 So, now we will see this other part some more kind of thumb fundage and then we start application ok. So, next little bit of a important concept is this derivative along the track trees ok let me just say it more ok . So, as you can see here the derivative along the track trees is in like it is a concept for develop for the same function V ok. So, if V is given as the function in the most right of time t and vector R n vector belonging to R n space and it is a real I mean it is a real valued function here ok taking be you know finally, getting the value to be in the real space it is continuously differentiable with respect to all arguments this is one condition that is necessary here and it should be continuously differentiable with all arguments and delta del V is denoting its gradient with respect to x ok this is written as a row vector that is another kind of a small thing then the function V dot defined by this way ok. So, what we are doing here if you see the V being function of time and this vector is differentiation will have a time part plus like you know this vector part and multiplied by del V by del del x or dx by dt this is a complete derivative. So, del V by del t ok plus del V by del x into dx by dt and del V by del x is actually this del V ok the gradient del V ok. So, this part is fine, but what we do here is for dx by dt which is x dot we are actually substituting our system ok. So, this is a very important part here that we are taking the derivative along the trajectories of a system ok. So, if you generally see this V function it can have derivative in many different directions which is a directional derivative ok. So, since V is a function of this vectors like you know R in space this derivative is a directional derivative you know it is a vector function. So, it will have a derivative which is directional derivative and we are taking it in the direction of the trajectories direction given by these trajectories of your function or of your system ok. The system is defined as x dot is equal to f of t comma x ok because we are substituting that x dot here we get this derivative which is along the trajectories of the system for this function ok. So, let me explain you this concept a little bit more in detail here. So, so if you have this Vx now like let us consider like you know the two state systems the x 1 and x 2 are only two states for the system we have defined as V to be positive definite function in some way ok. So, I am considering this kind of a bowel nature for this V and if I am at any point on this V ok whatever point I am then at this point I can define derivative in many directions on this x space here ok. So, along some direction. So, if you map this point on the plane you will get some kind of a directions for the derivative ok. So, I can ask what is my derivative along derivative of this function along some direction here in the x 1 x 2 space ok. Now, this particular direction that I am choosing to define is trajectory evolution of this system on this space plane ok. So, x dot p is actually giving me a derivative of this kind of a directories which are evolving ok. So, so when we put you know we take a derivative along this we substitute this x dot into our equation for V then actually we are kind of getting the derivative or this the way the function is evolving or the derivative function is going or the tangent to the to this V you know surface is a tangent to the surface along the direction of these directories ok. So, that is a kind of a notion that that gets into and if you see for these things in terms of ok. So, this so in terms of system dynamics if you see the system dynamics evolving here in some kind of a way like when that system dynamics trajectory is mapped like you know are predicted on this V surface then along the system to trajectory direction what is happening to V is what we are actually eventually getting. So, that is a kind of a concept for you know derivative along the trajectory. Now, how do we see the stability of a system based on these condition. So, suppose like you know we are considering the same kind of a system x dot is f of x t and then the equilibrium for the system is 0 and total energy is 0 at origin and positive otherwise. So, that is we know this energy function has these properties that we want for this say function candidate V. So, V is considered as an important function in the Lyapunov theorems eventually. So, you say that ok there exists such a V in such a way that some conditions will come ok. So, this V Lyapunov function candidate is important and energy is one of the possibilities for considering for this V ok. So, we see this kind of a idea in first in terms of the energy and then like you know we can argue some of the things. Now, system is perturbed from origin and we start observing. So, system is perturbed at some point from origin at this point and now we see the system kind of trajectory is evolved and we see what is happening to my V or what is happening to my energy E. So, in first case say energy is non-increasing ok. The energy remains there. So, we have I am perturbing here and like now I am like know the energy is same ok. So, V is equal to 0 ok. What it means is that the system trajectories will not leave like you know this V level and they will keep on kind of the system will keep on oscillating in the as predicted on the this face plane and see that the trajectories are going in the circles ok. Then if you see the other condition this is a this is kind of a stable also in the sense of Lyapunov definition ok. Remember that epsilon delta definition it is this is this system is stable in that sense ok. Although we may consider from the perspective of linear system this is like a marginally stable kind of a behavior that one can see. Although other thing is like you know this what we are talking is of the equilibrium stability of the equilibrium ok. In the Lyapunov we do not see the system stability because the system may have many equilibria ok. So, in the non-linear systems in linear systems we have only one equilibrium ok. So, we do not have any problem, but here in the non-linear systems there are many equilibria that are possible and we talk of the stability of the equilibrium rather than the system that is another important point to make sure. Then if you have a energy of a system monotonically decreasing ok. So, from here like no energy at some this is along the trajectories and it is decreasing along the trajectories the energy of a system is kind of decreasing and eventually going to 0 as you see this these trajectories are. So, so what we are doing is we are actually getting conditions from without the solution in place ok. So, what we are looking at is just a v dot ok v dot is less than 0 or not ok. So, we are not really like no looking as a at it as if it is like no solution is available and we are projecting the solution and we are seeing that no. What we are getting these conditions directly based on the v dot computation as per this previous kind of a formula ok. So, this is a v dot we can compute to v dot directly and conclude something based on this about v dot right. We do not need really the solution of this system to be there ok. That is the beauty of this theorem that without solving a system we can kind of you know do some kind of a predictions about the stability part ok. So, so this I am showing you the solution just for the sake of you know understanding here that if it is monotonically like decreasing energy along the trajectories. So, some kind of a evolution will happen here we do not know that we do not we need not know as long as we know that v dot is getting like no less than 0 happening along the trajectories we are fine ok. So, if that is the case that means like no some way the trajectories are kind of approaching like no this equilibrium position 0 and then if it is increasing function. So, this this becomes then the stable and attractive kind of a equilibrium and that is what gives you this asymptotic stability ok and when this is increasing you can see that like if this is increasing in the in the direction along the surface of this v then we are your trajectories are going away from like no it are going out of the bounds probably bounds of like you know whatever they are not coming back to the equilibrium 0 and the system is having that unstable kind of a behavior ok. So, these are the basic foundations you know the fundamentals for this Lyapunov theory ok. So, we can so this derivative of energy type positive definite function is 0 the equilibrium is stable then if it is strictly less than 0 then so this derivative the derivative of energy type function ok if this is strictly less than 0 then the equilibrium of the system is stable and attractive which means it is having asymptotic stability and there is this so this there is definite relationship between the stability and the property level ok. So, this is like know some kind of a sketchy kind of analysis to say that ok and there is more formal kind of a you know mathematical formulation Lyapunov has proposed and come up with the theorems on stability. So, the idea here is not to restrict ourselves to only energy type of functions you can define any function which is pdf is another kind of a beauty of Lyapunov theory that you would not restrict yourself to only energy type of function any function you can take if you are able to find these. So, these are so Lyapunov gives all the necessary conditions ok. So, if you are able to find like know whatever like know satisfying the conditions then you have a result. So, if you are not able to find we cannot say ok those are kind of conditions that typically the theorem has ok. So, these are like one way kind of a arguments ok. So, if you are able to give me like know these conditions satisfying be your other kind of a function whatever it is then I can guarantee you that your system will be stable or asymptotically stable or whatever their stability definitions will come for the system ok. So, that is a idea that Lyapunov has proposed ok. So, we will see one or two theorems and then like know we will close for now ok. So, this first theorem of stability proof ok you know proof here like know we are just going to see these theorems now ok the proof is there in the in the books if you want to get into, but you know as a part of this class we are see this otherwise like know this again Lyapunov theory and it is all these theorems becomes like a matter of discussion long discussion in maybe a half a semester kind of a course can be there on that and all the details nitty-gritty of you know definitions and mathematics ok. So, we are we are right now looking at more mainly application perspective for these. So, this is why we are not getting into too many mathematical these are, but just the essential kind of a details. So, that we can look at these theorems and apply them for a case that it is at our hand ok that is a whole idea of development here. Ok. So, this theorem on stability says that equilibrium 0 of the system ok. So, the system as we saw earlier that we define the system such that the equilibrium is 0 and this equilibrium 0 of the system is stable if there exists LPDF like know like locally positive definite function which is continuously differentiable at least once. V which is again defining based on this t and x here and real valued function ok and constant r greater than 0 such that this V dot of x is less than 0 less than or equal to 0 look this sign is not less than or equal to 0 and for all t greater than t 0 and for all x belonging to ball of radius r ok. So, this r constant is defined in the way that we have this condition valid at least for some distance or some kind of a norm around the origin ok and for that this is valid then we have the equilibrium of the system. So, V dot is evaluated along the trajectories of the system then this equilibrium of the system is uniformly asymptotically stable. So, this is the equilibrium of the system is stable sorry V dot is evaluated along the trajectories of the system then this equilibrium is stable ok. So, there are these conditions which are giving putting a condition on V dot to be less than or equal to 0 ok. So, this is important condition here. So, you define some kind of a V which is positive definite function of. So, it needs to have all the components of x represented in the function ok to be LPDF as we have seen some examples ok. So, you define some function ok it need not be energy function energy functions is typically the first candidate that we know we should try, but it is not mandatory that it has to be energy function no. So, when we do our Lagrangian formulation already we have a energy expressions available. So, that becomes like a naturally the first kind of a candidate to consider there ok. So, that is why I like know this is connected that D D matrix is kind of very important for these Lyapunov theorems when we start using V as a energy function for our mechanical systems ok. Then we have this definition of stability when V dot is less than or equal to 0 ok. So, you need to establish that V dot is less than or equal to 0 by same kind of a arguments of you know say either quadratic function or your basic definition of LPDF and thing like that ok. Alright now we have this uniformly asymptotically stable kind of a definition coming up here, where now we have a this exists decrescent LPDF V. So, now this is decresency condition is coming in addition. So, this function beta of class k is an upper bound you remember that decresency we talked in the last part is about existence of this function beta of class k such that the V is bounded from above also by this function ok. So, that is a bounded from above by this function and the bound from below is actually coming by the definition of LPDF for V ok. So, if that kind of a thing exists then it is uniformly ok. So, this is a condition on V and for asymptotic stability we need this minus V dot is an LPDF ok. So, this is now a little stronger condition than V dot is less than or equal to 0. So, now we want this V dot minus V dot to be an LPDF function. So, you check this minus V dot function and see that it has this property of LPDF where you remember the properties like we have a we. So, this minus V dot should be 0 here it should have some evolution such that it is bounded from below by some function of class k. That is what will these properties of this V dot will be minus V dot will be ok. So, you need to make sure that LPDF properties are satisfied for this minus V dot ok. LPDF at least in some you know ball of radius r where r is greater than 0 ok. From the origin you take some kind of a norm up to the point r and then like in that to that thing it should be valid ok. Then that equilibrium is called uniformly asymptotically stable ok. So, there are more kind of these definitions. So, what we need to bother about is this asymptotically stable system and a stable system. These are the two main kind of points we we take from here. There are more kind of finesse to these definitions asymptotic stability uniformly asymptotically stable system or exponentially stable system all those kind of different definitions are theorems are there which we are not getting into more details. What we are interested in to see is that this system is stable or asymptotically stable. The asymptotic stability is a stronger condition to have because in asymptotic stability we we actually look at the the the system trajectory is going to 0. So, like that we will we will see. So, we will talk about these and like know more examples from the next class onwards and application from the next class onwards ok. So, you can meanwhile try to apply it to this case ok and see whether you are able to kind of see through the the. So, do not look at this directly you can apply yourself to our simple kind of a case of mass lying on the surface to get a hold of like you know get a hang of how things work here and we will anyway discuss in the next class ok.