Technology / Technical guide
PI & PID Control: Proportional, Integral & Derivative Roles
PI and PID describe ways to combine present error, accumulated error and rate information. Their names identify mathematical roles, not a complete control design. Understanding those roles helps explain why two controllers can react differently to the same present error.
The present error is only part of a controller’s state.
Integral memory and derivative response add different information to proportional action; neither is a universal improvement by itself.
01 / Parallel form
Three terms contribute to one requested output.
A conceptual continuous parallel PID can be written u = Kp e + Ki ∫e dt + Kd de/dt, with a specified integral state.
A conceptual continuous parallel PID can be written u = Kp e + Ki ∫e dt + Kd de/dt, with a specified integral state. PI omits the derivative term. The error e is the difference between a declared reference and its compared observation.
MathWorks distinguishes weighted controller forms. These symbols identify roles, not settings. Other parameterisations place a proportional gain around combined terms, so similarly named gains need not be interchangeable.
02 / Proportional role
Present error creates a present contribution.
Proportional action scales the current error.
Proportional action scales the current error. With the same Kp and current error, its contribution is the same regardless of earlier error history. Whether that contribution is sufficient to remove an offset depends on the controlled system and disturbances.
Increasing a proportional gain is not simply equivalent to making every response better. The effect passes through the plant, feedback dynamics and constraints. A strong immediate request can coexist with oscillatory or limited physical response.
03 / Integral role
Accumulation gives the controller memory.
The integral term depends on error over time and the initial stored state.
The integral term depends on error over time and the initial stored state. A persistent same-sign error can keep changing this contribution even when the current error magnitude is unchanged. When the error returns to zero, the accumulated contribution need not instantly disappear.
Hypothetical integral area: an error of 2 arbitrary units maintained for 3 seconds contributes 6 unit·s to the integral. Converting that area to output requires Ki and the existing integral state. The example specifies neither a controller gain nor a physical actuator command.
04 / Derivative role
Rate information differs from prediction.
The ideal derivative term responds to the error’s rate of change.
The ideal derivative term responds to the error’s rate of change. It is zero for a truly constant error but can be large for rapid changes. Differentiation can also emphasise high-frequency measurement variation, which is why derivative implementation and filtering matter.
Derivative action does not predict an unknown future wind event. A reference discontinuity and a rapidly changing measurement can contribute differently depending on the controller form. Practical systems may differentiate a measurement rather than the full error; that is an implementation distinction.
05 / Term units
Gains must convert different error dimensions.
Let error have units E and requested output have units U, with time in seconds.
Let error have units E and requested output have units U, with time in seconds. In the stated parallel form, Kp has U/E, Ki has U/(E·s), and Kd has U·s/E. Each term then contributes the same output unit despite using different error quantities.
Sampling, filtered derivative, limits and state handling affect a practical controller beyond the ideal expression. The saturation guide explains why an accumulated term can disagree with achievable output.
06 / Product scope
The mathematical vocabulary does not identify an algorithm.
A feedback diagram or electric-yaw label does not establish that a MARSTON controller implements PI, PID or a particular gain structure.
A feedback diagram or electric-yaw label does not establish that a MARSTON controller implements PI, PID or a particular gain structure. Loop variables, sampling and actuator constraints belong to the actual implementation, which is not assigned by this explanation.
No gains, tuning steps, derivative filter settings or operating modifications are provided here. The yaw feedback guide supplies the separate distinction between requested and achieved quantities.
Continue the review with Control Saturation, Integral Windup & Anti-Windup.
Continue the review with Yaw Feedback: Setpoint, Measurement & Closed-Loop Error.
Continue the review with Wind Measurement Filtering: Bandwidth, Delay & Latency.
Continue the general control review with transfer functions, poles and zeros, gain and phase margins, and discrete control and zero-order hold. These explanations do not identify an algorithm implemented in a particular controller.
General background: MathWorks: PI/PID term definitions, controller forms and filtered derivative. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| P contribution | Kp e | Uses present error in the declared form. |
| I contribution | Ki times accumulated error | Initial state and elapsed history matter. |
| D contribution | Kd times error rate | Ideal derivative and practical filtering differ. |
| Gain dimensions | U/E; U/(E·s); U·s/E | Units refer to the specified parallel form. |
| Physical output | Requested u versus achieved response | Plant dynamics and limits remain separate. |
Engineering questions.
Does PI include derivative action?
No. The stated PI structure contains proportional and integral terms.
Does zero current error erase integral history?
No. Its stored contribution can remain.
Does a derivative predict future wind?
No. It responds to the defined signal’s rate information.
Can gains be copied between controller forms?
Not automatically. Definitions, units and implementation can differ.
Is more proportional gain always better?
No. Dynamics and constraints affect the result.
Are MARSTON PI/PID gains confirmed here?
No algorithm implementation or tuning values are assigned.
Technical enquiry
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PI and PID describe ways to combine present error, accumulated error and rate information. Share the model and applicable operating or design information for an engineering discussion.