Steady-state offset among classical controllers: For a unit-step load/change on a stable process, which controller structure tends to have the largest steady-state offset (residual error)?

Introduction / Context:
Offset (steady-state error) is a key performance metric for regulatory control. Controller structures differ in how they remove persistent error after a step disturbance or set-point change. Understanding which structures inherently eliminate offset guides practical selection in plants.

Given Data / Assumptions:

• Stable plant, unity feedback conceptual model.
• Step-type change and steady-state considered.
• Reasonable tuning, no actuator saturation or integrator windup.

Concept / Approach:
A pure proportional controller leaves nonzero steady-state error for typical process types because the controller output must balance the process load, requiring a finite error to generate a finite output. Adding integral action (PI or PID) introduces a pole at the origin in the open loop, driving the steady-state error to zero for step inputs. PD alone does not change the steady-state gain (derivative term vanishes at steady state), so it behaves like P with respect to offset. Hence, among the listed, P shows the largest offset; PI and PID remove it (ideally), PD behaves similar to P for offset.

Step-by-Step Solution:

Verification / Alternative check:
Final value theorem applied to standard unity-feedback loops confirms that integral action guarantees zero steady-state error for step inputs when the closed loop is stable.

Why Other Options Are Wrong:

Common Pitfalls:
Confusing transient improvement from derivative action with steady-state accuracy. Only integral action addresses offset.

Final Answer:
Proportional (P) controller