All language subtitles for 02_dynamics-and-1-d-linear-control.en

Now let's talk a little bit about control.

Again, I'm gonna focus on
the vertical direction.

So let's think about controlling height.

What you would like to do is to drive
the robot to a desired vertical position

either up or down.

Let's use x to measure
the vertical displacement.

Clearly, the acceleration is given by
the second derivative of position.

If you look on the left hand side,
you'll see the sum of the forces.

Let's call u the sum of
the forces divided by the mass.

So you have now a very simple second order
differential equation with an input u and

a variable x, such that u is equal
to the second derivative of x.

Our goal to control this vehicle
is to determine the function u,

such that the vehicle goes
to the desired position x.

So here is the control problem.

The system we have is
a very simple system.

It's a second order linear system.

You're trying to figure out what u of t,
what function of t that is u,

drives x to a desired position x desired.

If you have a desired trajectory,

in other words x desired is a function of
time, you want to synthesize the control

input u of t that allows the vehicle
to follow the desired trajectory.

In order to do that let's define an error.

The error function is essentially the
difference between the desired trajectory

and the actual trajectory.

So the larger the error, obviously,

the further the deviation from the actual
trajectory, from the desire trajectory.

What you'd like to do is to take
this error and decrease it to zero.

More specifically we want this
error to go exponentially to zero.

In other words, we wanna find u of t,

such that the error function satisfies
the second order differential equation.

Why this differential equation?

Well, in this differential equation,
there are two unknowns, Kp and Kv.

If I select appropriate values of Kp and

Kv, more specifically, if I ensure
that these values are positive,

I can guarantee that this error
will go exponentially to zero.

The control input that achieves that
is given by this very simple equation.

Again, the only reason I'm
pulling out this control equation

is because I want the error
to go exponentially to zero.

And that'll ensure that
x tends to x desired.

There are two variables in this equation,
one is K sub p, and the other is K sub v.

You'll see that K sub p
multiplies the error, and

adds the error times Kp
to the control function.

K sub v multiplies
the derivative of the error, and

adds that to the control function.

So one is called the proportional gain,
the other is called the derivative gain.

And in addition, you need some knowledge
of how you want the trajectory to vary.

So you're feeding forward the second
derivative of the desired trajectory.

This is often called the feedforward term.

And this completes your control law or

the control equation at then you
can use to drive your motors.

Here's a typical response of what
the error might look like if you use such

an approach to control.

The error starts out being non zero, but
quickly settles down to the zero value.

The error might undershoot, go from
a positive value to a negative value, but

eventually you're guaranteed
that it'll go to zero.

To summarize,
we've derived a very simple control law.

It's called the proportional
plus derivative control law,

which has a very simple form.

It has three terms, a feedforward term, a
proportional term, and a derivative term.

Each of these terms has a significance.

The proportional term acts like
a spring or a capacitance.

The higher the proportional term is,

the more springy the system becomes and
more likely it is to overshoot.

The higher the derivative term,
the more dense it becomes.

So this is like a viscous dashpot or
a resistance in an electrical system.

By increasing the derivative gain,
the system essentially gets damped,

and you can make it overdamped so that
it never overshoots the desired value.

In exceptional cases,
you might consider using

a more sophisticated version of
the proportional plus derivative control.

Here you have an extra term, which is
proportional to the integral of the error.

You often do this when you
don't know the model exactly.

So, for instance,
you might not know the mass, or

there might be some wind resistance
that you need to overcome, and

you don't know a priori how
much this wind resistance is.

The last term essentially
allows you to compensate for

unknown effects caused by
either unknown quantities, or

unknown wind conditions, or disturbances.

The downside of adding this additional
term is that your differential equation

now becomes a third-order
differential equation.

The reason for that is you've suddenly
added an integral in the mix and

if you want to eliminate the integral you
have to differentiate the whole equation

one more time introducing
a third derivative.

However, the benefit of this is
that this integral term will

make the error go to zero eventually.

So here are three examples
of the system based

on what values you pick for the
proportional gain and the derivative gain.

If both the gains are positive,

you're guaranteed stability,
as you see on the left side.

If K sub v is equal to 0,
then you're guaranteed marginal stability.

While the system will not drift,

you'll find it'll oscillate
about the desired value.

Of course, if one or

the other gain is negative, then you
essentially get an unstable system.

You can similarly explore the effect of
the integral gain which we haven't done in

this picture.

I now want to deal with a complete
simulation of the quadrotor.

Because we'll now require
three independent direction

we're gonna now introduce x,
y, and z coordinates.

And this time the z coordinate points up.

So here's a simulation of
the quadrotor and again,

we're using a proportional derivative
control to control height.

For the moment,
we're ignoring the other variables.

We're adding terms to make sure that the
lateral displacement is zero, the roll and

pitch stays zero, and the yaw stays zero.

But we don't have to worry about that for
the present moment.

We're only construing the proportional
derivative control of height.

And you can see that the error
starts out being non-zero and

then eventually settles down.

There is an overshoot, the red curve
overshoots the desired blue curve, but

eventually settles back down so
that the red and the blue curve coincide.

And here's an experiment that
demonstrates the same idea.

The robot is asked to hover and

in this case someone displaces the robot
from the nominal hover position.

And the robot fights to
overcome the disturbance.

Using a combination of proportional and
derivative gains, the robot is able to

compensate for that disturbance and then
settle back down into the hover position.

If you increase the value of Kp as I said
earlier, the system gets more springy.

So you can see that
the system now overshoots.

The red curve overshoots the blue step and
then settles down eventually.

Again, in this video you'll
see the same phenomenon.

The robot is hovering, but
when it's displaced, when it recovers from

the displaced position it overshoots and
comes back to the original position.

And this happens because the proportional
gain has been increased.

If you turn down this proportional gain,
you lose the overshoot but

instead you get a very soft response.

And then, of course,

if you turn up the derivative gain,
the system becomes overdamped.

So the overshoot disappears but

the system takes also a longer time in
order to get to the desired position.

And once again,
this video illustrates this.

The vehicle is displaced.

And it takes a longer time to get
back to the original position.

In order to get a feel for
these different terms in the controller.

Here's a simple exercise.

You have a simulator of
the system that you just saw.

Try to play around with the two gains,
K sub p and K sub v.

To achieve a simple goal, which is to get
a desired response in which the rise time,

in other words the time taken to get to
the desired position, is reasonably short.

And the overshoot is kept
below some modest value.