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
Consider a sixth order system with characteristic equation s6 + 2s5 + 8s4 + 12s3 + 20s2 + 16s + 16 = 0, The control system is:
Assertion (A) : According to Routh-Hurwitz criterion, the system represented by characteristic equation F(S) will be unstable, if the first column of the array contains no sign change.
Reason (R) : If any row of the Routh's table is multiplied or divided by a positive integer, then the system stability will not be affected.
Select your answer using the codes given below.
Assertion (A) : The roots of the characteristic equation must lie in the left hand s-plane for the system to exhibit a stable time response.
Reason (R) : Routh-Hurwitz criterion is a necessary and sufficient criterion for stability.
Select your answer using the codes given below.
A fourth order system is characterised by the equation S4 + 8S3 + 18S2 + 16S + 5 = 0. This is a :
According to Routh's stability criterion the necessary conditions for the system to be stable are defined as :
(a) All the coefficients of the characteristics equation should be missing.
(b) None of the coefficients should be real and should have different sign.
(c) None of the coefficients of the characteristics equation should be missing or zero.
(d) All the coefficients should be real and should have the same sign.Which of the above statements are correct ?
