JP2021015475A - Parameter design method and feedback control method - Google Patents

Abstract

To make it possible to easily design a feedback control system controller having robust stability and disturbance suppression characteristics.SOLUTION: A parameter design method is a method for designing parameters of a controller 1 that performs feedback control of a control target 2, wherein the controller includes a PID control unit 11 and a filter unit 12 separately installed from the PID control unit. The method is determined based on a transfer function of the control target. The method includes a setting process of setting a constraint condition for a mixed sensitivity problem, including robust stability and disturbance suppression characteristics for each of a plurality of frequencies obtained by discretizing a frequency domain including a frequency to be controlled and a decision process of determining the parameters of the controller, which are composed of the parameters of the PID control unit and the filter unit, respectively, by solving the mixed sensitivity problem under the constraint condition by an optimization method.SELECTED DRAWING: Figure 6

Description

The present invention relates to a method of designing parameters of a controller that performs feedback control, and a footback control method using a controller in which parameters designed using this method are set.

A practical feedback control system requires various performances. Specifically, stability, disturbance suppression, structural simplicity, adaptability, feasibility in hardware, ease of design, ease of maintenance, etc. The control system includes PID control, robust control, adaptive control, model prediction control, etc., but each control system has advantages and disadvantages and is not universal.

PID control has a simple structure, can be given adaptive functions by table lookup, etc., and has high performance but difficult to maintain hardware (industrial PCs, etc.), but also has limited performance. It can be easily realized by hardware such as general-purpose DCS (Distributed Control System) and PLC (Programmable Logical Control), which are easy to maintain. In addition, there are few parameters, and it is possible to manually adjust the site while trying control. On the other hand, due to its simple structure, its performance may be limited.

Further, in order to improve the performance of PID control, phase lead compensation, phase delay compensation, and phase lead / delay compensation may be added. However, these adjustment guidelines are not always clear quantitatively. In addition, when the control target is not modeled, adjustment is performed while observing the actual response.

H∞ control, which is one of the robust controls, can design a controller having robust stability and disturbance suppression specified in the frequency domain. If the model is accurate including uncertainty, it is possible to design a high-performance controller after reducing the number of trials and errors as compared with the PID control system. However, the structure becomes complicated, the controller becomes a high-order controller, and pole-zero cancellation may occur in the controller. In some cases, hardware capable of high-precision calculation is required. Moreover, it is difficult to adapt. In addition, it is difficult to realize with general-purpose controller hardware.

As H∞ control, for example, Non-Patent Document 1 discloses a technique for H∞ control of the molten metal level of a slab continuous casting machine.

Akira Murakami, 4 others, "Continuous casting machine hot water level H∞ control", [online], Kobe Steel Technical Report / Vol. 48 No. 2 (Sep. 1998), p. 54-57, [Search on June 3, 1991], Internet <URL: https://www.kobelco.co.jp/technology-review/pdf/48_2/054-057.pdf

Illustrate the problems that arise in the design of practical, robust and adaptive control systems. The design of robust control such as H∞ control is as follows. The order is high. Special special algorithms such as solving Ricatch inequalities and linear matrix inequalities are required. Since there is pole-zero cancellation, it tends to be numerically unstable and high-performance hardware is required (single-precision floating-point arithmetic is not sufficient, double-precision floating-point arithmetic is required, etc.). Difficult to adapt.

In reality, a high-order controller was calculated using dedicated software, and control was performed using high-performance control hardware. As for adaptation, a method of preparing a plurality of H∞ control controllers and switching the controllers during control was used.

On the other hand, the PID control system is mainly applied in the actual process, but has the following problems. With PID control, it is difficult to design in consideration of robustness. In addition, with PID control alone, the degree of freedom of control is small, and there are cases where the control capability is limited. Moreover, the limit of controllability is difficult to understand. That is, I did not know whether it was the limit of parameter adjustment or whether there was still an adjustment allowance.

An object of the present invention is a parameter design method capable of easily designing a feedback control system controller having robust stability and disturbance suppression characteristics, and a feedback control method using a controller in which parameters designed by this design method are set. Is to provide.

The parameter design method according to the first aspect of the present invention includes a PID control unit and a filter unit provided separately from the PID control unit, and designs the parameters of the controller in a controller that feedback-controls the control target. It is a method, which is determined based on the transfer function of the controlled object and the transfer function of the controller, and for the mixed sensitivity problem including robust stability and disturbance suppression characteristics, the frequency domain including the frequency to be controlled is separated. By solving the setting step of setting the constraint condition of the mixing sensitivity problem for each of the obtained plurality of frequencies and the mixing sensitivity problem under the constraint condition by an optimization method, the PID control unit and the filter unit The present invention includes a determination step of determining the parameters of the controller composed of the respective parameters.

The PID control unit is a part that executes PID control. The control executed by the PID control unit is a control including at least P control. That is, control without I control or D control (including control in which the gain of I control or D control is 0 and there is substantially no I control or D control) may be used. For example, P control, PI control, PD control, or PID control may be used. Further, it may be a proportional leading type, a differential leading type, or the like. Further, the PID control unit may include an inexact differential or a filter for removing noise. The frequency to be controlled is, for example, from frequency 0 to the Nyquist frequency in digital control. The filter unit has a function of adjusting the gain and the phase. For example, a quadratic filter unit (a filter unit composed of a quadratic rational expression) can be mentioned. Solving the mixed sensitivity problem by an optimization method means, for example, an evaluation function (for example, an equation) represented by a predetermined variable (variable K _W1H ) included in a weighting function (for example, equation (12)) constituting the mixed sensitivity problem. The minimum value of (16)) is obtained under constraints such as a mixing sensitivity problem (for example, equations (18) and (19)).

The parameter design method according to the first aspect of the present invention has the following advantages.

(1) Parameters can be mounted on a device (for example, DCS, PLC) on which the PID control unit and the filter unit can be mounted, and special hardware is not required.

(2) The order is smaller than that of H∞ control. In addition, since it does not cause pole-zero cancellation, high-precision arithmetic is not required (for example, double-precision floating-point arithmetic is not required, and single-precision floating-point numbers are sufficient).

(3) H∞ control requires dedicated software such as a solver for Ricatch equations and linear matrix inequalities, but the parameter design method according to the first aspect of the present invention does not require dedicated software and is general-purpose non-linear. Optimization software can be used.

(4) Robust stability and disturbance suppression characteristics can be specified.

(5) Since it is a combination of PID control and a filter, on-site adjustment is possible. In addition, the conventional classical control adjustment know-how (eg, phase lead / delay compensation know-how) can be used.

(6) In PID control, the control performance may not be sufficient (for example, the control target is a vibration system). By adding a filter unit to the controller separately from the PID control unit, the degree of freedom of the controller is increased and a controller with high performance can be obtained.

As described above, according to the parameter design method according to the first aspect of the present invention, it is possible to easily design a feedback control system controller having robust stability and disturbance suppression characteristics.

In the above configuration, the control target is a continuous casting machine, and the control amount is the molten metal level.

This configuration defines an example of a controlled object. The control target is not limited to this, and may be, for example, a vibration system.

The feedback control method according to the second aspect of the present invention is stored in the storage step of storing the respective parameters of the plurality of controllers designed by the parameter design method according to the first aspect of the present invention in advance and the storage step. Further, the present invention includes a change step of changing the controller by determining a parameter from each of the parameters of the plurality of controllers by interpolation and setting the control target to the parameter of the controller for feedback control. In the change step, the controller is changed so that the control characteristics of the controller are continuously changed.

Note that "determined by interpolation" means that the parameters are stored as table values and obtained by interpolation such as linear interpolation. Further, it includes equivalent processing such as function-approximate of parameters like a response surface and determination using the function. That is, it suffices that the controller can continuously change the parameters and can determine the continuous values of the parameters.

The feedback control method according to the second aspect of the present invention is based on the respective parameters of the plurality of controllers designed by the parameter design method according to the first aspect of the present invention. Depending on the state of the controlled object and the state of disturbance, one controller may not be sufficient. Therefore, in consideration of the state of the controlled object and the state of disturbance, a plurality of controllers having different control characteristics are prepared, and the controllers are changed according to the situation. Specifically, the parameters of a plurality of controllers having a fixed structure can be stored in a table, and the controllers can be continuously changed by a simple method of table lookup and interpolation. Therefore, robust stable and adaptive feedback control can be realized by general-purpose control (PID control and filter), and can be realized by general-purpose hardware such as DCS and PLC. As a result, the feedback control method according to the second aspect of the present invention can be applied to the molten metal level control of the continuous casting machine. It can also be applied to other equipment that can be modeled other than continuous casting machines.

Since the controller is changed so that the control characteristics of the controller change continuously, it is possible to prevent fluctuations in the control amount and the operation amount.

When the control target is a continuous casting machine and the control amount is at the molten metal level, the surface scratches on the cast slab are reduced, so that the surface grinding work is reduced. In addition, since the entrainment of powder is reduced, the quality of the final product such as a steel plate is improved.

According to the present invention, a parameter design method capable of easily designing a feedback control system controller having robust stability and disturbance suppression characteristics, and a feedback control method using a controller in which parameters designed by this design method are set are provided. Can be provided.

It is a block diagram of the feedback control system to which the feedback control method which concerns on embodiment is applied. It is a schematic diagram of a continuous casting machine. It is a block diagram of the controller shown in FIG. It is the Bode diagram of the H∞ controller and the graph which shows the frequency shaping result obtained by solving the general mixing sensitivity problem in the H∞ controller. It is a Bode diagram and the graph which shows the frequency shaping result of the controller which set the parameter obtained by solving the general mixing sensitivity problem in the controller provided with the PID control part and the secondary filter part. It is a flowchart of the design method of the controller which concerns on embodiment. It is a Bode diagram and the graph which shows the frequency shaping result of the controller which set the parameter obtained by solving the general mixing sensitivity problem in the controller provided with the PID control part and the primary filter part. It is a Bode diagram and the graph which shows the frequency shaping result of the controller which set the parameter obtained by solving the 2-Disc mixing sensitivity problem in the controller provided with the PID control part and the secondary filter part. Each determined parameters _{(K P, T I, a} 1, a 2, b 1, b 2) and shows _K f / A and the gain-type (H, M, L) the relationship between the tables, and, on the table It is a figure which shows the example of the change of the controller in. It is a figure which shows the example of the interpolation method of a parameter K _P. It is a diagram obtained by the graph the value of the table of each parameter _{(K P, T I, a} 1, a 2, b 1, b 2) shown in Figure 9A. It is a figure which shows the outline of the adaptation simulator. 3 is a graph showing setting conditions 1 to 3 and results 1 to 2 when adaptation simulation is performed for the controller 1 of Example 4. It is a graph which shows the result 3-8 when the adaptation simulation was performed about the controller 1 of Example 4. It is a figure which shows the example of the fluctuation at the time of switching at the time of the test of another control system. Each determined parameters _{(K P, T I, a} 1, a 2, b 1, b 2) and shows _K f / A and the gain-type (H, M, L) the relationship between the tables, and, on the table It is a figure which shows another example of the change of the controller in. 3 is a graph showing setting conditions 1 to 3 and results 1 to 2 when adaptation simulation is performed under the conditions shown in FIG. 11D for the controller of Example 4. It is a graph which shows the result 3-8 when the adaptation simulation was performed with respect to the controller of Example 4 under the condition shown in FIG. 11D. It is a flowchart of the adaptive operation of the controller of Example 4.

Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings. In each figure, the configurations with the same reference numerals indicate that they are the same configuration, and the description of the configurations already described will be omitted.

FIG. 1 is a block diagram of a feedback control system 100 to which the feedback control method according to the embodiment is applied. The feedback control system 100 includes a controller 1 and a control target 2. The control amount (output) y of the control target 2 is negatively fed back, and the deviation e between the target value r and the control amount y is input to the controller 1. The controller 1 calculates and outputs the operation amount u based on the deviation e. The operation amount u is input to the control target 2. In the case of the proportional leading type, the differential leading type, etc., a part of the manipulated variable u may be calculated based on the controlled variable y instead of the deviation e.

The control target 2 is, for example, a continuous casting machine 200. FIG. 2 is a schematic view of the continuous casting machine 200. The continuous casting machine 200 is an apparatus for producing a long slab by pouring molten steel into a mold 23 and pulling out molten steel whose side surfaces have solidified from the mold 23. The continuous casting machine 200 includes a tundish 21, a nozzle 22, a mold 23, and a plurality of sets of rolls 24, and are arranged from the upstream side to the downstream side in this order.

The tundish 21 is a device in which molten steel flows from a ladle (not shown) and removes predetermined inclusions from the molten steel. Enclosures are removed by ascending and separating. A through hole is formed in the bottom wall of the tundish 21, and a nozzle 22 is attached to the through hole.

The mold 23 is a device in which molten steel flows from the tundish 21 through the nozzle 22 and cools the molten steel to form steel having a predetermined shape. The mold 23 is water-cooled, and the molten steel in contact with the mold 23 is solidified from the outside to form a relatively thin solidified shell, and a steel having a predetermined shape is formed.

The roll 24 is a device that supports the steel while drawing the steel from the mold 23 at a predetermined speed. The rolls 24 are a set of two arranged so as to be in contact with both sides of the steel, and a plurality of rolls 24 are arranged on the next stage of the mold 23 with a predetermined interval from the upstream side to the downstream side. The plurality of sets of rolls 24 are arranged so that the casting direction (pulling direction) faces from the vertical direction to the horizontal direction.

The continuous casting machine 200 includes a slide valve 31, an actuator 32, and a vortex sensor 33, and controls the molten steel level in the mold 23. The eddy current sensor 33 measures the molten steel level in the mold 23 (Level in the figure. For control, the value converted to a positive height with the vertical upward direction is used as the molten steel level). .. The molten metal level measured by the vortex sensor 33 is output as a control amount y. The deviation e between the control amount y and the target value r is input to the controller 1. The controller 1 calculates the operation amount u and outputs it to the actuator 32. As described above, in the case of the proportional preceding type, the differential leading type, or the like, the control amount y may be input to the controller 1. The actuator 32 operates the slide valve 31 by the operating amount u. When the controller is a speed type, the actuator 32 operates the derivative or difference of the manipulated variable u.

FIG. 3 is a block diagram of the controller 1 shown in FIG. The controller 1 includes a PID control unit 11 and a filter unit 12. The filter unit 12 is provided separately from the PID control unit 11. The deviation e is input to the PID control unit 11, and the output of the PID control unit 11 is input to the filter unit 12. The output of the filter unit 12 becomes the operation amount u. As described above, in the case of the proportional preceding type, the differential leading type, or the like, the control amount y may be input to the PID control unit 11.

For example, an example of the transfer function K _PID (s) when the PID control unit 11 executes PID control is as follows. In the following description, the PID control unit having this structure will be used. The PID control unit that executes the equation (1) is originally one form of the "PID control unit", and should be described as "PID control unit that performs PID control of the structure of the equation (1)". However, since it is a long notation, it will be referred to as "PID control unit" hereafter. The K _PID (s) is a transfer function of the PID control unit 11, not a transfer function K (s) of the controller 1. The transfer function K (s) of the controller 1 will be described later (Equation (15)).

The basic parameters of the PID controller 11 of this embodiment, _{K P} (proportional gain), _{T I} (integral time) is _{T D} (derivative time). η is a coefficient for inexact differential and is often set to 0.1. When the equation (1) is modified, it becomes as follows.

Constant gain proportional gain, the gain of the integrator integral gain, if the gain of the differentiator and the differential gain, proportional gain is K _P, the integral gain is K _{P /} T _I, the derivative gain K _P · the T _D.

The filter unit 12 is a portion added to the PID control unit 11 to form the controller 1. PID control has a function of finely adjusting the gain and phase when sufficient control performance cannot be obtained. For example, in the case of the control target 2 of the vibration system, it may be difficult to control by PID control. In such a case, vibration may be suppressed by adding the filter unit 12 and increasing the controller gain of a specific frequency, for example. Alternatively, as another example, by reducing the controller gain of a specific frequency, it is possible to prevent the control from becoming unstable due to vibration. Depending on the value of the parameter of the filter unit 12, the filter unit 12 may be, for example, a phase lead compensator or a phase lag compensator.

It is assumed that the controlled object 2 is represented by the transfer function P (s) in the embodiment. It may include wasted time.

The sensitivity function S (s) and the quasi-complementary sensitivity function R (s) are defined as follows. These functions are defined using K (s) and P (s) and are dealt with in the mixed sensitivity problem.

Further, assuming that the additive uncertainty is Δ (s), the transfer function P (s) of the controlled object 2 can be expressed as follows.

Here, P ₀ (s) is the nominal control target 2.

The mixing sensitivity problem will be described. Let the weighting functions be W _1H (s) and W _2H (s). The subscript H indicates that it is a weighting function for designing the high gain type controller 1.

The mixed sensitivity problem of H∞ control is shown as follows. Here, P ₀ (s) is substituted for P (s) in the sensitivity function, the quasi-complementary sensitivity function, and the complementary sensitivity function to be used later. The same applies to any of the following mixing sensitivity problems.

This can be solved by using a known Ricatch inequality or linear matrix inequality solver (The MathWorks, Inc. "Robust Control Toolbox" (registered trademark)) hinfsyn function or the like.

In the embodiment, additive uncertainty is dealt with, but multiplicative uncertainty may be dealt with. In this case, in the mixed sensitivity problem, the sensitivity function S (s) and the complementary sensitivity function T (s) are dealt with. Let the weighting functions be W _1H '(s) and W _2H '(s).

Here, the mixed sensitivity problem of the formula (6) or (7) is referred to as a "general mixed sensitivity problem" in the embodiment to distinguish it from the 2-disk problem. For the 2-disk problem, the constraint is one of the following:

In the embodiment, the "mixing sensitivity problem" refers to a problem in which any of the equations (6) to (9) is included in the constraint condition. The mixed sensitivity problem is solved using an optimization technique. This will be shown in a concrete example. The control target 2 is a continuous casting machine 200.

Here, P ₀ (s) is a component of equation (5), T _SC is the stepping cylinder time constant, L _SC is the stepping cylinder waste time, K _f is the flow coefficient, and A is the mold. It is the cross-sectional area, T _CD is the level clock time constant, and L _CD is the level clock waste time. With reference to FIG. 2, the stepping cylinder is the actuator 32, the mold cross-sectional area is the cross-sectional area of the mold 23, and the level meter is the vortex sensor 33.

T _SC = 0.2, T _CD = 0.3, L _SC = 0.1, L _CD = 0.12. K _f and A vary depending on the casting conditions, and the range is 0.6 <K _f / A <2.1. First, in this example, the following case is considered as the first high gain type controller 1.

The weight function is set as follows. As shown by the evaluation function described later, the variable K _W1H of the weight function W _1H (s) is the value to be maximized.

By increasing the variable K _W1H as much as possible, the disturbance suppression property can be maximized.

The uncertainty Δ (s) is calculated assuming that the time constant and the dead time all change by ± 20%. The uncertainty Δ (s) constitutes equation (5). As described in Non-Patent Document 1, the weights W _2H (s), W _2M (s), and W _2L (s) described later are set so as to cover Δ (s) in the high frequency region.

As a comparative example, a design method of the H∞ controller will be described. An H∞ controller is obtained by solving a general mixing sensitivity problem. At this time, the structure of the H∞ controller is automatically determined and cannot be specified. FIG. 4 is a Bode diagram of the H∞ controller obtained by solving a general mixing sensitivity problem and a graph showing the frequency shaping result. It can be seen that the H∞ controller has a large gain at low frequencies and a substantially constant gain near 0.1 [Hz] on the high frequency side, and is basically PI control. Furthermore, the gain decreases near 1.0 [Hz] on the high frequency side, and it can be interpreted that a filter is added. As disclosed in Japanese Patent No. 3091061, the H∞ controller is obtained by the method of solving the equation (6), which is a general mixing sensitivity problem, as follows.

By comparing the poles and zeros, we can see the following.

zH1 ₁ and ₂ and pH ₁ and ₂ and ₃ cause pole-zero cancellation.

zH1 ₃ and pH1 ₄ is close to the pole-zero offset.

zH1 ₄ and pH1 ₅ is, close to the pole-zero offset.

zH1 ₁₀ , ₁₁ and pH 1 ₁₁ , ₁₂ cause pole-zero cancellation.

If we include the fact that it is close to the pole-zero offset in the pole-zero offset, we can see that there are six pole-zero offsets (zH1 _1,2 and pH1 _2,3 , zH1 _10,11 and pH1 _11,12 , respectively. Extreme zero offset). In this case, it is considered that at least the 6th order controller can be excluded from the 13th order controller, and the 7th order controller can approximate the H∞ controller. In reality, since the remaining 7th order also contains poles and zeros that have a small effect, it is considered that the H∞ controller can be approximated by a lower order controller.

In the embodiment, it is considered that the structure of the controller 1 is fixed to the PID control unit 11 with the filter unit 12 added to achieve performance close to that of the H∞ controller. Here, it is assumed that the structure is fixed means that only the value of the variable parameter is different. For example, the transfer functions 1 / (2 · s + 1) and 1 / (3 · s + 1) have a fixed structure of 1 / (T · s + 1) with a variable parameter of T. On the other hand, the transfer functions 1 / (2 · s + 1) and s / (3 · s + 1) have different molecules at 1 and s, and are not fixed. As an approximation method, there is also a method of lowering the dimension by realizing equilibrium, but it is not necessarily adopted because a controller having a fixed structure like the PID control unit 11 cannot be obtained.

The parameter design method of the embodiment is a parameter design method of the controller 1 including the PID control unit 11 and the filter unit 12. As this design method, there are the following Examples 1 to 3.

Example 1: Consider that the PID control unit + the secondary filter unit and the mixing sensitivity controller 1 are approximated by the PID control unit 11 + the secondary filter unit 12. The phase lead / lag compensation is often a first-order rational expression × a first-order rational expression, but in the embodiment, it is a quadratic rational expression having a wider range. Even if it is a first-order rational expression, trial and error and readjustment are required to determine the parameters for phase lead / delay compensation, but in the embodiment, even if it is a second-order rational expression, it is based on the model. , Find the parameters by optimization without trial and error and readjustment. The transfer function K (s) of the controller 1 of Example 1 is shown below.

The transfer function K (s) is composed of the transfer function K _PID (s) of the PID control unit 1 shown in Equation 1 and the transfer function of the second-order filter unit 12. Parameters of PID control unit 11 three _{(K P, T I, T} D) is a parameter of the filter unit 12 is four _{(a 1, a 2, b} 1, b 2). It is necessary to determine these seven parameters as the parameters of the controller 1. Controller 1 is not linear. It is difficult to determine these parameters because it is necessary to balance robust stability and disturbance inhibition. Even if the filter unit 12 of the primary, it is necessary to determine five parameters _{(K P, T I, T} D, a 1, b 1) a. At the same time, it is necessary to meet the needs of maintaining stability, increasing the disturbance suppression band as much as possible, reducing the fluctuation of the molten metal level, and improving the quality of slabs.

Frequency from _{f min (= 0.001) [Hz} ], f max (= 10) [Hz] to, the logarithmically spaced N ω ₍₌ 200) points, the frequency of the variable _f i (i = 1 , ..., N _ω ) is obtained. Here, when digital control is performed, the frequency to be controlled is from 0 to the Nyquist frequency, but in this example, the sampling period is 0.1 [s]. Therefore, the frequency ranges from 0 [Hz] to the Nyquist frequency 5 [Hz]. As for the lower limit, since the frequencies are logarithmically evenly spaced, it cannot be set to 0, so a small value of 0.001 [Hz] is set. The upper limit is set to 10 [Hz], which is a large value near the Nyquist frequency. Further, to obtain the _{f i,} multiplied by 2 [pi, variable _omega i angular frequency (i = 1, ···, N ω) a.

The evaluation function to be minimized is set as follows. The gain of the weighting function is made as large as possible to maximize the frequency band of disturbance suppression. The variable K _W1H is described by the equation (12).

Next, the constraints at each angular frequency are obtained. The norm constraint of the above-mentioned mixing sensitivity problem is as follows.

Here, σ () indicates the maximum singular value.

In order to ensure the accuracy at the time of optimization calculation, the constraint condition converted to dB is obtained as follows.

Here, the parameters of the PID controller 11 to a negative feedback, there are _{respectively, K p> 0, T I} > 0, T D ≧ 0 is essential constraints. Furthermore, since this optimization problem is non-convex and may fall into a local solution, the following constraints are set for each variable in order to facilitate convergence. _{Incidentally, K p> 0, T I} > 0, essential non constraint constraint T _D ≧ 0, which has provided for convenience to avoid the case where Recessed a local solution, not essential. For example, in the multi-start local search method of this example, the optimum solution can be obtained without any convenient restrictions by increasing the number of initial values to be tried at the time of optimization.

The last two equations limit the damping coefficient of the filter unit 12. That is, the last two equations are provided to search for the optimum value in a region where the zeros and poles are less likely to vibrate and are less likely to approach the imaginary axis.

In addition, in this example, the multi-start local search method is used for the optimization method, and for the initial value, the initial value is tried at multiple points, and the optimization result from the next initial value so as not to fall into a local solution. It was adopted.

The optimization problem is in the form of finding the minimum value of the evaluation function of the equation (16) under the constraints of the equations (18) and (19).

When the optimization problem is solved as a global optimization problem, the setting of constraint conditions excluding the constraint of the mixed sensitivity problem and the constraint of the PID parameter may be included in the global optimization or may not be necessary. The reason is that, in the embodiment, a kind of multi-start local search method is used, and the constraints excluding the constraint of the mixed sensitivity problem and the constraint of the PID parameter are convenient so that the convergence in the local search is easy. This is because it is provided in.

For the local search part, a public nonlinear optimization method is used (the fmincon function of "Optimization Toolbox" (registered trademark) of The MathWorks, Inc.). The following results were obtained.

PID control unit 11, the derivative time _{T D} is 0, it is understood that the substantially PI control. FIG. 5 is a Bode diagram and a graph showing the frequency shaping result of the controller 1 in which the parameters obtained by solving the general mixing sensitivity problem in the controller 1 including the PID control unit 11 and the secondary filter unit 12 are set. Is. The parameters here are the parameters shown in the equation (21). Since the filter is PI control + 2nd order, the order is 3rd order, but it can be seen that controller 1 having almost the same performance as the H∞ controller is obtained.

The controller 1 fixed to the structure of the PID control unit 11 + secondary filter unit 12 can be obtained by calculation instead of adjustment by using a model.

The design method of the controller 1 according to the embodiment (Example 1 and the controllers 1 of Examples 2 and 3 described later) will be briefly described. This design method includes a parameter design method according to an embodiment. FIG. 6 is a flowchart of the design method of the controller 1 according to the embodiment. This includes a method of designing parameters according to the embodiment. First, the designer sets the transfer function P (s) of the controlled object 2 (P ₀ (s) is used as P (s) as described above) (S1). Next, the designer determines the structures of the PID control unit 11 and the filter unit 12 (S2). Next, the designer sets the evaluation function (S3). Next, the designer sets the constraint condition (S4). Finally, the designer uses a computer to solve the global optimization problem (S5). As described above, the parameter of the controller 1 is determined by the process S5.

Example 2: PID control unit + 1st-order filter unit, mixing sensitivity Next, an example of the primary filter unit 12 is shown. Here, the parameters of the PID controller 11 to a negative feedback, there are _{respectively, K P> 0, T I} > 0, T D ≧ 0 is essential constraints. Furthermore, since this optimization problem is non-convex and may fall into a local solution, the following constraints are set for each variable in order to facilitate convergence. Here, for a ₂ and b ₂ , the optimization calculation is performed with a constraint that the value is set to 0. _{Incidentally, K P> 0, T I} > 0, T D ≧ 0 is indispensable except constraint constraints, which has provided for convenience to avoid the case of falling into a local solution, not essential. For example, in the multi-start local search method of this example, the optimum solution can be obtained without any convenient restrictions by increasing the number of initial values to be tried at the time of optimization.

As for the initial value, the initial value was tried at multiple points, and the optimization result from the next initial value was adopted so as not to fall into a local solution.

In the case of this linear filter, the evaluation function of Eq. (16) was optimized under the constraints of Eqs. (18) and (22), and the following results were obtained.

FIG. 7 is a Bode diagram and a graph showing the frequency shaping result of the controller 1 in which the parameters obtained by solving the general mixing sensitivity problem in the controller 1 including the PID control unit 11 and the primary filter unit 12 are set. Is. The parameters here are the parameters shown in the equation (24). The order is quadratic because of the PI control + 1st order filter. Although it is difficult to understand in FIG. 7 (it can be seen by enlarging FIG. 7), the sensitivity function S is on the low frequency side as compared with FIG. 5 (controller 1 that executes PI control + quadratic filter). From this, it can be seen that the disturbance suppression performance is lower than that of the controller 1 that executes the PI control + secondary filter and the H∞ controller. This is probably because the degree of freedom in design is not sufficient.

Example 3: PID control unit + secondary filter unit, 2-disk
Next, an example of the 2-disk problem is shown. In this example, the constraint of the mixing sensitivity problem is relaxed, so that the disturbance suppression band is widened. Further, in the case of one input / output system, there is an advantage that it is not necessary to calculate a singular value. It is the same as the above-mentioned case that the controller 1 is approximated by the PID control unit 11 + the second-order filter unit 12. The evaluation function is the same as in the above case.

Next, the constraints at each angular frequency are obtained. Unlike the above-mentioned mixing sensitivity problem, the restrictions of the 2-disk problem are as follows.

In order to ensure the accuracy at the time of optimization calculation, the constraint condition converted to dB is obtained as follows.

Here, the parameters of the PID controller 11 to a negative feedback, there are _{respectively, K P> 0, T I} > 0, T D ≧ 0 is essential constraints. Furthermore, since this optimization problem is non-convex and may fall into a local solution, the following constraints are set for each variable in order to facilitate convergence. _{Incidentally, K P> 0, T I} > 0, T D ≧ 0 is indispensable except constraint constraints, which has provided for convenience to avoid the case of falling into a local solution, not essential. By increasing the number of initial values to be tried at the time of optimization, the optimum solution can be obtained without any convenience restrictions.

As for the initial value, the initial value was tried at multiple points, and the optimization result from the next initial value was adopted so as not to fall into a local solution.

The optimization problem is in the form of finding the minimum value of the evaluation function of the equation (16) under the constraints of the equations (26) and (27).

The local search part was optimized using a publicly available nonlinear optimization method (the fmincon function of The MathWorks, Inc.'s "Optimization Toolbox"®) to obtain the following results.

The differential time T _D is a non-zero value. FIG. 8 is a Bode diagram and a graph showing the frequency shaping result of the controller 1 in which the parameters obtained by solving the 2-Disk mixing sensitivity problem in the controller 1 including the PID control unit 11 and the secondary filter unit 12 are set. Is. The parameters here are the parameters shown in the equation (29). Although it is difficult to understand in FIG. 8 (it can be seen by enlarging FIG. 8), it can be seen that the sensitivity function S is on the high frequency side and the disturbance suppression performance is high as compared with FIG.

However, it can be seen that the gain of the manipulated variable u in the vicinity of 0.2 [Hz] is large, and the manipulated variable saturation is likely to occur. This example is not used in "Example 4: Adaptation" described below.

Example 4: Consider that the controller 1 (PID control unit 11 + secondary filter unit 12) of the application example 1 is made to correspond to a change of the control target 2 (continuous casting machine 200). Since the structure is fixed, the control characteristics of the controller 1 can be continuously changed as described later.

In the control target 2, K _f / A is designed from 0.6 to 2.1 in increments of 0.1. Further, in the description of Example 1, the high gain type controller 1 is designed, but the middle gain type and low gain type controllers 1 which are lower gain types are also designed. Therefore, there are three types of gain to design. Therefore, the number of controllers 1 to be designed is 48. This is shown in the table below. The high gain type is indicated by "H", the middle gain type is indicated by "M", and the low gain type is indicated by "L".

Since the gain of the controlled object 2 changes more than twice, it is conceivable that the performance of one controller 1 deteriorates or becomes unstable. In the official PID control, the PID parameter may be written in the table and the PID parameter may be changed according to the control target 2 or the change of the disturbance. This time, it is considered that the parameters of the controller 1 are tabulated and the parameters are continuously changed according to the change of the control target 2 and the disturbance. By continuously changing the parameters, the control characteristics of the controller 1 are continuously changed.

K _f / A is designed from 0.6 to 2.1 in increments of 0.1 in the same manner as in Example 1 (in the case of 0.6, it has already been designed in Example 1).

Since H1 has already been calculated, H2 to H16 can be calculated in the same manner as in Example 1 by changing K _f / A in the equation (10) to obtain the controller 1 of Example 1.

Next, for the middle gain types M1 to M16, the weighting function and the evaluation function are set as follows to perform optimization. As shown by the evaluation function described later, the variable T _W1M of the weighting function W _1M (s) is the value to be minimized. W _2M (s) is the same as W _2H (s).

The evaluation function to be minimized for the middle gain type is set as follows. The evaluation function maximizes the frequency band of disturbance suppression by making T _W1M as small as possible and, as a result, increasing the gain of the weighting function W _1M (s).

Further, for the low gain types L1 to L16, the weighting function and the evaluation function to be minimized are set as follows to perform optimization. As shown by the evaluation function described later, the variable T _W1L of the weighting function W _1L (s) is the value to be minimized.

The evaluation function of the low gain type was set as follows. As small as possible T _W1L, as a result, by increasing the gain of the weighting function W _{1L (s),} is an evaluation function that maximizes the frequency band of the disturbance suppression.

As described above, the parameters of each of the 48 controllers 1 shown in the above table are obtained. The 48 controllers 1 have the same structure, but have different parameters and different control characteristics. In this example, the control characteristics here are determined by the combination of the K _f / A value and the gain type (H, M, L).

Figure 9A, each determined parameter _{(K P, T I, a} 1, a 2, b 1, b 2) and _K f / A and the gain-type (H, M, L) of the table showing the relationship between It is a figure. T _D is 0, so is omitted. This table is stored in advance in the storage unit of the controller 1. The 48 controllers 1 shown in the above table have a common structure, and these controllers 1 are realized by changing the parameters. This will be described in detail. The gain type H, _and the _{parameters K} f / A is identified by _{0.6 (K P, T I,} a 1, a 2, b 1, b 2) is selected by the interpolation in the controller 1 (Set to controller 1), controller 1 with controller number H1 is realized.

The controller 1 of Example 4 continuously changes its control characteristics (combination of K _f / A value and gain type). Here, H16 is the case of the high gain type K _f / A ₌ 2.1, and M3 is the case of the middle gain type K _f / A ₌ 0.8.

When the controller 1 is continuously changed, it is necessary to obtain it by interpolation from the table of FIG. 9A. In this example, The MathWorks, Inc. If you use the "n-D Lookup Table" block of "Simulink" (registered trademark), set the number of dimensions of the table to "2", and use "linear points and gradients" as the interpolation method. Good.

The logic for realizing the same method as this "n-D Look Table" with DCS or PLC is as follows. Now, the gain type is 1.4 and K _f / A ₌ 0.72. Here, it is assumed that the gain type is 1, H is 1, M is 2, and L is 3. That is, the gain type 1.4 is between H and M at a ratio of 0.4: 0.6. K _f / A = 0.72 is the ratio of 0.02: 0.08 = 0.2: 0.8 of the controllers * 2 and * 3 (* indicates any of H, M, and L). Show that it is in between. In this case, showing the interpolation method of _{K p} in FIG 9B.

The value of K _{p to} be obtained is
H2 (gain type = 1, K _f / A = 0.7) 0.928,
0.845 of H3 (gain type = 1, K _f / A = 0.8),
0.435 of M2 (gain type = 2, K _f / A = 0.7),
0.396 of M3 (gain type = 2, K _f / A = 0.8),
Is between.

First, TMP1 is taken at the point where H2 and M2 are internally divided into 0.4: 0.6. The value of K _p at this point is (0.4 × 0.435 + 0.6 × 0.928) = 0.731.

Next, TMP2 is taken at the point where H3 and M3 are internally divided into 0.4: 0.6. The value of K _p at this point is (0.4 × 0.396 + 0.6 × 0.845) = 0.665.

Next, RESULT is set at the point where TMP1 and TMP2 are internally divided into 0.2: 0.8. The value of K _p at this point is (0.2 × 0.665 + 0.8 × 0.731) = 0.718. The value of this RESULT becomes the value of K _p obtained by interpolation.

T _I, for even _{a 1, a 2, b 1} , b 2, it is possible to obtain the values by the same interpolation.

Here, the logic of this interpolation and The MathWorks, Inc. And "n-D Lookup Table" block interpolation logic of "Simulink" (registered trademark) is strictly there is a possibility that there is an error not identical to, the value of such _{K p} is later Figure Since the curved surface is smooth as shown in 10A, the error is considered to be small.

By continuously changing each parameter, the control characteristics of the controller 1 can be continuously changed. Figure 10A is a diagram in which the numerical values in the graph of the table of each parameter _{(K P, T I, a} 1, a 2, b 1, b 2) shown in Figure 9A. Each graph is shown in three dimensions, the first axis shows K _f / A, the second axis shows the gain type (H, M, L), and the third axis shows the parameters.

Consider operating the controller 1 of Example 4 adaptively. As shown by the broken line arrow in FIG. 9A, first, the gain type remains H, and the K _f / A of the control target 2 is continuously reduced from 2.1 to 0.6. Next, the gain type is continuously changed from H to M to L. This applies, for example, to reducing the casting speed and then changing the gain type, for example, to reduce the controller gain.

At this time, the controller 1 continuously changes from H16, H15, H14, ..., H2, and H1 according to the value of K _f / A. It is a parameter of the controller _{1 K P, T I, T} D, a 1, a 2, b 1, b 2 reads from the above table is calculated by interpolation.

Subsequently, when the gain type is changed from H to M to L, H is replaced with 1.0, M is replaced with 2.0, and L is replaced with 3.0, and the gain type is continuously changed from 1.0 to 3.0. There is. At this time, K _f / A is 0.6. _{K P, T I, T D} , a 1, a 2, b 1, b 2 are parameters of the controller 1, similarly, read from the above table is calculated by interpolation.

Looking at the operation of the controller 1 described above from the viewpoint of the method, the following storage process and change process are provided. In the storage process, each parameter (for example, FIG. 9A) of the 48 controllers (for example, FIG. 9A) having the same structure but different control characteristics is stored in advance in the storage unit of the controller. In the change process, the controller 1 is changed by determining the parameter from each of the parameters of the 48 controllers stored in the storage process by interpolation and setting the control target 2 as the parameter of the controller 1 for feedback control. To do. In the changing step, the controller 1 is changed so that the control characteristics of the controller 1 change continuously.

The controller 1 continuously changes the control characteristics based on the value of K _f / A and the value of the gain type (real number) during the feedback control of the continuous casting machine 200.

As an example, it is conceivable to calculate K _f / A for each sampling cycle and adopt the controller number (real number) corresponding to K _f / A. As for the gain type (real number), it is conceivable to change the gain type to the low gain side when the actuator is likely to be saturated. These are performed by the upper system of the PID control + filter control system (note that it means a higher level in terms of software, and the actual calculation may be performed inside the same DCS or PLC as the PID control unit + filter unit). Will be. That is, the method of determining the controller number and the gain type is performed in the upper system, but in this embodiment, the controller number is changed based on the calculated K _f / A, and the gain type is given. It is supposed to be. As a method for calculating the value of K _f / A, the formula (6) described in the following document may be used.

Murakami et al., "Continuous Casting Machine Hot Water Level H∞ Control", Journal of the System Control Information Society, Vol. 10, No. 11, pp. 607-615, 1997
FIG. 10B shows an outline of the adaptation simulator. First, specify the change of K _f / A as the simulation condition. Here, K _f / A is input to the parameter table of FIG. 9A, assuming that it can be calculated accurately in the upper system. The gain type is specified as being set in the upper system. Also, a level disturbance is applied. This simulator is a linearized deviation system. The dynamic characteristics of the continuous casting machine, which is originally a non-linear control target, are linearized around the operating point, and the molten metal level fluctuation caused by changes in the casting speed, TD weight, and mold width is fed by the method of Patent No. 6108923. It is assumed that it is compensated by forward. A simulation is performed using this simulator.

FIG. 11A is a graph showing setting conditions 1 to 3 and results 1 to 2 when adaptation simulation is performed for the controller 1 of Example 4. FIG. 11B is a graph showing the results 3 to 8 when the adaptation simulation is performed for the controller 1 of Example 4. In FIGS. 9A and 10A, the location in the table is changed as shown by the broken line arrow. By adapting not only the PID control unit 11 but also the filter unit 12, the performance equivalent to that of H∞ control is realized, and further, it has an adaptive function. As described in Non-Patent Document 1, when the controller is switched from H16 to H15 in steps rather than continuously, conventionally, when the level fluctuation at the time of switching is large, a transient response is used. In some cases, the control amount of the molten metal level y and the operation amount u temporarily increase.

Here, for reference, FIG. 11C shows an example at the time of testing with another control system. This is the result when the method of Non-Patent Document 1 of the prior art is applied as a test to an object different from the embodiment. From the top, the controller number (integer), gain type (integer), molten metal level, which is the control amount, and slide valve opening. The slide valve opening has a difference between the primary delay of the actuator and the dead time, but in the low frequency region, the operation is almost the same as the operation amount.

This is a case where the H∞ controller of M4 is switched to the H∞ controller of H4 at time 40 [s] and is switched to the H∞ controller of M4 at time 93 [s]. As a result of switching from H4 to M4 H∞ controller at 93 [s], fluctuations in the molten metal level and opening (corresponding to the amount of operation, excluding the difference in the dynamic characteristics of the actuator) are large. I understand. The M4 and H4 controllers in FIG. 11C are H∞ controllers designed separately for other control systems, and have different control characteristics from the H∞ controller described in the embodiment and the H∞ controller of Non-Patent Document 1. However, I think that it shows the same tendency qualitatively. That is, in the case of control involving switching, the transient response immediately after switching may cause a temporary change in the controlled variable or manipulated variable.

On the other hand, in the simulation result examples of FIGS. 11A and 11B, the control amount y and the operation amount u do not fluctuate. This will be described in detail below.

With reference to FIGS. 11A and 11B, this simulation is performed under setting conditions 1 to 3. In the graph showing the setting condition 1, the horizontal axis shows the time t and the vertical axis shows K _f / A. At t = 50-80 seconds (30 seconds), K _f / A is continuously reduced from 2.1 to 0.6. Decrease in K _f / A is decreased TD weight (weight of the molten steel in the tundish 21), which corresponds to the deceleration or the like of the casting speed _{V C.}

In the graph showing the setting condition 2, the horizontal axis indicates the time t, and the vertical axis indicates the gain type (H, M, L). At t = 100 to 130 seconds (30 seconds), the controller 1 is continuously changed into a high gain type, a middle gain type, and a low gain type.

In the graph showing the setting condition 3, the horizontal axis shows the time t and the vertical axis shows the disturbance with respect to the molten metal level. This disturbance is a sine wave with a period of 4 seconds and an amplitude of 10 mm. As for the disturbance, for evaluation, the level fluctuation is set to be large with a 4-second cycle, which is the most difficult to control. The reason is that in the case of the conventional switching control, when the level fluctuation is large at the moment of switching, a temporary level fluctuation is likely to occur.

The results of the simulation are shown in Result 1 and Result 2. In the graph showing the result 1, the horizontal axis shows the time t, and the vertical axis shows the molten metal level of the controlled amount. Results As shown in 3-8, the parameter _{K P, T I, T D} , _{by a 1, a 2, b 1} , b 2 are changed continuously, the control characteristics of the controller 1 is continuously varied Therefore, when the control characteristics of the controller 1 are changed stepwise (for example, when the controller 1 of the controller number H16 is switched to H15 or M16), a transient response that may occur is a temporary rise level. There is no fluctuation.

In the graph showing the result 2, the horizontal axis represents the time t and the vertical axis represents the manipulated variable. Results As shown in 3-8, the parameter _{K P, T I, T D} , _{by a 1, a 2, b 1} , b 2 is continuously changed, the control characteristics continuously change the controller 1 Therefore, there is no temporary change in the amount of operation that occurs when the control characteristics of the controller 1 are changed stepwise. Since the fluctuation of the operation amount can be reduced, the actuator is less likely to be saturated.

FIG. 11E and FIG. 11F show another simulation result executed under different conditions when the controller 1 is changed as shown in FIG. 11D. Compared to the first example, the only difference is that the K _f / A of setting condition 1 and setting condition 2 and the time for changing the gain type are both changed to 50 seconds of t = 50 to 100 seconds. .. Similar to FIGS. 11A and 11B, a parameter _{K P, T I, T D} , is _{a 1, a 2, b 1} , b 2, continuously changed, stable, rapid melt-surface level and the operation amount varies It can be confirmed that there is no control.

The simulations shown in FIGS. 11A and 11B, and FIGS. 11E and 11F alone do not theoretically guarantee the stability of the adaptation, but the adaptation is compared with the response of the feedback control system 100. It is expected that it can be adapted stably when the speed is slow. In fact, even in the simulation results, it can be confirmed that the control is stable and there is no problem in practical use.

Since the structure of the controller 1 is fixed and the controller 1 reads the parameters from the table, the control characteristics of the controller 1 are continuously changed, so that the adaptation is easy. In addition, compared to the case of H∞ control, there are few parameters, and there is no part that tends to be unstable in numerical calculation such as pole-zero cancellation, which is considered to be one of the reasons why stable adaptation can be achieved. Although not described in this embodiment, in another simulation result, it was confirmed that stable adaptive operation can be performed even if the variable type is changed from a double precision floating point number to a single precision floating point number. There is.

Further, since the control characteristics of the controller 1 are continuously changed, there is a problem that the control amount y and the operation amount u may temporarily increase when the controller 1 is switched, as in the case where the controller is switched. It has not occurred.

A flowchart of the adaptive operation of the controller 1 according to the fourth embodiment will be briefly described. FIG. 12 is a flowchart showing this. This flowchart shows one sampling cycle and is repeated for each sampling cycle. If the output of the operation amount u is a speed type, bump press switching with manual operation or the like is possible.

The controller 1 reads information for determining the controller 1 (S11). The information is the value of K _f / A and the gain type (real numbers 1 or more and 3 or less). "1" corresponds to the gain type H, "2" corresponds to the gain type M, and "3" corresponds to the gain type L.

The parameter determined in the process S12 is set in the controller 1, and the deviation e between the control amount y and the target value r is input (S13). Then, the controller 1 calculates the operation amount u based on the deviation e (S14) and outputs it to the control target 2 (S15). In addition to the deviation e as described above, the control amount y may be used.

When the controller 1 is realized by a digital device, the PID control unit 11 + the filter unit 12 are discretized by a known method. For this, for example, a difference may be used, or a bilinear transform may be used. The former discretizes by substituting the difference [s = (1-z ^-1 ) / ΔT] with z ^-1 as the delay operator and ΔT as the sampling period. The latter is discretized by substituting the bilinear transform [s = 2 / ΔT · (1-z ^-1 ) / (1 + z ^-1 )].

In the embodiment, a kind of multi-start local search method is used as a local nonlinear optimization method (sequential secondary design method, etc.), but the optimization method is a genetic algorithm (GA) or a particle swarm optimization method (). Other global optimization methods such as PSO) and Simulated Annealing (SA) may be used. Further, although the filter unit 12 shows an example of secondary (including the case of primary), it may be tertiary or higher.

The transfer function of the PID control unit 11 is not limited to the equation (1). As another example of the transfer function of the PID control unit 11, for example, there are the following three.

These transfer functions are taken from the following literature.

Seiichi Miyazaki and one other person, "Practical Science of Automatic Control Learned on a Personal Computer", CQ Publishing Co., Ltd., published in 1991, p. 114
As described above, it should be considered that the embodiments disclosed this time are exemplary in all respects and are not restrictive. The scope of the present invention is shown by the scope of claims rather than the above description, and is intended to include all modifications within the meaning and scope equivalent to the scope of claims.

1 Controller 2 Control target 11 PID control unit 12 Filter unit 21 Tundish 22 Nozzle 23 Mold 24 Roll 31 Slide valve 32 Actuator 33 Swirl flow sensor 100 Feedback control system 200 Continuous casting machine

Claims (3)

It is a method of designing the parameters of the controller in the controller which includes the PID control unit and the filter unit provided separately from the PID control unit and feedback-controls the control target.
A plurality of frequency domains including the frequency to be controlled for the mixed sensitivity problem, which is determined based on the transfer function of the controlled object and the transfer function of the controller and includes robust stability and disturbance suppression characteristics. A setting process for setting the constraint conditions of the mixing sensitivity problem for each frequency, and
Parameter design including a determination step of determining the parameters of the controller composed of the parameters of the PID control unit and the filter unit by solving the mixing sensitivity problem under the constraint conditions by an optimization method. Method. The parameter design method according to claim 1, wherein the control target is a continuous casting machine, and the control amount is the molten metal level. A storage step of preliminarily storing the respective parameters of the plurality of controllers designed by the parameter design method according to claim 1 or 2.
A change for changing the controller by determining a parameter from the respective parameters of the plurality of controllers stored in the storage step by interpolation and setting the control target to the parameter of the controller for feedback control. With the process,
The change step is a feedback control method in which the controller is changed so that the control characteristics of the controller are continuously changed.