Poles located exactly on the unit circle indicate the system is:

• A. Unstable
• B. Asymptotically stable
• C. Overdamped
• D. Marginally stable

Answer: (D)

When poles of a system's transfer function are located exactly on the unit circle in a discrete-time control system, it indicates that the system is marginally stable. This means that the system may oscillate indefinitely without either growing or decaying. Specifically, poles on the unit circle correspond to purely oscillatory behavior, where the system responds to certain inputs by producing sustained oscillations.

Marginal stability is characterized by the fact that the system's output remains bounded over time. However, any input or disturbance that causes energy to be added or taken away can lead to a different behavior. Unlike asymptotically stable systems, which return to an equilibrium point after disturbances, marginally stable systems do not have this property. Instead, they can remain in oscillation indefinitely if perturbed.

In contrast, the other options do not apply because unstable systems would have poles outside the unit circle, leading to growing signals. Asymptotic stability would require poles inside the unit circle, which indicates that disturbances are diminished over time. Overdamped systems are characterized by specific damping ratios in second-order systems, rather than the pole locations directly defining stability types. Therefore, the presence of poles on the unit circle signifies that the system operates under conditions of marginal stability.