the formation of folds. If a fold is formed, the strip may enter the next stand with triple strip thickness. The result may be a breakage of a roll or of a spindle. Thus, a loop control must be installed. For technological reasons, the strip tension should be kept constant between two stands, independent of the actual strip stock. Usually, the major mechanical and hydraulic components of each looper as shown in Fig. 2 are one hollow idler roll, approximately located midway between two consecutive stands: one pivoting arm which holds the roll and one rotating hydraulic actuator double.
Strip tension dynamics
The interstand strip tension σ(t) described here is close to [7,8,9], which is defined by Young’s modulus E of the strip and the strip stretch as the following:
where θ is the looper angel; L is the distance between two consecutive stands; L^{′}(θ) is the geometric strip length between stands i and i + 1 considering the motion of the looper; and (L + ξ(t) ) is the stored strip length, which changes as the result of mass flow difference ξ(t) between stands i and i + 1. It is worth to mention that ξ(t) is rather small compared with L. So (L + ξ(t)) is approximated to L. The equation (1) is rewritten as,
The geometric length between stands L^{′}(θ) is evaluated as follows:
where ℓ_{1}(θ), ℓ_{2}(θ) are the strip lengths depicted in Fig. 2. It can be computed according to the following geometrical formulas:
The interstand strip deviation \(\dot{\xi}(t)\) with respect to the interstand length L is described by the following equation:
where v_{i}(t) is the speed of the strip leaving the stand i , V_{i + 1}(t) is the strip speed entering the downstream stand (i + 1), and ω_{ξ}(t) represent unmodeled dynamics.
The strip speed is controlled by automatic speed regulator. It can be modelled as firstorder systems with time constant τ_{v} as the following:
The relation between the speed of the strip leaving the stand v_{i}(t) and the angular velocity ω_{i} of the work rollers is evaluated as:
where \({S}_f^i\) represents forward slip, and R^{i} represents work roller radius.
The mass flow is conserved through the stand and assuming the strip width is constant, the speed V_{i}(t) of the strip entering the ith stand can be evaluated as the following:
where H_{i}(t) and h_{i}(t) are the stand entry thickness and the exit thickness, respectively.
Finally, the derivative of interstand tension σ(t) is given as:
So, \(\dot{\sigma}(t)\) can be evaluated as:
where α and β are the angles between the strip and the pass line and given by
Looper dynamics
Looper dynamics is obtained by applying Newton’s law of motion to the looper system. Given the looper inertial momentum to hydraulic actuator J, the hydraulic actuator torque on the looper T_{u}(t), load torque on the looper T_{load}(θ), and unmodeled dynamics ω_{ω}(t), the dynamics can be expressed as:
where T_{load}(θ) represents the loads caused by strip tension T_{σ}(θ), strip weight T_{s}(θ), and looper weight T_{l}(θ). It can be evaluated as follows:
The looper actuator torque T_{u}^{i}(t) controlled by automatic torque regulator and modelled as firstorder system with time constant τ_{t} as the following:
The stand model
There are several mathematical models available in the technical literature and choice of the most appropriate is not always easy. The Sims’s model [10] is one of the most model presented in the literature as being useful for control development. The model description follows closely that in [11]. The gap of the mill stand can be estimated by starting from evaluation of the rolling force relationship as the following:
where F is the roll force, κ is the constrained yield stress of the material, Q_{p} is a geometrical factor which represents the contribution of the friction and inhomogeneities of deformation [11], (σ_{in} + σ_{out})/2 is the mean tension stress of the strip, and \(\sqrt{R^{\prime}\delta }\) is the contact arc between the strip and work rolls given in terms of the radius of the flattened roll R^{′} and the reduction of δ = (H_{i} − h_{i}).
The deformed work roll radius R^{′} is obtained using the Hitchcock equation as the following [12]:
where R represents the undeformed work roll radius, υ is Poisson’s ratio, and E is Young’s modulus.
The geometrical factor Q_{p} as expressed in Sims’s paper is a nonlinear function of the work roll radius (R), entry thickness H_{i} and exit thickness h_{i}:
and the reduction r is given by:
where Y is an approximation evaluation of the strip thickness at the neutral plane (shown in Fig. 3) and ϕ_{n} presents the angle between the centre line of the stand and the neutral plane.
The angle ϕ_{n} is approximated as:
The strip thickness h_{i} at the mill exit is evaluated using the linearized relation as:
where S_{i} is the position of the roll bite position actuator, F_{i} is the total rolling force (equal to P times W, where W is the strip width), and M_{i} is the mill modulus, which represents the amount of deformation of the mill stand under rolling force F_{i}.
The forward slip \({S}_f^i\) is a measure of the relative difference between the speed of the strip exiting the roll bite and the peripheral speed of the roll. It is defined as:
where v_{i} is the exit strip speed and V_{i} is entry strip speed [13]. In this model, the forward slip is expressed as [11]:
According to [14], the friction coefficient μ presented as empirical formula obtained for flat hot rolling mill in terms of workpiece temperature on degrees F and given as:
The hydraulic cylinder controlled by AGR are modelled as firstorder system with time constant τ_{s} as the following:
While the exit gauge h^{i} which depends on the mill modulus M, roll force F^{i}, and roll gap S^{i} [15] is given as:
The deviation of the rolling force from desired point is approximated by the following linear expression.
where H^{i} is the thickness of the strip entering the stand, \({\sigma}_b^i\) is the strip back tension, \({\sigma}_f^i\) is the strip front tension, and F_{di}is the disturbance of rolling force [16].
Control approaches
Many methods have been proposed and discussed for looper tension control. In this section, some of the applicable control architectures for the finishing mill are reviewed as shown in Fig. 4. The most common traditional PID schemes represented in the “PID controller” section. In the “Fuzzy controller” section, the control system based on fuzzyPID controller is proposed to replace the convention PID regulation in the “PID controller” section.
Nonlinear sliding mode control will be introduced in the “SMC technique” section and it is shown that in order to solve the sliding mode common problem (referred to as “chattering phenomenon”), the super twisting algorithm in implementations of sliding mode control techniques has been suggested. In the “Linear quadratic regulator” section, multivariable controller is decentralized with respect to the stands which will result in a separate LQR design for each stand/interstand zone.
PID controller
Two control loop configurations are studied in this work. The configurations are illustrated in Figs. 5 and 6. In the first control scheme, the conventional solution for regulation of thickness, angle, and tension is shown in Fig. 5. In this scheme: the looper angle deviation \(\Delta {\theta}^i(t)={\theta}_0^i{\theta}^i(t)\) is regulated by ASR based on a PID controller acting on the speed reference ω^{i, ref}(t) , the tension deviation \(\Delta {\sigma}^i(t)={\sigma}_0^i{\sigma}^i(t)\) is regulated by ATR based on PID controller that acts on the reference torque of the looper \({T}_u^{i, ref}(t)\), and the force deviations \(\Delta {F}^i(t)={F}_0^i{F}^i(t)\) is regulated by AGR based on PID regulator action on roll gap reference S^{i, ref}(t) of the capsule [9, 17, 18].
In the second control scheme (Fig. 6), the solution for regulation of angle, tension, and thickness proposed works oppositely to first control schemes. The ASR based on PID controller is used to regulate the tension deviations \(\Delta {\sigma}^i(t)={\sigma}_0^i{\sigma}^i(t)\) by trimming reference speed ω^{i, ref}(t) of upstream stand, whereas the ATR based on PID controller is used to regulate looper angle deviation \(\Delta {\theta}^i(t)={\theta}_0^i{\theta}^i(t)\) by trimming the reference torque \({T}_u^{i, ref}(t)\) of the looper arm. The AGC uses PID controller to regulate the force deviations \(\Delta {F}^i(t)={F}_0^i{F}^i(t)\) by changing the capsule position S^{i, ref}(t).
Fuzzy controller
The PID controller techniques are widely used in practice, due to the simplicity in tuning and implementation. However, the convention schemes have a problem that high gain causes overshoot and instability whereas low gain gives a low response. So fuzzyPID controller is proposed which gives dynamic gain based on the error and the rate of error change.
The twodimensional fuzzy controller as shown in Fig. 7 is the most commonly used controller form in fuzzy control. The controller has a structure that is quite similar to the PID. The fuzzy inference provides the tuning to PID parameters K_{p}, K_{i}, and K_{d} using a nonlinear mapping from the error and its derivative
The fuzzification process
Fuzzification is the process which transforms numerical form input variables into linguistic variables. In other words, it is mapping inputs to the FLC into fuzzy set membership. In this paper, the domain of language variable of input, system error e, and the rate of error ec are {−1, −0.6665, −0.3332, 0, 0.3332, 0.6665, 1}, and the fuzzy sets are {NB, NM, NS, Z, PS, PM, PB}. The elements of the subset stand for the following: negative big, negative medium, negative small, zero, positive small, positive medium, positive big. Figure 8 shows the fuzzy membership function for e and ec, where the number of membership functions and type of membership function are chosen to be such as triangular.
The domain of the language variable of the output,∆K_{p},∆K_{i},∆K_{d} are {1, 2.503, 3.952, 5.5, 7.03, 8.47, 10}, and the fuzzy sets are {out1, out2, out3, out4, out5, out6, out7}. The elements of the subset stand for output1: output7. Figure 9 shows the fuzzy membership function for∆K_{p},∆K_{i},∆K_{d}, where the number of membership functions and type of membership function is chosen to be such as triangular.
The fuzzy rule base consists of a set of consequent linguistic rules of the form ifthen statements (e.g. Table S1).
The K_{p} coefficient is to improve the regulation accuracy and speed up the time response of the system, but if K_{p} is relatively high, it results in overshoot and consequently system instability; the K_{i} coefficient is to eliminate the steadystate error of the system, but if K_{i} is relatively high, it leads to integrator windup and leads to overshoot; the K_{d} coefficient is to increase the stability of the system, reduce the overshoot, and improve the transient response, but if K_{d} is relatively high, it produces a highly oscillatory response due to noise amplification. Based on the above discussion, the relationship between deviation e, the rate of deviation ec, and the parameters K_{p}, K_{i} , K_{d} could be summarized as the following:
When the absolute value of the error is large, K_{p} should be large and K_{d} should be small to achieve a better fasttrack performance and the integration action should be limited to avoid a big overshoot.
When the absolute value of the error is in the middlesize, K_{p} should be smaller and K_{i} should be appropriate to avoid the system overshoot; in this case, the value of K_{d} term will generate a great impact on the system response.
When the absolute value of the error is small, K_{p} and K_{i} should be big to achieve a good steadystate performance, the value of K_{d} should be appropriate to avoid system settings near the oscillation, and the value of K_{d} should be appropriate. Table S1 translates the general discussion above to a set of implementable fuzzy control rules.
Defuzzification
The centre of area method is used to convert the fuzzy set to a crisp output value and the controller is obtained.
where ∆K_{p}, ∆K_{i}and∆K_{d} are scaling factors (fuzzy tuning output); K_{Pi}, K_{Ii}, and K_{Di} represent the outputs variables; μ is the membership function of fuzzy set; and n represents the number of the singlepoint set.
According to the fuzzy control model of each parameter and the membership assignment table of each fuzzy subset, the fuzzy matrix can be obtained for PID parameters as the following:
where K_{p}, K_{i}, and K_{d} are online calculated PID gains and \({K}_p^{\prime }\), \({K}_i^{\prime }\), and \({K}_d^{\prime }\) are the PID parameters of conventional PID.
FuzzyPID for looper tension control
According to the abovementioned design method and the control architecture illustrated in Fig. 10, the fuzzyPID controller is applied to the first scheme in the “FuzzyPID control using the first scheme (I)” section and the second scheme in the “FuzzyPID control using the second scheme (II)” section.
FuzzyPID control using the first scheme (I)
The tension error (44) and its derivative (45) are controlled by a fuzzyPID controller that acting on the reference torque of the looper \({T}_u^{i, ref}(t)\) (see Fig. 11).
The looper angle deviation (46) and its derivative (47) are controlled by a fuzzyPID acting on the speed reference ω^{i, ref}(t) .
Thus, force error (48) and its derivative (37) are controlled by a fuzzyPID controller that operates on the roll gap reference S^{i, ref}(t) .
According to the abovementioned design method and to control architecture illustrated in Fig. 7, the tension error and the derivative of tension (13) are controlled by a fuzzyPID controller that acts on the reference torque of the looper \({T}_u^{i, ref}(t)\). The looper angle deviation and the angle derivative (48) are controlled by a fuzzylike PID acting on the speed reference ω^{i, ref}(t) relative to the ith stand, the force error and the force error rate of change controlled by a fuzzylike PID controller that operates on the roll gap reference S^{i, ref}(t) of the hydraulic capsule
FuzzyPID control using the second scheme (II)
In this scheme, the tension error (44) and its derivative (45) are controlled by a fuzzyPID controller that acts on the reference speed of the looper ω^{i, ref}(t) . The looper angle deviation (46) and its derivative (47) are controlled by a fuzzyPID acting on the torque reference \({T}_u^{i, ref}(t)\). The force error (48) and its derivative (37) are controlled by a fuzzyPID controller that operates on the roll gap reference S^{i, ref}(t) of the hydraulic capsule relative to the ith stand where the force reference is decided according to the exit target thickness (see Fig. 12).
SMC technique
The inaccuracy between the mathematical model and the actual plant during formulation of any control problem arises from simplified representation of the system’s dynamics and/or unknown plant parameters [19]. The robust control methods which are supposed to achieve the desired performance in the presence of disturbance/unmodeled dynamics are the socalled sliding mode control technique.
Let us consider the singleinput dynamic system
where the scalar x is the output of interest, the scalar u is the control input, and \(\mathrm{x}={\left[x,\dot{x},\dots, {x}^{\left(n1\right)}\right]}^T\) is the state vector. The function f(x) is, in general, nonlinear and not precisely known but its norm is assumed to be upper bounded by a known continuous function of x; similarly, the control gain b(x) is, in general, nonlinear and partially or fully unknown but is of known sign and is bounded by known, continuous function of the x.
The control problem is to get the state x to track desired timevarying state trajectory \({\mathbf{x}}_d={\left[{x}_d\ {\dot{x}}_d\dots {x}_d^{\left(n1\right)}\right]}^T\) in the presence of modelling uncertainties in f(x) and b(x).
Let us consider \(\overset{\sim }{\mathbf{x}}=\mathbf{x}{\mathbf{x}}_d\) be the tracking error in the variable x, and let \(\overset{\sim }{\mathbf{x}}=\mathbf{x}{\mathbf{x}}_d={\left[\overset{\sim }{x}\kern0.5em \overset{\sim }{\dot{x}}\dots .\tilde{x}^{\left(n1\right)}\right]}^T\)be the tracking error vector. Now, the sliding surface S(t) can be defined as a timevarying surface in state space R^{(n)} by the scalar equation s(x; t) = 0 and λ is a strictly positive constant.
Given the initial condition x_{d}(0) = x(0), the tracking problem x = x_{d} is similar to remaining on the sliding surface s(t) for all t > 0; indeed, s = 0 represents a linear differential equation whose unique solution is \(\overset{\sim }{x}=0\). Thus, keeping the scalar s quantity at zero will simplify the tracking problem of ndimensional vector x_{d} to be firstorder problem.
Keeping the scalar s = 0 can be achieved by suitably choosing the control law u in (49) such that
where ε is a strictly positive constant. Condition (51) (called sliding condition) makes s(t) an invariant set. It is worth to mention that condition (51) forces the state trajectories to the surface s(t) and, once on the surface, the system trajectories remain on the surface itself.
The block diagram of our control scheme is shown in Fig. 13. The angle deviation \(\Delta {\theta}^i(t)={\theta}_0^i{\theta}^i(t)\) is controlled by trimming the reference torque \({T}_u^{i, ref}(t)\), tension deviations \(\Delta {\sigma}^i(t)={\sigma}_0^i{\sigma}^i(t)\) by a PID controller acting on the reference ω^{i, ref}(t) of the upstream stand, and the AGC uses a PID controller to regulate the force deviations \(\Delta {F}^i(t)={F}_0^i{F}^i(t)\) by changing the capsule position S^{i, ref}(t)
The SMC is applied to the looper angle loop as shown in Fig. 13. The procedure described here is similar to [8]. In order to apply the SMC, the equation of the looper angle need to be written in standard form as follows
where \(x={\left[{\theta}^i(t),{\dot{\theta}}^i(t)\right]}^T\). In order to apply SMC, the \({T}_{load}^i\left({\theta}^i,t\right)\) is needed to be approximated. The measured value of the hydraulic torque \({\hat{T}}_u^i(t)\) represents an approximation of f(x) since the angular acceleration of the looper is small.
The looper load torque \({T}_{load}^i\left(\theta \right)\) can be approximated by the measure of the hydraulic torque \({\hat{T}}_u^i(t)\) and looper actuator torque can be approximated by \({T}_u^{i, ref}(t)\). The upper bounds of the approximation errors are as follows:
The sliding surface is given by \(\mathrm{S}(t)={\overset{\sim }{\dot{\theta}}}^i(t)\lambda {\overset{\sim }{\theta}}^i(t)\), where
where \({\dot{\theta}}_d^i(t)\)is the target angular speed and \({\alpha}_d^i(t)\)is the angular acceleration of the looper. According to the standard SMC procedure [19], the control law can be obtained as the following
By choosing K ≥ f_{1} + f_{2} + εJ, we can now guarantee that (51) is verified. In order to eliminate the wellknown chattering, Sims [10] suggests replacing the following discontinuous
With
Also, Hongwei et al. [7] suggest a high slope saturation function instead of the signum function as a solution for chattering phenomena. The control low can be expressed by
where the saturation function sat(∙) is defined by
In this paper, we suggest to use a secondorder SMC algorithm the socalled super twisting sliding mode control.
Secondorder sliding mode control algorithms are a powerful alternative that completely solves the chattering issue without compromising the robustness properties as well. Since the control law is a continuous function of time as it is shown in Fig. 14 and can be written as
where W is a positive constant. The control low (61) can be written as
Linear quadratic regulator
The proposed multivariable controller is decentralized with respect to the stands which will result in a separate LQR design for each stand/interstand zone.
The objective is to regulate stand speed, looper torque, and roll gap for each stand/interstand to guarantee system stability in the presence of the disturbances affecting that zone. We consider a system described by the following state equation:
where A ∊ ℝ^{nxn}, B ∊ ℝ^{nxm}, C ∊ ℝ^{pxn} are state matrix, input matrix, and output matrix, respectively.
x ∊ ℝ^{n} is a vector, whose elements represent the individual state variables, and u ∊ ℝ^{m} is a vector whose elements represent the individual control variables. y ∊ ℝ^{m} is a vector whose elements represent the individual output variables.
The optimal control problem is defined in terms of minimizing the performance index:
where Q = Q^{T} is positive (semi) definite matrix. The problem solution is obtained by solving Riccati equation:
where P ∊ ℝ^{n} is the solution of algebraic Riccati Equation which results in the control law.
The feedback gain K is called the LQR (linear quadratic regulator) gain or optimal regulator gain and can be written as
For the controller design, the nonlinear model was linearized about the operating values and for the simplicity Q(x) = I, R(x) = ρI,where, ρ > 0. The variables represented by the elements of the state, control, and output vectors are shown in Table 1 for Stand#1:5 and Table 2 for Stand#6.
Simulation model
The dynamic simulations using MATLAB/Simulink were carried out to evaluate and compare the controllers’ performance. Each controller is simulated individually to illustrate its performance. The simulation runs for 20 s where the disturbance (71) presented after 2 s from simulation running time. Here, for the sake of completeness, the parameters were obtained from a real plant for the purpose of making the simulation models behave as exactly as possible.
The values of the parameters used in the dynamic model of the looper and tension system are reported in the “Operation values” section. The comparisons, based on the mean value, standard deviation, and the error in each stand, are presented in the “PID controller parameters” section.
Looper characteristics
The simulation result based on the looper mechanism depicts in Fig. 2 with geometry data listed in Table 3.
Operation values
The dynamic simulations using MATLAB/Simulink are performed with the operation values shown in Table 4.
PID controller parameters
One of the tested controllers is PID controller. The PID controller’s parameters, listed in Tables S2 and Table S3, are obtained using automatic PID tuner in MATLAB/Simulink software for scheme I and scheme II respectively.
Simulation scenario
After 2 s from simulation running time, the unmodeled dynamics [8] mentioned in (13) presented as a summation of two sin functions with phase shift multiplied by the percentage of the stand entry speed as the following,
In both control schemes proposed, the simulation is carried out for 20 s with disturbance presence after 2 s from the running time. The mean value, summation of error square of the states, and standard deviation are calculated for comparison purposes. The data are shown in Table S4, Table S5, and Table S6 respectively. The simulation results of the stand exit thickness, roll gap, the tension between stands, looper angle, and looper torque are shown in Figures S1, S2, S3, S4, S5, S6, S7, S8, S9, S10, S11, S12, S13, S14, S15, S16, S17, S18.
In light of the figures, one can notice that the first control scheme’s response to the disturbance presented at t ≥ 2 s from simulation running time is better than the second control scheme in terms of strip thickness and gap position overshoot. Moreover, in both control schemes, the overshoot decreased by replacing traditional PID by fuzzylike PID. The SMC had by far the lowest overshoot percentage followed by LQR.
The first control scheme’s results show the strip exit thickness and gap position tend to be close to the mean value compared to the results from the second control scheme, while the second control scheme’s results of looper position and looper torque are close to the mean value compared to the first control scheme. The standard deviation values become closer to the mean value when traditional PID is replaced by FuzzyPID. The standard deviation values show that the overall SMC results are better than LQR results. It is also noticeable the significant difference between the error summation results of SMC and LQR on the one hand and traditional control and Fuzzy on the other hand. However, The SMC has presented better results than LQR.
Conclusions
In this paper, two control schemes have been studied with different control techniques. It was shown that the Automatic Gage Regulator compensates the disturbances much better than the second control scheme, while the second control scheme compensates the disturbances in looper position better than first control scheme. Although, in both control schemes, the performance improved when PID was replaced by fuzzylike PID.
A multivariable controller is proposed; the benefit of using LQR controller reduces the interaction and provides stability in the presence of disturbance and unmodeled dynamics better than traditional PID even after being replaced by fuzzylike PID. However, the effectiveness of looper and tension control approach based on the super twisting sliding mode algorithm, thanks to its robustness and structure simplicity, makes the interaction between the looper angle and tension less serious than usual and achieved much better compensation for the disturbances.
Availability of data and materials
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 T _{u}(t):

Actuator torque on the looper
 ϕ _{ n } :

Angle at neutral plane
 AGR:

Automatic gap regulator
 ASR:

Automatic speed regulator
 ATR:

Automatic torque regulator
 κ :

Constrained yield stress
 B :

Control matrix
 u(t):

Control signal
 R(x):

Control weighting matrix
 R ^{′} :

Deformed work roll radius
 K _{ d } :

Derivative gain
 ξ(t):

Deviation of interstand strip length with respect to L
 ℓ _{1}(θ):

Distance between stand i and looper pivot
 ℓ _{2}(θ):

Distance between looper pivot and stand i + 1
 a :

Distance between the pivot point and upstream stand
 β :

Downstream strip angle
 e(t):

Error
 \({S}_f^i\) :

Forward slip
 μ:

Friction coefficient
 L ^{′}(θ):

Geometric looper length between stands
 Q _{ p } :

Geometrical factor
 g :

Gravitational constant
 K _{ i } :

Integral gain
 L :

Interstand length
 ILQ:

Inverse linear quadratic
 LQR:

Linear quadratic regulator
 T _{load}(θ):

Load torque on the looper
 θ :

Looper angle
 \({\alpha}_d^i(t)\) :

Looper angular acceleration
 l :

Looper arm length
 M _{ r } :

Looper arm weight
 \({\dot{\theta}}_d^i(t)\) :

Looper desired angular speed
 J :

Looper inertial momentum to the hydraulic actuator
 LTCB:

Looper torque calculation block
 r :

Looper roll diameter
 M _{ a } :

Looper roller weight
 T _{L}(θ):

Looper weight torque
 d :

Pivot point distance from the pass line
 υ :

Poisson’s ratio
 S :

Position of the roll
 P _{ v } :

Process variable
 K _{ p } :

Proportional gain
 S _{ p } :

Setpoint
 SMC:

Sliding mode control
 S(t):

Sliding surface
 K(x):

Solution to Riccati equation
 ω ^{i}(t):

Stand speed in rad/s
 Q(x):

State weighting matrix in LQR
 A(x):

Statedependent matrix
 C(x):

Statedependent output matrix
 a(x):

Statedependent vector
 ρ :

Steel density
 V(t)_{i} :

Strip entry speed to the ith stand
 v _{i}(t):

Strip exit speed from the ith stand
 H ^{i} :

Strip entry thickness
 h ^{i} :

Strip exit thickness
 T _{σ}(θ):

Strip tension torque
 Q(x):

State weighting matrix in LQR
References
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The authors would also like to thank ALLAH (Almighty) who strengthens us and supports us during this work.
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AG developed the theoretical formalism, performed the analytic calculations, and performed the numerical simulations. Both AG and ME contributed to the final version of the manuscript. Both ME and HAF supervised the project. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.
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Gaber, A., Elnaggar, M. & Fattah, H.A. Looper and tension control in hot strip finishing mills based on different control approaches. J. Eng. Appl. Sci. 69, 100 (2022). https://doi.org/10.1186/s4414702200145w
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DOI: https://doi.org/10.1186/s4414702200145w