# Matched and unmatched uncertainties

In control theory, according to the definition of matched uncertainty, the uncertainty enters the system through the same channel as the control.

If we consider the system: $$\dot x = F(x)+G(x)u+Δ(x, t),$$

where $F$ and $G$ are known functions comprising the nominal system and $Δ$ is an uncertain function known only to lie within some bounds. For example, we may know a function $ρ(x)$ so that $| Δ(x, t) |≤ ρ(x)$.

If we require that the uncertainty $Δ$ is of the form: $$Δ(x, t) = G(x) · \barΔ(x, t)$$

for some uncertain function $\barΔ$, this form is called the matching condition because it allows the system to be written as: $$\dot x = F(x)+G(x)[u+ \barΔ(x, t)]$$

where now the uncertainty $\barΔ$ is matched with the control $u$, that is, it enters the system through the same channel as the control.

What are the physical examples for matched and unmatched uncertainties?

(Levine, William S., ed. The control handbook. CRC press, 1996.)