CAD.2:4.2 - Derive the error instead of assuming tracking

For a useful linear branch, express both models at the fast sampling instants:

x^+=Ax+Bu+w,\qquad z^+=A_pz+B_pv.

Here w is an admitted bounded mismatch or disturbance, not a name for all unknown behavior. Select an interface law

u=K(x-Pz)+Qz+Rv.

Substitute x=Pz+e before simplifying. The actual error equation is

e^+=Fe+Hz+Gv+w, \quad F=A+BK,\quad H=AP+BQ-PA_p,\quad G=BR-PB_p.

This calculation makes three design choices visible. K changes error correction; P and Q determine whether the nominal state evolution can match; R determines input matching. Solving H=0 is useful when possible but is not automatically possible or necessary. If it fails, retain Hz and bound or redesign it. A stable F does not remove forced error from H, G or w.

Use MMP.9 to construct and qualify a reduced model before relying on it here. A fitted upper model with unexamined hidden-state memory is not made adequate by a convenient matrix dimension.