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.

Actuator lag is delayed adjustment of an actuator's actual output to its commanded value. A simple dynamic model iswhere $u_{\mathrm{cmd}}$ is the command, $u$ the delivered quantity, $K$ the steady-state conversion and $\tau$ the time constant. A first-order actuator model with an estimated time constant appears in MathWorks's aircraft-parameter estimation example.^[ref]

For a step command of amplitude $U$ from zero initial output,The response reaches about63.2percent of its final value at one time constant and95.0percent at three time constants. In an original example with $K=1$, $U=10$ output units and $\tau =0.20\mathrm{s}$, the delivered value at0.20seconds is6.32, and its95percent rise takes about0.599seconds. The command itself has already reached10 at time zero.

The same model's frequency response is $K/(1+i\omega \tau )$. At $\omega \tau =1$, its magnitude is $|K|/\sqrt{2}$ and its phase lag is45degrees for positive $K$. Thus an actuator that is accurate for a constant command can still attenuate and phase-shift a rapidly changing command.

A direct identification experiment records the command and the actual output on a shared time base, applies a small step around an operating point, and estimates the final change and time to63.2percent. This is an explanatory procedure for the model, not a claim about a particular measured actuator. Sensor filtering, delay and an unknown initial state can alter the fitted response and must be included when relevant.

In a control system, treating $u=u_{\mathrm{cmd}}$ discards this state. The example in unmodeled dynamics shows how doing so changes closed-loop motion from a single exponential to a damped oscillation. Physical actuator saturation and rate limits are additional effects; the linear first-order lag equation does not represent them automatically.