What is the difference between PD and PID?

PID adds an integral contribution based on accumulated error; PD uses only proportional and derivative terms.

Can derivative action be useful when current error is zero?

Yes. If the system is moving through the target, the error derivative can be nonzero and influence the command.

Does PD require a model of every physical detail?

The formula is simple, but selecting and validating gains requires enough understanding of the plant and implementation to support the intended performance.

AI Glossary / Graphs and symbolic AI / Intermediate

PD controller

Definition

A PD controller is a feedback controller that combines a proportional response to the current error with a derivative response to how that error is changing. PD stands for proportional - derivative. It has no integral term for accumulating error over time.

What it means

The controller compares a desired value with a measured value. If a robot joint is short of its target angle, proportional action produces a command related to that position error. Derivative action responds to the error's rate of change and can provide damping in a suitable control design.

This helps distinguish being far from the target from approaching it rapidly. The same position error can produce different commands depending on motion.

A PD controller is a classical control method, not inherently a machine-learning system. It can nevertheless be part of an AI robot, with a planner or learned policy specifying targets that the controller tracks.

How it works

For an ideal continuous-time controller, define error e(t)=r(t)−y(t), where r is the desired value and y is the measured value. The command is u(t)=Kp e(t)+Kd de(t)/dt. The gains determine the contribution of position-like error and its rate.

For a constant reference, the derivative of error is the negative of the measured output's rate. In position control, the derivative term can therefore oppose velocity toward the target as the error shrinks. Actual behavior depends on the plant dynamics and gain choices.

Digital implementations estimate derivatives from sampled measurements or use velocity sensors. Differentiation can amplify high-frequency noise, so filtering is commonly needed. Applying derivative action to the measurement rather than the full error is another implementation choice, particularly around abrupt reference changes.

Stability, saturation, delay and sampling effects require analysis. Increasing gains indiscriminately can worsen behavior. PD control also does not generally eliminate steady-state error caused by constant disturbances or uncompensated loads.

Three practical examples

A simulated robot joint uses position error and measured angular velocity to regulate a constant target angle.
A positioning stage combines proportional correction with derivative damping to reduce oscillatory motion under an appropriate model.
A robotics experiment compares proportional-only and PD feedback while measuring settling, overshoot and command noise.

Think of it this way

Think of moving a trolley toward a marked position. You respond both to how far it remains from the mark and to how quickly it is approaching. Moving quickly near the target calls for a different response from moving slowly at the same location.

The analogy is limited because a real controller uses measured signals, defined gains and actuator commands. Human anticipation is not the mathematical derivative term, and a stabilizing design depends on the system rather than intuition alone.

Worked example

Consider an invented joint controller with desired angle 1.0 radian and measured angle 0.8 radian. The constant target gives error e=0.2 radian. Suppose measured angular velocity is +0.3 radians per second, so de/dt=−0.3 radians per second.

Choose Kp=10 newton-meters per radian and Kd=2 newton-meter-seconds per radian. Proportional torque is 10×0.2=2 newton-meters. Derivative torque is 2×(−0.3)=−0.6 newton-meters. The combined command is 1.4 newton-meters.

At the same angle but velocity −0.3 radians per second, the error is increasing. The derivative contribution becomes +0.6 and the command becomes 2.6 newton-meters.

These calculations specify instantaneous commands only. They do not prove stability, account for gravity or select gains suitable for a real motor. If an actuator saturates, its delivered torque can differ from the requested value.

Illustrative case study

The fictional Willow Motion team adds derivative feedback to a simulated positioning system. The initial derivative estimate comes from raw differences between noisy position samples, producing rapidly changing commands.

Willow examines the measurement process, sampling interval and derivative filtering before interpreting the result as a failure of the basic PD idea. It also checks actuator limits and evaluates the revised design against the plant model.

This fictional case illustrates why implementation details matter. The ideal formula is a starting point; useful behavior depends on reliable signals, suitable gains and the actual system being controlled.

Common misconceptions and mistakes

The derivative term predicts the future perfectly. It responds to a measured or estimated rate of change.
More derivative gain is always better. Noise, delays and system dynamics can limit useful gain.
PD always removes steady-state error. Persistent loads may require other measures, such as integral action or appropriate feedforward compensation.
A correct arithmetic command proves closed-loop stability. Stability concerns behavior over time with the plant.
PD is a learning algorithm. Its gains can be tuned through various methods, but the controller itself need not learn.

Business use

PD control is relevant to robotics and other systems where measured state must track a target. For a project using AI planning, it clarifies the distinction between choosing a target and executing controlled motion toward it.

Ask which signals are measured, how derivatives are obtained and filtered, what units the gains use, and which actuator limits apply. Evaluate transient response, residual error, noise sensitivity and robustness to plausible model changes.

A controller that performs well in one simplified simulation should not be described as universally stable. The claim should identify the modeled dynamics, operating range, sample rate and relevant validation evidence. The glossary calculation is educational, not a tuning prescription.

Check your understanding

Why is the derivative torque negative when the joint moves toward the fixed target at +0.3 radians per second?

Answer: With e=r−y and constant r, increasing y makes the error decrease. Its derivative is −0.3, so positive Kd contributes a negative torque term under the stated sign convention.

Related terms

PID controller
Feedback control
proportional control
derivative action
damping
tracking error
actuator saturation
derivative filter

Learn next

Calculate PD commands for the same position error at three velocities: approaching, stationary and retreating. State the units and sign convention. Then use a plant model to distinguish an instantaneous command calculation from a closed-loop response analysis.

Closed-loop

Sources

Lynch and Park explain PD joint control, damping and practical gain limitations: https://modernrobotics.northwestern.edu/nu-gm-book-resource/11-4-motion-control-with-torque-or-force-inputs-part-1-of-3/

MathWorks documents PID-family controller forms and derivative filtering: https://www.mathworks.com/help/control/ug/proportional-integral-derivative-pid-controllers.html

The torque arithmetic and Willow Motion case are original illustrations, not validated settings for physical equipment.

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