CAD.1:4 - Solution
CAD.1:4.1 - Formulate the unsplit question
State the controlled quantities, objective or preference, hard constraints, operating domain, relevant disturbances, measurements and available actions. Distinguish three time quantities: the dynamics and disturbance times, the horizon over which a choice matters, and the time needed to obtain and execute that choice. A fast update is useful only if observations, computation and actuation support it.
Use a centralized mathematical formulation when it helps reveal coupling, even if that formulation is too expensive to run. It can be a finite-horizon objective with dynamics and constraints, a reach-and-avoid problem, or a reference-following requirement. It is the common comparison basis, not a promise that one optimizer must solve everything.
Retain a feasible simple controller as a baseline where available. State what it cannot achieve at acceptable effort: for example obstacle look-ahead, preview-dependent economy or long-horizon allocation. If the baseline’s actual limitation is unknown, test that uncertainty before adding layers.
When a qualified existing loop has adequate tracking and disturbance rejection but requested commands violate constraints, compare adding a reference or command governor before redesigning that loop. Use the joint plant-controller model, admissible initial conditions and qualified command-update rule described in CAD.Reference:4.5.1. Check whether the resulting command delay or distortion still meets the task. If it does, retain that candidate; if it does not, identify the unmet requirement that a different interface, planner or controller must address.
CAD.1:4.2 - Separate decisions because their information and effects differ
For each consequential decision, ask what it changes, which information it needs, how long its effect persists, how quickly a wrong choice becomes harmful, and what resources its computation consumes. Cluster decisions only when those answers support one useful calculation or law.
A slow planning decision might choose a route, reference segment or allocation. A faster decision might reject local disturbance or track that reference. This does not imply that the faster model is less detailed: a low-level model may retain actuator dynamics omitted by a planning model.
Generate materially different candidates. Compare a single loop, a planning-and-feedback split, or another separation motivated by the problem. Do not add a third layer merely because the source diagram contains three. A communication boundary can matter more than a time-scale boundary, and a faster processor can make a proposed split unnecessary.
CAD.1:4.3 - Expose what the split removes
A useful way to derive a candidate is to introduce separate variables for decisions that were coupled in the whole problem. Initially constrain them to agree. For example, let a planner choose a nominal trajectory z while the plant state is x; keep the relation that says what plant behavior z represents. Only then relax that relation to obtain separately solvable subproblems.
For each relaxation, recover its price. Replacing equality by tracking allows an error; removing full dynamics requires a realizability condition; replacing continuous interaction by periodic exchange introduces stale information and intersample evolution. Removing future disturbance information requires a prediction or uncertainty treatment. These are concrete missing relations, not a generic “integration risk”.
Use MMP.9 when reduction is involved: derive the retained evolution and identify terms depending on omitted state, past inputs or unresolved quantities. A term does not disappear because the upper model has fewer states. Decide whether it is retained through memory, bounded as mismatch, supplied by a lower closed loop, or left as an unresolved limitation.
The first candidate is complete enough to continue when every removed interaction has a proposed receiving treatment in CAD.2 or CAD.3. A treatment may fail; naming it does not establish it.
CAD.1:4.4 - Make the slow decision account for fast consequences
Do not optimize the slow part as if the faster part were free. Eliminate or approximate the fast response inside the slow objective when the model permits it.
For example, suppose a particular linear design starts at x = 0 and keeps x at zero by choosing a fast action f after observing w, while a slow action s remains fixed for a block. If the fast input matrix is invertible, the required correction is
f=-B_f^{-1}(B_s s+w).
Substitution gives a slow objective containing both the cost of s and the anticipated cost of these corrections. With a block of k forecast values, a possible norm-cost objective is
k\|s\|_s+\sum_{i=1}^{k}\|B_f^{-1}(B_s s+\widehat w_i)\|_f.
This is a derivation under a particular model, not a general recipe for plants with saturation or delay. When fast actions are limited, retain their feasibility condition as well as their cost. When an exact elimination is unavailable, use a qualified response model, a bound or joint optimization; label the remaining approximation and compare the actual combined result.
CAD.1:4.5 - Preserve the whole question through the next design move
Pass each candidate’s controlled behavior, separation reason and lost interactions to CAD.2. A command’s type alone is insufficient: a target position, a timed trajectory and a velocity segment imply different lower-level problems.
Keep candidates comparable under the same task, disturbance class and resource limits. A slower update with a smaller demanded reference change may look better only because the task was weakened. Retain a whole alternative until its decisive incompatibility or disadvantage is understood.