what are the differences among all the methods ...please explain in brief
You are right. All these methods are complementary and do the same thing. The main difference is that Bode plots, Nichols Chart and Nyquist criterion are very useful in the case when yo don'know the exact math model of the system. In this case from frequency response of the open loop system obtained experimentally, you will be able to verify the stability of the closed loop system. Also when you want to suppress the noise present in the system Bode, Nichols and Nyquist plots are very useful in controller design.
Thank you for the nice question
For the same reason we have crescent wrenches, box wrenches, socket sets, and vice grips. No matter what your specialty, the more different tools you carry in your toolbox, the more versatile and capable you will be.
Like everybody here explained before these tools complement between. To complement the answers, in practice sometimes you need more "practical" methods to achieve the control goals. The theory works for every case but sometimes, in some work fields your automation requirements are low or you don´t have all the exact transfer function model or even the data of some parts of the control plant so like Gezim explains, you can achieve the control goals applying Bode, Nicholson,and Nyquist. If you have more information about the mathematic model or you want to get in a high-level your control goals, accuracy and other features you can apply other methods that give you more advantages to analyze the stability system and you can design better controllers.
because every method has its own merits and demerits for achieving stability
One of the major purpose for stability analysis is to know the limits of your system to be controlled. An unstable open-loop system normally means you can not reach to some performance goals even with the best controller. Or some systems are not possible for some performance goals (e.g. delay and right half plan zeros in open loop system will significantly limit your reachable closed loop bandwidth - you will have so small gain margin).
Bode plots tells us how many resonant peaks are there if I am not wrong (magnitude response). Phase response is absolutely important.
Root locus tells about response when gain changes (stability is clear from response).
Linear and Nonlinear systems stability can be checked by Lyapunov Method as well.
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