Proportional (P) Controller: Step Response and the Effect of Gain (Kp)

The Proportional (P) controller is the simplest form of feedback control. It calculates the control output as:u = Kp × error
(where error = setpoint − actual value)When we apply a step input (sudden change in setpoint from 0 to 1), the system's response depends heavily on the gain Kp.Key Effects of Increasing Kp

• Faster response: The output rises toward the setpoint more quickly.
• Reduced steady-state error: The final offset gets smaller, but never zero.

For a typical first-order system and unit step, the steady-state error is:ess = 1 / (1 + Kp)
Final output = Kp / (1 + Kp)Step Response Comparison![Step Response Graph]
(Embed the graph here: setpoint jumps to 1.0; outputs shown for Kp = 0.5, 1.0, 2.0, 5.0, 10.0)

• Kp = 0.5 → final ≈ 0.33 (error 0.67) – slow, large offset
• Kp = 1.0 → final ≈ 0.50 (error 0.50)
• Kp = 2.0 → final ≈ 0.67 (error 0.33)
• Kp = 5.0 → final ≈ 0.83 (error 0.17)
• Kp = 10.0 → final ≈ 0.91 (error 0.09) – fast, small offset

In this first-order example, there's no overshoot. Higher-order systems can oscillate at high Kp.LimitationPure P control always leaves some steady-state error for step changes. To eliminate it, add an integral term (PI/PID controller).When to Use P-OnlyGreat for simple applications where a small offset is acceptable and fast response matters (e.g., basic fan speed or level control).Higher Kp gives you speed and accuracy—but push it too far and stability suffers.

Results

Theoretical steady-state values for unit step input:
(Formula: final output = Kp / (1 + Kp), error = 1 / (1 + Kp))
Kp =  0.5 → final output = 0.3333, steady-state error = 0.6667
Kp =  1.0 → final output = 0.5000, steady-state error = 0.5000
Kp =  2.0 → final output = 0.6667, steady-state error = 0.3333
Kp =  5.0 → final output = 0.8333, steady-state error = 0.1667
Kp = 10.0 → final output = 0.9091, steady-state error = 0.0909