 Hello, welcome to yet another session for NPTEL on nonlinear and adaptive control. I am Srikanth Sukumar from Systems and Control, IIT. We are well into our fifth week and we have been looking at rather interesting notions on how to prove convergence of parameter identification systems. We are of course motivated by these very nice background images that come up behind us and this one is on a spacecraft which is orbiting the Earth and we hope that the algorithms that we design and the algorithms that we are analyzing help us to develop autonomy for systems such as these. So without delaying further, let's sort of look at what we were doing until the last session. So we moved on to trying to prove exponential stability of a standard parameter identification system. So this system 7.1 that you see here is actually very, very standard in parameter update loss. So parameter update loss and the derivatives typically look something like this in a lot of applications. So we are trying to use persistence of excitation of the signal 5 in order to conclude that you have in fact stable parameter convergence. That is parameters converge to the true values if you have some persistently exciting signal and so on. So in order to prove that we of course took a Lyapunov candidate, a valid Lyapunov candidate and then we completed a V dot which we found was only negative semi-definite. But since we are interested in using this alternate exponential stability theorem, we are not too worried. So we actually take the integral of this V dot and this is what is this expression. So I am going to highlight this again so that we can come back to this later. So we are going to come back to this later. So our subsequent attempts, like I said, aren't there to try to get a bound on this one. How did we begin? We started from connecting the persistence of phi to UCO of this system. So we showed that this system is in fact UCO because of the persistence of phi itself. So this is where we went. So we want to first start from here. And so when we mark this, this is lecture 5.5. So now that we have established the UCO of a particular system, we want to use the idea of UCO being invariant under output injection in order to get to our original system. That's really the idea here. So what do we do for that? We consider a rather interesting output injection term, which is basically this k of t is minus alpha phi. So of course this is a vector. This is a vector of dimension n. Therefore this k, the gain itself is also a vector of dimension n. So then if we compute, so one of the requirements for the output injection theorem was we had to compute a moving average bound. Suppose we do that. So and that is what it is basically integral from t to t plus cap t of this quantity. And I remember that we do have some kind of persistence bound on this quantity. There is an upper bound on this quantity. And that's what we seek to use. It is well known that, I mean we use a rather well known equality which is that non or phi transpose phi or any vector for that matter is the same as trace of phi phi transpose. And this of course is non phi squared. Correct. This is a very standard equality. So here we have a norm of phi square because the alpha square can be really pulled out. It's just a scalar quantity. And the norm of phi square can be written as the trace of phi phi transpose. Why do we do that? Because we have an outer product here. We want to see an outer product here. That's it. So trace is just the sum of the diagonal elements. All right. So now if you look at this guy, you know that this is upper bounded by mu 2i. So the trace of this is upper bounded by mu 2 times n. Right? So why? Because again we use the fact that trace of mu 2 times the identity matrix is just n mu 2. Because this is an identity matrix of dimension n. Yeah. So if you just sum mu 2 on all the diagonals, you just get n times mu 2. And that's what we get here. Alpha squared n times mu 2. All right? So now that we have a nice upper bound on this sort of the output gain. Yeah. We know that A plus Kcc is also used here. Right? So in this case, I mean in this case basically K is just the K of t, the small K of t. And C is basically just phi transpose of t. Yeah. So if I substitute. Right? So what do I get? A was 0, of course. Right? Just from here. A is 0. C is phi transpose. And K is minus alpha phi. So if I just substitute here, just from my injection, output injection result, it says that if you have a nice moving average bounded K, then you cannot destroy the UCO property. If you do this output injection on the dynamics, then essentially what we get is A plus Kcc is UCO. And if I just substitute all these quantities in here, I will just get that minus alpha phi phi transpose, comma phi transpose is in fact UCO. Okay? So what do we do? So we go back to this nice integral. I am going to make this smaller so we can see both. We go back to this integral in equation 7.3 and that's what is rewritten here. Yeah. It's rewritten in the entire form. Here I had just, here it's just written as a phi transpose x whole square. But here we expand it. Here we expand it. Right? That's all. And not just expand it, what else do we do? We write x in terms of the state transition matrices. Right? So which is what this is. I have written x as phi st x t. Right? So x is phi st times x t. This is basically standard way of computing solutions for linear time varying systems using state transition matrices. Right? So if you want the solution at S, you take the state transition matrix from S to T and then multiply it by the value at x t. Now the important thing to remember is that, and I do that on both sides of course. Right? So the important thing to remember is that these two quantities are now independent of the integration variable. And therefore I can move them outside the integration. That's what we do. And what you're left with inside is again, our favorite UCO gradient. Yeah. So this is what we like. We want to see the UCO gradient everywhere. Right? Why? Because we are proving some kind of UCO condition and we're going to leverage it to bound this thing. Yeah. So remember, so this is what we have for our expression with the UCO gradient. And I also have that this system is now uniformly completely observable. Yeah. Which was obtained by an output injection. Right? Through a nice game. Yeah. Great. Now what is this system? So that's what we want to do. We want to write this system out. Yeah. What is this dynamical system? This dynamical system is just this and this. And because this is the A, this is the new C. This is the new A matrix. This is the new C matrix. And it's not the old AC, but this is the new A and new C matrix. And so this is the dynamical system and this is the output matrix. And what do I have? I have a system which looks like this. I have a system which looks like this. Now notice what is this system? This is in fact the system that we are trying to prove exponential stability of. Right? So this system is exactly this thing. The same system. All right. It's exactly the same system. Okay. And this is an output. This output is of course, you can see that this is not a real measurement. Yeah. I mean, I don't think anybody would be able to claim that if I use the sensor, I would measure something exactly like this. So this is not like a sensor measurement. This is sort of an artificially created output so that you can complete your stability. That's it. Yeah. Because we need an output injection. All right. That's all the purpose of using this one. Yeah. This why is not necessarily real measurement from a sensor. So let's not sort of, you know, we find ourselves and try to think that this is possibly some real measurement. It's not. I promise that it's any kind of a real measurement whatsoever. All I would say is it's just a construction per purpose of analyzing stability here. Yeah. So the only thing we need to focus on is sort of this system and that's good because this system is matching the system for which you want to prove stability. Now, what is the new CEO Grammy and corresponding to this? I know that this system is already you see. Yeah. With constants beta one bar and beta two bar. So what is the use your Grammian for this? The use your Grammian is T to T plus gap T. P transpose s tau. Sorry. P transpose SD C transpose, which is five s C, which is five transpose s and the state transition matrix again. Yes. Yeah. Let us look at this. And here the five is corresponding to the system. The five corresponds to the system. Yeah. The five is the state transition matrix of the system. Right. Now, if you look at this carefully, this expression. Right. So what do I anyway, before we do that, we know from the use your criteria that this is less than equal to some beta one bar identity and this is than equal to beta two bar identity. And so I have a bound on both sides. Yeah. Because of the use your condition. Now, if you look at the Grammian expression here, you see that it is exactly identical to here. Yeah. This is exactly the same. Yeah. Because five here is the state transition matrix corresponding to the original dynamics, which is this. And so is this. This five is also the state transition matrix corresponding to the same dynamics. Right. And the matrix in the middle of five and five transpose is of course the same. Yeah. So now the fact that these two are exactly identical helps us to bound this guy because of the use your condition. I have a lower bound on this. And because there's a negative sign, that is what we need the lower bound. So what do we have? We use this lower bound to actually upper bound V dot as minus two alpha beta one bar. No, it makes the square. There's one number in this. So to earlier try me or maybe I will just number it. Seven point three five. Let me call this equation seven point three five seven point three five. All right. So great. So I have an upper bound on the integral of V dot Yeah. And this exactly of the form that we require in an alternate exponential stability theorem. Right. And so great. I have in fact that the equilibrium X equal to zero for this sort of parameter identifier system is in fact exponentially stable. Right. And this of course we have leveraged persistence of excitation. You see you and you see you under output injection. Yeah. So and as I have mentioned before, the why that we actually see is purely for analysis purposes. Yeah. There's no role of why or there is no real measurement which can possibly resemble why. All right. So it's purely for the purpose of analysis. So yeah, just think of it as an artificial construction. Yeah. Now there are also extensions of these, which is again something which appears very commonly in model reference adaptive control parameter identifiers. Right. And that is this. This is not a time invariant system. So this is linear time varying system. Yeah. We look at this linear time varying system, which is something that is very common in model reference adaptive control. Error here is basically some kind of a tracking error or because it's a model reference system. So he is the error with the model reference typically. So that's what is mentioned here is that called. No. Yeah. So tracking error E and then you have a parameter estimation error theta table. Yeah. And usually this is the dynamics that you find. You will find like a sort of interesting dynamics where you have an a is basically coming from the original system itself. No problem. B times five is basically the dependence on the parameter. Yeah. Because because there's a parameter error that is an unknown parameter. So there's no way I can cancel the parameter term. So I sort of do the best. And so I get something in the parameter error and I get something in the update law, which is depending on the E on the tracking error. And so this is very standard, very standard in model reference adaptive control. When we get to that stage, you will see that our system turns out to be of this form. Okay. Now further, if you consider, so therefore we are considering this linear time varying system, right? Further, if we have AP to be a controllable pair, very standard assumption and AC to be an observable pair, again a standard assumption. Further, if there exists, you know, for a given positive definite symmetric Q, there exists a positive definite symmetric pieces that this Lyapunov equation is satisfied and this PB equal to C transpose if both of these happen. And phi is absolutely continuous in phi, phi dot. Then you have that the origin of the system is uniformly globally exponentially stable if and only if phi is persistently excited. Okay. So this, a lot of different points here. So these two, I would say are like standard assumptions and linear systems. Standard assumption in linear observer and control. So nothing special about this. This is very standard assumption in linear systems observer and control design. So it's not like there's something too new that we are introducing here. The next condition here is like exactly like a Lyapunov equation corresponding to AB whole bits. Yeah. So again, a matrix is a Horowitz matrix. Yeah, that's a stable matrix. Then this sort of an equation is always satisfied. Yeah. If A is a stable matrix, a Horowitz matrix, which is usually the case here. Yeah. Then this sort of an Lyapunov equation can always be satisfied. This is the sort of converse stability theorem for linear systems. Yeah. Converse Lyapunov theorem for linear systems. Now this condition is basically like a standard, what I would call a matching condition. This is sort of additional. This is a matching condition. And of course this, we have not really defined what absolute continuity is. So I would ask you to look up the definition. Yeah. But the purpose of feeding absolute continuum, absolutely continuous is just so that persistence, excitation, et cetera are well defined. Yeah. Because in order to do persistent excitation, you need to take an integral and so on and so forth. So essentially we require phi to be absolutely continuous and we also require phi and its derivative to be bounded. So this is a standard. This is again like a more, I would call a regularity assumption. So that things are very defined. That's all. Yeah. So that things are very defined. All right. So then if we have these four conditions, I would say relatively reasonable conditions. Then you can actually claim that Z equal to zero. That is the zero equilibrium is in fact uniformly globally exponentially stable if and only if phi is persistently excited. So this you can see is an extension of the previous result. The nice cool thing here is that you, when you say Z, Z is basically both these states together. You are saying not only do the tracking errors converge, but the parameters also converge. Yeah. So this is not very common adaptive control. Yeah. Let me be honest. In a lot of cases when we solve adaptive control problems, you and we will do that very soon. You will start to see that parameter convergence is not usually guaranteed. Yeah. All an adaptive control theorist will tell you is that in the presence of an uncertainty by designing a parameter estimator, what we can guarantee is that your tracking error, that is your control objective, which is say for example, your robot wants to move with some kind of sinusoidal shape. Yeah. So basically this kind of a tracking objective will be precisely met. No problem. Even with the parameter uncertainty. But identification of the set parameter may not happen. Yeah. And that is where there is this requirement of persistence of excitation and so on. And also here to the theorems that you have and the conditions that you have or the theorems that you have are rather limited in terms of, you know, what kind of systems they can handle. Because if you notice the two results that we saw are in fact only for linear systems. Yeah. It is not really that easy to come up with such conditions when your dynamics is nonlinear. All right. And that's what we are usually dealing with. We're usually dealing with nonlinear systems. Yeah. So there is still a lot of open problems in trying to talk about parameter convergence for nonlinear systems. Yeah. So adaptive identification of parameters. It's precise identification of parameters is not guaranteed. Yeah. There are only these few cases where in fact you have some results which under which typically work under persistence of excitation and they guarantee that, you know, your parameters will also converge to the true value. Right. On top of tracking error is going to zero. So tracking error is going to zero is guaranteed for adaptive control. And that's what most practitioners are interested in anyway. Yeah. They don't care about really identifying and noting down parameters. Sure. But in some cases, yes, you are also interested in doing that because once you identify the parameters precisely enough and if they stay fixed, you can implement simpler controller subsequently and not need to implement adaptive controllers. Yeah. So this is sort of what, you know, is the conundrum in adaptive control. Of course. Yeah. It provides a nice solution, but it also leaves something out there open to work. Yeah. Great. So one of the other points, the final points on this additional result is that this condition three that you have, that is this sort of condition, it actually helps you to, you know, ensure that this Lyapunov function works. Okay. And it's not very difficult to verify. I mean, we can actually take a very nice, simple derivative here. So, so V dot from here becomes E transpose P with E transpose P E dot. So this is, I believe, a constant previous, right? Because A is absolutely, A is a constant matrix. So we have, yeah, you will have, well, I'm going to write it out. E transpose P E dot plus E dot transpose P E plus theta dint of transpose theta dint of dot. Okay. And this, if I substitute for E dot from this guy here, right? So what will I get? I get E transpose P A E plus B phi transpose theta dint of plus same thing, E transpose A transpose plus theta dint of transpose phi B transpose times P E plus theta dint of transpose and theta dint of dot is just minus phi C. So minus theta dint of transpose phi times C. As usual, I have sort of gotten rid of all the time arguments, right? So if you look at just this two together, this becomes E transpose P A plus A transpose P E. Yeah. And then I'm left with, right? All right. Then I'm left with plus twice. In fact, I'm going to combine the theta dint of transpose phi B transpose E E times 10 E here, right? So I don't think we should have an half here. And so there should be a minus twice here. And so this is minus twice theta dint of transpose phi times C times E, right? So this, of course, is nice minus E transpose Q E from early Avalon equation. And these two, in fact, cancel out, right? How? This is by virtue of the fact that we have this set of a condition P B equal to C transpose. So if you see this, P B equal to C transpose. So C is equal to B transpose. So these two are the same. So they actually cancel. So we are left with this kind of a minus E transpose Q E. And this is our usual starting point, if you remember. This is where you have B dot is negative semi-definite. And you start to integrate B dot. And you go on with the proof. So if you are not completing the proof for this case, yeah, you can look up the proof. This is, this work is by Narendra, sometime in 1977, 78 and the year. Yeah. So you can look up the proof if you wish. But this is starting point, if you remember. We have a negative semi-definite V dot. We start to integrate the V dot. Start to use P, UCO and all such conditions on phi. And then we sort of try to go on from there. So I would strongly recommend it to try to see how you can complete this. Great. So what did we look at today? We completed the proof for parameter identifier convergence under persistence of excitation. We saw how we could leverage the PE, UCO under output injection, alternate exponential stability theorem. So basically all the neat little results that we looked at from the beginning of this week, we could leverage that to actually prove that you have this parameter identifier convergence. We also saw an extension of it, where you have this sort of a system which contains the tracking method E from the model reference adaptive control setting, which you will look at in the future. And also the theta tilde. So you have both the E dynamics and the theta tilde, that is the parameter error dynamics. And we claim that theta and E tilde are exponentially, uniformly, globally, exponentially stable at zero. If this, again, this phi type of a gain term is phi T even for persistent real estate. And we saw what kind of conditions are required for that. So anyway, we will continue more on this way. I mean, try to wrap up the sort of material in the next session. And yeah, that should be an interesting special. All right. Thank you.