 Unmodeled dynamics

Unmodeled dynamics are time-dependent behaviors omitted from the nominal model of a controlled plant or its interconnection. Examples include an omitted actuator lag, an additional flexible mode or a delay. They matter when the neglected behavior acts on time scales relevant to the controller. This differs from changing only the numerical parameters in a fixed equation model.

For an original example, a nominal plant satisfies $\dot{x}=u$, and state feedback control commands $u=-kx$. The nominal prediction is $\dot{x}=-kx$, with decay rate $k>0$. Now suppose the actual applied input follows a first-order actuator:Eliminating $u$ givesWith $k=1$ and $\tau =1$ in chosen consistent units, the exponents are $(-1\pm i\sqrt{3})/2$: the actual response can oscillate while decaying, whereas the nominal model predicts a single exponential. The additional state changes the transient even though the nominal static command remains the same.

The omitted actuator also admits a frequency-domain description. Relative to an ideal actuator with gain1, its transfer function is $1/(1+\tau s)$. The corresponding multiplicative uncertainty isThe relative discrepancy is small when $|\omega |\tau$ is small but approaches1 at high frequency. A low-frequency fit alone therefore need not justify a high-bandwidth design.

Dynamic uncertainty models can describe a family by bounds on its frequency response, with a shaping filter expressing which frequency ranges are poorly known. MathWorks's ultidyn documentation describes this construction for uncertain linear time-invariant systems.^[ref] Such an uncertainty family must be chosen to cover the omitted behavior; its mere presence in a model does not prove that every physical discrepancy has been included.