Consider a sixth order system with characteristic equation s6 + 2s5 + 8s4 + 12s3 + 20s2 + 16s + 16 = 0, The control system is:
Limitedly stable
To determine the stability of a sixth order system with the characteristic equation \(s^6 + 2s^5 + 8s^4 + 12s^3 + 20s^2 + 16s + 16 = 0\), we'll apply the Routh-Hurwitz stability criterion.
The Routh-Hurwitz criterion states that a system is stable if and only if all the elements in the first column of the Routh array are of the same sign and non-zero. Let's construct the Routh array for this characteristic equation.
| Row | Elements | |||
|---|---|---|---|---|
| 1 | \(s^6:\) | 1 | 8 | 20 |
| 2 | \(s^5:\) | 2 | 12 | 16 |
| 3 | \(s^4:\) | \(\dfrac{(2\times8) - (1\times12)}{2} = 2\) | \(\dfrac{(2\times20) - (1\times16)}{2} = 12\) | 0 |
| 4 | \(s^3:\) | \(\dfrac{(2\times12) - (2\times12)}{2} = 0\) | \(\dfrac{(2\times0) - (2\times0)}{2} = 0\) | |
| 5 | \(s^2:\) | 12 | 16 | 0 |
| 6 | \(s^1:\) | \(\dfrac{(12\times0) - (0\times16)}{12} = 0\) | 0 | |
| 7 | \(s^0:\) | 16 | ||
Analyzing the Routh array, we notice that the third row (\(s^4\)) results in a zero. To circumvent this problem and check for balance, we replace the zero with a small positive number \( \epsilon \). After making this replacement, we'll keep our calculations by checking further rows' composition of non-zero elements but ensure they don't significantly alter system behavior still keeping elements non-zero across the stability check.
Since there is a zero in the first column of the Routh array indicating possible oscillations, and other calculations show varying signs once progressed: the system is "Limitedly Stable". This classification means that while generally stable, critical inputs or conditions might push the system into oscillations leading overarching behaviors towards instability.
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 :
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 ?
