Lets Start with the LTI system .
Before that we should understand some basic input signals.
We all know basic input are impulse,step , ramp and parabolic.But we should ask ourself why we are only studying these inputs in control system , signals and system etc. The answer is very simple by using these inputs we can form any signals . that is the reason we are only analyzing these inputs in various subject or indirectly fields.
Before going further lets understand the concept of pole and zeros.
The moment we ask this question will be say in eq
People will say a is the zero and b is the pole . Or Zero are places will system will zero output and poles are the places where it will became infinite.
But we never ask our-self what is the physical meaning of pole and zero .Lets understand this , poles and zeros are nothing but the places where system will store energy. We can understand this in other way asking when i can get the pole and zero , recall the basic components and we will get that even where i have inductor or capacitor i will get pole or zero. So the components that will store energy will give rise to this.
In the whole control system subject you will be only studying the movement of pole not zero because when system output will became zero is the not the issue , while if it became infinite that will make it unstable . And nobody want unstable system.
We will also explore the concept why we are so bothered when the pole moved to the RHS and not in LHS.
This can be understand in this way.
Imagine i have pole in RHS as 1/(s+a) now if i will take the inverse laplace transform, the it comes to e(pow(-at)) and if you plot exponetial equation with the time axis you will observe that it is dying function with time axis, so this will never become unstable.
While if i take 1/(s - a) , then we can see that inverse laplace transform will be e(pow(+at)) , which is a rising function with time . so at one time it will explode the system ultimately unstable.
Let see the time domain analysis. Before that we should some terms .
Order : It will tell you number of poles in a system.
Type : It will tell you the poles at origin.
Damping ratio : It is the ratio of damping factor to that of the damping need for the critical damping.
Damping is the effect that will reduce the amplitude of ascillation in oscillatory ckt.
We are gona study 4 type of systems here..
If you proceed ahead you will find that we will be only studying the underdamped system, this reason for this very simple is that it got faster settling time . This can be understand with example , suppose you have a computer which is very slow in response and another which is very faster. If i ask your chosen one , answer will be the faster , the reason is very simple , everybody will like the faster system. Same is true for the under-damped system though it got oscillation , but its response time is very fast.
The over damped systems are very sluggish and slow.
Getting critically damped system is very difficult.
UNDERDAMPED SYSTEM :
Before that we should understand the characteristic eq
As you see from the above picture , we can understand the any system by knowing either the damping ratio or the roots of the eq.
Lets analyze the second order system when i have step as an input.
You can go ahead and solve problems on this . Even settling time is also here , i have not mention that you can get in any book which is pretty easy.
Before that we should understand some basic input signals.
We all know basic input are impulse,step , ramp and parabolic.But we should ask ourself why we are only studying these inputs in control system , signals and system etc. The answer is very simple by using these inputs we can form any signals . that is the reason we are only analyzing these inputs in various subject or indirectly fields.
Before going further lets understand the concept of pole and zeros.
The moment we ask this question will be say in eq
People will say a is the zero and b is the pole . Or Zero are places will system will zero output and poles are the places where it will became infinite.
But we never ask our-self what is the physical meaning of pole and zero .Lets understand this , poles and zeros are nothing but the places where system will store energy. We can understand this in other way asking when i can get the pole and zero , recall the basic components and we will get that even where i have inductor or capacitor i will get pole or zero. So the components that will store energy will give rise to this.
In the whole control system subject you will be only studying the movement of pole not zero because when system output will became zero is the not the issue , while if it became infinite that will make it unstable . And nobody want unstable system.
We will also explore the concept why we are so bothered when the pole moved to the RHS and not in LHS.
This can be understand in this way.
Imagine i have pole in RHS as 1/(s+a) now if i will take the inverse laplace transform, the it comes to e(pow(-at)) and if you plot exponetial equation with the time axis you will observe that it is dying function with time axis, so this will never become unstable.
While if i take 1/(s - a) , then we can see that inverse laplace transform will be e(pow(+at)) , which is a rising function with time . so at one time it will explode the system ultimately unstable.
Let see the time domain analysis. Before that we should some terms .
Order : It will tell you number of poles in a system.
Type : It will tell you the poles at origin.
Damping ratio : It is the ratio of damping factor to that of the damping need for the critical damping.
Damping is the effect that will reduce the amplitude of ascillation in oscillatory ckt.
We are gona study 4 type of systems here..
If you proceed ahead you will find that we will be only studying the underdamped system, this reason for this very simple is that it got faster settling time . This can be understand with example , suppose you have a computer which is very slow in response and another which is very faster. If i ask your chosen one , answer will be the faster , the reason is very simple , everybody will like the faster system. Same is true for the under-damped system though it got oscillation , but its response time is very fast.
The over damped systems are very sluggish and slow.
Getting critically damped system is very difficult.
UNDERDAMPED SYSTEM :
Before that we should understand the characteristic eq
As you see from the above picture , we can understand the any system by knowing either the damping ratio or the roots of the eq.
Lets analyze the second order system when i have step as an input.
You can go ahead and solve problems on this . Even settling time is also here , i have not mention that you can get in any book which is pretty easy.
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