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So, let's talk about global stability quickly.

Â So, we have this control,

Â this is Lyaponuv function.

Â We've already argued this one is definitely readily unbounded.

Â So sigma goes to infinity,

Â the Omega's go to infinity,

Â this always keeps going off to infinity.

Â It's good for any initial states and that is good.

Â And the V dot was minus P Delta Omega squared,

Â also globally negative semi-definite.

Â So we do have a globally stabilizing system at this point.

Â Right? But it turns out we have made no arguments about convergence.

Â We don't know if it converges or not.

Â It will, but we have to do this.

Â So now, switching was something we talked about.

Â Right? So we do have to switch,

Â and this was the classic thing.

Â There's two possible MRP sets.

Â One describes a short rotation, one the long.

Â We like to switch when Sigma squared goes to

Â one because then two at least my V function will be continuous.

Â So I have to switch to a little bit different theory from the Lyaponuv.

Â There's something called Switched Lyaponuv theories,

Â where you have completing Lyaponuv cost functions of your control- tracking errors.

Â And then you can switch between them to develop these controls.

Â I'm not going to go through all the details of that,

Â but the continuity is one of the important parts.

Â And so switching here at 180 from a control is very, very nice.

Â But the other issue is we need it with the Lyaponuv stability continuous derivatives.

Â Right? We don't have that when we switch,

Â we're jumping with those so we violate that agreement.

Â But with additional arguments until I tumble 180 degrees,

Â I know I'm perfectly stable.

Â In fact, I've guaranteed that and I've decreased energy.

Â And we can also show we will be asymptotic in that sense too in a second.

Â But we know we're going to decrease our Lyaponuv cost measure,

Â at some point time V one, we've reached this.

Â Now we switch our V function,

Â which is continuous, which is the key, is good,

Â so the cost error is the same but all of a sudden you might have

Â a discontinuous V function because we switched to derivatives.

Â That's no longer smooth, but you can switch

Â over and then it basically becomes a new problem.

Â We're resetting our control.

Â We ended up with plus 180.

Â I do a switch non-controlling,

Â get minus 180 because I'm going to complete the revolution that way.

Â And then once we made the switch,

Â your stability argument still hold for the next set of it.

Â So by breaking up the total control of performance into discrete steps of,

Â you know, up to the next switch,

Â then the next switch, the next switch,

Â all those arguments hold.

Â And through continuity of the V function that we have here,

Â we can still guarantee global stability of this system.

Â So that's with the switching.

Â I can show you graphically why this is important.

Â If you have a V function,

Â I just have Sigma as a scalar and V,

Â and I'm starting out here and I'm making it smaller.

Â You can have something where V dot is always negative.

Â I'm always dumping error,

Â and dumping error, and dumping error and then I switch.

Â If you don't make this continuous,

Â you could have something that switches to a higher level and

Â then you make it smaller and you switch again to a higher level.

Â And you could actually - well,

Â I start it up too high - but you could actually

Â increase and make it worse and worse and worse.

Â So every time after switching you guaranteed to make it

Â better but the switches itself could make these measures much worse.

Â And how you describing to mathematically this can happen.

Â But in our system we're guaranteed that my V function is continuous.

Â So I haven't made my error measure any worse.

Â And I may have a kink and then I go here and then there's another one.

Â And then it's going to settle in.

Â And because you have a finite amount of energy,

Â because approve with passivity,

Â this thing will continue to dump energy, dump energy,

Â dump energy, and it will converge.

Â And the resulting controls also look linear? I mean, continuous?

Â No, the control would be discontinuous.

Â So the control here, if you plotted this,

Â you would have some control and then all of a

Â sudden it does something else and then it does something else.

Â So, if you're worried about this continuous control.

Â What we typically do in control applications is

Â all the control signal goes through low pass filter,

Â that kind of smooth things out because of it get

Â an errant signal in a sensor or who knows what else,

Â and that will immediately kind of smooth out.

Â These are finite duration regions,

Â so it doesn't impact my overall stability.

Â It just makes the transition smoother,

Â so I don't excite fuel slosh and flexing and so forth.

Â So yeah, a little smoothing would be required to implement it,

Â but you have that smoothing anyway,

Â so just take advantage of it.

Â Those that are derivative became negative thoughts or negative definite?

Â Negative semi-definite still.

Â We haven't proven- the derivative we have here

Â is... Do I have? No.

Â This one is just negative semi-definite because it's

Â only negative definite in terms of the rates,

Â but the states don't appear.

Â Sigma's don't appear. So we have to look at extra arguments now

Â to talk about convergence.

Â Here's a, where was I?

Â I'm jumping all over the place. Okay.

Â So, we did this, this is kind of the you break- this outlining

Â again - this part you wouldn't be responsible for in an exam

Â - but I've outlined extra steps.

Â We can deal with the discontinuity and

Â all of the MRP description and still argue stability,

Â but you need additional arguments to go there.

Â So let's do an application because if D tumbling,

Â here I'm showing you the control.

Â Now, I'm just showing you one of the principal angles,

Â the 3D tumble, but the principal angles an easy one D measure.

Â I give it a huge initial rate of 60 degrees per second.

Â That's kind of like in skydiving, remember that.

Â There was once a student I had and I was in

Â California working for the summer and helping the local jump masters,

Â and this is formation flying with AFF.

Â So to the John Masters holding onto the student,

Â they're supposed to go up,

Â down, gently step off.

Â It was more of an up, down,

Â oh shit, you know, and off they go,

Â tumble, pushed off so hard two of us,

Â had no chance to recover.

Â We didn't have enough gas, so I just remember going, 'oh,

Â great', looked across the student as we're going upside down.

Â He's kind of smiles at I go, 'okay, let's do it'.

Â We completed the flip, stabilized it.

Â Soon had no clue, because you have so much sensory overload,

Â you have no idea you actual did a tumble,

Â but we both just go 'okay', you know.

Â Sometimes in life it's easier just to keep going and get where you want to get going,

Â you know, don't fight the error.

Â And that's what happens with this MRP control.

Â We give it a huge rate because it is switching it automatically will go, 'okay,

Â I'm going to fight this rate up to a point but once I'm past 180,

Â okay, I will stabilize the short way around'.

Â Now, I don't want to unwind the student and flip him 360 just out of principal, you know.

Â And that's kind of- sometimes we do but anyway.

Â But that's how it's going to stabilize. But that's the attitude rare.

Â So, we do get this discontinuity happens

Â once you research a stability arguments things work.

Â From an angular velocity you can see there's a kink,

Â because that's where the control changes.

Â I fight really hard and then I go, 'you know what?

Â I've got this momentum moving in that direction just I'm going to exploit

Â that momentum and finish up the revolution until I stabilize where I want to stabilize'.

Â So, that's a discontinuity in the control that you were asking about.

Â And in real life you would put

Â a little low pass filter or something to smooth out a little bit.

Â Again, that's something we have all the time so.

Â Ok, but that's an application where you see, now,

Â we've only talked about global stability.

Â The next step is going to be convergence,

Â so I'll do that in the next lecture.

Â But we do have global tracking.

Â You can put anything in there.

Â You can spell out Coca-Cola with a laser in the sky, whatever you wish.

Â That's your reference, you throw in the right reference trajectory and off it goes.

Â So it is extremely powerful and it has a lot of

Â nice properties that we'll be exploring over the next few lectures.

Â