 Robust linear constraints for bounded covariance models

Robust linear constraints for bounded covariance models require a linear inequality to hold for every error state admitted by the selected bounded model. A covariance matrix fixes second moments; the robust guarantee additionally depends on the support of a probability distribution or other specified uncertainty set. This distinction matters when a physical limit must never be crossed within the model.

Let $E=LS$ as in finite-support realization of a covariance matrix, where all coordinates of $S$ are independent fair signs. Then $\mathrm{Cov}(E)=L{L}^{\mathsf{T}}$. For nominal input $\mu$, require $a^{\mathsf{T}}(\mu +E)\ge b$ for every sign state. Writing $c=L^{\mathsf{T}}a$, direct minimization givesEach sign can independently oppose its coefficient, including when some coefficients vanish. Thus the exact necessary and sufficient condition is $a^{\mathsf{T}}\mu -\Vert L^{\mathsf{T}}a{\Vert }_1\ge b$, using the L1 norm. The same extremum holds over the full box $s_j\in [-1,1]$ because a linear function attains an extreme at a vertex. This is a support function calculation in robust optimization; the general worst-case formulation and its dependence on the uncertainty set are illustrated in the primary optimization text.^[ref]

For an original two-input example in volts, let $E_1=(1\mathrm{V})Z$ and $E_2=(1\mathrm{V})(0.6Z+0.8W)$, with independent fair signs $Z,W$. The covariance matrix is $\left(\begin{array}{cc}1& 0.6\\ 0.6& 1\end{array}\right){\mathrm{V}}^2$. To keep the actual first input at least as large as the second, let the nominal difference be $d={\mu }_1-{\mu }_2$. ThenA nominal difference of $1.0\mathrm{V}$ fails in the state $Z=-1,W=1$, despite being positive nominally. A difference of $1.3\mathrm{V}$ leaves a guaranteed $0.1\mathrm{V}$ margin under this model. Using only the separate bounds $|E_1|\le 1\mathrm{V}$, $|E_2|\le 1.4\mathrm{V}$ would demand $d\ge 2.4\mathrm{V}$, because that larger box admits joint states the sign model forbids.

The standard deviation of $E_1-E_2$ is instead $\sqrt{0.8}\mathrm{V}$. It is not the hard bound $1.2\mathrm{V}$. A different bounded law with the same covariance matrix can change the hard bound, as nonuniqueness of bounded uncertainty laws with a fixed covariance explains. For several linear inequalities, apply the support calculation to each row. This verifies an assumed model's feasible states; it does not establish that the chosen support contains every error of an actual instrument.