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.

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

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.

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.