 So, we know enough about the zero dynamics. So, the idea is this that if the zero dynamics is stable when the linear part is zero and the linear part is also stable then the combination is stable. I mean when I say stable I mean asymptotically stable ok. So, that is the general case as like I said cascaded systems literature pretty strong actually talks about such results only if you remember I mean we spoke of a very specific case where we had a stable system to begin with and then it was being driven by a passive system yeah. But you know that what is the purpose of passivity right because of passivity you got essentially some kind of nice stability right because once you have v dot is u transpose y then you construct the u as some phi y and then you have actually stability right you have something nice. So, this is also something similar you are saying that you have a system this kind of a system the non-linear system which is sort of you can think of it as being driven by this linear part and if this linear part is going to zero past enough or whatever going to zero then you have asymptotic stability for the combination what I have not written I have not written this, but this is something I wanted you guys to look at ok exercise yeah. So, I want you to complete this proof yeah because what am I saying I am saying that the system is locally asymptotically stable or asymptotically stable whatever with psi equal to zero. So, you have a v eta right it is given to you such that partial of v with respect to eta times q 0 eta is negative definite ok and then you have this to be Hurwitz. So, therefore, with this also you have some psi transpose p x psi as your possible candidate Lyapunov function from and the p you get from this Lyapunov equation nice Lyapunov equation. So, what I want to do is to investigate stability of the entire system with this new candidate Lyapunov function which is basically just the sum of this and this ok. I want to see what how you can do it or what you can do yeah you can make assumptions if you want alright you can make some assumptions if you want alright that is fine yeah if you want to make some assumptions you can make, but see if you can get something interesting yeah. So, again the motivation is cascaded systems things like this motivation is cascaded systems that look like this alright. So, great so that is the idea we can do nice get nice stability right that is the whole point I do not know let us see that is before going to Robinius let us go back to our example right. So, this was our DC motor the transform dynamics right the Z dynamics transform here what was the psi psi is this guy the first two and this is the eta ok. So, you see it has this this has this nice structure right if psi is going to 0 or if psi is equal to 0 means what Z 1 Z 2 is 0 right notice Z 1 is X 3 Z 2 is theta X 1 X 2 if Z 1 Z 2 is 0 means X 3 is 0 and X 1 or X 2 is 0 right which means most importantly this term is 0 ok and when this term is 0 what can I say about this dynamics exponentially stable yeah Z 3 dot is minus B Z 3. So, this is a minimum phase system yeah the 0 dynamics is exponentially stable even more than asymptotically stable it is in exponentially stable alright. So, this is a rather nice system right why all I have to do is check when psi is 0 when psi is 0 what happens to the eta dynamics ok. So, when psi is 0 these guys are 0 which means this guy is 0 this is gone. So, what am I left with Z 3 dot is minus B Z 3 it is exponentially stable very nice 0 dynamics. So, we can actually use these results all I have to do is design a U. So, that this system becomes exponentially stable that is also easy because I know that this is non-zero this is non-zero I mean that is the assumption. So, I will just divide by this theta X 2 whatever I mean I invert the theta X 2 and I can give it some minus k 1 Z 1 minus k 2 Z 2 right then basically this becomes a spring mass damper right this whole thing will become a spring mass damper and then. So, that is also exponentially stable right and then output of that basically is just going into this guy yeah. So, you can immediately apply these results to get a stabilizing feedback for the DC motor with a as you can see we started with a relatively complicated non-linear dynamics right it was not like too straight forward I do not know where it is yeah these nodes are a bit difficult to read this guy yeah. So, we started with relatively complicated dynamics right I mean there is enough non-linearity here yeah for you to be able to immediately design a control right from here right if I had asked you to design a control for this system directly it would not have been easy for you yeah would not have been easy for you I mean I even I cannot see how I would do it because control appears only here right. So, I can do nice things with this guy sure no problem, but what about these very non-linear terms I mean cannot even wrap my head around how I would have designed a control back stepping nothing will work here I do not see what will work here I mean backward back stepping something I do not know like start from here and then go here and then go here or something messy like that. But nothing is obvious to me passivity definitely not obvious to me no way I do not see I do not see what is I mean this how I will get passivity with this guy and this guy I really do not know yeah no idea not obvious what would be a storage function for this right not at all obvious if I look at started with this system which is a standard DC motor model I do not know how to design a control okay. But with some whatever playing around like we did we have the system where I now have a very good handle on how to do things I I own I do not even have to do design anything here this is what it is all I have to do is design stabilizing controller here yeah very straight forward right I mean cannot get easier than this I do not see it and I am not saying it is impossible I am not saying it is impossible see this is the output given to me this is the input you need to find a storage functions is that V dot is less than equal to U transpose Y see it is a fact that every system is passive or not almost a fact that everything that you see around you is a passive system but you have to pick the right output and input I do not see it as of now what is the output and what is the input okay see with standard mechanical systems what do people do they take energy as the storage function and work with it here there is no obvious storage function available to me. So that is why I am saying it is not obvious to me and so these are the only methods I know back stepping and passivity I do not know any other method and to be honest there are not that many other generic methods either okay of course if the CLF base method then the question is can you find a CLF for the system I mean is it obvious what is the CLF for the system again not so much to me yeah I do not know yeah the first thing I would do obvious is to which is what we did I think with xz3 z3 was I think x3 minus k3 yeah z3 was x2 minus k by b that seems like an obvious transformation because that will make this equal to z3 minus z3 right this is quite obvious that is one thing I would do that I would make this z3 or whatever some other variable instead of x2 say some other variable instead of x2 and so the derivative remains the same but I get something nice here but that messes this guy as a new variable will mess up with this one. So honestly not obvious even how to construct a controllable function yeah I mean I do not know again folks in electrical who do DC motor stabilization might be knowing what is a valid V for this but I do not know okay so off the top of my head I do not see how the methods that I have learned apply here yeah but you see feedback visualization gives a clean answer here yeah it is not even evident that this system is you know will give you get you to something nice like this okay all right I mean not evident to me all right okay all right folks so good good so feedback visualization is relatively powerful yeah it because you are doing a state transformation by using some method it almost look like some bunch of magic potions or some you know incantations right I did some lgh lfh and a bunch of lglfh and lflgh and whatever and add fh add fgh and so on and so forth and I read some fun place right where I can get some nicer looking system okay so quite powerful actually yeah good logic to why it works but it is still quite powerful all right great so this I want you to verify as an exercise that what happens if I use this V to this combined system let us see what you can get now we move on to a more involved theory topic also on feedback visualization this is where we start to answer the question when does such an output y equal to hx exist when do you have an output such that you can get linearization in fact in this case you look at full state linearization not partial linearization when can you actually when can you guarantee existence of such an output y okay so those are the questions that are being answered here this is why things get a V bit involved now so this is what is the context of Frobenius theorem okay again I am freely using nodes that Vivek had used a while ago I have of course made some nodes here and there but we start with the notion of distributions okay what is the distribution distribution is a pretty straightforward idea you already seen vector fields right you have seen f and g which is f is the drift vector field g is the control vector field typically your system looks not like typical dynamics is your typical dynamics is like this right you have multiple control not just one control right so you have a drift vector field and you have multiple control vector fields right we keep calling them vector field because at every point it gives you a direction every point in the state space what does it do it gives you a direction like this what it is drawn what is drawn here right you take a point f 1 gives you this direction say f 2 gives this direction what is the distribution distribution is just a span of this is the vector space that is formed by this okay so if you have some k vector fields then a distribution is basically assigned to each point this subspace this is the span okay so if you have two vector two vector fields okay and if you take the span of these f 1 and f 2 right then basically you can get any vector in this more or less in this plane because this is linearly independent vectors so you get anything in this plane but of course if you think higher dimensions you may not get everything in the plane and so on and so forth right so this is what it is its distribution is just a way of representing a span of these vector fields okay so which means that any element in the distribution yeah so p is of course you need a base point because you these vector fields are being evaluated at a particular point right that is why it is called a vector field and not a vector yeah because it is being evaluated at every point it is a function yeah so if you plug in this particular point p then at that point you take the span of all of this that is called the distribution delta of p and it is also notationally accurate to write delta as span of f 1 to f k how do you find the dimension just take the rank right but the point is you can see that the dimension depends on p right as you can see because I plugged in a point p right the way we have been doing dimensions you take the vector span and then you take the rank of that right here each of these depend on point p therefore the rank also depends on the point p okay that can be a problem it can vary with p but that is not what we look at we usually look at distributions which have constant rank yeah which means dimension remains the same for all your entire space of interest yeah p in x and such distributions are called non-singular distributions okay such distributions are called non-singular so the concept is pretty simple instead of vectors your vector field so you have to plug in a point to get a vector at that point right and then the distribution is just the span of all these vectors at that point right rank is basically just taking the rank of those vectors at that point if the rank is independent of the point you plugged in then it is a non-singular distribution okay we will only work with non-singular distributions alright and why what is the purpose of all this mess just if you forget the f let us suppose there is no f then you have a system which looks like this alright and you see at every instant by choosing the control where can you move at every instant what is the direction of your velocity it is in the span of f i right with whatever control you choose instantaneously where can you move you can only move in the span of f i right you cannot move anywhere else that is the whole relevance of all this yeah talking about distribution yeah because this is dictating the velocity so instantaneously you can only move in that direction cannot move in any arbitrary direction okay so if I create a vector field for you which is well known right you have a car it is a standard car you cannot move instantaneously your velocity cannot be orthogonal to the wheel wheel base okay if your car is pointing forward all wheels are forward you cannot instantaneously move this way okay this is a constraint of the system okay instantaneously remember I said instantaneously right so I cannot move sideways okay because that system prevents me if you write the dynamics carefully and you compute all this f i's you will find the distribution does not contain this or that okay anyway that is the motivation for talking about distributions and non-singular distributions we say that a vector field belongs to a distribution f and written as f delta if f of p belongs to delta of delta at p for all p okay a vector field is said to belong to a distribution if f of p is in delta of p for all simple further a distribution is said to be involutive if any two vector fields belonging in the distribution implies that the re bracket belongs in the distribution okay okay so we are asking for more things now okay we already talked about distribution and non-singularity of the distribution now we are saying what it means for a vector field to belong to a distribution that is pretty straightforward right that if you plug in any point then f of p is belongs to delta of p right and then you say that distribution is involutive if there are two vector fields belonging to the distribution means their little bracket also has to belong to the distribution notice the lee bracket is not some linear combination or anything right linear lee bracket is a sort of a bracket operation okay but we know that the lee bracket also gives a new vector field so what are we saying if two vector fields belong to a distribution this modified new vector field also belongs in the distribution okay so this is actually asking for a lot this is called involutivity yeah entire results on controllability are based on involutive distributions yeah remember I told you right once again all of that was very vague because we did not prove anything or use it in anywhere but I told you right that you do not just move along linear combinations of these you also move along lee brackets of these you can also move along lee brackets of these okay so that is the cool thing yeah so that is why we are interested in involutive distributions in general alright so we have a very nice result which really helps us to identify which distribution is involutive in a rather easy way okay what is the result it says as a non-singular distribution generated by smooth vector fields this is involutive if and only if the lee bracket between f i and f j is in the distribution now you might say I it seems like I have asked for everything but actually I have not okay actually I have not okay if I just go by this definition right if any two vectors belong to the distribution any two vector fields belong to the distribution their lee bracket must belong to the distribution that is what is involutivity so if I have f 1 to f k there are k such vectors can you imagine how many combinations I have to check many many yeah yeah because it is not that simple it is not enough to just check this much if I was looking going just by the definition so f 1 comma f 2 has to be in the distribution right f 1 comma f 1 comma f 2 has to be in the distribution f 3 comma f 1 comma f 2 has to be in the distribution you see how it is going and this is only the second successive lee bracket I have to go further further further right many many such successive lee brackets because every time I do a lee bracket I get another vector field right so you can imagine you are going just by the definition checking all of these impossible would make my life help all right so that is what this lemma does it gives you a very nice simple characterization it says that I do not have to check all of these lee brackets and the successive lee brackets and so on and so forth I only have to check this for every possible combination of these k vector fields okay