Robust stability
= Robust stability
\b[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.<robstab documentation in MATLAB>[Description, modeled uncertainty and stability-margin bounds]{ref}
For an original continuous-time linear family, consider
$$
\dot x=A_\theta x,\qquad
A_\theta=\begin{pmatrix}-1&\theta\\-\theta&-2\end{pmatrix},\qquad
\theta\in[-10,10].
$$
A common <Lyapunov function> is $V(x)=x_1^2+x_2^2$. Along every member of the family,
$$
\dot V=2x_1(-x_1+\theta x_2)+2x_2(-\theta x_1-2x_2)
=-2x_1^2-4x_2^2\leq-2V.
$$
The uncertain cross terms cancel because the same $\theta$ occurs with opposite signs. Consequently,
$$
V(t)\leq e^{-2t}V(0),\qquad
\|x(t)\|_2\leq e^{-t}\|x(0)\|_2.
$$
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 $\left(\begin{smallmatrix}-1&10\\10&-2\end{smallmatrix}\right)$ 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.