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:
| Package | Observation available before (u_k) | Error bound | Sensor-resource cost |
|---|---|---|---|
| F | Current (x_{1,k}) only | (0.01) | 1 |
| S | Both coordinates at (k-2), at every tick after startup | (0.005) per coordinate | 1 |
| H | Both current coordinates | (0.005) per coordinate | 3 |
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.