Исследование частотных характеристик САУ¶
В данном примере рассмотрим возможности среды Engee для исследования частотных характеристик САУ.
Структурные преобразования¶
Модель САУ представляет собой структурную схему, которая включает блоки типовых динамических звеньев. Передаточную функцию САУ находят, оперируя передаточными функциями этих звеньев.
При последовательном соединении передаточная функция цепочки звеньев равна произведению их передаточных функций:
$$
W(p)=\prod_{i=1}^n W_i(p)
$$
Передаточная функция группы параллельно соединенных звеньев равна сумме их передаточных функций:
$$
W(p)=\sum_{i=1}^n W_i(p)
$$
Если звено с передаточной функцией $W_1(p)$ охвачено обратной связью, в которой присутствует звено $W_{ос}(p)$, то передаточная функция замкнутого контура $W(p)$ определяется по формуле:
$$
W(p)=\frac{W_1(p)}{1 \pm W_1(p)W_{oc}(p)}
$$
Знак плюс или минус в знаменателе соответствует отрицательной и положительной обратной связи, соответственно.
Рассмотрим функции библиотеки ControlSystems.jl
, которые позволяют делать структурные преобразования схемы.
Out[0]:
TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
1.0
-------------------
0.5s^2 + 1.5s + 1.0
Continuous-time transfer function model
Out[0]:
TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
1.5s + 2.0
-------------------
0.5s^2 + 1.5s + 1.0
Continuous-time transfer function model
Для определения передаточной функции контура состоящего из звена охваченного обратной связью существует функция ControlSystemsBase.feedback
.
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
По умолчанию при использовании функции ControlSystemsBase.feedback
образуется соединение с отрицательной обратной связью. Однако, можно указать, что связь должна быть положительной, передав в качестве второго аргумента -1.
Out[0]:
TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
1.0
----
0.5s
Continuous-time transfer function model
Получение и исследование частотных характеристик САУ¶
Зададим передаточную функцию САУ, показанную на картинке ниже.
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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
Звенья расположены последовательно, поэтому получим общую передаточную функцию разомкнутой системы с использованием уже знакомой ControlSystemsBase.series
.
Out[0]:
TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
0.3
--------------------------------------
2.0e-6s^4 + 0.003s^3 + 1.001s^2 + 1.0s
Continuous-time transfer function model
Построим графики АФЧХ и ЛАФЧХ.
Также построим ЛАФЧХ разомкнутой системы с отображением запасов устойчивости по амплитуде и по фазе. Для этого используем функцию ControlSystemsBase.marginplot
.
Запасы устойчивости отображаются в названии графиков. Запас по амплитуде 1111.76 дБ, запас по фазе 73.89 градусов. ЛФЧХ пересекает 180 градусов на частоте 18.26 рад/с, а частота среза равна 0.29 рад/с. Также можно навести на график курсор и посмотреть значения амплитуды, фазы и частоты в любой выбранной точке.
В данном примере мы познакомились с функционалом Engee для исследования частотных характеристик САУ. Также изучили, как преобразовать структурную схему с помощью функций библиотеки ContolSystems.jl
. Узнать больше о построении графиков частотных характеристик можно в разделе Функции построения графиков.
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