 So, weird condition one is the passivity condition already unusual and then there is a zero state observed condition which is not too unusual. Let me see if I can try to sort of justify this and let me see I am not sure if I can. Let us go to the sort of linear system case and also we will test this condition in an example. So, you will anyway you know figure it out how to use it yeah it is almost like this Lassalle invariance condition. It looks like that if you see you have a set which is some function equal to 0 and you are saying it does not contain anything but the trivial solution that is the zero solution. So, and what is it a solution of? It is a solution of this system the uncontrolled dynamics. So, this whenever you put the control to 0 you are saying it is the uncontrolled dynamics no solution of this uncontrolled dynamics is here except the equilibrium the zero solution. Equilibrium itself is a solution it is a trajectory well not this but this still a trajectory. Is that clear? Now, let me try to sort of at least make sense of it and try to connect it with our linear system idea yeah this is our uncontrolled linear system LTI system and our output typical right. Of course, we assume things like you know X in R n Y in R p and all that same D typically p will be less than n yeah less measurements than states yeah great. Now, let us see how do you write the solution in this case it is X t is e to the power a t X 0 correct just the using the exponential. So, this will give me the output is c e to the power a t X 0 right. So, how do we why do we have the observability matrix condition? Because if I expand this yeah what is it identity plus a t plus a squared by 2 factorial t squared and so on multiplying X 0 right and of course, you can already start to see I can write this actually I can write this in infinite series if you want this is c c a c a squared multiplied by some something here yes yeah and what you have here is the controllability matrix sorry observability matrix yeah I it does not have to be infinite length because of I can make the infinite length into finite length why this again do not need to check this infinite matrix and only need to check the finite matrix why just 2 words name of a person a theorem why why can I shrink this infinite because this is infinite right I hope you believe that this is an infinite series. So, obviously it is an infinite length matrix, but your typical controllability typical observability matrix is only till what c c a squareds until c a n minus 1 ok why yeah there is due to Cayley Hamilton theorem yeah because all the higher powers are anyway you can write them as smaller powers. So, no need to write all the powers ok. So, this is the observability condition all right great now why this so let us see what is the set this set in this case what is the set in this case it is in the linear case I am saying that x in R n h x is equal to c x is equal to 0 ok. So, if I use my solution this is actually equal to the set of x 0 in R n such that let us see c c a c a n minus 1 ok because everything else is now independent of states right all this quantity this is all independent of states it contains what it contains time and this matrix right because once I use Cayley Hamilton I will just have some complicated functions of time here is that fine just by the way each of these will be infinite series inside this, but we do not care about all that. The idea is this in fact I do not think this is writable like this we have to write it in a different way this is because this is not compatible operation anymore x 0 is in R n and this is the what is the dimension of the Gramian sorry the control observability matrix number of rows is number of rows is p number of columns is n it is a p cross n matrix it is a p cross n matrix right. So, I think you will have to write this not like this I apologize I am going to erase this for those who copied already this is just doing vector math I am not doing anything magical this is x 0 transpose c c a c a n minus 1 transpose times this vector equal to 0 yeah you will get something like this ok you will get something like this why I just did if you notice this is now compatible right compatibility is huge result why this is a 1 by n right this is n by p right and this is whatever this is whatever dimension this is of course appropriate to make it a scalar right ok. So, this is fine this is now compatible and you can see that this is now this is the set now tell me something what is the set of x 0's that will make this 0 by the way this is to be for all t and all t cannot play a spoilsport here right because no in in in when I say that the set is 0 it has to be 0 t there can be no t and all happening there it has to be for all t t cannot mess it up. So, I am looking at set of all initial conditions right such that this product is 0 how did we get this product just from here plugging in the solution for x ok just from here we get this just plugging in the solution for x nothing magical and expanding the exponential and rewriting it in an appropriate matrix product form that is it I mean if you even if you are not convinced about this you can go back and think about it that is not a problem this is it will come out to be like this now tell me something when will this be 0 for what x 0's will this be 0 anybody yes good call for x 0 equal to 0 yes can you give me a slightly more complicated difficult answer like is that the only choice of x 0 for which this will be 0 see because this can do nothing right like you can pretty much forget about this because this is all a function of time and all this 0 thing has to hold for all time. So, obviously this product has to become 0 this product has to become 0 when will this product become 0 yeah somebody was saying something whoever said that word null space null space whenever x 0 is in the null space of this matrix what is the null space of this matrix called I think you guys have either not done this linear system very well or you forgotten unobservable subspace unobservable subspace the null space of this matrix is called the unobservable subspace alright okay no problem okay and this is called the unobservable subspace okay. So, whenever x 0 belongs to the kernel of the observability matrix or the basically x 0 belongs to the unobservable subspace only for those x 0 this is 0 okay alright and once you can find any one x 0 like that you have created a trajectory right because yeah you understand right as soon as a given initial condition I have solved this I obtained a trajectory right. So, for every x 0 in unobservable subspace I get one such trajectory right now if you say there is nothing but the trivial solution what are you saying then you are saying that there is no x 0 in the null space other than 0 itself okay so null space is empty is how we say it 0 we do not count yeah. So, we are saying the null space of the observability matrix is empty which means system is observable okay system is also what we have just codified in this slightly more complicated language is just the fact that system is observable okay what we are calling this 0 state observability is actually the observability condition that you have for linear systems at least for linear systems it boils down to that for non-linear systems you can have slightly more complicated notions of everything yeah which is why it is called 0 state observability yeah. So, basically you are saying that if you if you write this condition out for linear time invariant systems all you are saying is that the system is observable that is it okay all right make sense no confusion all right this ah okay okay your question is why do we say that the system is unobservable okay why do we say that the I defined observability as being able to identify the initial conditions from the outputs okay that was the idea. Now if you look at this expression yeah or I mean or basically this guy is just this guy if you look at this expression the question is can I reconstruct x 0 from y that is the question I am asking okay now if the system is observable let us look at the good case system is observable means this is maximal rank observability means it is maximal rank that is only then the kernel will be 0 right. So, what is the maximal rank p right because it is it is an p by n matrix wait a second did I get this correct this is a p by n matrix or not c is p by n c is p by n this is also p by n this is not a p by n matrix ridiculous I was wrong this did not correct me this is a this matrix is a what p by n times p matrix is that correct or not see this c itself is a p by n matrix right. So, this is also a p by n matrix right wait a second wait a second wait a second c is p by n first of all this is all messed up anyway that is fine it is supposed to be written in that way c is p by n this guy is also p by n everything is p by n and how many such entries do I have but this seems wrong to me it is very wrong yeah actually this is not the observability matrix itself this transpose is the observability matrix this is the observability matrix itself actually. So, it is c c a that is why it is all coming out to be messy in my head yeah because actually I am sorry it is just vector arranging the vector in matrix multiplication nothing magical this product like I said can be written as this guy and this is the observability matrix the transpose with the transpose. So, now the dimension there ok what is the dimension of this it is what p by n n p cross n right n p cross n thank god it is becoming n p cross n did I get this right up we get but like that n p cross n then though this product is not compatible I did not get this right then it is some function of t times c c a c a n minus 1 times x 0 equal to 0 ok this is now correct yeah because this was an n p cross n matrix the observability matrix is an n p cross n matrix and the x 0 is an n by 1 vector this is now correct all right big mess. But the basic point is this expression here can be written as this ok now if this can be written as this yeah and we are saying that this guy is full rank or maximal rank what is the maximum rank of this guy n yeah because p is of course, greater than equal to 1. So, it is n. So, it will be an n ranked matrix which means what which means what so basically you have some n ranked matrix multiplying x 0 it is a solvable system of linear equations it is a solvable system of linear equations ok the simple if you want to think more simply assume p is 1 single output it is a single output system ok if it is a single output system then this is an n by n matrix. So, basically what you have is a n by n matrix multiplying x 0 which means it is an invertible matrix right. So, I can invert it and get a my initial condition ok. So, that is the whole idea I need to be able to compute the initial condition from the given data which what is the data data is the measurements that is observability. Therefore, if you have anything in the unobservable subspace yeah. So, these are all funny it is a funny thing see just think about this ok if I have multiple points in the unobservable subspace x 0 1 and x 0 2 for both of them this is equal to 0 correct for both initial conditions my measurement was 0 for both initial conditions my measurements along the trajectory are 0. So, I cannot distinguish the two initial conditions anymore ok which is why this condition there can be no trajectory, but the trivial trajectory in this set ok. So, anyway this is itself a very nice subtle topic it is not that easy to follow which is why and also I am not teaching this I am more it is more haphazard. So, I was just trying to make a comparison yeah in linear systems the idea of observable unobservable controllable uncontrollable this in itself is a relatively involved topic I mean you have to sort of wrap your head around it takes a little bit of time ok. But the idea is if you have unobservable subspaces then you will have initial conditions which are indistinguishable by measurements you can take you know measurements and you will not be able to distinguish between the two initial conditions ok. So, I mean you can even say simply something like let me see if you have y equal to C e A t x 0 1 and also equal to C e A t x 0 2 these are the same this is the measurement right and two different initial conditions I have the same measurements then then basically these two are indistinguishable and if these are indistinguishable what did I just prove that C e A t notice x 0 1 and x 0 2 are different I just proved that x 0 1 minus x 0 2 is in the null space and x 0 1 minus x 0 2 is non-zero not a trivial vector. So, it is in the null space of this which means observability matrix is not maximal rank. So, you can go both ways yeah and you know sort of understand that you observability that is being able to construct initial conditions from measurements requires that observability gramian or observability matrix be maximal rank ok and that condition is what for non-linear systems is codified in this sort of zero state observable idea yeah again why it has this funny name is because observability has multiple there are multiple observability and controllability notions in non-linear systems ok. So, more complicated ok. So, that is the only reason why we have these multiple notions any questions. So, two important definitions passivity this we have not sort of connected to linear system we might later on as of now no need to connect it it is a just a general motion or general property of systems and the zero state observability which we have connected to observability of linear systems all right ok.