CAD.5:5 - Archetypal Grounding

CAD.5:5.1 - A rapid partial observation and a delayed complete observation

Consider two normalized deviation coordinates:

x_{k+1}= \begin{bmatrix}1.2&0\\0&0.8\end{bmatrix}x_k+u_k+w_k,\qquad |w_{1,k}|\le0.04,\quad |w_{2,k}|\le0.02.

The initial state is known to be zero. At every sample the requirement is (|x_1|\le0.08), (|x_2|\le0.06), with (|u_1|\le0.12), (|u_2|\le0.08). Inputs are executed without delay and their executed values are available. All specified packets and computations finish before the relevant input selection. These are assumptions of a mathematical design case, not measurements of a physical installation.

The offered sensor packages are:

The sensor budget is 2. Estimator execution and transport capacity are assumed separately qualified for F+S; the sensor cost is not a claim about total implementation cost. Compare F, S and F+S under this budget, and retain H as the simpler alternative if the budget changes.

With F+S, use F for the first coordinate. Propagate S’s second coordinate using the executed inputs:

\hat x_{2,k}=0.8^2y_{2,k-2}+0.8u_{2,k-2}+u_{2,k-1}.

The required internal state includes the two applicable input values and the delayed packet’s sample time. The uncertainty bound is

r_1=0.01,\qquad r_2=0.8^2(0.005)+(1+0.8)(0.02)=0.0392.

Before the first delayed packet arrives, propagate the known initial state and inputs; the second-coordinate uncertainty is at most (0.02) after one update and (0.036) after two. The same bound (0.0392) therefore covers startup. Incorporating S’s first-coordinate information as an additional intersection is permissible but unnecessary for the following guarantee.

Choose (u_i=-a_i\hat x_i), with (a_1=1.2,a_2=0.8). Then (x_i^+=a_i\eta_i+w_i), so

|x_1^+|\le1.2(0.01)+0.04=0.052<0.08, \qquad |x_2^+|\le0.8(0.0392)+0.02=0.05136<0.06.

Whenever the required state bounds hold,

|u_1|\le1.2(0.08+0.01)=0.108<0.12, \qquad |u_2|\le0.8(0.06+0.0392)=0.07936<0.08.

The initial state is admissible, and these inequalities establish the state and input requirements for every subsequent tick and every disturbance and measurement error within the stated bounds. This is a continuing conditional guarantee, not an inference from a plotted trajectory.

One particular run makes the internal information visible. Let measurement errors be zero and take (w_1=0.04,w_2=0.02) at every update. At (k=1), (x_2=0.02), but no S packet has arrived and its prediction is zero. At (k=2), (x_2=0.036), while S reports (x_{2,0}=0). At (k=3), S reports (x_{2,1}=0.02); the propagated center is (0.0128), not the current (x_{2,3}=0.0488). The difference (0.036) is due to the two unobserved disturbances. The controller applies (u_{2,3}=-0.01024). On the next tick its prediction includes this executed input. Reusing the old observation without that input would no longer implement the qualified update.

CAD.5:5.2 - Why the individual channels do not suffice

With F alone, the second coordinate is never observed and does not influence the observed first coordinate. Consider two runs with opposite second-coordinate disturbances and otherwise identical information. The controller chooses the same second-coordinate commands in both. After five updates, half their separation is

0.02(1+0.8+0.8^2+0.8^3+0.8^4)=0.067232>0.06.

No common center can place both states inside the required interval. This excludes every causal controller using F alone under the stated assumptions, not just (u=-A\hat x).

With S alone, the latest two first-coordinate disturbances are unobserved when the next command is chosen. Even with perfect older observations, the next first-coordinate interval has unavoidable half-width

0.04(1+1.2+1.2^2)=0.1456>0.08.

The same indistinguishability argument rules out the required all-time bound. Improving the old measurement’s accuracy cannot recover these disturbances before their consequences matter.

F+S is therefore sufficient where either offered channel alone is insufficient, under the same sensor budget and timing assumptions. This establishes a task-dependent benefit of combining channels. It does not establish that all heterogeneous sensing is superior or that the offered catalog is globally optimal.

CAD.5:5.3 - Increase the delay, then change the design or the demand

Suppose bus contention changes S’s age from two ticks to four, while F and the requirements remain unchanged. The reconstructed second-coordinate radius becomes

r_2(4)=0.8^4(0.005)+0.02(1+0.8+0.8^2+0.8^3)=0.061088.

The same controller’s next-state bound is (0.8r_2(4)+0.02=0.0688704), which exceeds (0.06). More decisively, four unobserved disturbances plus the next one yield the half-width (0.067232) from :5.2 even if S’s old measurement is exact. F still reveals nothing about the second coordinate. Thus a different filter or gain using the same available information cannot restore this hard bound.

The engineer must change a consequential condition. Restore the qualified two-tick transport arrangement, obtain a timely observation of the second coordinate, or revise the task or physical dynamics. If the sensor budget becomes 3, H alone supports the same feedback with next-state bounds (0.046) and (0.024); its input bounds are (0.102) and (0.052). H needs no delayed-observation reconstruction. It is now a simpler sufficient choice.

A different task can also justify less information. If the allowed second-coordinate deviation becomes (0.12), retain F and set (u_2=0). The unobserved stable second coordinate has the invariant bound

|x_2|\le\frac{0.02}{1-0.8}=0.1<0.12.

S, its delayed packet processing and its input-history reconstruction are then unnecessary. This is sufficient for the changed task; it is not a cheaper solution to the original (0.06) requirement.

CAD.5:5.4 - What an internal-path ablation does and does not establish

For the scalar noiseless plant (x_{k+1}=a x_k+u_k), with a one-tick delayed observation (y_k=x_{k-1}), the predictor

\hat x_k=a y_k+u_{k-1},\qquad u_k=-a\hat x_k

uses the executed-input history. Removing that input term while retaining the remaining gains gives (u_k=-a^2x_{k-1}). At (a=1.2), the resulting characteristic polynomial is (\lambda^2-1.2\lambda+1.44); its roots have modulus (1.2), so this ablation is unstable.

It does not prove that every controller lacking that input-history path is unstable. Retuning the memoryless delayed-output law to (u_k=-0.6x_{k-1}) gives roots of (\lambda^2-1.2\lambda+0.6), both with modulus (\sqrt{0.6}<1). The latter establishes stabilization of this noiseless model, not the original predictor’s performance or a bounded-disturbance constraint guarantee.

Use the first comparison to explain why a particular realization depends on its internal path. Use an impossibility argument such as :5.2–:5.3 when claiming that the available information cannot meet a requirement under any controller in the stated class.

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.