 Namaste, welcome to the session stability of systems and poles on s-plane. At the end of this session students will be able to explain the relation between stability and poles on s-plane. In this session we are going to discuss conditionally stable system, asymptotic stability, relative stability, relation between stability and poles on s-plane, conditionally stable system. In the last session we have completed types of systems where the types of systems are stable system, unstable system, marginally which is also known as critically stable system. So moving further to the types of system one of the system is also known as conditionally stable system. A linear time invariant system is said to be conditionally stable when it use bounded output for the condition of certain parameters of the system. So if condition of certain parameters of the system are not satisfied then the system becomes unstable. So if you remember in the last session we have seen a video on the car system it goes uncontrollable and sometimes it becomes unstable. So a car where it is a system it depends upon certain parameters where those may be braking system or acceleration system. So a car may be a conditionally stable system because it depends upon parameters, condition of the parameters. So that is why conditionally stable system is also important factor as in static stability. Zero input response, the output response of system that is driven by no any external input but initial state of the system only is called zero input response. So before proceeding further now take a pause here and recall what do you mean by BIBO system which we have seen in the last session. So I think you have remembered and recall what do you mean by BIBO system. So when output of a particular system is bounded with respect to bounded input then that system is known as a BIBO stable system. But if output is not bounded for the bounded input then it becomes unstable system. So for any bounded input if the output is bounded then system is said to be BIBO stable. Zero input stability or asymptotic stability if the zero input response c of t of a system subjected to a finite initial condition is bounded and reaches to zero as k tends to infinity. Then the system is said to be asymptotically stable. So a system having zero input stability so a system is called as a zero input stable or asymptotic stable when it fulfills these two criteria where m is greater than or equal to response of a system and less than infinity. And second condition is if response of the system tends to infinity then its output will be zero. So simply a system is asymptotically stable or it is also known as a internally stable when it is having bounded output as said in the first condition and it has a value zero when time reaches to infinity. So let us see relative stability. For a system whether it is relatively more stable or unstable depends on how fast its transient settles. Relative stability is related to settling time of a system which is decided by poles on s-plane. If poles are away from left half of imaginary axis is considered to be relatively more stable compared to a system having poles close to imaginary axis. Let us observe relative stability using poles on s-plane. So for example here we have a pole which is close to imaginary axis shown by blue colour and let us have one more pole which is on real part which is away from the imaginary axis shown by red colour. Then its response will be you can see the response for blue coloured pole which is near to the imaginary axis it takes more time to settle as compared to a pole which is shown by red colour which is away from the pole it settles easily and quickly. So we can see here red coloured pole which is away from the imaginary axis takes very less time to settle as compared to the blue coloured pole which is close to imaginary axis. So one more example if you have complex conjugate pole on the left hand side of the which is close to imaginary axis and let us have two more poles which are having complex conjugate property on the left hand side but away from the imaginary axis. Then response will be you can see here it is a decaying sinusoidal shown by black colour graph here for the black poles and red coloured response for the red poles which are away from the imaginary axis which settles quickly which settles earlier as compared to black coloured pole. So you can see here the poles which are away from the imaginary axis settles earlier. So here you can see we can conclude that relative stability improves as poles moves away from imaginary axis on left half of s plane. Now let us see relation between stability and poles on s plane. So for absolutely stable system the conditions are first the poles should be real and negative on left hand side of the s plane. So here we have an example. So the pole which is on left hand side of the s plane having the response step response which is bonded so it is absolutely stable and when poles are complex conjugate and real negative on left hand side of the s plane then step response will be decaying sinusoidal which is also response of absolutely stable system. Now for unstable system the conditions are poles should be real positive and should be on the right hand side of the s plane as shown in figure here. So step response will be which is which reaches to infinity and second condition is poles should be complex conjugate real positive on the right hand side of the s plane. So the step response will be increasing sinusoidal signal. One more condition for unstable system is if there are repeated pair of poles on imaginary axis with no pole on right hand side of the s plane. So as you can see here shown by blue colored poles which are pair and one more which are shown by red colored then step response will be increasing sinusoidal. For marginally or critically stable system the condition is the poles should be non-repeated on imaginary axis with no pole on right hand side of the s plane. So I can see here the poles are there on the imaginary axis which are not repeated and there is no pole on the right hand side of the s plane. So response will be bonded one where the sinusoidal signal is neither increasing nor decaying. So thank you.