 Hello, welcome to yet another session of our NPTEL on non-linear and adaptive control. I am Srikant Sukumar from Systems and Control IIT Bombay. So we have in our background a usual motivating image which is essentially a rover on Mars and these sort of vehicles are driven autonomously because it is not usually possible to have sort of humans controlling the equipment at all times on these extraterrestrial locations. So we want to be able to of course analyze and learn to develop algorithms that are going to drive systems such as these. So until last time what we were doing was essentially discussing the notion of positive definiteness. So this is what we were doing until last time. We were talking about positive definite functions. So this was one of the building blocks in the Lyapunov theorems. So we talked about what is the positive definite functions and after that we sort of wanted to look at a few easier tests for positive definiteness. So we saw the definition but then it's not always easy to apply the definition and so we wanted to look at some additional tests and we looked at these easier sort of tests that can be applied and we sort of also saw the connection between positive definite matrices and positive definite functions through an example and so through this example and of course we saw another example for positive definite function also. So we sort of go ahead in this vein of discussion and the next sort of function is a radially unbounded function. So let's sort of start our lecture 4.2 here. So we are starting to talk about the radially unbounded functions. So radially unbounded functions are sort of the next level of functions. As you can see you have a scalar value continuous function again. Of course this should be R. I mean I guess we use R here too. Let me see yeah we were using R here so I guess we can continue to use R here although R plus would suffice in all these cases. We don't necessarily have to use R itself but R plus is good enough. So non-negative reals for time is usually what we have. So we are of course looking at a function of time and a space variable which maps to real numbers. This has always been the setup for these functions that we denote as V which are essentially going to be our Lyapunov functions subsequently. So what do we require? We require that the function be zero at zero. So at zero value of state the function has to take the zero value. So this was the same condition for positive definiteness also. The big difference happens here. We want the existence of a class kr function which this function VTX dominates. So in the case of positive definiteness this was a class k function. So what's the difference between a class k and a class kr function? We have in fact seen several examples. So if you have a class k function it can be and we are allowed to have something like this function which sort of grows, grows, grows but stays bounded below this. So it's always increasing but stays bounded below this. On the other hand a class k infinity function has to be something like this which keeps drawing n goes to infinity. Yeah, as t goes to infinity. So here of course here you have the x and you have sort of phi x here. And since of course these functions take as arguments non-negative reals obviously you use a norm. This is what we have been doing. So just to compare again with the definition of positive definiteness it was the same sort of expression phi norm of x. The only difference being that here it was a class k function and now in the case of a radial unboundedness we require V to dominate a class kr function. So this naturally sort of implies that this domination has to be for all time. However notice that just like before we do not require if you think about it we don't require the V to be a continuously increasing function. Like a class k function is of course an always increasing function always strictly increasing function. But all we want for V is to dominate this class k function. And notice that again unlike the definition for positive definiteness here this lower bounding has to happen for all x and rn. In case of positive definiteness it was sufficient if it happened for x in some ball. Therefore we had this nice picture where beyond this ball the V was allowed to drop below the class k function. So essentially the idea is that it can give us local stability properties. So the idea of positive definiteness being defined in this manner is that it helps us conclude local stability properties. On the other hand radial unbounded is typically used to conclude global stability properties. And so we require this function V to dominate this class k function which is this guy in this particular case here for all time and for all values of the state. So this is much more stricter condition as you can imagine. However this still does not require that my function V be increasing or anything. It can be oscillating like this as long as it is bounded above its the second as long as it is bounded above your class k r function you are fine. Okay so this is sort of an example right this picture sort of gives you an example. The only thing is there is no crossing happening anymore. It always has to lie above this class k r function curve right and in such cases V is said to be radially unbounded. And just as we already stated before radial unboundedness is related to notions of global stability. Okay just like positive definiteness is connected to local stability radial unboundedness is connected to global stability. Okay what would be examples very simple examples I mean say I have x in r2 and denoted as x1 comma x2. So V x1 x2 equals to x1 square plus x2 square by 2 is in fact a radially unbounded function. Yeah why is it radially unbounded? Because if you look at this function itself and this in itself is a class k r function right this in itself is a class k r function why because it's 0 at 0 right. If you plug if you plug 0 value of this state that is x1 and x2 are both 0 which is what it means for x to be 0 right I get a 0 here and you know this is strictly increasing and going to infinity as any x goes to infinity. So in any direction if you go to infinity this is going to go to infinity all right. So therefore this is itself a class k r function therefore V is in fact a radial unbounded function also every class k r function will also be a radially unbounded function. Of course I have I can have you know a slightly more involved examples like you know I can take something like V with T also x1 x2 as T by 2 x1 square plus x2 square and here of course T is greater than equal to 0 right. So here of course this is you know this is greater than equal to half x1 square plus let me be careful here let me say this is not T but this is T plus 1 so this is greater than equal to half x1 square plus x2 square because T plus 1 is greater than equal to 1. So therefore this is greater than equal to half x1 square plus x2 square so implies V is again radially unbounded okay. So we can also have let's see let me try to think we can also have some kind of oscillating examples like the one that we showed in the picture and that can be constructed say by again introducing a time variable as 1 plus say sine square T divided by 2 x1 squared plus x2 squared right. So this is again greater than equal to half x1 squared plus x2 squared because sine square T this is of course greater than equal to 0. So this is this is fine and this is radially unbounded but what this term is going to do right this what this term is expected to do is to sort of make things oscillate around a bit yeah make things oscillate around a bit. So this term is going to contribute to some oscillation but in time right here if you notice the oscillation was with the state itself right that can also be constructed right I am not doing it here right but this can that can also be constructed okay. So I would really urge you to give it a thought as to you know try to try to construct such a Vx okay try to construct such a Vx right where it's you know x and it's oscillating with respect to x itself but it's still dominating a class k function right still dominating a class k function it's it's not too difficult honestly speaking yeah it's not too difficult I mean you can think of I mean the simple ideas are like you know I can add an oscillatory signal to x squared yeah something simple like that yeah as long as I do that I'm also still okay all right okay so these are all examples of radial unbounded functions again relatively simple examples okay but again when things become more complicated we do require easier conditions just like before just like the case of positive definiteness we do require easier conditions to verify radial unboundedness also okay so what is it so again if I if I think of the case when V is not a function of state and remember that when sorry V is not an explicit function of time and remember whenever V is only a function of state we have been using Wx as the notation right do not get confused at all and using V and W almost interchangeably it's just the purpose of the function is what is important and the notation of the function itself so whenever V does not depend on time explicitly I am denoting it with Wx okay so whenever it's just a function of the state right then I have three requirements right first is that W0 has to be 0 Wx has to be strictly positive for all non-zero states again I'll come to the difference between the positive definiteness case soon and finally Wx has to go to infinity as the state goes to infinity okay so this is important it has to go to infinity in any possible direction it has to go to infinity any possible direction so what is the difference first of all let's try to see what's the difference from the easier condition for positive definiteness okay so let's keep this in mind yeah and and remember that this third condition did not even exist for positive definiteness there was no requirement for the function to go to infinity at all so obviously this third condition is completely distinct right so we are not even going to discuss it let's look at the first two conditions only yeah if you look at the first two conditions you require the same first condition that the function itself be 0 when the state is 0 there's no difference there this is pretty much uniformed along all definitions concerning Lyapunov functions yeah that the function needs to be 0 at 0 why because 0 is where we are interested to go to right so think of the Lyapunov functions or these v functions as some kind of energy functions right and we want the system to settle at the origin that is x equal to 0 therefore we want the energy the so-called energy or this notional energy function to also be 0 or minimum value at 0 at the origin right doesn't make sense otherwise right yeah if it's not at the minimum value when it's at the origin then it could possibly go lower right and therefore we will move from the origin doesn't make sense since we want to stay at the origin right so all these functions invariably require that they be 0 when the state is sealed so this condition is identical no difference second condition is where there is a small change it is positive for all x but in a ball removing the origin it only has to hold true in a ball okay of course removing the origin but here we need it to happen for all rn this is also because how radial and boundedness is defined right that that this domination has to happen for all rn right everything has to happen for all rn because we are talking global notions here okay so this is the difference so the first two conditions I find me I would like to call it some kind of global global positive definiteness it is still positive definiteness the only thing is we have we are requiring something more stringent that this positive difference hold for every value of state other than the origin and not just in a ball around the origin okay so this is a little bit more stringent and beyond that we require that the function goes to infinity as the state goes to infinity so what we want to do to look at this you know sort of easier condition thing is to look at a lot of counter examples right so and that really helps us to understand what is not a really unbounded function right so I am going to label this counter examples okay and 1 v sorry v x1 x2 is equal to half x1 plus x2 squared okay so the question is is this positive definite is this radially unbounded okay so we have to answer all these questions so the first thing that you need to notice is that along x2 equals minus x1 what is this if I look at the state space I am going to draw the state space right this is x1 this is x2 ah you know why that's happening right this is x1 this is x2 and what is x2 is minus x1 I believe it is something like a line which looks like this ah this is this x2 is minus x1 is this line right so this was x2 equal to minus x1 so what happens along this line along this line so I cannot of course draw it because it's I like to do it in three dimensions v takes zero value okay so v equal to zero all along this line okay and that is a problem right that is a problem right why because since v not greater than zero for all x1 x2 not equal to zero in fact there is a line which can go all the way from zero to infinity and minus infinity where v is in fact not positive at all it's in fact exactly zero therefore v is not even satisfying the first two conditions right so this is I you can you can very safely say that not positive definitely when yes so forget radial unboundedness this function is not even positive definite okay so definitely doesn't work next counter example so I'm sorry I should have used w but okay that's fine I've used v but again let's remember we are using v and w interchangeably and it's the purpose that it serves is what is valuable to us and not the notation okay so let's not get too confused or too hung up with the notation okay all right so let's look at v but w now I will use w right x1 x2 is equal to half x1 square plus one fourth x14 okay is this positive definite is it satisfying so of course w zero is zero great okay but what happens let's see w of all zero comma alpha is also zero because notice that is no x2 here only one state appears so whenever you have a function which has only one state what happens I can just say w zero alpha is equal to zero which implies w not positive definite and if it's not positive definite implies w not radially unbounded yeah because I I need minimum positive definiteness to even go to radial unboundedness okay so no answer is no again okay so if you have any function which has only some of the states appearing and not all of the states it is immediately not a positive definite function and therefore not radially unbounded okay so you don't even have to do any calculation okay because you can always make an argument like I made here okay great let's look at a slightly better example w x1 x2 well I let me use something better we have already seen this example w x yeah where x can be in rn yeah so what about this does this satisfy the first two conditions one is is zero it's zero yes w zero is in fact zero yeah what else w x is in fact positive for all x not equal to zero it's obvious because if x is not zero norm x is positive we already made this argument so so this immediately implies norm x is positive and I'm done right if norm x is positive right then this is positive this is again some positive denominator so it doesn't matter so w x is positive so of course it satisfies the first two conditions great yeah excellent we are happy at least we have something that's positive definite right which we already proved had did prove in the previous lecture okay so this is not a different example what about radial unbounded that is what about the third condition does it go to infinity as x goes to infinity as you can see the answer is no right why this is where I had written a sort of expression for w last time this is where it will come to be of use to us right so w x can be written as 1 minus 1 over 1 plus norm x square okay so as x goes to infinity w x goes to 1 right because as x goes to infinity this goes to infinity so this whole thing goes to 0 and so on I'm left with this one so this is one of those functions which is a nice positive definite function so so from here I of course had this is this positive definite but because of this implies w not radially unbounded iub is the notation for radial unboundedness okay so it's a positive definite function but it is not radially unbounded because it doesn't go to infinity as the state goes to infinity okay so this is rather critical okay rather critical distinction so function like this cannot be used to prove global stability okay all right it will not help us do global stability properties okay then we'll see later on of course but yeah so this is not a radially unbounded function all right excellent so all right these are sort of the you know the easier condition corresponding to radial unboundedness the second condition is when we have a function of both state and time I think both time and state appear exactly you know exact parallel to what we did for positive definiteness we had one condition for when we had a function of the state only and another condition when we had a function of both the state and the time explicitly all right in such cases we just use the previous easier condition right we just again just like positive definiteness we just use the previous condition what do we say it still has to be zero for zero state values and all time okay um and but we said d in r plus here of course so fine that's non-negative time I mean anyway to be honest just to prevent ambiguity which better to say it is r plus yeah for all definitions of v time in r plus is quite okay okay because we never deal with non-negative with negative time okay so we have v t zero equal to zero is of obviously our standard requirement but the next requirement is simply you know using our previous positive radial unboundedness definition is that this v tx has to dominate a radially unbounded function of the state only okay v tx has to dominate wx which is radial unbound which is a radially unbounded function of the state only where wx is radially unbounded okay so wx is a radially unbounded function of the state only okay in this case of course constructing examples are really easy I can simply take a v v tx as again something like t plus 1 by 2 and norm x squared and we know that this is greater than equal to half norm x squared which is of course radially unbounded so this implies that this is radially unbounded okay so it's not very difficult to construct such radial unbounded examples of both state and time all right great great so what did we look at today we essentially did the exact same thing that we did for positive definiteness or the case of radial unboundedness which are functions which addition in addition to positive definiteness globally also go to infinity as the state goes to infinity we looked at some easier conditions to verify this radial unboundedness also right so again of course we will continue with the next set of functions in the upcoming lectures so that's it thank you for joining