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.
(A) is false, but (R) is true.
To address the given assertion and reason regarding the Routh-Hurwitz criterion, let's analyze each statement:
Understanding the Routh-Hurwitz Criterion:
- The Routh-Hurwitz criterion is a method used in control systems to determine the stability of a system. It does this by analyzing the characteristic equation of the system.
- A system is considered unstable if there are any sign changes in the first column of the Routh array.
- Conversely, a system will be stable if there are no sign changes in the first column.
Analysis of Statements:
- 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.
- This statement is false because a system is stable (not unstable) if there are no sign changes in the first column of the Routh array. The presence of sign changes indicates instability.
- 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.
- This statement is true. Multiplying or dividing a row by a positive constant will not change the sign pattern of the row, hence it does not affect stability.
Conclusion:
- Therefore, the correct answer is (A) is false, but (R) is true.
- (R) does not explain (A), but it is independently true.
Similar Questions
Which of the following statements are correct in respect of Routh's stability criterion ?
(A) This stability orientation applies to polynomials with only finite number of terms
(B) When this criterion is applied to a control system, the information about absolute stability can be obtained directly from the coefficients of the characteristic equations
(C) The Routh's stability criterion tells whether or not there are unstable roots in polynomial equation without actually solving for them
(D) Routh's criterion is not related to system stabilityChoose the most appropriate answer from the options given below :
The number of sign changes in the first column of Routh array reveals.
Routh-Hurwitz criterion applicable for
A. absolute stability
B. the number of roots lying on the right half of the S-plane
C. the gain margin and phase margin
D. the phase crossover frequency
E. the gain crossover frequency
Choose the most appropriate answer from the options given below :
Consider a sixth order system with characteristic equation s6 + 2s5 + 8s4 + 12s3 + 20s2 + 16s + 16 = 0, The control system is:
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 ?
What is the relation between k and T such that a unity feedback control system whose open loop transfer function below is stable :
\(G(s)=\dfrac{k}{s\left[s(s+10)+T\right]}\)
Assertion (A) : Routh Hurwitz criterion gives both absolute as well as relative stability using characteristic equation.
Reason (R) : For a system to be stable all roots of characteristic equation must lie in left half of s-plane.
Select your answer using the codes given below :
Important Questions from Routh-Hurwitz Stability Criteria
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
The disadvantage of the Routh's criteria are
The open loop transfer function of a unity gain negative feedback system is given by
\(\rm G(s) = \frac{k}{s^2 + 4s - 5}\)
The range of 𝑘 for which the system is stable, is
Which of the following is NOT the advantage of Routh-Hurwitz criterion of control systems?
The characteristic polynomial of a linear system is given as s4 + 3s3 + 5s2 + 6s + K + 10=0. What should be the condition on K so that the system is stable ?
