 Hello everyone, welcome to yet another session of our NPTEL on nonlinear adaptive control. I am Shri Khan Sukumar from Systems and Control IIT Bombay. So we just started our 11th week of lectures on nonlinear adaptive control. By now we have covered a large variety of methods to design and analyze algorithms that will drive autonomous systems robustly such as the SpaceX satellite that you see in the panel. So I hope you have found the exposition of nonlinear adaptive control interesting and I hope you will continue to be with me until the end of the course. So what we were doing last time is that we started to look at a particular method of adaptive control which allows us to sort of relax the persistency of excitation condition. So as we sort of discussed a little bit last time, persistency of excitation is a rather strong requirement for identification which well until recently was thought to be sacrosanct. So now what we want to do is to do this kind of real-time parameter learning without having persistence of excitation but with a weaker sort of vendors. So in order to do that we I mean in order to understand that of course the reference was this work by Cheyenne Roy, Shubhendra Vasim and Indra Khar. They of course have a lot of subsequent work in this direction also which I would strongly urge all of you to look at very interesting stuff. So one of the well I mean we started to look at a very simple setup of course for illustration. We like to see these basic problems but we've also seen in the past and I hope you are convinced by now that even if I give you a vector problem things are not going to be significantly different. So we started looking at a single integrator system and a single integrator tracking problem honestly speaking we didn't even go to the tracking aspect of things until now. The first thing we did was we wrote everything in a standard regressor parameter form that is y theta equal to u and because we wanted to write it in this form we had to resort some to over parameterization so we had to introduce one also as part of the unknown parameter vector and there was also x dot in the regressor. Now what essentially this x dot signifies is the derivative of the state which is typically not assumed to be measured but then we also showed that this filter itself is implementable by using a simple integration by parts type of a scheme and we even wrote out a nice formula for this right this integration by parts kind of a scheme okay. Then very reminiscent to what we were doing in you know projection based adaptive controllers all of these were motivated by slot teams work in the 80s is that we sort of tried to connect the filtered variable so we sort of wanted to write the equation in terms of the filtered variables and again in a very you know reminiscent of our projection based adaptive control scheme the filter equation turns out to be very similar to the original equation yeah and this also looks like a standard regressor parameter form which is that the y and u have been replaced by the yf and the uf. So slotting already showed that these do improve performance we also saw the same the creation of an attractive invariant set and we did the projection based adaptive control but it does not allow us to relax the persistence assumption which is why these authors who we are referring to now proposed addition of a second layer filter alright so this is where we sort of start today alright so the second layer filter we saw the structure already pretty standard it's like a you know yi f dot is governed by minus yf transpose yf with zero initial conditions and you yf dot is governed by yf transpose what uf with zero initial conditions again so one thing to note is that yi f is always going to be positive semi definite okay I think we have to change there's a sign issue here but it is actually a plus otherwise it's not positive semi definite but negative semi definite right by construction this is positive semi definite right because you start with a zero matrix and then you just keep adding a non-negative definite matrix non-negative definite symmetric matrix as the derivative and therefore you of course have positive semi definite yi f we do a similar exercise now we try to come up with the equation in terms of the yi f's and the ui f's right so in terms of second layer filtered variables right and in order to do that I as usual multiply both sides here by theta right and if you look at this this yf theta you know from our previous analysis is already uf by substitute the same here and again you notice that yi f theta zero is zero right and if you see this equation and this equation are exactly the same it's same initial conditions just the variables are differently named so here you have uif here you have yif theta right so therefore the by uniqueness of solutions of ordinary differential equations the solutions of these two also have to be the same which means that yif theta is exactly equal to uif alright so this is again very similar to the previous equation in fact as you can imagine the authors were smart enough to construct this so that such property such a property does hold okay that's the whole idea anyway okay so the construction is precisely so in order for this kind of an equality to happen okay great now that we have designed these two layer filters we are now going to actually look at the control problem which we sort of neglected until this point right great so what is the error dynamics its e dot is ex plus u minus r dot right and of course because we have new parameter theta we want to write everything in terms of theta so we write this as z times theta plus you minus and now in accordance with our standard certainty equivalence type methods we propose our control as minus z theta cap plus r dot and a nice negative term k right with some positive gain k right and with this what we will get is that e dot is minus k e minus z theta tilde alright so as usual we get this nice error term right now the interesting thing is the way we specify the parameter update law has no connection to any Lyapunov analysis right so this is again similar to the projection Bayes method but there it was motivated in a different way the choice of the update law here the choice of the update law is of course motivated in a different way in fact it's completely decoupled we just have two terms so here mu f and mu i f are some positive gains some positive scalars if you may and then you have y f transpose u f minus y f theta cap and u i f minus y i f theta cap right now if you use the fact that u f is y f theta and u i f is y i f theta yeah in fact the filters were constructed very smartly so that such a thing holds and if you substitute this here and this guy here you will get this mu f y f transpose y f theta tilde from here and mu i f y i f theta tilde from here right where of course your theta is sorry theta tilde is theta minus theta cap okay so this is a rather cool thing why why is it a cool thing one thing you already see is that i did not have to i mean even in the sigma epsilon modification we introduce some term in theta cap dot right so by the way i mean before i go further here we had a theta cap dot and here we have a theta tilde dot which is minus theta cap dot okay so that is why you have a negative sign here all right that's it that's why we have a negative sign going from here to here as simple as that so now um in the earlier sigma and epsilon modification type designs if you notice theta cap dot and also theta tilde dot of course did contain a term in theta cap right that was the whole idea behind it sigma modification and epsilon modification that you have this a cap type term yeah of course if i write a tilde dot also i will still have an a cap term okay but look at what happened here by virtue of our very very neat filter constructions i don't just have a theta cap i actually have a theta tilde here yeah theta tilde dot equation contains a theta tilde from this term and also from this term and not just that i already know why is already positive semi definite so this is already a non positive term here so something really nice so unlike sigma epsilon modification where i only had a theta hat in theta tilde dot equation here i get a theta tilde and a theta tilde dot equation right which makes this like a very very nice evolution with a very high chance of being asymptotically stable right and why was this possible this was possible or this was made possible only because of this kind of a relation if we did not write it in this regressor parameter form initially which was why theta equal to u we did not do this over parameterization i would not get u f equals y i f theta or u i f equals y i f theta right and because i did that and notice the u f is implementable because it is just a one filter one layer filter of u and u i f is just filtering u u f so obviously these are all implementable quantities so i have used exactly implementable quantities nothing unimplementable but because of the regressor parameter form standard structure y theta equal to u which i started with i could get theta tilde zero and now this is looking like a very nice system it looks like theta tilde dot equals minus k theta tilde just in a simple case right this anybody i mean most of you have seen non linear control now for almost 11 weeks more than 11 weeks we'll understand that this looks like a very promising system okay which can be asymptotically stable alright excellent yeah excellent this is i mean so that's why i marked this this is the magic this is where the magic has happened yeah and this magic has happened because we started with the regressor parameter form and then we constructed smart filters yeah not me but he strokes from ideally constructed smart filters okay so great then we of course i mean do some diapunov analysis yeah um and how do we do it be this is very standard i take an e squared by two yeah i have nothing to choose anymore i just need to do the analysis nothing remains to be chosen anymore only the analysis part remains correct great so i hope you understand so then i put us put a constant lambda this again something that you should remember from your projection based adaptive controller analysis also i put some arbitrary constant lambda just for the analysis remember it is not appearing in the control law it's not appearing in the update law because all of that has already been chosen yeah so there's nothing to be chosen anymore this lambda is only of use for the purpose of the stability analysis excellent so i just substitute for e uh this is from here i get e dot and i get this guy and then from here i get lambda times theta tilde transpose theta tilde dot so which is this with this nice negative sign and then uh i know for a fact that this is positive semi definite why because it's a inner product it's like x transpose x the y of transpose y of this positive semi definite right so this is and so this is positive semi definite at least yeah because the symmetric matrix so real eigenvalues it's a symmetric matrix constructed out of a inner product so like a non so again non negative definite all right um so we know that so we use this fact to sort of get rid of this term we don't even use this in the analysis so one might ask why introduce this term yeah if we did not use it in the analysis so in in this next step you see that i have dropped this term and i only left with this term kept this term so these two come in as it is right so this term comes in like this this term comes in like this i have replaced this with norm bounds that's it but this term i have removed why we keep this term in the update laws because it improves the numerical performance this has been shown and proven so this definitely improves the new this term in the update law does improve the numerical performance of the adaptation law and so it makes sense to keep this yeah all right excellent so uh so now if you look at it i am left with these three terms um and now we talk about the initial excitation property so um we say that this yf is initially exciting i.e. with constants t sigma 1 positive if this inner product yf transpose yf integrated from 0 to t is greater than equal to sigma 1 identity which is a positive definite matrix so what does it mean it means that even if yf transpose yf instantaneously is not guaranteed to be positive definite in fact impossible right because if you remember yf is r1 cross 2 yeah so if you multiply so it's it's at most rank one so yf is at most rank one if i multiply and again yf transpose is also at most rank one so uh it's it should be obvious to you that um at each instant rank is at most one right because the product rank of the product of matrices is the smallest of the rank of each of the matrix involved all right so here yf is the only matrix involved it has rank one so the rank of the product can be at most one okay but what we are claiming is that if i integrate from 0 to t there is sufficient rotation this is the same thing that we talked about in persistence excitation yeah if i but the thing is that here i integrate only for a finite from time 0 to cap t okay as opposed to this if i wanted to write yf persistent excitation yf is p with the same constants if i said with t sigma 1 positive if this doesn't look like a y yeah if uh integral t to t plus capital yf transpose tau yf tau greater than equal to sigma 1i greater than 0 for all t yeah so the difference so i have sort of made a slightly different kind of definition of persistence than what you would remember from class because i don't put a lower bound and there it's an outer products yf yf transpose instead of yf transpose yf but it doesn't matter yeah first of all i we discussed this even when we talked about uh you know persistence excitation that the lower bound the upper bound is not so critical yeah the upper bound is only for the purpose of talking about boundedness of the signals yeah the lower bound is what you will find in all definitions of persistence for sure so lower bound is the only key thing to be honest um and then um the fact that we use inner product versus outer product doesn't matter because i can always talk about persistence of the transpose yeah no problem yeah as simple as that so but if you look at the big difference the big difference here is that you need this condition to hold there's this positive definiteness condition to hold for all small t here there is no small t at all no small t right so you wanted to hold only for some initial window here you need to hold for all sliding windows if i if i take a window of time of size t and i keep sliding it in every window if i integrate this inner product it has to be positive okay so that's a very stringent requirement compared to the initial excitation requirement okay now what is well known see if if you have this condition if you have this initial excitation condition what do you know if you look at the solution yeah let's look at the very unit let's look at um the solution of the y if right so y if dot was simply y if transpose y if with initial condition at zero to be equal to zero so what is y if of t it is actually zero to three y if transpose tau y if transpose tau y if tau d tau and if you look at these two you match these two they're exactly the same they're exactly the same integral so what does it mean means that if i have initial excitation on y f then if the small t is greater than equal to capital y if has to be greater than equal to sigma one identity just by comparing these two to compare these two you see that if this small t becomes larger than this capital t then y if for that value of small t has to be greater than equal to sigma one n right because this integrand is always non-negative always makes a non-negative contribution cannot reduce yeah if you have a value of y if capital t to be sigma one i which is what you will have from here then the value of y f t beyond capital t time will also have to be greater than equal to sigma one n yeah just by the virtue of how this evolves as simple as that so because of this initial excitation which is a significantly less stringent requirement than persistent excitation yeah that that's it's evident in the names itself persistent means always exciting and initial means only initially exciting yeah so it can be boring later on completely fine all right so so for initially exciting signals you will have y i f to be greater than equal to sigma one i when t greater than equal to t okay so in this v dot expression i can have i i replace this quantity by sigma one i right i replace this quantity by sigma one i and of course then i also you know use this fact i mean i use the sum of squares so this is less than equal to half in squared plus z squared by t theta delta square so of course i use the sum of squares also and i get this expression beyond time t greater than equal to t it doesn't matter what happens until time equal to t less than t because until the it's only it's a it's only a finite time therefore the system would have expanded only a finite amount it doesn't matter how much but a finite amount right and beyond that i have this nice kind of a result right now you would ask you would sort of think that this is you know this is time varying and because it contains the state right what is z z contains the part of the regressor right that contains this that is not a constant so now how do we deal with that that's where the lambda comes in that's where the lambda come so here of course if k one is greater than half i'm done here i just need the lambda to be large enough to dominate this right because mu i f is of course in our hands also you can also choose mu i f to be large but sigma one is not really in our control now depends on the initial excitation properties of the segment right but if you choose lambda large enough suppose you fixed mu i f and fixed sigma one is out of control but if you choose lambda large enough then you can dominate this and once you dominate this guy you know that v dot is less than equal to zero so v dot is non-increasing sorry so v is non-increasing v is non-increasing states are bounded states are bounded then this bound continues to hold with some large lambda and the most important point is this lambda is not required for the implementation it's just for the analysis so this domination is given it's pretty standard pretty straightforward okay so once we have this we have nice negative terms in e and theta tilde yeah this is under the initial excitation condition so this is a big difference from our certainty equivalence control where you never get a term here like this theta tilde this term happens comes about very in a very straightforward way when you have initial excitation because of the fact that this update law contains the theta tilde right for the persistence excitation based control this theta tilde term doesn't show up very easy yeah there you have to prove results using you know uco conditions and all that we already saw that yeah so it's a little bit more complicated theta tilde does not show up in the adaptive controller here it does and because it does the liapunov analysis also becomes more straightforward all right so this is of course nice you have nice negative definiteness negative definite v dot and which immediately means that you have some nice exponential decay in fact because your v was also I mean quadratic in e and theta tilde and v dot is also negative quadratic in theta tilde so you have a nice exponential decay so pretty strong outcome I would say and therefore you can have convergence of both e and theta tilde exponent right so that's what I would say both e theta tilde converge exponential right so pretty strong result here so so that's that's what it is right the certainty equivalence sorry the initial excitation based adaptive controller right great so what did we look at we sort of continued our discussion on the initial excitation based adaptive controller we saw the second filter layer of course then we constructed the update law which interestingly brought about theta tilde terms nice negative looking terms and we know that if there's an initial excitation condition which is significantly less stringent than the persistent excitation condition we have nice negative definite terms in the theta tilde dot term and this helps us to prove exponential stability of the system right so we have exponential stability of e theta tilde dynamics which means that both e and theta tilde are going to converge exponentially in time yeah so of course all this happens for t greater than equal to t so this something we have to remember so in initial time there may be some finite expansion of the system which is okay it's still finite yeah this behavior can also of course be governed by choosing the ends appropriately and so on all right so in the next upcoming session we'll continue our discussion of course on initial excitation based adaptive control we'll look at higher order systems and things like that and so I hope to see you all there in the subsequent session thank you