a) The Routh stability criterion is a mathematical test used to determine the stability of a linear time-invariant (LTI) system based on the coefficients of its characteristic equation. The criterion states that for a system to be stable, all the coefficients of the characteristic equation must be positive.
The Routh stability criterion can be applied by constructing a Routh array using the coefficients of the characteristic equation. The first row of the array consists of the coefficients of the even powers of 's', and the second row consists of the coefficients of the odd powers of 's'. The subsequent rows are calculated based on the values of the previous rows. The stability of the system can be determined by checking the sign changes in the first column of the Routh array.
b) The characteristic equation is given as:
s^4 + 3s^3 + 3s^2 + s + K = 0
To determine the range of values of K for which the system is stable, we need to apply the Routh stability criterion. Constructing the Routh array:
Row 1: 1, 3, K
Row 2: 3, 1
Row 3: (3 - 3K)/3
For the system to be stable, all the coefficients in the first column of the Routh array must be positive. Since the coefficients are obtained from the original characteristic equation, we have the following conditions:
1 > 0
3 > 0
(3 - 3K)/3 > 0
Simplifying the third inequality:
3 - 3K > 0
-3K > -3
K < 1
Therefore, the range of values for K that ensures system stability is K < 1.