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Robust stability means that a closed-loop control system remains stable for every allowed realization of its control model uncertainty. The intended notion of stability and the uncertainty family must be stated. A finite list of stable simulations does not by itself establish the universal statement. Robust-analysis software reports margins relative to the uncertainty encoded in the model, rather than relative to every imaginable model error.^[ref]

For an original continuous-time linear family, considerA common Lyapunov function is $V(x)=x_1^2+x_2^2$. Along every member of the family,The uncertain cross terms cancel because the same $\theta$ occurs with opposite signs. Consequently,This proves exponential stability uniformly over the whole interval using the Euclidean norm, without sampling possible values of $\theta$. The inequality even remains valid for a bounded measurable $\theta (t)$ in this particular differential equation, because its derivation is pointwise and does not differentiate $\theta$.

The preserved parameter dependence is essential to this certificate. If the off-diagonal entries are instead independent values both equal to10, the matrix $\left(\begin{array}{cc}-1& 10\\ 10& -2\end{array}\right)$ has determinant $-98$ and eigenvalues of opposite signs. That different uncertainty family contains an unstable system.

Robust stability addresses growth and decay of internal motion. Robust performance additionally imposes a response bound, such as disturbance attenuation. A system can satisfy the first requirement while allowing an unacceptably large response to a bounded control disturbance. Conversely, a claimed performance guarantee is meaningless outside the uncertainty family for which its stability assumptions hold.