 Welcome to control of nonlinear dynamical systems. We are already in the second week right and we have already started some more serious material. So, last time we started looking at stability ok and how we started talking about stability was of course first by fixing a system initial conditions defining what is the solution. We also spoke little bit just to spark our interest on the notion of the flow yeah which essentially sort of tells you how a bunch of initial conditions map to some final conditions after a certain amount of time ok. So, beyond that we started talking about the notion of equilibrium alright and how we sort of care about isolated equilibrium yeah and we of course give an example of non isolated equilibrium very easy to construct yeah I also showed you this pen which is rolling on a surface yeah it is completely non isolated equilibrium because every point is an equilibrium. So, without disturbance you do not move from any point at all ok. So, once we have the notion of equilibrium all notions of Lyapunov stability are defined with respect to an equilibrium alright no equilibrium no notions of stability ok. So, this is how Lyapunov stability works yeah there is of course Lagrange's notions which we do not really talk about, but there the you do not particularly need notions of equilibrium there you are talking about boundedness ultimate boundedness and uniform boundedness and things like that ok. So, slightly different notions which do not particularly talk about stability in the sense of Lyapunov here we need an equilibrium. So, we started with the definition of stability. So, obviously you start seeing the equilibrium point appearing everywhere here yeah and stability essentially is an epsilon delta definition the way we understand it yeah it just codifies the fact that if you start close to the initial condition sorry if you start close to the equilibrium you are expected to remain close to the equilibrium for all time alright and that is codified as given an epsilon there is a delta such that if you have if you start within a delta ball you remain within an epsilon ball ok. So, we of course understood this a little bit better hopefully yeah we sort of try to compare it with boundedness and we sort of understood that it does not really compare with uniform boundedness well at all yeah one does not imply the other then we went on to talk about uniform stability which is just t0 removal ok. So, the delta does not depend on the initial time anymore and we clearly said that whenever we talk about uniformity in this course we are always you know talking about uniformity with respect to time ok. So, whenever we say uniformity time is involved ok. So, that is the idea alright and then we of course looked at this very very nice example why is it a nice example because we can actually construct a solution here yeah and we can look at some interesting properties and we in fact saw that this system is stable but not uniformly stable ok. So, we will of course explore more properties of this system as we go along and make more definitions alright. So, we will explore further properties of what this is but for now we know that this system that we have looked at is stable but not uniformly stable ok. So, what we do today is we continue to talk about more properties and we move towards asymptotic convergence or asymptotic stability ok. So, that is the idea for today's lecture at least yeah. So, we start by assuming that the equilibrium is the origin yes please. So, the equilibrium is simply any point in the state space where from which you never move under ideal circumstances ok that is you do not deviate from an equilibrium unless there is a disturbance present or anything like that ok. So, the equilibrium is simply a point on the state space from which there is no movement ok the states never move from there ok and how we compute it is pretty straightforward we just equate the right hand side to 0 that is it ok that is how we compute it that is how we did it for this example also alright ok great. So, for the rest of the presentation we assume that the equilibrium is in fact the origin yeah it is not difficult to shift the origin. So, that you can ensure that your equilibrium is always the origin you just do a simple change of coordinates like this yeah we are very used to doing this whenever we are talking about the tracking problem for example right whenever we are doing tracking we always do some kind of a subtraction to make sure that we are always talking about going to the origin ok. This is what we are comfortable it is as simple as that yeah also makes our notation simpler I do not have to keep writing x e everywhere alright. So, from now on assume that the equilibrium is the origin ok 0 0 in the state space alright great. So, we talk about the notion of attractivity now ok because this is the next important notion alright. So, what is attractivity for all t 0 there exists a delta again possibly depending on t 0 such that if you are within a delta ball of the origin or the equilibrium in this case as you can see then as t goes to infinity you approach the origin ok. So, this is simply attractivity the way you would understand it the only difference is you can you can see that it is defined locally yeah it is a local definition because it is saying that if you give me a initial time or t 0 I will give you a ball of certain radius within which you have to start if you want to get here ok. If you start beyond it we cannot guarantee anything ok. So, that is the important thing to remember the notion is local because I start within a delta ball I have to start within a delta ball that is it ok. Then obviously we try to strengthen these notions right there is the notion of uniform attractivity remember that I said that whenever we talk about uniformity it is always with respect to time. So, the only thing that depends on initial time here is this right. So, if a uniform attractivity this delta is independent of t 0 that is it ok as simple as that. So, very similar to stability uniform stability the delta was depending on t 0 and then it does not depend on t 0 exactly the same thing happening here ok I hope that is clear alright make sense ok. Then specialize further so so attractive strengthen to uniform attractive further strengthen to globally uniformly attractive what is that the delta goes away completely you can start at any initial condition any initial time and you approach the origin as t goes to infinity ok. So, you are strengthening the definitions as you go down here ok. So, remember that when we spoke about stability one of you even asked I think that is there is it local or global stability has no notion of local or global stability is just stability if you notice there is no if you give me an epsilon I give you a delta ok it is not local or global there is nothing local global, but here there is ok very clearly convergence is local or global. So, it is or attractivity in this case it is a parallel notion to convergence for series. So, attractivity is local or it is global ok that is it stability is a there is no local global there remember this ok. So, we have strengthened sufficiently I guess now the rest of the definitions are very simple it is just a combination of these two properties right what is the combination first we again start with the weakest notion asymptotic stability ok acronym AS alright we use these acronyms a lot of times because they are very long sentences to say yeah. So, asymptotic stability requires a combination of stability and attractivity ok that is it you already know what is stability you already know what is attractivity if you have both properties for a system it is asymptotically stable ok see I no longer require any more epsilon delta definition you know in books of course you can if you go to Vidya Sagar and you go to some other text they will probably formally tell you the definition of each of these, but then it is not required ok you have already defined stability you already defined attractivity if you have both the properties then it is asymptotically stable ok. And unfortunately different books have slightly different definitions yeah I would stick to what we are talking about ok in most cases they are identical ok you can prove that one implies the other and so on and so forth alright. So, do not worry about the slight differences for example if you look at Khalil you might find a slightly different definition if you look at the Vidya Sagar book you might see a slightly different definition yeah let that not you know sort of worry you one typically implies the other alright then we have uniform asymptotic stability here I just qualify each property with uniformity here ok. So, I need uniform stability and I need uniform attractivity ok. So, this is just uniform asymptotic stability alright. So, this is a pretty strong property yeah in fact one of the strongest properties you can have for non-linear system more often than not this is where you stop yeah then neither of these or none of these conditions actually talk about any rate of convergence alright you can never in fact in most non-linear systems you cannot actually say how fast you are going towards the origin ok it may be linear it may be sub linear it may be longer it may whatever it could be very slow ok. So, you cannot actually guarantee, but in some cases where you can you can define the notion of exponential stability why exponential and nothing else exponential is the holy grail because linear systems give you exponential stability right any linear system if you say it is stable it is exponentially stable it is nothing less ok alright well linear time vary time invariant systems alright. So, what is exponential stability there exist constants r a and b positive such that this sort of a equation is followed ok again vector norms are basically norm of xt is less than equal to a times norm of x0 times e to the power minus bt minus t0 ok. So, the typical exponential decay and this is this is to hold for all t t0 greater than equal to 0 and for all x0 less than r ok. In fact, you can probably write this slightly better and say that this is t greater than equal to t0 greater than equal to 0 yeah and for all initial conditions which are starting within a r ball ok. So, this is actually a local definition right whenever you are requiring your initial conditions to start within some ball of some radius ok then it is a local condition ok because you are requiring initial conditions within some set ok only then you converge is what you are saying here. If you start beyond that set you are not guaranteed anything all such properties are local property where your initial conditions are in fact, required to start within some kind of a ball. You can again strengthen this to global uniform sm tot oh sorry I apologize this is actually a strengthening of this guy yeah strengthening of this is the global uniform sm totic stability remember that there is no global local here. So, this remains as it is yeah, but this property there is possibility of a global counterpart. So, you say that you require global uniform attractivity ok. So, all these other properties beyond stability uniform stability and attractivity they are just a combination of these properties ok which is what makes things relatively easy in terms of at least writing the definitions alright. So, slightly off sequence I guess, but anyway you had sm totic stability then you went to uniform sm totic stability then you went to global uniform sm totic stability this is the strengthening ok. Why the exponential stability in between is also makes sense yeah because exponential stability is also local ok. So, now I can move from exponential stability to global exponential stability. What would be the difference this will go away right that is the only thing that is sort of keeping things local for you right. So, this will go away. So, that is what you see there exist constants now only 2 constants because R is no longer required right only 2 constants here such that the same thing happens again I would say yeah t greater than equal to t0 greater than 0 same thing happens, but for all initial conditions now ok no longer requiring any restriction on the initial conditions and hence global ok. So, what one of the exercise that is mentioned here is essentially to prove that exponential stability is stronger than US ok that is exponential stability implies uniform sm totic stability is an exercise yeah and similarly global exponential stability implies global uniform sm totic stability ok. So, you have to prove that this guy implies this guy and that this guy implies this guy only one way not the other way of course they are not equivalent ok. So, what we are trying to say is that exponential stability gives you something more than uniform sm totic stability and similarly global exponential stability something more than global uniform sm totic stability. So, that is what you have to prove ok. So, you have to start by assuming that you have this kind of a condition and then you have to prove uniform stability you have to prove uniform attractivity ok alright ok. Any questions yes yes if whenever global is not written you will assume that we are talking about local ok many books do write local the use the word local, but more often than not we do not we do not say local uniform sm totic stability local sm totic stability and all that. We just say sm totic stability if the qualifier global does not appear then you assume it is a local requirement ok that is yeah that is standard yes we will see. So, it is a good question if there is an attractive system that is not stable stable system not attractive stable system not attractive is very easy any example of stable system not being attractive oscillator yes standard oscillator spring mass damper no damper spring mass standard spring mass system oscillator they are all you know non attractive they are stable non attractive yeah linear oscillator let us stick to linear oscillator non linear oscillator is funny things might happen, but what you are asking is the other way around if there is an attractive system that is not stable. So, we will see examples we will see very interesting examples yeah alright I will immediately went to the example I want to go back first to this system ok the system we considered ok. Now let us look at this system and I really hope that you at least remember what happened here if you look at this system we wrote the solution in this form right where gamma is of course a function of initial time right and this is the initial condition yeah, but this is the basic evolution ok can you tell me if this system has any of the so you already know that it is stable and not uniformly stable ok does it have any attractivity property is the system attractive or does it have any attractivity property because that is what we need to claim asymptotic stability or something like that right, but do you think it has any attractivity property just looking at this solution what does attractivity require? If you start with a norm less than delta then you converge to the origin ok. So, if you start with norm less than delta you converge to the origin do you think that is that is going to happen here with this system yes why? T squared absolutely yeah once you fix the initial time yeah then because of T squared because T squared we discuss this right in fact this is the picture beyond a certain time whatever is in the exponent this guy is going to become negative and negative exponential means what? Decay right you are going to go to the origin. So, exponential of negative quantity in fact fast decreasing negative quantity is going to go to the origin ok great what is delta in this case what is the restriction on initial condition? I said norm x 0 less than delta implies you converge to the origin as T goes to infinity right. So, you are right as T goes to infinity obviously converges to 0 what is the delta what is the bound on initial condition that is required m, m is somehow the bound on this guy right. So, m is you are saying the supremum of this but why how do you but it is on an exponential right first of all you are taking exponential of this guy first of all what is f now there is no f I have an exponential of this guy remember I do not have any general function here it is a very specific function here and m is the upper bound of this term here. So, let us look at the definition again you want this for attractivity right start in a delta ball if you start in a delta ball you converge to the origin what is delta? Absolutely there is no delta it is a trick question all right where there is no delta you give me any initial condition how does it matter you give me any x T 0 the exponential this exponential is once you fix T 0 so gamma is fixed this exponential is always going to push me to 0 irrespective of what my initial condition was you can give me a x T 0 as 10 to the power 10 irrelevant this exponential is definitely going to go to it is a constant whatever it is it is a constant this is also a constant so both of these are just some constants even if they are huge it is irrelevant right because this exponential is going to go down really fast as T goes to infinity it is going to go to 0 so it is going to get rid of whatever these guys are if you get 10 to the power 10 here it will become less than 10 to the power minus 10 after a certain time you can always find that time also all right ok so in fact this is then what globally attractive ok great so it has global attractivity ok excellent what about global uniform attractivity yes no because there is no delta so uniformity only required delta to be independent of T 0 and all that but there is no delta requirement at all so it is globally uniformly attractive ok so what so what property does this system have now it has stability and global uniform attractivity so what does the combination give me GAS does not give me GUAS ok because I do not have uniform stability this is not there in fact well let us look at yeah so this property is not there so uniformity no instability no but we have this guy ok so best property we have is something that I have not written here it is something that I usually write here somewhere in between GAS is stable plus globally attractive ok so it has a property that I have not mentioned here in this list but you understand how it is coming it is not such a complicated thing all right so basically it is stable plus globally attractive ok so it is globally asymptotically stable that is it it is not globally uniformly asymptotically stable because that would require uniformity for stability which I do not have all right that is the idea great any questions so that is what it says here I believe did I say it somewhere here yeah yeah so I have actually said it here yeah globally asymptotically stable and not globally uniformly asymptotically stable all right