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 Choose the most appropriate answer from the options given below :
(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 stability
(A), (B) and (C) Only
Statement (D) denies the very purpose of the criterion, so it must be excluded — and the other three are all correct. Option 4.
(C) states what the criterion is for, and it is the whole point. Routh's test counts the roots of the characteristic equation lying in the right half plane without finding them. That mattered enormously before computers: factorising a fifth-order polynomial by hand is impractical, but building the array from its coefficients takes a minute. The rule is simply
\(\text{number of RHP roots}=\text{number of sign changes in the first column}\)
so a system is stable if and only if every entry in that column has the same sign.
(B) is correct, and "directly from the coefficients" is the operative phrase. Nothing but the coefficients of
\(a_{0}s^{n}+a_{1}s^{n-1}+\cdots+a_{n}=0\)
enters the array. A necessary preliminary check is immediate: if any coefficient is zero or of opposite sign to the others, the system is certainly unstable, and the array need not be built at all. The criterion gives absolute stability — a yes or no — but says nothing about relative stability, that is how much margin there is, which is what gain and phase margins measure.
(A) is correct and is a genuine restriction. The method operates on a finite table of coefficients, so it requires a polynomial with finitely many terms — a rational transfer function. A system containing a pure time delay has
\(e^{-sT}\)
in its characteristic equation, which is transcendental and has infinitely many terms in its series. Routh's criterion cannot be applied to it directly; the delay must first be approximated by a Padé rational function.
(D) is simply false, and its inclusion is what makes the question answerable at a glance: it contradicts (B) and (C) in the same list, so it cannot stand alongside them. That eliminates option 3 at once, and since (A) is also sound, the set of three is the answer.
Two special cases the method handles with small modifications: a zero appearing in the first column is replaced by a small \(\epsilon\), and an entire row of zeros signals symmetric roots — often a conjugate pair on the imaginary axis — whose location comes from the auxiliary equation formed from the row above.
Hence, the correct statements are (A), (B) and (C).
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 :
\(G(s)=\dfrac{k}{s\left[s(s+10)+T\right]}\)
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
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
