For a continuous-time linear system to be stable, where must all the poles of its transfer function be located in the complex s-plane?

• A. Strictly in the right-half plane
• B. On the imaginary axis
• C. At the origin
• D. Strictly in the left-half plane

Answer: (D)

For a continuous-time linear system to be stable, all the poles of its transfer function must be located strictly in the left-half plane of the complex s-plane. This requirement ensures that the system's impulse response will decay over time, leading to a bounded output for any bounded input.

When poles are located in the left-half plane, they correspond to exponential terms that decrease as time progresses, which is essential for stability. Each pole contributes a term to the system's response, and if these poles are situated in the left-half plane, they will naturally result in a system response that converges to zero, confirming the stability.

In contrast, poles situated in the right-half plane would correspond to terms that grow unbounded over time, indicating instability. Poles on the imaginary axis suggest oscillatory behavior that may not converge, representing a marginally stable system. A pole at the origin implies a constant gain, which typically leads to a system that is marginally stable as well but can become unstable depending on the context and inputs. Therefore, having all poles strictly in the left-half plane is a necessary and sufficient condition for the stability of a continuous-time linear system.