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.
Both (A) and (R) are true, but (R) is not the correct explanation of (A)
Examine the assertion — true, and it is the definition of stability. Each root \(s_i=\sigma_i+j\omega_i\) of the characteristic equation contributes a term \(e^{s_it}=e^{\sigma_it}e^{j\omega_it}\) to the natural response. The oscillatory factor has unit magnitude, so only the sign of \(\sigma_i\) matters:
| Root location | Contribution |
|---|---|
| \(\sigma \lt 0\) — left half plane | Decays to zero — stable |
| \(\sigma=0\) — on the axis | Sustained oscillation — marginal |
| \(\sigma \gt 0\) — right half plane | Grows without bound — unstable |
A single root in the right half plane is enough to make the whole response diverge, so every root must lie in the left half plane.
Examine the reason — true. Routh–Hurwitz is indeed both necessary and sufficient: the number of sign changes in the first column of the Routh array equals exactly the number of roots in the right half plane. No sign change means no such root, and the system is stable. Its great merit is that it answers the question without factorising the polynomial — only the coefficients are needed.
Does the reason explain the assertion? No, and this is the judgement the question turns on. The assertion states where the roots must be, and the justification for that is the exponential behaviour of \(e^{\sigma t}\) — nothing to do with Routh–Hurwitz. The reason supplies a test for checking whether the condition holds. A method of verification is not an explanation of the underlying requirement: the left-half-plane condition would remain exactly as true if Routh and Hurwitz had never devised their array. Both statements are true but the link fails, so the code is 2.
The general test for these items. Ask "why is the assertion true?" Here the answer is "because a positive real part makes the exponential grow", not "because a table of coefficients says so". A reason offering a tool where a cause is wanted is the classic signature of code 2.
How the criterion is applied. For \(F(s)=a_0s^{n}+a_1s^{n-1}+\dots+a_n\), a quick necessary check comes first — every coefficient must be present and of the same sign, or the system is already unstable. That alone is not sufficient, so the full array is built, each entry being
\(b_1=\dfrac{a_1a_2-a_0a_3}{a_1}\)
and so on down to the s0 row.
The other tools for the same question. The root locus shows where the roots move as gain changes; the Nyquist criterion counts encirclements of −1 and works even for systems with delay; and Routh–Hurwitz is the purely algebraic route. All three test the same left-half-plane condition.
Hence, both (A) and (R) are true, but (R) is not the correct explanation of (A).
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) : 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 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 ?
