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 :
(A) is false, but (R) is true.
The reason is a correct statement of the stability condition; the assertion overstates what the Routh-Hurwitz criterion delivers.
What the criterion does give. Building the array from the characteristic equation and inspecting the first column answers one question completely:
• No sign change → every root is in the left half plane — the system is stable.
• k sign changes → exactly k roots lie in the right half plane.
That is absolute stability: a yes-or-no verdict, plus a count of the offending roots. It is obtained without factorising the polynomial, which is the method's whole appeal.
What it does not give. Relative stability asks a different question — not whether the system is stable but how stable, how far the roots sit from the imaginary axis, how much overshoot and ringing to expect, how much gain or phase margin remains. The Routh array says nothing about any of that: two systems with wildly different damping can produce identical sign patterns.
| Question | Answered by |
|---|---|
| Is it stable? | Routh-Hurwitz |
| How many unstable roots? | Routh-Hurwitz |
| How close to instability? | Bode gain and phase margins, root locus, Nyquist |
| How much overshoot? | Damping ratio from the root locations |
The one extension worth mentioning is the shifted-axis trick: substituting \(s=z-\sigma\) and applying the criterion to the new polynomial tests whether all roots lie to the left of \(-\sigma\), which is a crude measure of relative stability. But that is an application of the criterion with extra work, not something the criterion supplies of itself — and it is why some texts describe the assertion as partly defensible, which is why this answer is flagged for confirmation.
The reason, meanwhile, is simply correct. A root at \(s=-\sigma\pm j\omega\) contributes \(e^{-\sigma t}\) to the response, which decays only if \(\sigma\gt0\) — that is, only if the root lies in the left half plane. Since it is a true statement while the assertion is not, the code is 4.
Hence, (A) is false but (R) is true.
Similar Questions
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) : 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 ?
What is the relation between k and T such that a unity feedback control system whose open loop transfer function below is stable :
\(G(s)=\dfrac{k}{s\left[s(s+10)+T\right]}\)
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 :
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 number of roots of s 3+ 5s 2+ 7s + 3 = 0 in the left half of the s-plane is
