 Namaste, welcome to the session stability of control system. At the end of this session, students will be able to explain basic concept of stability and classify types of stability. In this session, we are going to see basic concept of stability, stability analysis from closed loop transfer function and types of stability. Before starting this session, just watch this video and observe it carefully. So in this video, you have seen there is a car which is a system which goes uncontrollable at certain times and finally it get stable at certain moment. So it is very important factor for any system it should be get controlled and it should give stable output. So that is why stability is important for any system. So stability analysis is because every system has to pass through a transient change for some amount of time. So finally whether the system will reach to its desired steady state or not that will decide whether it is stable or unstable. So let us see concept of stability through certain examples. So first we have a deep container in which we have a solid ball and let us consider that we want to take it outside with a stick. So as it is a deep container, if you apply power which is beyond limit then also what will happen is the ball will stabilize finally after some oscillations. So in this case ball after certain oscillations it goes to the bottom of the container and it becomes stable. Suppose we have a cone here where we want to stabilize a ball on the top so which is not possible. So whenever we try to stabilize a ball which is solid ball here on the top of the cone it will collide from either side. So such a system is known as an unstable system. Now let us consider one more container which is shallow container as compared to the previous deep container and let us consider that again we have a solid ball in the container and if you want to take it outside again with the help of stick. So what will happen? If force applied to take it outside which is less than required then what will happen? Again ball will oscillate in the container itself and after some time it stabilizes to the bottom of the container. So here requirement is the force applied should be greater than force required to take the ball outside. So when we apply force which is more than required then a ball get outside. So through this example of systems what we can find is first system is stable system but also known as absolutely stable because whatever force we applied to the ball finally it gets stable at the bottom of the container. In the second system we have seen we tried to make the ball stable on the top of the cone which is not possible so it is unstable system and the last container we where we have applied force which is greater than required is also known as a force applied critical condition then only ball comes outside such a system is known as a marginally or critically stable system. Now the following definition of stability are applicable to linear time invariant system. Now before proceeding further take a pause here and recall what do you mean by linear time invariant system. So I think you have completed with this so what is linear time invariant system? So a system whose output is linear with respect to the input and which has a property of superposition and homogeneous that system is linear system and system having time invariant property also that system is known as a linear time invariant where time invariant system is the output depends upon input irrespective of the time. So here you can have the example that if you want to switch on the bulb so that will switch on whenever you are providing a supply. So if you switch it on after sometime again it will glow it if you switch on the bulb same bulb after 2 days again it will give you same output as per the input and if you switch on the same bulb after few months then also it will give same output. So here output of the system does not depend upon the time so it will give always output based on the input. So such a system is known as a time invariant system. So the having both properties of linearity and time invariant is known as a LTI system. So stable system is a system whose natural response approaches to 0 as time approaches to infinity, unstable system if the natural response of a system approaches to infinity or grows without bound that system is known as an unstable system. And a system is a marginally stable if its natural response neither decays nor grows but remains constant or oscillates within bound that system is known as a marginally stable. Let us see BIBO system or given by BIBO system. A system is a stable if for every bonded input it gives a bonded output such a system is known as a BIBO stable system. Let us see through the block diagram here. In this block diagram if input is bonded input signal and if output is also bonded then it is known as a BIBO stable system. So for example if input is given here for first order you can see here output is like this which is bonded output and for second order you can see here it is decaying sinusoidal signal. So such a system is known as BIBO stable system, unstable system a system is unstable if for any bonded input if it gives an unbounded output. So in this block diagram you can see input is always bonded signal but output may be unbounded. So for a same bonded input which we have given in previous system if output is like this which is growing to the infinity for first order and for second order you can see if output is growing sinusoidal signal then such a system is known as an unstable system. Now let us see stability analysis from closed loop transfer function. For stability analysis of a system total response of a system is the sum of transient and steady state responses. Now c of t is equal to c transient of t plus c steady state of t. So c transient of t is nothing but it is the response of the system from initial state to the final state as time changes and steady state is nothing but it is the response of the system as time reaches to infinity. Now based on this let us define stability again. So stable system is a system where the system having closed loop transfer function with poles only in the left of of the s plane and unstable systems are those systems which have closed loop transfer function with at least one pole in the right half and or there may be poles of multiplicity greater than one on imaginary axis of the s plane such systems are known as unstable system. Marginally stable systems are those systems where those have closed loop transfer function with only on imaginary axis pole of multiplicity of one and or poles on the left half of the s plane. Thank you for watching the video.