 This part of the course will begin with non-linear control design . So, we will have some small small chunks in which I will post this so that you can look at this little bit of a thing in and ponder over ok . You want to kind of continue in one chunk everything . So, we will begin with some preliminaries of design for non-linear control systems . So, let me get to the slides ok . Now, in this we will basically look at Lyapunov theory ok one of very important rules in the non-linear control analysis . So, you see that this Lyapunov theory is an analysis tool ok it is not a synthesis tool . So, you need to propose a control and check out whether it will be working . So, this will be the some kind of a iterative design you propose a control and find like know some kind of a stability proof for that if you do not get propose another control like that till the time you cannot get some stability proof ok and this Lyapunov theory has been extensively used in many many different kind of new control techniques design . So, it is form some kind of a fundamental foundation for non-linear control ok and here we are looking at mainly application perspective . So, we will present some kind of a preliminary mathematics which is at most essential for development of some of these control strategies . But mainly our focus is going to be on application rather than the proofs of the theorem for the for what Lyapunov has proposed . So, I will present you this some of these theorems, but we will not get into the details of the proof of them because our main focus is going to be on application ok . So, this theory yeah as we see it. So, so first we get into some fundamental mathematical plane memories and define like various notions of stability as I mentioned earlier also that from linear systems to non-linear systems there are different notions of stability that one needs to get into ok . So, let me just check things are getting recorded properly and then come back on just one point I think this should be fine . So, let us see start with these preliminaries now ok. So, first is a notion of stability . So, this Lyapunov like you know if you consider this all these Lyapunov's theorems and stability notions etcetera are developed based on the this state form of non-linear equation . So, x dot is equal to f of t comma x ok, where x is a vector and u if it is function it is a function of x or it is a function of time whatever way it is it is incorporated in this function f here . Now, the equilibrium of such a system is obtained by setting x dot is equal to 0 the very first important concept that equilibrium you want to know what is the equilibrium of a system you set x dot is equal to 0 whatever you get will be equilibrium of a system ok. And we assume that this equilibrium when you set x dot is equal to 0 in this kind of a form is 0 ok. So, if it is not 0 what you do is basically shift the new x you said new x 1 is equal to some x minus that equilibrium state and in that when you now see your equilibrium of a system that will be that will be 0 ok. So, it is important to make sure that you know all the equilibrium that we are talking about in this you know when you do this analysis is is considered to be 0 . So, now, we assume that ok this whatever system where it has been defined has equilibrium 0 ok that means, when we put x dot is equal to 0 we get x to be equal to 0 ok. So, that equilibrium 0 in the Lyapunov's notion first notion of stability is a stable kind of a system is said to be stable if for each epsilon greater than 0 and each t belonging to this positive real line there exists a delta which is function of epsilon and t such that this condition is satisfied. This S is a solution trajectory of this equation hm near equilibrium ok. So, so what it means is if I have the initial state ok developing in the in this ball of radius delta ok. So, what you have here is this norm of x 0 ok. So, this x 0 is a initial condition ok. So, we start off with x 0 as an initial condition at time t 0 ok and we see the evolution of this trajectory. Now, this evolution of a trajectory is less than epsilon for all t greater than t t 0 ok. Then we say that the system has a stability notion this stability notion. So, but it it says that if for each epsilon you are able to find delta ok. So, you are first given this epsilon bound on the trajectory ok as it evolves in time ok. So, once you have given this bound on the trajectory norm ok. So, this is when you say this this like of a second norm or like you know you can consider this as a as a square root of all all squares of the term of this vector ok these both the norms ok norm 2 kind of you can use whatever norm you want to use, but typically norm 2 is is used. Now, so you start off with this epsilon radius. So, this is an example on the in some kind of a you know planar or a two state kind of a system. So, x consist of a two state x 1 and x 2 kind of a states. So, you have this ball of radius epsilon around the equilibrium ok. So, so this epsilon ball is where your solution trajectory is should restrict. Then you are able to find some delta which is greater than 0 ok. So, their delta should not be 0. So, if you are starting at 0 state then you know that by the definition of equilibrium you will remain at equilibrium only, but you need to have little bit kind of a you know away from 0. So, this delta is greater than 0 such that for x 0 beginning into the. So, delta is some smaller kind of a region. So, suppose my x 0 evolves in this starts off in this delta region then my solution trajectories all are rest constrained to this ball of radius epsilon. Ok, if you are able to find such data this is like a base fundamental definition of stability for the Lyapunov's sense ok. So, for the non-linear system this kind of a base stability definition is used and then like you know you derive many other kind of conditions based on the this fundamental definition of stability ok. Then we have a other kind of a definitions coming up here. So, for the same system this equilibrium same now is 0 is our equilibrium is attractive if for each time greater than time belonging to R plus real line there is eta which is greater than 0 such that this solution trajectory tends to 0 as t tends to infinity ok. So, what it means is like the solution trajectory is not only remain now in the ball of epsilon, but they actually tend to go to 0 as t tends to infinity then the equilibrium is attractive. So, x 0 constrained in some kind of a ball of radius eta ok. So, you begin with this eta is greater than 0. So, again you are not supposed to be on the at the equilibrium point you can you can go little bit. So, your initial state should be away from equilibrium point by some small amount and if under that situation you know your solution trajectory is actually go to 0 eventually as t tends to infinity then your equilibrium is attractive that is a definition. So, so here like you can construct some kind of a ball of radius eta and you start of those solution they will do not come out of this eta, but they actually go to 0 ok. So, the solution trajectories are actually attracted to the equilibrium and then the system becomes attractive ok. So, it is see look here it is said that the solution trajectories not necessarily remain in the ball of radius eta ok. So, they may go out of that ball, but they may come back eventually and you know go to 0 that is what is this definition says ok. And for asymptotic stability ok this is a we have to define some notions ok. No, no there are these different notions are for next thing, but for asymptotic stability it needs to be stable and attractive ok. So, the this is a other notion of stability where it is not only stable, but it is attractive also the equilibrium is attractive also and it equilibrium is stable in this sense ok. So, stable and attractive both are properties are there then like you know you have a asymptotic stability ok. So, these are the basic fundamental notions here. So, we will introduce some more notions for the definition and some part and some analysis purposes. So, that we will be able to apply Lyapunov theory ok. So, this some of you may be familiar with the function ok. This function is defined on the time and vector r n ok taking to r. So, this particular kind of a function is coming up in the Lyapunov theory later as a Lyapunov candidate they call it for a stability purposes ok. So, this is defined over time and this vector r n. So, same as like you know what we have in for this definition of f time and x ok. So, it is a function of like general function of time and vector r n and it said to be locally positive definite function ok namely like LPDF. If it is continuous then it needs to have the property that v of t comma 0 is equal to 0. So, whenever this vector state is 0 you give the get the value of the function to be 0 for all t greater than 0 this should be valid ok. So, this is another property and then there exists a constant r greater than 0 and a function alpha of class k now we will see what is a class k function such that the this function is bounded by from below it is like the lower bound for this function bounded from below by this function of class k. So, this class k function we will see first and then like you know we will come back to this definition ok. So, this class k function is a function which is monotonically increasing function ok. So, it is continuous strictly increasing this class k function if you see this it is strictly increasing sorry I will get a pointer right here. So, this is strictly increasing function here ok continuous function and positive for x other than 0 ok. So, it is a 0 for x is equal to 0 same as like v function and this has no time dependency class k function is only function of state ok there is no time dependency of class k function ok. So, if you see this alpha is only of x ok there is no time coming here ok. So, for such a this alpha is a function by which this v is bounded from the lower side. So, v is always greater than alpha of class k then this function is called locally positive definite function. This is like you know again most fundamental definition of this Lyapunov function candidates ok LPDF function for LPDF you have to have these conditions. So, that means this function v of t comma x cannot touch the x axis here ok it cannot come to 0 again ok once it is 0 only at x is equal to 0 and now you know you see for any other value it cannot come to 0 again within the ball of radius r ok. So, this is important. So, this is for locally local definition you need to have this ball of radius r at least for some kind of a norm of like you know limit on the norm of x it is valid it is ok then that is a locally positive definite function ok. So, this is a base definition for the locally positive definite function and if this r is extended now to the entire space r tends to infinity then like you know this will become a positive definite function we do not say globally, but it is positive definite function ok. So, that is a main kind of a definition here ok. So, again the same thing the positive definite function the function is said to be pdf if it is continuous same condition as before then it has v is equal to 0 for r t when x is 0 this r n vector which we are having it function of and then there exists a function of alpha of class k such that now this function is bounded from below by this function alpha of class k for all r n space and for all t greater than 0 of course which was there already for the pdf case ok. So, if you just plot I mean this kind of like you know this showing these plots some kind of a schematics to kind of make sure the understanding happens really well it is actually r n space and we may be able to like you know see this function like this only if it is a function of x alone if you have some kind of a time axis we are not putting the time axis on this plot. So, you may see that the function may have some kind of a time variation also at the same x norm possibilities there, but we are not kind of capturing that part in here ok. So, this is just to kind of see some kind of a graphical illustration of what it means ok. So, now let us see some examples ok. So, we need to check whether these conditions are valid for these functions or not ok. So, remember the definition you need to have a function of class k and we should be continuous. So, these are the conditions continuous and we of 0 is 0 and function condition on these for non 0 x ok class k function condition ok. So, we need to like to strictly follow these these properties for the for checking whether these functions are positive definite or not. So, if you see this function for example, what do you think just do a little bit of a thinking whether this is continuous yes it is continuous whether we this if I put x is equal to 0 straight here I get v is equal to 0 for any time yes that is valid here. Now, it is bounded from below by class k function and you see that. So, you find some function by which it is bounded from below and I think this is done ok. So, this is satisfying all the conditions and that is why it is a positive definite function. Now, if I see this function ok this is a function of x 1, x 2, x 3 and this state is also given as x 1, x 2, x 3 ok. So, when you have first thing you need to check whether all the straight components are showing up in the function or not. So, if you see these conditions is it continuous yes it is continuous then this suppose I put x 1, x 2, x 3, 0 x is equal to 0 in the in the v then v is indeed is equal to 0 and now this class k function condition is is satisfying that for all other values of v other than 0, x this function is not taking value 0 any time. So, this is also a positive definite function, but now if I define this function x 1 there is to 4, x 2 raise to 2 and I have this as a state then we need to see whether this is positive definite function ok. So, see although the form looks similar to this function hmm first condition gets satisfied, second condition gets satisfied, but the third condition you see there is a some problem ok see this x 3 component is not showing up here. So, for even if x 3 is positive ok then the norm x is not equal to 0 norm x is some kind of a finite value here, but if x 1 and x 2 are 0 then this function is taking value 0 ok this is not allowed. So, that is why this is not a positive definite function if the state is x 1, x 2, x 3 if the state is given to be x 1 and x 2 alone then this becomes a positive definite function. So, that is a catch here that may be need to be careful about ok when x 1 is defined as this state for this function then it is not positive definite function, but if x 1 is defined as a state consisting of x 1 and x 2 alone then this is a positive definite function clear. This is what is a main crux we need to look for in the examples ok. Typically we tend to miss because like ok we know ok these are all kind of terms which are positive and then we do not see really ok what state we are looking for ok. So, that is a that is a kind of important condition. So, these other definitions are again based on if you understand class k functions then this is putting some more conditions on these. These are the conditions that you know typically Lyapunov theory it puts on some of these. So, these are more from the mathematical perspective, but many of the examples we do not need to really worry about these conditions ok most of the application examples in the control actual control. So, there are some kind of academic examples one can give which need like you know these to be satisfied and things like that, but from application perspective we need not bother too much about this condition, but we will like you know see this for the sake of completeness. So, v is decreasing function if there exists a constant r greater than 0 and function beta of class k such that v is lesser than beta ok. So, you remember like know the alpha of class k function alpha was bound bounding this v from below and now this is bound from above ok. So, v is always less than this beta is guaranteed for any time t greater than 0 the. So, in time the function cannot go to you know infinity. So, that is what is condition that it is bounded from above also by class k function. So, it I think it is we have a pictorial thing no I think there is no there. So, that is a condition like know put for v for for bounding from above also now by a function of say function beta of class k ok. So, there is a bound coming from above also that is what is a decrease in function. Then v is radially unbounded if it is bounded from below my function alpha and for some continuous function alpha with additional property that alpha r tends to infinity as r tends to infinity ok. So, now this function alpha actually tends to infinity it is not a function which is so, if you see this function here alpha of class k is actually going to infinity as x tends to infinity ok. So, it is not getting saturated here somewhere ok. So, it has to go to infinity as it tends to infinity and it has again this bounds from the top and bottom ok. So, this is an additional condition this is additional property this alpha with additional property, but this radially unbounded condition will have this decrease and see also or not I am not very sure about ok. So, we will check that. But radially unbounded function will have definitely this property it needs to be decreased and probably not ok. So, the function is decreased with you know bound from above ok and radially unbounded if this bound from below goes to infinity as t tends to infinity. And negative definite function if it is if minus v is n positive definite function or negative LPDF if the positive like minus v is LPDF ok. So, these are the kind of conditions that we we have more definitions ok. So, I think these are the main kind of a fundamental definitions here and then we start using the other concept of quadratic functions. So, this quadratic function you may be some of you may be aware about this concept and we have partly seen it when we did the proof that ok your you know kinetic energy was considered as a quadratic function right in terms of the velocities or the generalized coordinate velocities ok. So, so this concept we will see and probably we will stop for this mathematical preliminaries part here and then next part we will look at the proofs not proofs like the theorems of Lyapunov and this application ok. So, these quadratic functions typically will have this kind of a quadratic form ok. So, you are familiar with this form where you have n vector state like say vector belonging to R n space and you are developing this form for all these n components coming up in the in this form ok. If this is a quadratic terms only here we do not have cubic and other kind of a terms linear terms also are not there only all the terms are quadratic in nature. So, x 1, x 2 you can have multiplications of whatever components, but no degree no more than 2 ok. Then this form can be expressed as a x transpose A x and where A is this matrix of the coefficients here ok. So, you see this cross coupling terms A 1 2, A 1 2 they are kind of going to be same here ok. I can split these two A 1 2 into one part which will go up here and other part which will come up here ok. So, this is a kind of a quadratic form of. So, for matrix or given a matrix I can write a quadratic form in this kind of a fashion ok. So, either way and for such a form to be positive definite or those kind of properties it can hold there are certain conditions that mathematically have been derived ok. So, we can one can talk about the positive definiteness of such a form based on there are many ways one can see one of the ways is like look at like minimum value of the quadratic function and see that value if it is greater than 0 then the form is having this property of you know pdf or lpdf or things like that we can talk about ok. So, for that there are by using that such kind of a concepts of you know mathematical you know optimization or things like that people have come up with this some kind of tangible conditions based on these theorems ok. So, they have proposed these theorems to give some kind of a tangible conditions for this form of quadratic nature. Now, why we are bothered about this quadratic form because we will use this kind of a form many places for proposing the V's that we talked about in the definition ok. So, and then use them for some kind of a type of theorem applications ok. So, this successive minors if you take you know determinants of successive minors. So, this is a first minor of a matrix A then there is a second minor of matrix A and like that you can develop successive minors of this matrix A and get their determinants and if their determinants are positive then you say this quadratic form is positive definite this quadratic form yeah is a positive definite function yeah ok. So, it has continuity we can have we see that continuity property is existing when it is the state is 0 you will have this form is equal to 0 that is also existing. So, only thing that we need is this take you know the positiveness or alpha function kind of a property that will come if the minimum of this function is guaranteed to be having value greater than 0 then minimum with respect to x state of force then this form will be positive definite. So, using that condition one can get this other more tangible properties we will not get into the proofs of these things in too much detail right now ok. So, you need to see all the successive minors of A to be positive to get the positive definiteness property then this is other kind of a theorem which says that quadratic functions are positive definite functions if the eigenvalues of A are all positive ok. So, these are all eigenvalues of A you can find out and if they are positive then this quadratic form has a positive value ok. So, we will now like now see this thing in the in the next part of the course ok. So, derivative along the trajectory is another concept that we develop and that is where like you know our function will come into picture and then we will see what is the F1 of theorem for stability and think like that ok. So, we will stop at this point here.