A Reframing for Velocity-Driven Systems — New Year 2026

🎆 Happy New Year 2026.
This article is not written to criticize PID, but to re-examine the meaning of the Integral (I) term when the true control objective is velocity, not position.
From that re-examination, a cleaner and more physical viewpoint naturally emerges: Energy-Based Control

🔧 1. The real problem is not PID — it is the control objective

Classic PID implicitly assumes:

The controlled variable is position.

This assumption:

• works well for RC servos and clean lab systems
• breaks down in industrial motion systems where:
  • commands are velocity
  • position is merely a time integral

In modern tracking and following systems, the true control object is:

⚙️ Velocity, not angle or distance.

Applying classic PID blindly leads to:

• PID output interpreted as position
• followed by an ad-hoc conversion to velocity

There is no rigorous mathematical foundation for that conversion.

🔁 2. Why the classical I-term “chokes” in practice

The classical integral term is defined as:

$$I(t) = \int_{t_0}^{t} K_i\, e(\tau)\, d\tau$$

Consider a simple case:

• position command: 0° → 120°
• physical limit: 0° → 180°

Within the first few steps:

• the integral term already exceeds physical bounds
• even though the system is far from settled

As a result:

• engineers are forced to make Ki extremely small
• which renders the integral ineffective later

This is not a tuning problem — it is a mismatch between mathematics and physics.

⚡ 3. Reframing the system: velocity is the controlled variable

Let us explicitly control velocity:

• $V_{target}$Vtarget​: target velocity
• $V_{actual}$Vactual​: actual servo velocity

Velocity error:$$E_v = V_{target} – V_{actual}$$

Integrate over time:

$$\int_{t_0}^{t} E_v(\tau)\, d\tau = \int_{t_0}^{t} V_{target}\, d\tau – \int_{t_0}^{t} V_{actual}\, d\tau$$

Recognizing that:

• $\int V_{target} = pos_{cmd}(t) – pos_{cmd}(t_0)$
• $\int V_{actual} = pos_{cur}(t) – pos_{cur}(t_0)$

📐 4. A simplifying assumption without loss of generality

🔹 Assumption:
The target starts at the servo’s initial position:

$$pos_{cmd}(t_0) = pos_{cur}(t_0) = pos_0$$

Then:$$\int_{t_0}^{t} E_v(\tau)\, d\tau = pos_{cmd}(t) – pos_{cur}(t)$$

🎯 Which is exactly the position error:

$$e = pos_{cmd} – pos_{cur}$$

✨ 5. A key and elegant conclusion

🔥 The integral of velocity error equals position error.

This means:

• the integral term does not need to be computed
• it already exists naturally as e

The “I” term is no longer a blind accumulator —
it represents stored energy in the system.

🔋 6. Energy-Based interpretation of the I-term

In this framework:

• e = accumulated energy in the system
• e large → system is far from the target, more energy is required to close the gap
• e small → system is near the target, energy injection should be reduced

If e changes abruptly:

• energy is injected or removed abruptly
• the system enters an irregular transition state

We quantify this irregularity as:

float r = fabsf(e - e_last) / (fabsf(e_last) + Efloor);

Key properties:

• normalization by historical energy
• no circular dependency on the current state
• r is later saturated to [0..1]

👉 r is not a control variable, only a state indicator.

🧠 7. Re-interpreting a fuzzy rule correctly

Consider the rule:

IF E far AND R stable THEN Ki high

With the energy view, this becomes clear:

• the system is stably tracking
• but still far from the target
  → it is safe and efficient to inject more energy

No heuristics. No PID dogma.

🏭 8. Why this approach appeared so late

Not because it is incorrect — but because it was hard to see:

1. RC servos → simple, forgiving → PID survived
2. Industrial servos → black boxes + severe noise
3. A single parity or CRC failure:
   • no error message
   • silent system
   • no observability

In such environments, physics-based reasoning is difficult to validate, so simpler abstractions dominated.

🌱 9. Closing remarks — New Year 2026

Classic PID:

• is not wrong
• but not usable for velocity-driven systems when taken literally

Energy-Based Control:

• does not reject PID
• it restores the physical meaning of the I-term
• simpler, cleaner, and mathematically grounded

🎆 Happy New Year 2026.
May we continue to question what seems “obvious”,
and follow the mathematics all the way back to physics.