 If you remember in the last week's lectures we were looking at stability, we ended with interesting examples of stability that is attractive but not stable asymptotically stable, globally asymptotically stable, non-uniform stability and so on and so forth. And then finally what we had done was we had sort of proved some simpler results for linear systems. So what were the simpler results? It sort of characterized stability in the form of bounds on the state transition matrix. But in spite of all of this we sort of explained and I hope all of us understood that this sort of characterization is still a definition of stability. It should be considered a definition of stability and not necessarily a test of stability because using these definitions is not very easy. For time-mating linear systems you will still require to solve the system in order to get state transition matrices in order for us to be able to talk about boundedness. So these are still to be seen as definitions and not as tests and that is where we are going to move on to this week. So we start with our material on Lyapunov theorems. You can see that I am using my earlier NPTEL nodes. So these parts are, there is some commonality here. Of course there is a, I mean we will look at a little bit more detail on some proofs but that will come probably later this week or actually won't come this week it probably come next week. So like I said I am out on Friday so we will not have a class on that. So let us first look at the results. We just look at the results how to use them and we will look at the proof of one or two of these results later. That is the idea. That way we have some handle on what we are even trying to do. So as I had mentioned very clearly I think that there is a problem with stability definitions we have to solve the system of equations. Now this is has to be the case because if you are using epsilon delta of any form you will have to solve. We whatever examples we do we actually solved the system and which is not possible for most nonlinear systems. And so we are looking at actually not quantitative but I would say more qualitative methods. But anyway I leave this word as it is because depends on how you think about it. So before we go on to talk about the Lyapunov theorems we need some background. So we first talk about function classes. All of you already know a few function classes by the way these are characterized by the LP norms the capital LP norms. So you already are aware of some certain function classes. These are additional function classes that we are going to talk about. So I hope you keep in mind that there are several kinds of categories of functions that we tend to invoke in all our analysis. So what are these function classes? The first one is a class k function and all of these functions are defined from r plus to r plus. So nonnegative reals to nonnegative reals r plus denotes nonnegative reals. So 0 is included in this set. So a function phi is of class k if it is continuous strictly increasing and it is 0 at 0. That is it. Only these three conditions. Examples are like as it is mentioned here 1 minus e minus x and x. Why you can verify when x is 0 this is 0. So here also I have mentioned if you take phi x equal to x this is one of the simplest examples because it is 0 at 0 it is definitely strictly increasing. You can take a partial and see and it is definitely continuous. In fact it is smooth. Similarly you can take any other polynomial. You can imagine that any polynomial would work. Polynomials are in fact analytic functions. So way more than smooth. They are even beyond smooth. So polynomials all work as long as the coefficients are positive and all the nice things happen. Otherwise it might be decreasing. That is the problem. So typically if you take x square is also fine. It is also an increasing function because we are looking only at 0 and beyond. Arguments are nonnegative reals. So therefore this works. So this also works on the negative side. But if I take x cubed and all then there is a problem. Then finally we have a function which is of this kind 1 minus e minus x. I keep using this example even for supremum I use this example. It actually turns out to be very nice interesting function. 1 minus e minus x if I take partial with respect to x it is what? E minus x which is positive. Whenever x is nonnegative and as long as x is not infinity and infinity is anyway not part of reals. So we are fine. So therefore this is also a continuous function. In fact smooth again also a strictly increasing function and it is 0 at 0. So all three functions satisfy these criteria. Is that clear? This is what is the class K function. Remember arguments are always scalar valued nonnegative reals and the output is also or the image is also scalar valued nonnegative reals. So scalar valued functions are all we are talking about. Do not ever design a vector valued function or a vector argument function and think of it as class K. No. There is no such character. Alright. Class L function. It is lot of a flipped version. A function phi again you see that this never changes is class L if it is continuous strictly decreasing and it is the initial value is finite. An example is 1 over x plus 1 because at 0 this is of course 1. It is strictly decreasing is evident because as x keeps increasing again remember x is in R plus. As x keeps increasing this keeps decreasing obvious and it is continuous. Continuity you can also verify continuity because it is there is only continuity issue at x equal to minus 1 but that is not a valid argument. So it is fine. One of the things that we need to note is that if a function is class K then it does not mean that the negative is class L. I hope this is sort of obvious because as soon as I take a negative my image lies in R minus which in itself is not allowed. So as a definition it is not allowed. So negative of class K function is not a class L function. So negative valued functions we do not use. This is just a why we talk about these functions is these are what make up Lyapunov functions. And Lyapunov functions if anybody has ever seen these are actually like energy like function. And we never talk about negative energy not in our community. I mean maybe in quantum we can talk about it but I am not an expert there. So I cannot comment on it. But here we do not obviously have notions of negative energy. So Lyapunov functions typically have connotations of energy. So obviously we do not allow negative valued quantities here. So class K and class L two characterizations. Now we have another characterization which is a sort of a stronger characterization. It is a class KR function. A function again same argument is class KR if it is class K and it goes to infinity as the argument goes to infinity. That is it. This is the only additional requirement. So one of the examples we considered for class K deliberately was this. And that function looks like this. Strictly increasing by the way. No problem. But it maxes out at one. Never exceeds one for whatever value of the argument. Because plot looks like this. It is getting closer and closer to one but never actually hits one. So this is not class KR. Because even though the argument becomes infinite or tends to infinity, we are all talking about tending to infinity, your function value will tend to one in fact. So not infinity. So these are, there is a certain limitation about these functions. That is why they are only class K and not class KR. But if you take these polynomial functions that we looked at, they have all the nice properties. It goes to infinity as the argument goes to infinity. So these are in fact class KR function. So as we move on, you will see that class K functions are connected with notions of local stability or stability. Class KR functions are connected to notions of global stability. And class L functions are connected to uniformity. So these are the three classes of functions. Then the three kind of stability definitions we have seen. Everything else is a sort of combination of all this. Because you say that you have some kind of stability, uniform stability, global stability, global uniform stability. And then there is the qualifiers of asymptotic and so on and so forth. That is okay. That also we will see how they are connected. But local stability typically class K tested via class K functions, global stability via class KR functions and uniformity using class L functions. So one of the key things that anyway I have already mentioned is that we assume an equilibrium of 0. We assume the equilibrium to be 0. Say that again. No, R is just, instead of using X, I have just used R. Just the argument. I have just changed the labeling on the argument. So it is a class KR function, not that R. This is just some notation for the argument. I have just replaced X by this. Great. Now we are in a position to talk about definiteness. Once we have defined this class K function, class KR function, we can talk about definiteness. Before I even go to definiteness, I hope all of you, we just looked at it a couple of classes ago. For matrices, symmetric matrices, you have very clean and clear notions of definiteness. Here you have this notation, by the way. Whenever I use this notation, this means that the matrix is positive definite. Because there is no idea of positivity of matrix otherwise. So whenever I use this notation A greater than 0, it means I am saying that the matrix is positive definite. You also know that it means that the quadratic forms are always positive for non-zero X. You also know that the eigenvalues of A are always positive and all principal minors have positive determinant. We exactly looked at these three characterizations for positive definiteness of matrices. Now what we want to do is generalize positive definiteness of matrices to functions. Because again, those of you who had exposure to doing internal stability for linear systems, you know that you use something like the Lyapunov equation. And the Lyapunov equation essentially looks like what? It says that where q is and q and p are symmetric matrices. So the statement of course goes more formally like given a q, there exists a p which solves this Lyapunov equation. Why this characterizes stability is because you choose your v, a Lyapunov candidate in fact, a Lyapunov function in fact, as X transpose TX for the system. And if you take a derivative of this v dot and your system is governed by X dot equal to AX because this is the system for which you are studying the stability. Then v dot is actually turns out to be X transpose PA plus A transpose PX. Alright, makes sense? And this is in fact, in fact I did not write it completely, this is positive definite, this is positive definite. You actually require that given any positive definite symmetric q, you can obtain a positive definite symmetric P which satisfies this Lyapunov equation. And once we have that essentially what we are saying is that this becomes a positive definite function. It is a sort of extension of positive definiteness from matrices to functions. This is now a function of X. So, this is what we want to connect. Now we want to say that we may have more general forms. Then X transpose PX, something quadratic, something simple. Nonlinear systems have a lot of different structure. You cannot necessarily say that for every nonlinear system I can use a quadratic Lyapunov function. That is in itself a big assumption. If you say that there exists a quadratically Lyapunov function for a nonlinear system, that is a very big assumption. Because you are somehow saying that you can use linear notions to analyze this nonlinear system. So, anyway, so the point is we want to have a more general characterization of positive definiteness for functions. So, that is where we are going, that is where we are going, that is the aim here. So, definiteness, what does it mean? It says that positive definiteness requires that you have a scalar valued continuous function which is of time and of some states belonging to a ball of radius r. That is this guy. This is what is the ball of radius r. We have already spoken about it. If you use different norms, you will get different shapes here. You can get in general you can think of it as an ellipse or a circle, but you can also get square and rhombus and what not depending on the norm. But it is a ball. We keep calling it a ball of radius r. Otherwise you have to say neighborhood and all that. So, that makes life a little bit more complicated. All right. Great. So, we want this function v and this is the most standard notation you will ever find of time and states belonging to a local region and it maps to real numbers. It is always scalar valued. So, energy like. Again, energy is the most if you think kinetic energy, potential energy, scalar value. Taking the states give you some scalar value. That is the most obvious characterization of v. But remember in more often than not we do not use the energy of the system as v. In several cases we do, but many more times we do not. What do you require from this? This is as of now I have only defined the domain and the range. What do I require? I require that it is 0 for 0 states. The function has to be 0 valued when the states are at 0. It is almost like a norm condition. Remember. It has to be 0 when states are 0 for all t in r plus. It does not matter what t is. If I put 0 states, it is almost like saying that at equilibrium if my right hand side of the system is 0, then this function which I am trying to analyze use to analyze the system cannot be non-zero when the states are 0. Does not make sense. So, it has to be 0 when the states are 0. There has to exist a class k function phi such that this function dominates phi norm of x. Notice how the class k function has been used. I mentioned very clearly that the class k functions have domain and range as non-negative reals. But we are talking about states. So, how do we compare? We use the norm of the state as the argument. The argument of a class k function will always be the norm of the state. It can be a weighted norm, no problem. But it has to be a norm. So, why? Because the norm is always non-negative. So, by virtue of taking the norm, I made the argument non-negative. All right, great. So, important thing to remember is that this does not mean that v is strictly increasing. v only has to dominate a strictly increasing function. Notice this, look at this picture. v does not have to be strictly increasing itself. If I think of phi x as this function, a strictly increasing continuous function which is 0 at 0 and v itself is also 0 at 0, v does not have to be strictly increasing. It can oscillate, but it has to remain above this line. Also, remember, my states were always required to be in this ball, bounded ball. Therefore, this domination also does not need to last forever. It only needs to last until the norm of the states. In fact, this is not x, but this is norm of x. This only has to last until the norm is less than r. Beyond that, it does not need to last. So, two things. This characterization number 2 does not mean that v is strictly increasing, only requires to be bounded by a strictly increasing function and does not have to be bounded for all states. It has to be bounded only for certain states in a certain ball of radius r. That is how we are doing this characterization and that is it. This is what is positive definite function. This is how we define it. We have still not connected it with the matrix definition for linear system. We will go there. So, of course, like I said positive definiteness will connect to asymptotic stability. We will look at how. Do not worry about it. Now, if you look at, if you think of this norm x and then you know norm x is something like this. This is the two norm, for example. Then a phi norm x is basically some, if you can think of phi norm x as something like this, this is a class k function. Remember, I hope you are convinced this is a class k function. I am sort of using this, I mean going reverse and constructing a v, a positive definite v by the way. Usually that is not the case. First you are given a v, then you have to think of a phi. I am sort of going the opposite side just to give an example. You know the two norm, just the Euclidean distance and then suppose I construct this class k function. This is a class k function. I hope you are convinced. Is anybody not convinced that this is a class k function? All you have to verify is that it is 0 at 0. It is strictly increasing. Do you believe it is strictly increasing? This can be written as 1 minus 1 over norm x square plus 1. It is strictly increasing and it has to be continuous or it is continuous in the norm. Not difficult to see that it is continuous in the norm. Only issue could have happened here, but there is a square here. So, obviously not an issue. So, this is continuous strictly increasing 0 at 0 valid class k function. So, if my v tx is something like this, this guy. Then this is going to dominate phi norm x. For all t greater than t in R plus. For all t in R plus, this is going to dominate this guy. So, this v is a positive definite function. Notice how different it is from your linear system sort of positive definite function. Characterization is rather different. I hope that is clear.