Mold level control device and method

Abstract

To provide a molten metal surface level controller and a molten metal surface level control method which can suppress periodic bulging disturbance while maintaining a wide control bandwidth of feedback control.SOLUTION: A molten metal surface level controller CT according to the present invention, controls a molten metal surface level of a continuous casting machine CM by using feedback control to control an adjustment device that adjusts a molten steel quantity supplied to a casting mold of the continuous casting machine CM. The molten metal surface level controller CT includes: an FF compensator 12 that determines a feed-forward compensation amount composed of a sinusoidal wave component having an amplitude, a period, and a phase for eradicating outer disturbance to be applied to a molten metal surface level on the basis of the molten metal surface level, a manipulation amount of the adjustment device, and a transfer function of the continuous casting machine CM related to the molten metal surface level: and an adder 13 that, when performing control by using the feedback control, adds the feed-forward compensation amount determined by the FF compensator 12 to a feedback compensation amount in the feedback control.SELECTED DRAWING: Figure 2

Description

[Technical Field]

[0001] The present invention relates to a molten metal level control device and a molten metal level control method for controlling the molten metal level in a mold in a continuous casting machine. [Background technology]

[0002] In continuous casting machines that continuously solidify molten steel to produce slabs, which are semi-finished products with a uniform shape, a phenomenon known as bulging occurs. This bulging occurs when the solidified shell bulges outward due to the static pressure of the molten steel inside, and then this bulge collapses when the slab is withdrawn during casting, resulting in a cyclical repetition of these bulging and collapse cycles. This bulging causes the surface of the molten steel (molten metal level) in the solidified shell to rise and fall (fluctuations in the molten metal level). Therefore, suppressing fluctuations in the molten metal level due to bulging and maintaining a constant molten metal level in the mold is important for maintaining good quality of the slab (steel billet). Techniques for maintaining a constant molten metal level are disclosed, for example, in Patent Documents 1 and 2.

[0003] The mold level control device for a continuous casting machine disclosed in Patent Document 1 is a device that measures the molten metal level in a mold in a continuous casting machine for molten metal, and controls the opening of a means for pouring molten metal into the mold in accordance with an opening command calculated using the deviation between the measured molten metal level and a target molten metal level, so as to maintain the molten metal level at the target molten metal level. The device includes an opening control unit that adjusts the opening of the pouring means based on the measured molten metal level and the target molten metal level, a disturbance estimation unit that estimates disturbances that occur when the molten metal is poured into the mold based on the measured molten metal level and the past opening of the pouring means, and calculates an estimated disturbance amount value, and a bulging visualization monitoring unit that calculates a fluctuation period of unsteady bulging based on the estimated disturbance amount calculated by the disturbance estimation unit and the casting speed of the continuous casting machine. The bulging visualization monitoring unit adjusts the opening of the means for pouring molten metal into the mold in accordance with the measured molten metal level and the target molten metal level. and the casting speed; an oversampling unit that converts time series data of the disturbance amount estimate value acquired by the data acquisition unit over a predetermined period into pre-oversampling data based on the casting length, which is an integral value of the casting speed, and oversamples the pre-oversampling data based on the casting length to generate post-oversampling data; an FFT processing unit that performs fast Fourier transform analysis on the post-oversampling data based on the casting length to calculate a distance frequency spectrum of an unsteady bulging occurrence roll pitch at which unsteady bulging occurs; and a peak detection unit that detects a peak in the distance frequency spectrum to calculate the unsteady bulging occurrence roll pitch, and calculates a fluctuation period of the unsteady bulging based on the unsteady bulging occurrence roll pitch and the casting speed.

[0004] The mold level control device for a continuous casting machine disclosed in Patent Document 1 estimates disturbances using a disturbance estimation unit (disturbance estimation observer). At this estimation stage, the estimation of the disturbance is not based on the casting length, and the disturbance estimation observer itself has dynamic characteristics, so a phase lag occurs in the estimated disturbance. Therefore, even if the frequency (or period), amplitude, and phase of the disturbance based on the slab length are accurately obtained, the phase lags and the amplitude becomes small, making it impossible to accurately obtain the phase and amplitude.

[0005] On the other hand, the molten metal level control device for a continuous casting machine disclosed in Patent Document 2 detects the molten metal level in the mold of the continuous casting machine, determines the deviation between the detected molten metal level and a predetermined target level, calculates the amount of change in the opening of the inlet for molten metal into the mold based on the determined deviation, changes the opening in accordance with the calculated amount of change, and controls the molten metal level to maintain the target level.The device is equipped with a frequency detection unit that detects the frequency of periodic level fluctuations contained in the molten metal level, an oscillator that oscillates a signal at the frequency detected by the frequency detection unit, a phase / amplitude calculation unit that calculates a phase and amplitude so as to cancel out the periodic level fluctuations, and an adder that adds the signal oscillated by the oscillator and having the phase and amplitude calculated by the phase / amplitude calculation unit to the calculated amount of change, thereby correcting the amount of change in the opening.

[0006] The mold level control device for a continuous casting machine disclosed in Patent Document 2 adds a signal separate from the feedback control signal, potentially enabling it to suppress high-frequency disturbances and disturbances with multiple frequencies that are difficult to control with feedback control. However, Patent Document 2 detects only the frequency of fluctuations "contained in the mold level signal," and does not include the frequency of fluctuations in the manipulated variable or the frequency of fluctuations in the flow rate regulator opening. Therefore, even if the disturbance is suppressed and vibration-related level fluctuations disappear, the mold level control device for a continuous casting machine disclosed in Patent Document 2 cannot be applied. Furthermore, there is a possibility that level fluctuations caused by the manipulated variable may be interpreted as level fluctuations due to a disturbance. In other words, the influence of the manipulated variable is not taken into account. [Prior art documents] [Patent documents]

[0007] [Patent Document 1] Japanese Patent Application Laid-Open No. 2011-143414 [Patent Document 2] Japanese Patent Application Laid-Open No. 2002-059249 Summary of the Invention [Problem to be solved by the invention]

[0008] Conventionally, mold level control has had the following problems. The first problem is the limitations of feedback control. Because the control target in mold level control is an integral system, there is a 90-degree phase delay. For this reason, feedback control requires that the phase be advanced and the gain be increased at the frequency of the mold level fluctuations caused by bulging. While this can suppress disturbances of specific frequencies, it has been confirmed that it narrows the control band in order to ensure stability. For this reason, it is difficult to respond when disturbances of multiple frequencies are applied, when the frequency changes, or when high-frequency disturbances are applied.

[0009] There are two possible countermeasures. The first is to narrow the control band at the expense of the other, and suppress disturbances at specific frequencies, although it becomes more difficult to suppress disturbances at low frequencies. The second is to ensure a wide control band and suppress disturbances at low frequencies, although there are limits to suppressing disturbances at specific frequencies.

[0010] This second measure (method) was able to keep fluctuations in the molten metal level small by mainly suppressing low-frequency disturbances, even when multiple frequency disturbances were applied, the frequency changed, or high-frequency disturbances were applied. However, it was not possible to suppress disturbances caused by bulging to nearly zero, and it is desirable to respond to recent stricter quality standards and the need to ensure production capacity.

[0011] The present invention has been made in view of the above circumstances, and an object of the present invention is to provide a molten metal level control device and a molten metal level control method that can suppress periodic bulging disturbances while maintaining a wide control band of feedback control. [Means for solving the problem]

[0012] As a result of various investigations, the present inventors have found that the above object can be achieved by the present invention described below. That is, a mold level control device according to one aspect of the present invention is a device for controlling the mold level of a continuous casting machine by using feedback control to control an adjustment device that adjusts the amount of molten steel supplied to a mold of the continuous casting machine. The device includes an FF compensation unit that calculates a feedforward compensation amount consisting of sinusoidal components having an amplitude, period, and phase for canceling out disturbances applied to the mold level, based on the mold level, an operation amount of the adjustment device, and a transfer function of the continuous casting machine that relates to the mold level, and an adder that adds the feedforward compensation amount calculated by the FF compensation unit to the feedback compensation amount of the feedback control when controlling using the feedback control. When the mold level and the operation amount of the adjustment device are multiple values ​​in the time domain that are arranged in a chronological order at predetermined time intervals, the FF compensation unit converts the values ​​of the mold level and the operation amount of the adjustment device in the time domain into values ​​in a spatial domain whose coordinate axes are in the longitudinal direction of the slab produced by the continuous casting machine, and converts the transfer function from the time frequency domain to the spatial frequency domain before calculating the feedforward compensation amount.

[0013] This type of mold level control device calculates not only the amplitude and period but also the phase in order to cancel out disturbances, and therefore can suppress fluctuations in the mold level due to disturbances through feedforward compensation. Since the mold level control device adds the feedforward compensation amount to the feedback compensation amount, it is less likely to affect the feedback control, and therefore the feedback control band can be maintained wide. Therefore, the mold level control device can suppress periodic bulging disturbances while maintaining a wide control band of feedback control.

[0014] In the above-described mold level control device, when the mold level and the manipulated variable of the adjusting device are a plurality of values ​​arranged in time series at predetermined time intervals in the time domain, the FF compensation unit converts the values ​​of the mold level and the manipulated variable of the adjusting device in the time domain into values ​​in a spatial domain having coordinate axes in the longitudinal direction of the slab produced by the continuous casting machine, and converts the transfer function from the time frequency domain to the spatial frequency domain, before determining the feedforward compensation amount.

[0015] When the casting speed is changed, the intervals between values ​​on the slab are also changed in the time domain, which affects the casting speed. However, since the above-described mold level control device changes from the time frequency domain to the spatial frequency domain, it is possible to appropriately determine the feedforward compensation amount and appropriately control the mold level without being affected by such an effect.

[0016] In another aspect, in the above-mentioned molten metal surface level control device, the FF compensation unit obtains a plurality of the sine wave components for a plurality of frequencies different from each other, and obtains the amount of cancellation of the disturbance for each of a first phase of a first sine wave component having a peak from among the plurality of sine wave components, a second phase of a second sine wave component on the low frequency side relative to the first sine wave component, and a third phase of a third sine wave component on the high frequency side relative to the first sine wave component, and selects the phase that most cancels the disturbance from among the first to third phases as the phase of the feedback compensation amount.

[0017] The feedforward compensation amount may be composed of a first sine wave component having a peak, but the first sine wave component may have an opposite phase to the desired phase for canceling the disturbance. The above-mentioned molten metal level control device selects the phase that best cancels the disturbance from the first to third phases of the first to third sine wave components as the phase of the feedforward compensation amount, thereby enabling selection of a more appropriate phase. The above-mentioned molten metal level control device can also handle cases where disturbances of multiple frequencies are applied. For example, when peaks are adjacent, the phase also changes significantly near the frequency of the adjacent peaks. However, by determining the amount of cancellation of the disturbance and determining the phase that cancels the disturbance, the above-mentioned molten metal level control device can select an appropriate phase.

[0018] In another aspect, the above-mentioned molten metal surface level control device further includes a gain type changing unit that changes the gain of the feedback control to be smaller than the current value when the absolute value of the rate of the manipulated variable of the adjustment device is larger than a predetermined first threshold, and changes the gain of the feedback control to be larger than the current value when the absolute value of the rate of the manipulated variable of the adjustment device is smaller than a predetermined positive second threshold that is smaller than the first threshold.

[0019] Such a molten metal level control device can perform feedback control with an appropriate gain according to the absolute value of the rate of the manipulated variable of the adjusting device, and can control the molten metal level within the rate limit (within the operating speed) of the actuator that drives the valve. Furthermore, the above-mentioned molten metal level control device can prevent the control system from becoming unstable due to modeling errors, etc.

[0020] In another aspect, in these above-mentioned molten metal level control devices, feedforward compensation is performed in a time-frequency domain that is lower than the time frequency at which the phase difference of the modeling error of the controlled object is π / 3 and lower than the time frequency of the surface wave of the molten metal surface in the mold.

[0021] In such a mold level control device, the feedforward compensation is robust against modeling errors.

[0022] A mold level control method according to one aspect of the present invention controls the mold level of a continuous casting machine by using feedback control to control an adjustment device that adjusts the amount of molten steel supplied from a tundish of the continuous casting machine. The method includes: an FF compensation step of determining a feedforward compensation amount consisting of sinusoidal components having an amplitude, period, and phase for canceling out disturbances applied to the mold level, based on the mold level, an operation variable of the adjustment device, and a transfer function of the continuous casting machine that is related to the mold level; and an adding step of adding the feedforward compensation amount determined in the FF compensation step to the feedback compensation amount of the feedback control when controlling using the feedback control. When the mold level and the operation variable of the adjustment device are multiple values ​​in the time domain that are arranged in a chronological order at predetermined time intervals, the FF compensation step converts the values ​​of the mold level and the operation variable of the adjustment device in the time domain into values ​​in a spatial domain whose coordinate axes are in the longitudinal direction of the slab produced by the continuous casting machine, and converts the transfer function from the time frequency domain to the spatial frequency domain before determining the feedforward compensation amount.

[0023] This type of mold level control method determines not only the amplitude and period but also the phase in order to cancel out disturbances, so that fluctuations in the mold level due to disturbances can be suppressed by feedforward compensation. Since the above mold level control method adds a feedforward compensation amount to the feedback compensation amount, it is less likely to affect feedback control, so the bandwidth of the feedback control can be maintained wide. Therefore, the above mold level control method can suppress periodic bulging disturbances while maintaining a wide control bandwidth of feedback control. Since the above mold level control method changes from the time frequency domain to the spatial frequency domain, it can appropriately determine the feedforward compensation amount without being affected by such disturbances, and can appropriately control the mold level. [Effects of the Invention]

[0024] The molten metal level control device and molten metal level control method according to the present invention can suppress periodic bulging disturbances while maintaining a wide control band of feedback control. [Brief explanation of the drawings]

[0025] [Figure 1] FIG. 1 is a schematic diagram showing a configuration of a continuous casting machine according to an embodiment. [Figure 2] FIG. 2 is a block diagram of the continuous casting machine relating to control of the molten metal surface level. [Figure 3] FIG. 3 is a block diagram showing the configuration of an FB compensator shown in FIG. 2. [Figure 4] 10 is a graph showing a Bode diagram and a frequency shaping result of an FB compensator including a PID control unit and a second-order filter unit, the Bode diagram being obtained by solving a mixed sensitivity problem. [Figure 5] 10 is a graph showing a Bode diagram of an H∞ controller and a frequency shaping result obtained by solving a mixed sensitivity problem in the H∞ controller. [Figure 6] 10 is a lookup table showing each parameter in the FB compensator for each Kf / A and each gain type. [Figure 7] FIG. 10 is a diagram illustrating an example of an interpolation method for each parameter Kp in the FB compensator. [Figure 8] FIG. 1 is a diagram illustrating a transformation from the time domain to the space domain. [Figure 9] As an example, this figure shows the simulation results of a value (a value equivalent to a disturbance manipulated variable) obtained by back-calculating an unknown disturbance up to the manipulated variable. [Figure 10] FIG. 10 is a diagram illustrating a gain for reducing the amount of feedforward compensation for high-frequency disturbances. [Figure 11] 3 is a flowchart showing the operation of the molten metal surface level control device in the embodiment. [Figure 12] 12 is a flowchart showing a process for calculating a feedforward compensation amount shown in FIG. 11. [Figure 13]FIG. 10 is a diagram showing a simulation result in the case of a constant casting speed. [Figure 14] FIG. 10 shows simulation results for the case of variable casting speed. [Figure 15] FIG. 10 is a diagram showing a simulation result in a comparative example. [Figure 16] FIG. 10 is a diagram showing a simulation result when modeling errors are taken into consideration. DETAILED DESCRIPTION OF THE INVENTION

[0026] Hereinafter, one or more embodiments of the present invention will be described with reference to the drawings. However, the scope of the invention is not limited to the disclosed embodiments. In addition, components with the same reference numerals in each drawing indicate the same components, and their description will be omitted as appropriate. In this specification, when referring to a general term, a reference numeral without a subscript is used, and when referring to an individual component, a reference numeral with a subscript is used.

[0027] The mold level control device in this embodiment controls the mold level of a continuous casting machine by using feedback control to control a valve that adjusts the amount of molten steel supplied from a tundish of the continuous casting machine to a mold. In this embodiment, the mold level control device further includes an FF compensation unit that calculates a feedforward compensation amount consisting of a sinusoidal component having an amplitude, period, and phase to cancel out disturbances applied to the mold level based on the mold level, the valve opening or operation amount of the valve, and a transfer function of the continuous casting machine related to the mold level. Also included is an adder that, when controlling using the feedback control, adds the feedforward compensation amount calculated by the FF compensation unit to the feedback compensation amount of the feedback control. That is, the mold level control device in this embodiment controls the mold level by adding a feedforward compensation amount for canceling out disturbances to the feedback compensation amount. This mold level control device will be described in more detail below.

[0028] Fig. 1 is a schematic diagram showing the configuration of a continuous casting machine according to an embodiment, and Fig. 2 is a block diagram of the continuous casting machine relating to control of the molten metal surface level.

[0029] The continuous casting machine CM, in which the molten metal level is controlled by the molten metal level control device CT in the embodiment, includes, for example, a tundish TD, a mold MD, and multiple sets of rolls RL, which are arranged in this order from upstream to downstream, as shown in FIG.

[0030] The tundish TD is a device into which molten steel flows from a ladle (not shown) and into which predetermined inclusions are removed. The inclusions are removed by floating up and separating. The ladle is a device into which molten steel after secondary refining, in which the composition has been finely adjusted, is poured and which stores the molten steel, and is disposed at the top. Note that in the ladle, some of the inclusions are also removed by floating up and separating. The tundish TD is disposed next to the ladle.

[0031] The tundish TD has a through hole formed in its bottom, and a nozzle NZ is attached to the through hole to guide the molten steel to the mold MD. A valve VL is attached to the nozzle NZ to adjust the amount of molten steel supplied from the tundish TD to the mold MD. The valve VL is, for example, a slide valve. An actuator AC is provided on the valve VL to drive the valve VL. The actuator AC is, for example, a stepping cylinder. Although the example shown in FIG. 1 uses the valve VL, it may be another device, such as a stopper, that can adjust the flow rate of molten steel and obtain a transfer function.

[0032] The mold MD is a device into which the molten steel flows from the tundish TD through a nozzle NZ, and the molten steel is cooled to form a cast slab Ob of a predetermined shape. The mold MD is water-cooled, and the molten steel that comes into contact with the mold MD solidifies from the outside to form a relatively thin solidified shell, which is then formed into the cast slab Ob of the predetermined shape. The mold MD is disposed next to the tundish TD.

[0033] A level measurement unit 1 is provided near the mold MD to measure the height of the surface of molten steel (molten steel level) in the mold. The level measurement unit 1 is, for example, an eddy current sensor. The molten steel level is controlled by a height value with a vertically upward direction being positive. The level measurement unit 1 outputs the measured molten steel level to the mold level control device CT as a controlled variable y (y(s)). Based on the molten steel level (controlled variable y) and other factors, the molten steel level control device CT generates a manipulated variable u (u(s)), as described below, and outputs this manipulated variable u to an actuator AC. The actuator AC drives the valve VL so that the valve opening v (v(s)) corresponds to the manipulated variable u. This adjusts the cross-sectional area of ​​the flow path of molten steel flowing through the nozzle NZ, thereby adjusting the amount of molten steel supplied from the tundish TD to the mold MD and controlling the molten steel level.

[0034] The rolls (rollers) RL are devices that support the slab Ob while withdrawing the slab Ob from the mold MD at a variable speed (casting speed). The rolls RL are arranged in pairs so as to contact both sides of the slab Ob, and are arranged at predetermined intervals from the upstream side to the downstream side next to the mold MD. The predetermined intervals may be equal or may be unequal, such as being wider downstream than upstream. The multiple sets of rolls RL are arranged so that the casting direction (withdrawing direction) is oriented from the vertical direction to the horizontal direction. The multiple sets of rolls RL are each composed of one or more drive rolls and multiple driven rolls.

[0035] The continuous casting machine CM is further equipped with various known sensors (not shown), such as a sensor for determining the mass of molten steel in the tundish, a sensor for determining the mold width, and a sensor for determining the casting speed.

[0036] It has been confirmed that when bulging occurs in such a continuous caster CM, its period is (roll pitch at the location where bulging occurs) / (casting speed). Therefore, in the time domain, the bulging period [s] is inversely proportional to the casting speed. On the other hand, in the spatial domain where the slab is fixed, the period [m] is the value of the roll pitch and is independent of the casting speed.

[0037] In a block diagram, for example, as shown in Figure 2, the continuous casting machine CM for controlling the molten metal level comprises a molten metal level control device CT, an actuator AC, and a block VL_MD_1 which combines a valve VL, a mold MD, and a level measurement unit 1, and the molten metal level control device CT comprises a subtractor 10, an FB compensator 11, an FF compensator 12, and an adder 13.

[0038] The mold level control device CT is configured with, for example, a computer (including a PLC (Programmable Logic Controller)) equipped with a CPU (Central Processing Unit), memory, an input / output interface, and peripheral circuits, and the subtractor 10, FB compensator 11, FF compensator 12, and adder 13 are functionally configured in the CPU by executing a predetermined program. The memory includes, for example, a ROM (Read Only Memory), which is a nonvolatile memory element, an EEPROM (Electrically Erasable Programmable Read Only Memory), which is a rewritable nonvolatile memory element, and a RAM (Random Access Memory), which serves as the CPU's working memory and stores data generated during program execution. The input / output interface inputs and outputs predetermined data to and from various sensors and devices higher than the mold level control device CT.

[0039] The subtractor 10 receives the target value of the molten metal surface level and the controlled variable y(s), calculates the deviation (difference) between these target values ​​and the controlled variable y(s), and outputs the calculated deviation to the FB compensator 11.

[0040] The FB compensator 11 determines the amount of feedback compensation according to the deviation input from the subtractor 10 as described below, and outputs the determined amount of feedback compensation to the adder 13 .

[0041] The FF compensator 12 determines a feedforward compensation amount consisting of a sine wave component having an amplitude, period, and phase for canceling out the disturbance applied to the molten metal level, based on the molten metal level, the valve opening of the valve VL, and a transfer function of the continuous casting machine CM related to the molten metal level, as will be described later, and outputs the determined feedforward compensation amount to the adder 13.

[0042] When performing control using feedback control, the adder 13 adds the feedforward compensation amount calculated by the FF compensation unit 12 to the feedback compensation amount of the feedback control input from the FB compensator 11, and outputs the result of this addition to the actuator AC, which is a stepping cylinder AC in this embodiment.

[0043] A stepping cylinder AC, which is an example of an actuator AC, drives a valve VL, which in this embodiment is a slide valve VL, so that the valve opening v(s) corresponds to the manipulated variable u(s). This adjusts the cross-sectional area of ​​the flow path of molten steel through the nozzle NZ, thereby adjusting the amount of molten steel supplied from the tundish DT to the mold MD and controlling the molten steel level. This molten steel level is measured by a level measurement unit 1. When a disturbance d(s) is applied (superimposed) to the molten steel level, the molten steel level affected by the disturbance d(s) is measured by the level measurement unit 1 as a controlled variable y(s). This measured controlled variable y(s) is input to a subtractor 10 of the molten steel level control device CT. While the disturbance d(s) is normally applied before the level measurement unit 1, in FIG. 2 the value after passing through the level measurement unit 1 is used as the disturbance d(s).

[0044] In the block diagram of the continuous casting machine CM for controlling the molten metal level, the transfer function P(s) in the block of controlled objects consisting of the actuator AC, valve VL, mold MD, and level measurement unit 1 is expressed as follows:

[0045]

number

[0046] where T SC is the time constant of the stepping cylinder AC (time constant of the actuator AC), and L SC is the dead time of the stepping cylinder AC (dead time of the actuator AC), and K f is the flow coefficient [m 2 / s] (the amount of molten steel flowing per unit time [m 3 / s]), A is the cross-sectional area of ​​the mold MD, and T CD is the time constant of the level measurement unit 1, in this embodiment the eddy current sensor 1, and L CD is the dead time of the level measurement unit 1 (eddy current sensor 1). SC =0.2 [s], T CD =0.3 [s], L SC =0.1 [s], L CD =0.12[s]. Of course, T SC , T CD , L SC and L CD The values ​​of the flow coefficient K are changed depending on the continuous casting machine CM. f The cross-sectional area A of the mold MD varies depending on the casting conditions, and the range is, for example, 0.6 <K f / A<2.1. f / A can be calculated from the mass of molten steel in the tundish, the mold width, and the casting speed by a known calculation method. The calculation method is disclosed, for example, in "Akira Murakami, Yoshiharu Nishida, Hisashi Miki, Toru Matsuura, and Masaru Nakao, "H∞ Control of Molten Steel Level in a Continuous Casting Machine," Transactions of the Institute of Systems, Control and Information Engineers, Vol. 10, No. 11, pp. 607-615, 1997" (Reference 1).

[0047] Feedforward compensation is used for known disturbances, such as changes in casting speed, tundish molten steel mass, and mold width. It is also possible to consider changes in the molten steel surface level (SV) as needed. Specific techniques are disclosed in Japanese Patent Application Laid-Open Publication No. 2014-200830. Even during unsteady conditions, such as changes in the casting speed, level fluctuations can be suppressed for known disturbances. While the amount of feedforward compensation for these known disturbances is added as part of the manipulated variable, this is omitted from the drawings and formulas in the following description of the embodiments.

[0048] Transfer function P from valve opening v(s) to molten metal surface level y(s) v2l (s) is expressed as the following equation 2.

[0049]

number

[0050] On the other hand, as described above, the value after passing through the level measurement unit 1 (eddy current sensor 1) is considered to be the disturbance d(s), so the molten metal surface level y(s) is expressed as in the following equation 3, and by rearranging the equation, the disturbance d(s) becomes the following equation 4.

[0051]

number

[0052]

number

[0053] The value obtained by back-calculating the unknown disturbance d(s) to the manipulated variable u(s) (hereinafter referred to as the "disturbance manipulated variable equivalent value") u d (s) is expressed as the following equation 5.

[0054]

number

[0055] The compensation amount (feedforward compensation amount) ff to cancel this unknown disturbance dist (s) is basically the disturbance control amount equivalent value u d The FF compensator 12 uses this feedforward compensation amount ff dist (s) is sought.

[0056]

number

[0057] Here, the manipulated variable u(s) means a value equivalent to the output of the position-type mold level control device CT that controls the stepping cylinder AC, etc., and is the sum of compensation amounts such as feedback compensation amount and feedforward compensation amount. If the mold level control device CT is a velocity-type device, the manipulated variable u(s) is the integrated or added value of the sum of the feedback compensation rate, feedforward compensation rate, etc.

[0058] Note that the operation amount u(s) is used instead of the valve opening v(s), and the transfer function P v2l Instead of (s), a transfer function P(s) may be used.

[0059] The amount of feedback compensation determined by the FB compensator 11 and the amount of feedforward compensation determined by the FF compensator 12 will be described in more detail below.

[0060] (Calculation of feedback compensation amount) Fig. 3 is a block diagram showing the configuration of the FB compensator shown in Fig. 2. Fig. 4 is a graph showing the Bode diagram of the FB compensator obtained by solving a mixed sensitivity problem in an FB compensator including a PID control unit and a second-order filter unit, and the frequency shaping result. Note that in this example, the mixed sensitivity problem for additive uncertainty described in Literature 1 is used. Fig. 5 is a graph showing the Bode diagram of the H∞ controller obtained by solving a mixed sensitivity problem in an H∞ controller, and the frequency shaping result.

[0061] The FB compensator 11 includes, for example, a PID control unit 111 and a filter unit 112 as shown in FIG.

[0062] The PID control unit 111 performs PID control, including at least P control, based on the deviation between the target value input from the subtractor 10 and the controlled variable y(s). More specifically, the PID control unit 111 performs any of P control, PI control, PD control, and PID control. The PID control unit 111 outputs its output to the filter unit 112. The PID control unit 111 may be a proportional-leading type, a differential-leading type, or the like. Furthermore, the PID control unit 111 may have an imperfect derivative or a filter for noise removal. The frequency at which control is performed ranges from frequency 0 to the Nyquist frequency in digital control, for example.

[0063] The filter unit 112 adjusts the gain and phase. Examples of the filter unit 112 include a first-order filter (a filter expressed by a first-order rational expression) and a second-order filter (a filter expressed by a second-order rational expression). The filter unit 112 outputs its output to the adder 13 as a feedback compensation amount.

[0064] As an example, when the filter unit 112 is a second-order filter, the transfer function K(s) of the FB compensator 11 is expressed by the following equation F1.

[0065]

number

[0066] The transfer function K(s) is the transfer function K of the PID control unit 111 shown in formula F1. PID (s) and the transfer function of the second-order filter unit 112. The PID control unit 111 has three parameters (K P , T I , T D), and the filter unit 112 has four parameters (a1, a2, b1, b2). The FB compensator 11 determines these seven parameters as follows, by solving a mixed sensitivity problem using an optimization method, for example, instead of a linear equation, taking into consideration the balance between robust stability and disturbance suppression.

[0067] Solving a mixed sensitivity problem using an optimization technique involves, for example, finding the minimum value of an evaluation function represented by predetermined variables included in a weighting function that constitutes the mixed sensitivity problem, under constraints such as the mixed sensitivity problem.

[0068] Let frequency be f min (=0.001) [Hz] to f max (=10) [Hz], logarithmically spaced N ω (=200) points, frequency variable f i (i=1,...,N ω ) is set. When digital control is performed, the controlled frequency ranges from 0 to the Nyquist frequency, but in this example, the sampling period is set to 0.1 [s]. Therefore, the frequency ranges from 0 [Hz] to the Nyquist frequency of 5 [Hz]. The lower limit frequency cannot be set to 0 logarithmically, so it is set to a small value of 0.001 [Hz]. The upper limit frequency is set to a large value of 10 [Hz] near the Nyquist frequency. Frequency f i is multiplied by 2π to obtain the angular frequency variable ω i (i=1,...,N ω )

[0069] The evaluation function J to be minimized is W1H The evaluation function J in equation F2 is set as the weighting function W for the mixed sensitivity problem expressed in equations F3 and F4 below. 1H (s), W 2H This function maximizes the frequency band for disturbance suppression by making the gain of (s) as large as possible.

[0070]

number

[0071]

number

[0072]

number

[0073] As a constraint condition at each angular frequency, the norm constraint of the mixed sensitivity problem is expressed as in the following equation F5. Here, σ() indicates the maximum singular value. To ensure accuracy during optimization calculation, this equation F5 is converted to dB and becomes the constraint condition of the following equation F6.

[0074]

number

[0075]

number

[0076] The subscript "H" indicates a high-gain type, and similar calculations will be performed later for the middle-gain type "M" and the low-gain type "L."

[0077] In addition, when substituting formula F3 and formula F4 into formula F5 or formula F6, W 1H (s) is substituted into W1(s), and W 2H (s) is substituted into W2(s). Similar calculations are performed for the middle gain type and low gain type described below.

[0078] Here, the parameters of the PID control unit 111 are set as follows to achieve negative feedback: K p >0, T I >0, TD≧0 are required. This optimization problem is non-convex and may fall into a local solution, so to make it easier to converge, constraints such as the following equation F7 were set for each variable. p >0, TI Constraints other than the constraints TD > 0 and TD ≥ 0 are provided for convenience in order to avoid falling into a local solution, and are not essential. For example, in the multi-start local search method of this example, an optimal solution can be found without any convenient constraints by increasing the number of initial values ​​tried during optimization.

[0079]

number

[0080] The last two equations in the constraint condition equation F7 impose restrictions on the attenuation coefficient of the filter unit 112. That is, the last two equations are set in order to search for an optimal value in a region where the zeros and poles are unlikely to oscillate and are unlikely to approach the imaginary axis.

[0081] For example, a multi-start local search method is used as the optimization method, and the initial value is tested at multiple points, and the optimization result from the initial value that does not fall into a local optimum is adopted. W1H =0.1, K P =0.2, T I =17.6, T D =1.0, a1=1.0, a2=1.0, b1=1.0, b2=1.0. Here, K f / A was set to 0.6 (K f / A=0.6).

[0082] The optimization problem is to find the minimum value of the evaluation function J in formula F2 under the constraints in formulas F6 and F7. By using a publicly available nonlinear optimization method (for example, the fmincon function in the "Optimization Toolbox" (registered trademark) of The MathWorks, Inc.), W1H =0.559485, K P =1.031486, T I =17.579967, T D The results obtained were: =0.0, a1=0.604110, a2=0.229163, b1=0.495033, b2=0.074074.

[0083] The PID control unit 111 calculates the differential time T D is 0, which is essentially PI control. The Bode diagram and frequency shaping results of the FB compensator 11 using this result are shown in FIG. 4. For comparison, the Bode diagram and frequency shaping results of the H∞ controller obtained by solving the mixed sensitivity problem are shown in FIG. 5. The calculation method for this H∞ controller is disclosed in, for example, the above-mentioned document 1. Comparing FIG. 4 with FIG. 5 reveals that the FB compensator 11 has almost the same performance as the H∞ controller.

[0084] The FB compensator 11 thus obtained is designed in advance in such a way that its control characteristics change continuously in order to adapt to changes in the CM of the continuous casting machine, and a plurality of such compensators are stored in the memory. f Each FB compensator 11 is calculated by changing / A from 0.6 to 2.1 in increments of 0.1. f Depending on / A, real numbers from 1 (1.0) to 16 (16.0) are assigned in order, and these are called FB compensator numbers. f When / A=0.6, the FB compensator number is 1 (1.0), and K f When / A=2.1, the FB compensator number is 16 (16.0). In the above example, K f A high-gain type FB compensator 11 was obtained when / A=0.6, but a middle-gain type FB compensator 11 having a lower gain than the high-gain type FB compensator 11 can also be obtained, and a low-gain type FB compensator 11 having a lower gain than the middle-gain type FB compensator 11 can also be obtained. Therefore, the FB compensator 11 can be obtained by f There are 16 types of / A and 3 types of gain type, for a total of 48 types, which are shown in Table 1 below. In Table 1, high gain types are represented by "H", middle gain types are represented by "M", and low gain types are represented by "L".

[0085] [Table 1]

[0086] The above result is the FB compensator 11 for H1, and the FB compensators 11 for the remaining H2 to H16 are similarly determined.

[0087] The middle gain type FB compensator 11 can be similarly obtained by using the following equations F8 to F10 instead of the above equations F2 to F4. W1M as small as possible, and as a result, the weight function W in Equation F9 1M By increasing the gain of (s), the evaluation function maximizes the frequency band of disturbance suppression. When substituting formulas F9 and F10 into formula F5 or F6, W 1M (s) is substituted into W1(s), and W 2M (s) is substituted into W2(s).

[0088]

number

[0089]

number

[0090]

number

[0091] The low-gain type FB compensator 11 can be similarly obtained by using the following equations F11 to F13 instead of the above equations F2 to F4. W1L as small as possible, and as a result, the weighting function W in Equation F12 1L By increasing the gain of (s), the evaluation function maximizes the frequency band of disturbance suppression. When substituting formulas F12 and F13 into formula F5 or F6, W 1L (s) is substituted into W1(s), and W 2L (s) is substituted into W2(s).

[0092]

number

[0093]

number

[0094]

number

[0095] Figure 6 shows the K f FIG. 6A is a lookup table showing the parameters of the FB compensator for each K / A and gain type. f Parameter K by / A and gain type p The leftmost column indicates the FB compensator number. This is the same for Fig. 6B and subsequent figures. Fig. 6B shows the K f Parameter T by / A and gain type I FIG. 6C is a lookup table showing the values ​​of K f FIG. 6D is a lookup table showing the values ​​of the parameter b1 for each K / A and gain type. f FIG. 6E is a lookup table showing the values ​​of the parameter b2 for each / A and gain type. f FIG. 6F is a lookup table showing the values ​​of the parameter a1 for each / A and gain type. f This is a lookup table showing the values ​​of the parameter a2 for each / A and gain type. D is omitted because it is 0.0.

[0096] The parameters K of the 48 types of FB compensators 11 obtained in this way P , T I , a1, a2, b1, b2 are shown in the form of a look-up table in Figure 6. For example, the FB compensator 11 of M2 is P =0.435, T I=11.608, a1=0.49, a2=0.159, b1=0.517, b2=0.078. The look-up table shown in Fig. 6 is stored in the memory of the molten metal level control device CT.

[0097] In this embodiment, as shown in FIG. 2, the FB compensator 11 is configured to have a system (higher system) SY higher than the FB compensator 11, and to have K f The FB compensator 11 is specified and configured by specifying the value of / A and the value of the gain type. For this reason, the high gain type H is defined as 1.0, the middle gain type M is defined as 2.0, and the low gain type L is defined as 3.0, and the gain type can be continuously changed from 1.0 to 3.0.

[0098] The host system SY may be a system configured within the molten metal level control device CT, or may be a system configured external to the molten metal level control device CT. In the example shown in Fig. 2, the host system SY is a system configured within the molten metal level control device CT. In this embodiment, the host system SY changes the gain of the feedback control to be smaller than the current value when the absolute value of the rate of the manipulated variable of the valve VL is larger than a predetermined first threshold, and changes the gain of the feedback control to be larger than the current value when the absolute value of the rate of the manipulated variable of the valve VL is smaller than a predetermined positive second threshold that is smaller than the first threshold. This host system SY corresponds to an example of a gain type changing unit.

[0099] The lookup table shown in Figure 6 shows the K f / A, so K given by the higher system SY f If the value of / A and the value of the gain type are not in the lookup table shown in Figure 6, fThe FB compensator 11 corresponding to the value of / A and the value of the gain type is generated by interpolation from the lookup table shown in FIG. 6. For this interpolation, for example, the "nD Lookup Table" block of "Simulink" (registered trademark) by The MathWorks, Inc. can be used. In the settings of this block, the number of dimensions of the table is set to "2" and the interpolation method is set to "linear point and slope." As a method similar in principle, interpolation is performed as follows.

[0100] FIG. 7 shows an example of the parameters K p For example, if the gain type is 1.4 and K f Assume that / A is 0.72. Gain type = 1.4 is between gain type 1 (i.e., high gain type H) and gain type 2 (i.e., middle gain type M) at a ratio of 0.4:0.6, as shown in Figure 7. f / A=0.72 is between the FB compensator 11 (*2) of FB compensator number 2 and the FB compensator 11 (*3) of FB compensator number 3 in the ratio of 0.02:0.08=0.2:0.8 (* indicates either H, M, or L). In this case, the required K p The value of is as shown in Figure 7, H2 (gain type = 1, K f / A=0.7) 0.928, H3 (Gain Type=1, K f / A=0.8) 0.845, M2 (Gain Type=2, K f / A=0.7) 0.435, M3 (Gain Type=2, K f / A=0.8) is between 0.396.

[0101] First, TMP1 is found at the point where H2 and M2 are divided into 0.4:0.6. p The value of is (0.4 0.435 + 0.6 0.928) = 0.731. Next, TMP2 is found at the point where H3 and M3 are divided internally at 0.4:0.6. K at this point pThe value of is (0.4 0.396 + 0.6 0.845) = 0.665. RESULT is then found at the point where TMP1 and TMP2 are divided internally at 0.2:0.8. K at this point p The value of RESULT is (0.2 0.665 + 0.8 0.731) = 0.718. This RESULT value is the K obtained by interpolation. p The other parameter T I , a1, a2, b1, and b2 can also be obtained by interpolation in the same way. In this case, the FB compensator number is 2.2.

[0102] This allows K f Based on the real value of / A and the real value of the gain type, an FB compensator 11 can be formed in which the control characteristics are continuously changed. Then, the feedback compensation amount is calculated from the deviation of the molten metal surface level (the difference between the target value and the controlled variable y).

[0103] (Calculation of feedforward compensation amount) FIG. 8 is a diagram illustrating the conversion from the time domain to the space domain. The feedforward compensation amount is calculated based on the molten metal level (control variable y), the valve opening v (or manipulated variable u) of the valve VL, and the transfer function of the continuous casting machine CM related to the molten metal level. Because the molten metal level y and the valve opening v (or manipulated variable u) are sampled at predetermined time intervals, they are represented as multiple values ​​arranged in a time series at predetermined time intervals in the time domain. The present is set to 0, and the value is a function of the past time t' (t'≧0; for example, t'=1.0 for 1.0 s ago). For example, as shown in FIG. 8A, the values ​​of the molten metal level y and the valve opening v (or manipulated variable u) at time 0 s are the values ​​at the current molten metal level, and the values ​​of the molten metal level y and the valve opening v (or manipulated variable u) at past time t' s are the values ​​at a position x' away from the current molten metal level in the slab Ob. While the following description focuses on the valve opening v, the same explanation can be applied to the case of the manipulated variable u. If the casting speed is constant, the values ​​of the molten metal level y and the valve opening v are spaced at equal intervals on the slab Ob, and the feedforward compensation amount can be calculated in this time domain. However, because the casting speed is variable, the FF compensator 12 first converts the values ​​of the molten metal level and the valve opening v of the valve VL in the time domain into values ​​in the spatial domain, whose coordinate axes are aligned in the longitudinal direction of the slab Ob produced by the continuous casting machine CM, as shown in FIG. 8C. This is because bulging is thought to occur at a substantially constant frequency or period in the spatial domain.

[0104] More specifically, first, the values ​​of the molten metal surface level y and the valve opening degree v in the time domain are associated with positions on the slab Ob. For example, the position x'(t') on the slab Ob at time t' can be found by integrating the casting velocity Vc(t) (where t means the time t [s] ago, t ≥ 0) from the current time 0 toward the past time t' (t' ≥ 0) (x'(t') = ∫Vc(t)dt, where the integral ∫ is from 0 to t'). Since Vc(t) ≥ 0, x'(t') ≥ 0.

[0105] Next, as shown in Fig. 8B, N positions X'(i) (i = 0, 1, . . . , N-1) on the slab Ob, which are equally spaced on the coordinate axis X' along the longitudinal direction of the slab Ob, with the current molten metal level position X'(0) as the coordinate origin, are associated with values ​​of the molten metal level y in the time domain and the valve opening v. Depending on the casting speed Vc, the sampling period, and the number of past data points stored in memory, there may be cases where values ​​of the molten metal level y and the valve opening v in the time domain corresponding to the position X'(i) (i = 0, 1, . . . , N-1) do not exist. In such cases, values ​​of the molten metal level Y'(i) (i = 0, 1, . . . , N-1) and the valve opening V'(i) (i = 0, 1, . . . , N-1) in the spatial domain corresponding to the position X'(i) (i = 0, 1, . . . , N-1) are found by interpolation or extrapolation.

[0106] Specifically, it is as follows: When the position X'(i) is monotonically increasing, the time t X’i '(t X’i '≧0) is obtained by interpolation or extrapolation from x'(t X’i ') = X'(i). At this time t X’i The time t' does not necessarily coincide with an integer multiple of the sampling period, but is a real multiple. The relationship between the time t', the molten metal level y, and the valve opening v is known, so the obtained time t X’i ', the molten metal level y(t X’i ') and valve opening v(t X’i ') can be obtained by interpolation or extrapolation. These become Y'(i) and V'(i).

[0107] Here, the interpolation may be, for example, linear interpolation.

[0108] Extrapolation is necessary when, for example, in the case of a constant velocity, Vc, sampling period, and number of accumulated data points are the maximum value of the distance in spatial coordinates, and this maximum value is smaller than X'(N-1). For example, if there is a t' corresponding to X'(1000), but no t' corresponding to X'(1001) and beyond, then you can use the t' corresponding to X'(1000) as the t' corresponding to X'(1001) and beyond. This allows you to find values ​​from Y'(1001) onwards and V'(1001) onwards by extrapolation.

[0109] Next, although this is acceptable, we will address the fact that the time is reversed, which makes it difficult to understand. Therefore, to simplify the handling, the sign of the coordinate axis X' is converted as shown in Figure 8C. Specifically, it becomes X(i) = X'(N-1-i) (i = 0, 1, . . . , N-1). For example, the coordinate axis X' becomes the coordinate axis X, whose coordinate origin is the position X'(N-1) = X(0) on the slab Ob, time N-1 [s] before.

[0110] Because the axis has changed, the molten metal level and valve opening also change to Y(i) = Y'(N-1-i) (i = 0, 1, , N-1) and V(i) = V'(N-1-i) (i = 0, 1, , N-1).

[0111] Note that the positive and negative signs of the coordinate axes are reversed for simplification and are not essential. However, if the positive and negative signs are not reversed, the coordinates used in the subsequent calculations will differ from those in this embodiment.

[0112] As a result, the values ​​of the molten metal level y and the valve opening v of the valve VL in the time domain are converted into values ​​in the spatial domain having a coordinate axis X in the longitudinal direction of the slab Ob produced by the continuous casting machine CM. For example, N=1024 (=2 10 ), the position X on the coordinate axis X = {X0,X1,X2,...,X 1021 ,X 1022 ,X 1023}, the melt surface level in the spatial domain Y={Y0,Y1,Y2,···,Y 1021 ,Y1022 ,Y 1023}, the valve opening in the spatial domain V={V0,V1,V2,···,V 1021 ,V 1022 ,V 1023} is obtained. Here, for simplicity, numerical values ​​are replaced with variables. Note that in this example, N=1024, but this is not limiting and may be an appropriate number such as 512 or 2048.

[0113] The molten metal level Y and the valve opening V of the valve VL are determined by their deviations from the equilibrium point, and each component of the molten metal level Y [m] and the valve opening V [m] (in this embodiment, the valve VL is a slide valve VL, so the unit of the valve opening V is [m]) is initialized to 0. This makes it possible to use this method even when the casting speed is very slow and 1,024 data points cannot be obtained. The SV (Setting Value) itself, which is the target value of the molten metal level Y, can be used as the equilibrium point of the molten metal level Y. The value obtained by applying a low-pass filter with a time constant of 10 [s] to the measured value of the valve opening V can be used as the equilibrium point of the valve opening V.

[0114] The FF compensator 12 then performs a fast Fourier transform (FFT) on the molten metal level Y[m] and the valve opening V[m] to determine the frequency characteristics of the molten metal level Y[m] and the valve opening V[m]. The FFT method is disclosed, for example, in "C Language Digital Signal Processing," by Akizuki, Matsuyama, and Yoshie, Baifukan, 1989, pp. 121-127 (Reference 2). Alternatively, the fft function of "Matlab" (registered trademark) by The MathWorks, Inc. can also be used.

[0115] The discretized frequency f[1 / m] is f={(i / 1024)·(1 / 0.004)}={f0,f1,f2,···,f 1021 ,f 1022 ,f 1023} (i=0,1,···,1023), and the discretized period L [m] is L={(1024 / i)·0.004}={L0,L1,L2,···,L 1021 ,L 1022 ,L 1023} (i=0,1,...,1023). Here, as above, numerical values ​​are replaced with variables for simplification. The same applies below. Note that △X, the interval in the spatial domain, is set to 0.004 [m] in consideration of the frequency resolution and the number of FFT points, but other values ​​may be used.

[0116] The FFT results for the molten metal level Y [m] and the valve opening V [m] are values ​​corresponding to the frequency f or the period L, and the FFT result for the molten metal level Y [m] is Y FFT ={Y FFT0 ,Y FFT1 ,Y FFT2 ,···,Y FFT1021 ,Y FFT1022 ,Y FFT1023}, and the result of FFT of the valve opening V [m] is V FFT ={V FFT0 ,V FFT1 ,V FFT2 ,···,V FFT1021 ,V FFT1022 ,V FFT1023}.

[0117] Each element is a complex number, Y FFTi (i=1,2,···,510,511) is Y FFT(1024-i) It is conjugate with V. FFTi The same applies to Y FFT0 = 0, and V FFT0 = 0. Note that 511 is a value calculated as N / 2-1.

[0118] In this embodiment, there are four domains, namely, first to fourth domains. The first domain is the time domain (amplitude [m], frequency [Hz] or period [s], phase [rad]). The second domain is the time-frequency domain (amplitude [m], frequency [Hz] or period [s], phase [rad]). The third domain is the spatial domain (amplitude [m], frequency [1 / m] or period [m], phase [rad]). And the fourth domain is the spatial frequency domain (amplitude [m], frequency [1 / m] or period [m], phase [rad]). The time domain of the first domain is transformed into the time-frequency domain of the second domain by its FFT. Therefore, the time-frequency domain of the second domain is transformed into the time domain of the first domain by its inverse Fourier transform (inverse FFT). The spatial domain of the third domain is transformed into the spatial frequency domain of the fourth domain by its FFT. Therefore, the spatial frequency domain of the fourth domain is transformed into the spatial domain of the third domain by its inverse FFT.

[0119] In FIG. 8, the first time domain is transformed into the third space domain.

[0120] Next, the FF compensator 12 calculates the transfer functions P(s), P(s) of the continuous casting machine CM related to the molten metal level y. v2l (s) from the time frequency domain to the spatial frequency domain, i.e., the same frequencies f={f0, f1, f2, . . . , f 1021 ,f 1022 ,f 1023}, the frequency response (complex number) is calculated.

[0121] More specifically, the frequency response (complex number) is calculated as follows: FFT and valve opening V FFT Since the unit of frequency is [1 / m] in the spatial frequency domain, some ingenuity is required when calculating the value of the transfer function at the above frequency f or period L. If the reference casting speed is V c_n Then, the converted frequency f tf is used. f tf ={(i / 1024)·(V c_n / 0.004)}={f tf0 ,f tf1 ,f tf2 ,···,f tf1021 ,f tf1022 ,f tf1023}[Hz] (i=0,1,···,1023).

[0122] From the above, N = 1024 is the number of FFT points, and △X = 0.004 [m] is the data interval in the spatial domain with coordinate axis X. Note that the frequency in the time frequency domain, which is the time domain transformed by FFT, is {(i / 1024)·(1 / △t)} [Hz] (i = 0, 1, , 1023), so the above formula is a virtual sampling period, △t = 0.004 / V c_n This is equivalent to changing it to

[0123] V c_n can be set appropriately to the current value, the average value while the slab Ob travels a certain distance in the past, or the like, but here it is set to the current value. In other words, this is equivalent to regarding the current casting speed Vc as having been maintained constant over the past time.

[0124] ω tfi =2πf tfi (i=0,1,···,1023), for i=0 to 512, P(jω tfi ) and P v2l (jω tfi ) and for i=513 to 1023, the value is the conjugate of the value of 1024-i, and P tf ={P tf0 ,P tf1 ,P tf2 ,···,P tf1021 ,P tf1022 ,P tf1023}, P v2ltf ={P v2ltf0 ,P v2ltf1 ,P v2ltf2 ,···,P v2ltf1021 ,P v2ltf1022 ,P v2ltf1023} is obtained. Each component is a complex number. So far, the transfer function has been changed from the time-frequency domain, which is generally replaced by s = jω, to s = jωtf By replacing it with, the change to the spatial frequency domain is made. tf , P v2ltf The frequency response of is obtained. Here, the discretized frequency and discretized period are the same values ​​as f and L described above, making it possible to perform calculations within the spatial frequency domain. Note that 512 is a value calculated using N / 2.

[0125] Then, the FF compensator 12 calculates a feedforward compensation amount consisting of a sine wave component having an amplitude, a period, and a phase for canceling out the disturbance applied to the molten metal surface level.

[0126] More specifically, since the frequency characteristics and frequency response of each frequency have been obtained in this way, the disturbance D FFT That is, D FFT ={Y FFTi -P v2ltfi ·V FFTi}={D FFT0 ,D FFT1 ,D FFT2 ,···,D FFT1021 ,D FFT1022 ,D FFT1023}. i = 0 to 512 are found using Equation 4, and for i = 513 to 1023, the conjugate value of 1024-i is used. In order to eliminate stationary components and low-frequency components, in this example, values ​​corresponding to frequencies equivalent to periods of 1.0 [m] or more in a spatial domain having coordinate axis X are forcibly set to 0.

[0127] Next, using Equation 5, the disturbance control amount equivalent value UD FFT In other words, UD FFT ={D FFTi / P tfi}={UD FFT0 ,UD FFT1 ,UD FFT2 ,···,UD FFT1021 ,UD FFT1022 ,UD FFT1023 As mentioned above, the disturbance control amount equivalent value UD FFT By inverting the sign of , the amount of feedforward compensation for the disturbance is essentially determined.

[0128] As mentioned above, the manipulated variable may be used instead of the valve opening. In this case, the transfer function P v2l Instead of (s), the transfer function P(s) is used.

[0129] Next, UD FFT The amplitude of each component of ampUD FFTi and phaseUD FFTi and can be calculated using the following equations 7 and 8.

[0130]

number

[0131]

number

[0132] As a result, the disturbance control amount equivalent value, its amplitude, and frequency at each frequency are obtained.

[0133] Next, UD of these 512 points (N / 2) FFTi By selecting the peak amplitude from the component values ​​of the signal, the frequency, amplitude and phase of the feedforward compensation amount are determined.

[0134] UD of 512 points FFTi From the component values ​​of, for example, a frequency that satisfies the following condition is selected. The discretized period L [m] is in the range of 0.1 to 1.0 [m]. The amplitude is 1.0 mm (= 0.001 m) or more. -It has the largest amplitude at seven discrete frequencies, including before and after.

[0135] This allows the frequency, amplitude, and phase of the disturbance control amount equivalent value, which has a peak in its amplitude, to be determined. Note that there may be zero or multiple peaks.

[0136] It may be possible to perform feedforward compensation using the frequency, amplitude, and phase obtained in this way as is, but in reality this is difficult because the obtained phase may be shifted by approximately π [rad] and the frequency is only obtained discretely.

[0137] FIG. 9 shows, as an example, the results of a simulation of a value (equivalent to the disturbance manipulated variable) obtained by back-calculating an unknown disturbance up to the manipulated variable. The equivalent disturbance manipulated variable serves as the basis for the feedforward compensation. FIG. 9A shows the numerical values ​​of the simulation results, FIG. 9B shows a graph of the amplitude represented by the numerical values ​​in FIG. 9A, and FIG. 9C shows a graph of the phase represented by the numerical values ​​in FIG. 9A. The horizontal axis in each of FIGS. 9B and 9C is i, the vertical axis in FIG. 9B is the amplitude, and the vertical axis in FIG. 9C is the phase. FIG. 9 shows the results of a simulation in the case where the casting speed is constant and there is feedforward compensation for the disturbance. FIG. 9 shows the discrete frequency of the FFT, the discrete period of the FFT calculated in Equation 7, and UD at time 200 [s] in the simulation of FIG. 13. FFT The amplitude and phase of each component are shown as a fraction from 0 to 30.

[0138] The phase shift is about π [rad] at the frequency before and after the amplitude peak. This can be on the high frequency side of the peak frequency or on the low frequency side of the peak frequency. In Figure 9, the first u d The amplitude peak is at i=12, and the phase changes between i=12 and i=13 on the high frequency side. The second amplitude peak is at i=25, and the phase changes between i=25 and i=24 on the low frequency side. If the phase is shifted by approximately π [rad], the fluctuation will be amplified, so it is necessary to select an appropriate phase.

[0139] On the other hand, with discrete frequencies, errors may occur in the amount of feedforward compensation, so it is necessary to increase the frequency precision.

[0140] Therefore, the frequency, amplitude, and phase at each peak of the amplitude of the disturbance manipulation amount equivalent value can be calculated as follows: Note that, for simplicity, the case of the first peak (i=12) in Fig. 9 will be explained here, but the same applies when there are multiple peaks.

[0141] First, the FF compensator 12 calculates the frequency of the feedforward compensation amount (the frequency of the value corresponding to the disturbance control amount) f peak More specifically, the frequency with the largest amplitude is selected from the frequencies adjacent to the peak frequency, and the frequency f peak is calculated by weighting the frequency of the peak and the selected frequency by the amplitude of the peak and the amplitude of the selected frequency. 12 (=2.92969[1 / m], its amplitude = 0.00386[m]) and the adjacent frequency f 11 , f 13 Among these, the frequency f with the largest amplitude 13 (=3.17383[1 / m], its amplitude = 0.00290[m]) is selected, and the frequency f peak =((0.00386·2.92969+0.00290·3.17383) / (0.00386+0.00290))=3.03[1 / m]. This allows us to calculate a more accurate frequency, rather than being limited to the discrete frequency values ​​themselves.

[0142] Next, the FF compensator 12 calculates the amplitude of the feedforward compensation amount (the amplitude of the value equivalent to the disturbance control amount) amp in the spatial frequency domain. peak In the above example, the peak frequency f 12 Amplitude at amp peak =0.00386. This value is the amplitude at the location of the manipulated variable u, and is filtered and multiplied by the gain of the controlled object to obtain the amplitude at the location of the disturbance d. Therefore, this amplitude amp peak = 0.00386 is different from the amplitude of disturbance d, 0.001 [m], which will be described later.

[0143] Although not used in this example, the amplitude can also be calculated by other methods. Specifically, this method involves taking the square root of the sum of the squares of the three amplitudes at the peak and the frequencies before and after it. The frequency of the peak is f 12 (=2.92969[1 / m], its amplitude = 0.00386[m]), and the frequencies before and after it are f 11 (=2.68555[1 / m], its amplitude=0.00104[m]), f 13 (=3.17383[1 / m], its amplitude = 0.00290[m]). The amplitude in this case is √(0.00386 2 +0.00104 2 +0.00290 2 ) = 0.00494. It has been confirmed that this method can accurately determine the amplitude when the disturbance is a single-cycle sine wave and is stationary. However, control simulations have shown that robustness tends to decrease when a disturbance consisting of multiple sine waves is applied or when the disturbance is non-stationary. Therefore, in this example, the value of the peak amplitude itself, which is a smaller value as described above, is used instead, and it has been confirmed that stable control is possible.

[0144] Here, this amplitude amp peak is likely to change in the unsteady state or when a sudden disturbance is applied, so in this embodiment, a low-pass filter with a time constant (for example, 5 [s]) is further applied, and the result of application is the amplitude amp peak This prevents the frequency, amplitude, and phase of the disturbance control amount equivalent value from changing frequently. peak = 0.00386, and the amplitude amp peak =0.00378.

[0145] Next, the FF compensator 12 calculates the phase of the feedforward compensation amount in the spatial frequency domain. More specifically, for the disturbance manipulation amount equivalent value, for each of three phases at the peak frequency and two adjacent frequencies, the amount of disturbance cancellation is calculated by calculating the inner product in the spatial domain on the coordinate axis X, and the phase that most effectively counteracts the disturbance is selected.

[0146] More specifically, it is calculated as follows: First, the frequency number of the peak is set to j, the frequency adjacent to it on the low frequency side is set to pre, and the frequency adjacent to it on the high frequency side is set to post. In the above example, j=12, pre=11, and post=13.

[0147] First, let us explain the case of peak frequency j. d The phase of the disturbance d at the application point (later stage of P(s)) is 2π (frequency) (distance from the center of the slab length at point N) + π + (u d (phase of P(s)) + (phase lag of P(s)). Here, π is added because the sign is inverted. -u d When the wave reaches the location where the disturbance d is applied, the amplitude ff of the wave at the position X(i) (i=0,1,...,N-1) of the coordinate axis X is _at_dj is expressed as the following equation 9, ignoring the gain of the controlled object.

[0148]

number

[0149] Here, since the gain of the controlled object is positive and is calculated commonly at frequency j, frequency pre, and frequency post, the gain of the controlled object is ignored to reduce the amount of calculation. Since the high-frequency component is large at the location of the manipulated variable u, making accurate evaluation difficult, the inner product is calculated at the location where the disturbance d is applied. This is because the continuous casting machine CM is essentially an integrator, and the value at the location of the manipulated variable u corresponds to the differentiated value of the disturbance.

[0150] Next, to calculate the inner product, the disturbance D FFT The disturbance d can be obtained by performing the inverse FFT on FFT )={d0,d1,d2,···,d 1021 ,d 1022 ,d 1023}. Each component represents the amplitude of the disturbance at position X(i) (i=0, 1, . . . , N-1) on the coordinate axis X. The inverse FFT technique is disclosed on pages 127 to 130 of the above-mentioned document 2. The fft function of "Matlab" (registered trademark) from The MathWorks, Inc. can also be used. Note that the imaginary component becomes negligibly small, and only the real component is extracted as the disturbance d, and each element of the disturbance d becomes a real number.

[0151] And the amplitude ff _at_dj and the dot product IP with the disturbance d in this spatial domain j is calculated using the following equation 10. That is, for i=0 to 1023, the product of each component is calculated and the sum of these is calculated.

[0152]

number

[0153] Similarly, for the frequency pre, the amplitude ff of the wave at position X(i) (i=0,1,...,N-1) on the coordinate axis X is _at_dpre is calculated, and the dot product IP pre Similarly, for the frequency post, the amplitude ff of the wave at the position X(i) (i=0,1,...,N-1) of the coordinate axis X is _at_dpost is calculated, and the dot product IP post is required.

[0154] If the dot product IP is smallest and negative, it is considered to cancel out the disturbance d most effectively. j , dot product IP pre and the dot product IP post , the phase of the smallest and most negative inner product IP is the phase of the feedforward compensation amount of the disturbance peak That is, the inner product IP j If is minimum and negative, the phase of the feedforward compensation of the disturbance peak =phaseUD FFTj The inner product IP oreIf is minimum and negative, the phase of the feedforward compensation of the disturbance peak =phaseUD FFTpre The inner product IP post If is minimum and negative, the phase of the feedforward compensation of the disturbance peak =phaseUD FFTpost In cases other than these three, the disturbance is not canceled, so the amplitude is set to 0, and as a result, feedforward compensation of the disturbance at this peak is not performed. In the above example, peak =phaseUD FFTj =-0.02916.

[0155] This allows the appropriate phase to be selected. This method can also be used in unsteady cases where the casting speed Vc changes or the characteristics of the controlled object change.

[0156] From the above, for one peak (peak at frequency j = 12), the frequency f peak , amplitude amp peak and phase peak If there are other peaks, the frequency f peak , amplitude amp peak and phase peak is required.

[0157] Although not used in this example, the phase can also be calculated using other methods. Specifically, when the i of the amplitude peak of the disturbance manipulated variable equivalent value is odd, the phase of the peak is calculated by adding π to the phase of that peak. When the i of the amplitude peak of the disturbance manipulated variable equivalent value is even, the phase of the peak itself is used. For example, in the case of Figure 9B, the first peak is at i = 12, which is an even number, so the phase is taken as -0.02916. On the other hand, the second peak is at i = 25, which is an odd number, so π is added, resulting in a phase of 2.83821 + π = 5.9798. We have confirmed that this method can be used to calculate the phase when the disturbance is a single-cycle sinusoidal wave in the steady state. However, it has been found that in the unsteady state, the phase may occasionally be shifted by π. This is undesirable because it amplifies the disturbance. Therefore, in this example, the above-mentioned method of selecting a phase by an inner product is used instead, and it has been confirmed that an appropriate phase can be selected even in an unsteady portion.

[0158] The FF compensator 12 then determines the rate of the feedforward compensation amount in the next sampling period or the feedforward compensation amount itself in the time domain from the frequency, amplitude, and phase of the feedforward compensation amount of the disturbance corresponding to each peak in the spatial frequency domain.

[0159] More specifically, the amount of disturbance feedforward compensation output at the next sampling period in the time domain is the value at the location where "current casting speed Vc x sampling period" is advanced in the spatial domain of coordinate axis X. When the mold level control device CT is a velocity type, the rate dff / dt [m / s] (time derivative of dff) of the amount of disturbance feedforward compensation for one peak is expressed by the following equation 11.

[0160]

number

[0161] Here, gainmp1 is the gain multiplied by the sampling period for calculating the amount of high-frequency compensation, and is represented by the line graph shown in FIG. 10, for example. FIG. 10 is a diagram showing the gain for reducing the amount of feedforward compensation for high-frequency disturbances. The horizontal axis of FIG. 10 represents the period [m], and the vertical axis represents the gain. gainmp1 shown in FIG. 10 has the characteristic that the gain increases proportionally from 0 to 1.0 when the period is from 0 to 0.33 [m], and saturates at 1.0 when the period is 0.33 [m] or more. Therefore, when the period of the disturbance is 0.33 [m] or more, the gain is 1, and the amount of feedforward compensation for the disturbance required after one sampling period is used as is. This corresponds to the first peak in the example above.

[0162] When the molten metal level control device CT is a velocity type, the rate of the feedforward compensation amount for the disturbance is multiplied by a gain. The rate of the feedforward compensation amount decreases in proportion to the period. As a result, the feedforward compensation amount also decreases in proportion to the period. This is because the higher the frequency, the less robust the device is against phase changes, making it easier for phase shifts in the feedforward compensation to occur, which limits the feedforward compensation amount. Note that the second peak in the example above corresponds to this.

[0163] Feedforward compensation amount ff for disturbance at position-type sampling k+1 k+1 is the feedforward compensation amount ff at the kth sampling k By using the above formula, it is expressed as in the following formula 12. Note that other methods such as trapezoidal integration may also be used.

[0164]

number

[0165] Although the above description has been given for one peak, when there are multiple peaks, similar calculations are performed for each peak, and the rates or compensation amounts of disturbance feedforward compensation for all peaks are summed up to obtain Equation 11, which is the rate of the final disturbance feedforward compensation amount, or Equation 12, which is the final disturbance feedforward compensation amount.

[0166] Next, the operation of this embodiment will be described. Fig. 11 is a flowchart showing the operation of the molten metal surface level control device in this embodiment. Fig. 12 is a flowchart showing the calculation process of the feedforward compensation amount shown in Fig. 11.

[0167] When the power is turned on, the mold level control device CT configured as above initializes the necessary parts and starts operation. The computer constituting the mold level control device CT executes a program to functionally configure a host system SY, a subtractor 10, a feedback compensator 11, a feedback compensator 12 and an adder 13.

[0168] In Fig. 11, when the sampling timing arrives at every sampling period Δt, the molten metal level control device CT acquires a predetermined casting state from the host system SY and determines the FB compensator number (S1). f The mass of molten steel in the tundish, the width of the mold, and the casting speed are used to calculate K / A, and the molten steel level and the valve opening are used to control the molten steel level. The host system SY calculates K from the mass of molten steel in the tundish, the width of the mold, and the casting speed using the known calculation method as described above. f The upper system SY calculates the gain type adjusted in the process S5 described later in the molten metal surface level control executed at the previous sampling timing, and calculates the calculated K fThe FB compensator number corresponding to / A is selected and determined, and this determined FB compensator number is output to the FB compensator 11. At the first sampling timing after startup, any of the high gain type, middle gain type and low gain type may be used. For example, if the FB compensator number is a middle gain type and the above-obtained K f The FB compensator number corresponding to / A is selected and determined.

[0169] Next, the molten metal level control device CT calculates the parameter K according to the gain type input from the upper system SY and the determined FB compensator number by the FB compensator 11. P , T I , T D (=0.0), a1, a2, b1, and b2 are determined based on the look-up table stored in the memory, and the feedback compensation amount (FB compensation amount) is calculated (S2).

[0170] Next, the molten metal level control device CT uses the FF compensator 12 to determine a feedforward compensation amount (FF compensation amount) consisting of a sinusoidal wave component having an amplitude, period, and phase to cancel out the disturbance applied to the molten metal level, based on the molten metal level y, the valve opening v (or operation amount u) of the valve VL, and the transfer function of the continuous casting machine CM related to the molten metal level (S3).

[0171] More specifically, in step S3, as shown in FIG. 12, the FF compensator 12 first converts the values ​​y and v of the molten metal level y and the valve opening v of the valve VL in the time domain into values ​​Y and V in the spatial domain having a coordinate axis X in the longitudinal direction of the slab Ob produced by the continuous casting machine CM (S31).

[0172] Next, the FF compensator 12 performs FFT on each of the molten metal level Y and the valve opening degree V to determine the frequency characteristics of each of the molten metal level Y and the valve opening degree V in the spatial frequency domain (S32).

[0173] Next, the FF compensator 12 calculates the transfer functions P(s), P(s) of the continuous casting machine CM related to the molten metal level y. v2l(s) from the time frequency domain to the spatial frequency domain, and v2l The FF compensator 12 calculates the frequency response of the molten metal level Y and the valve opening V in the spatial frequency domain, and the transfer functions P(s) and P(s) of the continuous casting machine. v2l From the frequency response in the spatial frequency domain of (s), a disturbance control amount equivalent value is calculated by back-calculating the unknown disturbance up to the control amount (S33).

[0174] Next, the FF compensator 12 selects the peak of the amplitude in the value equivalent to the disturbance control amount in the spatial frequency domain (S34).

[0175] Next, the FF compensator 12 calculates, for each of the one or more peaks selected in step S34, the frequency of the feedforward compensation amount (the frequency of the value equivalent to the disturbance control amount) f peak (S35).

[0176] Next, the FF compensator 12 calculates the amplitude of the feedforward compensation amount (the amplitude of the value equivalent to the disturbance control amount) amp for each of the one or more peaks selected in step S34. peak is calculated (S36).

[0177] Next, the FF compensator 12 calculates the phase of the feedforward compensation amount in the spatial frequency domain for each of the one or more peaks selected in step S34. peak As described above, if a phase that cancels out the disturbance cannot be found, the amplitude of the peak is set to 0, and feedforward compensation of the disturbance at the peak is not performed.

[0178] Then, the FF compensator 12 determines the rate of the feedforward compensation amount in the next sampling period or the feedforward compensation amount itself in the time domain from the frequency, amplitude, and phase of the feedforward compensation amount of the disturbance corresponding to each peak in the spatial frequency domain (S38), and then this process S3 is terminated.

[0179] When each of the steps S31 to S38 shown in FIG. 12 is completed, returning to FIG. 11, the molten metal level control device CT then adds, by the adder 13, the feedback compensation amount calculated by the FB compensator 11 in step S2 and the feedforward compensation amount calculated by the FF compensator 12 in step S3 (S4).

[0180] Next, the molten metal level control device CT adjusts the gain type through the host system SY (S5). More specifically, if the absolute value of the rate of the manipulated variable is greater than a predetermined first threshold, the host system SY reduces the gain of the feedback control from its current value. If the absolute value of the rate of the manipulated variable is less than a predetermined positive second threshold that is less than the first threshold, the host system SY increases the gain of the feedback control from its current value. More specifically, in this embodiment, as described above, the high gain type H is defined as 1.0, the middle gain type M is defined as 2.0, and the low gain type L is defined as 3.0. Therefore, if the molten metal level control device CT is a velocity type, the host system SY adds a preset first value (e.g., 0.01 or 0.02) to the numerical value of the current gain type if the absolute value is greater than the first threshold. This changes the gain type from the current gain type to the low gain type, and prevents saturation of the rate (operation speed) due to the rate limit (operation speed limit) of the actuator AC (stepping cylinder AC in this embodiment). If the absolute value is smaller than the second threshold, the host system SY subtracts a second predetermined value (e.g., 0.01 or 0.02) set in advance from the numerical value of the current gain type. This changes the gain type from the current gain type to the high gain type. The first threshold is equal to or smaller than the rate limit of the actuator AC and is appropriately set in advance from a plurality of samples. The second threshold is a positive value smaller than the first threshold and is appropriately set in advance from a plurality of samples. The first and second predetermined values ​​may be the same or different from each other.

[0181] Next, the molten metal level control device CT outputs the addition result obtained in step S4 to the actuator AC (stepping cylinder AC in this embodiment) (S6). The actuator AC (stepping cylinder AC in this embodiment) drives the valve VL (slide valve VL in this embodiment) in accordance with the addition result input from the molten metal level control device CT. In this way, the molten metal level at the current sampling timing is controlled.

[0182] The molten metal level control device CT then determines whether or not the control of the molten metal level has ended (S7). If the result of this determination is that the control of the molten metal level has ended (Yes), for example, due to turning off the power or receiving an input to end the control of the molten metal level, the molten metal level control device CT ends the control of the molten metal level. On the other hand, if the result of the determination is that the control of the molten metal level has not ended (No), the molten metal level control device CT returns the process to step S1.

[0183] The molten metal level control device CT controls the molten metal level through such an operation.

[0184] Next, the simulation results will be described. Fig. 13 shows the simulation results for a constant casting speed. Fig. 14 shows the simulation results for a variable casting speed. Fig. 15 shows the simulation results for a comparative example. Figs. 13A, 14A, and 15A show the casting speed, with the horizontal axis representing time [s] and the vertical axis representing the casting speed [m / s]. Figs. 13B, 14B, and 15B show the disturbance, with the horizontal axis representing time [s] and the vertical axis representing the disturbance [m]. Figs. 13C, 14C, and 15C show the molten metal level, with the horizontal axis representing time [s] and the vertical axis representing the molten metal level [m]. Figs. 13D, 14D, and 15D show the opening of the slide valve, with the horizontal axis representing time [s] and the vertical axis representing the opening [m]. 13E, 14E, and 15E show the types of FB compensators 11, with the horizontal axis representing time [s], the left vertical axis representing the gain type, and the right vertical axis representing the FB compensator number.P , T I , a1, a2, b1, b2, and the horizontal axis represents time [s], and the left vertical axis represents the parameter K P , a1, a2, b1, b2, and the right vertical axis represents the parameter T I 13G, 14G, and 15G show the feedback compensation amount, with the horizontal axis representing time [s] and the vertical axis representing the feedback compensation amount (FB compensation amount) [m]. 13H and 14H show the feedforward compensation amount of disturbance in the position type, with the horizontal axis representing time [s] and the vertical axis representing the feedforward compensation amount of disturbance (FF compensation amount of disturbance) [m]. As mentioned above, the feedforward compensation amount for known disturbances such as changes in casting speed is omitted. 13I and 14I show the period of the component of the feedforward compensation amount of disturbance (FF compensation amount of disturbance), with the horizontal axis representing time [s] and the vertical axis representing the period [m] of the component of the FF compensation amount of disturbance. 13J and 14J show the amplitude of the disturbance feedforward compensation amount (disturbance FF compensation amount) component, with the horizontal axis representing time [s] and the vertical axis representing the amplitude [m] of the disturbance FF compensation amount component. 13K and 14K show the phase of the disturbance feedforward compensation amount (disturbance FF compensation amount) component, with the horizontal axis representing time [s] and the vertical axis representing the phase [rad] of the disturbance FF compensation amount component.

[0185] The mold level control device of the comparative example is configured by removing the FF compensator 12 and the adder 13 from the mold level control device CT of the embodiment shown in Fig. 2. In other words, the mold level control device of the comparative example is configured with a host system SY, a subtractor 10, and an FB compensator 11, and also has a function for adjusting the gain type by the host system SY.

[0186] In the simulation, the mass of molten steel in the tundish was 40.0×10 3The casting speed is set to [kg], the mold width is set to 1.80 [m], and the casting speed is set to a constant (uniform speed) of 0.02 [m / s] in the cases shown in Figures 13 and 15 (see Figures 13A and 15A), and in the case shown in Figure 14, it is set to a constant (uniform speed) of 0.02 [m / s] from 0 [s] to 150 [s], changes from 0.02 [m / s] to 0.01 [m / s] between 150 [s] and 160 [s], and then set to a constant (uniform speed) of 0.01 [m / s] (see Figure 14A). The disturbance applied to the molten metal level as a disturbance due to bulging is formed in the spatial domain on the coordinate axis X by superimposing a first wave with a period of 0.33 [m] and an amplitude of 0.010 [m] and a second wave with a period of 0.165 [m] and an amplitude of 0.0025 [m] (see FIGS. 13B, 14B, and 15B). Of course, this disturbance is not measured (cannot be observed) by the molten metal level control device CT of the embodiment or the molten metal level control device of the comparative example. The molten metal level control device CT of the embodiment performs back-calculation as described above.

[0187] Under these simulation conditions, in the comparative example of the mold level control device, as shown in FIG. 15C, the mold level PV (Process Variable) fluctuates significantly due to disturbances (disturbances due to bulging). This indicates that the comparative example of the mold level control device has difficulty suppressing disturbances (disturbances due to bulging). The mold level PV is a measured value, and the mold level SV is a target value. In the comparative example of the mold level control device, as shown in FIG. 15E, immediately after the start of the simulation, it is determined that saturation at the rate limit of the actuator AC is unlikely, and the gain type is continuously adjusted from the middle gain type 2.0 to the high gain type 1.0. The FB compensator number is constant because the casting conditions (casting conditions) of the mass of molten steel in the tundish, the mold width, and the casting speed are constant, as described above. Meanwhile, the parameters of the FB compensator 11 change with the change in gain type, as shown in FIG. 15F. Even if the parameters change in this way, as shown in FIG. 15C, the molten metal surface level PV remains steady, although it fluctuates, and is stable in this state.

[0188] In contrast, in the case of the mold level control device CT of the embodiment in the case of a constant casting speed, as shown in FIG. 13C, the mold level PV gradually fluctuates even when there is a disturbance, indicating that the FF compensator 12 effectively controls the mold level. Comparing FIG. 13D and FIG. 15D, it can be seen that the amplitude of the valve opening is approximately the same as in the comparative example, but the phase delay is smaller than in the comparative example. This is thought to be why the mold level fluctuations gradually decrease, as shown in FIG. 13C. The gain type, FB compensator number, and parameters of the FB compensator 11 are approximately the same as in the comparative example, as shown in FIGS. 13E and 13F. As shown in FIG. 13G, the feedback compensation amount gradually decreases, while the feedforward compensation amount gradually increases, as shown in FIG. 13H. Therefore, it is thought that the feedforward compensation amount gradually increases to counteract the disturbance (disturbance due to bulging), and the feedback compensation amount gradually decreases accordingly. As shown in FIG. 13I, the periods of the components of the feedforward compensation amount are 0.330 [m] and 0.165 [m] in the spatial domain on the coordinate axis X, and can be accurately estimated (back-calculated). As shown in FIG. 13J, the amplitudes of the components of the feedforward compensation amount change continuously. Also, as shown in FIG. 13K, the phases of the components of the feedforward compensation amount are temporarily and discontinuously selected to suppress level fluctuations in the unsteady portion before 120 [s] in the first half, while suppressing level fluctuations. Then, in the nearly steady portion after 120 [s] in the second half, while suppressing level fluctuations, while suppressing level fluctuations, while suppressing level fluctuations in the almost steady portion after 120 [s] in the second half ...

[0189] On the other hand, in the embodiment of the mold level control device CT for variable casting speeds, the disturbance is formed by the first and second waves defined in the spatial domain on the coordinate axis X, as described above. However, as the casting speed changes as described above, its period changes in the time domain, as shown in FIG. 14B. As shown in FIG. 14C, the mold level PV gradually fluctuates less even with changes in the casting speed, demonstrating that the FF compensator 12 effectively controls the mold level. In particular, after the casting speed is decelerated, the mold level PV fluctuates less. As described above, because feedforward compensation for changes in the casting speed is performed, the mold level PV does not rise sharply when the casting speed is decelerated. It can be seen that the fluctuation of the mold level PV gradually decreases due to feedforward compensation of the disturbance. Furthermore, the slower the casting speed, the more effective the feedback control is. As shown in FIG. 14D, the valve opening decreases as the casting speed is decelerated, and the valve VL is driven in the closing direction. As shown in Figures 14E and 14F, the gain type, FB compensator number, and parameters of the FB compensator 11 are almost the same as those in the comparative example before the casting speed is reduced. However, when the casting speed is reduced, the K fBecause / A also changes, the FB compensator number and the parameters of the FB compensator 11 also change. After the casting speed is reduced, the FB compensator number and the parameters of the FB compensator 11 remain constant. Even when the FB compensator number changes, fluctuations in the molten metal level PV are suppressed, and molten metal level control remains stable. As shown in Figure 14G, the feedback compensation amount gradually decreases before and after the casting speed is reduced, while as shown in Figure 14H, the feedforward compensation amount gradually increases. Therefore, it is believed that the feedforward compensation amount gradually increases to counteract disturbances (such as bulging), and the feedback compensation amount gradually decreases accordingly. It can be seen that the period of the feedforward control amount in the time domain is elongated (lengthened) after the casting speed is reduced. As shown in Figure 14I, the periods of the components of the feedforward compensation amount are 0.330 [m] and 0.165 [m] in the spatial domain on the coordinate axis X, indicating accurate estimation (back-calculation). Even if the casting speed changes, the time domain is converted to the spatial domain, so it can be confirmed that the change in the casting speed is not affected. As shown in Figure 14J, the amplitude of each component of the feedforward compensation amount changes before and after the casting speed is slowed down because the transfer function of the controlled object changes before and after the casting speed is slowed down. As shown in Figure 14K, the time change rate of each phase of each component of the feedforward compensation amount in the time domain decreases after the casting speed is slowed down. This corresponds to the decrease in the casting speed and the lengthening of the period of the disturbance in the time domain.

[0190] In addition, the robustness of the feedforward compensation is required, that is, the level fluctuations do not increase even when modeling errors exist. To achieve this, it is particularly effective to limit the period of the sine wave of the feedforward compensation. A specific example will be explained below.

[0191] Stepping cylinder time constant T SC =0.2[s], dead time L SC =0.1 [s], time constant T of the eddy current sensor CD =0.3 [s], dead time L CD= 0.12 [s] respectively fluctuates by 20%, the time corresponding to the phase difference of the modeling error is approximately (0.2 + 0.1 + 0.3 + 0.12) × 0.2 = 0.144 [s].

[0192] To cancel out a sine wave by adding a sine wave with the same amplitude but a different phase, the phase must be shifted by π (180 degrees) from the original sine wave. If the phase shift exceeds the range of π±π / 3, the amplitude of the resulting signal will be larger than the original sine wave signal, resulting in amplification. Here, π / 3 corresponds to 1 / 6 of one period.

[0193] Therefore, if the amplitude is accurately estimated, the period of the sine wave for which feedforward compensation is performed must be longer than 0.144 x 6 = 0.864 [s], and this value is one of the conditions for the lower limit of the period of the sine wave for which feedforward compensation is performed.

[0194] Furthermore, surface waves on the molten metal surface in the mold exist as modeling errors. The fundamental wave of the surface wave is a standing wave with its antinodes at both ends of the long side, which is the width of the mold. The eddy current sensor is installed away from the center of the mold, so it detects the surface waves. However, level control, which operates the valve opening, can control the average level of the entire molten metal surface, but cannot control the surface waves. If the period of the surface waves is also controlled, spillover can cause the control to become unstable.

[0195] Here, the period of the fundamental wave of the surface wave is expressed as √[2π (2 mold width) / gravitational acceleration], which is 1.5 [s] in this example. This value is one of the conditions for the lower limit of the period of the sine wave that performs feedforward compensation.

[0196] Based on the above two conditions, the lower limit of the period of the sine wave for which feedforward compensation is performed is at least 1.5 [s]. However, taking into account a margin, in this example, the lower limit of the period of the sine wave for which feedforward compensation is performed is set to 4.0 [s].

[0197] As a result, feedforward compensation does not destabilize control even if there are modeling errors in the time constant or dead time.Furthermore, feedforward compensation does not output a manipulated variable with the period of the surface waves, and does not excite waves with the period of the surface waves, so spillover does not occur and control does not become unstable.

[0198] The following explanation will be given using the simulation results in FIG. 16. FIG. 16 is a diagram showing the simulation results when modeling errors are taken into consideration. FIG. 16A shows the casting speed, with the horizontal axis representing time [s] and each vertical axis representing the casting speed [m / s]. FIG. 16B shows the disturbance, with the horizontal axis representing time [s] and the vertical axis representing the disturbance [m]. FIG. 16C shows the molten metal level, with the horizontal axis representing time [s] and the vertical axis representing the molten metal level [m]. FIG. 16D shows the opening of the slide valve, with the horizontal axis representing time [s] and the vertical axis representing the opening [m]. FIG. 16E shows the type of FB compensator 11, with the horizontal axis representing time [s], the left vertical axis representing the gain type, and the right vertical axis representing the FB compensator number. FIG. 16F shows the parameter K of the FB compensator. P , T I , a1, a2, b1, b2, the horizontal axis is time [s], the left vertical axis is parameter K P , a1, a2, b1, b2, and the right vertical axis is the parameter T IFIG. 16G shows the feedback compensation amount, with the horizontal axis representing time [s] and the vertical axis representing the feedback compensation amount (FB compensation amount) [m]. FIG. 16H shows the feedforward compensation amount of a position-type disturbance, with the horizontal axis representing time [s] and the vertical axis representing the feedforward compensation amount of a disturbance (FF compensation amount of a disturbance) [m]. As mentioned above, the feedforward compensation amount for known disturbances such as changes in casting speed is omitted. FIG. 16I shows the period of the component of the feedforward compensation amount of a disturbance (FF compensation amount of a disturbance), with the horizontal axis representing time [s] and the vertical axis representing the period [m] of the component of the FF compensation amount of a disturbance. FIG. 16J shows the amplitude of the component of the feedforward compensation amount of a disturbance (FF compensation amount of a disturbance), with the horizontal axis representing time [s] and the vertical axis representing the amplitude [m] of the component of the FF compensation amount of a disturbance. FIG. 16K shows the phase of the component of the disturbance feedforward compensation amount (disturbance FF compensation amount), with the horizontal axis representing time [s] and the vertical axis representing the phase [rad] of the disturbance FF compensation amount component.

[0199] Figure 16 shows the simulation results in which feedforward compensation was applied to only sinusoidal waves with a period of 4.0 s or more when a modeling error was present, i.e., when the time constant and dead time of the stepping cylinder and eddy current sensor of the continuous casting machine increased by 20% and a surface wave with a large amplitude of 5 mm was applied in this example.

[0200] As shown in FIG. 16A, the casting speed was 0.020 [m / s], similar to FIGS. 13 and 15, the mass of molten steel in the tundish was 40.0×10 3 [kg], and the mold width was 1.80 [m].

[0201] As shown in FIG. 16B, the disturbance is formed by superimposing a first wave, which is a fundamental wave with a period of 0.33 [m] and an amplitude of 0.010 [m], and a second wave, which is a harmonic wave with a period of 0.165 [m] and an amplitude of 0.0025 [m], as in FIGS. 13 to 15, and further adding a surface wave with an amplitude of 5 [mm] and a period of 1.5 [s], as described above.

[0202] As shown in Figure 16C, the amplitude of the molten metal level PV is stably controlled, with the level fluctuations due to the first and second waves reduced. Here, as expected, the level fluctuations due to the surface waves with a period of 1.5 [s] are not excited, and the amplitude of the level fluctuations with a period of 1.5 [s] remains almost constant. This shows that the feedforward compensation is robust against modeling errors. This is because, as shown in Figures 16I to 16K (described later), the sine wave with a period of 1.5 [s], i.e., the sine wave with a period of 1.5 × 0.020 = 0.03 [m] in the spatial domain of the slab length, is not included in the feedforward compensation.

[0203] In this example, under the above conditions (the discretized period L [m] is in the range of 0.1 to 1.0 [m], the amplitude is 1.0 [mm] (= 0.001 [m]) or greater, and the maximum amplitude is among the seven discrete frequencies, including the preceding and following points), the period L [m] is in the range of 0.1 to 1.0 [m], and the casting speed is 0.02 [m / s]. When converted to the time domain, the period is 5 to 50 [s], and the above conditions are stricter and take priority. However, if the casting speed is 0.04 [m / s], for example, the period condition converted to the time domain is 2.5 to 25 [s], and the feedforward compensation robustness condition of "a period of 4.0 [s] or greater" takes priority. Therefore, period (frequency) constraints in the time domain are essential.

[0204] FIG. 16D shows the valve opening.

[0205] The gain type in Figure 16E is set to low gain (L) (value 3.0 in the graph). This indicates that because the surface wave was set to a large value, the rate of the manipulated variable due to feedback compensation often exceeded the specified value, and the gain type was changed to lower the feedback gain. In other words, the effectiveness of feedback control is weakened. The FB compensator number remains unchanged because conditions such as the casting speed are constant.

[0206] Figure 16F shows the change in the parameters of the PI+ filter, which can be seen to be changed depending on the gain type.

[0207] Figure 16G shows the amount of feedback compensation, and Figure 16H shows the amount of feedforward compensation. It can be seen that as the amount of feedforward compensation increases, the amount of feedback compensation decreases. Also, the feedback compensation contains a small amount of surface wave periodic components, while the feedforward compensation does not contain any surface wave periodic components.

[0208] 16I to 16K respectively show the period, amplitude, and phase of the sine wave of the feedforward compensation amount component, and as mentioned above, no component with the period of the surface wave is included, and only sine waves corresponding to the fundamental wave of the bulging disturbance with a period of 0.33 [m] and its harmonic with a period of 0.165 [m] are included. And, since there is no component with the period of the surface wave, it can be seen that the molten metal level fluctuation with the period of the surface wave is not excited.

[0209] From the above, it can be seen that feedforward compensation is robust against modeling errors. Specifically, feedforward compensation is robust against modeling errors when performed in a time-frequency domain that is lower than the time frequency at which the phase difference of the modeling error of the controlled object is π / 3 and lower than the time frequency of the surface waves on the surface of the molten metal in the mold. In the above example, the period corresponding to the time frequency at which the phase difference of the modeling error of the controlled object is π / 3 is 0.144 [s] × 6 = 0.864 [s], and the period corresponding to the time frequency of the surface waves on the surface of the molten metal in the mold is 1.5 [s]. Therefore, the time-frequency domain is lower than the period of 1.5 [s]. In the above simulation, feedforward compensation is performed on the lower frequency side of the period of 4.0 [s], taking into account a margin.

[0210] As explained above, the mold level control device CT and mold level control method implemented therein in the embodiment calculate not only the amplitude and period but also the phase in order to cancel out disturbances, and therefore can suppress fluctuations in the mold level due to disturbances through feedforward compensation. The mold level control device CT and mold level control method add a feedforward compensation amount to the feedback compensation amount, which is unlikely to affect the stability of the feedback control and can maintain a wide feedback control band. Therefore, the mold level control device CT and mold level control method can suppress periodic bulging disturbances while maintaining a wide control band of feedback control.

[0211] In the case of the disturbance estimation observer disclosed in Patent Document 1, a filter is used, which causes a phase delay and limits the effect of reducing fluctuations in the molten metal level. If feedback compensation is used to deal with periodic disturbances caused by bulging, the control band will become narrow.

[0212] When the casting speed is changed, the intervals between values ​​on the slab are also changed in the time domain, which has an effect. The above-described mold level control device CT and mold level control method change the time domain to the spatial domain, so that the feedforward compensation amount can be appropriately calculated without being affected by such an effect, and the mold level can be appropriately controlled.

[0213] The feedforward compensation amount may be composed of a first sine wave component (a component of frequency j) having a peak, but the first sine wave component may have an opposite phase to the desired phase for canceling the disturbance. The above-mentioned mold level control device CT and mold level control method select the phase that best cancels the disturbance from the first to third phases of the first to third sine wave components (components of frequency j, pre, and post) as the phase of the feedforward compensation amount, thereby enabling selection of a more appropriate phase. The above-mentioned mold level control device CT and mold level control method can also handle cases where disturbances of multiple frequencies are applied. For example, when peaks are adjacent, the phase also changes significantly near the frequency of the adjacent peaks. However, by determining the amount of cancellation of the disturbance and determining the phase that cancels the disturbance, the above-mentioned mold level control device CT and mold level control method can select an appropriate phase.

[0214] Since it is determined whether or not to cancel the disturbance within a fixed period (time and space) at the sampling timing, the above-mentioned molten metal level control device CT and molten metal level control method are unlikely to provide overly sensitive compensation even in unsteady cases where the disturbance or the controlled object changes.

[0215] The above-mentioned molten metal level control device CT and molten metal level control method adjust the gain type, so that feedback control can be performed with an appropriate gain according to the absolute value of the rate of operation of the valve VL, and the molten metal level can be controlled within the rate limit (within the operating speed limit) of the actuator that drives the valve. In other words, the above-mentioned molten metal level control device CT and molten metal level control method can make the performance of feedback control as high as possible without causing instability, and can continue control.

[0216] Therefore, the above-mentioned molten metal level control device CT and molten metal level control method operate adaptively to changes in the controlled object and external disturbances, maintain stability, maintain a wide control band, and use feedforward compensation in combination with feedback compensation, making it possible to suppress fluctuations in the molten metal level due to external disturbances, even if the external disturbances contain multiple frequency components.

[0217] In order to express the present invention, the present invention has been properly and sufficiently described above through the embodiments with reference to the drawings, but it should be recognized that those skilled in the art can easily change and / or improve the above-mentioned embodiments. Therefore, unless the changes or improvements made by those skilled in the art are at a level that causes departure from the scope of the claims described in the claims, such changes or improvements are interpreted as being included in the scope of the claims. [Explanation of symbols]

[0218] CT Molten metal level control device SY Upper system CM continuous casting machine TD Tundish AC Actuator VL Valve MD mold RL Role 1 Level measurement section 10 Subtractor 11 FB compensator 12 FF compensator 13 Adder 111 PID control unit 112 Filter section

Claims

1. A molten steel level control device that controls a molten steel level in a continuous casting machine by using feedback control to adjust a flow rate of molten steel supplied to a mold of the continuous casting machine, an FF compensation unit that calculates a feedforward compensation amount consisting of a sine wave component having an amplitude, a period, and a phase for canceling out a disturbance applied to the molten metal level, based on the molten metal level, an operation amount of the device, and a transfer function of the continuous casting machine related to the molten metal level; an adder that adds a feedforward compensation amount calculated by the FF compensator to a feedback compensation amount of the feedback control when performing control using the feedback control, When the molten metal level and the operation amount of the device are a plurality of values ​​arranged in time series at predetermined time intervals in the time domain, the FF compensation unit converts the values ​​of the molten metal level and the operation amount of the device in the time domain into values ​​in a spatial domain having coordinate axes in the longitudinal direction of the slab produced by the continuous casting machine, and converts the transfer function from the time frequency domain to the spatial frequency domain, and then calculates the feedforward compensation amount. Water level control device.

2. the FF compensation unit obtains a plurality of the sine wave components for a plurality of frequencies different from each other, obtains a cancellation amount of the disturbance for each of a first phase of a first sine wave component having a peak from among the plurality of sine wave components, a second phase of a second sine wave component on the low frequency side relative to the first sine wave component, and a third phase of a third sine wave component on the high frequency side relative to the first sine wave component, and selects, from the first to third phases, a phase that cancels out the disturbance most as the phase of the feedback compensation amount. The molten metal level control device according to claim 1 .

3. The control system further includes a gain type changing unit that changes the gain of the feedback control to be smaller than the current value when the absolute value of the rate of the manipulated variable of the device is larger than a predetermined first threshold, and changes the gain of the feedback control to be larger than the current value when the absolute value of the rate of the manipulated variable of the device is smaller than a predetermined positive second threshold that is smaller than the first threshold. The molten metal level control device according to claim 1 or 2.

4. Feedforward compensation is performed in a time-frequency domain that is lower than the time frequency at which the phase difference of the modeling error of the controlled object is π / 3 and is lower than the time frequency of the surface wave of the molten metal surface in the mold. The molten metal level control device according to claim 1 or 2.

5. A method for controlling a molten steel level in a continuous casting machine by using feedback control to control a device capable of adjusting a flow rate of molten steel supplied from a tundish of the continuous casting machine to a mold, comprising: an FF compensation step of determining a feedforward compensation amount consisting of a sine wave component having an amplitude, a period, and a phase for canceling out a disturbance applied to the molten metal level, based on the molten metal level, an operation amount of the device, and a transfer function of the continuous casting machine related to the molten metal level; an adding step of adding the feedforward compensation amount calculated in the FF compensation step to a feedback compensation amount of the feedback control when performing control using the feedback control, When the molten metal level and the operation amount of the device are a plurality of values ​​arranged in a time domain at predetermined time intervals in a time series, the FF compensation step converts the values ​​of the molten metal level and the operation amount of the device in the time domain into values ​​in a spatial domain having coordinate axes in the longitudinal direction of the slab produced by the continuous casting machine, and converts the transfer function from the time frequency domain to the spatial frequency domain, and then calculates the feedforward compensation amount. Method for controlling the melt level.

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