All language subtitles for 09_supplementary-material-dynamical-systems.en

In this segment, we'll formally
define a dynamical system and

look at the mathematical models of
some example dynamical systems.

A dynamical system is a system in
which the effects of input actions

do not immediately affect the system.

For example, if you turn on
the thermostat in a cold room,

the temperature in the room will not
immediately rise to the set temperature.

It will take some time for
the room to actually heat up.

Similarly, if you push the gas pedal
in your car, it takes time for

your car to speed up to
the desired velocity.

Every dynamical system is defined by its
state, which is a collection of variables

that completely characterizes
the motion of a system.

The most common states,
are the positions and

velocities of physical
components of the system.

The states of a system are commonly
denoted by the variable x.

As we've seen in lecture,

we use the notation x of t to describe
the values of a system's states over time.

This function x of t
is often characterized

by a set of governing ordinary
differential equations.

The order of a system refers to
the highest derivative that appears

in governing equations.

In lecture, we analyzed the dynamical
system, x double dot equals u.

We see that the second derivative of x is
the highest derivative that appears in

this equation.

Therefore, x double dot equals u
represents a second order system.

Next, we'll see a few more
examples of dynamical systems and

how they are modeled.

An example of a one-dimensional
dynamical system is a mass on a string.

The mass can only move backwards and
forwards in the y direction.

The mass's position is governed by the
following ordinary differential equation.

We see that the highest
derivative of y to appear

in the equation is the second derivative,
making this a second order system.

The state of the system is the position
and velocity of the mass along the y axis.

The input to this system is
an additional force on the mass.

Inverted pendulum on a cart is an example
of a two dimensional dynamical system.

Here, the cart is allowed to
drive along the y direction

while the pendulum is
simultaneously allowed to fall.

The motion of this system to be
modeled by the following set

of coupled ordinary
differential equations.

We see that once again the second
derivatives of the cart position and

pendulum angle are the highest
derivatives to appear in the equations,

making this a second order system.

There are four states in this system.

The first two are the positions
of the cart and the pendulum and

the last two are their velocities.

The input is an additional
force on the cart itself.

Here, we assume that we cannot directly
apply a force to the pendulum.

Finally, considered the Quadrotor.

In this weeks lecture,

we only look at the motion of
the Quadrotor in the Z direction.

As a result, we were able to model
it as a one dimensional system.

However, it turns out that to completely
characterize a motion of the Quadrotor,

we need to know its xyz position and

angular orientation, as well as
its linear and angular velocities.

We'll talk more about the dynamic
equations of the quadrotor

in the coming weeks.