- Email: [email protected]
Supervisory control of automatic pouring machine
ARTICLE IN PRESS Control Engineering Practice 18 (2010) 230–241
Contents lists available at ScienceDirect
Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Supervisory control of automatic pouring machine Ken’ichi Yano a,, Kazuhiko Terashima b a b
Department of Human and Information Systems Engineering, Gifu University, 1-1 Yanagido, Gifu 501-1193, Japan Department of Production Systems Engineering, Toyohashi University of Technology, Hibarigaoka 1-1, Tempaku-cho, Toyohashi, 441-8580, Japan
a r t i c l e in f o
a b s t r a c t
Article history: Received 19 January 2009 Accepted 1 November 2009 Available online 28 November 2009
This paper presents supervisory control of a total molten-metal pouring system to improve the productivity of the factory, the safety of workers, and the quality of the product. Through the pouring processes model, a forward tilting control input was calculated by an adaptive feedforward control system to hold the liquid in the sprue cup at a constant level considering the change effected by the accumulating slag in the ladle. A backward tilting input was obtained by means of the hybrid shape approach applied to suppress the slosh. The supervisory control system reasonably switches from the forward tilting motion to the backward tilting motion by using model predictive control to achieve the accurate poured quantity. The validity of the proposed total control system was demonstrated through experiments. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Automatic process control Supervisory control Predictive control Intelligent manufacturing systems Casting process
1. Introduction The environment in a casting factory is very dangerous, creating poor working conditions for workers, primarily because of the handling of molten metal. In contrast to other kinds of industry, however, many casting factories have not yet benefited from advanced automation (Lerner & Laukhin, 2000; Lindsay, 1983). Manual handling is still generally used in the pouring process. It is very difficult to measure and control the behavior of molten metal in the casting process; further, many tasks must be controlled simultaneously in one batch of the process. A system with the following objectives should be designed: (1) To pour molten metal precisely into the sprue cup or the pouring basin of a mold by controlling the position and angle of a ladle in 3D space. (2) To keep the sprue cup filled to the specified high level of molten metal from the start of the pour to the end, in order to finish pouring faster and to avoid trapping dust and a vortex in the mold, which induces casting defects. (3) To pour the precise amount of metal required by the mold by predicting the flow quantity overspill from the ladle during back-tilting in order to avoid over-pouring and to cut scrap loss. (4) To suppress sloshing (liquid vibration) when the ladle is tilted backward when finishing the pouring in the specified time, and especially, to eliminate the residual vibration of molten metal at the end point, in order to avoid overflow and to Corresponding author. Fax: + 81 58 293 2509.
E-mail addresses: [email protected] (K. Yano), [email protected] (K. Terashima). 0967-0661/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2009.11.001
prevent any deterioration in quality due to contamination by sloshing. (5) To guarantee robustness for changes in pouring conditions due to the accumulated slag in the ladle produced by impurities or oxides. These tasks have generally been employed by a teaching and playback method on usual automatic pouring systems. However, such a method generates some problems because of inappropriate tilting velocity and lack of robustness for changes in pouring conditions; namely, molten metal overflows from the mold or the quality of the product decreases because the supply of the molten metal into the mold comes short and slags are entrapped in the mold. This especially occurs when the flow pattern of molten metal from the ladle is varied by the slag attached to the inside of the ladle. The conventional systems do not include advanced controls such as sloshing suppression or prediction of flow quantities. In addition, it is often required to adjust the pouring input, which leads to poor productivity. On the other hand, an expert worker engaging in a casting factory handles these difficult tasks by his hunch and experience. It is said that the quality of products made by manual operation is superior to that by automatic pouring. Also, long term training is required to master the expert skill of a pouring operation. Therefore, the development of an automatic pouring system that can replicate experts’ skills is in high demand. With respect to the automation of the pouring process, control systems that include the suppression of sloshing in liquid container transfer have been largely studied (Feddema et al., 1997; Yano & Terashima, 2001, 2005; Yano, Higashikawa, & Terashima, 2002). Sloshing suppression in tilting-type automatic
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
pouring machines has also been studied. Yano, Toda, and Terashima (2001) has studied high-speed transfer in threedimensional space using an automatic pouring machine with four degrees-of-freedom, as well. Terashima and Yano (2001) has studied on suppressing sloshing of a pouring machine in the case of the backward tilting of a two-dimensional thin-type ladle. However, pouring process analysis and control in pouring motion are still insufficient for the purposes. There are no studies on control models for the molten metal fluidity from the tilting motion of a ladle to the liquid level in a mold, nor are there studies on the pouring flow quantity and its control for the total pouring process. Furthermore, no study has considered adhesion to the ladle of the accumulated slag produced by impurities or oxides in the molten metal. Therefore, the previous systems cannot guarantee robustness for changes in pouring conditions due to the accumulation of slag. On the other hand, Kitada, Kondo, Kusachi, and Sasame (1998) has reported liquid level control in molds for continuous casting by H1 control. Also, model-based control of the mold level under various model uncertainties (Barron, Aguilar, Gonzalez, & Melendez, 1998) has been conducted for continuous casting. But no studies on liquid level control in the sprue cup have been reported for the batch process, despite its importance. The purpose of this study is to control the pouring process series in a way that replicates the experts’ skill, and thereby to improve the productivity of the factory, the safety of workers, and the quality of the product. A mathematical model of a pouring process series was built; based on the model, a forward tilting control input was designed to hold the liquid in the sprue cup at a constant level. Yano and Terashima (2005) obtained the backward tilting input by the hybrid shape approach without direct feedback for sloshing to suppress the sloshing after backward tilting. To supply the accurate pouring quantity into the mold, the controller is switched at the appropriate timing from forward tilting input to backward tilting input, based on a predicting method implemented by a supervisor placed at the upper level of the total system. Furthermore, an intelligent control system that can recalculate the feedforward control input by estimating the change in the pouring flow-rate model due to slag adhering to the nozzle tip of the ladle has been developed. These controllers are switched by the supervisory control. To achieve the goal of pouring quickly while maintaining safety, the liquid in the sprue cup must reach the desired level quickly and be kept at or near setpoint by feedforward control. When a difference between the real flowrate curve and the desired flowrate curve occurs due to slag adhering to the nozzle tip of the ladle, the feedforward control input is recalculated by a learning control. By this operation, the risk of overflow can be avoided. During pouring, at every sampling time, the supervisor is able to calculate the total quantity of molten metal that would overspill from the ladle after back-tilting, and thereby decide the time to switch from forward tilting input to backward tilting input. Thus, the system enables the pouring machine to supply the accurate flow volume to the mold, and the amount of wasted materials is reduced. The paper is organized as follows. Section 2 presents the details of the automatic pouring machine. In Section 3, the problem statement, design problem and the design procedure are described, and the proposed control system is constructed. In Sections 4 and 5, a mathematical model of the pouring process series was built, and a forward tilting control input was designed to hold the liquid in the sprue cup at a constant level. Then, in Sections 6 and 7, the backward tilting input was obtained by the hybrid shape approach and the predicting control method implemented by the supervisory system. Finally, in Section 8, an intelligent control system that can recalculate the feedforward
231
control input by estimating the change in the pouring flow-rate model is discussed.
2. Automatic pouring machine Figs. 1 and 2 show an illustration and a photograph of the tilting-type automatic pouring machine. The fan-type ladle has a radius of 0.34 m and a height of 0.36 m. The capacity of each mold is 1:19 103 m3 . The rotary direction of the Y-axis is driven by an AC servomotor. The center of the ladle’s rotation shaft is placed near the ladle’s center of gravity. When the ladle is rotated around the center of gravity, the tip of the ladle nozzle (or mouth) moves in a circular trajectory. It is difficult to pour the molten metal into a mold by the tilting of the pouring mouth. Thus, the tip position of the ladle nozzle is controlled invariably during pouring by means of a synchronous control of Y- and Z-axes for rotational motion around the Y-axis of the ladle (Terashima, Yano, Sugimoto, & Watanabe, 2001). X-, Y- and Z-axes are also driven by AC servomotors. The driving force of each of these motors is amplified through a ball and screw mechanism. Each axis can be independently moved. The level of liquid in the sprue cup is measured by a laser sensor above the relief sprue. This laser sensor measures the distance by a triangulation
Laser sensor
Z
Ladle
Y Spurecup Relief spure
Θ
X
Load cell
Mold
AC servo motors and encoders Fig. 1. Illustration of the tilting-type automatic pouring robot.
Fig. 2. Photograph of the automatic pouring robot.
ARTICLE IN PRESS 232
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
method. The liquid level can be measured by the reflection of laser beam from a float on the relief sprue as shown in Fig. 1. The laser sensor used in this paper can measure liquid levels higher than 0.036 m from the bottom of a cup (0 m). The liquid weight in the ladle is measured by a load cell installed in the ladle. The weight of the ladle with no liquid is set to be 0 kg. At every sampling time, the present outflow quantity is calculated by subtracting the present weight of a ladle from the initial ladle weight. The amount of sloshing in the ladle is measured using an electric resistance-type sensor composed of two stainless bars. This sensor is not used for the purpose of feedback, but to evaluate the sloshing suppression.
An ideal pouring pattern has three parts, as drawn in Fig. 3: (1) the rising part, (2) the equilibrium part, and (3) the cutting part. First, in the rising part, the liquid level in the sprue cup must be raised to the set point as quickly as possible. Second, the liquid level must be held constant in the sprue cup while the mold is being filled with metal. Finally, in the cutting part of the pouring pattern, backward tilting after pouring must be carried out quickly without residual vibration. A series of steps in the process shown in Fig. 3 is needed to maintain the liquid level in the sprue cup at a set point. The pouring control system has been designed considering the risk of overflow and/or insufficiency of molten metal due to slag adhering to the nozzle tip of the ladle and/or the mold gate. In this control system, a predictive control is applied to avoid the risk of overflow or insufficiency of molten metal, and a 2-degree-offreedom control designed by a H1 control is applied for overflow emergencies. For details on the 2 degree-of-freedom control, refer to the literature (Terashima, Yano, & Sugimoto, 2003). Thus, one cycle of the pouring process is so complicated that it is difficult to control the system by a single controller. Therefore, in this study, the supervisory control system conducts a switch from the level control system to back-tilting control system based on the amount of accumulated slag, as shown in Fig. 4. Here, u is the input voltage to the ladle, y is the output (y: the tilting angle, M: the liquid weight, hc : liquid level in the cup) as shown in Eq. (1), w is the disturbance, and s is the switching signal. By switching the controllers at the optimal timing, as described in the next sections, the pouring robot can pour the molten metal into the mold successfully:
3. Pouring process and pouring control system Feedforward control can achieve a high degree of accuracy when it is used in the appropriate situation. Systematic pouring operations that fill a mold with molten metal and hold the liquid level constantly in the sprue cup are in high demand. Since this pouring process is a batch process that handles the liquid from a transient state to a steady state, a system by feedback control alone is insufficient because of the time delay. In order to shorten the cycle time and maintain steady control (e.g., a constant level in the sprue cup), it is strongly required to have a feedforward control in which the flow pattern is specified beforehand.
Flow rate [m3/s]
(1) Rising part (2) Equilibrium part
yT ¼ ½y M hc
ð1Þ
(3) Cutting part 4. Pouring process model The object liquid is water. Even if real plants in the casting industry are considered, it is possible to use water in this study, because the Reynolds number of water at room temperature is almost the same as that of molten iron or molten aluminum at high temperature considering the similarity law in fluid dynamics. Then the fluid behavior of molten metal may be fairly well predicted by water analysis. Terashima and Yano (2001) has also clarified this phenomenon by CFD simulation. A schematic diagram of the pouring model is shown in Fig. 5, where oðtÞ is the tilting angular velocity of the ladle, qðtÞ is the flow rate into the sprue cup, hc ðtÞ is the liquid level in the sprue
Time [s] q in
q q qout in > out (1) Rising part
q in
q out q in = q out (2) Equilibrium part
q in = 0 (3) Cutting part
Fig. 3. Pouring flow rate curve.
σ
σ
Fig. 4. Structure of the supervisory control system.
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
233
L
R
ϕ
Ln Rn
ϕ ϕ
hf
ϕ
Fig. 6. Illustration of the fan-type ladle.
Fig. 5. Illustration of a pouring model.
where R is the length of the ladle (0.36 m), L is the width of the ladle (0.24 m), Rn is the length of the nozzle (0.085 m), and Ln is the width of the nozzle (0.065 m):
Table 1 Motor parameters.
Y-axis Z-axis Y-axis
Kf ¼
Km
Tm
Vmax
0.0830 m/sV 0.0828 m/sV 0.4290 rad/sV
0.006 s 0.007 s 0.006 s
0.5 m/s 0.5 m/s 2.62 rad/s
Amax 2
1:0 m=s 1:0 m=s2 26:2 rad=s2
cup, hm ðtÞ is the liquid level in the mold, hs ðtÞ is the liquid level in the mold over the gate, AC is the cross-sectional area of the cup, AM is the cross-sectional area of the mold, and AG is the crosssectional area of the gate. The units in the figure are millimeters. 4.1. Motor model: PM The relationship between the angular velocity of the tilting ladle and the input voltage to the motor related to the rotation of the ladle is described in the following equation: doðtÞ 1 K oðtÞ þ m uðtÞ ¼ Tm dt Tm
R2 LpR2n ðLLn Þp 2p
It is supposed that molten metal does not flow out when the ladle starts to tilt, but that the liquid above the nozzle tip increases. This liquid increment Vi ðtÞ can be expressed by the ladle’s angular velocity oðtÞ as Eq. (4). On the other hand, liquid flows out from the ladle if liquid exists in the part above the nozzle tip. Then, outflow qðtÞ is obtained by Bernoulli’s theorem, where cl is the flow coefficient of the ladle and hf ðtÞ is the liquid level above the nozzle tip: Z
Vi ðtÞ ¼ Kf
t
oðtÞ dt
ð4Þ
0
The value of hf ðtÞ was obtained by dividing the liquid volume above the nozzle tip by the cross-sectional area Al ¼ RLRn ðLLn Þ of the ladle, as shown in Eq. (6). The value cl ¼ 0:91 is obtained by an experiment: qðtÞ ¼
Z
hf ðtÞ
cl Ln
0
ð2Þ
where o ðrad=sÞ is the angular velocity, u ðVÞ is the input voltage for Y-axis, Kym ðrad=ðsVÞÞ is the gain of the motor, and Tym ðsÞ is the time constant of the motor. The motor models of the X-, Y- and Z-axes are all described by Eq. (1). Table 1 shows the identification results of the parameters including the maximum velocity Vmax and maximum acceleration Amax of the motors on each axis. This model is described as PM in this paper.
ð3Þ
hf ðtÞ ¼
1 Al
Z 0
t
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 3=2 2 2gh dh ¼ cl Ln 2g hf ðtÞ 3
dV i ðtÞ qðtÞ dt dt
ð5Þ
ð6Þ
Finally, the flow-rate model can be described by Eqs. (7) and (8) from Eqs. (4)–(6) as follows. This model is described as PF in this paper: 3=2 h_ f ðtÞ ¼ af hf ðtÞ þbf oðtÞ
3=2
4.2. Flow-rate model: PF
qðtÞ ¼ cf hf
In this paper, a fan-type ladle as shown in Fig. 6 is used for the pouring. The surface area of the liquid in the fan ladle becomes constant when the ladle is rotated at a constant angular velocity; therefore, pouring outflow per 1 rad Kf can be expressed as Eq. (3),
where
af ¼
ð7Þ
ðtÞ
2cl Ln pffiffiffiffiffiffi 2g ; 3Al
ð8Þ
bf ¼
Kf ; Al
cf ¼
pffiffiffiffiffiffi 2 cl Ln 2g 3
ð9Þ
ARTICLE IN PRESS 234
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
4.3. Liquid level model: PL The liquid level model is built by considering mass balance, Bernoulli’s theorem and a continuous equation as follows (Sugimoto, Yano, & Terashima, 2002): dhc ðtÞ 1 AG pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðtÞcðtÞ 2gðhc ðtÞ þ h1 hs ðtÞÞ ¼ AC dt AC
ð10Þ
dhm ðtÞ AG pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðhc ðtÞ þ h1 hs ðtÞÞ ¼ cðtÞ AM dt
ð11Þ
where c is the flow coefficient (-) and g is gravitational acceleration ðm2 =sÞ. In this paper, a time-variant flow coefficient cðtÞ was introduced in order to model the liquid level from the beginning to the end of the pouring, as shown in Eq. (12). Namely, the initial value of the flow coefficient is set to 0, and shifted up gradually to reach the final flow coefficient ce by the end time of filling the cup, where hc ðtÞ is the liquid level in the cup and href is the target level, set to 0.06 m in this study. A final flow coefficient of ce ¼ 0:51 was obtained by conducting pouring experiments. This model is described as PL in this paper: cðtÞ ¼
ce hc ðtÞ href
ð12Þ
Finally, the pouring process model from the input voltage uðtÞ to the servomotor on the Y axis to the liquid level in the sprue cup hc ðtÞ is derived by connecting the motor model Eq. (2), the flowrate model Eqs. (7) and (8), and the level model Eqs. (10)–(12).
5. Feedforward control of liquid level In this section, the feedforward input that keeps the liquid level constant in the sprue cup is calculated. Suppose that the liquid level in the sprue cup rose with a sinusoidal curve and maintained the target level href shown by Eq. (13) when the ideal flow rate shown in Fig. 3 inflows into a mold. Then this liquid level curve is defined as the target liquid level curve hcref . The target liquid level curve hcref can be calculated if the rising time trise is decided. 8 > < href sin 2p t þ href t ð0 rt o trise Þ 2p trise trise ð13Þ hcref ðtÞ ¼ > : href ðtrise r tÞ Eventually, the feedforward control input for the liquid level control is calculated by substituting the target liquid level curve 1 1 1 PF PL as shown in Fig. 7 and Eq. hcref for the inverse model PM (14), 1 ðsÞPF1 ðsÞPL1 ðsÞhcref u ¼ PM
ð14Þ
Namely, the output hðtÞ can correspond completely to the target hcref by Eq. (15): h ¼ PðsÞu ¼ PL ðsÞPF ðsÞPM ðsÞu
ð15Þ
The feedforward control input is obtained by theoretical analysis using the inverse model of the liquid level model, the inverse flow-rate model and the inverse motor model, as shown below. In this paper, the target level in the sprue cup href was set to 0.06 m and the rising time was set to 2.57 s for level control. Inverse liquid level model: PL1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dhcref ðtÞ ð16Þ qðtÞ ¼ Ac þ cðtÞAG 2gðhcref ðtÞ þ h1 hs ðtÞÞ dt Inverse flow-rate model: PF1 !2=3 3 pffiffiffiffiffiffi qðtÞ hf ðtÞ ¼ 2cl Ln 2g
oðtÞ ¼
pffiffiffiffiffiffi Al dhf ðtÞ 2cl Ln 2g 3=2 hf ðtÞ þ 3Kf Kf dt
uðtÞ ¼
Tmy doðtÞ 1 oðtÞ þ Kmy dt Kmy
PF-1
Inverse Motor Model
Inverse Flowrate Model
PM-1 Inverse Level Model
ð19Þ
The obtained feedforward control input for the forward tilting motion is shown in Fig. 8. In Fig. 8, the liquid level in the sprue cup, the outflow, the tilting acceleration, the input voltage, and the pouring quantity are shown from above. It is confirmed that the obtained outflow is like an ideal flowrate as shown in Fig. 3. Next, the experimental result with the feedforward input is shown in Fig. 9. As a result, the liquid level in the sprue cup was held around setpoint. At this time, the error of liquid level was less than 0.005 m. Thus, the appropriateness of the pouring process model and the validity of feedforward input was confirmed.
6. Back-tilting control with sloshing suppression Yano and Terashima (2005) has designed this system to suppress sloshing using a hybrid shape approach. This approach is a design method for starting control and vibration damping, and satisfies hybrid specifications in both the frequency domain and the time domain by using real-time feedback of only the position data. In fact, in a liquid container transfer system, sloshing is caused by self-transfer movement under circumstances in which there is no disturbance. To suppress sloshing, it is important to shape the frequency response of the controller to be notch-typed at the resonance frequency. It is theoretically possible to suppress vibration by controlling acceleration; therefore, a dynamical model of sloshing is not necessarily required in the control design. Fig. 10 shows the target closed-loop system. In this approach, sloshing can be damped by furnishing the controller with a notch filter at the resonance frequency of the sloshing. Refer to the literature Yano and Terashima (2005) for the details.
u
PL-1
ð18Þ
1 Inverse motor model: PM
Angular Velocity
Input Voltage
Liquid Level Curve hcref
ð17Þ
PM
Motor Model
FF Controller Fig. 7. Block diagram of the inverse model.
Liquid Level
Flow rate q PF
Flowrate Model
PL Level Model
1 s Pouring Process
h Outflow
M
ARTICLE IN PRESS
Liquid Level Curve [m]
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
235
0.09 0.06 0.03 0 1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4 Time [s]
5
6
7
8
0 -4
2
Angular Velocity [rad/s]
0 0.2 0.1 0 -0.1 -0.2
Input Voltage [V]
Flow Rate [m3/s]
x 10 4
0.5 0 -0.5 0
Outflow [m3]
-3
2 1.5 1 0.5 0
x 10
0
Fig. 8. Simulation results in forward tilting.
0.09
Table 2 Sloshing natural frequencies.
Liquid Level [m]
0.08 1st pouring 2nd pouring 3rd pouring 4th pouring
0.07
1st mode
3rd mode
1.27 Hz 1.17 Hz 1.07 Hz 0.97 Hz
– – – 3.02 Hz (18.97 rad/s)
(7.98 rad/s) (7.35 rad/s) (6.72 rad/s) (6.09 rad/s)
0.06 In order to design a controller that does not excite sloshing, it is necessary to specify the natural frequency of the sloshing. Table 2 shows natural frequencies of the sloshing after each pouring. By considering these results, the controller obtained by hybrid shape approach is as follows:
0.05
0.04 0.036
KðsÞ ¼
0.03 0
1
2
3 4 Time [s]
5
Fig. 9. Experimental result by the feedforward control.
6
2 s2 þ2zo s þ o2 Kp Y nj nj Tl s þ 1 j ¼ 1 s2 þ onj s þ o2nj
ð20Þ
where on1 ¼ 7:04 rad=s, on2 ¼ 18:97 rad=s, z ¼ 0:0001, and Kp ¼ 0:0807; Tl ¼ 0:052. Kp and Tl of the controller were determined by a simplex method that minimized the settling time of the back-tilting of the ladle. In Fig. 11, the frequency response of the controller is shown. The controller gain becomes lower in notch-shape around the natural frequency of sloshing. Fig. 12 shows the sloshing after backward tilting. Sloshing is suppressed better by the hybrid shape approach than by proportional control.
7. Predictive control of flow quantity
Fig. 10. Target closed-loop system.
Residual pouring quantity (RPQ) is defined as the liquid weight flowing out of the ladle from the start of the cutting part to the
ARTICLE IN PRESS 236
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
−20
M1 (t0) Weight [kg]
−30
Gain [dB]
−40
−50
0 −60
t0
Time [s]
−70
M1 (t0) −80 0 10
1
2
10 10 Angular frequency [rad/s]
M2
Fig. 11. Frequency response of the controller.
Fig. 13. Static outflow of the over weight in the ladle.
x 10−3 5 Controller by Hybrid Shape approach Propotional control
4
Weight [kg]
3
sloshing [m]
2 1 0
M2 (t)
0
−1
Time [s]
t0
−2 −3 14
15
16
17 Time [s]
18
19
20
Fig. 12. Sloshing after backward tilting.
end, during back-tilting of the ladle. In order to conduct the control of flow quantity from the ladle to the mold exactly, it is necessary to predict the RPQ. In order to predict the RPQ, let us consider the state of the liquid inside the ladle as shown in Fig. 13. Then, the liquid inside a ladle is considered as two parts, comprised the upper part of the liquid above the nozzle and the lower part under the nozzle. The upper part is called the ‘‘over weight part, M1 ’’ , and lower part the ‘‘equilibrium weight part, M2 ’’ . The total weight inside the ladle becomes M1 þ M2 . When the pouring is stopped in the state as shown in Fig. 13, the metal weight in the ladle finally becomes only the equilibrium weight. Let us consider the decrease of the over weight M1 with the increase of time. The relationship between the tilting angular velocity of the ladle o and the liquid level in the ladle hf is described in Eq. (7). In this case, Eq. (7) can be rewritten as follows because o ¼ 0: h_ f ðtÞ ¼ Af hf ðtÞ3=2
ð21Þ
M2 Fig. 14. Change of the over weight by the back-tilting motion.
The liquid level hf can be calculated to solve the differential equation as Eq. (22), where hf 0 is the initial liquid level in the ladle: 1=2 2
hf ðtÞ ¼ ð12Af t þ hf 0
Þ
ð22Þ
Then, the over weight M1 is calculated by multiplying the surface area Al ¼ RLRn ðLLn Þ by hf , as shown in Eq. (24). Eq. (24) is then transformed into a discrete-time equation as shown in Eq. (24): 1=2 2
M1 ðtÞ ¼ Al ð12Af t þ hf 0
Þ
M1 ½kþ j ¼ Al ð12Af t½j þ hf ½k1=2 Þ2
ð23Þ ð24Þ
Next, let us consider the change of M2 with the increase of time. During back-tilting, the ladle is tilted backward at the center of
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
r ¼ 1000 kg=m3 , it follows that
gravity of the ladle to reduce the sloshing because of decreasing inertia. As shown in Fig. 14, the equilibrium weight increases with back-tilting. M2 ðtÞ can be calculated by the tilting angle yðtÞ of the ladle, as shown in Fig. 15. The ladle is tilted backward by the hybrid-shape controller described in the previous section. When the closed-loop system from the reference r to the tilting angle y in Fig. 10 is described as Gcl ¼ fA2 ; B2 ; C2 ; 0g, the discrete-time state equation becomes as follows: xd2 ½k þ1 ¼ Ad2 xd2 ½k þBd2 ud2 ½k
ð25Þ
y½k ¼ Cd2 xd2 ½k
ð26Þ
M2 ½k þj ¼ rKf Cd2 Ajd2 xd2 ½k þ
j1 X
! Cd2 Aj1i Bd2 yr ½k þi d2
ð29Þ
i¼0
The liquid weight over the nozzle in the ladle becomes M1 ðtÞ minus M2 ðtÞ while back-tilting as shown in Fig. 15. M1 ðtÞ equals M2 ðtÞ at time tn . Therefore, after tn , the fluid does not flow out of the ladle, even if the ladle is tilted further backward. Hence, the prediction value Q^ p ðt0 Þ of RPQ at time t0 becomes Q^ p ðt0 Þ ¼ M1 ðt0 ÞM1 ðtn Þ
ð30Þ
If the following condition is satisfied, the supervisor sends a signal
Then, it follows that
s to the controller, and the controller is switched from the level
y½k þ1 ¼ Cd2 xd2 ½kþ 1 ¼ Cd2 ðAd2 xd2 ½k þBd2 ud2 ½kÞ
controller to the back-tilting controller. Switching condition:
ð27Þ
Further, it follows that
y½k þj ¼ Cd2 Ajd2 xd2 ½k þ
237
j1 X
Qref ¼ Q^ p ðt0 Þ þ Mðt0 Þ Cd2 Aj1i Bd2 yr ½k þ i d2
ð28Þ
ð31Þ
where Mðt0 Þ is the total metal weight flowing out of the ladle up to time t0 which can be measured by a load cell, and Qref is the reference pouring weight. Fig. 16 shows experimental results by the supervisory control with prediction of the residual pouring quantity (RPQ). Supervisory control with prediction of flow quantity shows better results than the results without prediction. For a reference weight of 1.274 kg, the proposed method attained 1.282 kg, and the conventional method without prediction attained 1.304 kg. Experiments have been done with various tilting velocities and angles. In any case, the proposed method showed good results at a similar precision as that shown in Fig. 16.
i¼0
Because the increment M2 ½k þ j ðkgÞ of the equilibrium weight can be calculated by multiplying the trajectory y½k þj ðdegÞ during back-tilting with the flow gain Kf ðm3 =degÞ and the density
M1(t0) M2(t) Weight [kg]
Qˆ p 8. Adaptive feedforward control of pouring process considering a disturbance
M1(t) In conventional methods, the slag attached to the inside of the ladle is not considered. Therefore, it cannot be guaranteed that the change in the outflow pattern of molten metal will not affect the outcome. In other words, if a nominal control input is applied to the disturbed ladle, the liquid level in the sprue cup becomes unstable and the supply of the molten metal into the mold falls
0 t0
Time [s]
Input [V]
Fig. 15. Predictive value of the outflow.
0.5 0 0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
10 5
CupLevel [m]
0 0.08
Outflow [kg]
Angle [deg]
-0.5
1.5 1 0.5 0
0.06 0.04
Supervisory Control Without prediction Set point
0
1
2
3
4
5
6
Time [s] Fig. 16. Experimental result by the flow-rate control.
7
8
9
ARTICLE IN PRESS 238
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
These are obtained by calculating Eqs. (7) and (8) by the Runge– Kutta method, where the sampling time is set to 0.01 s and a ¼ 1000 and l ¼ 0:97 in this study.
short. As a result, impurities such as dust, air bubbles, and slag are entrapped in the mold, and the quality of the product decreases. Furthermore, molten metal overflows the mold if the control input changes inappropriately. Therefore, it is desirable to construct a control system that realizes liquid level and flow rate control in any kind of situation. The control system developed in this study estimates changes in flow-rate-model parameters between each pouring cycle and updates the feedforward input step by step.
8.2. Renewal of feedforward control input When the flow-rate model is varied by a disturbance such as accumulated slag, it is necessary to renew the feedforward control input for every pouring motion. Whenever pouring is repeated, this disturbance accumulates. The pouring system that considers this disturbance is shown in Fig. 17. In Fig. 17, the control system consists of the motor model PM , the flow-rate model PF , the level model PL and the estimator EF . This system estimates the parameters of the flow-rate model, and the feedforward control input for the liquid level control is renewed for the next mold. As for the feedforward controller, it is derived from the inverse model of the flow-rate model and the motor model. The controller renews the liquid level control input by substituting the estimated parameters for an inverse model of the flow-rate model. u, o, q, and hc are the input voltage (V) to the servo-motor, the angular velocity (deg/s), the flow quantity ðm3 =sÞ, and the liquid level (m), respectively. In addition, qref is the reference flow quantity ðm3 =sÞ in order to control the liquid level in the sprue cup.
8.1. Estimation of flow-rate-model parameters Because cf is cf ¼ af Al as shown in Eq. (9), cf can be calculated by af , where Al is the cross-sectional area of the ladle. Therefore, an estimate of the flow-rate-model parameters is performed on af and bf in Eq. (7). The recursive least-square method as shown in the following equations is used so that the parameters are estimated online during the pouring motion: ^ y^ F ðkÞ ¼ Pðk1Þ þ
Gðk1ÞcðkÞ eðkÞ l þ cðkÞT Gðk1ÞcðkÞ
ð32Þ
eðkÞ ¼ yðkÞcðkÞT y^ F ðk1Þ GðkÞ ¼ "
c¼
1
(
l 3=2
hf
Gðk1Þ
ð33Þ
Gðk1ÞcðkÞcðkÞT Gðk1Þ l þ cðkÞT Gðk1ÞcðkÞ
) ð34Þ
8.3. Verification by experiments
# ð35Þ
o
A gate was established on both sides of the nozzle as a disturbance of the flow-rate model, as shown in Fig. 18. The nozzle width was changed from 65 to 41 mm. The experimental result with the nominal pouring control input for the disturbed system is shown in Fig. 19. In the experiment, a model predictive control was applied to the back-tilting control considering the sloshing suppression control and the flow quantity as stated in the previous sections. As a result of Fig. 19, the liquid level could not be kept at the target level because of the disturbance. The tilting angular velocity and the liquid level in the ladle, which are the input
where y^ F is the estimation value consisting of parameters a^ f and b^ f in the flow-rate model, l is the oblivion coefficient, e is the estimated error, and G is the error covariance matrix. The output yðkÞ becomes yðkÞ ¼ h_ f . In addition, the initial values of the estimation value y^ F and the error covariance matrix G are set as Eq. (36): " # a^ f ð0Þ a 0 ^y ð0Þ ¼ ; ð36Þ G ð0Þ ¼ F b^ f ð0Þ 0 a
Flow rate model parameter qF hcref
PL-1
qref
PˆF-1
Level inverse model
Control input wref
Flow rate inverse model
Angular velocity
u
q
w
PM-1
PF
PM
Motor inverse model
Motor model
qref
Level inverse model
PˆF-1 Flow rate inverse model
hc
level model 1st Pouring
EF
qF
PL-1
PL
Flow rate model
Flow rate model parameter
hcref
Cup level
Flow rate
Estimator
u
wref -1 M
P
Motor inverse model
PM
w
Motor model
q PF
PL
Flow rate model
level model 2nd Pouring
EF
Fig. 17. Intelligent control system for the accumulating slag.
hc
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
12
41
and output for estimating flow-rate-model parameters, are shown in Fig. 20. Fig. 21 shows the estimated results of the parameters, where the liquid level in the ladle was calculated by the tilting angular velocity measured by an encoder and the ladle weight measured by a load cell. As a result of estimation, the flow-rate-model parameters were obtained as a^ f ¼ 0:0520 and a^ f ¼ 3:316. Then, the control input was renewed with these new parameters. The renewed feedforward input based on the updated parameters is shown in Fig. 22, where the chained line shows the control result without a disturbance and with the nominal control input, the dashed line shows the control result with a disturbance and with the nominal control input, and the solid line the result with a disturbance but with the renewed control input. As seen from the figure, it was possible to control the liquid level in the sprue cup and the pouring flow quantity by renewing the pouring control input with an estimation of the change in the pouring flow-rate model due to the accumulated slag attached to the inside of the ladle of the same quality as is the case without disturbance. Actually, in a real pouring factory, slag forms slowly in the ladle; such a large amount of slag as in this experiment does not form suddenly. Therefore, it can be controlled very nicely over time because the control input is renewed step by step from the start time.
12
Dam
65 Unit : mm
Nozzle Tip
Input Voltage [V]
Fig. 18. Disturbance of the flow-rate model.
0.4 0.2 0 −0.2 −0.4 0
1
2
3
4
5
6
7
8
9
0.08
9. Conclusions
0.06
This paper presents a supervisory control system for an automatic molten-metal pouring process that replicates expert pourers’ skill in order to improve the productivity of the factory, the safety of workers, and the quality of the product. Feedforward control inputs for the pouring process were obtained, and the liquid level in the sprue cup was held to around the set point. In order to avoid overflow of liquid from the sprue cup during backtilting, a method to predict the quantity flowing out of the ladle during back-tilting was proposed. It was shown that by
0.04 0.02 0 0
1
2
3
4 5 Time [s]
6
7
8
9
Fig. 19. Experimental results of FF input with a disturbed model.
Angular Velocity [rad/s]
0.2 0.1 0 -0.1 -0.2
Liquid Height in Ladle [mm]
Liquid Level [m]
239
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4 5 Time [s]
6
7
8
9
30 25 20 15 10 5 0
Fig. 20. Experimental results of the angular velocity and liquid height in a ladle with the disturbed model.
ARTICLE IN PRESS 240
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
0
Parameter af
−0.02 −0.04 −0.06 −0.08 −0.1 0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
5
Parameter bf
4 3 2 1 0 Time [s] Fig. 21. Experimental results of the estimated flow-rate-model parameters.
Renewed FF Input Nominal FF Input Without Disturbance
Input Voltage [V]
0.4 0.2 0 −0.2 −0.4 0
1
2
3
4
5
6
7
8
9
0
1
2
3
4 5 Time [s]
6
7
8
9
Liquid Level [m]
0.08 0.06 0.04 0.02 0
Fig. 22. Experimental results of renewed FF input with a disturbed model.
supervising the predicted information on the flow quantity in real time, the switching from a level control system to a back-tilting control system with damping of sloshing can be well executed. In the end, an intelligent control system that can recalculate the feedforward control input by estimating the change in the pouring flow-rate model due to slag was presented. As a result, a pouring control system was realized that could control the liquid level in the sprue cup and the pouring flow quantity by renewing the pouring control input with an estimation of the change in the
pouring flow-rate model due to the accumulated slag attached to the inside of the ladle.
References Barron, M. A., Aguilar, R., Gonzalez, J., & Melendez, E. (1998). Model-based control of mold level in a continuous caster under model uncertainties. Control Engineering Practice, 6, 191–196.
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
Feddema, J. T., et al. (1997). Control for slosh-free motion of an open container. IEEE Control Systems Magazine, 17, 29–36. Kitada, H., Kondo, O., Kusachi, H., & Sasame, K. (1998). H1 control of molten steel level in continuous caster. IEEE Transactions on Control Systems Technology, 6, 200–207. Lerner, Y., & Laukhin, N. (2000). Development trends in pouring technology. Foundry Trade Journal, 11, 16–21. Lindsay, W. (1983). Automatic pouring and metal distribution system. Foundry Trade Journal, 151–165. Sugimoto, Y., Yano, K., & Terashima, K. (2002). Liquid level control of automatic pouring robot by two-degrees-of-freedom control. In Proceedings of IFAC World Congress (pp. 21–26). Terashima, K., & Yano, K. (2001). Sloshing analysis and suppression control of tilting-type automatic pouring machine. Control Engineering Practice, 9, 607–620. Terashima, K., Yano, K., & Sugimoto, Y. (2003). Modeling and robust control of liquid level in a sprue cup for batch-type casting pouring processes. In Proceedings of the IASTED international conference on intelligent systems and control (pp. 33–38).
241
Terashima, K., Yano, K., Sugimoto, Y., & Watanabe, M. (2001). Position control of ladle tip and sloshing suppression during tilting motion in automatic pouring machine. In Proceedings of IFAC symposium on automation in mining, mineral and metal processing (pp. 182–187). Yano, K., & Terashima, K. (2001). Robust liquid container transfer control for complete sloshing suppression. IEEE Transactions on Control Systems Technology, 9, 483–493. Yano, K., & Terashima, K. (2005). Sloshing suppression control of liquid transfer systems considering 3d transfer path. IEEE/ASME Transactions on Mechatronics, 10, 8–16. Yano, K., Higashikawa, S., & Terashima, K. (2002). Motion control of liquid container considering an inclined transfer path. Control Engineering Practice, 10, 465–472. Yano, K., Toda, T., & Terashima, K. (2001). Sloshing suppression control of automatic pouring robot by hybrid shape approach. In Proceedings of IEEE conference on decision and control (pp. 1328–1333).
Contents lists available at ScienceDirect
Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Supervisory control of automatic pouring machine Ken’ichi Yano a,, Kazuhiko Terashima b a b
Department of Human and Information Systems Engineering, Gifu University, 1-1 Yanagido, Gifu 501-1193, Japan Department of Production Systems Engineering, Toyohashi University of Technology, Hibarigaoka 1-1, Tempaku-cho, Toyohashi, 441-8580, Japan
a r t i c l e in f o
a b s t r a c t
Article history: Received 19 January 2009 Accepted 1 November 2009 Available online 28 November 2009
This paper presents supervisory control of a total molten-metal pouring system to improve the productivity of the factory, the safety of workers, and the quality of the product. Through the pouring processes model, a forward tilting control input was calculated by an adaptive feedforward control system to hold the liquid in the sprue cup at a constant level considering the change effected by the accumulating slag in the ladle. A backward tilting input was obtained by means of the hybrid shape approach applied to suppress the slosh. The supervisory control system reasonably switches from the forward tilting motion to the backward tilting motion by using model predictive control to achieve the accurate poured quantity. The validity of the proposed total control system was demonstrated through experiments. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Automatic process control Supervisory control Predictive control Intelligent manufacturing systems Casting process
1. Introduction The environment in a casting factory is very dangerous, creating poor working conditions for workers, primarily because of the handling of molten metal. In contrast to other kinds of industry, however, many casting factories have not yet benefited from advanced automation (Lerner & Laukhin, 2000; Lindsay, 1983). Manual handling is still generally used in the pouring process. It is very difficult to measure and control the behavior of molten metal in the casting process; further, many tasks must be controlled simultaneously in one batch of the process. A system with the following objectives should be designed: (1) To pour molten metal precisely into the sprue cup or the pouring basin of a mold by controlling the position and angle of a ladle in 3D space. (2) To keep the sprue cup filled to the specified high level of molten metal from the start of the pour to the end, in order to finish pouring faster and to avoid trapping dust and a vortex in the mold, which induces casting defects. (3) To pour the precise amount of metal required by the mold by predicting the flow quantity overspill from the ladle during back-tilting in order to avoid over-pouring and to cut scrap loss. (4) To suppress sloshing (liquid vibration) when the ladle is tilted backward when finishing the pouring in the specified time, and especially, to eliminate the residual vibration of molten metal at the end point, in order to avoid overflow and to Corresponding author. Fax: + 81 58 293 2509.
E-mail addresses: [email protected] (K. Yano), [email protected] (K. Terashima). 0967-0661/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2009.11.001
prevent any deterioration in quality due to contamination by sloshing. (5) To guarantee robustness for changes in pouring conditions due to the accumulated slag in the ladle produced by impurities or oxides. These tasks have generally been employed by a teaching and playback method on usual automatic pouring systems. However, such a method generates some problems because of inappropriate tilting velocity and lack of robustness for changes in pouring conditions; namely, molten metal overflows from the mold or the quality of the product decreases because the supply of the molten metal into the mold comes short and slags are entrapped in the mold. This especially occurs when the flow pattern of molten metal from the ladle is varied by the slag attached to the inside of the ladle. The conventional systems do not include advanced controls such as sloshing suppression or prediction of flow quantities. In addition, it is often required to adjust the pouring input, which leads to poor productivity. On the other hand, an expert worker engaging in a casting factory handles these difficult tasks by his hunch and experience. It is said that the quality of products made by manual operation is superior to that by automatic pouring. Also, long term training is required to master the expert skill of a pouring operation. Therefore, the development of an automatic pouring system that can replicate experts’ skills is in high demand. With respect to the automation of the pouring process, control systems that include the suppression of sloshing in liquid container transfer have been largely studied (Feddema et al., 1997; Yano & Terashima, 2001, 2005; Yano, Higashikawa, & Terashima, 2002). Sloshing suppression in tilting-type automatic
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
pouring machines has also been studied. Yano, Toda, and Terashima (2001) has studied high-speed transfer in threedimensional space using an automatic pouring machine with four degrees-of-freedom, as well. Terashima and Yano (2001) has studied on suppressing sloshing of a pouring machine in the case of the backward tilting of a two-dimensional thin-type ladle. However, pouring process analysis and control in pouring motion are still insufficient for the purposes. There are no studies on control models for the molten metal fluidity from the tilting motion of a ladle to the liquid level in a mold, nor are there studies on the pouring flow quantity and its control for the total pouring process. Furthermore, no study has considered adhesion to the ladle of the accumulated slag produced by impurities or oxides in the molten metal. Therefore, the previous systems cannot guarantee robustness for changes in pouring conditions due to the accumulation of slag. On the other hand, Kitada, Kondo, Kusachi, and Sasame (1998) has reported liquid level control in molds for continuous casting by H1 control. Also, model-based control of the mold level under various model uncertainties (Barron, Aguilar, Gonzalez, & Melendez, 1998) has been conducted for continuous casting. But no studies on liquid level control in the sprue cup have been reported for the batch process, despite its importance. The purpose of this study is to control the pouring process series in a way that replicates the experts’ skill, and thereby to improve the productivity of the factory, the safety of workers, and the quality of the product. A mathematical model of a pouring process series was built; based on the model, a forward tilting control input was designed to hold the liquid in the sprue cup at a constant level. Yano and Terashima (2005) obtained the backward tilting input by the hybrid shape approach without direct feedback for sloshing to suppress the sloshing after backward tilting. To supply the accurate pouring quantity into the mold, the controller is switched at the appropriate timing from forward tilting input to backward tilting input, based on a predicting method implemented by a supervisor placed at the upper level of the total system. Furthermore, an intelligent control system that can recalculate the feedforward control input by estimating the change in the pouring flow-rate model due to slag adhering to the nozzle tip of the ladle has been developed. These controllers are switched by the supervisory control. To achieve the goal of pouring quickly while maintaining safety, the liquid in the sprue cup must reach the desired level quickly and be kept at or near setpoint by feedforward control. When a difference between the real flowrate curve and the desired flowrate curve occurs due to slag adhering to the nozzle tip of the ladle, the feedforward control input is recalculated by a learning control. By this operation, the risk of overflow can be avoided. During pouring, at every sampling time, the supervisor is able to calculate the total quantity of molten metal that would overspill from the ladle after back-tilting, and thereby decide the time to switch from forward tilting input to backward tilting input. Thus, the system enables the pouring machine to supply the accurate flow volume to the mold, and the amount of wasted materials is reduced. The paper is organized as follows. Section 2 presents the details of the automatic pouring machine. In Section 3, the problem statement, design problem and the design procedure are described, and the proposed control system is constructed. In Sections 4 and 5, a mathematical model of the pouring process series was built, and a forward tilting control input was designed to hold the liquid in the sprue cup at a constant level. Then, in Sections 6 and 7, the backward tilting input was obtained by the hybrid shape approach and the predicting control method implemented by the supervisory system. Finally, in Section 8, an intelligent control system that can recalculate the feedforward
231
control input by estimating the change in the pouring flow-rate model is discussed.
2. Automatic pouring machine Figs. 1 and 2 show an illustration and a photograph of the tilting-type automatic pouring machine. The fan-type ladle has a radius of 0.34 m and a height of 0.36 m. The capacity of each mold is 1:19 103 m3 . The rotary direction of the Y-axis is driven by an AC servomotor. The center of the ladle’s rotation shaft is placed near the ladle’s center of gravity. When the ladle is rotated around the center of gravity, the tip of the ladle nozzle (or mouth) moves in a circular trajectory. It is difficult to pour the molten metal into a mold by the tilting of the pouring mouth. Thus, the tip position of the ladle nozzle is controlled invariably during pouring by means of a synchronous control of Y- and Z-axes for rotational motion around the Y-axis of the ladle (Terashima, Yano, Sugimoto, & Watanabe, 2001). X-, Y- and Z-axes are also driven by AC servomotors. The driving force of each of these motors is amplified through a ball and screw mechanism. Each axis can be independently moved. The level of liquid in the sprue cup is measured by a laser sensor above the relief sprue. This laser sensor measures the distance by a triangulation
Laser sensor
Z
Ladle
Y Spurecup Relief spure
Θ
X
Load cell
Mold
AC servo motors and encoders Fig. 1. Illustration of the tilting-type automatic pouring robot.
Fig. 2. Photograph of the automatic pouring robot.
ARTICLE IN PRESS 232
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
method. The liquid level can be measured by the reflection of laser beam from a float on the relief sprue as shown in Fig. 1. The laser sensor used in this paper can measure liquid levels higher than 0.036 m from the bottom of a cup (0 m). The liquid weight in the ladle is measured by a load cell installed in the ladle. The weight of the ladle with no liquid is set to be 0 kg. At every sampling time, the present outflow quantity is calculated by subtracting the present weight of a ladle from the initial ladle weight. The amount of sloshing in the ladle is measured using an electric resistance-type sensor composed of two stainless bars. This sensor is not used for the purpose of feedback, but to evaluate the sloshing suppression.
An ideal pouring pattern has three parts, as drawn in Fig. 3: (1) the rising part, (2) the equilibrium part, and (3) the cutting part. First, in the rising part, the liquid level in the sprue cup must be raised to the set point as quickly as possible. Second, the liquid level must be held constant in the sprue cup while the mold is being filled with metal. Finally, in the cutting part of the pouring pattern, backward tilting after pouring must be carried out quickly without residual vibration. A series of steps in the process shown in Fig. 3 is needed to maintain the liquid level in the sprue cup at a set point. The pouring control system has been designed considering the risk of overflow and/or insufficiency of molten metal due to slag adhering to the nozzle tip of the ladle and/or the mold gate. In this control system, a predictive control is applied to avoid the risk of overflow or insufficiency of molten metal, and a 2-degree-offreedom control designed by a H1 control is applied for overflow emergencies. For details on the 2 degree-of-freedom control, refer to the literature (Terashima, Yano, & Sugimoto, 2003). Thus, one cycle of the pouring process is so complicated that it is difficult to control the system by a single controller. Therefore, in this study, the supervisory control system conducts a switch from the level control system to back-tilting control system based on the amount of accumulated slag, as shown in Fig. 4. Here, u is the input voltage to the ladle, y is the output (y: the tilting angle, M: the liquid weight, hc : liquid level in the cup) as shown in Eq. (1), w is the disturbance, and s is the switching signal. By switching the controllers at the optimal timing, as described in the next sections, the pouring robot can pour the molten metal into the mold successfully:
3. Pouring process and pouring control system Feedforward control can achieve a high degree of accuracy when it is used in the appropriate situation. Systematic pouring operations that fill a mold with molten metal and hold the liquid level constantly in the sprue cup are in high demand. Since this pouring process is a batch process that handles the liquid from a transient state to a steady state, a system by feedback control alone is insufficient because of the time delay. In order to shorten the cycle time and maintain steady control (e.g., a constant level in the sprue cup), it is strongly required to have a feedforward control in which the flow pattern is specified beforehand.
Flow rate [m3/s]
(1) Rising part (2) Equilibrium part
yT ¼ ½y M hc
ð1Þ
(3) Cutting part 4. Pouring process model The object liquid is water. Even if real plants in the casting industry are considered, it is possible to use water in this study, because the Reynolds number of water at room temperature is almost the same as that of molten iron or molten aluminum at high temperature considering the similarity law in fluid dynamics. Then the fluid behavior of molten metal may be fairly well predicted by water analysis. Terashima and Yano (2001) has also clarified this phenomenon by CFD simulation. A schematic diagram of the pouring model is shown in Fig. 5, where oðtÞ is the tilting angular velocity of the ladle, qðtÞ is the flow rate into the sprue cup, hc ðtÞ is the liquid level in the sprue
Time [s] q in
q q qout in > out (1) Rising part
q in
q out q in = q out (2) Equilibrium part
q in = 0 (3) Cutting part
Fig. 3. Pouring flow rate curve.
σ
σ
Fig. 4. Structure of the supervisory control system.
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
233
L
R
ϕ
Ln Rn
ϕ ϕ
hf
ϕ
Fig. 6. Illustration of the fan-type ladle.
Fig. 5. Illustration of a pouring model.
where R is the length of the ladle (0.36 m), L is the width of the ladle (0.24 m), Rn is the length of the nozzle (0.085 m), and Ln is the width of the nozzle (0.065 m):
Table 1 Motor parameters.
Y-axis Z-axis Y-axis
Kf ¼
Km
Tm
Vmax
0.0830 m/sV 0.0828 m/sV 0.4290 rad/sV
0.006 s 0.007 s 0.006 s
0.5 m/s 0.5 m/s 2.62 rad/s
Amax 2
1:0 m=s 1:0 m=s2 26:2 rad=s2
cup, hm ðtÞ is the liquid level in the mold, hs ðtÞ is the liquid level in the mold over the gate, AC is the cross-sectional area of the cup, AM is the cross-sectional area of the mold, and AG is the crosssectional area of the gate. The units in the figure are millimeters. 4.1. Motor model: PM The relationship between the angular velocity of the tilting ladle and the input voltage to the motor related to the rotation of the ladle is described in the following equation: doðtÞ 1 K oðtÞ þ m uðtÞ ¼ Tm dt Tm
R2 LpR2n ðLLn Þp 2p
It is supposed that molten metal does not flow out when the ladle starts to tilt, but that the liquid above the nozzle tip increases. This liquid increment Vi ðtÞ can be expressed by the ladle’s angular velocity oðtÞ as Eq. (4). On the other hand, liquid flows out from the ladle if liquid exists in the part above the nozzle tip. Then, outflow qðtÞ is obtained by Bernoulli’s theorem, where cl is the flow coefficient of the ladle and hf ðtÞ is the liquid level above the nozzle tip: Z
Vi ðtÞ ¼ Kf
t
oðtÞ dt
ð4Þ
0
The value of hf ðtÞ was obtained by dividing the liquid volume above the nozzle tip by the cross-sectional area Al ¼ RLRn ðLLn Þ of the ladle, as shown in Eq. (6). The value cl ¼ 0:91 is obtained by an experiment: qðtÞ ¼
Z
hf ðtÞ
cl Ln
0
ð2Þ
where o ðrad=sÞ is the angular velocity, u ðVÞ is the input voltage for Y-axis, Kym ðrad=ðsVÞÞ is the gain of the motor, and Tym ðsÞ is the time constant of the motor. The motor models of the X-, Y- and Z-axes are all described by Eq. (1). Table 1 shows the identification results of the parameters including the maximum velocity Vmax and maximum acceleration Amax of the motors on each axis. This model is described as PM in this paper.
ð3Þ
hf ðtÞ ¼
1 Al
Z 0
t
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 3=2 2 2gh dh ¼ cl Ln 2g hf ðtÞ 3
dV i ðtÞ qðtÞ dt dt
ð5Þ
ð6Þ
Finally, the flow-rate model can be described by Eqs. (7) and (8) from Eqs. (4)–(6) as follows. This model is described as PF in this paper: 3=2 h_ f ðtÞ ¼ af hf ðtÞ þbf oðtÞ
3=2
4.2. Flow-rate model: PF
qðtÞ ¼ cf hf
In this paper, a fan-type ladle as shown in Fig. 6 is used for the pouring. The surface area of the liquid in the fan ladle becomes constant when the ladle is rotated at a constant angular velocity; therefore, pouring outflow per 1 rad Kf can be expressed as Eq. (3),
where
af ¼
ð7Þ
ðtÞ
2cl Ln pffiffiffiffiffiffi 2g ; 3Al
ð8Þ
bf ¼
Kf ; Al
cf ¼
pffiffiffiffiffiffi 2 cl Ln 2g 3
ð9Þ
ARTICLE IN PRESS 234
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
4.3. Liquid level model: PL The liquid level model is built by considering mass balance, Bernoulli’s theorem and a continuous equation as follows (Sugimoto, Yano, & Terashima, 2002): dhc ðtÞ 1 AG pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðtÞcðtÞ 2gðhc ðtÞ þ h1 hs ðtÞÞ ¼ AC dt AC
ð10Þ
dhm ðtÞ AG pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðhc ðtÞ þ h1 hs ðtÞÞ ¼ cðtÞ AM dt
ð11Þ
where c is the flow coefficient (-) and g is gravitational acceleration ðm2 =sÞ. In this paper, a time-variant flow coefficient cðtÞ was introduced in order to model the liquid level from the beginning to the end of the pouring, as shown in Eq. (12). Namely, the initial value of the flow coefficient is set to 0, and shifted up gradually to reach the final flow coefficient ce by the end time of filling the cup, where hc ðtÞ is the liquid level in the cup and href is the target level, set to 0.06 m in this study. A final flow coefficient of ce ¼ 0:51 was obtained by conducting pouring experiments. This model is described as PL in this paper: cðtÞ ¼
ce hc ðtÞ href
ð12Þ
Finally, the pouring process model from the input voltage uðtÞ to the servomotor on the Y axis to the liquid level in the sprue cup hc ðtÞ is derived by connecting the motor model Eq. (2), the flowrate model Eqs. (7) and (8), and the level model Eqs. (10)–(12).
5. Feedforward control of liquid level In this section, the feedforward input that keeps the liquid level constant in the sprue cup is calculated. Suppose that the liquid level in the sprue cup rose with a sinusoidal curve and maintained the target level href shown by Eq. (13) when the ideal flow rate shown in Fig. 3 inflows into a mold. Then this liquid level curve is defined as the target liquid level curve hcref . The target liquid level curve hcref can be calculated if the rising time trise is decided. 8 > < href sin 2p t þ href t ð0 rt o trise Þ 2p trise trise ð13Þ hcref ðtÞ ¼ > : href ðtrise r tÞ Eventually, the feedforward control input for the liquid level control is calculated by substituting the target liquid level curve 1 1 1 PF PL as shown in Fig. 7 and Eq. hcref for the inverse model PM (14), 1 ðsÞPF1 ðsÞPL1 ðsÞhcref u ¼ PM
ð14Þ
Namely, the output hðtÞ can correspond completely to the target hcref by Eq. (15): h ¼ PðsÞu ¼ PL ðsÞPF ðsÞPM ðsÞu
ð15Þ
The feedforward control input is obtained by theoretical analysis using the inverse model of the liquid level model, the inverse flow-rate model and the inverse motor model, as shown below. In this paper, the target level in the sprue cup href was set to 0.06 m and the rising time was set to 2.57 s for level control. Inverse liquid level model: PL1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dhcref ðtÞ ð16Þ qðtÞ ¼ Ac þ cðtÞAG 2gðhcref ðtÞ þ h1 hs ðtÞÞ dt Inverse flow-rate model: PF1 !2=3 3 pffiffiffiffiffiffi qðtÞ hf ðtÞ ¼ 2cl Ln 2g
oðtÞ ¼
pffiffiffiffiffiffi Al dhf ðtÞ 2cl Ln 2g 3=2 hf ðtÞ þ 3Kf Kf dt
uðtÞ ¼
Tmy doðtÞ 1 oðtÞ þ Kmy dt Kmy
PF-1
Inverse Motor Model
Inverse Flowrate Model
PM-1 Inverse Level Model
ð19Þ
The obtained feedforward control input for the forward tilting motion is shown in Fig. 8. In Fig. 8, the liquid level in the sprue cup, the outflow, the tilting acceleration, the input voltage, and the pouring quantity are shown from above. It is confirmed that the obtained outflow is like an ideal flowrate as shown in Fig. 3. Next, the experimental result with the feedforward input is shown in Fig. 9. As a result, the liquid level in the sprue cup was held around setpoint. At this time, the error of liquid level was less than 0.005 m. Thus, the appropriateness of the pouring process model and the validity of feedforward input was confirmed.
6. Back-tilting control with sloshing suppression Yano and Terashima (2005) has designed this system to suppress sloshing using a hybrid shape approach. This approach is a design method for starting control and vibration damping, and satisfies hybrid specifications in both the frequency domain and the time domain by using real-time feedback of only the position data. In fact, in a liquid container transfer system, sloshing is caused by self-transfer movement under circumstances in which there is no disturbance. To suppress sloshing, it is important to shape the frequency response of the controller to be notch-typed at the resonance frequency. It is theoretically possible to suppress vibration by controlling acceleration; therefore, a dynamical model of sloshing is not necessarily required in the control design. Fig. 10 shows the target closed-loop system. In this approach, sloshing can be damped by furnishing the controller with a notch filter at the resonance frequency of the sloshing. Refer to the literature Yano and Terashima (2005) for the details.
u
PL-1
ð18Þ
1 Inverse motor model: PM
Angular Velocity
Input Voltage
Liquid Level Curve hcref
ð17Þ
PM
Motor Model
FF Controller Fig. 7. Block diagram of the inverse model.
Liquid Level
Flow rate q PF
Flowrate Model
PL Level Model
1 s Pouring Process
h Outflow
M
ARTICLE IN PRESS
Liquid Level Curve [m]
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
235
0.09 0.06 0.03 0 1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4 Time [s]
5
6
7
8
0 -4
2
Angular Velocity [rad/s]
0 0.2 0.1 0 -0.1 -0.2
Input Voltage [V]
Flow Rate [m3/s]
x 10 4
0.5 0 -0.5 0
Outflow [m3]
-3
2 1.5 1 0.5 0
x 10
0
Fig. 8. Simulation results in forward tilting.
0.09
Table 2 Sloshing natural frequencies.
Liquid Level [m]
0.08 1st pouring 2nd pouring 3rd pouring 4th pouring
0.07
1st mode
3rd mode
1.27 Hz 1.17 Hz 1.07 Hz 0.97 Hz
– – – 3.02 Hz (18.97 rad/s)
(7.98 rad/s) (7.35 rad/s) (6.72 rad/s) (6.09 rad/s)
0.06 In order to design a controller that does not excite sloshing, it is necessary to specify the natural frequency of the sloshing. Table 2 shows natural frequencies of the sloshing after each pouring. By considering these results, the controller obtained by hybrid shape approach is as follows:
0.05
0.04 0.036
KðsÞ ¼
0.03 0
1
2
3 4 Time [s]
5
Fig. 9. Experimental result by the feedforward control.
6
2 s2 þ2zo s þ o2 Kp Y nj nj Tl s þ 1 j ¼ 1 s2 þ onj s þ o2nj
ð20Þ
where on1 ¼ 7:04 rad=s, on2 ¼ 18:97 rad=s, z ¼ 0:0001, and Kp ¼ 0:0807; Tl ¼ 0:052. Kp and Tl of the controller were determined by a simplex method that minimized the settling time of the back-tilting of the ladle. In Fig. 11, the frequency response of the controller is shown. The controller gain becomes lower in notch-shape around the natural frequency of sloshing. Fig. 12 shows the sloshing after backward tilting. Sloshing is suppressed better by the hybrid shape approach than by proportional control.
7. Predictive control of flow quantity
Fig. 10. Target closed-loop system.
Residual pouring quantity (RPQ) is defined as the liquid weight flowing out of the ladle from the start of the cutting part to the
ARTICLE IN PRESS 236
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
−20
M1 (t0) Weight [kg]
−30
Gain [dB]
−40
−50
0 −60
t0
Time [s]
−70
M1 (t0) −80 0 10
1
2
10 10 Angular frequency [rad/s]
M2
Fig. 11. Frequency response of the controller.
Fig. 13. Static outflow of the over weight in the ladle.
x 10−3 5 Controller by Hybrid Shape approach Propotional control
4
Weight [kg]
3
sloshing [m]
2 1 0
M2 (t)
0
−1
Time [s]
t0
−2 −3 14
15
16
17 Time [s]
18
19
20
Fig. 12. Sloshing after backward tilting.
end, during back-tilting of the ladle. In order to conduct the control of flow quantity from the ladle to the mold exactly, it is necessary to predict the RPQ. In order to predict the RPQ, let us consider the state of the liquid inside the ladle as shown in Fig. 13. Then, the liquid inside a ladle is considered as two parts, comprised the upper part of the liquid above the nozzle and the lower part under the nozzle. The upper part is called the ‘‘over weight part, M1 ’’ , and lower part the ‘‘equilibrium weight part, M2 ’’ . The total weight inside the ladle becomes M1 þ M2 . When the pouring is stopped in the state as shown in Fig. 13, the metal weight in the ladle finally becomes only the equilibrium weight. Let us consider the decrease of the over weight M1 with the increase of time. The relationship between the tilting angular velocity of the ladle o and the liquid level in the ladle hf is described in Eq. (7). In this case, Eq. (7) can be rewritten as follows because o ¼ 0: h_ f ðtÞ ¼ Af hf ðtÞ3=2
ð21Þ
M2 Fig. 14. Change of the over weight by the back-tilting motion.
The liquid level hf can be calculated to solve the differential equation as Eq. (22), where hf 0 is the initial liquid level in the ladle: 1=2 2
hf ðtÞ ¼ ð12Af t þ hf 0
Þ
ð22Þ
Then, the over weight M1 is calculated by multiplying the surface area Al ¼ RLRn ðLLn Þ by hf , as shown in Eq. (24). Eq. (24) is then transformed into a discrete-time equation as shown in Eq. (24): 1=2 2
M1 ðtÞ ¼ Al ð12Af t þ hf 0
Þ
M1 ½kþ j ¼ Al ð12Af t½j þ hf ½k1=2 Þ2
ð23Þ ð24Þ
Next, let us consider the change of M2 with the increase of time. During back-tilting, the ladle is tilted backward at the center of
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
r ¼ 1000 kg=m3 , it follows that
gravity of the ladle to reduce the sloshing because of decreasing inertia. As shown in Fig. 14, the equilibrium weight increases with back-tilting. M2 ðtÞ can be calculated by the tilting angle yðtÞ of the ladle, as shown in Fig. 15. The ladle is tilted backward by the hybrid-shape controller described in the previous section. When the closed-loop system from the reference r to the tilting angle y in Fig. 10 is described as Gcl ¼ fA2 ; B2 ; C2 ; 0g, the discrete-time state equation becomes as follows: xd2 ½k þ1 ¼ Ad2 xd2 ½k þBd2 ud2 ½k
ð25Þ
y½k ¼ Cd2 xd2 ½k
ð26Þ
M2 ½k þj ¼ rKf Cd2 Ajd2 xd2 ½k þ
j1 X
! Cd2 Aj1i Bd2 yr ½k þi d2
ð29Þ
i¼0
The liquid weight over the nozzle in the ladle becomes M1 ðtÞ minus M2 ðtÞ while back-tilting as shown in Fig. 15. M1 ðtÞ equals M2 ðtÞ at time tn . Therefore, after tn , the fluid does not flow out of the ladle, even if the ladle is tilted further backward. Hence, the prediction value Q^ p ðt0 Þ of RPQ at time t0 becomes Q^ p ðt0 Þ ¼ M1 ðt0 ÞM1 ðtn Þ
ð30Þ
If the following condition is satisfied, the supervisor sends a signal
Then, it follows that
s to the controller, and the controller is switched from the level
y½k þ1 ¼ Cd2 xd2 ½kþ 1 ¼ Cd2 ðAd2 xd2 ½k þBd2 ud2 ½kÞ
controller to the back-tilting controller. Switching condition:
ð27Þ
Further, it follows that
y½k þj ¼ Cd2 Ajd2 xd2 ½k þ
237
j1 X
Qref ¼ Q^ p ðt0 Þ þ Mðt0 Þ Cd2 Aj1i Bd2 yr ½k þ i d2
ð28Þ
ð31Þ
where Mðt0 Þ is the total metal weight flowing out of the ladle up to time t0 which can be measured by a load cell, and Qref is the reference pouring weight. Fig. 16 shows experimental results by the supervisory control with prediction of the residual pouring quantity (RPQ). Supervisory control with prediction of flow quantity shows better results than the results without prediction. For a reference weight of 1.274 kg, the proposed method attained 1.282 kg, and the conventional method without prediction attained 1.304 kg. Experiments have been done with various tilting velocities and angles. In any case, the proposed method showed good results at a similar precision as that shown in Fig. 16.
i¼0
Because the increment M2 ½k þ j ðkgÞ of the equilibrium weight can be calculated by multiplying the trajectory y½k þj ðdegÞ during back-tilting with the flow gain Kf ðm3 =degÞ and the density
M1(t0) M2(t) Weight [kg]
Qˆ p 8. Adaptive feedforward control of pouring process considering a disturbance
M1(t) In conventional methods, the slag attached to the inside of the ladle is not considered. Therefore, it cannot be guaranteed that the change in the outflow pattern of molten metal will not affect the outcome. In other words, if a nominal control input is applied to the disturbed ladle, the liquid level in the sprue cup becomes unstable and the supply of the molten metal into the mold falls
0 t0
Time [s]
Input [V]
Fig. 15. Predictive value of the outflow.
0.5 0 0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
10 5
CupLevel [m]
0 0.08
Outflow [kg]
Angle [deg]
-0.5
1.5 1 0.5 0
0.06 0.04
Supervisory Control Without prediction Set point
0
1
2
3
4
5
6
Time [s] Fig. 16. Experimental result by the flow-rate control.
7
8
9
ARTICLE IN PRESS 238
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
These are obtained by calculating Eqs. (7) and (8) by the Runge– Kutta method, where the sampling time is set to 0.01 s and a ¼ 1000 and l ¼ 0:97 in this study.
short. As a result, impurities such as dust, air bubbles, and slag are entrapped in the mold, and the quality of the product decreases. Furthermore, molten metal overflows the mold if the control input changes inappropriately. Therefore, it is desirable to construct a control system that realizes liquid level and flow rate control in any kind of situation. The control system developed in this study estimates changes in flow-rate-model parameters between each pouring cycle and updates the feedforward input step by step.
8.2. Renewal of feedforward control input When the flow-rate model is varied by a disturbance such as accumulated slag, it is necessary to renew the feedforward control input for every pouring motion. Whenever pouring is repeated, this disturbance accumulates. The pouring system that considers this disturbance is shown in Fig. 17. In Fig. 17, the control system consists of the motor model PM , the flow-rate model PF , the level model PL and the estimator EF . This system estimates the parameters of the flow-rate model, and the feedforward control input for the liquid level control is renewed for the next mold. As for the feedforward controller, it is derived from the inverse model of the flow-rate model and the motor model. The controller renews the liquid level control input by substituting the estimated parameters for an inverse model of the flow-rate model. u, o, q, and hc are the input voltage (V) to the servo-motor, the angular velocity (deg/s), the flow quantity ðm3 =sÞ, and the liquid level (m), respectively. In addition, qref is the reference flow quantity ðm3 =sÞ in order to control the liquid level in the sprue cup.
8.1. Estimation of flow-rate-model parameters Because cf is cf ¼ af Al as shown in Eq. (9), cf can be calculated by af , where Al is the cross-sectional area of the ladle. Therefore, an estimate of the flow-rate-model parameters is performed on af and bf in Eq. (7). The recursive least-square method as shown in the following equations is used so that the parameters are estimated online during the pouring motion: ^ y^ F ðkÞ ¼ Pðk1Þ þ
Gðk1ÞcðkÞ eðkÞ l þ cðkÞT Gðk1ÞcðkÞ
ð32Þ
eðkÞ ¼ yðkÞcðkÞT y^ F ðk1Þ GðkÞ ¼ "
c¼
1
(
l 3=2
hf
Gðk1Þ
ð33Þ
Gðk1ÞcðkÞcðkÞT Gðk1Þ l þ cðkÞT Gðk1ÞcðkÞ
) ð34Þ
8.3. Verification by experiments
# ð35Þ
o
A gate was established on both sides of the nozzle as a disturbance of the flow-rate model, as shown in Fig. 18. The nozzle width was changed from 65 to 41 mm. The experimental result with the nominal pouring control input for the disturbed system is shown in Fig. 19. In the experiment, a model predictive control was applied to the back-tilting control considering the sloshing suppression control and the flow quantity as stated in the previous sections. As a result of Fig. 19, the liquid level could not be kept at the target level because of the disturbance. The tilting angular velocity and the liquid level in the ladle, which are the input
where y^ F is the estimation value consisting of parameters a^ f and b^ f in the flow-rate model, l is the oblivion coefficient, e is the estimated error, and G is the error covariance matrix. The output yðkÞ becomes yðkÞ ¼ h_ f . In addition, the initial values of the estimation value y^ F and the error covariance matrix G are set as Eq. (36): " # a^ f ð0Þ a 0 ^y ð0Þ ¼ ; ð36Þ G ð0Þ ¼ F b^ f ð0Þ 0 a
Flow rate model parameter qF hcref
PL-1
qref
PˆF-1
Level inverse model
Control input wref
Flow rate inverse model
Angular velocity
u
q
w
PM-1
PF
PM
Motor inverse model
Motor model
qref
Level inverse model
PˆF-1 Flow rate inverse model
hc
level model 1st Pouring
EF
qF
PL-1
PL
Flow rate model
Flow rate model parameter
hcref
Cup level
Flow rate
Estimator
u
wref -1 M
P
Motor inverse model
PM
w
Motor model
q PF
PL
Flow rate model
level model 2nd Pouring
EF
Fig. 17. Intelligent control system for the accumulating slag.
hc
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
12
41
and output for estimating flow-rate-model parameters, are shown in Fig. 20. Fig. 21 shows the estimated results of the parameters, where the liquid level in the ladle was calculated by the tilting angular velocity measured by an encoder and the ladle weight measured by a load cell. As a result of estimation, the flow-rate-model parameters were obtained as a^ f ¼ 0:0520 and a^ f ¼ 3:316. Then, the control input was renewed with these new parameters. The renewed feedforward input based on the updated parameters is shown in Fig. 22, where the chained line shows the control result without a disturbance and with the nominal control input, the dashed line shows the control result with a disturbance and with the nominal control input, and the solid line the result with a disturbance but with the renewed control input. As seen from the figure, it was possible to control the liquid level in the sprue cup and the pouring flow quantity by renewing the pouring control input with an estimation of the change in the pouring flow-rate model due to the accumulated slag attached to the inside of the ladle of the same quality as is the case without disturbance. Actually, in a real pouring factory, slag forms slowly in the ladle; such a large amount of slag as in this experiment does not form suddenly. Therefore, it can be controlled very nicely over time because the control input is renewed step by step from the start time.
12
Dam
65 Unit : mm
Nozzle Tip
Input Voltage [V]
Fig. 18. Disturbance of the flow-rate model.
0.4 0.2 0 −0.2 −0.4 0
1
2
3
4
5
6
7
8
9
0.08
9. Conclusions
0.06
This paper presents a supervisory control system for an automatic molten-metal pouring process that replicates expert pourers’ skill in order to improve the productivity of the factory, the safety of workers, and the quality of the product. Feedforward control inputs for the pouring process were obtained, and the liquid level in the sprue cup was held to around the set point. In order to avoid overflow of liquid from the sprue cup during backtilting, a method to predict the quantity flowing out of the ladle during back-tilting was proposed. It was shown that by
0.04 0.02 0 0
1
2
3
4 5 Time [s]
6
7
8
9
Fig. 19. Experimental results of FF input with a disturbed model.
Angular Velocity [rad/s]
0.2 0.1 0 -0.1 -0.2
Liquid Height in Ladle [mm]
Liquid Level [m]
239
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4 5 Time [s]
6
7
8
9
30 25 20 15 10 5 0
Fig. 20. Experimental results of the angular velocity and liquid height in a ladle with the disturbed model.
ARTICLE IN PRESS 240
K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
0
Parameter af
−0.02 −0.04 −0.06 −0.08 −0.1 0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
5
Parameter bf
4 3 2 1 0 Time [s] Fig. 21. Experimental results of the estimated flow-rate-model parameters.
Renewed FF Input Nominal FF Input Without Disturbance
Input Voltage [V]
0.4 0.2 0 −0.2 −0.4 0
1
2
3
4
5
6
7
8
9
0
1
2
3
4 5 Time [s]
6
7
8
9
Liquid Level [m]
0.08 0.06 0.04 0.02 0
Fig. 22. Experimental results of renewed FF input with a disturbed model.
supervising the predicted information on the flow quantity in real time, the switching from a level control system to a back-tilting control system with damping of sloshing can be well executed. In the end, an intelligent control system that can recalculate the feedforward control input by estimating the change in the pouring flow-rate model due to slag was presented. As a result, a pouring control system was realized that could control the liquid level in the sprue cup and the pouring flow quantity by renewing the pouring control input with an estimation of the change in the
pouring flow-rate model due to the accumulated slag attached to the inside of the ladle.
References Barron, M. A., Aguilar, R., Gonzalez, J., & Melendez, E. (1998). Model-based control of mold level in a continuous caster under model uncertainties. Control Engineering Practice, 6, 191–196.
ARTICLE IN PRESS K. Yano, K. Terashima / Control Engineering Practice 18 (2010) 230–241
Feddema, J. T., et al. (1997). Control for slosh-free motion of an open container. IEEE Control Systems Magazine, 17, 29–36. Kitada, H., Kondo, O., Kusachi, H., & Sasame, K. (1998). H1 control of molten steel level in continuous caster. IEEE Transactions on Control Systems Technology, 6, 200–207. Lerner, Y., & Laukhin, N. (2000). Development trends in pouring technology. Foundry Trade Journal, 11, 16–21. Lindsay, W. (1983). Automatic pouring and metal distribution system. Foundry Trade Journal, 151–165. Sugimoto, Y., Yano, K., & Terashima, K. (2002). Liquid level control of automatic pouring robot by two-degrees-of-freedom control. In Proceedings of IFAC World Congress (pp. 21–26). Terashima, K., & Yano, K. (2001). Sloshing analysis and suppression control of tilting-type automatic pouring machine. Control Engineering Practice, 9, 607–620. Terashima, K., Yano, K., & Sugimoto, Y. (2003). Modeling and robust control of liquid level in a sprue cup for batch-type casting pouring processes. In Proceedings of the IASTED international conference on intelligent systems and control (pp. 33–38).
241
Terashima, K., Yano, K., Sugimoto, Y., & Watanabe, M. (2001). Position control of ladle tip and sloshing suppression during tilting motion in automatic pouring machine. In Proceedings of IFAC symposium on automation in mining, mineral and metal processing (pp. 182–187). Yano, K., & Terashima, K. (2001). Robust liquid container transfer control for complete sloshing suppression. IEEE Transactions on Control Systems Technology, 9, 483–493. Yano, K., & Terashima, K. (2005). Sloshing suppression control of liquid transfer systems considering 3d transfer path. IEEE/ASME Transactions on Mechatronics, 10, 8–16. Yano, K., Higashikawa, S., & Terashima, K. (2002). Motion control of liquid container considering an inclined transfer path. Control Engineering Practice, 10, 465–472. Yano, K., Toda, T., & Terashima, K. (2001). Sloshing suppression control of automatic pouring robot by hybrid shape approach. In Proceedings of IEEE conference on decision and control (pp. 1328–1333).