The first two rows in the Routh table for the characteristic equation of a certain closed-loop control system are given as The range of $K$ for which the system is stable is
To determine the range of \( K \) for which the system is stable, we need to construct the Routh array using the given table and apply the Routh-Hurwitz stability criterion.
The first two rows of the Routh array are given as:
The Routh array is constructed as follows:
| \( s^3 \) | 1 | \( 2K + 3 \) |
| \( s^2 \) | 2K | 4 |
| \( s^1 \) | \(\frac{(2K)(2K+3) - 4 \cdot 1}{2K}\) | 0 |
| \( s^0 \) | 4 |
We need the first column elements of the Routh array to be positive for stability:
- For \( s^3 \), the coefficient is 1, which is already positive.
- For \( s^2 \), the coefficient is \( 2K \). \(2K > 0\) implies \( K > 0 \).
- Now, calculate the coefficient for \( s^1 \):
The coefficient for \( s^1 \) is given by:
\[\frac{(2K)(2K+3) - 4 \cdot 1}{2K} = \frac{4K^2 + 6K - 4}{2K} = 2K + 3 - \frac{2}{K}\]For stability, \( 2K + 3 - \frac{2}{K} > 0 \).
- Simplifying, we get:
- Solving the quadratic inequality \( 2K^2 + 3K - 2 = 0 \) using the quadratic formula:
Thus, \( K = \frac{1}{2} \) or \( K = -2 \).
The solution to the inequality involves testing intervals determined by roots \( K = -2 \) and \( K = \frac{1}{2} \). The expression \( 2K^2 + 3K - 2 > 0 \) is satisfied for:
\[K > \frac{1}{2} \, \text{or} \, K < -2\]Since \( K > 0 \) is also a requirement for stability, the valid range for stable \( K \) is:
0.5 < K < \(\infty\)
Therefore, the correct answer is 0.5 < K < \(\infty\).
Important Questions from Routh-Hurwitz Stability Criteria
Match List I with List II:
List I
(Coefficients of s 2+ a 1s + a 2= 0)
List II
(Nature of Roots)
(A) a \(_1^2\) > 4a 2 (I) Negative real and equal (B) a \(_1^2\) = 4a 2 (II) Conjugate Imaginary (C) a \(_1^2\) < 4a 2 (III) Negative Real and Unequal (D) a 1= 0
a 2≠ 0
(IV) Conjugate Complex (Real part negative) Choose the correct answer from the options given below:
A closed-loop control system has a characteristic equation given by s 3 + 2.4s 2 + 1.8s + 0.5 = 0. Find out the value of a, b, c, and d using the Routh Hurwitz criterion.
s 3
1
1.8
s 2
2.4
0.5
s 1
a
c
s 0
b
d
Determine the stability of system:
S 3+ S 2+ S + 4
Which of the following is NOT the advantage of Routh-Hurwitz criterion of control systems?
The characteristic equation of given system is 6s + K = 0. Determined the range of K for which the system to be stable.
