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.