# MEE 421, Quarantine Spring

In the spring of 2020, Dynamic Systems & Control II has gone to distance learning. The videos below present course content following spring break. In most cases, you can find accompanying Matlab and Simulink files posted on Piazza.

**A List of Links:**

- Review of Segway Simulink Model
- Segway Simulink Visualization
- Segway Simulink Equilibrium
- A Graphical Look at a System of Nonlinear ODEs
- A Practical Review of Taylor Series
- Linear Approximation of Nonlinear ODEs
- Dynamics of the Linearized System
- Segway Linearization Part I
- Segway Linearization Part II
- Project: Moment about CG due to Tension
- Segway Reference Signal Part I
- Segway Reference Signal Part II
- Craft Animation – Local Python
- Craft Animation – Google Colab

#### Review of Segway Simulink Model

Here is a brief review of the Segway Simulink Model. We will build off this Simulink model to design and implement a controller.

#### Segway Simulink Visualization

Sometimes it gets difficult to interpret the behavior of the system just by looking at the Simulink plots. So I’ve created a small Python program to generate an animation.

#### Segway Simulink Equilibrium

In a previous analysis, we found conditions for equilibrium. Now, let’s verify the equilibrium with Simulink. While doing this, we’ll start a Matlab LiveScript to perform our control analysis.

#### A Graphical Look at a System of Nonlinear ODEs

Here, we take a graphical look at the following system of differential equations

\( \dot x = f_1(x,y) = x ( \mu_0 – x^2 + b \, y^2) \),

\( \dot y = f_2(x,y) = y ( \mu_1 + c \, x^2 + y^2) \).

This is the cubic order normal form for a co-dimension 2 Hopf-Hopf bifurcation. We show plots of \( f_1(x, y) \) and \( f_2(x,y) \) and discuss equilibria.

#### A Practical Review of Taylor Series

Before we go on to approximate nonlinear differential equations by linear ones, it would be good to review Taylor Series. Here is my “Practical Review”.

#### Linear Approximation of Nonlinear ODEs

What it means to approximate nonlinear differential equations with linear ones.

#### Dynamics of the Linearized System

How do the dynamics of the linearized system compare to those of the original nonlinear system?

#### Segway Linearization Part I

To linearize the Segway equations is not trivial. Here we pull out the linear parts and properly cancel out the constant parts.

#### Segway Linearization Part II

We complete the linearization and compute it in the Matlab Live Script.

#### Project: Moment about Center of Mass due to Tension

Here I address a common difficulty that students appear to be having.

#### Segway Reference Signal Part I

When you send a reference signal through the control input, the stabilized equilibrium gets shifted. Here we begin showing how to systematically use the reference signal to specify the position of the segway.

#### Segway Reference Signal Part II

Here we finish the mathematics of the reference signal then implement it in Simulink.

#### Craft Animation – Local Python

Here is a video that shows you how to get run the Python program for generating an animation of your Simulink data.