CAD.5:5.5 - Recover two coupled coordinates from observations at different times

The general construction in :4.5 is not a coordinate-by-coordinate delay correction. Consider

A=\begin{bmatrix}6/5&1/5\\1/10&4/5\end{bmatrix},\qquad B=I,

with disturbance radii ((.04,.02)) and measurement-error radius (.005). At time (k=2), use a first-coordinate observation sampled at time 0 and a second-coordinate observation sampled at time 1. Both samples and the actually applied (u_0,u_1) are available before this estimate is used. After subtracting the known input response,

r=\begin{bmatrix}y_{1,0}\\y_{2,1}-[0\ \ 1]u_0\end{bmatrix}, \quad O=\begin{bmatrix}1&0\\1/10&4/5\end{bmatrix}, \quad J=O^{-1}=\begin{bmatrix}1&0\\-1/8&5/4\end{bmatrix}.

Thus (\hat x_2=A^2Jr+A u_0+u_1). The second measurement contains (w_{02}), the second coordinate of the first disturbance, so it is related to the uncertainty in the subsequent propagation. Writing (w_0=(w_{01},w_{02})), (w_1=(w_{11},w_{12})), and the two measurement errors as (v_1,v_2), substitution gives

\eta_1=\frac65w_{01}-\frac3{10}w_{02}+w_{11} -\frac{141}{100}v_1-\frac12v_2,

\eta_2=\frac1{10}w_{01}-\frac1{40}w_{02}+w_{12} -\frac{47}{400}v_1-\frac{33}{40}v_2.

The exact coordinate radii for the stated box of original uncertainties are ((.10355,.0292125)). Using the separate Minkowski enclosure in :4.5 gives the larger radii ((.11155,.0612125)): the same (w_{02}) was allowed to vary separately in two occurrences. Both enclosures are valid; retaining the shared variable can save a margin that a conservative test would spend.

This calculation supplies a finite-window error set, not a controller or an all-time guarantee for this new plant. The next use is :4.6 with the required actions and continuation, or :4.7 when a planning-and-feedback interface is already selected. Repeated use also needs the other allowed sample windows and startup to be covered.