So, let's talk about global stability quickly. So, we have this control, this is Lyaponuv function. We've already argued this one is definitely readily unbounded. So sigma goes to infinity, the Omega's go to infinity, this always keeps going off to infinity. It's good for any initial states and that is good. And the V dot was minus P Delta Omega squared, also globally negative semi-definite. So we do have a globally stabilizing system at this point. Right? But it turns out we have made no arguments about convergence. We don't know if it converges or not. It will, but we have to do this. So now, switching was something we talked about. Right? So we do have to switch, and this was the classic thing. There's two possible MRP sets. One describes a short rotation, one the long. We like to switch when Sigma squared goes to one because then two at least my V function will be continuous. So I have to switch to a little bit different theory from the Lyaponuv. There's something called Switched Lyaponuv theories, where you have completing Lyaponuv cost functions of your control- tracking errors. And then you can switch between them to develop these controls. I'm not going to go through all the details of that, but the continuity is one of the important parts. And so switching here at 180 from a control is very, very nice. But the other issue is we need it with the Lyaponuv stability continuous derivatives. Right? We don't have that when we switch, we're jumping with those so we violate that agreement. But with additional arguments until I tumble 180 degrees, I know I'm perfectly stable. In fact, I've guaranteed that and I've decreased energy. And we can also show we will be asymptotic in that sense too in a second. But we know we're going to decrease our Lyaponuv cost measure, at some point time V one, we've reached this. Now we switch our V function, which is continuous, which is the key, is good, so the cost error is the same but all of a sudden you might have a discontinuous V function because we switched to derivatives. That's no longer smooth, but you can switch over and then it basically becomes a new problem. We're resetting our control. We ended up with plus 180. I do a switch non-controlling, get minus 180 because I'm going to complete the revolution that way. And then once we made the switch, your stability argument still hold for the next set of it. So by breaking up the total control of performance into discrete steps of, you know, up to the next switch, then the next switch, the next switch, all those arguments hold. And through continuity of the V function that we have here, we can still guarantee global stability of this system. So that's with the switching. I can show you graphically why this is important. If you have a V function, I just have Sigma as a scalar and V, and I'm starting out here and I'm making it smaller. You can have something where V dot is always negative. I'm always dumping error, and dumping error, and dumping error and then I switch. If you don't make this continuous, you could have something that switches to a higher level and then you make it smaller and you switch again to a higher level. And you could actually - well, I start it up too high - but you could actually increase and make it worse and worse and worse. So every time after switching you guaranteed to make it better but the switches itself could make these measures much worse. And how you describing to mathematically this can happen. But in our system we're guaranteed that my V function is continuous. So I haven't made my error measure any worse. And I may have a kink and then I go here and then there's another one. And then it's going to settle in. And because you have a finite amount of energy, because approve with passivity, this thing will continue to dump energy, dump energy, dump energy, and it will converge. And the resulting controls also look linear? I mean, continuous? No, the control would be discontinuous. So the control here, if you plotted this, you would have some control and then all of a sudden it does something else and then it does something else. So, if you're worried about this continuous control. What we typically do in control applications is all the control signal goes through low pass filter, that kind of smooth things out because of it get an errant signal in a sensor or who knows what else, and that will immediately kind of smooth out. These are finite duration regions, so it doesn't impact my overall stability. It just makes the transition smoother, so I don't excite fuel slosh and flexing and so forth. So yeah, a little smoothing would be required to implement it, but you have that smoothing anyway, so just take advantage of it. Those that are derivative became negative thoughts or negative definite? Negative semi-definite still. We haven't proven- the derivative we have here is... Do I have? No. This one is just negative semi-definite because it's only negative definite in terms of the rates, but the states don't appear. Sigma's don't appear. So we have to look at extra arguments now to talk about convergence. Here's a, where was I? I'm jumping all over the place. Okay. So, we did this, this is kind of the you break- this outlining again - this part you wouldn't be responsible for in an exam - but I've outlined extra steps. We can deal with the discontinuity and all of the MRP description and still argue stability, but you need additional arguments to go there. So let's do an application because if D tumbling, here I'm showing you the control. Now, I'm just showing you one of the principal angles, the 3D tumble, but the principal angles an easy one D measure. I give it a huge initial rate of 60 degrees per second. That's kind of like in skydiving, remember that. There was once a student I had and I was in California working for the summer and helping the local jump masters, and this is formation flying with AFF. So to the John Masters holding onto the student, they're supposed to go up, down, gently step off. It was more of an up, down, oh shit, you know, and off they go, tumble, pushed off so hard two of us, had no chance to recover. We didn't have enough gas, so I just remember going, 'oh, great', looked across the student as we're going upside down. He's kind of smiles at I go, 'okay, let's do it'. We completed the flip, stabilized it. Soon had no clue, because you have so much sensory overload, you have no idea you actual did a tumble, but we both just go 'okay', you know. Sometimes in life it's easier just to keep going and get where you want to get going, you know, don't fight the error. And that's what happens with this MRP control. We give it a huge rate because it is switching it automatically will go, 'okay, I'm going to fight this rate up to a point but once I'm past 180, okay, I will stabilize the short way around'. Now, I don't want to unwind the student and flip him 360 just out of principal, you know. And that's kind of- sometimes we do but anyway. But that's how it's going to stabilize. But that's the attitude rare. So, we do get this discontinuity happens once you research a stability arguments things work. From an angular velocity you can see there's a kink, because that's where the control changes. I fight really hard and then I go, 'you know what? I've got this momentum moving in that direction just I'm going to exploit that momentum and finish up the revolution until I stabilize where I want to stabilize'. So, that's a discontinuity in the control that you were asking about. And in real life you would put a little low pass filter or something to smooth out a little bit. Again, that's something we have all the time so. Ok, but that's an application where you see, now, we've only talked about global stability. The next step is going to be convergence, so I'll do that in the next lecture. But we do have global tracking. You can put anything in there. You can spell out Coca-Cola with a laser in the sky, whatever you wish. That's your reference, you throw in the right reference trajectory and off it goes. So it is extremely powerful and it has a lot of nice properties that we'll be exploring over the next few lectures.
