 So, we have you know sort of looked at a lot of these preliminaries now yeah. So, we very quickly jumped to the real stuff yeah we talk start talking about stability ok and I have written as you can see separately in the sense of Lyapunov ok. So, this was of course all of these notions are due to the notions that we study are due to A. M. Lyapunov pressure and mathematician probably non-linear control the way we know it will not exist without it ok. So, maybe somewhere in 1860s, 70s and so on he wrote couple of articles which you know which sort of delineated what is the notion of stability and how do you know ensure that stability is achieved ok. So, very very I would say important contributions to the field you know like I said the field itself may not have existed because this stability is basically the notion that we are always hunting as control guys ok whatever you do whatever system you are trying to dive any anybody who is working with any dynamical system eventually most of your feedback and most of the control that you talk about be autonomous cars be it you know aerial vehicles be it smart grids you are always trying to hunt for stability yeah because the basic idea is and you all already know from linear system you have the notion of input output stability right and then you have internal stability, but most people hardly study internal stability in linear system usually talk about input output stability yeah, but it is still a notion of stability the idea being that external disturbances do not make your system deviate significantly from its operating point ok. So, once you achieve an operating point for example you know your robot has a converge to a trajectory that you wanted to follow yeah you do not want it to deviate you know if there are any disturbance ok. So, notion of stability is how you classify almost all problems in dynamical systems and control ok. So, there is no of course there is notions of optimality which is a separate sort of line of thought in itself where you do not talk about you know being resistant to disturbance being robust to disturbance and all that, but then it is open loop what we call open loop ok in the sense that any optimal trajectory or anything optimal that you come up with and again trajectory is again a very general word does not have to be something a robot is following or something a car is following no even a smart grid biological system you can always create trajectories ok. For us trajectory is just a bunch of a smooth curve in state space any smooth curve in state space is a trajectory for us ok. So, these trajectories that are designed with some considerations may be optimality considerations yeah for example, if you want a satellite you are launching a satellite and you wanted to go to the moon once it is the first thing you do it is you put it in a low earth orbit then you know it tends to expand its orbit expand its orbit then after a certain point it has something like a swing ok between the earth when the moon gets closest to earth there is a swing which is which means that there is a thrust extra thrust that is applied at a particular trajectory at a particular time. So, and once it applies the trajectory it escapes the earths trajectory and it sort of starts rotating about it moves in straight line then it goes to the dark side of the moon then it starts rotating about the moon ok. So, there is like a particular escape route it takes from the earths trajectories by doing a thrust maneuver at a very specific time ok. This is very optimal this is like one of the purest applications of optimality you will see ok here it is mostly its open loop there is no feedback or anything at that particular time when you see that you are you have this earth moon sort of a very nice appropriately located earth and moon you do the burn you do the burn maneuver you create the thrust you escape the earths trajectory earths orbit and then you move on to the moons orbit ok. So, this is more or less standard way of how we fly satellites because otherwise we will never have enough fuel if you if you did some random ridiculous things you will never have enough fuel to reach anywhere ok. So, I mean we have only limited fuel it is not like we have you know petrol pump or something that is going to fill gas for us in between. So, you just have enough. So, this is where all the optimality questions come open loop optimality is purely open loop. However, once you get into orbit or you think of the lander problem you need feedback because there will be disturbances which are trying to once you have an optimal trajectory you try to follow the optimal trajectory ok. You need feedback why do you need feedback because you want to keep following the optimal trajectory what if you deviate what if you have I know suppose you had some solar panel and then some serious solar radiation happened at that time ok because some particular moment when you know some sunspot exploded there was some extra solar radiation coming through. So, you started you know reorienting and tumbling in your orbit and doing some crazy things which you do not want to do ok. You want to be pointed in a particular space then you need feedback right. I mean you have sensors which are picking up that ok I am starting to tumble. So, now I do a detumbling control or any general attitude control orientation control. So, I sort of make sure that I come back to a particular orientation and I do this ok. So, that is where stability comes in ok this is feedback this stability optimal open loop ok. Both are very important these days there is of course concepts of doing them both together also and people are trying to derive stable laws by running optimal engines. So, that is also one way but it is numerical way of doing things maybe something that is more research than reality at this point. But yeah our field relies completely on stability optimality obviously there are quite a few courses yeah in SISCON and otherwise also you can I would say you should always get exposure to both sides of the coin ok. All right very very big preamble I think very motivating all right stability ok. We are always talking about a system which looks like this x dot is f t x ok with some initial condition yeah whenever I specify a dynamical system I specify a initial condition ok without initial condition yeah does not make any sense. Then there is a solution once I plug in the initial condition usually denoted by a different symbol ok most mathematically precise textbooks like Vidya Sagar for non-linear systems you will find the notation for the solution is different from notation of the state although in a lot of my notes you will find ok so fundamental matrix is more a linear system motion yeah so the state transition matrix comes from the fundamental matrix. So, that is little bit more of a linear system motion in non-linear systems the terminology is different the notation looks similar ok. So, this is called a solution if I wrote this as phi t x0 ok just this just change the notation to and put the t as the subscript ok this is called the flow this is called the flow of this dynamical system as stress ok why is it called the flow it is very beautiful it is amazing yeah I mean how we have made everything very geometrically intuitive yeah so just think about this here it looks like just a solution right I mean I plug in some initial condition initial time mostly whenever you talk about flows you sort of do not talk about the initial time you sort of forget the initial time technically you should remember the initial time also but most often more often than not you forget the initial time you say that it is some fixed time t0 and then you keep changing the x0 ok. So, here we are just talking about plugging the initial condition getting a solution. So, this is a solution function of time but the flow is something way more interesting right it gives you something more interesting when I look at it in this form why say I have a bunch of initial conditions say these initial conditions come from a ellipse which I call capital X0 ok now by virtue of this differential equation solution once I plug in one x0 ok and I flow it flow it for time t ok just like you can think of flow in the river you put one leaf at one point another leaf at another point in the river another leaf you put a bunch of leaves from this ellipse into the river and it flows along this solution right because once I plug in a x0 and I plug in a time t I move in a certain way right. So, what is this I move here say I move here I move here. So, it may so happen that I make it a little bit distorted right and basically what I am saying is this is time t so this is time t. So, basically all these leaves imaginary leaves that I put in this flow they of course move differently right they cannot all be even though the average velocity of your stream may be similar and all that but overall because of obstacles or whatever everything flows differently and you may have a distorted shape now yeah you may have start with an ellipse you may have a distorted shape ok. So, this is the notion of a flow ok and a lot of controllability and observability notions are based on flows ok we do not again we are not sure if we will talk about those in the nonlinear context in this class I am not sure if we will have the time and it is also deeply more intense mathematically. So, I do not know how much we will be able to prepare ourselves for it but that is the notion of a flow ok. We basically just talk about the solution ok why because we are at a lot of times interested in this function of time ok because we want to look at this as a function of time all right. Once you put in the initial condition and initial time we have a function of time here ok and we look at it sometime we just call it x of t by the way yeah I do not actually specify this I write it as x of t ok. So, whenever I write it as x of t please understand that we are talking about the solution all right great great yeah I know it seems like we are talking too much about just some notation but it is not because the solution is fixed only by the initial condition once I change this everything changes all right great. Once I have a system like this I need to talk about equilibrium what is the equilibrium the equilibrium is the state from which you never move ideally ideally yeah in reality you will always move but ideally it is a solution from which you never move yeah very simple if I have rolling objects like these I mean in fact every point is an equilibrium for this right this is a very interesting example right every point is an equilibrium because once I put it here it is fixed put it here do not disturb it fixed right. So, this sort of a system everything is an equilibrium this is an example of a non-isolated equilibrium ok because every point in x is actually an equilibrium all right. So, equilibriums are class how do you compute the equilibrium you compute it by equating the right hand side to 0 because that is what makes sure that x dot is 0 if x dot is 0 states are not going moving anywhere. So, you are fixed in state space means yeah that is essentially what you want you are at equilibrium. So, equilibrium is computed by equating this to 0 ok what is an isolated equilibrium equilibrium is isolated if there is no equilibrium arbitrary close to it ok I do not write it as a definition deliberately yeah because there is no need to make it mathematical all you want is that cannot if you have one equilibrium no equilibrium should be arbitrarily close to it ok then it is an isolated equilibrium this is an example of a non-isolated equilibrium this is an equilibrium this is an arbitrary close to this have equilibrium everywhere ok. And this is also an example if you look at this right here x 1 dot is x 1 x 2 x 2 dot is x 1 square what is the equilibrium equate these two 0 all I need is x 1 to be 0 right all I need is x 1 to be 0 x 2 can be anything right I hope this is clear yes I am equating x 1 x 2 to 0 and x 1 square to 0. So, once x 1 is 0 both are 0 nothing moves. So, x 2 is arbitrary. So, I mean equilibrium look like this and what is that if I draw it on the x y axis is the entire y axis ok the entire y axis is the equilibrium this is a non-isolated equilibrium we do not like this all right we do not like this because all our results are based on convergence ok. Now so stability asymptotic stability these are all properties which somehow connect to convergence. Now if you tell me that I am talking about the origin for convergence you cannot because when a when the trajectory comes very close here yeah. So, this is also arbitrary close to the origin right. So, basically what I am saying is there will always be a point which is so close to the origin that the talking about convergence of the origin and convergence of that point is identical yeah you will never be able to talk about convergence to the origin because you will always have a point so close to it equilibrium so close to it that talking about convergence of origin and talking about convergence of this other equilibrium is exactly the same okay. So, you want to in a lot of cases you can transform the system so that your equilibrium becomes isolated all right you may be able to do it yeah if not you cannot talk about stability in the sense of Lyapunov in these cases okay. I hope that is clear you will not be able to talk about stability in the sense of Lyapunov if you do not have an isolated equilibrium okay. So, please always verify that your system has an isolated equilibrium if not figure out a transformation if possible to convert the equilibrium to isolated equilibrium if not sorry Kandor Lyapunov stability you will have to figure out other notions of stability okay all right and that brings us to the first notion of stability let us see if we can highlight this all right so this is the notion of Lyapunov stability just called Lyapunov stability okay for stability in the sense of Lyapunov all right okay great remember we are going back to epsilon delta definitions yeah what does Lyapunov stability try to classify in terms of solutions it says that if you start close to the equilibrium you will remain close to the equilibrium that is it this is what is remain Lyapunov stability yeah in words it just says if you start close to the equilibrium that is if your trajectories are initialized close to the equilibrium that is x0 is close to the equilibrium then x of t that is the solution will remain always remain close to the equilibrium always for all time okay that is Lyapunov stability in when you talk about when you say Lyapunov stability there is no notion of local or global it is just Lyapunov stability there is no notion of local or global all you are not talking about convergence notice I did not say if I start close to the equilibrium I will go to the equilibrium no I just said if I start x0 close to the equilibrium my solutions x t will always remain close to the equilibrium that is it okay it is actually Bebo stability in typically it is sort of comparable to Bebo stability it is bounded input bounded output stability comparable not the same okay comparable to bounded input bounded output stability from the typical linear system sort of motions okay alright how do we put it mathematically we put it as a challenge solution always like this yeah give an epsilon find a delta yeah remember when we talked about convergence we talked about given an epsilon find an n okay here for all epsilon this is the notation for all epsilon positive there exists a delta which can potentially depend on the initial time and epsilon itself positive such that whenever x0 is delta close to the equilibrium x t is epsilon close to the equilibrium okay so always start with epsilon okay please never try to flip this I always get this question first you are given an epsilon then you find a delta not the other way around okay although the way I said it in words seemed like the other way around if I start close I remain close but that is not how the mathematical challenges or mathematical definition mathematical definition says first you predefined how far you are allowed to go from the equilibrium then I will give you how small my initial condition should be okay first you give me an epsilon then I give you a delta such that if you start in a delta ball around the origin you remain in an epsilon ball around the epsilon sorry equilibrium clear by the way whenever I talk about this I may very instead of saying norm and norm different x0 minus xc and all that I will keep saying delta close or delta ball yeah please get used to this because we have already spoken about what is the you know norm what does the two norm x two norm of x less than equal to one look like looks like a ball okay so whenever I say ball doesn't have to be a ball can be a square can be a rhombus depending on the norm you choose yeah here depending on the norm I choose notice I have not mentioned any norm here yeah these are all vector norms this is also vector norm but I did not specify one norm two norm you can choose any norm okay doesn't matter norms are comparable just don't change the norm so important thing is you are given an epsilon then you find a delta okay and I keep using the word delta ball epsilon ball just to indicate norm x less than something norm x less than one norm x less than delta norm x less than epsilon okay please be aware can anybody tell me if epsilon ball will be larger or delta ball will be larger or epsilon greater than delta epsilon less than delta epsilon equal to delta does this this definition indicate any relation between epsilon and delta epsilon can be larger than delta that is a very vague answer delta should be equal to epsilon no doesn't necessary not necessary huh epsilon should be smaller so you are saying that if I start in a larger initial condition ball I will remain in a smaller final condition forever I will remain at a smaller ball okay let's let's look at all cases what happens if let's look at cases right I mean what happens if epsilon is less than delta suppose you give me an epsilon and I give you a delta which is larger than epsilon what happens can you check both conditions this condition this condition is obviously satisfied because I gave you the delta right so you will this has to be satisfied what about this condition actually this is by the way I am sorry it is not evident unfortunately the way I made this this is included in this definition yeah I will just do this yeah that's fine yeah for all t greater than equal to t0 is already obviously included okay so now if epsilon is less than delta what happens for to this guy what happens to this guy not satisfied at t0 if I put t0 here the distance between these guys is delta which is larger than epsilon so this is not satisfied at initial time itself so there is a problem if epsilon is less than delta okay if epsilon is equal to delta no problem yeah yeah but so this is not possible no this is not possible okay so epsilon has to be greater than equal to delta it makes intuitive sense also right my initial condition ball will be smaller than where I want to remain for all time yeah I mean if you if I tell you that I want to remain in say you know I mean in in a in a 5 centimeter ball for all 5 centimeter radius for all time my initial condition definitely has to be smaller than that I mean much smaller for me to be able to because I have to allow for some expansion I can't just assume that the system you know remain inside you know even equal is difficult to achieve in most cases okay all right great so this is sort of the picture here that you that I typically show so this will be the epsilon ball the larger ball corresponding to it you will always find a smaller delta ball so that your trajectory start here allow for it to get out obviously I mean or remain inside but definitely can't go inside instantly right yeah so delta has to be less than equal to epsilon okay so that's the picture