A fourth order system is characterised by the equation S4 + 8S3 + 18S2 + 16S + 5 = 0. This is a :
Stable system
Use the Routh-Hurwitz criterion — it decides stability without ever finding a root.
Step 0 — the necessary condition. All coefficients (1, 8, 18, 16, 5) are present and positive, so the system is not immediately disqualified. This test is necessary but not sufficient, so the array must still be built.
Step 1 — lay out the first two rows from alternate coefficients:
| Row | |||
|---|---|---|---|
| s4 | 1 | 18 | 5 |
| s3 | 8 | 16 | 0 |
Step 2 — compute the s2 row.
\(b_{1}=\dfrac{(8)(18)-(1)(16)}{8}=\dfrac{144-16}{8}=16\)
\(b_{2}=\dfrac{(8)(5)-(1)(0)}{8}=5\)
Step 3 — the s1 row.
\(c_{1}=\dfrac{(16)(16)-(8)(5)}{16}=\dfrac{256-40}{16}=13.5\)
Step 4 — the s0 row is simply the last coefficient, 5.
| Row | Entries |
|---|---|
| s4 | 1 18 5 |
| s3 | 8 16 |
| s2 | 16 5 |
| s1 | 13.5 |
| s0 | 5 |
Read the first column : 1, 8, 16, 13.5, 5. There is no sign change, so no root lies in the right half of the s-plane and the system is stable — option 1.
What the other options would have required. An unstable verdict needs at least one sign change in the first column, and the number of changes equals the number of right-half-plane roots. A marginally stable system produces an entire row of zeros, signalling a pair of roots exactly on the imaginary axis. Conditional stability applies only when the equation carries an adjustable gain K, so that stability holds over some range of K; there is no such parameter here.
A confirming factorisation. The polynomial happens to factor neatly:
\(s^{4}+8s^{3}+18s^{2}+16s+5=(s+1)(s+1)(s+5)\left(s+1\right)\) — more precisely its roots are all real and negative, at approximately −0.47, −1, −1.53 and −5. Every root has a negative real part, which is exactly what the Routh array predicted.
Hence, the system is stable.
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.
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 :
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
