A detailed introduction to the influence of the zero and pole points of the transfer function on the system:
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Effect of Zero: The zero point of the transfer function is the point at which the numerator of the transfer function is zero. The zero point affects the frequency response and stability of the system. in particular:
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Frequency Response: The position of the zero affects the gain and phase characteristics of the system at different frequencies. When the zero of the transfer function corresponds to a certain frequency on the frequency axis, it causes the gain of the system to increase or the phase to advance at that frequency. Therefore, by adjusting the position of the zero point, the gain and phase characteristics of the system at different frequencies can be adjusted.
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Stability: For a linear time invariant (LTI) system, the system will be stable if all zeros lie in the left half plane. If there are zeros in the right half plane, the system may become unstable. Therefore, in the design of the control system, it is necessary to ensure that the real parts of all zeros lie in the left half plane to maintain the stability of the system.
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Influence of Pole: The pole of the transfer function is the point that makes the denominator of the transfer function zero. The poles have an effect on the frequency response, stability and dynamic characteristics of the system. in particular:
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Frequency Response: The location of the poles affects the gain and phase characteristics of the system at different frequencies. When the pole of the transfer function corresponds to a certain frequency on the frequency axis, it causes a gain drop or a phase delay of the system at that frequency. Therefore, by adjusting the position of the pole, the gain and phase characteristics of the system at different frequencies can be adjusted.
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Stability: For a linear time invariant (LTI) system, the system will be stable if the real parts of all poles lie in the left half plane. If there are poles in the right half plane, the system may become unstable. Therefore, in the design of the control system, it is necessary to ensure that the real parts of all poles lie in the left half plane to maintain the stability of the system.
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Dynamic characteristics: The position of the pole will affect the dynamic characteristics of the system, such as response time, overshoot and damping ratio, etc. By adjusting the position of the pole, faster response time, less overshoot or better damping characteristics can be achieved.
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To sum up, the zeros and poles of the transfer function affect the frequency response, stability and dynamic characteristics of the system respectively. In system analysis and control design, it is very important to analyze and adjust the zero and pole points of the transfer function.
