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. Which of the above statements are correct ?
(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.
(c), (d)
The two necessary conditions are stated in (c) and (d), so the answer is option 4. Statements (a) and (b) are their negations and are simply false.
Where the conditions come from. If every root of the characteristic equation has a negative real part, the polynomial factorises into terms of the form \((s+a)\) and \((s^{2}+bs+c)\) with a, b, c all positive. Multiplying out such factors can only ever produce positive coefficients — there is no subtraction anywhere in the expansion. So:
• every coefficient must be present, since a missing term means a zero coefficient where a positive one was required — statement (c);
• every coefficient must be real and of the same sign — statement (d).
These conditions are necessary but not sufficient, and that is the crucial qualification. Their usefulness is negative: they let an unstable system be rejected instantly, without building the array. For example
\(s^{3}+2s^{2}+5=0\)
is unstable on inspection — the \(s^{1}\) term is missing. But passing the test proves nothing:
\(s^{3}+s^{2}+2s+8=0\)
has all coefficients present, positive and real, yet its Routh array shows two sign changes and two roots in the right half plane. Only the full array is sufficient.
| Test | What it establishes |
|---|---|
| All coefficients present, same sign | Necessary — failure proves instability |
| No sign change in the Routh array's first column | Sufficient — proves stability |
Reading the array completes the picture: the number of sign changes in the first column equals the number of roots in the right half plane, so the method not only decides stability but counts the offending roots. Two special cases need handling — a zero appearing in the first column, dealt with by substituting a small ε or by the reciprocal-polynomial trick, and an entire row of zeros, which signals symmetrically placed roots and is handled through the auxiliary equation, whose roots are the ones lying on the imaginary axis.
Why the criterion is valuable at all is that it answers the stability question without factorising the polynomial — and for a system containing an unknown gain K it yields directly the range of K over which the closed loop remains stable.
Hence, the correct statements are (c) and (d).
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 :
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 :
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
