Consider the following statements regarding stability in terms of characteristic equation of a control system: 1. If any oscillations set up in a system in consequence to application of an input are damped out with respect to time, the system is said to be stable. 2. If the magnitude of the oscillations is sustained, the system is unstable. 3. For a stable system, there should be no change of sign in the first column of Routh array formed from the coefficients of the characteristic equation expressed in polynomial form. Which of the above statements are correct?
Control System Stability Analysis
This question examines the conditions for control system stability based on the characteristic equation and related analysis methods like the Routh array.
Statement Evaluation
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Statement 1: Oscillations and Stability
This statement correctly describes asymptotic stability. A system is considered stable if any initial oscillations or disturbances eventually decrease and die out over time, returning the system to its equilibrium state.
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Statement 2: Sustained Oscillations
This statement is incorrect. If oscillations are sustained (maintain a constant magnitude), the system is classified as marginally stable, not unstable. Instability occurs when oscillations grow exponentially over time.
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Statement 3: Routh Array First Column Rule
This statement is correct. According to the Routh-Hurwitz stability criterion, for a linear time-invariant (LTI) system to be stable, all the roots of its characteristic equation must lie in the left half of the complex s-plane. This translates to the condition that all elements in the first column of the Routh array must be non-zero and possess the same sign (typically positive, assuming the leading coefficient of the characteristic polynomial is positive). Any sign change in the first column indicates the presence of unstable poles (roots in the right-half plane).
Correct Statements Identification
Based on the analysis of each statement:
- Statement 1 is correct.
- Statement 2 is incorrect.
- Statement 3 is correct.
Therefore, the correct statements regarding control system stability are 1 and 3.
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 characteristic equation of given system is 6s + K = 0. Determined the range of K for which the system to be stable.
