# Lec-37 Mapping of Control in the Complex-Plane

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### Lec-37 Mapping of Control in the Complex-Plane We have been looking at the root locus method for getting some idea of the transient response of the feedback control system or even open loop control system and we saw that some of the ideas associated with the root locus can be used to determine the value of the gain beyond which the system may become unstable Now we look at another approach to the same problem which is based on what is called frequency response Historically, the root locus method was invented in the 1950’s whereas the frequency response approach to the feedback control system analysis and design was developed in the 1930’s and 40’s. As I told you earlier, people who have working with telephone network, first came across this problem of stabilizing of amplifier and Harold Black invented the idea of or rather applied the idea of feedback to stabilize amplifiers and because of this people who were studying amplifiers till that time started looking at feedback in amplifiers and this is how the frequency response method was developed. In the case of an amplifier, whether it is useful speech which was the main application transmission of voice signal over long distances, over a large telephone network. This was the first application, today of course one can talk about other communication networks wired or wireless Now in these applications, frequency responses are very important concept a signal being sinusoidal is a very meaningful signal for example, a pure tone if one looks at it on the oscilloscope looks like almost a perfect sinusoidal signal and parts of speech also look like periodic waveforms and therefore, we know from Fourier series approach that it can be a thought of has consisting of number of sinusoidal signals because of this, it is quite natural to think of frequency response when studying amplifier performance and so, the frequency response techniques were developed in the 1930’s and 40’s for not only studying the performance of amplifiers but also studying the effect of feedback on the behavior of the performance of amplifiers So we will take a look at it now, although it seems that the root locus method which was invented later does a better job for this purpose than the frequency response base analysis for method. Now, we will start with the very simple result which is as follows you have all studied functions of a complex variable in your mathematics courses. So, suppose z is a complex variable that is z stands for any complex number, what so ever then you might have a function of z which is also a complex number that is its complex value of function of a complex variable and typically in the maths courses, the symbol z and q are use for the 2 complex numbers. So w equal to f z represents what is called a mapping or a transformation of the complex plane into the complex plane and in order to show this diagrammatically, one shows the 2 numbers z and w, start with the complex number z apply the function f to it, you get the value f of z that is another complex number w So these 2 numbers z and w are shown into 2 different planes that is you think of instead of 1 single complex plane, you think of 2 different complex planes. So one is call the z plane, other is call the w plane. So let us say here is the z plane, in other words in this plane or in this part of the diagram, I am going to represent the complex number z and here is the other plane. Let us say, we call it the w plane that is in this part of the diagram, I am going to represent the value of the function at a complex number z So for example if I choose a complex number z, here it may be some complex number like, let us say 2 plus j 3 g, right. It is a complex number in the first quadrant of course, I am not put the real and imaginary axes heading here. So if I do that this is the positive real axis, this is the positive imaginary axis for the z plane likewise, I will have the real and imaginary axis that is real and imaginary parts of w, in the w plane. So let us say there is some function f such that its value it 2 plus j 3 is let us say minus 1 plus j 3. So where is the point minus 1 plus j 3 all that will be a point 1 plus j 3 all that will be a point somewhere here, here is the point minus 1 plus j 3, the real part is minus1 the imaginary part is plus 3 and so, one says that this point 2 plus j 3 is mapped into the point minus 1 plus j 3 under the function or mapping f and this function or mapping f, we will do different things to different points in the complex plane, different points in the complex plane will be mapped in the complex plane, z plane will be mapped into or taken into different points in the w plane and one uses this terminology of mapping z is transformed into w under the action of f or f maps a point in the z plane into point in the w plane So this is the way in which one looks at such functions and what they do to complex numbers Now, let us consider a very simple complex function namely w, which is f z is simply given by let us say Z minus 1. So, if I start with the complex number z then what this function f does it, it simply subtracts from it the number 1 and I get a new complex number and that is the value of the function, it is a function as simple as that Now, let us look at the following. Here is the z plane, in the z plane I draw simple closed curve and a very special one, I will draw what is called the unit circle. So suppose this represents the unit circle in the z plane, what does this mean to say that this is the unit circle in the z plane, it means that it is a set of all points z such that modular sub z is equal to 1, it is in way is equation of the circle unit circle in the z plane, mod z equal to 1. So, here is a point which is on the unit circle so here is the complex number z and this is the corresponding vector drawn from the origin to the complex number z or this vector can also represent the complex number z Then, the modulus of this z which is the length of this vector must be 1 and so all the points which lie at distance one from the origin will satisfy this condition and conversely, any points satisfying this condition must lie at a distance of 1 from the origin. So this is a curve in the complex z plane, it is a simple closed curve and you should go back to your complex variables or complex functions course and find out, what is the simple closed curve. Now one goes a little further, one thinks of tracing the curve or moving along the curve in a particular direction For example, I will put the arrow here this way which means now, I am going to trace the curve or move along the curve in the counter clockwise direction Now, such a curve with an arrow on it which indicates the direction in which you want to trace the curve is called a contour. So, I have chosen a contour in the complex plane and it is conventional to use the Greek letter capital gamma to denote any such contour So in this particular case, I have a closed curve gamma in the z plane which is traveled along or travels in the counter clockwise direction. I have this curve but there is a direction given a or marked on it which is the direction in which I am going to travel     is not the only thing that you will come across when we use this, for our study of the feedback, effect of feedback on control system performance Okay, let me get back to the problem then, here is the z plane, here is the w plane and I have a simple function w equal to f z, where f z is given by a simple formula z minus 1 Now I choose to start with a simple closed curve in the z plane and let us go back and again choose the unit circle and trace it in a particular sense or direction. Let us say I have chosen the counter clockwise direction, so as the point moves around this curve this origin of the z plane is circled exactly one, the number of encirclements is plus 1. Now think of the corresponding point w that is for each point z here, I calculate equal to z minus 1 that gives me a point in the w plane, do that for each point on the curve I will get a curve in the w plane. This curve is called the image of the original curve gamma under the action of the function f and because this is gamma, points on gamma are being acted upon by f to produce points on this image curve, sometimes this image this curve is represented as f gamma. Actually, in your complex functions course probably you have come across something like but at that you time you may not have paid much attention to it Well, now is the time to either revise it or pay attention because this is going to be useful in our study of performance closed loop feedback control system. In particular investigations of stability of a feedback control system. So, let us look at the simple problem of a mapping of the z plane in to the w plane that is we have a function w given by f of z of a complex number or variable z, w also is a complex number and let us take it to be the simple function z minus 1. Now looking upon it as a mapping of the z plane into the w plane, we would like to see what the mapping does to a contour or simple closed curve, I have chosen this simple closed curve here which is the unit circle around the origin in the z plane, I have called it gamma and I have chosen the orientation as shown by this arrow the radius of the circle is 1 So the points on this circle where do they go in the w plane. We can choose a few simple cases for example, take the point 1 j 0 or 1. So if z is 1 what is w, w is z minus 1 w is 0, so in other words the point by 1 j 0 is mapped to the origin of the w plane, this is now we talk about it A point in the z plane namely 1 plus j 0 is mapped into a point in the w plane namely w equal to 0 by the function f z, in this case the function is z minus 1. Let us take another point the other extremity of the diameter through the origin minus 1 plus j 0. So z is minus 1, so what is w? W is z minus 1 and therefore it is minus 2 or minus 2 plus j 0. So this point minus 1 plus j 0 will be mapped to the point minus 2 plus j 0 and in fact, you can now see what is going to happen, the action of the function is to subtract the 1 from the number z Now subtracting 1 does what to the complex number. It does not affect the imaginary part because your are subtracting only the real number 1 but it affects the real part, what happens to the real part, since you are subtracting 1 the real part is reduced by 1. Therefore, you can see now that this circle gamma will be mapped into a circle, gamma prime or as I said we can call it f of gamma. It is called the image of the gamma under the function f as this curve gamma is traversed in this counter clockwise direction, the curve gamma prime or f gamma will also be traversed in the counter clockwise direction and the circle image mapped into a circle that is happing because this transformation or mapping is a very simple one, namely it is really a mapping which in mechanics you would call a displacement the circle is displaced by one unit to the left or in the complex plane there is a shift by minus 1 So the circle goes into a circle, now the contour gamma encircles the origin once as we saw earlier and of course one can see it immediately, what about the contour gamma    