Consider the following statements related to stability of the control system : 1. Poles in right half-plane (rhp) yield pure exponential decaying natural response. 2. If poles of multiplicity greater than one are present on the imaginary axis, then the system is marginally stable, 3. If one pole is present in right-half plane, the system is unstable. 4. A system is stable if the natural response approaches zero as time approaches  infinity. Which of the above statements is/are not correct   ?

Option 2 : 1 and 2 only

Concept:

Stability Definition:

• A system is Stable if the natural response approaches zero as time approaches infinity.
• A system is Unstable if the natural response approaches infinity as time approaches infinity.
• A system is Marginally Stable if the natural response neither decays nor grows but remains constant or oscillates within a bound.

 

BIBO Stability:

• A system is Stable if every bounded input yields a bounded output.
• A system is Unstable if any bounded input yields an unbounded output.

 

Stability Analysis from Closed Loop Transfer function:

• Stable systems have closed-loop transfer functions with poles only in the left half-plane.
• Unstable systems have closed-loop transfer functions with at least one pole in the right half plane and/or poles of multiplicity greater than one on the imaginary axis.
• Marginally Stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity 1 and poles in the left half-plane.

 

 

Sufficient Conditions for Instability and Marginal Stability:

• A system is Unstable if all signs of the coefficients of the denominator of the closed loop transfer function are not same.
• If powers of s are missing from the denominator of the closed loop transfer function, then the system is either Unstable or at best Marginally Stable

 

Solution:

1. Statement (1) states that 'Poles in right half-plane (rhp) yield pure exponential decaying natural response' is incorrect and the correct statement is that 'Poles in right half-plane (rhp) yield exponential rising natural response'.
2. Statement (2) states that 'If poles of multiplicity greater than one are present on the imaginary axis, then the system is marginally stable' is incorrect and the correct statement is that 'Marginally Stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity 1 and poles in the left half-plane.
3. Statement (3) states that 'If one pole is present in right-half plane, the system is unstable' is correct. 
4. Statement (4) states that 'A system is stable if the natural response approaches zero as time approaches  infinity' is correct.

 

As the incorrect statements are statement (1) and (2), so the correct option is option (2). 

Stability of closed loop system with gain variation:

As gain is increased from 3 to 7, the system becomes unstable.