Proportional Control and Stability

How hard should the robot correct?

A rover faces 0° and must turn to 90°. A tiny correction is safe but slow. A strong correction moves quickly—but may carry the rover past the target and send it back again.

Predict: If the rover keeps overshooting left and right, should you always increase its gain?

The proportional rule

Define signed error:

e = setpoint - measurement

Then compute the controller output:

u = Kp × e

Kp is proportional gain. It converts error units into command units. With a 90° setpoint, a 60° measurement, and Kp = 0.02 command/degree:

e = 90° - 60° = +30°

u = 0.02 × 30 = +0.60

The positive sign tells the rover which way to turn. The magnitude tells it how strongly.

The actuator gets the final word

Suppose the motor command must remain between −0.50 and +0.50. The controller asks for +0.60, but the applied command is:

u_applied = clamp(+0.60, -0.50, +0.50) = +0.50

This is saturation. The controller equation and physical command are not always identical. Any evaluation that ignores command limits can recommend impossible behavior.

Read a response, not a slogan

A low gain often gives a slow response. A moderate gain may balance speed and overshoot. An excessive gain can amplify delayed corrections, cause repeated overshoot, keep the actuator saturated, or destabilize some plants. These are tendencies, not universal guarantees: the plant, sampling, friction, delay, and limits matter.

Worked gain comparison

Three simulated runs use the same rover, target, starting angle, update period, and command limit.

• Kp = 0.005: small commands; slow approach; little overshoot.
• Kp = 0.02: faster approach; bounded correction; settles inside tolerance.
• Kp = 0.08: reaches the target quickly but overshoots, reverses, and spends more time saturated.

The defensible choice is not “the biggest gain.” It is the gain whose full response meets the task constraints.

Stability intuition

For this lesson, call an equilibrium locally stable when a small disturbance produces a response that remains bounded and returns toward the target. One successful crossing of 90° does not establish stability. Inspect what happens afterward and repeat with a disturbance.

Retrieval pause

At a setpoint of 90°, the measured heading is 100°, and Kp = 0.03. What are the signed error and unsaturated command?

Answer after calculating: e = -10°, so u = -0.30. The command reverses direction because the rover passed the target.

Misconception checks

• “More gain always improves control.” No. More gain changes the closed-loop dynamics and control effort; it may worsen overshoot, saturation, or stability.
• “Zero error means the robot is safe.” No. Velocity, obstacles, actuator state, and sensor validity still matter.
• “A simulation proves hardware behavior.” No. It supplies evidence about a stated model, not a guarantee about a different physical plant.

Summary

Proportional control maps signed error to command through u = Kp e. Gain should be selected from response evidence—speed, overshoot, settling, residual error, saturation, and boundedness—under realistic limits and disturbances.

Further learning

• WPILib PID Controller
• University of Michigan: Introduction to PID Controller Design
• MIT Underactuated Robotics: Lyapunov Analysis