 So, in the last class what we were talking about was a circle criterion. So, the general idea was that we know if you have a passive system and you interconnected a passive linear system and you have a non-linearity which is passive that means something which lies in the zero infinity sector, then when you interconnect the two you get asymptotic stability. Now one could think of a class of non-linearity which is not really the zero infinity sector but some other let us say K1, K2 sector. So, in the last lecture we went through these various transformations that you can do and these transformations change the given non-linearity which is in some K1, K2 sector into a non-linearity in the zero infinity sector. Now when you have a linear plant interconnected with this given non-linearity and we want to talk about the stability of this closed loop system, we could change the non-linearity from the given K1, K2 sector to the zero infinity sector. But you see that is on the feedback loop. So, then appropriate changes need to be done on the linear plant and when you do the appropriate changes on the linear plant, then the interconnection between this new linear plant and the new non-linearity is completely equivalent to the interconnection between the old linear plant and the old non-linearity. The only difference in this whole process is that now the new non-linearity is in the zero infinity sector which means it is passive and because that is passive, the corresponding new linear system if that is passive, then we know that in this new system with the new linearity and the new non-linearity that is asymptotically stable and because the two systems are equivalent, therefore the old linear system interconnected with the old non-linearity is stable. So, this was the essential idea that was used. Now, transforming a given non-linearity in the K1, K2 sector to a non-linearity in the zero infinity sector, we can do that using these loop transformations. But then if one does not want to do this loop transformation, but you are given a linear plant, you are given a linear plant, the old linear plant and the non-linearity, then by loop transformation the linear plant is converted to some new linear plant which must be passive. But without doing this conversion of the linear plant, if one can predict whether the new linear plant along with becomes passive or not by looking at the Nyquist plot of the old linear plant, then there is some advantage in this. And the circle criterion is one thing which lets us do that. So, perhaps I will just repeat a bit about what we have already discussed earlier. So, perhaps the situation where you are looking at a non-linearity in the K1, K2 sector, and we will see what the linear plant changes to and so on. So, suppose we consider a non-linearity, which is in the K1, K2 sector. So, what we mean by that is if you think of psi as the input to the non-linearity and phi as the output to the non-linearity, well, there is this line with slope K1 and there is this other line with slope K2. And what we are saying is that the non-linearity is such that it lies in the K1, K2 sector. And of course, the other way to talk about this is that phi by psi is, this thing is greater than K1 and it is less than K2. So, this is another way that you can rewrite this, characterize this non-linearity. Now, if one is looking at a linear plant G of S and interconnected with this non-linearity NL in this feedback form. So, suppose we have this particular situation, then we want to talk about the asymptotic stability. I mean under what conditions on GS, I mean what should be the characteristics of GS such that when GS is interconnected with a non-linearity in the K1, K2 sector, the resulting system is asymptotically stable. And then what we had discussed in the last class is that this non-linearity in the NL, I mean this non-linearity NL, this can be converted into non-linearity. So, we can go through it in two steps. So, first you have this NL and you first convert it into let me call it NL1, which is something in the, so NL1 belongs to, so this is a non-linearity in the 0 K sector where this K is K2 minus K1. So, you can do one transformation like this and then this can be followed by another transformation. The second transformation is when you convert something in the 0 K sector to this second non-linearity, which is a passive non-linearity that means it is in the 0 infinity sector. Now, when one does this, then the linear plant which we had here G or G of S that also gets transformed in a certain way and we have talked about it earlier. So, the way it gets transformed is when you take the non-linearity NL to NL1, then the linearity gets transformed to G upon 1 plus K1G. So, this becomes the new linear plant with this non-linearity. So, the interconnection of these two is equivalent to the original intersection that we were interested in and then this conversion from the 0 K sector to the 0 infinity sector makes a conversion here which makes this KG upon 1 plus K1G plus 1. But then we saw that this is equivalent to 1 plus K2G upon 1 plus K1G. So, now this non-linearity with this plant which is 1 plus K2G upon 1 plus K1G, this interconnection feedback interconnection between this linear plant and this non-linearity is exactly the same as the interconnection between this original plant and this non-linearity. And because this non-linearity is in the 0 infinity sector, we now can use the passivity theorem. And so, if this resulting plant from G given G and use K1 and K2 and make this new plant and if this new plant is passive and stable or in other words the Nyquist plot of this new thing lies in the right half plane and it is stable, then this interconnection is asymptotically stable and that translates to this original interconnection being asymptotically stable. But one would like to check that, so what we want to check is the following. So, given G and therefore the Nyquist plot of G, we want to check whether this given 1 plus K2G upon 1 plus K1G, whether this is positive real. And then in the last class, I sort of demonstrated how we do this checking. So, one thing you do is if you look at the denominator, this gives, I mean this gives that the pole of the system is like when G is equal to minus 1 by K1. So, of course K1 is less than K2 and since K1 is less than K2, therefore minus 1 by K1 let us say is this point and minus 1 by K2 is this point. And when you are trying to evaluate this transfer function given G j omega, so we said that suppose you have any point Z here and let me call it Z and you want to evaluate 1 plus K1 for the K2Z upon 1 minus 1 plus K1Z. Then the angle of this is essentially you draw these vectors here and here to here and look at these angles alpha and beta and the angle of this transfer function is going to be alpha minus beta. Now, asking for this transfer function's Nyquist plot to lie in the right half plane is the same as asking for alpha minus beta to be in this range between minus pi by 2 and pi by 2. And then we made the main statement of the circle criterion which was that you look at this circle. Now, inside this circle if you have any point Z, I mean along the boundary of the circle if you take any point Z then 1 plus K2Z upon 1 plus K1Z this has an angle which is precisely pi by 2. Yeah, pi by 2 if it is up there and minus pi by 2 if it is in the lower semicircle if it is in the upper semicircle it is plus pi by 2 in the lower semicircle minus pi by 2. If the point Z is outside then this alpha minus beta satisfies this condition and if the point Z is inside the circle then alpha minus beta in fact turns out to be I mean the modulus value of alpha minus beta turns out to be larger than pi by 2. And this is where we stopped the last time. So, what does this mean? This means suppose you have given this G and you have given this Nyquist plot of G and the Nyquist plot of G lies completely outside this circle. So, maybe this is the Nyquist plot of the circle this is the Nyquist plot of the plant so G j omega. Now this lies completely outside the circle. So, for every point along the Nyquist plot because of the argument that we had given earlier alpha minus beta this angle is going to be between pi by 2 and minus pi by 2. So, what it means is for this particular plant G if you calculate 1 plus k 2 G upon 1 plus k 1 G and plot the Nyquist plot of this new transfer function then that Nyquist plot is going to lie completely in the right half plane. But in the discussion that we had earlier we had said that given a linear plant if the Nyquist plot lies in the right half plane that means the real part of the of every point on the Nyquist plot is positive that does not necessarily mean that the transfer function the given transfer function is positive real. I had mentioned that if one also insists that not just the imaginary axis but all of the right half plane maps into the right half plane then that transfer function is certainly positive real. Now, how does one guarantee that given such a situation where so what we were looking at earlier. So, let us say we have this circle here and let us say this is minus 1 by k 2 and this is this here is minus 1 by k 1 and you have a Nyquist plot and let me think of the Nyquist plot like that. So, G j omega of course this G j omega does not enter into the circle and therefore when you transform G j omega into 1 plus k 2 G upon 1 plus k 1 G. So, let me call it G 1. So, if you draw G 1 of J omega then that will lie completely in the right half of the complex plane. But when you do this mapping of G 1 into this maybe in this particular case I would not know what the Nyquist plot of G 1 looks like but let us suppose that it looks let us say something like this. Now, of course this would be the other half of the Nyquist plot. Now, whether the right half plane under this map G 1 maps inside or outside how does one decide that because that will decide whether the resulting transfer function that you have got apart from being positive real it should also be stable or apart from being positive real it should also be stable or another equivalent definition is that apart from the Nyquist plot being on the right hand side all of the right half should map into the interior or rather into the right half plane. So, how can we now check that with respect to this original you know the Nyquist plot of the original plan which is G j omega. Now, it turns out that the way one does this is very similar to the Nyquist plot criterion that one uses for linear plans. So, let me now try and motivate this interpretation. So, here we go let me draw that thing once more. So, let us say this here is the Nyquist plot that we have and let me suppose that this here is the circle that we had obtained earlier. So, this is minus 1 by k2 and this is minus 1 by k1. Now, we are in this particular situation analyzing this closed loop system which has the linear plant and with a nonlinearity in feedback loop and this nonlinearity, this nonlinearity is a nonlinearity that lies in the k1 k2 sector. Now, what do we mean by this k1 k2 sector? Well, this is also clear this is the line with k1, this is the line with k2, this being the input to the nonlinearity, this being the output. So, what we mean when we say nonlinearity is lying in the k1 k2 sector is that the nonlinearity is something like this. Now, instead of thinking of a nonlinearity, let us think of a linearity, I mean let us think of a linear element that lies in the sector. So, let us say something like this. So, this has slope k where k is in the interval. So, k is in the interval from k1 to k2. So, the slope of this blue line here is k and so instead of the nonlinearity, let us assume the feedback instead of the nonlinearity is really a linear feedback with value k. So, let us assume that this is the portion which is connected and not the nonlinearity. Now, if k is connected instead of the nonlinearity, then the resulting transfer function is going to be g upon 1 plus kg. When can we say that this g upon 1 plus kg, when can we say when is g upon 1 plus kg stable? The way we decide when this transfer function is stable is again by looking at the Nyquist plot and what we should have is that the Nyquist plot should not encircle the point minus 1 by k. So, encirclements of minus 1 by k decide stability of g upon 1 plus kg and how is that done? That is done by using the Nyquist criterion, which is that suppose the original transfer function g was stable, then the number of encirclements of this point minus 1 by k must be 0. On the other hand, if g was unstable, if you recall that there was this theorem if the number of zeros in the right half plane of the transfer function is given by z and the number of poles in the right half plane of g j omega was given by p, then we had something like n is equal to z minus p, this kind of a formula in the Nyquist criterion and what that translates to is depending upon the number of right half zeros or right half poles of g of s, one can specify that this g of j omega should encircle this particular point minus 1 by k the appropriate number of times in the clockwise or the anti-clockwise direction depending upon whether the number of zeros is larger or the number of zeros the number of zeros in the right half plane is larger or the number of poles in the right half plane is larger. So, if one makes the assumption that g is a stable plan for example, then in that case, this g j omega should not encircle the point minus 1 by k and notice that this minus 1 by k is going to be some point here because the slope is between k 1 and k 2. So, now if you look at all these linear plans which can lie between k 1 and k 2 each time you will get some you know for stability. So, suppose you start with g of s which is stable then for the resulting closed loop system to be stable you would say that this g of j omega should not intersect some point minus 1 by k and this point minus 1 by k will vary here between the point minus 1 by k 2 which is what you will have if you take the slope of the linear part to be k 2 or minus 1 by k 1 if you take the slope to be k 1. So, as you vary this k you get all these points in the real part of inside the circle and it says that g j omega should not encircle any of them. That is of course if you start off with a g of s which is stable then this Nyquist plot should not encircle. On the other hand if you start off with some g s which is not stable that means it has poles or zeros in the right half plane then what you would get is that each of the times it should encircle the point minus 1 by k the appropriate number of times in the clockwise or the anti-clockwise. So, you could very well have a g of j omega which looks like that and then for each one of these points this guy might result in so let me call this g 1 g 1 j omega and this is such that for each of these points minus 1 by k it encloses it an appropriate number of times. So, if I also draw its reflection sorry which should go something like that which means that any point here minus 1 by k gets enclosed once and twice in the clockwise direction and so twice in the clockwise direction means the original transfer function suppose the original transfer function had had two poles in the right half plane then if because you have these two encirclements therefore the resulting transfer function is going to be stable. Now that was what the Nyquist criterion told us now that is when you have a linear feedback and for each one of these linear feedback what we are claiming is these points. But now we do not have a linear feedback but we have a nonlinearity and this nonlinearity you can think of as like a linear feedback with a linearity lying in between these slopes k 1 and k 2 with some perturbation. So, you could think of this nonlinearity as something linear like this k but with some perturbations. Now one way to view the circle is that this perturbations from this k are captured here within the circle and so the any nonlinearity in this k 1, k 2 sector can be thought of as a linearity with perturbations and that linearity with perturbations well for the linear parts you get this thing and the perturbation is the rest of the circle. So, if you avoid any point in this rest of the circle and you have a G of G omega which avoids that point but then the original G of S suppose it is stable then in fact this whole circle should not be encircled but if G of S had unstable unstable poles then this whole thing should be encircled the appropriate number of times. Now here is a very interesting way to think of it. Suppose you think of this k 1 going up that means this interval is such that it lies between k 1 and k 2 and this k 1 is allowed to go up. Now as k 1 is allowed to go up therefore the value of k 1 changes and say therefore this becomes so k 2 is kept constant so therefore this becomes another circle a smaller circle and then as it is allowed to go up and up finally let us say this k 1 is made larger and larger until finally k 1 is equal to k 2 then what would have happened is this circle would have shrunk until it becomes just this point minus 1 by k 2. Now if this interval is shrunk from k 1, k 2 to k 2, k 2 that means k 1 has become k 2 then there cannot be a nonlinearity the only feedback that you have is in fact the linear feedback with linearity being k 2. But what that would have meant is that this circle has shrunk down to this one point minus 1 by k 2 and then by the Nyquist criterion for linear plans we know that the number of encirclements of that minus 1 by k 2 by this g j omega would depend upon the open loop g j omegas I mean the open loop plan g s whether it is stable or not if it is stable for example then you should not have any encirclements of minus 1 by k 2 if it is not if it is unstable it has poles in the right half plane then there should be an appropriate number of encirclements of the minus 1 by k 2. So in some sense all this nonlinearity lying between k 1 and k 2 is captured by this circle and that circle shrinks down to a point when you shrink this interval down to making it a linear gain. And conversely if you start from a linear gain and you expand it out then as you expand it out the uncertainty comes out in the form of this circle here you know it expands out into that circle with the appropriate size and if the transfer function does the correct number of encirclements for that for that circle then the resulting system is asymptotically stable. So in a sense it is the generalization of the Nyquist criterion that one uses for the linear plans. So now you could have the various circle criterion for various different nonlinearities and so let us now look at what happens as you change the nonlinearity or the sector in which the nonlinearity is present. So suppose you take this nonlinearity and let us suppose this is slope k 2 this is slope k 1. Now as a result of this what you are going to get here the circle that you are going to get well the circle you are going to get is something like that like so well that might not look like a circle but let us just assume that this is a circle. So this is minus 1 by k 2 and this is minus 1 by k 1 and it is in the k 1 k 2 sector and this is the circle criterion that means the Nyquist plot should not enter the circle. So in some sense the forbidden region so the inside of the circle is the forbidden region so long as the Nyquist plot lies outside you are fine they transform Nyquist plot will lie on the right half. Now let us do one thing let this k 1 so the nonlinearity is lying in the k 1 k 2 sector let us move this k 1 downwards that means the k 1 k 2 the lower limit is made even lower so this is 1 by k 1. So as k 1 is made smaller minus 1 by k 1 becomes a larger thing and therefore the resulting circle is larger so as you have made it smaller so suppose you made it this small then you will end up with a circle which is larger sorry this might not look like a circle but you have to imagine this is a circle and as you keep lowering k 1 further and further this circle becomes larger and larger until until you lowered this k 1 so much that k 1 became equal to 0 that means now you are thinking of the nonlinearity in the sector 0 k 2. Now what is going to happen here this circle that point minus 1 by k 1 is becoming larger and larger until when k 1 becomes 0 this value of minus 1 upon k 1 becomes minus 1 upon 0 that is infinity so it is gone real far off and then the circle criterion essentially tells you that that particular circle is everything to the left which means the Nyquist plot in this particular case if you are looking at k 1 0 and k 2 that means if you are looking at a nonlinearity like this then the Nyquist plot should lie to the right. Now instead of pulling this k 1 down if you keep the k 1 constant and you push k 2 up then this was the original circle when you push k 2 up this gets extended until when you hit infinity it becomes a circle with minus 1 by k 1 here and 0 here so some circle like that if you expand k 1 if you bring k 1 down to 0 and at the same time take k 2 up to infinity and make it a 0 infinity sector well as you are taking k 2 up this point keeps expanding until you get a circle like that and as you keep expanding k 1 it goes off to infinity and so finally you have this imaginary axis and everything to the left of this is the forbidden area so your Nyquist plot should lie completely in the right half plane which is what essentially the result about positive reality is all about. So this result is in fact a more general kind of a result of which the positive real condition is a special case but now interestingly we can do more things. For example, keep k 2 like this and k 1 could be extended to such an extent that k 1 in fact becomes negative so this is k 1 so one is looking at a nonlinearity which can lie in this whole sector so this is the zero slope so in this whole sector so you could think of a nonlinearity like this now what would happen in this particular case so if you go back here you keeping k 2 constant and k 1 you are extending and it keeps going until it reaches infinity and so therefore you have this whole region is the forbidden region after that when k 1 becomes negative 1 upon k 1 minus 1 upon k 1 is in fact a positive quantity which is close to plus infinity. So then what is going to happen is when you are looking at k 1 k 2 interval with k 1 being less than 0 therefore minus 1 by k 1 is greater than 0 and so this minus 1 by k 1 is probably some point here minus 1 by k 1 and minus 1 by k 2 is here so you can take these two points and think of this circle here so you get a circle like this but there is a catch it turns out that now the bad portion is the outside of the circle in other words in other words the Nyquist plot should lie completely within this region and anywhere outside is the forbidden region so earlier we had the circle and the inside of the circle was forbidden and anywhere outside was allowed but now when this k 1 has become negative it turns out that you again get a circle but it is the outside which is forbidden and the Nyquist plot has to lie inside. Now one way to think of this is in the following way so suppose you have the complex plane so what you can do is you have this complex plane and all points which are the infinite points you can think of folding the complex plane up and all the points which are infinity think of them together as one point therefore now this complex plane has become like a sphere. Now this circle that we drew on the complex plane if you now translate it onto the sphere you end up getting a circle on the sphere on the surface of the sphere somewhere you have drawn the circle. Now if you draw a straight line on the complex plane then think about the straight line the straight line if you translate onto the sphere you will mark all the points in the complex plane the corresponding points on the sphere but you see all this infinity you collected up and you had this special point and so on the if you are thinking about the sphere think about the north pole of the sphere as the special point which is all the infinities collected together. So when you are looking at a straight line this straight line goes to plus infinity and minus infinity which means when you translate it into a curve on the sphere it will touch the the the north pole and so straight lines essentially translate into circles on the sphere which pass through the north pole. So if the circles pass through the north pole now this is the good part you had you see in this in this diagram you had the circle and the circle kept expanding that means you had the circle in the left half plane which contained the forbidden region and it kept expanding. Now it kept expanding until it it became a straight line that is when k1 became zero slope. So that gave you a circle which pass through the north pole so you see you had a small circle on the surface of the sphere and the circle kept expanding and the inside of this I mean on the sphere you are drawing the circle and the inside region of the circle is a bad region the outside region of the circle is the good region as far as the surface of the sphere is concerned and you translate that into paper this is what you get. Now as the circle keeps growing finally when it becomes the straight line that means the slope k1 is equal to zero then the circle has grown in such a way that it now passes through the north pole. Now when the slope is further reduced from k1 equal to zero then what happens is that this circle which pass through the north pole has got larger and the infinite point is a forbidden point. Now on the sphere if you have a circle that circle will translate either into a straight line if it passes through the north pole or into a circle. Now the point of at infinity will correspond to the infinite region I mean the point you know the outer region when you translate it into the map it translates to the outer region. So now when the circle expanded so that it became larger this point was the forbidden I mean the infinite point the north pole was a forbidden point so that is precisely what happens here when this keeps expanding out and comes to the other side then the point at infinity is a forbidden point so the forbidden part is the outer part and this part is the nice part. Now if you keep this k1 constant here and now start moving k2 so k1 has a negative slope and now you start moving k2. As you bring k2 down what is going to happen is this nice region this k2 down means this is going to go that way. So this nice region is something which is going to keep expanding because this minus 1 by k1 is constant and minus 1 by k2 keeps going further and further so it keeps expanding that way. Yeah so the nice region keeps expanding but the outside region is the bad region until when k2 hits 0. So when k2 hits 0 this minus 1 by k2 has gone off to infinity which means you have a straight line here and the good portion is this side and the back portion is on to the other side and then suppose k2 becomes negative then minus 1 by k1 is here and minus 1 by k2 would be further to the right and then you have would have a circle there and the interior of that circle has to be avoided whereas the exterior is the good portion. So maybe I would just draw a series of pictures with the circle and I will also draw which is the good region which is the bad region. So suppose you have a nonlinearity like this so the nonlinearity is in this shaded region so the nonlinearity is between k1 k2 where k1 is positive and less than k2 less than infinity. What this translates to is a circle this is the point minus 1 by k2 this is the point minus 1 by k1 and the forbidden region is the shaded part. This is what we first showed. Now if the nonlinearity is expanded such that the nonlinearity is in this sector like this so the nonlinearity now is in the 0 k2 sector then what this translates to here so here you have this outside is the good part. Now here what you will have is that minus 1 by k2 is still there let me probably draw these arrows for the original axis. So minus 1 by k2 but minus 1 by 0 is infinity so what you have is this line like this and this thing that I am shading is the forbidden region this is a nice region. So if you had a nonlinearity like this this circle is there and the circle is the forbidden region if you had the nonlinearity lying between 0 k2 then it is this thing this whole half plane in some sense is the forbidden region and the g j omega has to lie there I mean the Nyquist plot has to lie to the right of this particular thing. Now further if you have a nonlinearity so let me draw the axis and you have a nonlinearity which lies in this area so the nonlinearity is in the k1 k2 sector so this is k1 this is k2 where k1 is less than 0 is less than k2 then what the circle criterion really tells us is because this is less k1 is negative therefore minus 1 by k1 is up here and here somewhere is minus 1 by k2 and so you will have a circle and the forbidden region is the outside of the circle so unlike the earlier case now the forbidden region is the outside of the circle and then if you push further and you have a nonlinearity which lies in a sector like that so this here is k1 this here is k2 so what we have is k1 is less than k2 which in turn is less than 0 now in this case what you have is so you will have minus 1 by k1 and here you have minus 1 by k2 and here you have a circle and now it turns out that the forbidden region is again the inside of the circle so there are all these various interpretations that happen so one quick way to see it is if you have a nonlinearity in the range between k1 and k2 and therefore you can plot the points minus 1 by k1 and minus 1 by k2 and you can draw the circle which connects these two just like here or here or in the earlier two cases like here or when one of the slopes was 0 so that is infinity so these two. Now if the sign of k1 and k2 are both the same like in this case or in this case then whatever is the circle you got the interior of that circle is the bad part but if the sign of the two are different like in this case then it is the exterior that is the bad part so this sort of sums up the various situations that can happen of course k2 could be made equal to 0 or k1 could be made equal to 0 and you have these special cases k2 could go off to infinity which means that in the original axis it will hit 0 there the circle will hit 0 and the special case of 0 infinity sector being the you know the left half is the forbidden part and the right half is the nice part. So that gives the complete sort of interpretation for the various aspects of the circle criterion. Now the other thing is that of course we have been till now asking about global asymptotic stability but sometimes using circle criterion we can talk about local stability and we cannot talk about the global stability. Now what I mean by that is the following so it might so happen that the non-linearity has some characteristics which maybe look like this so let us say it has some characteristics like this so I mean one way that you could write out this characteristics is that the non-linearity is such that f of xi that is a non-linearity is equal to xi when let us say xi is between plus 1 and minus 1 so minus 1 less than xi less than plus 1 okay maybe I make it equal and this is equal to 1 so this is like a saturation so it is like linear and then it saturates it is equal to 1 when mod is xi is greater than 1. So you have some non-linearity like this. Okay so if you have a non-linearity like this now what is the sector under which this non-linearity lies well clearly this slope and the other slope being the zero slope so we could think of it as lying in that sector so you can think of this non-linearity as a non-linearity this non-linearity as a non-linearity that lies in the 0 1 sector because this 1 is 0 and of course this non-linearity lies completely within this particular sector. So now if you use that circle criterion to translate this what that means is so there is 0 and there is 1 so what that means is if you are looking at the Nyquist plot so corresponding to 1 there is this minus 1 and so the everything to the left of this is forbidden and you can have a Nyquist plot lying on the right and then such a plant so such a G when interconnected with this non-linearity will give us will give us global asymptotic stability. But now the following could happen maybe you had a G and that G had a Nyquist plot which looked something like that and let us say that this G was such that it had open loop poles in the right half plane. Now by circle criterion it might be that there is this circle here with the inside being forbidden and this Nyquist plot is such that for this circle the resulting close loops stability can be predicted. But now if this is the case so here this might be some minus 1 by k and this minus 1 by k might correspond to a slope like k here. Now if you look at the original non-linearity the original non-linearity is this one and the original non-linearity of course lies in the 0 infinity sector. But if you have to think of a non-linearity in the k 1 sector so if you have to think of this original non-linearity as lying in the k 1 sector then as far as this non-linearity is concerned it is only up to here up to this value let me call this value alpha and here maybe minus alpha. So it is between minus alpha and alpha that means for the input of the non-linearity lying between minus alpha and alpha can this non-linearity that we have originally drawn this non-linearity here we could think of this non-linearity as lying in the k 1 sector if you restrict yourself to minus alpha and alpha. Now this psi that means input to the non-linearity is essentially the value of the signal on this branch here and if this branch value is restricted to psi mod psi less than alpha then the non-linearity that we are considering this non-linearity has characteristics which lie in the k 1 sector and because it lies in the k 1 sector and the g j omega does not intersect this particular circle we can say for that so long as the psi is restricted to something less than alpha this given system is asymptotically stable. So it is not globally asymptotically stable but it is asymptotically stable so long as you look at only that portion of the phase space where the psi is less than the modulus value of psi is less than alpha and then for that restricted region of this phase space of course this includes the situation when psi is equal to 0 and when psi is equal to 0 the output is also equal to 0 and that in fact is the equilibrium point that we want to get things into so you have the phase plane you have the origin of this phase plane and for psi less than alpha you have an area surrounding it and what we can say from this Nyquist plot criterion is that so long as the psi is less than alpha and you start somewhere for this system with this psi value being less than alpha you guarantee that you reach the origin and so this is like local asymptotic stability as opposed to global asymptotic stability okay so there are various things that you can do with the circle criterion depending upon half how the original nyquist plot looks like yeah so so anyway so with that I guess I am out of time for this lecture so let me stop here now we will continue with the new topic in the next lecture