Question

Apply the Routh-Hurwitz criterion to determine the system's stability with the characteristic polynomial q(s):

a) If all coefficients of the first column have the same sign, the system is stable.
b) If any coefficient in the first column has a different sign, the system is unstable.
c) If there are no sign changes in the first column, the system is marginally stable.
d) If there are μltiple sign changes in the first column, the system stability is undetermined.

Answer 1

The Routh-Hurwitz criterion is used to determine the stability of a system based on the signs of the coefficients of the characteristic polynomial. By creating a Routh array and analyzing the signs in the first column, the stability of the system can be determined as stable, unstable, marginally stable, or undetermined.

Step-by-step explanation:

The Routh-Hurwitz criterion is used to determine the stability of a system based on the signs of the coefficients of the characteristic polynomial. Here are the steps to apply the Routh-Hurwitz criterion:

1. Write the characteristic polynomial in the form q(s) = a_ns^n + a_{n-1}s^{n-1} + ... + a_0.
2. Create the first column of the Routh array by writing the coefficients of the even powers of s (a_ns^n, a_{n-2}s^{n-2}, a_{n-4}s^{n-4}, ...).
3. Create the second column of the Routh array by writing the coefficients of the odd powers of s (a_{n-1}s^{n-1}, a_{n-3}s^{n-3}, a_{n-5}s^{n-5}, ...).
4. Continue creating the rest of the rows of the Routh array by using the following formulas:
   • For row i, column j, calculate the value using: a_{i-1}a_{j} - a_{i}a_{j-1}.
   • If there is a zero in the first column, replace it with a small positive value (epsilon) to avoid division by zero.
   • Repeat the calculations for each row until the entire Routh array is filled.
5. Determine the stability of the system based on the signs in the first column of the Routh array:
   • If all coefficients in the first column have the same sign, the system is stable.
   • If any coefficient in the first column has a different sign, the system is unstable.
   • If there are no sign changes in the first column, the system is marginally stable.
   • If there are multiple sign changes in the first column, the system stability is undetermined.