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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Analysis of Dynamic Systems
Schedule
Step1:  Almost every aspect of everyday life is affected by some kind of control system.
The basic elements of a control system are
Step2:  Example
Step3:  Uses feedback from the output to the input.
Step4:  For example, what happens in an elevator in both cases? And in the system of opening and closing of a door?
The feedback has the following goals
Step5:  It can be shown that its transfer function is
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Python Code:
from IPython.display import SVG
SVG('img/intro_fig1.svg')
Explanation: Analysis of Dynamic Systems
Schedule:
Getting started
Introduction
Mathematical bases
Bode diagrams
Modeling with linear elements
State variables
Block diagrams
Time response
Frequency response
Stability
Root Locus
Final project
Course evaluation
Introduction
Course Notebook
Download the master file, unzip it to a place on the hard disk easy to find (I recommend the root of the hard disk, or a folder called c:/temp), and from inside the folder run:
ipython notebook
You'll be fine from there...
Control Systems
A system can behave in many ways, output variables can have different goals, for example:
Get to the destination in the shortest time.
Get to the destination in the shortest time and with the minimum consumption.
End of explanation
from IPython.display import SVG
SVG('img/intro_fig2.svg')
Explanation: Almost every aspect of everyday life is affected by some kind of control system.
The basic elements of a control system are:
- Control goals
- System components
- Results
Purpose of the control: To control the outputs c in a predetermined way, by means of the inputs u and the control system.
Feedback: Its function is to reduce the error between the reference input and the output. It also ensures stability.
Plant: is the combination of process and actuator. A plant is often referred to with a transfer function (in the s-domain) which indicates the relation between an input signal and the output signal of a system without feedback, commonly determined by physical properties of the system.
Open-loop control systems
End of explanation
from IPython.display import SVG
SVG('img/intro_fig3.svg')
Explanation: Example: A washing machine. The wash cycle is determined by the user's criteria. There are no sensors that measure the degree of cleaning of clothes.
Closed-loop control systems
End of explanation
from IPython.display import SVG
SVG('img/intro_fig4.svg')
Explanation: Uses feedback from the output to the input.
End of explanation
from IPython.display import SVG
SVG('img/intro_fig5.svg')
Explanation: For example, what happens in an elevator in both cases? And in the system of opening and closing of a door?
The feedback has the following goals:
- Reduce error between reference input and output.
- Ensure stability.
Many times a physical plant can have feedback when analyzed in detail. When the variables of a system exhibit a closed sequence of cause and effect relationships, the system has feedback (input and output are related). For example, for the system:
End of explanation
from IPython.display import SVG
SVG('img/intro_fig6.svg')
Explanation: It can be shown that its transfer function is:
$$\frac{c}{r} = \frac{G}{1+GH}$$
Effect of feedback on gain: From the above equation it is observed that the gain of the system is affected by a factor of $1+GH$. $G$ and $H$ are functions of the frequency, so that the gain can increase in some frequencies and decrease in others.
Effect of feedback on stability: Stability is a concept that indicates the ability of a system to follow a control input. A system is unstable when its output is out of control or increases without limits. Again let's look at the equation when $GH = -1$. The output of the system is infinite for any finite input. Therefore, it can be said that feedback can cause instability in a system. It should be noted that this is not the only condition of instability.
If the plant has $GH = -1$, and it is inevitable, it is possible to add another feedback loop, obtaining a new transfer function. Therefore, the global system can be stable if $F$ is properly selected.
End of explanation
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