CAD.5:4 - Solution

Construct the possible current states from the information actually available at the action time. Determine whether one action, chosen with that information, works for all those states and admits the required continuation. Use failures of that construction to change the observation, its timing, the internal representation or the control demand. Compare complete alternatives under the same demand.

CAD.5:4.1 - Fix the action time and the control-relevant distinctions

Use the sampled plant model

x_{k+1}=A x_k+B u^a_k+w_k,\qquad w_k\in W,\qquad x_0\in X_0.

Here (u^a_k) is the input actually applied during the update, not merely the requested command. Specify state or output constraints, input limits and the duration for which they must hold. The discrete guarantee covers sample instants; physical constraints between samples require a qualified intersample model and bound. A tracking problem can use deviation coordinates about a known reference; changes of that reference must remain in the model.

For the direct construction below, (A,B,W,X_0) are known, (W) and (X_0) are bounded, and the executed input is available to the estimator. Known actuator queues can first be included in the plant state. Unknown actuation error must be bounded and included in the model; an unbounded discrepancy between requested and applied input defeats the stated guarantee.

State what is available before selecting (u_k). A packet that arrives after that selection cannot support that action. Include sampling phase, transmission, computation and actuation timing in this account. Specify a maximum age of usable observations and a maximum gap between required samples: a delivery-delay bound alone does not guarantee that a sensor samples often enough.

Ask which possible situations the controller must distinguish. If two situations consistent with all available information require disjoint admissible action sets, no deterministic causal controller using only that information can meet the requirement for both. Conversely, do not demand that every state coordinate be accurately observed if their remaining differences do not prevent an acceptable action and continuation.

Begin with the least elaborate plausible arrangement. Include a qualified current sensor, a fixed observer already meeting the task, or no observation of a harmless stable component. These are substantive alternatives, not merely baselines to be defeated.

CAD.5:4.2 - Form realizable observation and timing alternatives

For each offered channel (j), specify its observed quantity (C_jx), error model, sampling rule, delivery rule and resource demand. For a nonsaturating bounded-error channel, a packet contains

(j,\tau,y),\qquad y=C_jx_\tau+v,\qquad v\in V_j,

where (\tau) is the sample time and (k-\tau) is its age when used. The channel’s constraint on the sampled state is the set

M(j,\tau,y)=\{x_\tau:y-C_jx_\tau\in V_j\}.

A quantized symbol instead denotes its whole quantizer cell, including the noise bound. A saturated extreme cell may be unbounded; do not replace it with a small symmetric error interval. For adaptive thresholds, retain or communicate the thresholds used for that particular sample.

Construct a finite set of feasible channel-and-schedule alternatives. Include the resource coupling that can change their timing: two sensors sharing a bus, for example, need not retain their isolated delivery bounds. State which loss and delay combinations are allowed. A fixed periodic schedule or an explicitly bounded family of allowed modes supplies timing conditions for the error calculation. The number of cases and the cost of set propagation still have to fit the design and execution effort. Arbitrary online sensing policies require a separate policy construction.

A rapid narrow channel, a slow complete channel and their combination are useful candidates when the problem suggests them. They are not compulsory categories. Future-disturbance preview is different from a delayed observation of the current plant; give it its own prediction model rather than treating its timestamp as negative measurement delay.

Do not assume a universal law exchanging accuracy for speed. Obtain that relation from the proposed device, encoding, bandwidth, energy or other physical constraint. If a fast complete channel fits the same total budget, it may remove the reason for a heterogeneous arrangement.

CAD.5:4.3 - Construct an internal state that can be updated

Let (D) be the largest allowed measurement age. A general sufficient representation retains the compatible joint set of recent states

\xi_k=(x_k,x_{k-1},\ldots,x_{k-D}),\qquad \xi_k\in\Xi_k,

together with the applied-input history, sample identities and the schedule or arrival mode needed for subsequent updates. At startup retain only states at nonnegative times and use (X_0); do not invent a prehistory. The set (\Xi_k) represents states compatible with the initial condition, model, applied inputs and all observations already incorporated.

Predict the joint set by the map

T(\xi_k,u^a_k,w_k) =(A x_k+B u^a_k+w_k,\;x_k,\ldots,x_{k-D+1}).

For a packet arriving at the next action time, intersect this predicted set with the measurement constraint on its historical component (x_\tau). With several packets, intersect all their valid constraints. The projection onto the first component gives the possible current states (X_k=\pi_0\Xi_k).

This is a constructive sufficient internal state, not a claim of minimum order. It preserves the correlation between a past observation and the present through intervening dynamics and actions. Separate marginal intervals need not preserve that correlation; using them as a Cartesian outer approximation is safe only when the ensuing larger uncertainty is carried into control.

A practical realization can use a replay buffer, an observer with equivalent sufficient memory, or a recursive outer set such as an interval, polytope or zonotope. In each case, the represented set must contain every state compatible with the stated model and data. Approximation may enlarge that set; it may not discard a possible state merely to obtain a feasible control.

CAD.5:4.4 - Preserve the information needed by the next update

At each action time, use the following causal order:

1. Propagate the previous representation using the input actually executed and the allowed disturbance.
2. Incorporate each newly available packet at its sample time, then obtain the current-state projection.
3. Choose the estimate or set-based action from that projection, apply it by the qualified deadline, and retain the input information needed by the next update.

A state older than (D) can be projected out once no allowed future packet refers to it. This is different from throwing away the dependence between retained states. Duplicate packets do not supply additional independent information. In a bounded-set construction, independent noise need not be assumed: common error sources must be represented jointly or enclosed conservatively. With an uncertain timestamp, use the union of the admissible timestamp hypotheses, not their intersection.

If the compatible set becomes empty, the current model, bounds, packet, timestamp or executed-input record is inconsistent. Do not silently reset to a convenient point. Use a separately qualified fallback, obtain a new trusted initialization, or declare the demanded continuation unsupported. Packet loss or delay outside the allowed family likewise invalidates the relevant bound until a qualified recovery applies.

Reduce the internal representation when its update and control use remain sufficient. One exact reduction is available when (s=T_rx) has closed dynamics (T_rA=A_rT_r), the needed observations factor as (C_j=C_{rj}T_r), and the control law and constrained quantities depend only on (s). Then use (s^+=A_rs+T_rB u^a+T_rw) and the corresponding observation sets. If an omitted component influences a required future quantity, its influence must instead be retained or bounded.

Thus two histories may be merged only when the merged representation still supports every required future update and action under the allowed future observations and disturbances. Equal current estimates are not enough. A.3.3.PI develops this general sufficiency test; here the joint-state construction provides a direct control-specific realization.

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.

CAD.5:4.6 - Establish control under that information

First test whether the information permits a common action. In the following calculation, the selected (u) is executed as represented by the plant model; any bounded execution discrepancy must already be included in its disturbance or augmented state. For a required next-state set (R_{\mathrm{next}}), the robust admissible inputs are

\mathcal U(X_k)= \{u\in U:A X_k\oplus\{Bu\}\oplus W\subseteq R_{\mathrm{next}}\}.

The quantifiers are “choose one (u), then allow every (x\in X_k) and (w\in W),” not “for every possible state, choose a different action.” For a halfspace (h^\top x\le b) of (R_{\mathrm{next}}), the test is

\sup_{x\in X_k}h^\top Ax+h^\top Bu+\sup_{w\in W}h^\top w\le b.

This provides a direct calculation for polytopes and support-function representations. When the test fails on an outer approximation, refine the approximation or change the design before claiming impossibility. A negative result for one chosen controller is weaker still.

One feasible step does not establish a continuing guarantee. A sufficient construction for a specified feedback (u=K\hat x+g), with constant (g) in the selected operating mode, uses (F=A+BK):

x_{k+1}=F x_k+B g+w_k-BK\eta_k.

Find a set (R), containing the allowed initial states and contained in the required state set, for which

F R\oplus\{Bg\}\oplus W\oplus(-BK)V_\eta\subseteq R, \qquad K(R\oplus(-V_\eta))\oplus\{g\}\subseteq U.

Together with a valid uniform error bound, these conditions prove state and input admissibility by induction. For a stable (F), one constructive candidate is the reachable sum of the forcing set (D_f={Bg}\oplus W\oplus(-BK)V_\eta):

R_\infty=\bigoplus_{i=0}^{\infty}F^i D_f.

Use a finite outer approximation with a bounded tail, and enlarge it for the specified initial set if necessary; verify the inclusions after enlargement. For example, if an induced norm satisfies (|F|\le q_F<1) and (D_f) lies in a ball of radius (d_f), the ball of radius (d_f/(1-q_F)), enlarged to contain the initial ball, is invariant. A Euclidean norm larger than one does not prove that no other norm or set works.

These are sufficient conditions, not a universal synthesis algorithm. A stable hidden mode may be handled without measurement; an unstable or tightly constrained uncertain direction may exhaust the input or state margin. Failure can mean excessive approximation, a poor given feedback, an infeasible information arrangement, or an infeasible demand. Distinguish them.

A decisive impossibility witness consists of two admissible histories producing the same available packets and internal information but requiring disjoint actions, or an unavoidable uncertainty width larger than the permitted state interval. Such a witness rules out all causal controllers under those observation conditions, not merely the proposed feedback. Section :5 supplies both a positive construction and such a negative result.

CAD.5:4.7 - Connect the information result to a layered control construction

The method can be used without a layered architecture. When CAD.2 supplies a nominal state (z), command (v), projection (P) and control interface, replace its exact-state feedback by the implemented estimate:

u=K(\hat x-Pz)+Qz+Rv,\qquad e=x-Pz,\qquad \eta=x-\hat x.

For (x^+=Ax+Bu+w) and (z^+=A_pz+B_pv), the derived error dynamics are

e^+=F e+H z+G v+w-BK\eta,

where (F=A+BK), (H=AP+BQ-PA_p), and (G=BR-PB_p). The additional forcing (-BK\eta) is the effect of acting on the estimate.

Use the effective disturbance set (W\oplus(-BK)V_\eta) in CAD.3’s invariant-error construction. For every allowed nominal state and command, state admissibility requires ({Pz}\oplus E\subseteq X), while the implemented input requires

\{Qz+Rv\}\oplus K E\oplus(-K)V_\eta\subseteq U.

These sums are conservative enclosures, not assertions that tracking and estimation errors are independent. A joint set can retain their correlation when the separate bounds are too costly.

CAD.4 then compares whole arrangements using the estimator’s memory, communication, worst-case completion time, admissible packet timing and resulting error set. A new sensor changes (V_\eta), which can change the tracking-error set (E), the allowed nominal motion and the value of the whole architecture. A larger planner can also increase sensing or estimation latency through a shared resource, requiring the same calculation in reverse.

The added result is the construction of observation availability, internal information state and estimation uncertainty. CAD.1–4 continue to own decomposition, operational interfaces, tracking bounds and whole architectural comparison; this method does not replace them.

CAD.5:4.8 - Compare whole alternatives and return to the failed dependence

For each serious channel-and-schedule alternative, obtain the information update, its cost and delay, the error bound and the control conclusion. Keep the demanded behavior, initialization and resource accounting the same. Compare the alternatives that meet the demand; distinguish a proof of impossibility from failure of a sufficient bound.

Choose a sufficient arrangement with an explained cost and margin. A comparison can establish the best member of a stated finite family only when the objective is comparable and every potentially better member is evaluated decisively. Otherwise return the best qualified member found and the unresolved alternatives. A stochastic design can instead minimize an explicitly selected expected cost, but its covariance or a finite successful simulation is not a deterministic bound for this method’s hard constraints.

Return according to what failed:

CAD.5:4.9 - Make the result executable and reusable

Return the selected and rejected complete arrangements with the controlled task, plant and uncertainty conditions, channel meanings and timing, initialization, internal state and update, control law, and the admissibility calculation. Make the unresolved distinction explicit if no candidate has a sufficient guarantee.

For implementation, connect sample timestamps and identities, actually executed inputs and the estimator’s update to real producers and consumers. Qualify computation and transport deadlines under the proposed shared load. Select an already qualified behavior for missing data, inconsistent data and late completion; “hold the last command” is not automatically safe.

What changes in practice is the choice of the information the controller obtains and retains. The engineer can explain why a channel matters to an action, what must be remembered while waiting for another channel, and which changed condition invalidates the resulting control. A diagram, estimator score or list of sensors alone cannot supply that explanation.