Looper and tension control in hot strip finishing mills based on different control approaches

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 fuzzy-PID 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 multi-variable 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 semi-finished 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 250-mm 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.

Fig. 1

The schematic diagram of a traditional hot strip mill

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], neuro-fuzzy 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 fuzzy-like 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.

Fig. 2

Looper geometry [9]

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:

$$\sigma (t)=E\left[\frac{L^{\prime}\left(\theta \right)-\left(L+\xi (t)\right)}{L+\xi (t)}\right]$$

(1)

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,

$$\sigma (t)=E\left[\frac{L^{\prime}\left(\theta \right)-\left(L+\xi (t)\right)}{L}\right]$$

(2)

The geometric length between stands L^′(θ) is evaluated as follows:

$${L}^{\prime}\left(\theta \right)={\ell}_1\left(\theta \right)+{\ell}_2\left(\theta \right)$$

(3)

where ℓ₁(θ), ℓ₂(θ) are the strip lengths depicted in Fig. 2. It can be computed according to the following geometrical formulas:

$${\ell}_1\left(\theta \right)=\sqrt{{\left(a+l\ cos\theta \right)}^2+{\left(l\ sin\theta +\left(r-d\right)\right)}^2}$$

(4)

$${\ell}_2\left(\theta \right)=\sqrt{{\left(\left(L-a\right)-l\ cos\theta \right)}^2+{\left(l\ sin\theta +\left(r-d\right)\right)}^2}$$

(5)

The interstand strip deviation $\dot{\xi}(t)$ with respect to the interstand length L is described by the following equation:

$$\dot{\xi}(t)={v}_i(t)-{V}_{i+1}(t)+{\omega}_{\xi }(t)$$

(6)

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 first-order systems with time constant τ_v as the following:

$$\dot{v_i}\left(\mathrm{t}\right)=-\frac{1}{\tau_v}{v}_i\left(\mathrm{t}\right)+\frac{1}{\tau_v}{v}^{ref}\left(\mathrm{t}\right),\kern0.5em {v}_i(0)={v}_0^i$$

(7)

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:

$${v}_i(t)=\left(1+{S}_f^i\right){\omega}_i(t){R}^i$$

(8)

where ${S}_f^i$ represents forward slip, and Rⁱ 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:

$${V}_i(t)=\frac{v_i(t){h}_i(t)}{H_i(t)}$$

(9)

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:

$$\dot{\sigma}(t)=\frac{E}{L}\left[\frac{d}{dt}{L}^{\prime}\left(\theta \right)-\dot{\xi}(t)\right]$$

(10)

$$\frac{d}{d t}{L}^{\prime}\left(\theta \right)=\frac{d{L}^{\prime }}{d\theta}\frac{d\theta}{d t}$$

(11)

$$\frac{d{L}^{\prime }}{d\theta}=\frac{d{\ell}_1}{d\theta}+\frac{d{\ell}_2}{d\theta}$$

(12)

So, $\dot{\sigma}(t)$ can be evaluated as:

$$\dot{\sigma}(t)=\frac{E}{L}\left[l\left[\mathit{\sin}\left(\theta +\beta \right)-\mathit{\sin}\left(\theta -\alpha \right)\right]\dot{\theta}(t)-\left(\ {v}_i(t)-V{(t)}_{i+1}+{\omega}_{\xi }(t)\right)\right]$$

(13)

where α and β are the angles between the strip and the pass line and given by

$$\alpha ={\mathit{\tan}}^{-1}\left[\frac{l\ sin\theta +\left(r-d\right)}{a+l\ cos\theta}\right]$$

(14)

$$\beta ={\mathit{\tan}}^{-1}\left[\frac{l\ sin\theta +\left(r-d\right)}{\left(L-a\right)-l\ cos\theta}\right]$$

(15)

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:

$$J\ddot{\theta}(t)={T}_u(t)-{T}_{load}\left(\theta \right)+{\omega}_{\omega }(t)$$

(16)

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:

$${T}_{\sigma}\left(\theta \right)=\sigma hWl\left[\mathit{\sin}\left(\theta +\beta \right)-\mathit{\sin}\left(\theta -\alpha \right)\right]$$

(17)

$${T}_s\left(\theta \right)=\rho g{h}^iW\left({l}_1^i\left(\theta \right)+{l}_2^i\left(\theta \right)\right)l\ cos\theta$$

(18)

$${T}_L\left(\theta \right)= glcos\theta \left(\frac{M_a}{2}+{M}_r\right)$$

(19)

The looper actuator torque T_uⁱ(t) controlled by automatic torque regulator and modelled as first-order system with time constant τ_t as the following:

$${\dot{T}}_u^i\left(\mathrm{t}\right)=-\frac{1}{\tau_t}{T_u}^i\left(\mathrm{t}\right)+\frac{1}{\tau_t}{T_u}^{ref}\left(\mathrm{t}\right),{T_u}^i(0)={T}_0^i$$

(20)

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:

$$F=\left(\kappa {Q}_p-\frac{\sigma_{in}+{\sigma}_{out}}{2}\right)W\sqrt{R^{\prime}\delta }$$

(21)

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]:

$${R}^{\prime }=R\Big(1+\frac{16\left(1-{\upsilon}^2\right)P}{\pi E\Delta h}$$

(22)

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 non-linear function of the work roll radius (R), entry thickness H_i and exit thickness h_i:

$${Q}_p=\frac{\pi }{2}a\ {\mathit{\tan}}^{-1}\sqrt{\frac{r}{1-r}}-\frac{\pi }{4}- ab\ \ln \left(\ \frac{Y}{h_i}\ \right)+\frac{1}{2}\ ab\ \ln \left(\ \frac{1}{1-r}\ \right)$$

(23)

$$a=\sqrt{\frac{1-r}{r}}$$

(24)

$$b=\sqrt{\frac{R^{\prime }}{h_i}}$$

(25)

and the reduction r is given by:

$$r=\frac{H_i-{h}_i}{H_i}$$

(26)

$$Y={h}_i+{R}^{\prime }{\phi}_n^2$$

(27)

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.

Fig. 3

Roll bite area [13]

The angle ϕ_n is approximated as:

$${\phi}_n\cong \frac{1}{b}\left(\tan \left(\frac{\pi }{8}\frac{\mathit{\ln}\left(1-r\right)}{b}+\frac{1}{2}{\tan}^{-1}\sqrt{\frac{r}{1-r}}\right)\right)$$

(28)

The strip thickness h_i at the mill exit is evaluated using the linearized relation as:

$${h}_i={S}_i+\frac{F_i}{M_i}$$

(29)

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:

$${S}_f^i=\frac{V_i(t)-{v}_i(t)}{V_i(t)}$$

(30)

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]:

$${S}_f^i=\frac{R^{\prime }}{h_i}{\left({\beta}_n\right)}^2$$

(31)

$${\beta}_n=\frac{\phi_1}{2}-\frac{\delta \kappa +{\sigma}_{in}{H}_i+{\sigma}_{out}{h}_i}{4\kappa {R}_p\mu }$$

(32)

$${\phi}_1=\sqrt{\frac{\delta }{R_p}}$$

(33)

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:

$$\mu =2.7x{10}^{-4}\ {T}_F-0.08$$

(34)

The hydraulic cylinder controlled by AGR are modelled as first-order system with time constant τ_s as the following:

$${\dot{S}}^i\left(\mathrm{t}\right)=-\frac{1}{\ {\tau}_s}{S}^i\left(\mathrm{t}\right)+\frac{1}{\ {\tau}_s}{S}^{ref}\left(\mathrm{t}\right),{S}^i(0)={S}_0^i$$

(35)

While the exit gauge hⁱ which depends on the mill modulus M, roll force Fⁱ, and roll gap Sⁱ [15] is given as:

$$\Delta {h}^i(t)=\Delta {S}^i(t)+\frac{\Delta {\mathrm{F}}^i(t)}{M^i}$$

(36)

The deviation of the rolling force from desired point is approximated by the following linear expression.

$$\Delta {F}_i(t)=\left(\frac{\partial {F}_i}{\partial {H}_i}\right)\Delta {H}_i+\left(\frac{\partial {F}_i}{\partial {h}_i}\right)\Delta {\mathrm{h}}_i+\left(\frac{\partial {F}_i}{\partial {\sigma}_{i-1}}\right)\Delta {\sigma}_{i-1}+\left(\frac{\partial {F}_i}{\partial {\sigma}_i}\right)\Delta {\sigma}_i+\Delta {F}_{di}$$

(37)

where Hⁱ 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_diis 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 fuzzy-PID controller is proposed to replace the convention PID regulation in the “PID controller” section.

Fig. 4

Control architectures for finishing mill

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].

Fig. 5

Conventional PID control scheme (I) [9]

Fig. 6

Conventional PID control scheme (II) [9]

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 fuzzy-PID controller is proposed which gives dynamic gain based on the error and the rate of error change.

The two-dimensional 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

Fig. 7

Structure of fuzzy-like PID controller

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.

Fig. 8

The membership function of e, ec

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.

Fig. 9

The membership function of the outputs

The fuzzy rule base consists of a set of consequent linguistic rules of the form if-then 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 steady-state error of the system, but if K_i is relatively high, it leads to integrator wind-up 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 fast-track performance and the integration action should be limited to avoid a big overshoot.

When the absolute value of the error is in the middle-size, 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 steady-state 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.

$$\Delta {K}_p={f}_1\left(e, ec\right)=\frac{\sum_{i=1}^n{\mu}_i\left(e, ec\right){K}_{Pi}}{\sum_{i=1}^n{\mu}_i\left(e, ec\right)}$$

(38)

$$\Delta {K}_i={f}_2\left(e, ec\right)=\frac{\sum_{i=1}^n{\mu}_i\left(e, ec\right){K}_{Ii}}{\sum_{i=1}^n{\mu}_i\left(e, ec\right)}$$

(39)

$$\Delta {K}_d={f}_3\left(e, ec\right)=\frac{\sum_{i=1}^n{\mu}_i\left(e, ec\right){K}_{Di}}{\sum_{i=1}^n{\mu}_i\left(e, ec\right)}$$

(40)

where ∆K_p, ∆K_iand∆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 single-point 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:

$${K}_p=\Delta {K}_p\ast {K}_p^{\prime }$$

(41)

$${K}_i=\Delta {K}_i\ast {K}_i^{\prime }$$

(42)

$${K}_d=\Delta {K}_d\ast {K}_d^{\prime }$$

(43)

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.

Fuzzy-PID for looper tension control

According to the above-mentioned design method and the control architecture illustrated in Fig. 10, the fuzzy-PID controller is applied to the first scheme in the “Fuzzy-PID control using the first scheme (I)” section and the second scheme in the “Fuzzy-PID control using the second scheme (II)” section.

Fig. 10

The simulation block diagram of the fuzzy-PID controller

Fuzzy-PID control using the first scheme (I)

The tension error (44) and its derivative (45) are controlled by a fuzzy-PID controller that acting on the reference torque of the looper ${T}_u^{i, ref}(t)$ (see Fig. 11).

$$\Delta {\sigma}^i(t)={\sigma}_0^i-{\sigma}^i(t)$$

(44)

$$\dot{\sigma}(t)=\frac{E}{L}\left[\frac{d}{dt}{L}^{\prime}\left(\theta \right)-\dot{\xi}(t)\right]$$

(45)

Fig. 11

Fuzzy-PID control scheme (I)

The looper angle deviation (46) and its derivative (47) are controlled by a fuzzy-PID acting on the speed reference ω^{i, ref}(t) .

$$\Delta {\theta}^i(t)={\theta}_0^i-{\theta}^i(t)$$

(46)

$$\dot{\theta}\left(t\ \right)=\int \frac{1}{\mathrm{J}}\left[{T}_u(t)-{T}_{load}(t)\right]$$

(47)

Thus, force error (48) and its derivative (37) are controlled by a fuzzy-PID controller that operates on the roll gap reference S^{i, ref}(t) .

According to the above-mentioned design method and to control architecture illustrated in Fig. 7, the tension error and the derivative of tension (13) are controlled by a fuzzy-PID 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 fuzzy-like 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 fuzzy-like PID controller that operates on the roll gap reference S^{i, ref}(t) of the hydraulic capsule

$$\Delta {F}^i(t)={F}_0^i-{F}^i(t)$$

(48)

Fuzzy-PID control using the second scheme (II)

In this scheme, the tension error (44) and its derivative (45) are controlled by a fuzzy-PID 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 fuzzy-PID acting on the torque reference ${T}_u^{i, ref}(t)$. The force error (48) and its derivative (37) are controlled by a fuzzy-PID 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).

Fig. 12

Fuzzy-PID control scheme (II)

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 so-called sliding mode control technique.

Let us consider the single-input dynamic system

$${x}^{(n)}=f\left(\mathbf{x}\right)+b\left(\mathbf{x}\right)u$$

(49)

where the scalar x is the output of interest, the scalar u is the control input, and $\mathrm{x}={\left[x,\dot{x},\dots, {x}^{\left(n-1\right)}\right]}^T$ 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 time-varying state trajectory ${\mathbf{x}}_d={\left[{x}_d\ {\dot{x}}_d\dots {x}_d^{\left(n-1\right)}\right]}^T$ 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 $\overset{\sim }{\mathbf{x}}=\mathbf{x}-{\mathbf{x}}_d={\left[\overset{\sim }{x}\kern0.5em \overset{\sim }{\dot{x}}\dots .\tilde{x}^{\left(n-1\right)}\right]}^T$be the tracking error vector. Now, the sliding surface S(t) can be defined as a time-varying surface in state space R⁽ⁿ⁾ by the scalar equation s(x; t) = 0 and λ is a strictly positive constant.

$$s\left(\mathbf{x};t\right)={\left(\frac{d}{dt}+\lambda \right)}^{n-1}\overset{\sim }{x}$$

(50)

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 n-dimensional vector x_d to be first-order problem.

Keeping the scalar s = 0 can be achieved by suitably choosing the control law u in (49) such that

$$\frac{1}{2}\frac{d}{dt}{s}^2\le -\varepsilon \left|s\right|$$

(51)

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)

Fig. 13

SMC control structure

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

$$f(x)=-\frac{1}{J}\left({T}_{load}^i\left({\theta}^i,t\right)\right)$$

(52)

$$b(x)=\frac{1}{J}$$

(53)

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:

$$\left|{T}_{load}^i\left(\theta \right)-{\hat{T}}_u^i(t)\right|\le {f}_1$$

(54)

$$\left|{T}_u^{i, ref}(t)-{T}_u^i(t)\right|\le {f}_2$$

(55)

The sliding surface is given by $\mathrm{S}(t)={\overset{\sim }{\dot{\theta}}}^i(t)-\lambda {\overset{\sim }{\theta}}^i(t)$, where

$${\overset{\sim }{\theta}}^i(t)={\theta}^i(t)-{\theta}_d^i(t)$$

(56)

$${\overset{\sim }{\dot{\theta}}}^i(t)={\dot{\theta}}^i(t)-{\dot{\theta}}_d^i(t),$$

(57)

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

$${T}_{u}^{i, ref}(t)=\Big({\hat{T}}_{u}^i(t)+J\left({\alpha}_{d}^i(t)-\lambda {\overset{\sim }{\ddot{\theta}}}^i(t)\right)-K\ \mathit{\operatorname{sign}}\left[{S}^i(t)\right]$$

(58)

By choosing K ≥ f₁ + f₂ + εJ, we can now guarantee that (51) is verified. In order to eliminate the well-known chattering, Sims [10] suggests replacing the following discontinuous

$$u=-K\ \mathit{\operatorname{sign}}\left[{S}^i(t)\right]$$

(59)

With

$$u=-K{\left(\frac{S^i(t)}{\varnothing}\right)}^2{signS}^i(t)\kern3.25em \mathrm{if}\ \left|{S}^i(t)\right|<\varnothing$$

(60)

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

$${T}_u^{i, ref}(t)=\left({\hat{T}}_u^i(t)+J\right({\alpha}_d^i(t)-\lambda {\overset{\sim }{\ddot{\theta}}}^i(t)-K\ sat\left[{S}^i(t)/\epsilon \right]$$

(61)

where the saturation function sat(∙) is defined by

$$sat(x)=\left\{\begin{array}{c}x,\kern6.75em \left|x\right|\le 1\\ {}\mathit{\operatorname{sign}}(x),\kern3.5em \left|x\right|>1\end{array}\right.$$

In this paper, we suggest to use a second-order SMC algorithm the so-called super twisting sliding mode control.

Second-order 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

$$u=-\mathrm{K}\sqrt{\left|s\right|}\ \mathit{\operatorname{sgn}}(s)+w$$

(62)

$$\dot{w}=-W\ \mathit{\operatorname{sgn}}(s)$$

(63)

Fig. 14

Super twisting algorithm

where W is a positive constant. The control low (61) can be written as

$${T}_u^{i, ref}(t)=\Big({\hat{T}}_u^i(t)+J\left({\alpha}_d^i(t)-\lambda {\overset{\sim }{\ddot{\theta}}}^i(t)\right)-\mathrm{K}\sqrt{\left|s\right|}\ \mathit{\operatorname{sgn}}(s)+w$$

(64)

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:

$$\dot{x}= Ax+ Bu,\kern1.25em t\ge 0\kern3.5em x(0)={x}_0$$

(65)

$$y=\mathrm{Cx}$$

(66)

where A ∊ ℝ^nxn, B ∊ ℝ^nxm, C ∊ ℝ^pxn are state matrix, input matrix, and output matrix, respectively.

x ∊ ℝⁿ 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:

$$J=\frac{1}{2}\int_0^T\left({x}^T Qx+{u}^T Ru\right) dt$$

(67)

where Q = Q^T is positive (semi-) definite matrix. The problem solution is obtained by solving Riccati equation:

$${A}^TP+ PA-\mathrm{P}B{R}^{-1}{B}^TP+Q=0$$

(68)

where P ∊ ℝⁿ is the solution of algebraic Riccati Equation which results in the control law.

$$u(t)= Kx(t)$$

(69)

The feedback gain K is called the LQR (linear quadratic regulator) gain or optimal regulator gain and can be written as

$$K=-{R}^{-1}{B}^TP$$

(70)

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.

Table 1 Stand#i state, control, and output vectors

Table 2 Stand#6 state, control, and output vectors

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.

Table 3 The looper geometry main data

Operation values

The dynamic simulations using MATLAB/Simulink are performed with the operation values shown in Table 4.

Table 4 Operation values

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,

$${\omega}_{\xi }(t)=0.05\ {V}^i+0.01\left(\sin \left(12t+8\right)+\sin (6t)\right)$$

(71)

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 fuzzy-like 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 Fuzzy-PID. 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 fuzzy-like 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 fuzzy-like 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.