Video 1.9 - Glue Lecture 1 - Dynamical Models

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Table of Contents:
* E.1) Dynamical Models
* E.2) Understanding what is a model (ball example)
* E.2.1) Exercise in derivatives
* E.2.2) Equations in Action
* E.2.3) Working with Differential Equations
* E.2.4) Visualize the difference equation
* E.2.5) Final thoughts: Dynamical models
* E.2.5.1) Solving the Differential Equation

E.1) Dynamical Models

Q: What is a model?
Ans: A model is something that describes how a system changes (or evolves) with time.

The system could be: a robot

And the model could describe, for example:

• how the position changes with time
• or, how the angles change with time
• etc

Q: what is the idea behind controls?
Ans: The idea behind controls is that we are going to influence this change to make our system do something we want to do,

E.2) Understanding what is a model (ball example)

E.2.1) Exercise in derivatives

Imagine we want to model a ball,

and we know that it's position $x(t)$ is as a function of time,

and if we graph this function we get the following,

And now we can take the derivative of the position with respect to time, $\dot{x}(t)$

We also know that the derivative of position is velocity, and if we graph this function we get the following,

We can do this further on, and take one more derivative, $\ddot{x}(t)$.
Which we know is the acceleration.

and if we graph this function we get the following,

E.2.2) Equations in Action

In action...
What does these equations mean outside the graphs?

Let's draw the ball and see how its position changes a long some axis with time,

E.2.3) Working with Differential Equations

Given the differential equation,

How do we find $x(t)$, in order to describe (or plot) the position of the ball?

We have two options:

1. We can try to solve the differential equation and get $x(t)$,
   Spoiler Alert: we already know the answer,
   

2. We can discretize the world and use a Taylor Approximation, in order to get $x(k\delta t)$ as we've learned before,

Let's cut the time axis in discrete amounts of $\delta t=0.5$,

And let's add a counter "k" to jump to different instants of time,

Now, our differential equation is,

And the Taylor Expansion is,

And if he apply it to our example we get,

Therefore,

is the difference equation that approximates our differential equation.

E.2.4) Visualize the difference equation

But, how does this translate into motion?

Let's go through the difference equation step-by-step,

First, we know that $x_0=10$ and $t_0=0$ then,

when $k=0$,

when $k=1$,

when $k=2$,

and so on …

Notice one thing, the difference equation is really a linear approximation of the actual solution (which we know is an exponential). If we had added more terms of the Taylor series, we would have a discrete model closer to the continuous model.

E.2.5) Final thoughts: Dynamical models

So, to recap... our Dynamical Model is,

In general,

Punchline: Given the following dynamical model....

...we know how $x$ evolves (numerically)

But we didn't really get $x(t)$, the solution to the differential equation...

E.2.5.1) Solving the Differential Equation

To get the mathematical expression for $x(t)$, we integrate!

For example: Given the differential equation and its initial condition, find the exact solution.

Solution: Using Leibniz's notation,

We separate the variables,

And we integrate both sides,

And we plug in the initial condition,

And now we solve for $x(t)$,

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Z) 🗃️ Glossary

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