 Once again we are discussing the transient response systems and for that we are going to discuss the stability of the control system. I am Professor V.X. Sathayam of Vulture Institute of Technology, Sallapur. Like our previous video, I will tell you that after learning after going through this video what you will be able to do, then you will be able to identify whether the system is stable or not. See this is very important because as a control engineer, first of all you must know that whether the system is stable or not and if it is stable, no problem, you must know that where the particular say instability zone is present and if it is not how you can make it stable and if it is marginally stable then what to do. So these are the issues that we are going to discuss today. So in the last module we have seen that if my system is, that is a characteristic equation is a polynomial of degree 2 or degree 1, forget about degree 1 because degree 1 is always with a simple that is s plus 2, s plus 3 is equal to 0 is going to be stable provided both have got same signs. Now see here, if I write s plus 4 is equal to 0, now this is a polynomial of degree 1. In this, this is the degree 1 and a constant term that is degree s is to 0. So all the terms are present, so it is a complete polynomial and coefficient is positive here and positive here, so this system is guaranteed stable because its value is equal to minus 4 which is on the left hand of the s plane. So we are looking at this left side that is the s plane where the system is going to be stable and we also mentioned that when my equation is a quadratic equation that is x square is equal to 5 s plus 6 is equal to 0 and if it is a complete polynomial and coefficients are having similar sign that is positive, positive and positive then system is guaranteed stable. Now guaranteed stable has got under name which is called as asymptotically stable. Asymptotically stable means that we are going to discuss in detail in our root locus say videos and we will come to know that asymptotic stability means the root locus will not cross this imaginary line, so it will say be moving in this fashion on this side it will never come to the right side that is called as asymptotic stability. And that system is always say better because we know that system is not going to become unstable or marginally stable at any value of parameter that we change because system parameter that we can change in our control architecture are either k gain or it is omega, so there are two variables that we can change and for that different methodologies are there means if you change k value that is variable is k there that is given to us we call it as a root locus and when we change omega from 0 to infinity it is called as bode plot. Now these particular say analysis that we are studying in transient system in the next few videos we are studying in detail about the row stability criteria and type 1 and type 2 error and how to resolve them these are important blocks that you require when you complete your root locus problems because that is essential that you must know how to resolve this particular issues before you draw the root locus. Now come to our point that in the last lecture we have made it now clear that whenever system is up to order 2 that is second order system if it is a complete polynomial means all terms in powers of s are present and each coefficient is having same sign plus plus plus or minus minus minus then system is guaranteed stable. Now today we are taking another case in which there is a quadratic it is a cubic equation now see this is a cubic so I written a cubic s cube plus s square plus 2s plus 8 is equal to 0 now this is my say characteristic equation qs which is nothing but it is a denominator of closed loop transfer function with negative feedback which is equal to 1 plus gh means if open loop transfer function is given to you you can convert it into 1 plus gh find the equation make a polynomial and you can say that this is the polynomial of order 3 4 5 always you will get a polynomial because nothing will be there in the denominator because everything will go to the right and it will be made 0. So always remember this is a polynomial of nth degree but today we are taking it only up to cubic say cubic equation now what we have mentioned first thing is always remember mathematically whenever we make any comment mathematics is very particular it says that you tell me first necessary condition you first tell me necessary condition why because if necessary condition is satisfied then only we can go for further necessary condition satisfied then we can go for very simple analogy suppose you want to purchase a shirt okay and second is say you are going to have a particular size the first thing is whether you are going to purchase a shirt or not if it there is a necessary condition is satisfied then only the question of size comes so similarly here necessary condition is that it must be a complete polynomial it must be a complete polynomial that is necessary condition so this is satisfied because there is a term s cube there is a term s square there is term 2 s and there is a constant term so powers of s s to 3 s s to 2 s s to 1 and s s to 0 are available so necessary condition is satisfied second part of the necessary condition is that coefficients of all the terms must be of same sign sign so here 1 1 2 and 8 so all are positive so all the coefficients are positive so second condition is also satisfied that is coefficients have same sign all the coefficients have same sign means now I am looking at this equation from a different perspective yes now I have got a characteristic equation which is given by s cube plus s square plus 2 s plus 8 is equal to 0 and it is a complete polynomial and coefficients of each term are having same sign either plus or either minus so all necessary conditions are satisfied so we cannot say that this is stable this is stable because only necessary condition is satisfied what is the sufficient condition that we are going to make it clear sufficient condition this is necessary satisfied then what is the sufficient condition sufficient condition is as we have seen in our last video sufficient condition is that all my roots all my roots must lie in the left half of the s plane left half of the s plane it they should not lie on the imaginary axis so it should not lie on this but it should be on the left side of the imaginary axis ok means if this necessary condition is satisfied there are two conditions one is it must be a complete polynomial yes then all the coefficients must have the same sign yes so check it first check we make it complete polynomial ok so I will write here it is a complete polynomial complete polynomial satisfied then second check I will make whether all the coefficients have the same sign so I will write here same sign for all coefficients satisfied so necessary conditions both are satisfied here now the question is what is the sufficient condition sufficient condition is that all roots must be on the left hand of the s plane now if I ask a simple question what are the possibilities of the root of this cubicle equation there are three roots s1 s2 and s3 s1 s2 and s3 I am not going to calculate roots right now I am just making a prediction what are the possibilities so what is the possibility all these three roots may be real all these three roots may be real then all these three roots may be negative real numbers all these three roots may be positive real numbers and then one is real positive and these two are negative real numbers these two are positive this is negative so all combinations of these real numbers with positive and negative so with root will be somewhere on the this one root will be here two roots will be here three roots will be here no root here all these combinations that is first part second thing is s1 is real and this is complex conjugate this is complex conjugate now complex conjugate may be such that it is having plus or minus a plus or minus bi this is the format now the problem with this is if I want my root to be on the left hand of the s side it must be minus a plus or minus bi this is very important because minus a ensures that it is on the left side minus a ensures that it is on the left side real part and bi is here and minus bi is here that is why I was telling from the beginning that whenever I get the roots so this is my one root here and this is my second complex root for this problem you solve on your calculator you will see that you will get roots which are not lying on this side but they are lying on this side so the possibility for this particular roots that a complex pair may be present on the left side complex pair may be present on the right side or or a complex pair may lie on the imaginary axis that I am not making a comment right now ok it lying on this particular axis but for this problem what is happening the complex pair or complex roots on the right side of the x axis sorry s plane so as per the sufficiency condition 3 roots are there one root is on the left side which is real and 2 complex roots are there which are on the right side they are placed here as a complex conjugate so sufficiency condition is not satisfied so when necessary condition is satisfied but sufficiency condition is not satisfied system is unstable so without any doubt you say that system is unstable because I have got 3 roots one root is real which is on the left side negative number and 2 values are complex conjugates and the real part is positive so they are coming on the right side but suppose I have got suppose I get this roots on the left side and real number here the system is stable because all your roots are on the left hand side forget about whether it is a complex or not that is immaterial my root is lying on the entire this left plane I can call this as a stable system so with this you have got an idea about the necessary condition and sufficient condition which is separate for second order equation and higher order equation now for reference you can report these standard books Revan, Bakshi and Bhargapati thank you for your patience real say hearing in the third video we will go for further say points about transient systems thank you