Some systems don’t begin at rest.
Even when untouched, they move.
The air presses. Gravity pulls. The aircraft glides, climbs, or sinks—not because it was commanded, but because something is always acting.
This is the nature of drift.
And in control theory, we model it not as noise, but as structure—as part of the system’s identity.
In an affine formulation with drift, the system’s dynamics are expressed as:
ẋ = A(x)x + B(x)u + c(x)
Here, A(x)x captures the system’s internal behavior,
B(x)u represents the control input,
and c(x)—the drift term—represents forces that persist even when no input is applied.
This is what makes the model “affine” instead of strictly linear. It includes a constant shift: a built-in push, a bias, a quiet motion that never stops.
In aircraft systems, this drift can represent:
– Gravitational pull during flight, always urging descent.
– Trimmed thrust required to maintain altitude.
– Asymmetries in aerodynamic surfaces or structural loading.
– Environmental forces like steady wind or pressure gradients.
Even when the pilot—or autopilot—issues no command, the aircraft continues to move. Not because it’s unstable, but because its neutral behavior is not zero.
Affine models with drift are essential when modeling these realities.
They allow us to design controllers around the truth, not around simplifications. They help us account for the fact that sometimes, just to hold steady, we must push back. That control is not only about responding to change—but about countering what was already there.
In such systems, control design must explicitly handle this drift.
A PID controller must include integral action to eliminate steady-state error caused by it.
An MPC must plan its horizon knowing that even with zero input, motion will continue.
A feedback linearization strategy must include cancellation of the drift term to recover a stable origin.
This drift is not noise.
It is a signature of the system’s true resting motion.
And the best controllers do not ignore it. They work with it.
Because in intelligent flight, holding position often requires more than staying still.
It requires resisting the slow, constant forces that would otherwise carry you away.
An affine system with drift reminds us:
Even in silence, there is motion.
And even in equilibrium, there is effort.