A pole with |z| > 1 results in a system that is:

• A. Stable
• B. Marginally stable
• C. Critically damped
• D. Unstable

Answer: (D)

In control systems and signal processing, the stability of a system is often determined by the location of its poles in the complex plane. When analyzing the poles of a transfer function, if any pole has a magnitude greater than 1 (i.e., |z| > 1 in the z-domain), this indicates that the corresponding response to an input will grow exponentially over time, leading to instability.

For discrete-time systems, having poles outside the unit circle (|z| > 1) results in outputs that diverge as time progresses. This behavior contrasts with stable systems, where all poles must lie inside the unit circle (|z| < 1), leading to outputs that decay to zero. Thus, a pole with a magnitude greater than one indicates that the system's response will become increasingly uncontrollable, causing potential instability.

In summary, a system with a pole where |z| > 1 is considered unstable because such a pole contributes to an exponentially growing response, which does not converge back to zero over time. Stability is crucial in control systems to ensure that the output remains bounded and manages to return to equilibrium after disturbances.