CAD.5:4.5 - Obtain a usable error bound

Choose a representative (\hat x_k) and compute an enclosure (V_{\eta,k}) such that

x_k=\hat x_k+\eta_k,\qquad \eta_k\in V_{\eta,k}.

The enclosure can be obtained directly from (X_k-\hat x_k). To qualify a continuing controller, obtain a bound valid over the whole allowed run, including startup, admissible delays, sample gaps and loss modes. A bound from one favorable packet sequence is insufficient.

For scalar propagation from a measurement of age (d), with coefficient (a), measurement-error radius (\epsilon), disturbance radius (q) and known intervening inputs, a sufficient radius is

r(d)=|a|^d\epsilon+q\sum_{i=0}^{d-1}|a|^i.

The propagated center includes every executed input. The second term is uncertainty about unobserved disturbances; a more accurate old sample does not remove it.

For a multidimensional plant, a finite-window construction gives a sufficient bound without requiring every sensor to observe every coordinate. Take available samples with times (\tau_i\in[k-L,k]), subtract their known input responses, and stack them:

r_i=y_i-C_i\sum_{s=k-L}^{\tau_i-1}A^{\tau_i-1-s}B u^a_s,\qquad r=O x_{k-L}+n,\qquad O_i=C_i A^{\tau_i-(k-L)}.

The stacked noise has components

n_i=C_i\sum_{s=k-L}^{\tau_i-1}A^{\tau_i-1-s}w_s+v_i.

Obtain (N) by applying these linear maps to the allowed joint disturbance and measurement-error set; separate component bounds give a conservative enclosure if the joint set is unavailable. Retain shared disturbances when practical. If (O) has full column rank, choose a left inverse (J), (JO=I), and form

\hat x_k=A^LJr+\sum_{s=k-L}^{k-1}A^{k-1-s}B u^a_s.

A sufficient error enclosure is

V_\eta=\left(\bigoplus_{s=k-L}^{k-1}A^{k-1-s}W\right) \oplus(-A^LJ)N.

The Minkowski sum (\oplus) adds all allowed elements of its operands. This enclosure may be conservative because the same disturbances occur in both terms; it does not assume their independence. Enumerate the admissible window and schedule modes, include the initial transient from (X_0), and enclose their errors. Poor conditioning of (O) can make a full-rank window useless for the required control margin.

Full-state observability is sufficient for this particular construction, not necessary for the task. A stable hidden component may have a small enough reachable set; a reduced state from :4.4 may suffice. If the window has no left inverse, return to the compatible-set construction and the required actions rather than declaring every such design impossible.