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# Feedback Control Systems and Routh Criterion

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##### Solution Summary

The solution involves step-by-step calculations and explanations for finding positive proportional controller gain, the range of the positive proportional controller gain, and the effects of adding integral control with respect to system stability and step input tracking.

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G(s) = 1/[(s+1)*(s+2)]
H(s) = 4/(s+10)

Routh Criterion for closed loop system stability:

1. Proportional controller: K in the forward path

Transfer function = Y(s)/R(s) = k*G(s)/[1+kG(s)*H(s)]

Charcteristics function: [1+kG(s)*H(s)] = 0

Look at coefficients and arrange them in routh table and make sure that values in column do not change signs (+/-) for stability.

[1+kG(s)*H(s)] = 1 + k*1/[(s+1)*(s+2)]*4/(s+10)
(s+1)*(s+2)*(s+10) + 4k = 0
s^3 + 13 s^2 + 32 s + (20+4k) = 0

Develop the Routh table

s^3 1 32
s^2 13 (20+4k)
s^1 [32-{(20+4k)/13}] 0
s^0 (20+4k)

For stability, all the column 1 ...