KR100902536B1 - Method For Adjustment Of Control Parameter In Feedback Controller - Google Patents

Abstract

The present invention relates to a control coefficient adjusting method of a feedback controller, and more particularly, in adjusting the cutoff frequency of the low pass filter and the control coefficient of the controller to minimize the value of the objective function expressed as a function of error. The present invention relates to a control coefficient adjusting method of a feedback controller that can automatically determine a control coefficient without relying on experiments and to reduce the overshoot occurring to improve the control performance of the controller.

for teeth,

(a) initializing a design vector consisting of the design variables kp, kd, w;

(b) defining an objective function;

(c) calculating a gradient vector of the design vector;

(d) calculating a direction vector using the gradient vector;

(e) calculating an optimal point of an objective function using the direction vector;

(f) updating the design vector; and

(g) repeating steps (c) to (f);

It provides a control coefficient adjustment method of the feedback controller, characterized in that configured to include.

Feedback controller, PD controller, automatic control, optimal design, low pass filter, overshoot, gain, error rate, objective function, weight function, violation function, optimal point.

Description

How to adjust control coefficient of feedback controller {Method For Adjustment Of Control Parameter In Feedback Controller}

The present invention relates to a control coefficient adjusting method of a feedback controller, and more particularly, in adjusting the cutoff frequency of the low pass filter and the control coefficient of the controller to minimize the value of the objective function expressed as a function of error. The present invention relates to a control coefficient adjusting method of a feedback controller that can automatically determine a control coefficient without relying on experiments and to reduce the overshoot occurring to improve the control performance of the controller.

A hardware implementation of a general feedback controller will be described with reference to FIG. 1.

As a control system, XY table using a rotary motor is used.The controller composed of microprocessors has a quadrature count function that counts pulse signals obtained from motor encoders and converts them into position signals on a DAQ board. The control value is output for the detected error.

In addition, sensors that detect the position of the motor are typically encoders, but can be linear scale signals through potentiometers or multipliers. Sensor signals are limited by the resolution of the sensor or the resolution of the DAQ board, but equipment with resolutions beyond the accuracy required by the controller is used.

Fig. 2 shows a block diagram of a general control system in which the discrete time controller determines the appropriate controller output u for an error e which is the difference between the reference input yd and the actual position y detected via the sensor or the like. It outputs to the system through D / A of the DAQ board, and the position moved by the controller output is detected again through the sensor, and this method continuously controls the system.

The above system includes all control systems in which control needs are recognized, including the XY table described above.

When the control system is represented as a secondary system composed of mass, damper, and spring of FIG. 3, the mass, damper, and spring are respectively mass (m), damping coefficient (c), and elasticity. 3, f is a force applied to the system, x is the position of the system. In this case, the system may be represented by a differential equation of the form as shown in Equation 1 below.

 Since f is proportional to the current ia and the torque constant Kt, and the current ia is proportional to the coefficient Ka and the voltage v of the motor driver, Equation 1 can be expressed as follows. have.

The coefficient H is represented by the product of Kt and Ka.

,

인 경우, 라플라스 변환을 통하여 상기 수학식 2를 다음과 같이 나타낼 수 있다.In Equation 2, the initial condition is

,

In Equation 2, Equation 2 may be expressed as follows through the Laplace transform.

Here, when kp is a proportional gain of the feedback controller and kd is a derivative gain of the feedback controller, the characteristic equation of the closed loop system is expressed by Equation 4 below.

와 비교하여 보면, kd는 감쇄 비율(

)과 관련이 있고, kp는 고유 진동수(

)와 관련이 있음을 알 수 있다.Equation 4 is a characteristic equation of the secondary system

Compared with, kd is the decay rate (

) And kp is the natural frequency (

) Is related to

Therefore, in order to reach the desired trajectory with stiffness quickly, kp must be increased. However, when a certain threshold is exceeded, the performance of the controller is degraded due to vibration. The kp response characteristic of this feedback controller is shown in FIG.

In addition, kd acts as a damping effect, and thus has an effect of removing vibration. The kd response characteristic of this feedback controller is shown in FIG.

Therefore, a need for how to design the coefficients of the feedback controller has been raised, and an appropriate feedback controller coefficient adjustment method has been proposed. Representative examples thereof include the Ziegler-Nichols method and the Cohen-Coon method.

However, the controller coefficient setting method as described above,

First, iterative process takes a long time because iterative process is performed manually based on the response of predetermined coefficient.

Second, we need the help of an experienced engineer to analyze the response characteristics.

Third, when the system model is changed, there is a problem in that the engineer coefficient needs to be redesigned by an experienced engineer.

The present invention has been made in view of the above-mentioned point, and it is possible to automatically design a feedback coefficient without help by a human, and a control coefficient adjusting method of a feedback controller that can easily derive a controller coefficient even when a system model is changed. The purpose is to provide.

Control coefficient adjustment method of the feedback controller of the present invention for achieving the above object,

(a) initializing a design vector consisting of the design variables kp, kd, w;

(b) defining an objective function;

(c) calculating a gradient vector of the design vector;

(d) calculating a direction vector using the gradient vector;

(e) calculating an optimal point of an objective function using the direction vector;

(f) updating the design vector; and

(g) repeating steps (c) to (f);

Characterized in that comprises a.

According to the control coefficient adjustment method of the invention feedback controller as described above, it is possible to automatically design the feedback coefficient without the help of human, and to easily derive the control coefficient even when the system model is difficult to know or the system model changes easily. Therefore, it can be easily applied in actual industrial sites, and various economic and commercial effects are expected.

The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting of the present invention. A singular expression includes a plural expression unless the context clearly indicates otherwise. In this application, the terms “comprises” or “having” are intended to indicate that there is a feature, number, step, action, component, part, or combination thereof described in the specification, and one or more other It is to be understood that the present invention does not exclude the possibility of the presence or the addition of features, numbers, steps, operations, components, parts, or a combination thereof.

Hereinafter, the present invention will be described in detail with reference to the accompanying drawings.

In general, a feedback controller uses a proportional-integral-derivative controller (Proportional, Integral, Derivative, PID), but the present invention is limited to a proportional-differential controller (Proportional, Derivative, PD).

In particular, the present invention is characterized by further comprising a low pass filter 700 in the general system consisting of the controller 710 and the control system 720 to convert the input signal (input shaping).

In the case of a sinusoidal wave, since the reference trajectory is differentiable (differentiable trajectory), there is no big problem, but in the case of the step input, the performance of the controller is greatly reduced due to the initial differentiation point, and the overshoot as shown in FIG. (overshoot) will occur. If the system is used in a machine tool such as a CNC machine, the overshoot can be a big problem since it damages the workpiece and becomes the first priority when designing a controller.

Therefore, a lowpass filter is installed in front of the reference trajectory to remove the transient response due to the initial overshoot.

The reference trajectory desired by the user (eg, staircase input) becomes the input r of the actual feedback controller via the low pass filter 700.

The characteristics of the input r of the feedback controller passed through the low pass filter 700 are shown in FIG. 8 based on the cutoff frequency value, and the low pass filter is a Butterworth first low pass. A pass filter was used and a step function was used as the reference input.

As shown in FIG. 8, the lower the cutoff frequency, the lower the input of the low frequency region is cut off. Therefore, the step input is gradually increased, while the higher the cutoff frequency is cut off only for the input of the high frequency region. Increase.

In this case, the trajectory that increases excessively slowly has the disadvantage of increasing the speed of arrival, whereas if the reference trajectory increases excessively rapidly, the initial transient response may be induced as if no input shaping was performed. Therefore, a method for determining an appropriate cutoff frequency is needed.

Hereinafter, a method of designing three values of the proportional gain (kp), differential gain (kd), and cutoff frequency (w) as design variables will be described in detail.

Kp, kd, and w are defined as x ₁ , x ₂ , and x ₃ , respectively, and have the following constraints (constraint).

로 나타낼 수 있고, 상기 j는 1 이상 3 이하 값을 갖는다. 상기의 구속 조건 외에 제어기가 갖기를 원하는 성능을 갖도록 구속 조건이 추가될 수 있다.Therefore, the general constraint is as shown in Equation 5 above.

It may be represented by, wherein j has a value of 1 or more and 3 or less. In addition to the constraints described above, constraints can be added to have the performance the controller desires to have.

At this time, the cost function (J) representing the performance of the controller is defined as in Equation 6 below.

는 각 설계 변수에 대한 구속 조건을 나타내는 위반 함수 벡터, w(t)는 가중함수, e(t)는 오차를 나타낸다.Where N is the total number of samples, r is the coefficient of the violation function for the constraint,

Is a violation function vector representing constraints for each design variable, w (t) is a weighted function, and e (t) is an error.

Therefore, an object of the present invention is to obtain a design variable that minimizes the objective function J through a short trial in a short time.

는 다음과 같이 정의된다.Where the violation function vector

Is defined as

의 제곱만큼)에 비례하게 목적 함수가 증가함으로써 자연스럽게 구속 조건을 만족하지 못하는 방향으로는 최적화가 진행되지 않도록 한다. In other words, if the constraint is satisfied, it does not affect the objective function and conversely, if it is not satisfied,

By increasing the objective function in proportion to the square of, the optimization does not proceed in the direction that does not naturally satisfy the constraint.

Meanwhile, the weighting function w (t) may be defined differently according to the performance desired by the controller by the user. As an example, in order to reduce the steady state error, a weighting function w (t) may be defined that increases weight as time increases. As another example, an initial transient response may be defined as shown in FIG. To reduce, we can define a weighting function w (t) whose weight decreases rapidly with time.

The method to find the design variable that minimizes the objective function J is as follows.

를 초기화한다(1000). 이때 각 변수는 충분히 작은 값으로 설정하며, 예를 들어 x₁ = 0.001, x₂ = 0.001, ,x₃ = 0.001과 같이 설정할 수 있다.First, a vector of design variables x ₁ , x ₂ , and x ₃

Initialize (1000). In this case, each variable is set to a sufficiently small value, for example, x ₁ = 0.001, x ₂ = 0.001, and x ₃ = 0.001.

가 초기화된 이후 상기 벡터

를 이용하여 구배 벡터(gradient vector)

를 다음의 수학식 8과 같이 계산한다(1010).vector

The vector after is initialized

Gradient vector using

Is calculated as in Equation 8 below (1010).

값을 나타낸다.K is the kth loop, i.e.

Indicates a value.

에 대한 방향 벡터

에 비해서 큰

에 대한 방향 벡터

은 넓은 범위에서의 추세를 반영하므로 원하는 값 으로 빨리 찾아감을 확인할 수 있으므로, 구배 벡터를 구하기 위한

값은 반복 과정을 수행함에 따라 변화하는 설계 변수에 능동적으로 변화해야 한다. 따라서 설계 변수의 크기(

)에 비례하게

값을 결정한다. 이는 다음 수학식 9와 같이 나타난다.Relatively small as shown in FIG.

Direction vector for

Larger than

Direction vector for

Since reflects a trend over a wide range, we can quickly find the desired value,

The value should change actively as the design variable changes as the iteration proceeds. Therefore, the size of the design variable (

In proportion to

Determine the value. This is expressed by the following equation (9).

는 임의의 상수로서, 일 예로서 0.01로 설정할 수 있다.Above

Is an arbitrary constant and may be set to 0.01 as an example.

인 경우

이 되고, 설계 변수

인 경우에는

가 된다. Thus design variables

If

Design variables

If is

Becomes

값이 상대적으로 크면 방향 벡터가 잘못된 방향으로 결정될 수 있으므로,

값을

=

X 0.1 로 갱신하며, 일정 횟수 이상

값을 갱신하여도 목적 함수 값이 감소하지 않는 경우 최적점에 수렴하였다고 판단하고 설계 변수 계산 과정을 종료한다.In addition, as shown in FIG. 13, when the design variable almost reaches the optimum point

If the value is relatively large, the direction vector may be determined in the wrong direction,

Value

=

Renew to X 0.1, over a certain number of times

If the value of the objective function does not decrease even if the value is updated, it is determined that the optimal point has been converged and the process of calculating the design variable is terminated.

는 목적 함수가 가장 급하게 증가하는 방향을 나타내며, 본 발명에서 방향 벡터

는 최대 경사 강하법(steepest descent method) 또는 켤레 경사도법(conjugate gradient method)을 이용하여 계산한다.The gradient vector above

Denotes the direction in which the objective function increases most rapidly, and in the present invention, the direction vector

Is calculated using either the steepest descent method or the conjugate gradient method.

Using the maximum gradient descent method, the direction vector is calculated as follows.

On the other hand, when using the conjugate conjugate gradient method, the direction vector is calculated as follows.

이며, k가 2 이상일 때

로 나타난다. K represents the k-th loop, when k = 1

When k is 2 or more

Appears.

In the present invention, since the necessary value in the execution process for finding the optimal value is a direction vector having a size of 1, the size of the direction vector is made to be 1 by the following equation.

를 계산한 후, 상기 방향 벡터 방향으로 목적함수 J가 최소로 되는 지점을 계산한다(1030).Direction vector through the same method as above

After the calculation, the point at which the objective function J becomes the minimum in the direction vector direction is calculated (1030).

The minimum point calculation step 1030 is largely divided into two steps. The first iteration step is a step of determining a section in which an optimum point exists, and the second iteration step is a step of determining an optimum point in the section. A step of determining a section in which an optimum point exists as a first iteration step will be described with reference to FIG. 15.

값을 결정하여야 한다. 본 발명에서 n번째 제1 반복 반계에서의

값은 다음 수학식 13과 같이 나타난다.In order to determine the interval in which the optimum point exists, the initial value of the interval size

The value should be determined. In the nth iteration of the repeating system

The value is represented by the following equation (13).

The size of the interval can increase at a constant rate (m _(n) = 1), but the range of the interval can be determined quickly and the golden ratio (1.618) can be used to reduce the number of initial trials when using the golden split method in the second iteration. ) Was used. M _(n) is represented as follows.

값에 따른 임시 설계 변수 벡터는 다음과 같이 나타난다.remind

The temporary design variable vector by value is shown as follows.

는 k 번째 루프에서 구간의 크기가

인 경우의 임시 설계 변수 벡터이다.remind

Is the size of the interval in the kth loop

Temporary design variable vector when.

값에 대한 목적 함수값

을 계산하고, 이 값을

값과 비교하여,

값이 더 큰 경우 제1 반복 단계를 종료하고 제2 반복단계를 수행하며, 반대로

값이 더 큰 경우 n을 1 증가시켜 제1 반복 단계를 다시 진행한다.At this time

Objective function value

Calculate this value

Compared to the value,

If the value is larger, the first iteration step ends and the second iteration step is performed.

If the value is larger, n is increased by 1 to proceed to the first iteration step again.

값에 따라서 제1 반복 단계 횟수가 결정됨을 알 수 있다. 즉

값이 과도하게 작은 경우 구간을 결정하기 위한 반복 수행이 많아 질 수 있고, 반대로

이 과도하게 큰 경우 전체 구간이 커져 최적화를 통한 성능 증가를 기대하기 어렵게 된다. 따라서 적절한 값의

을 결정할 필요성이 있으며, 이는 설계 변수 벡터

의 크기에 따라서 다음과 같이 결정된다.Here, as shown in FIG.

It can be seen that the number of first iteration steps is determined according to the value. In other words

If the value is too small, there may be more iterations to determine the interval.

In this excessively large case, the entire interval becomes large, and it is difficult to expect an increase in performance through optimization. So of proper value

Needs to be determined, which is a design variable vector

Depending on the size of is determined as follows.

값은 임의의 양의 실수로서 제1 반복 단계의 수행 과정 횟수(n)에 따라서 결정된다. 즉, n의 값이 충분히 클 경우

를 증가시켜 수행 과정 횟수(n)을 감소시키고, 반대로 n의 값이 충분히 작을 경우

값을 증가시켜 수행 과정 횟 수(n)를 증가시킨다. 이러한 과정을 통해 제1 반복 단계의 수행 과정 횟수(n)가 일정한 범위의 값에 포함되도록 유지한다.Above

The value is determined by any positive real number, depending on the number of times n of performing the first iteration step. That is, if the value of n is large enough

Decreases the number of executions (n), and conversely, if the value of n is small enough

Increase the value to increase the number of executions (n). Through this process, the number of execution processes n of the first repetition step is maintained to be included in a predetermined range of values.

값이 결정되면, 설계 변수 벡터

를 다음과 같이 갱신한다.After determining the section including the optimal point through the first iteration step as described above, and performing the second iteration step to find the best point in the section. In this case, the method of finding the optimum point is assumed to be a half interval method generally used in engineering numerical analysis, a golden section method used in the first iteration step, or a point of a quadratic curve. Quadratic interploation, which is interpolated, may be used. Optimum points through this process

Once the value is determined, the design variable vector

Update to

값을 구하는 단계부터 설계 변수 벡터

를 갱신하는 과정까지 반복하여 수행한다. 이때 만약 구배 벡터

의 크기가 임의의 설정값

보다 작은 경우 (즉,

) 그때의 설계 변수 벡터

를 최적값으로 선정하고 반복 과정을 종료한다. Then increase the value of k by 1

Design variable vector from the stage of finding the value

Repeat to update the process. If the gradient vector

Is a random setting

Less than (i.e.

Design variable vector

Choose the optimal value and finish the iteration process.

Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings.

인 시스템 모델에 대하여 최대 경사 강하법(steepest descent method)을 사용하여 설계 변수와 목적 함수의 변화를 도 15에 나타내었다. 반복 단계가 진행됨에 따라 목적 함수(J)의 함수값이 감소함을 알 수 있다.

The variation of the design variables and the objective function is shown in FIG. 15 using the steepest descent method for the phosphorus system model. It can be seen that as the iteration step proceeds, the function value of the objective function J decreases.

In addition, referring to FIG. 16, it can be seen that the desired system response gradually appears as the number of iterations increases in the upper figure, and the error gradually decreases as the number of iterations increases in the lower figure. .

17 and 18 show the case of using the conjugate gradient method for the system model as described above, and the result is almost similar to the case of using the maximum gradient drop method.

In addition, in the case of the linear motor system modeled as a simple mass system as shown in FIG. 19, as a result of using the method of the present invention, as shown in FIG. It can be seen.

While the invention has been shown and described with respect to certain preferred embodiments, the invention is not limited to these embodiments, and those of ordinary skill in the art claim the invention as claimed in the appended claims. It includes all the various forms of embodiments that can be implemented without departing from the spirit.

1 is a diagram showing a hardware configuration of a control system;

2 is a view showing a control system according to the related art,

3 shows an m.c.k system,

4 is a diagram illustrating a response to a reference trajectory according to a kp value;

5 is a diagram illustrating a response to a reference trajectory according to a kd value;

6 is a diagram showing that overshoot occurs with respect to a step input;

7 shows a control system according to the invention,

8 is a view illustrating conversion of an input signal according to a cutoff frequency value;

9 is a view showing a weighting function according to an embodiment of the present invention;

10 is a view showing an optimization process according to the present invention,

11 is a view showing an optimization process according to an embodiment of the present invention;

값에 따른 구배 벡터와 방향 벡터를 나타낸 도면,12 is

A diagram showing a gradient vector and a direction vector according to a value,

값에 의한 효과를 나타낸 도면,13 is large when the close to the optimum point

Drawing showing the effect by value,

값에 따른 반복 횟수의 변화를 나타낸 도면,14 is

A diagram showing a change in the number of repetitions according to a value,

15 is a view showing a change in the value of J, x ₁ , x ₂ , x ₃ as the optimization process using the maximum gradient descent method,

16 is a view showing a change in response and error with respect to the reference trajectory as the optimization process using the maximum gradient descent method proceeds.

17 is a view showing a change in the value of J, x ₁ , x ₂ , x ₃ as the optimization process using the conjugate gradient method;

18 is a view showing a change in response and error with respect to the reference trajectory as the optimization process using the conjugate gradient method proceeds;

19 is a configuration diagram of a multi-task machine system,

FIG. 20 is a diagram illustrating a response change and a change of an error with respect to a reference trajectory as the optimization process using the maximum gradient method is performed for the multi-task machine.

Claims (17)

(a) initializing a design vector consisting of the control parameters kp, kd of the PD controller and w, the cutoff frequency of the low pass filter for input shaping of the PD controller; (b) defining an objective function; (c) calculating a gradient vector of the design vector; (d) calculating a direction vector using the gradient vector; (e) calculating an optimal point of an objective function using the direction vector; (f) updating the design vector; and (g) repeating steps (c) to (f); Control coefficient adjustment method of the feedback controller, characterized in that comprises a. The method according to claim 1, Objective function of step (b)

W (t) is a weighting function defined by the user, e (t) is an error over time, r is an arbitrary constant,

The control coefficient adjustment method of the feedback controller, characterized in that represents the violation function vector of the constraint for each design variable. The method according to claim 2, The constraint condition for the design variable is a control coefficient adjustment method of the feedback controller, characterized in that the design variables kp, kd, w will be greater than or equal to zero. The method according to claim 2, Gradient vector of step (c)

And x ₁ , x ₂ , and x ₃ are kp, kd, and w, respectively. The method according to claim 4, Delta value used to calculate partial derivative of the gradient vector of step (c)

Is the following equation

Determined by the above

Is an arbitrary constant. The method according to claim 5, If the value of the objective function does not decrease even after repeating step (g)

A control coefficient adjusting method of a feedback controller, characterized in that the value is reduced. The method according to claim 5 or 6, The direction vector of step (d) is

Control coefficient adjustment method of the feedback controller, characterized in that determined by. The method according to claim 7, In order to make the direction vector 1

Control coefficient adjustment method of the feedback controller, characterized in that for performing. The method according to claim 5 or 6, The direction vector of step (d) is

If k = 1,

When k is 2 or more,

Control coefficient adjustment method of the feedback controller, characterized in that. The method according to claim 9, In order to make the direction vector 1

Control coefficient adjustment method of the feedback controller, characterized in that for performing. The method according to claim 5 or 6, And (e) comprises a first iteration step of determining a section in which an optimum point exists and a second iteration step of determining an optimum point in the section. The method according to claim 11, The first iteration step, (e-1) section size

Calculating a value; (e-2) above

Temporary Design Vector Values Using Values

Calculating; (e-3) above

Objective function according to

Obtaining; (e-4) The objective function value

Is the previous objective function value

Determining whether a section is determined by comparing with the result; Control coefficient adjustment method of the feedback controller, characterized in that comprises a. The method according to claim 12, The step (e-1),

ego,

Wherein m _(n) is m _(n) = 1 or

Defined as any of The p is an arbitrary constant determined in consideration of the sizes of x ₁ , x ₂ , and x ₃ , and is a scale factor of a section size. The method according to claim 12, Step (e-2),

Control coefficient adjustment method of the feedback controller, characterized in that. The method according to claim 12, The step (e-4) is,

Value is

If greater than then

Is determined as the interval with the best point, and vice versa.

Value is

If less than or equal to the control coefficient adjustment method of the feedback controller, characterized in that repeating the steps (e-1) to (e-4). The method according to claim 13, If the value n of the number of iterations of the first repetition step is greater than the reference value,

Increase the value, and if the value of the number of repetitions (n) of the first repetition step is smaller than the reference value,

A control coefficient adjusting method of a feedback controller, characterized in that the value is reduced. The method according to claim 11, The second iteration step of the control coefficient control method of the feedback controller, characterized in that for calculating the optimum point using any one of the half interval method, golden section method, the second interpolation method.

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