When Point-Based Control Collides with Physical Reality

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Introduction

Classical PID control has existed for more than a century and is still widely taught today. In many simple systems, it appears to “work,” especially when applied to RC servos or low-performance applications.

However, once we enter the domain of modern servo systems, continuous tracking, and strict physical constraints, classical PID reveals a fundamental flaw:

Classical PID breaks down as soon as its mathematical demands exceed the physical capabilities of the system.

This article does not attempt to dismiss PID’s historical value. Instead, it explains why classical PID cannot be safely or reliably extended to modern servo control under physical limits.

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1. Classical PID Is Not Mathematically Wrong

The standard PID formulation is well known:$$u(t) = K_p e(t) + K_i \int e(t)\,dt + K_d \frac{de(t)}{dt}$$

From a purely mathematical standpoint:

• Each term is dimensionally consistent
• Position error is fully accounted for
• No internal contradiction exists

If $u(t)$ is treated as an abstract control signal, and the plant is assumed to “know” how to act on it, nothing is wrong yet.

👉 The failure does not originate in the equation itself.

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2. The Real Trap Lies in Actuation: Physical Limits

The problem appears when PID is applied to a real physical system, which inevitably has limits:

• Maximum velocity
• Maximum acceleration
• Finite torque
• Finite energy

No real system can accelerate infinitely.

Yet classical PID implicitly assumes that it can.

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3. Why Overshoot Is Inevitable (Not an Accident)

Consider a common scenario:

• A distant setpoint
• The system starts from rest
• The position error $e > 0$ remains large for multiple cycles

During this phase:

• The integral term accumulates a large positive value
• The proportional term is initially large and positive

As the system approaches the setpoint:

• The P term decreases
• But the I term remains large

To cancel a large positive integral term, classical PID requires the system to accelerate very aggressively early on, so that $e$ decreases rapidly—this mathematically demands a large negative D term.

However, at exactly this moment:

• The system hits its physical limits
• The required acceleration is impossible

➡️ A contradiction emerges between mathematical demand and physical capability

As a result:

Overshoot is no longer an accident—it becomes inevitable.

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4. Why RC Servos “Seem to Work”

RC servos expose this flaw clearly:

• Their speed limits are strict
• Infinite acceleration is impossible

They appear to work only because:

• Precision requirements are low
• Oscillation around the setpoint is tolerated
• Continuous jitter is accepted as “normal behavior”

In reality:

RC servos merely mask the problem by tolerating instability, not by solving it.

This is not a hardware limitation, but a methodological one.

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5. Tracking Does Not Save Classical PID

A common counterargument is:

“In tracking applications, the setpoint changes continuously, so PID should behave better.”

This is incorrect.

In tracking systems:

• Each frame introduces a new setpoint
• But the integral term is not reset
• Small errors accumulate frame after frame

The result:

• Large overshoot may disappear
• But stop–go, hopping motion appears instead

The system repeatedly:

• rushes to each frame’s target
• brakes abruptly
• then accelerates again

The motion resembles a frog hopping, not smooth tracking.

➡️ The problem persists—it merely changes form.

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6. The Core Failure: Point-Based Setpoints Under Physical Constraints

The root cause is fundamental:

Classical PID is built around point-based setpoints, while real systems are constrained by velocity, acceleration, and finite energy.

With distant setpoints:

• PID demands excessive speed and acceleration
• Physical limits prevent compliance
• Integral energy cannot be discharged
• Overshoot is unavoidable

With tracking:

• Setpoints change frame by frame
• The system is forced into repeated braking and acceleration
• Motion becomes discontinuous

👉 The flaw lies in the control model, not the actuator.

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7. Conclusion

Classical PID:

• Is historically significant
• Is mathematically consistent
• But fails under real physical constraints

Overshoot, oscillation, and hopping motion are:

• Not tuning mistakes
• Not hardware defects
• But unavoidable consequences of applying point-based control to energy-limited systems

A different approach is required—one that directly controls velocity and energy, allowing the system to shorten error by moderate acceleration while operating safely away from physical limits.

That model is known as Energy-Based Control (MotionEdge).
See: Classical PID vs Energy-Based Control.