Investigation of the frequency characteristics of ACS

In this example, let's look at the possibilities of the Engee environment for studying the frequency characteristics of ACS.

Structural transformations

The ACS model is a structural scheme that includes blocks of typical dynamic links. The transfer function of the ACS is found using the transfer functions of these links.

When connected in series, the transfer function of the chain of links is equal to the product of their transfer functions.:

The transfer function of a group of parallel connected links is equal to the sum of their transfer functions:

If the link has a transfer function it is covered by feedback, in which there is a link , then the closed loop transfer function It is determined by the formula:

The plus or minus sign in the denominator corresponds to negative and positive feedback, respectively.

Consider the library functions ControlSystems.jl, which allow you to make structural transformations of the circuit.

Pkg.add(["ControlSystems"])

#Подключим библиотеку для работы с САУ
using ControlSystems

#Зададим передаточные функции
W1 = tf([1],[0.5, 1]);
W2 = tf([1],[1, 1]);

For sequential multiplication of transfer functions, there is a function ControlSystemsBase.series.

#Последовательное перемножение систем
series(W1,W2)

TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
        1.0
-------------------
0.5s^2 + 1.5s + 1.0

Continuous-time transfer function model

The transfer function of a group of parallel links can be determined using the function ControlSystemsBase.parallel.

#Параллельное соединение звеньев
parallel(W1,W2)

TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
    1.5s + 2.0
-------------------
0.5s^2 + 1.5s + 1.0

Continuous-time transfer function model

To determine the transfer function of a contour consisting of a link covered by feedback, there is a function ControlSystemsBase.feedback.

#Если звено охвачено единичной обратной связью, вторым аргументом указывается 1
W3 = feedback(W1,1)

#Если обратная связь проходит через звено W2
W4 = feedback(W1,W2)

display([W3,W4])

2-element Vector{TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}}:
 TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
   1.0
----------
0.5s + 2.0

Continuous-time transfer function model
 TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
    1.0s + 1.0
-------------------
0.5s^2 + 1.5s + 2.0

Continuous-time transfer function model

By default, when using the function ControlSystemsBase.feedback A negative feedback connection is formed. However, you can specify that the relationship should be positive by passing -1 as the second argument.

#Звено охвачено единичной положительной обратной связью
feedback(W1, -1)

TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
1.0
----
0.5s

Continuous-time transfer function model

Obtaining and studying the frequency characteristics of ACS

Let's set the transfer function of the ACS shown in the picture below.

#Зададим параметры САУ
Ku = 10;
Tu = 0.001;
Km = 3;
Tm = 0.1;
Te = 0.02;
Kp = 0.01;

#Задаем передаточные функции звеньев САУ
W_1 = Ku*tf([1],[Tu, 1]);
W_2 = Km*tf([1],[Tm*Te, 1, 1]);
W_3 = Kp*tf([1],[1, 0]);

display([W_1, W_2, W_3])

3-element Vector{TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}}:
 TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
    10.0
------------
0.001s + 1.0

Continuous-time transfer function model
 TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
         3.0
---------------------
0.002s^2 + 1.0s + 1.0

Continuous-time transfer function model
 TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
0.01
----
1.0s

Continuous-time transfer function model

The links are arranged sequentially, so we get the general transfer function of an open-loop system using the already familiar ControlSystemsBase.series.

#Получаем передаточную функцию разомкнутой системы
W = series(W_1, W_2)
W = series(W, W_3)

TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
                 0.3
--------------------------------------
2.0e-6s^4 + 0.003s^3 + 1.001s^2 + 1.0s

Continuous-time transfer function model

Let's plot the AFCH and LAPCH charts.

#Получение АФЧХ
w = -100:0.1:100
nyquistplot(W,w,title="АФЧХ",label="W")

We will also construct an open-loop frequency response with a display of stability reserves in amplitude and phase. To do this, use the function ControlSystemsBase.marginplot.

#Определим запасы устойчивости по амплитуде и по фазе
marginplot(W,label="W")

Sustainability stocks are shown in the names of the charts. The amplitude margin is 1111.76 dB, the phase margin is 73.89 degrees. The LFCH crosses 180 degrees at a frequency of 18.26 rad/s, and the cutoff frequency is 0.29 rad/s. You can also hover the cursor over the graph and view the values of amplitude, phase, and frequency at any selected point.

Conclusion

In this example, we got acquainted with the Engee functionality for studying the frequency characteristics of ACS. We also learned how to transform a block diagram using library functions. ContolSystems.jl. To learn more about plotting frequency characteristics, see Plotting Functions.