Linear_Systems_Reduction

A gentle intro to reduction of linear systems block diagrams using series, parallel and feedback configuration

Why Python?

The guide of this lab uses MATLAB. However,compared to MATLAB python is a lot easier to get, setup, and requires minimal system resources. This makes it very convenient for any of us eager to see the control commands at work and not spend much wondering how to get started.

Setup

If you encounter any issue or unclarity during this setup, don't hesitate to reach out to @mugoh

Package

This part assumes you are unfamiliar with git and pip. If that's not you, go straight ahead to Minimal Setup.

1. Get a clone zipped-copy of this repo by clicking here
2. Extract the zip file and move the file to Downloads. This is important. We will be using the downloads folder.

Windows Platform

Dependencies

1. You will need python3 installed. Click here to install python3

• Ensure to check the box Add python to path on the first installation window.
• Open a command prompt window and checkout the Downloads folder using the command below

    cd Downloads

2. Install the package manager

[ Deprecated ] Skip this step(By default installing python installs pip for you).

• Download get-pip from here. Save the file to Downloads.

• Install pip (Ensure to have installed python first)

    python get-pip.py

3. Navigate to the cloned repo

    cd Linear_Systems_Reduction

4. Install the project dependencies

    pip install -r requirements.txt

Minimal Setup

Done the Windows Platform part? Skip this

1. Access a clone copy of the repo

    git clone git@github.com:hogum/Linear_Systems_Reduction.git

2. Navigate to the project directory

    cd Linear_Systems_Reduction

3. Install the project dependencies

    pip install -r requirements.txt

Usage

The LAB has five examples and exercises

1. Running the examples

• To run an example, pass the name of that example as an argument as shown below

Example No.	How to run	Block Activity
Example 1	python run.py example1	Series
Example 2	python run.py example2	Parallel
Example 3	python run.py example3	Unity Feedback
Example 4	python run.py example4	Non-Unity Feedback
Example 5	python run.py example5	Zeros and Poles

Sample Result

Feedback example ouput

Poles and Zeros

2. Running Exercises

Exercise No.	How to run
Exercise 1	python run.py exercise1
Exercise 2	python run.py exercise2
Exercise 3	python run.py exercise3
Exercise 4	python run.py exercise4

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Running the Examples section

• Find all the examples in this directory

Obective

The objective of this exercise will be to learn commands in MATLAB that would be used to reduce linear systems block diagram using series, parallel and feedback configuration.

Example 1: Series Configuration

Given the transfer functions of individual blocks generate the system transfer function of the block combinations.

Example 2: Parallel Configuration

If the two blocks are connected as shown below then the blocks are said to be in parallel. It would like adding two transfer functions.

Example 3: Feedback

Given a unity feedback system as shown in the figure, obtain the overall transfer function.

Example 4: FeedBack Non-Unity

Given a non-unity feedback system as shown in the figure, obtain the overall transfer function.

Example 5: Zeros and Poles

Given a system transfer function plot the location of the system zeros and poles

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Solutions to the exercises

• Find the solutions to the exercises in this directory.

Exercise 1:

For the following multi-loop feedback system, get closed loop transfer function and the corresponding pole-zero map of the system.

Exercise 2: Consider the feedback system depicted in the figure below

a. Compute the closed-loop transfer function using the ‘series’ and ‘feedback’ functions 
b. Obtain the closed-loop system unit step response with the ‘step’ function and verify that final value of the output is 2/5.

Exercise 3: A satellite single-axis altitude control system can be represented by the block diagram in the figure given. The variables ‘k’, ‘a’ and ‘b’ are controller parameters, and ‘J’ is the spacecraft moment of inertia. Suppose the nominal moment of inertia is ‘J’ = 10.8E8, and the controller parameters are k=10.8E8, a=1, and b=8.

a. Develop an m-file script to compute the closed-loop transfer function  
b. Compute and plot the step response to a 10o step input. 
c. The exact moment of inertia is generally unknown and may change slowly with time. Compare the step response performance of the spacecraft when J is reduced by 20% and 50%. Discuss your results.