 Alright, welcome to another class of non-linear control systems, control of non-linear dynamical systems whatever you want to call it, yeah. So last week we had started talking about the, you know, the central analysis method of non-linear control, which is this Lyapunov stability theorems, right. So we started with some, you know, preliminaries that is we talked about function classes, the class K, L, class K, R and so on. Then we went on to discuss definiteness and how they are connected to these function classes and also how to extend the notions of definiteness of matrices to definiteness of functions, right. So basically we understood or we figured that any positive definite or a definite matrix or positive definite matrix is going to lead to a positive definite function, okay. Once we construct a quadratic form of it, okay. So we saw some nice examples, of course we also had these easy conditions to test definiteness and so on, yeah. So again definiteness, positive definiteness and so on. So we had these relatively easier conditions, then we talked about radial unboundedness. So these were all properties and I think there is also of course once we had radial unboundedness then we also spoke about decrescence for which we did not give any easier characterization. This is because the easier characterization is not easy at all. I mean if you are interested you can look at Vidya Sagar's book to see this easier characterization but it is not very easy, yeah. So I would definitely say that it is, it would be very, very beneficial for all of you to go back and look at Vidya Sagar's book at least once in a while to see how things are going. I mean the chapter again, I forget the chapter number but it is very easy to find. You can see the Lyapunov stability analysis chapter in Vidya Sagar's book. A lot of the material that is here is being derived from there, yeah. It is one of the most comprehensive and very, very, you know, mathematically precise description of all this, yeah. So because now that you have seen this, I have made it of course a little bit distilled it and toned it down from the Vidya Sagar language. So now that you have seen this and you understand this material it will definitely be easier for you to follow what is in Vidya Sagar's book, okay, alright. And then finally we had semi-definiteness properties. So we had four properties positive definiteness, radial unboundedness, decrescence and semi-definiteness. We already said that positive definiteness was connected to stability, asymptotic stability. So local properties, radial unboundedness is connected to global properties, so global asymptotic stability, decrescence is connected to uniformity, so uniform stability properties and finally semi-definiteness is just connected to plane stability, okay, nothing more than that, alright. So we also started discussing the Lyapunov theorems themselves, alright, I believe that I had mentioned that we will look at the theorems first, understand them, maybe even try to apply them a little bit, then we look at the proofs of, you know, at least one bit, yeah, we look at some of it at least, yeah, so that we, once you see a proof of sort of one version of the theorem, everything else, you know, sort of follows, not difficult to conclude the rest, okay, alright. So we started with this structure of the dynamics, yes, and we of course assumed all the nice things that is 0 as the equilibrium point and f is locally Lipschitz, so that existence of unique solutions is not a problem at all, alright. And then we defined the notion of lead derivative or directional derivative, I mentioned very clearly that the function v of x or v of x comma t has no connection to a dynamical system as such, it is just a function of some variable x and some variable t, yeah. Then when you take the derivative or this directional derivative, that is when you bring in the dynamics of the system through this term, yeah, in fact you will see a lot of times that we use the same structure of v of x, yeah, to analyze many different systems, okay. So that is why this is very carefully defined, okay, it is a definition, whenever I use this notation, this notation implies I am defining something, okay, it is not just an equality, yeah, you might think it is an equality because all we did was compute a v dot, but it is not. By no stretch of imagination because when I define a function of x and t, I am not saying anything about dependence of anything on time at all, right, I am just defining functions of x and t. At max you can take the partial of v with respect to t, but there is no, there would be no notion of taking partial with respect to x because x is itself an independent variable once I define v x t, okay, but only when I take a derivative and I define it so that you have dependence of x also on t through this evolution, okay, it is as simple as that. So this, I am saying so many words, but this is just to sort of impress upon you that the function v has no connection to your dynamics, all right, and many, and many a times the same function v, we used to analyze different dynamical systems, all right, excellent. Then we did the, I believe the first two statements, we said that we first required to have a candidate Lyapunov function. What is a candidate Lyapunov function? It is a C1 function of state and time such that v is positive definite, this is the minimum requirement for it to be a candidate Lyapunov function and then you need some of these conditions to be satisfied, okay. If v dot is only negative semi definite, that is it is not definite just that it never takes positive values, so v dot never takes positive values, yeah, it takes only non positive values that is it is negative semi definite, then the equilibrium is just stable and on top of this, if the v that you started with is also decrescent, then you have uniform stability, okay. So these were the two statements that we did and you remember all the other stability definitions are sort of strengthened versions of this, right, you start with stability and then you move on to asymptotic versions and uniform versions and things like that, okay, exponential versions, the next one, again remember this is sacrosanct, without this you cannot use any Lyapunov stability theorem, okay. So be careful when you choose a v that satisfies a condition like this, okay, alright. So typically an energy function would satisfy this, yeah, typically energy of a system for a Lagrangian system will satisfy something like this, yeah, or a conservative system the way you know, yeah, okay. Next one, local asymptotic stability, all I need is the semi-definiteness is no longer enough, I need negative definiteness, okay and this is what I mentioned that definiteness is connected to stability, asymptotic stability. So I am clearly saying local asymptotic stability, although we typically do not use this word, right, we have not been saying this, you just say it is asymptotically stable and the acronym is also AS, yeah, there is no LAS, alright, okay. Again the specialization of this would be to start with a decrescent v and then I get uniform asymptotic stability, okay. So the results are very straightforward, once you have the ingredients the results look very easy, okay. You have stability, uniform stability, once you have v dot to be negative definite you have asymptotic stability, if you start with a v that was decrescent and you have v dot negative definite, you have local uniform asymptotic stability, again local is not something we state necessarily, alright, then say again I am going to state all of these before I go to the examples, alright. Then you have global stability notions, yeah, now for global stability I need the negative definiteness of course, but I now need v to be radially unbounded, yeah, a positive definite v is no longer enough. Remember that the arguments also change, this vr does not work anymore, yeah, v cannot be valid only on a ball around the origin, it has to be valid for all rn, okay, so therefore you need v to map all states not just in a ball to real number and then c1 positive definite, okay. So v is now required to be radially unbounded which means that its arguments have to take all possible states and v dot is negative definite then you have global uniform asymptotic stability, okay, actually sorry I also missed saying that it has to be decrescent. Of course if I remove the decrescence what do I get? What if you just globally asymptotic, uniformity is gone, as soon as decrescence is gone uniformity is gone, okay, alright, okay. Then if you remember we I am of course not stating all the intermediate versions because I understand that you understand that if I remove decrescent I remove uniformity and so on and so forth, yeah. You see which word is associated with which word, it is as simple, it is as simple as a word association, yeah. Of course when we do examples it is not as simple but for now the statements are very straight forward, alright, in fact it is almost like you can have a cheat sheet in your head, it is very easy, okay. Now finally when we want exponential stability the conditions are slightly different, you do not use the positive definiteness and you do not state them like that. What we say is if v is decrescent and there exist 3 class k functions all of the same order of magnitude such that v Tx is lower bounded by phi 1 norm x and upper bounded by phi 2 norm x and further v dot is lower bound is upper bounded by negative of phi 3 norm x, okay. Now if you notice this highlighted green thing implies positive definiteness, alright and then this highlighted yellow thing implies negative definiteness, okay. Now here I am already stating decrescents separately, okay, however if you look at sort of the right hand side it is not exactly decrescent but it is pretty close, yeah this is how we had stated decrescent, the only difference was there was an absolute value, yeah. Here we do not particularly need the absolute value because we have already assumed v to be lower bounded by a class k function which means that it is lower bounded by 0 because the class k function at x equal to 0 will be 0, right, which means the left hand side you may essentially implying that v is already positive semi definite by this assumption at least in fact positive definite by this assumption. So absolute value is not required because v is never going to the negative side at all so absolute value of v is irrelevant. So what you have here is effectively decrescents, okay. So we have stated all 3 requirements for which you had for global uniform asymptotic stability, right, there is no difference as such, we have stated all the requirements just in this mathematical form rather than writing the words, okay. What is the difference? The difference is these words, okay, so the difference is these words, same order of magnitude, okay and I will get to this soon, okay. So this gives me local exponential stability, why because they were only class k functions, the comparison functions were only class k and you can see that I carefully, I was very careful and I said all this is valid for x in some ball of radius r, alright. So if I want to go to the global version, what do you think I will need? What will happen? Radially unbounded, so I will need, well yeah, I will need all 3 to be readily unbounded I guess, yeah because they are the same order of magnitude, so all 3 will have to be readily unbounded and of course this will be all of r, okay, this will be the only difference. So you can see that I am already saying v is readily unbounded although I do not need to and then you have all 3 functions, there exist 3 functions in class k r, yeah, oh I see, such that this happens, alright, such that this happens and if you see I have also said what is the meaning of functions being of the same order, it means they are comparable by a constant. This should remind you of the ability to compare the norms, right, the vector norms are also comparable, this is almost a similar definition, any 2 functions are said to be of the same order of magnitude if they are comparable via constants, okay, so and you notice that if f and g can be written like this then g and f can be written in this way, right, so basically f and g are comparable functions, I mean examples are if you have one function which is x squared, phi 1 is x squared and phi 2 is x 4, x to the power 4 then they are not comparable because you will never be able to find a constant gamma 1, gamma 2 which will relate to you, okay, so simply saying, I am just giving a scalar example or you can even take a vector norm x square and norm x 4 not same order of magnitude, okay because they cannot be related by these constants, okay, great, so now we have seen pretty much all the Lyapunov theorems, okay, it is very quick, I mean once you create the setup actually it is very quick to state, very easy, you know how the specialization goes, right, start with negative semi-definiteness for v dot, you get stability, go to negative definiteness for v dot, asymptotic stability and if you go to radial unboundedness for v, you get global asymptotic stability, if you add decrescence on v, you get uniformity in all of these, okay and finally for exponential stability you need, remember we already it is part of your assignment, first assignment that exponential stability implies uniform asymptotic stability and similarly global exponential stability implies global uniform asymptotic stability, so exponential is by definition uniform, therefore you need the decrescence conditions also, right, whenever you are talking about exponential stability, so exponential stability requires existence of these 3 functions, okay, makes sense, excellent, examples, let us do examples, this is where, this is what is our you know bread and butter, right, if we cannot do examples we cannot do anything, the simple harmonic oscillator, the simplest example anybody will start with, what is the simple harmonic oscillator, it is just x1 dot is x2, x2 dot is minus x1, alright, for a system like this it should be obvious to you that, well or it should be obvious or you must have seen it before that the phase plane portrait that is the evolution of this in the phase plane looks like circles, okay, why, because you can think of x1 square plus x2 square and you take derivative of x1 square plus x2 square along this, yeah, this evolution makes x1 square plus x2 square constant, okay, just you can check it, it is very easy, anyway we will do it in the Lyapunov function anyway, right, so that is what we choose is a Lyapunov function, it is just half x1 square plus half x2 square, I mean I have just taken this and divided by 2, okay, in this case by the way this is a conservative system, conservative system, so in this case this is actually the energy of the system, okay, this is the potential energy plus the kinetic energy, okay and this is energy conserving system, therefore you see that it is just moving in circles, concentric circles, alright, alright, so I take my V as exactly the energy of the system, notice V is c1, in fact it is radially unbounded, I hope you are convinced that this V is radially unbounded, yeah, goes to infinity, I mean first of all it is, it is strictly positive whenever x norm x is non-zero, alright and it goes to infinity as norm x goes to infinity, in any direction it does not matter, alright, therefore V is positive definiteness, in fact V is radially unbounded, okay, so anyway this is a linear system, so V is radially unbounded, alright, so you can see that, that this V is radially unbounded, you can just focus here, now if I take the derivative V dot, there is no time argument, right, so uniformity is free, yeah, just like I said, there is no time argument in the system, no time argument in V uniformity is free, we do not even talk about uniform stability notions, okay, right, so partial of V with respect to x times the evolution, so what is partial of V with respect to x times this, it is just the way you take derivative, okay, after all this definition all you have to do is take derivatives, it is just x1 x1 dot plus x2 x2 dot, just taking derivatives and plugging in from the dynamics, if I plug in x1 dot from the dynamics it is x2, if I plug in x2 dot from the dynamics it is minus x1, right, so essentially it is 0, the sum is just 0, okay, by Leah and so what have we done, what have we shown, we have shown that V dot is always 0, exactly 0, which means it is only negative semi-definite, right, it is not negative definite, yeah, because even for non-zero values of the state, V dot will always be 0, okay, make sense, yes, alright, good, okay, fine, so by Leapon of stability theorem, I started with the V which was positive definite, readily unbounded in fact and V dot turned out to be only negative semi-definite, therefore my equilibrium that is the origin is stable, okay, this is all I have, yeah and this is the fact there is nothing more, you cannot get anything more for this system, because the phase plane portraits all look like this, yeah, so if you start at some point here you will just follow this circle, if you start at another point you will follow this circle, if you start at another point you will follow this circle, wherever you start you will just start tracing a circle of that radius in the phase plane, okay, it is as simple as that, it is one of the simplest systems to illustrate Leapon of theorems, whoo, next one is a complicated one, see and you start seeing how things get very complicated very soon, that is probably the aim of this example, yeah, yeah, I just played with this system a little bit, yeah, I just made it time varying, alright, so x1 dot remains x2, yeah and x2 dot is minus x1 divided by 1 plus t, right, I just made it time varying a little, now I want to see if I can do anything, so what do I do, I sort of choose my Leapon of function in a slightly more smart way because otherwise I think I will not be able to proceed at all, so I choose it as half x1 square, the first term remains the same and by the way many people ask me how do you choose Leapon of functions and such, there is no way, it is an art, okay, you either start with the energy of the system and then try to modify the term, it does not have to be the energy of the system or it is motivated by some literature, okay, it is not a guaranteed process that this is what will work and this is how I can get a Leapon of function, okay, no, you cannot do that, alright, great, so I play with the terms, alright, I take the same term as the first case but then in the second term I add this guy because I want to do some cancellation because of this guy because I want to do this time cancellation, right, so notice already, well before going there I should say something, what about V, is it positive definite, yes, yes, because it is greater than half x1 square plus half x2 square, right, so it is positive definite, in fact radially unbounded, so V is positive definite, in fact radially unbounded, is V decrescent, is V decrescent, no, you remember we did this example, right, whatever class K function you give me that needs to be, that needs to upper bound this guy, I will just dominate it by bumping up time because the class K function will have no argument of time, so once x is fixed, this is fixed, this is fixed, the class K function is fixed, I will just bump up time arbitrarily and I will beat any class K function that you give me, right, so no, not decrescent, okay, only this much, so then I go on and take the derivative, right, I have three terms x1 x1 dot, yeah, 1 plus T x2 x2 dot but then by the chain rule I have to take derivative of this guy also, so I have x2 square divided by 2, okay, so then I have x1 x2 here, plugging in x1 dot, I get x1 x2, plugging in x2 dot, this 1 plus T cancels out, I have minus x1 x2, right, so this term and this term cancels out and then I am left with x2 squared by 2, okay, something pretty bad happened, right, because v dot turned out to be x2 squared by 2 which is greater than equal to 0, yeah, does not mean anything because it may just be the case that I chose a bad v, does not mean that the system is unstable, this is not enough to say that the system is bad or unstable or whatever, remember, yeah, it may just be the case that I chose a bad v, I cannot choose a good v, so that is my problem, alright, great, so that is what I have said, cannot conclude on stability yet, okay, I thought about it a bit, I could not find any good v honestly speaking, yeah, which would let me conclude anything, but maybe you guys can try, I do not know, yeah, you can try what you can get, but as far as I could see the system is rather is unstable and why I conclude that is that if I just look at the dynamics of the system, see it is very difficult, it is not easy to solve the system, can I solve the system actually, no, it will be rather hard, yeah, because these two are coupled, yeah, if they were not coupled this would have been okay, but it is not going to be very easy to solve the system, I mean I may be able to use some time varying linear system tricks to solve it, but it is not obvious how to solve this, okay, because of this guy and the fact that this is a coupled system, okay, so what did I do, I thought about it and I tried to see the phase portrait, yeah, but for large values of time, okay, let us look at what happens for very small values of time, very small values of time, this is almost equal to 1, right, almost equal to 1, so looks like a harmonic oscillator, yeah, cycle, so for very small values of time it looks like this, this circle, okay, but as time increases, right, so first of all drawing a phase portrait for a time varying system is also unintuitive, because the time argument is not visible here, I cannot use the time argument here unless I draw a third axis and make something very complicated, yeah, but you see it is not possible to do a very good phase portrait based analysis for time varying systems also, okay, so you see with such a small change adding a time dependence, things can go rather messy, okay, but why conclude that it is possibly not stable, what happens for large time, this guy dominates, you can forget 1, this term is going to 0 almost, okay, so x2 dot is 0, x2 dot does not change, so wherever I start, I stay at that same level in the vertical axis, okay, but x1 dot keeps increasing, right, because if I started far away, that is why you see the size of my arrows, this is what is in, if I start close to the origin, small horizontal velocities, if I start further away larger horizontal velocities, further away very large horizontal velocities, similarly if I start here small negative velocities, larger negative velocities, very large negative velocities, okay, so I can see that there exist initial conditions which are never coming to the origin or doing anything nice, just think about it, if I make a ball, what is stability, you give me an epsilon ball, I have to give you a delta ball, can you do that, no, right, because that everything is going away, some initial conditions will push you this way, some initial conditions will push you this way and large time is where we are thinking of things happening, right, I mean we do not care about transient, I mean stability, sorry, asymptotic stability has no concern with transients, okay, so that is much newer results in nonlinear control where you start talking about transients, yeah, also one of the complaints of linear system folks that you do not care about transients, okay, so nonlinear systems all the analysis rotates around or converges to asymptotic results, yeah, so large time basically, so large time, in large time I can see that things will not work well, yeah, all my trajectories will start to explode in some sense, so I guess in some sense this was not so wrong, okay, but you see it took me a lot of intuition and effort to even get a result like this, yeah, for even this very timey system, okay.