Questions: Proportional-Integral-Derivative Control: Combined Action

Question 1

A temperature controller uses only proportional control (P-only) with gain Kₚ. The system reaches steady state with a constant 3°C error (offset) below the setpoint. What is the most direct way to eliminate this offset without changing Kₚ?

• A. Increase the setpoint by 3°C to compensate for the expected offset
• B. Add integral action, which accumulates the error over time until the output eliminates it
• C. Add derivative action, which detects the persistent error and adds corrective force
• D. Double Kₚ, which will halve the offset and eventually reduce it to zero

Answer: (B)

Pure proportional control requires a nonzero error to produce any output — it can only apply corrective force proportional to what error currently exists. A constant load disturbance requires a constant corrective force, and the only way P-only control generates that force is by maintaining a nonzero steady-state error (offset). Increasing Kₚ reduces the offset but cannot eliminate it without becoming infinite. Integral action solves this by accumulating the error: even a tiny persistent offset causes the integral term to grow until the output increases enough to eliminate the error entirely. Derivative action responds to the rate of error change, not accumulated error, so it cannot eliminate steady-state offset.

Question 2

During a large setpoint step in a PID-controlled process, the actuator fully saturates at its maximum position for an extended period. What is the likely consequence if no anti-windup scheme is implemented?

• A. The derivative term will saturate and become zero, leaving only P+I control active
• B. The integral term keeps accumulating error even though the actuator cannot respond, causing a large overshoot when the actuator finally desaturates
• C. The proportional term will dominate and the loop will become oscillatory at its natural frequency
• D. The controller will switch to pure proportional control until saturation ends

Answer: (B)

Integrator windup occurs when the actuator is saturated and cannot produce additional output, but the integral term keeps accumulating error as if it could. When the process variable finally approaches the setpoint and the actuator desaturates, the integral has accumulated a large value that continues to drive the actuator in the same direction, causing significant overshoot. Anti-windup schemes — such as clamping the integral when saturation is detected — prevent this by stopping or unwinding the integral accumulation during saturation.

Question 3

Derivative action in a PID controller contributes phase lead to the open-loop transfer function, which improves phase margin and damps oscillatory responses.

• T. True
• F. False

Answer: True

A derivative term contributes phase lead in the frequency domain, counteracting the phase lag introduced by the plant and other loop elements. This improves phase margin and moves the closed-loop poles away from the imaginary axis. In the time domain, derivative action provides 'anticipatory braking' — if the error is large but shrinking rapidly, the derivative term reduces the control effort before the error reaches zero, preventing overshoot and damping oscillations. This is why derivative action is sometimes called 'rate action' or 'anticipatory control.'

Question 4

Increasing the integral gain Kᵢ generally improves control performance by eliminating steady-state error faster.

• T. True
• F. False

Answer: False

Integral action adds a pole at the origin in the open-loop transfer function, which introduces phase lag. Increasing Kᵢ increases the integral action's contribution across frequencies, adding more phase lag and reducing phase margin. At some point, the reduced phase margin destabilizes the loop, causing oscillation or even instability. There is an optimal range for Kᵢ: enough to eliminate steady-state error at an acceptable rate without destabilizing the loop. Large Kᵢ also increases susceptibility to integrator windup during actuator saturation. Faster error elimination is not always better — Kᵢ must be tuned against process dynamics.

Question 5

Explain why pure proportional control cannot eliminate steady-state error in the presence of a constant load disturbance, even with very high proportional gain.

Model answer: Pure proportional control generates output u(t) = Kₚ·e(t). For the controller to produce any output at all, there must be a nonzero error e(t). A constant load disturbance requires a constant nonzero control output to counteract it. Therefore, the proportional controller must maintain a constant nonzero error at steady state — this residual error (offset) is what sustains the corrective force. Increasing Kₚ makes the required offset smaller (the same corrective force is generated by a smaller error), but offset can never be zero unless Kₚ is infinite, which would destabilize the loop. Eliminating offset requires integral action, which accumulates past error and can provide the needed constant force even as current error approaches zero.

A proportional controller lacks memory — it only knows the current error, not what it has been. To generate a sustained output, it needs a sustained input (error). An integrator, by contrast, accumulates past error and can maintain a nonzero output even when current error is zero. This is why the integrator is called 'reset action': it resets the steady-state operating point until offset is eliminated.