# Phase plane analysis and Matlab code toolbox

Phase plain analysis is a useful visualization tool to understand the characteristics of systems including not only linear system but also nonlinear system. For example, we can determine stability of the system from this phase plane analysis.

The attachment file <here> is Matlab toolbox to draw phase plain. The attached file includes a simple demo and the below is the result. You can draw phase plane, magnify where you are interest recursively. You can see how to use the Matlab code in the following Youtube video.

# How to draw?

Given,

$\dot{x}_1=f_1(x_1,x_2)$

$\dot{x}_2=f_2(x_1,x_2)$

we can find the below equation

$\frac{dx_2}{dx_1}=\frac{f_2(x_1,x_2)}{f_1(x_1,x_2)}$

From $\frac{dx_2}{dx_1}$, we can find the direction of the phase change at the point of $(x_1,&space;x_2)$.

1. It is an exact method. We can see the change of system’s state including transient response.
2. Simple graphical method. It is very intuitive and easy to understand its characteristics.

# Limit:

1. Limited to the 2nd order system. It is expandable, but hard to visualize.

# Reference

1. Lecture of Prof. Fernadez in Mech. Eng, The Univ. of Texas at Austin.

# Feedback linearization control (FBL) for nonlinear system control proof, practical implementation, and easy example part 2

This article explains about Feedback linearization control (FBL) method for control of a nonlinear system. By demonstrating a control strategy of the inverted pendulum problem, I am going to explain how to implement its algorithm into a real system. The basic idea is that we can cancel control input by manipulating control input. The below is its practical implementation method and example.

If you want to know the proof of feedback linearization control method. refer this.

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I am Youngmok Yun, and writing about robotics theories and my research.

My main site is http://youngmok.com, and Korean ver. is  http://yunyoungmok.tistory.com.

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# Feedback linearization control (FBL) for nonlinear system control proof, practical implementation, and easy example part1

This article explains about Feedback linearization control (FBL) method for control of a nonlinear system. This is one of the easiest strategy to control nonlinear systems, but pretty powerful. The basic idea is that we can cancel control input by manipulating control input. The below is its proof.

This is the most basic concept to explain easily. If the nonlinear system is not fully controllable, we should use another strategy which finds reduced order manifold (ROM). About the more advanced technique and examples will be explained in the next articles.

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I am Youngmok Yun, and writing about robotics theories and my research.

My main site is http://youngmok.com, and Korean ver. is  http://yunyoungmok.tistory.com.

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