7. Stability via Frequency Response

[Music] oh [Applause] [Music] [Applause] [Music] hi today I'd like to look at the stability problem in a somewhat different way we've found to this point that we have two methods for evaluating the stability of a feedback system one of them we looked at very briefly or principally via homework problems in particular the rowth test and what the Ralph test does is simply give us a a straightforward numerical way of determining the number of poles of a closed loop transfer function that lie on the right half of the S plane hi today I'd like to look at the stability problem in a somewhat different way we've found to this point that we have two methods for evaluating the stability of a feedback system one of them we looked at very briefly or principally via homework problems in particular the rowth test and what the rowth test does is simply give us a a straightforward numerical way of determining the number of poles of a closed loop transfer function that lie in the right half of the S plane and in that sense it simply gives us a binary answer to the stability problem in other words the system is either stable if if all of the poles of the closed loop transfer function lie in the left half of the S plane or it's not stable if one or more lie in the right half as an alternative to that sort of stability analysis we introduced the idea of a root Locust diagram and this gives us considerably more insight into the performance of the system we're able to determine relative stability in the case where system does have all of its closed loop poles in the left half of the S plane we're able to tell how close for example a dominant pole pair is to the right half plane and so this becomes a much more powerful tool for the design and feedback systems today we're going to look at another aspect of the stability problem in particular how can we determine the stability of a feedback system really based on frequency response measurements performed on the system and we'll see that this gives us yet another tool for looking at the stability of a feedback system and again a very powerful design tool why are we interested in an alternate technique root Locust certainly is a a fairly General way to look at the stability problem well one is just that there's certain systems which are more easily investigated by root Locus other systems are more readily investigated by frequency domain kinds of approaches and we can find different kinds of information via these two approaches uh but there's another reason and that is that uh there are certain sorts of systems that really don't lend themselves readily to analysis via root Locus techniques we realize that in order to perform a root Locus analysis we have to have the loop transmission of the feedback system expressed as a ratio of pols in s and in particular a finite ratio of polinomial in s a finite number of poles and zeros but not all transfer functions are expressable in that form for example a time delay has a transfer function e the minus s where to is the magnitude of the time delay and this transcendental function is not expressible as a ratio of polinomial where the polinomial polinomial themselves are finite so there's an example of a transfer function that really can't be used directly in our root Locus analysis another case of of major practical interest involves the case where the information concerning the loop transmission is available only in experimentally measured form in other words we may have measured the frequency response associated with Loop transmission we can if we want to curve fit and estimate that Loop transmission as a collection of poles and zeros but oftentimes that's a cumbersome operation and subject to inaccuracies so again it'd be helpful if we had a technique that could work directly with the measured frequency response of the loop transmission and would allow us to evaluate the stability of the system using that information we start again with a system in our standard form an input an output a forward path transmission small a of s and a feedback path transmission F ofs when we started looking at the stability problem we mentioned that we can give a physical idea of why we might have instability in some systems by considering what happens if we break the loop and look look at the Loop Transmission in other words we proposed a test where we apply a test generator to the system here and look at the signal that the the loop returns possibly V subr and we found out that for certain systems in the absence of an input we'd be able to get V subr to be identically equal to V subt under sosal steady state conditions for some frequency of excitation when that happens we said well we could then imagine oscillation because we'd be able to reconnect the loop reclose the loop take away our generator and the system would continue to to oscillate forever in particular we looked at this situation uh for a system which had a loop transmission that included three coincident poles on the negative real axis and we found out that for that sort of a system if we had a low frequency Loop transmission magnitude a FN of 8 the system in fact was capable of sustain constant amplitude oscillations I also mentioned that there were certain problems with that sort of analysis that it wasn't a completely General one and that it was possible that we could get into trouble with it and I'd like to go back and and look at another case of this and see where we might possibly get into trouble with that sort of simple-minded frequency response stability analysis again let's consider a system where we're making Loop transmission measurements and suppose we had an a of minus1 and an F of one at some frequency in other words when we make sosal steady state measurements on the system we make the input zero we find that uh for some frequency of excitation a has a magnitude of 10 and provides a 180 Dee phase shift F has a magnitude of one with no phase shift consequently the loop transmission which includes one more inversion or one more 180° phase shift at this point in the diagram has a magnitude of + 10 and we suspect that that something might be wrong with that uh if we put in a test signal here get out a signal that's exactly in Phase with our test signal uh but 10 times as large we might suspect that there's potential difficulty with that sort of a system well there may or may not be as it happens we can't really tell about the stability of a feedback system by looking at its frequency response at only one frequency we really have to look at the frequency response of the system over all frequencies in order to say anything definite about the stability of the system let's consider the system that we mentioned earlier where a is equal to minus1 and F is equal to 1 we simply went blindly ahead we could write the Clos Loop transfer function of the system at the particular Omega which gives us the a of minus 10 and the F of one as simply the forward path over one minus the loop transmission but the forward path is now minus 10 uh the loop transmission is itself + 10 so we get 1 minus 10 in the denominator and we conclude that the input to Output transfer function at that frequency would have a magnitude of 10 9ths that's self-consistent let's consider what happens when we put in an input for example of one unit our equation if it's correct tells us that the output will then be 10 9ths and let's see when we subtract the 10 9ths from the one unit we'll get minus 1 9th but at this particular frequency we've said that a of s is minus10 and when we multiply the - one 9th signal that exists at this point by minus1 we in fact get 10 9th so the arithmetic seems to be self-consistent and so it's at least superficially possible for this system to behave in the manner that I've described in other words behave in a stable manner even though at this particular frequency it has a loop transmission of + 10 well as I say we have to look really at the characteristics of the loop transmission at all frequencies in order to determine stability this simple evaluation at one frequency isn't enough and in fact can can give us erroneous results the way we determine the stability of a feedback system based on frequency response measurements in particular the frequency response associated with Loop transmission is called the Nyquist test and here as we started with root Locus we look at the characteristic equation for the system which is simply one minus the loop transmission and in our standard notation that's 1 plus a of s f of s and again exactly as in the root Locus development we recognize that closed loop poles of the system will lie at values of s where this term goes to zero the characteristic equation is of course the denominator polinomial of any closed loop transfer function of the system and so the closed loop Pooles of the system lie at values of s where this quantity goes to zero or correspondingly uh at values of s where a of s f of s equals minus one that's those of course are the values of s for which this quantity goes to zero our system will be unstable if the closed loop Poes exist for values of s with positive real parts and other words if there's there if the system has closed loopholes lying in the right half of the S plane so our stability test is uh to determine if there are any values of s with positive real parts for which a of s f of s equals minus one that's the case if there's one or more than one value of s with a positive real part for which a of s f of s equals minus one the system is then unstable how do we make this determination well what Nyquist what the Nyquist test suggest that we do is the following we perform a mapping we evaluate our the negative of our Loop Transmission in particular the AF product for values of s so what we do is a mapping that takes values of s in the S plane and transfers them to an AF plane remember that AF of s is simply a function of S and S is a complex variable in general with a a real part and an imaginary part and so when we substitute substitute into AF some specific value of s corresponding to a point in the S plane we get a specific value for the AF product again a complex number and we choose to write that AF product in a particular form we choose to use a so-called gain phase representation for the AF product in other words here we have a gain phase plot in the AF plane if you will the this axis is magnitude or gain generally for the same reasons that we do with a bod plot why we use a logarithmic presentation so here we might have unit magnitude a magnitude of 10 100 and so forth a tenth 1/ 100th again as I say a a logarithmic scale is usually most convenient because of the large dynamic range associated with the magnitude of the loop transmission and this axis is the angle associated with AF 0° in the middle plus 90° Plus 180° and so forth and then negative angles to the left in this gain phase plane again we'll see many uses of a presentation of the loop transmission or the negative of the loop transmission the AF product in a gain phase plane but the important point is of course we can go from a point in the S plane substitute in that value of s to the expression for AF and determine a magnitude and angle which we then plot in the AF plane and there's certain properties of this mapping that are useful for our investigation of stability in particular if we evaluate the AF product as s takes on values lying on a small circle in the S plane we find that we get a small circle in the AF plane we also find out that if we evaluate the AF product again for values of s that lie in a closed Contour in the S plane possibly here when we evaluate the AF product as s takes on this sequence of values we again get a closed Contour in the AF plane possibly this furthermore all values of s that lie to one side of this Contour in the S plane lie on one side of the Contour in the AF plane there's an issue of insided this outsid this here and we have to investigate that separately but for example if we found that a value of s here for example inside what we conventionally think of as being inside the Contour in the S plane uh for example lay here what appears to be outside the Contour in the AF plane then all value Val of s inside this Contour in the S plane would give rise to values of AF outside this Contour in the AF plane there's one additional property that we we'd like to use and that is that if we make a small right angle turn in other words we evaluate um AF four values of s that take on a right angle in the S plane uh we again get a right angle in the AF plane possibly this uh as it happens there's a because of the choice of coordinates uh when we make a right hand turn in the S plane in other words if we evaluated AF along this path and that direction such that we took a rightand turn why we'd find that we actually take a leftand turn in the AF plane and that simply has to do with the choice of axes for the angle had we chosen to plot positive angles to the left rather than it's conventionally done why the the right hand turns would have been maintained well how do we use all of this to investigate the stability of a system the secret is to evaluate the AF quantity along a contour that concludes the region of Interest we're interested in whether there's any values of s with positive real parts that give rise to minus ones of the AF product and so what we do is evaluate the AF product along a contour that includes the entire region of interest in particular the entire right half of the S plane all values of s with positive real parts of course lie in the right half of the S plane so what we do is choose a contour in the S plane that includes the values of f s of interest and the one that does that is is one that includes the imaginary axis and then is eventually closed with a very large semicircle that includes the entire right half of the S plane so what we do is evaluate s on a contour like this that includes the region of interest and what we want to know is is whether there are any minus ones of the AF product that correspond to values of s in this region well the thing that gives us some Leverage is the fact that the minus ones are very clearly discernable in the AF plane if we look at the AF plane let's consider how we identify a minus one in the AF plane well a minus one of course is a magnitude of one and an angle of 180° or minus 180° or in general odd integral multiples of 180° so we have a minus1 right here magnitude of one and angle of 180° we have another minus1 in the AF plane minus 180° with a magnitude of one and we have more and more of these we have one another 360° displaced from this at Plus 540° at minus 540 degre and so forth so the minus ones are very easily observed in the AF plane and that's the thing that that gives us leverage in or makes the Nyquist test a useful way of finding the stability of a system system let's look at how this might work here we show the Nyquist construction we again are looking in the S plane our usual axes the imaginary axis here J Omega the real axis and we choose a contour for our Nyquist test that includes the entire region of Interest usually we start at s equal 0 plus just a small amount of J in other words at at this point in the S plane evaluate the AF product at s equals 0 plus j0 plus if you will and then we continue evaluating the AF product as s takes on increasing values along this line an imaginary component but no real component we get magnitudes and angles for the AF product corresponding to the values of s along this line and of course those are simply uh the values that we'd get if we made frequency response measurements because evaluation of the AF product for S equals J Omega is exactly what we get when we evaluate the frequency response or measure the frequency response of our system so we can get the values of the AF product by experimental measurements and as I mentioned earlier that's one of the advantages of the nist test we can use experimentally determined data in order to look at the stability of a system uh we finally have reached a point in our construction where we have gone very very very far from the origin in the S plane in other words out to S equals still zero real part plus Jr where R is a large number we then continue the Contour into the region of interest uh along really almost any path but conventionally a large semicircle is chosen so we let s equal r e j Theta as Theta goes from plus 90° to - 90° and then we continue our Contour here uh going from s = 0us Jr uh back to S = 0 + j 0 minus so this is the Contour that we use to evaluate the AF product and when we do that we get some cont tour in the AF plane and I've assumed this is the one that we've gotten here's the value of the AF product for S equal 0 + j0 plus as we increase Omega let s equal 0 plus J Omega only an imaginary component the real component being zero we assume uh for this particular illustration that the curve follows this path in other words its magnitude gets smaller here we have the magnitude in the gain phase AF plane the magnitude gets smaller in this direction the angle gets progressively more negative for increasing Omega and eventually we get down to some very small magnitude for S equal plus Jr we then have to look and we'll we'll do this in detail with a specific example but we then have to look at what happens uh as we continue into the right half of the S plane what happens to the AF product as we continue into the right half of the S plane along our large semicircle let say We'll look at the specifics of that for a numerical example uh and we find out of course that since AF uh has to be a a physically realizable transfer function or has to correspond to a physically realizable transfer function why there has to be symmetry in this AF plot and that symmetry is is shown in other words once we've plotted the first half of the curve starting at s equal j0 plus going out to S equal J are and then continuing along a large semicircle back to a point that corresponds to the positive real axis at at a radius R from the origin once we've drawn that much of the curve the rest of the curve has to be a mirror image of this leftand portion uh reflecting the fact that the poles and zeros associated with the AF product have to occur in complex conjugate pairs well here we see the minus one points there's two of them and our only remaining problem is the determination of insided outsid andness in other words we look at our original construction and we readily identify this as the inside of of the Contour what appears to us to be the inside of the Contour in the S plane and if in fact we can find that there is one point in here that plots in here in the AF plane then we conclude in this particular figure that there are two values of s with positive real parts that give us minus ones of the AF product that's clear because here's a minus1 here's another minus one inside our Contour in other words if this was the region that corresponded to the Shaded portion of of the Contour in the last figure uh then in fact we found two minus 1es of AF that correspond to values of s inside that shaded region in other words values of s with positive real parts so our system would then have two closed loop Poes lying in the right half of the S plane we'd conclude that it was unstable let's apply this uh to a specific example where we assume that the transfer function is in fact expressable as a ratio of polinomial in s and let's see how that goes let's choose an AF product that's 10 3 uh over s + 1 * .1s +1 * 01 s +1 in other words we uh have an a FN a DC Loop transmission magnitude of 10 3 then we have three real axis poles here we show that situation uh in fact to Aid our Construction in the S plane why we show the three Loop transmission poles we'd also probably show zeros to help ourselves if any existed uh here we have a pole at minus1 a pole at Min -1 a third pole at s equal uh minus 100 and here's the nyis Contour that we're interested in and we can see at least quickly what's happening the the the general nature of the plot in the AF plane by considering vectors drawn from these three poles to some point on the imaginary axis and we notice of course that this function would have a maximum magnitude at s equal zero and then since the length of these three vectors to a test point on the imaginary axis increases for increasing values of Omega why we' we'd find that the magnitude of the AF product would decrease we know it's DC value we know its value for S equals z and the magnitude will then decrease as Omega increases furthermore as we go to very large values of Omega way out up here in the top of the plane why the angle approaches minus 270° we eventually get- 90 degrees of course from each of the three poles so the angle of the transfer function eventually approaches minus- 270 degrees 3 times Theus 90 once we're out to a very large radius and continue around why our magnitude stays fixed if R is very large compared to minus 100 then as we close the Contour why the lengths of these three vectors stay fixed uh as we go along a radius of capital r and we notice that the angle associated with the AF product must Return To Zero when our we're at this point on the Contour because again if we drew vectors from the three poles in question the angle associated with each of those vectors would be 0 deges they'd lie along the positive real axis and so that's sort of a an argument allows us to construct the corresponding plot in the AF plane and this is what happens when we do that here we start out at a value of a thand a magnitude of a thand 1 10 100 a th000 a magnitude of a thand for S equal j0 plus the as Omega increases the Ang Ang Le becomes negative ASM totically approaching - 270° for S equal plus Jr uh the magnitude then stays fixed as we circle into the right half of the S plane for our evaluation and the plot is of course symmetric if we are careful in this construction uh we find out that the minus1 points lie in the Shaded region for an a fnot of 1,000 and so our only final task is to determine whether this shaded region corresponds to this shaded region and the way to do that or at least one way to do it is to take a test detour uh incidentally we could do this by depending on the property that I mentioned earlier that when we take a right-and turn in the S plane we actually take a left left hand turn in the AF plane so here notice we've taken a right hand turn at this point in the S plane and we find that we take a corresponding leftand turn in the AF plane like so and that in fact tells us that the Shaded regions correspond but another way to see that and one that I find physically more appealing is to take a little test detour let's look at a specific point inside this Contour in the right half of the S plane we can do that here we can assume we're at this point and then move in this direction a small amount when we do that the magnitude must drop since the vectors from the poles all increase in length so the magnitude of our function must drop the angle remains 0 de if we walk along the positive real axis so if we evaluated AF at a point in here its angle must be 0 degre but its magnitude must be smaller than the magnitude that corresponds to our evaluation here and so we conclude that as we move along the test D tour shown in the previous figure we move in this direction consequently the Shaded regions in the two figures correspond and that tells us in this particular example that there are two values of s with positive real parts that will lead to minus ones of the AF product here's a minus one here's a minus one there must be two values of s in the right half plane that give rise to those values of the AF product consequently we conclude that this particular system had two closed loop poles in the right half of the S plane it's unstable we of course suspected that already we can we can get this this same development by another technique that we know we can look at this from a root Locus point of view and here we draw a root Locus diagram for the same system uh we have a pole at minus1 a pole at minus10 a pole at minus 100 and we find out that if we do the root Locus construction this is a familiar pattern these two branches come together meet at about the arithmetic mean somewhere around minus 52 and then Branch off eventually reach asmp tootes that make angles of plus and minus 60° and a third branch goes off into the left half plane and what that tells us is that for sufficiently large values of a fnot the system has two closed loop Poes with positive real parts here's one and here's one once the a fnot product gets large enough we see exactly that behavior in this diagram this apparently is an a fnot product that's large enough for this particular location of the three real axis poles or the particular relative location we find out that an a fnot product of a thousand is enough we can convince ourselves of that for example via a ralth kind of development but uh that's shown in the Nyquist construction that we have two minus ones of the AF product uh that result four values of s with positive real Parts if we lowered the a f product that would correspond to Simply sliding the Shaded curve downward we lowered it enough we'd finally get it down when we lowered it about this much we'd finally get the curve down to the point where the minus ones were outside the Contour then the system would be stable and that corresponds to the condition where we've lowered we started a high Loop transmission magnitude a high aot fnot product we have two poles in the right half of the S plane we lower a FN and eventually the system becomes stable so we see exactly that same behavior as we must in the root Locus presentation well again as in the case of of the root Locus development we're interested in addition to a binary answer to the stability problem sort of a relative measure of stability usually we're able to fairly easily convince ourselves that the system is stable or fairly easily get the system to be stable in an absolute sense but the question is sort of how close to instability is it or some of the what are some of the useful measures of relative stability uh what are the values of some of the useful measures of relative stability really how close to instability are we and so forth and we we saw we could get that very easily in a root Locust construction but since we had the location of all of the closed loopholes of the system and we could tell for example the damping ratio associated with the dominant pole pair and and any other measure of relative stability that we'd like to have we have the information available what we do in this sort of a development in other words when we evaluate stability based on frequency response uh is to really look at features of the closed loop frequency response of the system and in particular look at things for example like the amount of peaking that we get when we look at the closed loop response of the system we excite the system look at sidal steady state response and we see for example the amount of resonant Peak we get we feel that that a larger resonant Peak indicates a relatively less stable system the evaluation is most easily made if we consider a system where the Dynamics are all in the forward path so I'll I'll assume a system uh input output and we assume that all of the Dynamics of the system are concentrated in the forward path and furthermore again just to normalize things we'll assume that we have Unity feedback and then what we're going to do is go ahead and look for basically resonances or peaking in the sidal steady state response that links V out to VI let me point out that this really isn't any any restriction suppose we had a general system where we were interested in the frequency response we'd like to determine uh V out over VI for some value of Omega uh and we had in general the Dynamics distributed between the forward path and the feedback path well we could manipulate the block diagram as follows this block diagram is identically equal to the lower one where we have the AF product in the forward path Unity feedback and then follow that by a block that's 1 over F of J Omega and we can really evaluate the relative stability of our system by considering just this part of the system so we really don't lose any generality when our interest is the evaluation of stability we found out in a root Locus sense that depends only on the AF product again when we look when we uh base our stability analysis on frequency measurements once again we're only really concerned in the about the AF product and we can determine our stability information from this portion of the block diagram corresponding correspondingly this one so what we'd like to do is look at the behavior of go over 1 + G where G is a function of J Omega we consider the closed loop gain a of J Omega being equal to G of J Omega over 1+ G of J Omega and we can very easily evaluate that for limiting cases because we recognize that this quantity is just about uh equal to one when the G of J Omega product uh is very very large in other words I should have an equals one here when the G of J Omega product is very large under those conditions when G of J Omega is large compared to one uh the numerator denominator are just about equal and so the magnitude of this term is is just about one so we have one limit at the other end when G of J Omega is very small uh the denominator is dominated by the one term and under those conditions the ratio G over 1 + G is just about equal to G itself under those conditions G is not important in the denominator the difficulty in the evaluation comes when we're working at magnitudes of G of about one there the angle of G has a very profound effect on the quantity G over 1+ G for example if the magnitude of G is close to one but its angle is close to- 180° in other words G is about minus one we have a very very large magnitude for the G over 1 plus G product conversely if the magnitude of G is about 1 and its angle is 0 degrees we have a magnitude of about a half uh for the G over 1+ G product so the difficulty is finding out or the place we have to focus on is what happens for magnitudes of G close to one uh at the extremes for large magnitudes of G or for small magnitudes of G uh it's fairly clear what the answer is well we'd be able to effectively make a dictionary that relates G uh over 1 plus g to g itself G is a complex number we could express it for example in gain phase terms as a magnitude and an angle and then there's some mapping from G as a a gain phase quantity a magnitude and angle to G over 1 plus G as a magnitude in angle and we could simply make up a dictionary that listed possible combinations for the magn itude and angle of G uh and the corresponding magnitudes and angles for G over 1 plus G it happens that a way that's a little bit more convenient for our purposes is to not make a dictionary but rather use a device called the Nickels chart and here's a nickels chart it's a little bit confusing uh but what there is is a grid in the background which are gain phase coordinates here's the magnitude of G uh the angle of G along here here we have a magnitude of G running from 10us 3r up to 10 3 here we have an angle of G running from 0 degrees to minus 360° and this plane repeats every 360° so we we have all the information that we need and what we're able to do is go in uh for example we have a g of 10 and an angle of -60° that's a point in here somewhere and we can determine corresponding values for G over 1 + G from the coordinates the the curved lines on the figure so if we went into our Point magnitude of g equals 10 uh angle of G is -60° that's somewhere in here well let's see we we'd be about on this magnitude curve for G over 1+ G that's a magnitude of about 0.9 and we B if we cared on on about this angle contour for G over 1+ G uh the angle of G over 1 plus G would be about Min -10° in the Nickels chart the critical point is of course the minus one point a magnitude of one and an angle of- 180° and what happens uh as we approach that point here's a magnitude of one here's an angle of- 180° and as we approach that point the magnitude Contours get larger and larger and larger here we're starting at a magnitude just slightly above 1 and now we're getting closer to G equal 1 here's G = 10 and an angle of minus 180° and we have a magnitude of about uh 1.1 or 1.2 at that point something like so about 1.1 and now we're going down to a magnitude of the square root of 10 and we have a magnitude for G over 1 plus G of about 1.5 and here as we approach the minus1 point the magnitude gets larger and larger there'd be uh little ellipses in here with progressively larger magnitudes as we got closer and closer to the minus one point very nice way to look at this is to look at a three-dimensional Pres presentation of the Nickels chart something that we might call Mount nickels for example and here we have exactly that situation we are looking at precisely the diagram we have on the uh view graph uh where we have magnitude along this axis here we're at a magnitude of a tenth uh a magnitude of one in here and the magnitude increases in this direction for the magnitude of G and the angle of G runs along this axis uh here we have an angle of of 0 de and an angle ofus 360° over at this Edge and now the magnitude of G over 1 plus G is the height above the plane we have have sort of a contour map here actually a three-dimensional contour map and we get the value the magnitude of G over 1+ G as the height above this plane and so we can Envision uh uh for example drawing uh uh a series of values for the AF product as Omega changes evaluate AF of J Omega or G of J Omega for various values of Omega that'd be some Contour in the G plane it would have a magnitude and an angle for various values of of Omega and the height the magnitude of G over 1+ G would be the closed loop magnitude that corresponded at various frequencies when we had a a g at this point for example uh that's about a magnitude of one and and maybe an angle of U oh minus 150° or something like that why we'd find out that that we had a magnitude that was this much that's our relative elevation above the uh Unity Point and in particular the behavior that we see is that as we approach the minus1 point as we approach a magnitude of one in here right here and an angle of minus 180° right here the magnitude gets progressively larger and precisely at the minus one point of course the magnitude goes to infinity and we can emphasize that I think in a in a somewhat graphic Way by putting a flag on top of Mount nickels right at the minus one point where the magnitude gets to be very very close to Infinity uh incidentally uh this is just really a pitch for the equivalent of this subject that I teach at MIT 6302 is the number of a feedback system subject that I teach at MIT and we didn't change the flag for this particular presentation usually we don't need a nickels chart that extends over anywhere near as much area as the one that I've shown currently on The View graph we argued earlier that we were able to very easily estimate the behavior for large G magnitudes and for small G magnitudes so really we're only concerned with what goes on near a magnitude of g equals 1 and so normally when we're working with this Aid we use a an expanded nickels chart that focuses on the minus one point so here we have a nickels chart CH that covers only the range of angles from -60° uh down just slightly Beyond - 180 degrees and magnitudes again that Center about the minus one or or the magnitude of one point here's one a tenth 100 10 and so we focus on the part of the gain phase plane that's of interest in particular the portion of the gain phase plane that uh corresponds to magnitudes close to one for the loop transmission and uh angles of the AF product ofus 180 degre corresponding to a positive Loop transmission with a magnitude of one so we focus on on this corner of the a of the Nickles chart this corner of the gain phase plane which is of greatest interest to us when we're evaluating the stability of a system and we use this is our measure really of relative stability we can imagine plotting a gain phase plot for a loop transmission we can plot an AF plot really just the leftand portion of the Nickels of the Nyquist construction we could plot that in this plane Omega would be a parameter running along that curve and what we'd do is find uh we'd be able to get the Clos Loop transfer function by simply evaluating the magnitude and if we cared to the angle of G over 1+ G uh as we progressed along the Contour in the a in the gain phase plane next time we'll look at an actual example of how we might do this construction and then continue with with our evaluation or our stability evaluation based on frequency domain Concepts thank you