The systems with transfer functions G_1(s) = 10008/s + 10000 G_2 (s) = 1008/s +

Question

The systems with transfer functions G_1(s) = 10008/s + 10000 G_2 (s) = 1008/s + 100 G_3 (s) = 1/s + 1 are connected in series, and the resulting system - initially at rest - is given an impulse input. After some initial time, which of the following is true of the response? (a) It approximates the impulse response of the system with transfer function G_1(s) (b) It approximates the impulse response of the system with transfer function G_2(s) (c) It approximates the impulse response of the system with transfer function G_3(s) (d) It depends on the order in which the three subsystems are connected Explain your answer.

Explanation / Answer

Option (c) is correct because s= -1 is the dominant pole of the combined system

When a system is of high order we use approximate techniques to simplify the model and dominant pole comes into the picture.The dominant pole(s) of are those which satisfy below criterion:

1) Have the largest initial condition (i.e. most energy), and
2) Slowest decay rate (so they last the longest).

It turns out that the pole(s) closest to s=0 has the largest initial condition and the pole(s) closest to the jw axis
have the slowest decay rate.

So, to simplify complex system we can neglect non-dominant poles.

In our case when connected in series the resulting system will have three poles s=-100, s=-10, s=-1. From above argument, s=-1 is dominant pole and response would be similar to that of G3.

Also, it doesn't matter how you connect the system because in series transfer system will remain same.

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