How the Proportional, Integral, and Derivative Terms of a PID Controller Eliminate Steady-State Error

The common assumption in industrial automation is that simply increasing a system's responsiveness will eventually force it to its exact target. The mathematical models of control theory prove the opposite: a purely proportional response guarantees it will never arrive.

Whether steering a cargo ship, regulating a 2026 commercial espresso machine, or managing the cooling rods in a nuclear reactor, the mechanism keeping the system stable is almost certainly a Proportional-Integral-Derivative (PID) controller. First formalized in 1922 by Nicolas Minorsky for the US Navy, the algorithm remains the undisputed foundation of physical automation.

The proportional term is the foundation of the algorithm. It calculates the current difference between the desired setpoint and the actual measured value, applying a corrective force proportional to that error. If a drone is ten feet below its target altitude, the proportional term commands the rotors to spin fast; if it is one foot below, it commands them to spin slowly.

However, proportional control contains a fatal mathematical flaw when used alone. As the system approaches its target, the error shrinks, and therefore the corrective force shrinks.

Eventually, the shrinking corrective force exactly matches the system's ambient resistance—whether that is gravity pulling down on the drone, or ambient air cooling a heating element. At this equilibrium point, the system stops moving, leaving a permanent gap known as steady-state error.

The University of Michigan's control theory models quantify this gap precisely. For a standard step input, the steady-state error is defined by the ratio of 1 divided by 1 plus the proportional gain. No matter how high the proportional gain is cranked, the denominator ensures the error never reaches true zero.[3]

"A proportional controller will have the effect of reducing the rise time and will reduce but never eliminate the steady-state error," the University of Michigan researchers note in their PID design introduction.[3]

This is where the integral term becomes necessary. Instead of only looking at the present moment, the integral term accumulates past errors over time.

Even if the steady-state error is a microscopic 0.1 degrees, the integral term continuously adds that 0.1 to a running total. Over seconds or minutes, this accumulated sum grows large enough to generate the exact amount of force needed to push the system the rest of the way.

Electrical Technology's 2019 analysis of PI controllers confirms this mechanism. "The integral controller produces an output which is proportional to the integral of the error signal," they write, explaining that this continuous summation is what finally drives the steady-state error to exactly zero.[4]

But solving the steady-state error introduces a new problem: overshoot. Because the integral term accumulates a massive historical sum while the system is getting up to speed, it often holds onto that built-up force even as the system crosses the target line, causing it to blast past the goal.

Engineering LibreTexts highlights this exact vulnerability in their 2020 breakdown of PID downsides. They point out that integral windup can cause severe oscillations, where the system violently overcorrects back and forth across the setpoint.[2]

To tame this oscillation, the derivative term is added. The derivative term calculates the rate of change of the error, effectively predicting the future. If the system is approaching the target too quickly, the derivative term applies a negative force—a brake—to slow it down.

Zurich Instruments, in their 2023 principles guide, describes the derivative action as providing "a damping effect on the system response," which is crucial for preventing the overshoot caused by the integral term's historical baggage.[1]

While the mathematics of PID are elegant on a chalkboard, real-world implementation requires hard limits. Theoretical models allow for infinite gains, but physical hardware does not.

ABB's industrial controller specifications reveal these physical boundaries. Their hardware limits the integral time constant to a strict range of 0.01 to 9999 seconds, and caps the derivative time constant between 0.001 and 9999 seconds.[5]

These hardcoded limits mean that while a PID controller can mathematically eliminate steady-state error, the physical reality of the hardware dictates exactly how fast that elimination can safely occur without tearing the machinery apart.[5]