0:06

So now these are the definitions.

Â And this is what we're going to be using.

Â And we're going to actually derive.

Â We can't solve the equations and improve their properties,

Â because these are not analytically solvable.

Â So we will use all the tools that are open to do it.

Â Linearization is still a very powerful thing.

Â I got one slide on this I think you did some homework as well.

Â So the linearization allows you to say, hey,

Â this is the reference is great this is how I should be pointing here.

Â I got that reference history, you have also done this somehow in your homeworks.

Â It got your reference motions that you can integrate and

Â now I want to look departures relative to this.

Â So you can always do a linearized approach.

Â But immediately soon as you do linearizations you always just get

Â local arguments.

Â If the linearized system which is what we did with the dual spinner was stable,

Â we had it down to a linear form x plus some k times x essentially, right?

Â And with the right omega spin rates, that k became positive or

Â negative and so either stable or unstable.

Â But that argument of stability was actually only a local argument.

Â We didn't have any mathematical proof that for an arbitrary tumbling,

Â large departure motions that dual spinning would be stable.

Â It's just for very small neighboring ones.

Â Now, what is small that's very application specific?

Â For some applications smaller might behave plus,

Â minus 120 degrees which is not really small.

Â But that's what it is, right?

Â For others it might be plus, minus micro radiance or something and that's it.

Â 1:40

So, it depends on the application.

Â The linearization approach,

Â we've done some of this already in your last homework you did it as well.

Â You had this equation, you had to linearized around the 90 degree point.

Â There's a whole process of how you do this.

Â You've got your reference to linearize you have to define your states here

Â relative to the reference.

Â So introducing deltas.

Â That's our departure motion again.

Â Delta x is x in an actual state minus xr and xr could be fixed or vary with time.

Â And then my control, you might have a reference control.

Â Let's say, you know, that hovering, it wants to fly at 20 meters.

Â Great, well, it's going to have to produce a thrust that gives it one g

Â of acceleration, otherwise, it's going to start dropping again, right?

Â So there would be a certain, let's say, that's ten Newton,

Â that's a nominal thrust you have to produce to hold that reference.

Â 2:29

And then if you're off a little bit you might have to have more thrust to come, or

Â if you're too high, you might reduce the thrust slightly, right?

Â So this is the reference part, the ten Newton.

Â The delta u becomes the feedback part, because you can solve for the actual u.

Â The control is being u r plus delta u.

Â Defeat forward part plus defeat back part.

Â That's kind of the structure.

Â So the reference trajectories we have often then generate our feed forward part

Â of the control.

Â If UAV was to accelerate, well, to accelerate it has to have more thrust, but

Â you can compute that.

Â So nominally, yes ramp up from 10 to 20 newtons over the time period.

Â That'll give you the history that you want.

Â That will again be your reference.

Â So you as a user designed this and this is the reference that will achieve,

Â this is the reference control that would achieve the reference motion.

Â Hover, start to rise, stop.

Â You can come up with an open loop force history, that would have to happen.

Â So that's what we are linearizing about.

Â So if you look at an actual Dynamical System that's nonlinear,

Â we have control applied.

Â We want to linearize the departures, delta x is x minus xr.

Â So you put dots on everything and say, okay,

Â my delta x dot is going to be x dot minus xr dot.

Â And xr dot is just f of x r and u r.

Â Right, we've already got that one.

Â But the x dot, we are now going to linearize with the Taylor series.

Â So if you remember Taylor series, we have y is equal to f of x and

Â you want to linearize about x is equal to five.

Â You put in f for

Â five plus the first partial times the delta plus the next second partial.

Â And that's what we're seeing here.

Â We're doing first order stuff, so your Taylor series expansion of a function is

Â going to be the function evaluated at the reference plus the first partial

Â with respect to the states, evaluated at the states.

Â That's why it's xr and ur multiply times the small departure.

Â But we not only have departures in states, we also have departures in controls.

Â We will be putting in just the open loop control, we may have to stabilize it so

Â we have a delta u that happens as well, so we take the partial of

Â f with respect to your control variable and then apply small departures.

Â And everything else is higher ordered terms.

Â So if you do this, all you're left is this minus this always cancels and

Â you're left with some equation here that's basically delta x double dot plus this

Â partial times delta x plus another partial times delta u is equal to,

Â well, those two things are equal.

Â This form, some of you may not have seen.

Â Many of you will have seen this form, though.

Â Basically, you got a x dot equal to a x plus b u, right?

Â That's the classic, that's the plan matrix, that's the control ability matrix,

Â that's where you come together.

Â But for non linear system, this is how you find that A and that is the B.

Â It's the partial of the actual f, denominator f function,

Â with respect to the states after reference.

Â That's what gives you the A matrix.

Â And the other partial, that's what gives you the B matrix.

Â [COUGH] So I expect you to do Taylor series expansion.

Â it is also in vectorial form.

Â We've done this kind of a partials when we did that one over RQ thing in class with

Â a gravity gradients derivation and so forth.

Â So it's the same math as being applied here.

Â Take one of the hall marks as you solve this is as well in some systems.

Â So I'll let you guys work with that.

Â So but

Â this is, you can always do this as well and then you can argue linear stability.

Â But just not realize if you do this approach,

Â at best you've only argue local stability.

Â Not global.

Â