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Multiplicative uncertainty represents a model error relative to a nominal controlled plant. For a scalar transfer function, one useful form iswhere $G_0$ is nominal, $W$ shapes the permitted relative error across frequency, and $\Delta$ belongs to a specified bounded uncertainty class. Unlike an additive error, the correction is multiplied by the nominal plant. MathWorks's ucover reference distinguishes input multiplicative, output multiplicative and additive forms.^[ref]

For an original static-error example, take $G_0(s)=2/(s+1)$, $W=0.20$ and a real constant $\Delta \in [-1,1]$. The allowed family isThe uncertainty changes gain while preserving the pole in this family. A dynamic $\Delta (s)$ can instead introduce frequency-dependent magnitude and phase changes; its admissible dynamics must be stated separately.

An omitted actuator lag illustrates a dynamic relative error. If $G=G_0/(1+\tau s)$, thenAt $|\omega |\tau =0.1$, its magnitude is about0.0995; at $|\omega |\tau =10$, it is about0.995. A constant10percent relative-error bound would cover the first frequency but fail badly at the second. A shaping function should reflect that discrepancy rather than extrapolate a low-frequency fit.

For a matrix-valued plant, multiplication order matters. Input uncertainty is placed on the right of $G_0$, while output uncertainty is placed on the left; the identity matrices and uncertainty dimensions must match the corresponding signal spaces.^[ref] These forms generally cannot be interchanged when matrix products do not commute.

The scalar ratio $(G-G_0)/G_0$ also needs care near zeros of $G_0$. A tiny absolute discrepancy can be a large relative discrepancy there. Choosing multiplicative uncertainty is therefore a modeling decision about which relative errors and frequency ranges a robust control argument is intended to cover.

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.