Control Engineering Basics
Which of the following is the condition for a continuous-time linear system to be stable (its impulse response approaches zero with time)?
- AAll zeros of the closed-loop transfer function have negative real parts
- BAll poles of the closed-loop transfer function lie inside the unit circle of the z-plane
- CAll poles of the closed-loop transfer function have negative real parts
Show answer and explanation
Answer: C. All poles of the closed-loop transfer function have negative real parts
A pole p produces a component of the form e^(pt) in the response, which decays with time if its real part is negative and keeps growing if it is positive. The condition for stability is therefore that all poles lie in the left half of the s-plane. Poles on the imaginary axis cause an oscillation that continues without decaying, which cannot be called stable (marginal stability). The inside of the unit circle is the condition for discrete-time systems expressed with the z-transform.
- A. Zeros affect the shape of the response, but stability is determined by the poles.
- B. The inside of the unit circle is the stability condition for discrete-time systems (the z-plane).
- C. Correct. The system is stable if all poles lie in the left half of the s-plane.