= Parametric model uncertainty
\b[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.<ureal documentation in MATLAB>[Description, Range and Name properties]{ref}
An original <RC circuit> example uses the first-order <transfer function>
$$
H(s)=\frac1{1+RCs},\qquad
R\in[9,11]\,\mathrm{k\Omega},\qquad C\in[90,110]\,\mathrm{nF}.
$$
Assume every resistance/capacitance pair in those positive intervals is allowed. Since their product increases with either factor, the time constant has the exact range
$$
\tau=RC\in[0.81,1.21]\,\mathrm{ms}.
$$
The 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 $1/\sqrt2$ is $\omega_c=1/\tau$, giving approximately $826.4$ to $1234.6\,\mathrm{rad\,s^{-1}}$. 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.
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