= Unmodeled dynamics
\b[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:
$$
\dot x=u,\qquad \tau\dot u+u=-kx,\qquad \tau>0.
$$
Eliminating $u$ gives
$$
\tau\ddot x+\dot x+kx=0.
$$
With $k=1$ and $\tau=1$ in chosen consistent units, the exponents are $(-1\pm i\sqrt3)/2$: 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> is
$$
\Delta(s)=\frac1{1+\tau s}-1=-\frac{\tau s}{1+\tau s},\qquad
|\Delta(i\omega)|=\frac{|\omega|\tau}{\sqrt{1+\omega^2\tau^2}}.
$$
The 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>.<ultidyn documentation in MATLAB>[Description and uncertain-dynamics shaping-filter example]{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.
Back to article page
