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 :
A and B only
What the Routh–Hurwitz criterion is. It is a purely algebraic test carried out on the coefficients of the closed-loop characteristic polynomial
\(a_0s^{n}+a_1s^{n-1}+\ldots+a_n=0\)
The coefficients are arranged into the Routh array, and the first column is inspected. Its great advantage is that it decides stability without factorising the polynomial — no need to find the roots at all.
A — "absolute stability." TRUE. Absolute stability is the yes/no question "are all roots in the left half plane?" The criterion answers it directly: the system is stable if and only if every entry of the first column has the same sign (a necessary preliminary being that all coefficients are present and of like sign).
B — "the number of roots lying in the right half of the s-plane." TRUE. The count of sign changes down the first column equals the number of right-half-plane roots. So the test gives not just stability but the degree of instability.
C, D, E — gain margin, phase margin, phase-crossover frequency, gain-crossover frequency. FALSE. These are all relative-stability measures defined on the frequency response of the open-loop system: gain margin at the frequency where the phase is −180°, phase margin at the frequency where the magnitude is unity. They come from Bode, Nyquist or Nichols plots, which require evaluating \(G(j\omega)H(j\omega)\) over frequency. Routh's array never touches jω — it works only with the polynomial coefficients — so it cannot yield any of them.
A useful thing Routh can do that the options do not mention: by leaving a design parameter K symbolic in the array, the criterion gives the range of K for which the closed loop stays stable — a standard exam application.
Hence, the correct answer is A and B only.
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.
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 :
\(G(s)=\dfrac{k}{s\left[s(s+10)+T\right]}\)
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 ?
