Consider the unity negative feedback control system shown in the Figure. The value of gain $K$ ($>0$) at which the given system will remain marginally stable is __. (Answer in integer)
To determine the gain K for marginal stability in a unity feedback control system, the characteristic equation is derived from the denominator of the closed-loop transfer function.
The open-loop transfer function is given by:
G(s)H(s) = K/[s(s+7)(s+11)]
For a unity feedback system, the characteristic equation is:
1 + G(s)H(s) = 0 ⇒ 1 + K/[s(s+7)(s+11)] = 0
Which rearranges to the polynomial:
s(s+7)(s+11) + K = 0
This simplifies to:
s3 + 18s2 + 77s + K = 0
For marginal stability, the roots of this polynomial must lie on the imaginary axis. The Routh-Hurwitz stability criterion helps in determining the conditions for stability.
Constructing the Routh array:
| s3 | 1 | 77 |
| s2 | 18 | K |
| s1 | (18×77-1×K)/18 | 0 |
| s0 | K |
The system is marginally stable if there's a sign change at s1, ensuring the first row term becomes zero:
18×77 - K = 0
Solve for K:
1386 = K
Therefore, the value of gain K that keeps the system marginally stable is 1386.
This value lies within the given range of 1386 to 1386, confirming correctness.
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.
