But does this make sense?

At least a little bit?

It's this classic linear control which I'm assuming you've seen at some point or

at least familiar with the spring mast ampere system.

Or something like this you get the characteristic equation.

These are the roots.

And once you have the roots of a system you can look up on Wiki.

You can pull out all these standard form factors.

So that's really cool.

So we can now start with, I have this decay time.

I don't what this actually when I write code I often do it this way.

I don't say just pick a p(1) but I'm looking at my inertia and saying.

You know what this craft I should settle but within 20 seconds,

30 seconds it should be at least half as close, right?

I think that as a starting value and

then whatever the inertia is it sets the right gain.

And this could be done even in a self tuning way.

So I'm going to show you some examples here where we compare how linear or

how well we can predict the response.

Got my interias, got some large initial altitude errors and rates,

it's going to tumble.

I've got my games that I've picked and with the system and

you saw we decoupled the one access stuff.

The access stuff and the three access stuff but

making some diagonalizing assumptions on the p matrix and the i matrix.

So we can look at them individually and just got a sum squared, the F's along

is the mean route errors, root mean squares of sigma i's and omega i's.

That's what I'll be plotting here.

So the control, you can see it's a large maneuver it actually goes past

128 where it switches to the other one and then stabilizes.

I picked gains on purpose to have different kind of expected decay times.

Some of the axes decayed their errors very quickly.

This other axis which is sigma 2 it looks like is getting there but

it takes much longer.

So I'll give this different frequencies that should match up with my linearized

predictions.

The omega's same thing, as your sigma's settled the omegas, that's omega ones,

omega twos,

delta omega two's going to take longer control you get all your access.

Now here I can show,

you can see that that second axis is decaying at a much slower rate.

And the one and twos are decaying at a much faster rate.

From this figure plotting these epson on a log scale, I can extract out,

you can do kind of a curve fit and figure out what the mean trend is.

That's your exponential decay time and

I can compare it to my linearized predictions.

And keep in mind I'm doing a very large tumbling maneuver here,

this is not a small maneuver.

So what I get out of this is decay times what came out of my linearized

predictions was 15, 75, and 15 seconds, the half life.

The actual ones come out of the four non linear simulations are pretty darn

close to that.

That's kind of a nice it doesn't take much post tuning of the parameters to get those

decay times close to what you wanted.

So the differences here only a few percent.

And this is a down natural frequency.

That's why I'm looking at these oscillations.

So you get about ten of them.

And then divide by ten that gives you a good mean.

What is that to damp frequency?

And what my linearized predictions were were here.

And the actual response is here.

And you can see we've only got a few percent errors out of this stuff.

So this is kind of a nice thing then right.

So we've got some good feedback in selection technique that we can use.

Now dispute this wonderful lecture I just gave, I can promise you in the home works.

The last homework you do, you do a lot of controls.

And there's always people that comes to me going well I think everything's working

but and this is what their plot looks like.