# Control System - Frequency of Oscillation

A- For the feedback control system equation

s³ +(1+k) s²+10s+(5+15k)=0

The parameter k must be positive. What is the maximum value k can assume before the system becomes unstable? When k is equal to the maximum value, the system oscillates. Determine the frequency of oscillation.

B- For the feedback control system equation

s³ +(4+k) s²+6s+16+8k =0

The parameter k must be positive. What is the maximum value k can assume before the system becomes unstable? When k is equal to the maximum value, the system oscillates. Determine the frequency of oscillation.

C- The characteristic equation for a feedback control system is

(s+2)(s²+4s+8)+ k =0

Determine the range for k for which the system is stable.

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#### Solution Preview

We can check the system's stability by using Routh's stability criterion

Given polynomial 〖a_0 s〗^n+a_1 s^(n-1)+⋯+a_(n-1) s+a_n=0. Write the coefficients in the following Routh array:

s^n a_0 a_2 a_4 a_6 ...

s^(n-1) a_1 a_3 a_5 a_7 ...

s^(n-2) b_1 b_2 b_3 b_4 ...

s^(n-3) c_1 c_2 c_3 c_4 ...

:

s^2 d_1 d_2

s^1 〖 e〗_1

s^0 f_0

where b_1=(a_1 a_2-a_0 a_3)/a_1 , b_2=(a_1 a_4-a_0 a_5)/a_1 , ...

c_1=(b_1 a_3-a_1 b_2)/b_1 , c_2=(b_1 a_5-a_1 b_3)/b_1 , ...

The system's stability is determined from the signs of the coefficients in the first columns.

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A ) For the feedback control ...

#### Solution Summary

Frequency of oscillation is examined for a control system. Feedback control system equations are provided.