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Consider the standard second-order system of the form (ωₙ²)/(s²+2ζωₙ s+ωₙ²) with the poles p and p^* having negative real parts. The pole locations are also shown in the figure. Now consider two second-order systems as defined below:
System 1: ωₙ = 3 rad/sec and θ = 60°
System 2: ωₙ = 1 rad/sec and θ = 70°
Which one of the following statements is correct?
- Settling time of System 1 is more than that of System 2.
- Settling time of System 2 is more than that of System 1.
- Settling times of both the systems are the same.
- Settling time cannot be computed from the given information.
Correct answer: Settling time of System 2 is more than that of System 1.
Solution
The settling time of a second-order system is influenced by the natural frequency ( ext{n}) and damping ratio ( ext{z}). System 1, with a higher natural frequency of 3 rad/sec, will settle faster than System 2, which has a lower natural frequency of 1 rad/sec, despite the difference in damping angles.
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