Mold Oscillation Feedforward Control Algorithm for Sinusoidal Oscillation of Various Asymmetries

Abstract

A feedforward control algorithm for the mold oscillators of a continuous casting process is proposed for both sinusoidal and non-sinusoidal oscillations. As steel industries grow, processes allowing for faster continuous casting are required in order to keep up with production demands. To achieve a quicker, continuous casting process, while maintaining slab quality, requires mold oscillations that make use of improved powder consumption rate and reduced friction force. Recently, non-sinusoidal oscillations have been researched owing to their advantages in regards to lubrication and friction. There have been several mechanical approaches but no algorithmic approaches. In this paper, a feedforward control algorithm consisting of embedded input shape and feedforward velocity control algorithms is proposed. Although conventional feedback controls have been commercialized for technical applications, the input shape control algorithm can prevent the non-negligible phase delays or amplitude reduction problems caused by a slow response of systems. In addition, the proposed feedforward velocity control algorithm is derived from the relationship between the valve input and mold velocity, and it allows the mold to correctly follow the non-sinusoidal oscillation. To determine the feasibility of the proposed algorithm, a Matlab Simulink model is derived from the mathematical model. Additionally, 1/100 scale specimen of a typical industry mold oscillator is used. The tests on the specimen showed that the proposed feedforward control algorithm could achieve precise control for both sinusoidal and non-sinusoidal oscillations.

1. Introduction

In steel making industries, a continuous casting process is a widely used operation for transforming molten steel into slab form. During this process, the mold oscillates with a particular pattern, and it is the pattern that allows for the transformation. The upward and downward movements of the mold not only reduce the forces that cause the molten steel to adhere to the inner wall of the mold but also promote the dispersion of the mold powder.¹⁾ This oscillation is, therefore, a key element in the continuous casting process as it has a positive effect on the lubrication and friction aspects of the mold flux.²⁾

The powder consumption rate and friction force of mold flux have an influence on the slab quality and the possibility of breakout.³⁾ Consequently, research has been done to analyze the relationship between the oscillation parameters and the mold lubrication or friction.^{1,2,3,4,5,6,7)} The oscillation parameters can be divided into three particular parameters: frequency, stroke, and modification ratio.⁵⁾ In the steel industry factories, sinusoidal mold oscillation has been used.

Recently, there has been greater demands for a fast-speed continuous casting in response to increases in production.³⁾ One obstacle to overcome while maintaining slab quality is the reduction in powder consumption rate and the increase in friction forces.⁵⁾ The typical sinusoidal oscillation currently in use cannot produce a satisfactory powder consumption rate and friction force. As a result, an asymmetric sinusoidal mold oscillation, which has an additional degree of freedom, has been researched for fast-speed continuous casting processes.^3,8,9,10) As commonly done in many papers, we henceforth refer to asymmetric sinusoidal waves as non-sinusoidal waves for the remainder of this document. In the non-sinusoidal mold oscillation process, the mold moves upward slowly and downward quickly as compared to the sinusoidal oscillation process of the same frequency. Such oscillation changes improve the powder consumption rate and decrease the friction force due to a decrease in negative strip ratio and an increase in positive strip time.^3,9) Previous studies applying the non-sinusoidal mold oscillation have focused on the mechanical approaches.^8,9,10) Although these works have delivered meaningful results, they have problems relating to cost requirements and restricted applicability due to their requirement on facilities.

In this paper, an algorithmic approach for handling both sinusoidal and non-sinusoidal oscillation is proposed. The algorithm is of a combined structure employing conventional feedback control and feedforward control algorithm. Although feedback control algorithms have been widely used for sinusoidal oscillation, there can be a non-negligible phase delay or an amplitude reduction problem when the system response is slow. To resolve this issue, an input shape control algorithm is proposed that is based on the feedback information returned from the mold position. The reshaped imaginary reference input makes the mold properly follow the original reference input. Furthermore, a velocity feedforward control algorithm is proposed for the non-sinusoidal mold oscillation. Because the mold velocity is closely connected to the valve input, the proposed velocity feedforward control algorithm with appropriate gain is effective for the non-sinusoidal oscillation. The proposed control scheme was simulated on Matlab Simulink model as well as on a 1/100 scale model of an actual mold oscillator. The test results showed that the mold adequately follows the original reference input for both the sinusoidal and non-sinusoidal oscillations of various input parameters.

2. Model Description

In this paper, a mold oscillator based on a hydraulic servo system with dual cylinders is targeted. The basic configuration of a mold oscillator is shown in Fig. 1. The model dynamics for a single hydraulic actuator consist of the valve, pressure, and cylinder dynamics. Of course, there are additional components such as, for examples, pipelines that connect the valves, actuators, and power supply system. The pipeline dynamics are not introduced in this section because it has a negligible effect under a low-frequency operation.¹¹⁾

Fig. 1.

System configuration for mold oscillator with dual cylinder.

The dynamics of the servo valve can usually be approximated by a second-order model,¹¹⁾ such as   

$$\frac{1}{{\omega }_v^2}{\ddot{x}}_v+\frac{2D_v}{{\omega }_v}{\dot{x}}_v+x_v+f_{hs}\text{sign}\left({\dot{x}}_v\right)=u_v,$$(1)

where x_v is the normalized valve position, D_v is the damping coefficient, ω_v is the natural radian frequency, f_hs is the constant related to valve hysteresis and response sensitivity, sign(·) is a sign function, and u_v is the input signal applied to the valve actuator. As x_v varies according to the applied valve input, the oil flows into and out of the chambers of the cylinder through the valve orifices. Because the target hydraulic actuator is a cylinder with symmetric rods (resulting in both chambers have same pressure area), the flow can be described using a load pressure, which is the pressure drop across the load. Therefore, the load pressure dynamic¹¹⁾ is   

$${\dot{p}}_L=\frac{E}{V}\left(Q_L-2A_p{\dot{x}}_p\right),$$(2)

where p_L is the load pressure, E is the effective bulk modulus of the oil, V is the volume of each chamber, Q_L is the load flow, and A_p and x_p are the rod area and displacement of the cylinder, respectively. Q_L is defined as   

$$Q_L=c_v{x}_v\sqrt{2}\sqrt{p_S-p_T-sign\left(x_v\right)p_L},$$(3)

where c_v is a flow constant which represents the flow efficiency of the valve, and p_S and p_T are supply and tank pressures, respectively. The pressure-flow equation is derived under the assumption that the valves have identical orifices with the equivalent critical center. The cylinder dynamic ${\ddot{x}}_p$, which is based on the load pressure, is therefore defined as   

$${\ddot{x}}_p=\frac{1}{m_t}\left(p_L{A}_p-F_f\left({\dot{x}}_p\right)-F_{ext}\right),$$(4)

where m_t is the total mass of the oscillator, F_f (${\dot{x}}_p$) represents friction force, and F_ext is an external force applied by an external load.

3. Feedforward Control Algorithms

In this chapter, the feedforward control algorithms for mold oscillation are proposed. The feedforward control algorithm is composed of two separate algorithms; an input shape control algorithm and a velocity feedforward control algorithm. The overall control is a combined structure of a conventional feedback and proposed feedforward control algorithms as shown in Fig. 2. The reference input is reshaped to prevent phase delays and amplitude reductions according to the feedback information returned from the current position of the mold. Additionally, for the fast-speed continuous casting processes, non-sinusoidal mold oscillations are known to be more effective than typical sinusoidal oscillations.³⁾ The mold velocity feedforward control presented in this paper is a more effective control algorithm for non-sinusoidal mold oscillations.

Fig. 2.

Overall diagram for mold oscillator control algorithm.

3.1. Input Shape Control Algorithm

The sinusoidal wave, which has generally been applied to the reference input u(t) in steel industries, is determined by three variables; radial frequency w, phase ϕ(t) and amplitude A(t) such that y_r(t)=A(t)sin(ωt+ϕ(t)). As a solution to the phase delay or amplitude reduction problems of the mold position x(t), the y_r(t) is reshaped into $y_r^{\prime }\left(t\right)$ such that $y_r^{\prime }\left(t\right)$=A′(t)sin(ωt+ϕ′(t)). In other words, the input shape control algorithm is split into the phase and amplitude adjustment algorithms.

First, the idea behind the phase adjustment is to shift the reference input so that it is in the reverse direction of the phase lead-lag relationship. For example, if x(t) is delayed relative to the reference input, ϕ′(t) is increased from ϕ(t) until the phase of x(t) matches that of y_r(t). To achieve this, the following index is introduced:   

$$I_p\left(t\right)=\text{sign}\left(v_r\left(t\right)\right)\cdot \left(y_r\left(t\right)-x^{*}\left(t\right)\right),$$(5)

where v_r(t) is the velocity of the reference input and x^*(t) is the normalized mold position, i.e., $x^{*}\left(t\right)=\frac{A\left(t\right)}{A_x\left(t\right)}x\left(t\right)$. Here, A_x(t) is the amplitude of mold position measured from the previous period. When the mold position is delayed relative to the reference input, I_p(t) is positive for the intervals in which two waves are propagating in the same direction (Fig. 3). If the mold position becomes led due to an over-estimated ϕ(t), I_p(t) becomes negative. Therefore, the phase adjustment of the $y_r^{\prime }\left(t\right)$ is found to be   

$${\varphi }^{\prime }\left(t\right)=\varphi \left(t\right)+k_{I,ph}{\int }_0^t{I}_p\left(\tau \right)d\tau ,$$(6)

where k_I,ph is a gain constant used to obtain the phase of u′(t). From (6), the phase is controlled until the tracking error becomes zero. Note that (5) and (6) are calculated only when the two waves are propagating in the same direction with equivalent amplitudes. This is why x^*(t) is used rather than the x(t) in (5).

Fig. 3.

Curves for two sinusoidal waves with same amplitude and different phases. (Online version in color.)

Similar to the phase adjustment, the amplitude of the reference input is adjusted using the feedback information returned from the mold position. However, as opposed to the phase adjustment case in which the mold position is used directly, the index for amplitude adjustment is derived from the maxima and minima of the mold position in each period. The basic idea is to scale up or down the maxima and minima of the reference input by comparing those of the mold position with reference values. To achieve this, the following indices are introduced:   

$$I_{A_M}\left(t\right)=A\left(t\right)-{max}_{\left(k-1\right)T\le \tau <kT}\left(x\left(\tau \right)\right),$$(7)

  

$$I_{A_m}\left(t\right)={min}_{\left(k-1\right)T\le \tau <kT}\left(x\left(\tau \right)\right)-A\left(t\right),$$(8)

where t is in the range of (k−1)T≤t<kT. Here, T is the period of mold oscillation, and k is the period number. This means that the maxima and minima of mold position from the previous period are used to obtain (7) and (8). Furthermore, the amplitude is adjusted using a scale factor A′(t)=s(t)A(t) where s(t) is derived as   

$$s\left(\text{t}\right)=s_M\left(t\right)\text{sgn}\left(\frac{u\left(t\right)}{\left|u\left(t\right)\right|}\right)+s_m\left(t\right)\text{sgn}\left(\frac{-u\left(t\right)}{\left|u\left(t\right)\right|}\right).$$(9)

Here, sgn(a)=a if a>0, else sgn(a)=0. In addition, s_M(t) and s_m(t) are calculated from indices $I_{A_M}\left(t\right)$ and $I_{A_m}\left(t\right)$ such that   

$$s_M\left(t\right)=s_M\left(0\right)+k_{I,A}{\int }_0^t{I}_{A_M}\left(\tau \right)d\tau ,$$(10)

  

$$s_m\left(t\right)=s_m\left(0\right)+k_{I,A}{\int }_0^t{I}_{A_m}\left(\tau \right)d\tau ,$$(11)

where k_I,A is a gain constant for reshaping the amplitude of u′(t). From (5) through (11), the reshaped reference input is therefore   

$$u^{\prime }\left(t\right)=s\left(t\right)A\left(t\right)sin\left(wt+{\varphi }^{\prime }\left(t\right)\right).$$(12)

The effects of the input shape control algorithm are shown in Fig. 4. As seen, the imaginary reshaped input reference has larger maxima and minima, and led phase compared to the original input reference u(t).

Fig. 4.

Examples of original reference input and imaginary reshaped reference input. (Online version in color.)

3.2. Velocity Feedforward Control Algorithm

Recently, the continuous casting process requires a fast-speed operation to increase production. Because powder consumption rate is reduced and friction is increased as a result of increased casting speeds, more effective lubrication is required. Among the various parameters that affect slab quality, a sinusoidal wave with an asymmetric modification to sinusoidal wave has a more significant effect.⁵⁾ Therefore, a non-sinusoidal oscillation has been introduced to achieve a fast-speed continuous casting process. In this section, we propose a velocity feedforward control algorithm that is appropriate for non-sinusoidal mold oscillations.

This paper focuses on a proportional relationship between the valve input and cylinder velocity. The load flow Q_L is proportional to x_v, and it has the equivalent waveform with ${\dot{x}}_p$ under the assumption that p_L is a slow-varying parameter and $\frac{V}{E\cdot A_p}{\dot{p}}_L$ is relatively small compared to ${\dot{x}}_p$ in (2). This means that the cylinder velocity basically has the same waveform as that of the valve input without the scale difference. Further, the non-sinusoidal wave and its derivative have different waveforms and are given as   

$$y_r\left(t\right)=Asin\left(\omega t-C_1\text{sin}(\omega t+C_{2}sin\left(\omega t\right)\right)$$(13)

  

$$\begin{array}{l}{\dot{y}}_r\left(t\right)=A\left\{\omega -C_{1}cos\left(\omega t+C_{2}sin\left(\omega t\right)\right)\cdot \left(\omega +C_{2}wcos\left(\omega t\right)\right)\right\}\\ \hspace{1em}\hspace{1em}\cdot cos\left(\omega t-C_1\text{sin}\left(\omega t+C_{2}sin\left(\omega t\right)\right)\right)\end{array}$$(14)

where C₁ and C₂ are parameters decides an asymmetry of non-sinusoidal oscillation. In non-sinusoidal mold oscillations, the mold moves upward more slowly and downward more quickly as compared to the sinusoidal oscillation. For example, (13) and (14) with 60% and 70% asymmetries are plotted in Fig. 5. As seen, the two waves are very different from each other, and the velocity curve has a wide positive peak and a narrow negative peak. This trend becomes more obvious as the asymmetry increases.

Fig. 5.

Curves for non-sinusoidal oscillation with asymmetry 60%, 65%, 70% (a) Position curves; (b) Velocity curves. (Online version in color.)

This difference in non-sinusoidal oscillation has a clear effect on the control performance of the mold oscillator. With previous feedback control algorithms, the control reference did not have a wide enough positive and narrow enough negative peaks as compared to those of non-sinusoidal waves (Fig. 6(a)). This problem causes an imbalance between the maxima and minima of mold positions, i.e., the mold position is shifted upward. From the proposed input shape control algorithm, s_m(t) grows quicker than s_M(t) to compensate the shift of the mold position, and thus, the maxima and minima of mold positions become equivalent. However, this is a limited solution as the input shape control algorithm only modifies the scale of the wave.

Fig. 6.

Curves for control references (a) without the velocity feedforward control algorithm; (b) with the velocity feedforward control algorithm. (Online version in color.)

A more reasonable solution is to generate the control reference using the velocity of the non-sinusoidal wave. The ratio between the velocity feedforward and feedback control references is an important factor. Furthermore, the ratio is decided by the feedforward velocity gain of the non-sinusoidal wave. Including the velocity feedforward control, the total control reference is thus given as   

$$u_v=k_P\left(y_r^{\prime }\left(t\right)-x\left(t\right)\right)+k_I{\int }_0^t\left(y_r^{\prime }\left(\tau \right)-x\left(\tau \right)\right)d\tau +k_f{\dot{y}}_r^{\prime }\left(t\right),$$(15)

where k_p, k_I, k_f are the P feedback gain, the I feedback gain, and the velocity feedforward control gain, respectively. The feedback gains are heuristically determined constants while the feedforward gain is a variable. As mentioned previously, the minima control factor s_m (t) grows quicker than the maxima control factor s_M (t) in order to prevent shifts in the mold position. These two factors are equivalent when the control reference properly matches the velocity of non-sinusoidal oscillation. Therefore, the feedforward gain is adjusted until the s_m (t) and s_M (t) become equivalent such that   

$$k_f\left(t\right)=k_f\left(0\right)+k_{I,FF}{\int }_0^t\left(s_m\left(\tau \right)-s_M\left(\tau \right)\right)d\tau ,$$(16)

where k_f (0) is the initial feedforward gain and k_I,FF is a gain constant for tuning feedforward gain. Figure 6(b) indicates that the velocity feedforward control reference can, with a proper scale modification, produce the valve input that has the same waveform as that of the mold velocity.

4. Experimental Results

The proposed feedforward control algorithm was tested on a Matlab Simulink model and a scaled specimen of a typical mold oscillator used in industry. The Simulink model was derived from the mathematical model described in Section 2. The specimen was 1/100 scale of a typical mold oscillator and it was configurated as shown in Fig. 7. A sensor system was built in the specimen with the capability of measuring several aspects including the pressures of the top and bottom chambers, the position of cylinder piston, and the external load weight. With an implemented PLC code, the oscillation tests were performed with several input reference signals. Also, the discrete-time based controllers were employed in practice although the algorithms are described in continuous-time versions. The controller gains and specimen spec are shown in Table 1. Here, the controller gains were heuristically determined.

Fig. 7.

Test specimen made as 1/100 of real mold oscillator. (Online version in color.)

Table 1. Controller gains and specimen spec.

items	values
kp, kI	0.05, 0.005
kI,ph, kI,A, kI,FF	0.1, 0.05, 0.005
sM(0), sm(0), kf(0)	1.1, 1.1,0
Specimen weight	400 kg
Specimen size	1000 (W) × 600 (D) × 1000 (H) mm
Stroke	50 mm
Supply pressure	250 bar

First, the proposed feedforward control algorithm was tested on the Simulink model. Four cases were tested: (a) sinusoidal oscillation with only feedback control (b) sinusoidal oscillation with proposed input shape control (c) non-sinusoidal oscillation with input shape control and without velocity feedforward control (d) non-sinusoidal oscillation with velocity feedforward control. This model-based test was done to demonstrate the behavior of the proposed algorithm. Following this, the performance analysis was carried out on the specimen. As a representative example, tests were conducting using the following oscillation parameters: stroke = 2 [mm], frequency = 150 [cpm] and asymmetry = 60 [%]. In Figs. 8(a)–8(d), the original reference input and mold position for a steady state condition were plotted. As seen, the phase delay and amplitude reduction problems were eliminated with the addition of the input shape control algorithm (Figs. 8(a)–8(b)). Furthermore, the mold position under the proposed velocity feedforward control algorithm reflects more closely the reference input compared to the case without the proposed velocity feedforward control algorithm (Figs. 8(c)–8(d)).

Fig. 8.

Curves for the reference input and corresponding mold position (Simulink test) (a) with only the feedback control (b) with input shape control (c) with input shape control and without velocity feedforward control (d) with velocity feedforward control algorithm. (Online version in color.)

Following the Simulink simulation, the algorithm was then tested on the specimen. The test results for the sinusoidal oscillation with stroke=2,3,4 [mm] and frequency=150,180,210 [cpm] were summarized in Table 2. As seen, the proposed input shape control algorithm suitably works for the sinusoidal mold oscillation. For the subsequent experiment on the specimen, non-sinusoidal mold oscillations of various asymmetries, strokes and frequencies were tested. Also, the plant algorithm that is currently being used in actual mold oscillator of POSCO was employed. The plant algorithm is a blackbox software package. Two performance indices were chosen to verify the performance; the averaged downward velocity and the positive strip time.⁹⁾ Note that the averaged downward velocity is used instead of the negative strip ratio⁹⁾ because the casting speed can not be defined in the specimen test. Under the equivalent casting speed, the faster averaged downward velocity gives a lower negative strip ratio, which suggests, therefore, an improvement in the compressive force aspect. Additionally, the longer positive strip time results in an increased powder consumption rate and reduced friction forces.³⁾

Table 2. Tracking errors between the input reference and mold position (specimen test) when strokes=2,3,4 [mm] and frequencies=150,180,210 [cpm].

Stroke	2 mm
Frequency (cpm)	150	180	210
Tracking error	3.80%	2.11%	2.82%
Frequency		180 cpm
Stroke (mm)	2	3	4
Tracking error	2.11%	3.58%	4.47%

The test results using asymmetry = 60,65,70 [%], stroke = 2 [mm] and frequency = 150 [cpm] were summarized in Figs. 9(a)–9(b). As seen, the proposed algorithm results in improved performance in regards to averaged downward velocity and positive strip time as compared to the plant algorithm, especially for elevated asymmetries. Because higher asymmetry is required for faster casting speed, the proposed velocity feedforward control algorithm would be useful.

Fig. 9.

Curves for the comparison of plant algorithm and proposed algorithm (Specimen test): increases of (a) averaged downward velocity (b) positive strip time. (Online version in color.)

5. Conclusion

In this study, a feedforward control algorithm has been developed for both sinusoidal and non-sinusoidal mold oscillations. The proposed feedforward control algorithm consists of two components: an input shape control algorithm and a velocity feedforward control algorithm. The input shape control algorithm compensates for the phase delay and amplitude reduction of the mold position by generating an imaginary reshaped input reference. To handle the non-sinusoidal oscillation, a velocity feedforward control algorithm was introduced based on the relationship between the valve input and mold velocities. The proposed algorithm was tested on a model-based Matlab Simulink and a 1/100 scale specimen of a typical mold oscillator. For various input parameters including frequency, amplitude, and asymmetry, the mold position accurately followed the input reference using the proposed feedforward control algorithm.

Acknowledgement

This research was supported by the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the “ICT Consilience Creative Program” (IITP-2015-R0346-15-1007) supervised by the IITP (Institute for Information & Communications Technology Promotion)”.

The author would like to thank for POSCO that provides the specimen and helps the research.

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