 Hello, welcome to yet another session of our NPTEL on non-linear and adaptive control. I am Srikanth Sukumar from Systems and Control IIT, Bombay. We are now well into the third week of our course on non-need adaptive control and we are already well underway into learning the concepts that will help us analyze autonomous algorithms that drive systems such as what we see in our background. So without delay any further, let me sort of recap for us what we were looking at until last week. So last time we sort of tried to complete this example of a system which is a classical example by Masera, which is solved in the book by VPS Arvind. And for this particular system we show that it is stable, but it is not uniformly stable. So this is a rather interesting sort of system and we did a little bit of nice bit of work and analysis in order to try to see what kind of choice of delta must one make in order to remove the dependence on the initial time. And what we concluded, this is sort of the highlight of last time. What we concluded was that we are required to choose the smallest possible delta out of all the deltas that we get for different initial times. And that's what we need to do in order to make delta independent of zero and obtain uniform stability. We also sort of looked at this sort of very, very briefly looked at what is this van der Paul oscillator. So what we want to do is of course complete discussing this van der Paul oscillator system in a little bit more detail today before we go ahead. So what is this van der Paul oscillator? We already said last time that this is a very, very sort of classical nonlinear oscillator system. It's also immensely popular because we want, because there are certain scenarios in which we want to construct systems which have an oscillatory model, for example, pacemakers that are used in parts of individuals who have sort of arrhythmia and other inherent issues. All right. So what we want to do here is to look at whether it's stable or uniformly stable or unstable. So this is what we want to do. And the van der Paul oscillator system is given by this equation 1.6. And here we have the handle of choosing this particular mu. We can choose this constant mu and sort of defines or changes the behavior of the van der Paul oscillator. All right. So the first thing we of course do is write this 1.6 in the standard state space form. And for this we of course choose states x and y, as always. And so this equation 1.7 is what we will analyze. The first thing to observe is that 00 is an isolated equilibrium, right? So this is an isolated equilibrium point for this system. Okay. Once we do that, if you remember what I had suggested was that we just make phase plane plots, all right? We are not going to really try to solve this because it is rather difficult to write a closed form solution for this system. Okay. So what we do is we simply make a phase plane portrait of the system. And this is what is shown in 1 and 2, figures 1 and 2. So although it says for every mu, here it is plotted for particular value of mu, the phase plane portrait. And it so happens that for any initial condition your system actually converges to this sort of a limit cycle. You know this thing that you see here. This is the oscillatory behavior of the system if you may. Okay. For any oscillator the behavior is always such that it follows some sort of a periodic limit cycle at the end. Okay. There is a periodic limit set. And this is called limit cycle. Okay. So we already know, well, we have not already defined it yet. But anyway, so this is called a limit cycle behavior because it is a cyclical behavior in the phase plane. Right. So this is what happens. Wherever you start, it doesn't matter where you start. You will always end up into this following this sort of a limit cycle. And this is true for most values of mu. Okay. It's true for most values of mu. So now if I want to talk about stability of the origin, what do I require? I require that given any epsilon positive, I must be able to find a delta such that if initial conditions are within delta, then my solution lies within an epsilon ball of the equilibrium. Right. Now, if you, now we stated very carefully here, we say that if you choose an epsilon which is smaller than the sort of the radius of the limit cycle. Okay. And this is given by an example here. Right. So the limit cycle is this A set A, which looks something like this. And if we choose an epsilon ball, which is this guy, right, you can see that every solution is eventually converging to this cycle A. Therefore, it doesn't matter how small a delta I make out here. My solution is definitely going to get out of this epsilon ball and converge to this set A. And because of this, you know, sort of unusual behavior, the equilibrium zero zero is in fact unstable. Okay. Because I can choose any epsilon ball smaller than this, right. It has to lie inside the limit cycle, of course. So if I choose any epsilon, which lies inside the limit cycle, there is no possibility of finding a delta because for every initial condition inside this ball, you will always go to the set A. All right. And this is a problem. Okay. In terms of, I mean, it is a problem not in general, but it is a problem in terms of how we have defined stability. Okay. How we have defined stability. Right. So for stability, they require that if the user gives me any epsilon, right, it can be anything arbitrarily large, arbitrarily small, it doesn't matter for every possible epsilon that the user gives me, I must be able to you know, give or provide a delta says that initial conditions inside the delta ball will remain inside the epsilon. Now, in this case, this is absolutely not as you can see. And because anything inside this ball will always go to A. Okay. And we know that delta cannot be larger than epsilon. Right. That's obvious. Excellent. Right. So equilibrium 00 is in fact unstable. However, this is a very nice system. Right. I mean, this is a very nice and well behaved system in the sense that all trajectories seem to go to the limit cycle. In fact, trajectories for certain values of mu trajectories outside also go to the limit cycle. Okay. For certain values of you, the trajectories outside the limit cycle also converge to the limit cycle. And for all the values of mu trajectories in starting inside the limit cycle will converge to the limit cycle. Okay. So this is a rather nicely behaved system, but it is not stable in the sense of Lyapunov. Okay. So remember that although we have painstakingly defined these notions of, you know, stability in the sense of Lyapunov and all, but it is not, you know, the final poly grid. I mean, there are systems which behave nicely, which do not exhibit stability in the sense of Lyapunov. Okay. This is rather nice. Yeah. Of course, this might also lead us to thinking about more general definitions of stability. Right. This might, but then of course we don't look at that in this course. But yes, it does make sense to think about more general definitions of stability, etc. Okay. Great. Great. So now that we sort of have a decent handle on what is stability and uniform stability, we want to look at more additional properties that the dynamical system might possess. Yeah. And we've already sort of seen in this Bob Lutz Lemme analysis that we are very interested in convergence. And we always want things to converge to zero, functions to converge to zero. Right. So this is a property that we are rather really interested in. Right. So now this convergence property is formulated in systems theoretical language as attractivity. Okay. So this is the notion of convergence for dynamical systems. Right. So suppose, you know, in order to talk about attractivity, what I do is I assume that my equilibrium is in fact, at the origin. Right. It's not such a, you know, difficult assumption to satisfy. This does not happen. I simply do a change of coordinates, such as this. And once I do this change of coordinates, yeah, it should be, it will become evident to you that this equilibrium in this y coordinates is in fact, the origin. Okay. This is just to make our notation easy. Yeah. That's, and it's very standard practice. Yeah. So it's not something crazy. It's just very, very standard. It's not even a big deal. All right. We always consider origin to be our equilibrium. Yeah. All right. So what does it mean for the origin to be attractive? It means that for all initial times, there exists a delta depending on this initial time, only the initial time, such that initial conditions within the delta wall result in all state trajectories converging to zero as T goes to infinity. Okay. So this is essentially convergence. Essentially convergence the way we know. The only thing we have done is we have fixed some initial time and initial states. Yeah. So for attractivity, I require, I'm required to find a, you know, initial condition ball of size delta. This delta is of course, allowed to depend on T zero. Yeah. Unlike the stability case here, there is no epsilon. So there's no epsilon dependence. Right. But T zero is of course there. So it can depend on T zero. Right. And we require that if the initial conditions are within this delta wall that we have provided, then all my solutions, my solutions, the state trajectories in fact converts to zero as T goes to infinity. As usual, we also have the notion of uniform attractivity. Okay. So what is uniform attractivity? It is that for all T zero, there exists a delta. Now this delta doesn't depend on T zero either. Therefore, yeah. I mean, so let me be a little bit more precise. So here delta positive. Similarly, delta positive. This is important. Okay. Let's not miss this. So we need the delta to be of course a positive quantity. Now here the delta cannot depend on T zero either. So it's just a constant. Right. So for uniform attractivity, you need to be able to find a positive constant delta such that initial conditions within the delta ball guarantee that your state trajectories converts to the origin as T goes to infinity. Okay. So as before, as in the case of the definition of stability versus uniform stability, here too, the uniformity is with respect to the initial time. Okay. That is what we mean by uniformity in a system theoretic setting. All right. So I hope that's clear. Excellent. So now that we have these two new notions, why don't we test these for the example by Masera that we have already seen? Okay. So if you look at what I'm going to do is for the Masera system, I'm going to write this solution out again. Okay. So I will write out this solution again. So let's see if I can in fact do that. If I can copy it. Right. So it doesn't let me copy it. So I hope it lets me paste it too. Right. So here you go. So this was, I guess, and this was our example. Right. I mean, this was, yeah, this was the example for Masera example. Yeah. So this is what is our, was our solution. And what do we want to do? We want to sort of understand whether this is attractive and slash all uniformity attractive. Okay. So let's see. Is this an attractive system? So in this case, I just need to find a delta. Okay. So I just need to find a delta. Right. So find delta or convergence. All right. So that's all I need to do. I need to find a delta for convergence. Now the question is what sort of initial conditions are allowed? So let's note something. Okay. Let's notice something. So once I fix this t zero, right. So in fact, it doesn't matter whether I fix t zero or no, but, but let's notice something. This is, this is some value here, right. Some, you know, constant, right. Similarly, gamma is also some constant. Yeah. After t zero x zero fixed. Okay. For a particular choice of t zero and an x zero, it doesn't matter what these choices are in fact, but once I've chosen this t zero and x zero, these two are in fact constant quantities. All right. They're not changing with t. Like, notice, they're not functions of t. So there's no t dependence in either of these. t appears only here. And what do I know about this term? I know that this is going to be exponentially decaying. It's going to be exponentially decaying. Okay. So this guy, once t zero and x zero are fixed for a particular t zero and x zero, these two are fixed quantities. This guy is definitely exponentially decaying. So as t goes to infinity, this is going to zero. And then rather fast. It's an exponential decay, right. And therefore, it doesn't matter what these quantities are. And the important thing is they don't change with t. And because they don't change with t, this entire thing is going to go to zero as t goes to infinity. Therefore, I hope you believe me when I say that delta is infinity. So what I can say is for all x zero in Rn, I have limit as t goes to infinity, norm x t is in fact zero. Okay. All right. So in fact, it is not just, well, it is definitely uniformly attractive. That's for sure. It's uniformly, but it's also globally attractive. Okay. It's not just uniformly, but also globally attractive. And this is what, right. So this is anywhere from the previous example. This is what is our next definition. Globally uniformly attractive. That is for all t zero and x zero, limit as t goes to infinity, norm x t is zero. This is in fact our next, was our next definition. Okay. This was in fact our next definition. So this is a rather strong property, rather strong property that this Messera system that we were considering was globally attractive or uniformly globally attractive. However, you want to say it. Okay. So it was not just this, but something more because there is the delta can be pushed to infinity. So it's a rather powerful property. Yeah. Excellent. Excellent. Excellent. So this picture, I mean, I know we didn't sort of discuss this, but this is a illustration of what happens when for the Van der Poel oscillator, just spend a minute on it before going further. Yeah. For the Van der Poel oscillator, we had this phase plane for a particular mu, right? But for all muses, this is the only difference that happens is as you change mu, this limit cycle sort of seems to change in shape a little bit. Yeah. Otherwise, all trajectories starting at origin still converge to the limit cycles. Okay. So the only thing that changes is the shape of this limit cycle. So the origin continues to remain unstable as we stated before for the Van der Poel oscillator. All right. Excellent. Excellent. So this is, yeah. So as it's mentioned here, the limit cycle begins as a circle and with varying mu becomes increasingly sharp at the corners here. And here you see it becomes sharp. Okay. As you increase the mu. So the nature of the limit cycle changes. So depending on your application, you may choose a particular mu, but the fact that the origin is unstable continues to be as is. Okay. Excellent. All right. So we've already seen stability, uniform stability. And now we have seen three attractivity properties. There was no notion of notice that I want to say this note. No such thing as global stability. Yeah. As far as the stability definitions go, there is no such thing as global stability. We saw stability. We saw uniform stability. So two different definitions. Yeah, because we are always requiring us, we always are required to choose an epsilon and correspondingly get a delta. Okay. So and delta cannot be greater than epsilon. We've already know that. So therefore it's not possible to say that delta can be arbitrary. Delta cannot be arbitrary. Delta definitely has to be less than or equal to epsilon. Okay. So delta spanning all of RN is not allowed. Right. Therefore there is no such notion as global stability. It doesn't make sense at all. Yeah. However, in case of attractivity, we saw three different definitions. We saw attractivity. We saw uniform attractivity. And we also have global uniform attractivity. Okay. And that is what is this definition. Okay. That is, it doesn't matter what your initial condition and initial time are. Their trajectories do converge to zero st goes to infinity. And we in fact, saw that the mass error system that we have is globally uniformly attract. Okay. So now we'll continue to look at this mass error example and talk about each. Now the rest of the definitions are rather straightforward. The rest of the definitions are rather straightforward and incrementally strong. Yeah. So let's look at asymptotic stability. What is asymptotic stability? It is simply combining stability and attractivity. Okay. So we already know that, for example, about the mass error system that this table, we already know that it is attractive. Therefore, mass error system is asymptotically stable. Okay. So the mass error system is in fact asymptotically stable. Okay. Let's go to the next one. The next one is uniform asymptotic stability. So we have added the qualifier uniform. So therefore, the qualifier uniform also goes into both of these definitions. Okay. That is it is uniformly stable and uniformly attractive. Now the question is, is the mass error system uniformly stable and uniform? Is it uniformly asymptotically stable? The answer is no. The answer is no. Why? Because it is uniformly attractive. Sure. It just proved that it is uniformly attractive because there is no delta and delta is infinite. Right. But it is not uniformly stable. It is only stable. Therefore, the mass error system. So this I will, I mean, I will start to make these acronyms asymptotic stability, uniform asymptotic stability, exponential stability, not, so the mass error system is not uniformly asymptotic stable, asymptotically stable, because it at the origin of course. So whenever I say something is stable, uniformly asymptotically stable, and I don't mention the equilibrium. Yeah. I'm just trying to shorten the sentence. Okay. Whenever you're talking about any of these notions, you're always talking about a particular equilibrium. Okay. So the mass error system is asymptotically stable at the origin. The mass error system is not uniformly asymptotically stable at the origin. So when I write it out formally, I have to write the whole sentence. Okay. Please keep this in mind. You have to write out the whole sentence. I'm not saying the whole thing occasionally because it's clear from the context that I'm talking about the zero equilibrium. Okay. So the mass error system is not uniformly asymptotically stable because it fails on one count. And we need both of these. Okay. Let's skip this exponential stability and go to the, because exponential stability is the strongest property. In fact, so it should come in the end. So this is the next property is global uniform asymptotic stability. Okay. And what does it require? It requires global uniform asymptotic stability. So it requires uniform stability and global uniform attractivity. Okay. Now it should be obvious to you that because the mass error system is not uniformly asymptotically stable, it cannot be globally uniformly asymptotically stable either. Okay. However, there is of course a definition somewhere in between, which is not really mentioned here, but I will still put it for a reference. Well, let me put it say here, global asymptotic stability. So this is G. So this is also, this is G A S. And what does this require? It requires stability because there is no uniformity plus globally attractive. Okay. Stability plus globally attractive. Okay. And you will note now that mass error system is globally asymptotically stable. Yeah, because it is stable. And it also is globally attractive. So this is not the best property. So whenever we have a system, we're always trying to find the best property that is satisfied out of all of us. Okay. So asymptotic stability, yes, as their system is asymptotically stable, but it is something more because the system converges from any initial condition to the origin. That is something stronger than just asymptotic stability, which is a local notion. Therefore, this notion of global asymptotic stability also exists, which is that this mass error system also satisfies this, that is that the origin is stable and also globally attractive. Right. So the mass error system is in fact a globally asymptotically stable system. All right. I hope that is clear. Excellent. Excellent. So we've done some interesting things today and we sort of try to summarize what we have done today. So we of course started off with talking about the Van der Paul oscillator. We saw its face plane portrait to sort of judge whether it's stable, uniformly stable or unstable. What we found was that for different values of mute, the limit cycle shape changes. But overall, the stability question has the same answer. That is the origin is always unstable. It's neither stable nor uniformly stable. Okay. Then we went on to talk about additional definitions beyond stability and these are related to convergence. And so these properties are called attractivity properties. Okay. So we talked about attractivity. We talked about uniform attractivity, global uniform attractivity, global attractivity. Then we talked about asymptotic stability, global asymptotic stability, uniform asymptotic stability and global uniform asymptotic stability. So we will sort of do this again a little bit next time. But what we were able to conclude for this mass error system, this example that we've been talking about the whole lot is that it is the best property that it satisfies is that it is globally asymptotically stable. That is, it is stable and it is globally attractive together. All right. So that's it folks. So we will meet again next time. So thank you.