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Control model uncertainty specifies how the behavior of a controlled plant may differ from a nominal model. A robust control claim is made over this allowed family. The description includes which parameters may change, their ranges, and which occurrences must represent the same physical quantity. It can also include unmodeled dynamics, such as an omitted actuator response, that cannot be represented by varying one constant coefficient.

Parametric model uncertainty retains a chosen equation form while allowing its parameters to vary. For an original example, a damped mass obeysHere $v$ is velocity and $u$ the applied-force control input. In the form $\dot{v}=av+bu$, the coefficients areTheir numerical ranges in consistent SI units are approximately $a\in [-2.222,-1.818]$ and $b\in [0.909,1.111]$, but they are not independent. The combination $a=-2.222$, $b=0.909$ does not correspond to one allowed mass. Replacing the physically linked family by the full rectangle of independent coefficient values introduces additional possible models.

This dependency matters when claiming robust stability or robust performance: enlarging the family may produce a more conservative test, while accidentally narrowing it can remove a genuinely possible failure. MathWorks's uncertain-parameter representation tracks the identity of named uncertain elements when they recur in a model.^[ref]

A bounded uncertainty set does not by itself assign probabilities. Saying $m\in [0.9,1.1]$ does not say that the endpoints are unlikely or that all masses are equally likely. It states what the guarantee must cover. Conversely, fitting the nominal mass from data does not prove that this interval covers every relevant operating condition.

The assumed time behavior is also part of the specification. An unknown but fixed mass and an arbitrarily time-varying coefficient are different model families. Likewise, uncertain gain does not automatically cover an unmodeled delay. A useful uncertainty description connects allowed mathematical variations to the physical mechanisms and measurements that justify them.

Parametric model uncertainty allows specified parameters to vary while retaining a chosen mathematical model structure. It records a nominal value and an allowed set, such as an interval, for each uncertain quantity. Repeated occurrences of the same physical parameter must remain linked. MathWorks's uncertain-real-parameter model explicitly represents nominal values, ranges and named repeated parameters.^[ref]

An original RC circuit example uses the first-order transfer functionAssume every resistance/capacitance pair in those positive intervals is allowed. Since their product increases with either factor, the time constant has the exact rangeThe nominal product is1millisecond, so separate symmetric10percent component ranges give an asymmetric time-constant range of minus19percent to plus21percent. The angular frequency at which this model's magnitude falls to $1/\sqrt{2}$ is ${\omega }_c=1/\tau$, giving approximately $826.4$ to $1234.6\mathrm{rad}{\mathrm{s}}^{-1}$. Reciprocal conversion reverses the endpoints.

This family changes response speed while preserving the modeled DC gain $H(0)=1$. A guarantee of correct DC gain therefore does not establish a particular transient or bandwidth. Conversely, an extra pole from unmodeled dynamics is absent from every member of this first-order family, regardless of how widely $R$ and $C$ are allowed to vary.

Parameter dependencies can restrict the family further. If the same uncertain resistor appears in several coefficients, independently varying its appearances changes the physical model. The mass example in control model uncertainty derives one such coefficient relation. Parameter bounds are also not a probability distribution: an interval describes what must be covered, while probability assignments require additional information.

Robust stability and robust performance can then be assessed over this specified family. The result applies to the model structure and bounds actually supplied, not automatically to aging, temperature changes or other effects outside them.