 Hello everyone, welcome to another week of our NPTEL on nonlinear and adaptive control. I am Srikanth Pumat from Systems and Control IIT, Bonway. We have as always this very nice motivating image on our background of this SpaceX satellite that is orbiting the earth and we are well on our way to developing algorithms and analyzing them that will eventually help drive systems such as this satellite autonomous. So without delaying too much, we want to give a quick recap of what we were doing last week. So last week was sort of an excursion on, you know, so last week, if you see here was sort of an excursion on the notions of persistence of excitation, all right. So we wanted to use persistence of excitation in order to prove stability of certain parameter identification systems, right, and that is what was the focus of almost the entire week. But of course, it was not just a persistency of excitation itself, we introduced several new notions such as, you know, these alternate exponential stability theorems, then we, you know, sort of connected persistence of excitation, you know, so we wanted to make this connection to exponential stability. So we introduced the notion of uniform complete observability and of course connected that to exponential stability. And further we went on to talk about the notions of exponential stability for linear time varying systems, which is what commonly appears in parameter identification for linear system. So in model reference adaptive control, this is what you see, we will see this. And then of course you see under output injection and then we went on to prove, you know, how persistence allows us to guarantee that parameters converge to their true values, all right. So of course, we also saw more general results of integral Emma and on parameter varying systems, yeah, we sort of reface this by saying that such parameter varying systems are in fact more common than purely time varying systems in adaptive control. And we will of course, again, look at examples. So we wanted to show, you know, this, how this, you know, parameter varying systems can also be analyzed under this new notion of lambda uniform persistence of excitation, all right. So what we will do today is, I mean, starting today, of course, is start to look at our first adaptive control program, all right. So you'll spend this week actually introducing the notions of adaptive control to all of you, right. So that is the idea here, right. So but before we go on to begin this, I wanted to say a little bit more about how we were doing the proof in the very end of the last week sessions of stability of parameter varying systems. So I want to sort of give an alternate approach to doing this proof, which makes our life a little bit more easier. Yeah. So that is sort of what I want to look at for a few moments first. And then we will move on to our material on adaptive control. All right. All right. Excellent. So let's begin. Yeah. So if we go back and you look at the system we were analyzing, this was this very similar to this time varying system. The only thing additional thing here was there was an additional parameter lambda. Yeah. And we want all these nice exponential stability properties for the system, which do not depend on this lambda anymore. We want this lambda independent. So although this lambda is a parameter, which means once it's this once the value is known, it's fixed. But in a lot of circumstances, we want to make claims about a system for many different values of lambda or for a range of values of lambda, all right. And we don't want our stability properties or convergence properties to be impacted by values within this range. So even if the value changes from say one to five, for example, I don't want the stability properties to alter within this range, right. This is what is the purpose of doing para uniform parameter uniform properties. All right. So this is the system. It's a simple scalar system. We already analyzed this kind of a system where lambda was not present. But when lambda is present, we need this additional mechanism of this integral lemma and so on and so forth, lambda uniform persistency of excitation. These were already introduced. However, I want to slightly change how we analyze it. So the general integral lemma required two things, that the max of the infinity norm of the signal and some P norm is upper bounded by a constant times the norm of the initial state. Yeah. And that's what we are going to try to use this because this gives us lambda uniform local exponential stability, which means that uniform local exponential stability holds for all lambda in some domain. And of course, we can get a global version of course. Yeah, you can get a global version if this C exists for all initial conditions and all it as simple as that. So we want to we want to try to satisfy these sort of conditions for this system. So how do we go about it? So how we started about started doing it last time. Yeah, is that we sort of took a function V. Let's not worry about it being a Liapolo function and so on and so forth. Just a function V, which you can see is nice, radial and bounded and all that. But we're not going to comment on the Lyapunov theorem because we're not really using the Lyapunov theorems here. So we take a derivative and it's pretty straightforward. It just comes out to be minus a squared times x squared, which I know is negative semi definite. Now, because it's negative semi definite, I know that V is non-increasing. And this immediately means that the square of x z t is less than equal to square of x z 0. And from here, I immediately get a relationship between the infinity norm of x and the initial condition and the initial value. This is the first requirement. So the infinity norm of x I know is already bounded by norm of x 0. So that's what I have shown. So now, in order to go forward to the rest of the steps, I'm not going to use these. OK, I'm not going to use this. Yeah, I'm not going to use this. So I'm going to directly integrate this. I'm going to try to directly integrate. Yeah, all right. So I'm going to write it in a sort of different color so that this is evident. So I know that this is equal to minus twice a square times the V. So because V is x squared by 2, so V dot is minus twice a square times V. So now, this is a nice scalar system. And I can integrate it. What is the integral? So if I integrate it, I will get something like dV over V is equal to, and this is integral from 0 to, say, some V of t. Yeah, and this is integral from 0 to t. I will get something like a minus 2 squared d lambda dt. Notice lambda is fixed because it's a parameter. So once I've chosen the value of the parameter, it is fixed. So therefore, the integration is just with respect to time. So what do I have here? So if I solve it, I will get something like Vt is equal to, let's see, it's actually not 0, but V0. It's V0 t to the power minus twice integral 0 to t a squared t lambda dt. And this is where I start to use my measure lemma, is where I start to use my measure lemma. What does it say? I know that this a is lambda uniform p. And if there is a lambda uniform uniformly persistent signal, then I know from this measure lemma that there exists a lower bounded finite time over which the signal has a nice lower bound value. OK? So if I actually expand this, if I actually try to expand this, this will be something like this is less than equal to V0 e to the power minus twice 0 to sum dt is squared t lambda dt. So all I've done is I've written this small t as some kt, yeah, plus delta. So we did this even before. t is kt plus delta. And ignore delta. I can do this because I have used this unique quality here. And once I have done that, I use this on each such interval. On each 0 to t and t to interval, I use this kind of an integration condition. So I'm going to write it again. I'm going to write this again as less than equal to V0 again. e to the power minus 2 summation over i equal to 1 to k. Right? Integral i, and this will be i minus 1 t to i t a squared t lambda dt. OK? And this quantity is simply equal to V0. So I'll just retain less than equal to just. So that we're not confused. The inequality is, so this and this are just equal. So this is, well, actually, no, there is still a less than equal to. This is fine. So this is less than equal to e to the power minus 2 summation i equals 1 to k. Integral from i mu i is squared t lambda dt. OK? So notice what I'm doing. What I'm doing is I'm taking this interval right? See, notice in any interval of t to t plus cap t, OK? I'm saying that there is a small sub interval over which this quantity is at least this much value. And this is exactly what I'm going to use. OK? So I'm just going to look at the sub interval. And why is it OK to look at only the sub interval? Because this is a non-negative quantity. And so if I discard the rest of it, the rest of it is going to make at least some non-negative contribution, right? Therefore I can, even if I don't consider the rest of the interval, I'm OK, all right? So I'm OK with considering a sub interval. And now that I've considered the sub interval, I know that on this sub interval, I have this nice lower bomb. I have this nice lower bomb, all right? And that's what I'm going to use. So this is mu over 2t phi m and all this, whatever this quantity, OK? So this is basically going to be less than or equal to v0 e minus 2 sum over i equal to 1 to k. This is going to be, I hope you get this expression again, mu over 2t phi m whole squared, mu over 2t phi m whole squared times the measure of this, right? So basically this is a lower bound. This quantity is a lower bound. So I can pull this out. I take the lower bound, I pull this out. So the integral is just this dt, which means it's just the length of this interval. And that length of this interval is at least this much. So this, again, I copy 2t mu over 2t phi m square minus mu. So this is just t mu divided by 2t phi m squared minus mu. So this is what I get, OK? And notice, this summation means nothing much because there is no dependence on i here. So this summation can be erased and replaced with k because I have k copies of the same thing, right? Because I have k copies of the same thing, all right? I hope this sort of makes sense. I hope this sort of makes sense. I'll give you all a few moments. Just think about it. And that this sort of makes sense, OK? All right. So now if you see, I have a k and a t here. Yeah, this is k and a t, which is equal to t minus delta. This is equal to t minus delta. So this k times t is just t minus delta. So I can actually write this as such. This is just equal to v0 e to the power minus 2 times. I'm going to call this some k1 squared and the rest. Honestly, I'm going to call this as some constant, yeah? Gamma times t minus delta. So I have combined all everything else into one constant gamma. You can do that, all right? I can do that, OK? And this is equal to v0 e to the power gamma delta times e to the power minus gamma t, OK? So what have we shown? So what have we effectively shown? Let's look at this carefully. We have shown that x t squared, which is v of t, is less than equal to x0 squared from here times e to the power gamma delta, which is a constant, times e to the power minus gamma t, all right? Excellent. Now it's not difficult to integrate both sides. I'll integrate both sides with respect to time. So integral x t squared dt from 0 to infinity has to be less than equal to 0 to infinity x0 squared e gamma delta e minus gamma t dt. Here only this is depending on time, right? And what will I get here? I will just get e to the power gamma delta divided by gamma times 1 minus e, just a second, we can be careful. This is going to be e to the power minus gamma t, minus gamma, so this is just going to be 1. And I'm going to get x0 squared here, right? And what is this quantity on the left? This is nothing but the square of the L2 norm. So this is essentially what I've computed, is actually non x2 squared. Yeah, this is just non x2 squared is what I have computed, OK? So I have also gotten then what? So I'm sorry, so this is, yeah. So what I've proven is that norm x2 squared is less than equal to e to the power gamma delta divided by gamma norm x0 squared. Because x0 squared and norm x0 squared is the same, they are scalar quantities. So I've also gotten a relationship between a p norm and upper bound on the p norm, that is the 2 norm in this case, and the initial condition norm, which is, again, what is this requirement? So we've also gotten a c here. So I have bounded the infinity norm by some value multiplying the x0 norm. And the p norm, that is the 2 norm in this case, also by some multiple of the initial condition norm. And that is what we need for your integral lemma. And so this is enough to prove that you have, it's enough to prove it. So this highlighted quantity along with this to help satisfy the integral lemma. And this is enough to prove that your system is lambda uniformly, locally, exponentially stable, OK? In fact, in this case, I believe we can even claim lambda uniformly globally exponentially stable. Because I don't think it depends on, there is any, it's agnostic to the initial condition actually, all right? Excellent. So this is what sort of wraps up our discussion on persistence of excitation. All right, so this is sort of what was remaining. We had done it with a different method. And there were some queries left there on how we were integrating this and so on. So I wanted to give you an alternate approach. In fact, this is very similar to what we did for the purely time varying case where there was no parameter. So it is not too far from what we had done. All right? Excellent. Excellent. Excellent. So now we can sort of move forward and look at what is adaptive control? And how do we do adaptive control for a very basic system? And that's the idea. So basically, we had already mentioned that adaptive control is the notion of designing a feedback control for system when some parameters are unknown. And we, of course, want to achieve some tracking object. We definitely want to achieve some tracking objective. So you can think of, again, we can think of robotic systems or space systems that's like our nice motivational image. Suppose your spacecraft is you want the spacecraft to track a particular orientation trajectory or even a trajectory or a rate on the orbit. The idea is that you may not very well be able to model ever the entire spacecraft. It's not very easy to test a spacecraft while it's rotating on the Earth because of, in general, the sizes that are involved. So we usually rely on some kind of robustness in our controllers in order for our standard orientation controllers to work. So adaptive controller typically does away with this by adding an estimator or a parameter estimator. For example, if the inertia is unknown because of the fact that you did not model the spacecraft very well, you can apply an adaptive controller, which is agnostic to this error in the inertia parameter. And it will still achieve perfect tracking. And so this is pretty good, achieves perfect tracking. Yeah. So anyway, so this is where adaptive control shines. And we want to start off with like a first order nonlinear system. Yeah, so what is the first order nonlinear system? It's a system of the kind 2.1. Yeah, here there is an unknown parameter. So this theta star is typically an unknown parameter multiplied by some nonlinear function of state and time. And then there is also the control effort, which we get to design. So this U of t is what we get to design. So more and more we start to design things. And of course, there is some initial condition. We keep things very simple. It is a scalar system. So the states evolve in reals. The function f also maps the state and time to real numbers. And similarly, the control is also real number. And theta star is some unknown constant parameter. So this is the sort of setup you will always have an adaptive control. So basically, you make a few different assumptions. So we will of course talk about the assumptions subsequently. The first thing is that the control objective is typically for your state to track some smooth bounded trajectory r of t. Why does it have to be smooth? Because you don't want very jerky motion. For example, again, I think of a robot or a satellite. I don't want them to do a lot of jerks. Yeah, I want them to move smoothly. So it's not very unnatural to think of smooth trajectories. And of course, you want bounded trajectories because again, we don't want the states to become unbounded typically. I mean, those would be rather very unusual applications where you want the states to become unbounded. And those are usually considered on a case by case basis. So the objective is for the state x to track the trajectory r of t. So what are the standard assumptions? The first is that the model is linearly parameterized. What does it mean? It means that the unknown parameters appear linearly in the state space model. So in this example or the first order scalar system, you already see that the parameter is appearing linearly. That is the first assumption. Now, one may think that this is a rather restrictive assumption. But honestly, it's not. In a lot of examples, then we may look at some example. It's possible to deal with nonlinear parameters also with this assumption. Instead of identifying, for example, if the parameter appears as theta squared instead of a theta, what an adaptive control will do is to identify theta squared rather than trying to identify theta. And therefore, the system is still linear in the parameters. So a lot of cases can still be handled of nonlinear. So that's not such a big deal. So the second assumption is that the unknown parameters are constant. So this is, again, one might think of a rather restrictive assumption. Again, in a lot of cases, this works out quite well. When your unknowns are time or state depending, what we usually do is we consider basis functions. So we assume that this function is parameterized linearly as a linear combination of basis functions. And then we try to identify the weights of the basis functions, which are constant. So this is, in fact, the basis for neural networks, where you have basis functions and you just try to identify the weights using an adaptive control type design. And therefore, again, these weights are constant. So in a lot of these cases, we identify constant unknown parameters only. And that's what we will look at in the entirety of this course. Of course, there are results which are more research oriented on slowly time varying parameters. So it's very well known that if your parameters vary during the operation very slowly, or for example, the parameters change maybe once in a day or once in two days, an adaptive controller is still able to re-adapt to the new parameter values. This is, again, something rather crude, it's like a self-repair mechanism. If you have a system for which there's some kind of a fault, some issue, and then you have a parameter which suddenly goes bonkers and changes values, an adaptive controller, if implemented, will still adapt to this new value. And it has been shown in experiments and in theory that you still get good performance. This is something that cannot be claimed by conventional non-linear. So this is, again, something special. So we assume that the model is linearly parameterized, and we assume that the unknown parameters are constant. So this is the setup. So we will start from the next session of trying to design a controller, an adaptive controller for the system. What is an adaptive controller? It involves designing this controller to do tracking and also an estimator for this parameter feature stuff. All right, so that's the idea. Excellent. So what did we look at today? We spent most of our time, to be honest, trying to analyze this parameter in a varying system, right? Using motions of the general integral lemma and lambda uniform persistency of excitation. And we started off on the first-order scalar system for which we will start designing an adaptive controller in the subsequent session. All right, so this is where we stop today. Thank you very much for joining us.