 Hello everyone welcome to course on control of nonlinear dynamical systems we are in the second day of our lectures alright. So last time we mostly we introduced the course and then we moved on to see a few examples of nonlinear systems from I would hope from a relatively wide range of areas alright. You know you had these disease spread models, you had tractor trailer models, so aeromechanical system, biological systems, atmospheric models and so on. We also saw some of the cases which are not very amenable for what we are going to do in this course okay and this has got to do with existence and uniqueness of solutions right. So we saw some examples of when you may have non-existence of solutions and non-uniqueness of solutions alright and we sort of you know got a feel for how this can sort of create problems for us in terms of the analysis. Now at the end of it we sort of made a blanket assumption alright that we are not going to consider these funny and extreme cases. So we are simply going to assume pretty nice smoothness like assumption yeah which is called the global Lipschitz condition okay which essentially looks something like this right. I mean this is what the global Lipschitz condition says that the function of course is separately you require that the function be piecewise continuous in time right and on top of it you require something like a global Lipschitz condition. This is something like a differentiability but more than that yeah a little bit more than differentiability and so this is more or less enough for solutions to exist and be unique for all times which is greater than initiative alright. So this is the assumption blanket assumption we make going forward okay alright. So today we start with some basic preliminaries okay for so you can see that I am using some of my own adaptive control nodes. So this material is common between the two. So we are simply going to look at some preliminary material. First we talk about a few myths and temptations then we talk about more you know more basic things like norms yeah a lot of you might already have exposure to norms and so on but we are going to still repeat it yeah just to establish the notation we are going to use yeah most importantly this will basically set up the notation that we are going to use in this course. So later on it should not feel very alien to you okay alright great. So the first sort of thing that we look at is a few myths that a lot of us carry when we do asymptotic analysis okay when we do any kind of analysis for that matter alright. So what are these myths and temptations let us look at this first of all if I tell you that there is a function which is real value so it takes real numbers and outputs real numbers yeah and I tell you that the function converges to a constant this does not necessarily imply that the derivative converges to 0 okay. So a lot of us sort of thing that oh the function is converging to a constant the derivative should converge to 0 okay it is why it is very true that if the function is constant for all time then its derivative is 0 for all time this is true no doubt about this but here we are not talking about the function being constant for all time all we are saying is the function is converging to a constant as t goes to infinity yeah again I hope these notations are clear if not please go ahead and revise yeah your functions and continuity and limits and so on and so forth we are not going to discuss those okay we are saying that in the limit if the function converges to a constant then the derivative does not necessarily converge to 0 okay. So some very easy examples okay so this is one example so if you look at this function sin of t square divided by t okay and if you take the limit as t goes to infinity what happens to this function the limit goes to 0 right yeah everybody agrees because the numerator is just going to oscillate between minus 1 to plus 1 yeah and the denominator is basically going to go to infinity linearly right. So therefore the ratio is definitely going to 0 yeah this is and the other thing to remember is that this is a pretty nice smooth function everywhere except you know at t equal to 0 but since we are really talking about t going to infinity we do not really care so much about t what happens at t equal to 0 we are talking about large time rather than you know small time so we are not that worried about it is not so nice behavior at the origin at t equal to 0 not at t equal to 0 alright. Now let us look at the derivative alright very easy to compute right so I just use the if you want to use the product rule or yeah you can use the product rule or you can use the ratio rule yeah the derivative is just minus sin t square over t square when I take the derivative of 1 over t and then I have to take the derivative of this guy which gives me twice t cos t square and so that cancels with the time right. So I am just left with twice cosine t square okay I hope you are convinced that this is in fact correct okay alright. Now what happens this guy still goes to 0 right again numerator continues to oscillate but the denominator is actually going to infinity in fact much faster yeah but this term continues to oscillate right you do not know what is doing at infinity yeah in fact this function f dot of t does not have a limit as t goes to infinity okay this function does not have a limit as t goes to infinity I hope this is clear okay. Now can you give me another example I gave you one example can you give me one more very easy to construct with similar ideas I guess what do you think cos t square by t making your life really easy there alright yeah cosine t square by t alright. So you can construct similar ones yeah you can also construct different ones okay I mean it is not too difficult okay it is not too difficult to do okay let us see yeah yeah you can construct different ones also okay I am not going to tell you what but yeah so you can take cos t square by t as another example okay you can always do that yeah you can also do things like sin t cube by t square yeah so I mean so many different choices okay so there are so many functions I mean large swaths of functions which do not satisfy this kind of a mistake but I can promise you in a test I will definitely find one of you at least writing that because the function converges to constant the derivative converges to c okay this happens all the time okay so be very careful that is it absolutely absolutely all of it yes all I am saying is remember this is a myth yeah do not write in the test that my function converges to a constant so my derivative is going to 0 yeah that is wrong very wrong alright okay what about the converse that is not true either yeah the derivative of function converges to 0 the function does not converge to a constant either so the converse is also not true okay so all of this mess is created because of the limits yeah so the limit makes all of these go wrong yeah asymptotic because all I am saying is asymptotically something happens to the function doesn't mean that asymptotically the same something similar nicely happens to the integral or the derivative okay so again an example very easy yeah this is a much easier example if I take the function as log t log natural of t then the derivative is what 1 over t what happens goes to 0 yeah the derivative is going to 0 yeah right if I take the limit of the derivative going to 0 as t goes to infinity right just 1 over t right what happens to this guy as t goes to infinity blows up right really bad yeah blows up in fact yeah so sub linear growth it is a sub linear growth but it is still is going to go to infinity as t goes to infinity okay so really bad functions okay in that sense again another example can't use this one difficult now to continue this one because 1 over t square will not work will 1 over t square work 1 over t square if f dot of t is 1 over t square then fine it goes to z the derivative f dot of t is going to 0 what about the integral of 1 over t square also goes to 0 okay the well-known fact actually yeah I mean when you have a when the denominator has powers more than 1 strictly more than 1 then this is a convergent series yeah I think those of you who have seen series they would know that if I take a series 1 over n yeah then it's not necessarily convergent but 1 over n squared it is yeah so the similar idea actually comes through here also okay so this sort of an example will not work something else motivated by the previous example maybe from the previous like this one motivated by this can you construct something and not the one that I crossed out of this is wrong or any other for example where the derivative converges to 0 but the function itself does not converse to a constant t e minus t so the function the function is going to 0 s 2 e raise to minus t will dominate t you are saying yeah yeah exponential will dominate the linear growth so e to the power minus t will take you to 0 square root of t nice example alright square root of t what happens f of t square root of t f dot of t is did I get this right definitely did not get this this is minus right minus 1 over twice square the derivative right so now this guy is going to go to 0 as t goes to infinity right but this is again blowing up this is going to infinity yeah as t goes to infinity right this is okay right yeah yeah okay so anything with square root of t sin of square root of t this will also work alright so many examples again I mean no dearth of examples here yeah so please resist from ever saying anywhere that if functions converge to constant the derivatives converge to 0 derivatives converge to 0 functions converge to constant okay so be very very because we do asymptotic analysis all the time and I can tell you the serious temptation while you are writing things in the flow you tend to write it yeah I am done yeah but that is not okay you have to prove those things that it really happens that the function converges to 0 and the derivative also converges to 0 or something like that you have to actually prove it yeah it requires something more yeah alright alright we will probably at some stage also talk about what is this something more but not immediately alright so now we go on to a little bit of more pedantic material yeah I do not know how interesting you will find it but anyway I mean this is like I said the purpose is to establish notation and we will establish it alright so the first thing is vector and matrix norms right so we keep using norms all the time because this is more often than not we will be working with real vector space right so everything is real numbers or rn or rk and rp so we want to have a means of talking about distance lengths of vectors and so on yeah so how do I say that my state is 0 I can only say that in the norm sense right so or if I how do I say that the state is getting closer to 0 I mean staying 0 is still easy if all the components are 0 but how do I say that my states are coming close to 0 yeah has to be in a norm sense yeah so this is the only way I can measure some distance length and so on ok so we are always working with what is a norm linear space alright so this is the idea maybe talk a little bit more detail on this but for now remember that we are always working with a norm linear space yeah there are several notions connected to it again we will look at this in a little bit more detail yeah so this bit will be a little bit pedantic and mathematical so norm is basically a function which takes your element in the vector space in this case we are just saying rn yeah but norm is valid I mean you can generate a norm for any norm linear space or any norm vector space so this function is a valid norm if it satisfies these properties ok very standard properties non negativity yeah and then you have the scalar multiplication property and triangle inequality ok so these are really the three properties and of course this is a key addendum in some sense right if the norm is 0 the state or basically x has to be 0 ok so there is no two ways about it alright so what are the typical commonly used norms for vectors it is the infinity norm and the p norm what is the infinity norm it is just the largest element of the vector largest absolute value of the vector elements ok you take the absolute value take the max ok and what is the p norm the p norm is basically just take the absolute value to the power p and then take the summation and then take the pth root ok so the most common one amongst these is the what is the Euclidean distance right so if I have for example if I have a vector in r4 like this right this guy then what is the infinity norm it is 7 right because if I take the absolute value of all elements 7 is the largest one ok similarly what is the one norm it is just the you know the one norm is just basically you know it is just the sum of all the elements in this case right again because all the elements are positive right so the one norm is just the summation so that is 3 plus 2 plus 7 plus 5 that is this guy right and what is the two norm is this expression right take the squares basically take the squares of the absolute values take the summation and then take the pth root of this ok so this is the standard way of computing distance this is the Euclidean distance we are used to computing distance with the two norm all this yeah the other norms we do not let us you know I will not go there first the other norm we do not use like you know immediately depends on those are very special purpose norms you do not always use them so I will say that it is some of them are little bit more mathematical constructs but they are all still very very useful ok so the one norm is of course a very very important norm alright now one of the questions I typically ask is so suppose I go here to get some space suppose I tell you that x is in R2 ok two dimensional vector ok so this is the notation x belongs to R2 ok I hope values to this notation alright so now I ask you what does the set A1 which is basically norm of x infinitely less than equal to 1 what does this set look like square ok alright so you are saying ok so what you are saying it is a square centered at centered at origin ok alright so it is a square centered at origin and how big is the square 1 1 unit each ok so I do not know if I made a square alright this is the square ok so right you are right so this is basically a square of basically this is 1 comma 0 this is minus 1 comma 0 and this is 0 comma 1 and this is 0 comma minus 1 ok why is this a square can anybody sort of understand I mean figure why this is a square why why why distance of what so the infinity norm is what is the value in x and y alright ok then now what exactly exactly yeah so the so basically since the infinity norm is less than equal to 1 it means each component in absolute value is allowed to be less than equal to 1 right and so you can see that all of those are right here in the square ok you you basically if you get out of this square you are guaranteed to violate this condition ok as soon as you get out of this square you are guaranteed to violate this condition ok so that is the whole idea as simple as that ok great great ok good now what about a2 which is right this everybody knows very well right ok I can draw a circle I have assistance alright so now I just have to make some centered lines here alright it is what it is and the radius is 1 alright so circle is basically the equation of the circle is x square plus y square less than equal to 1 in this case so the disc not the circle but the disc so and that is exactly what we are doing alright great what ok this was actually wrong this should have been called a infinity right what is a1 1 norm less than equal to 1 it is two dimensional x is two dimensional it is a diamond yeah x is two dimensional yes it would be a line if it was one dimensional yeah so let us see let me try to make this one yeah why ok so basically you will get if you take the absolute value right so what you want is that the equation that you get here will be absolute value x1 plus absolute value x2 less than equal to 1 this breaks down to four lines right x1 plus x2 less than equal to 1 x1 minus x2 less than equal to 1 minus x1 plus x2 less than equal to 1 minus x1 minus x2 less than equal to 1 ok so therefore you will get the area as the intersection of these four right so the line the basically the and the rhombus edges are the lines the four straight lines right like this line is what is this line x1 plus x2 equal to 1 yeah I guess this one should be x1 minus x2 equal to 1 then so on and so forth yeah it is just the intersection of these lines and whatever is contained inside them because you have a less than equal to 1 yeah it was greater than equal to 1 then it would be everything outside ok good clearly you guys have seen all this before knowledge