4.9 Continuous and Discrete Architecture Variables

Parameter design assumes the architecture exists¶

Changing a gear ratio or controller gain is parameter design. Deciding whether a gearbox, encoder, current sensor, or communication link exists is architecture design. Let $a$ represent discrete architecture choices:

where the entries select a gearbox, position sensor, current sensor, and velocity sensor. Categorical variables may select among component types or topologies.

Architecture changes equations, dimensions, available information, and feasible controllers. If $a_v=0$, direct velocity feedback is unavailable unless an estimator reconstructs velocity. If $a_g=0$, gear-ratio variable $N$ should not remain active. The formulation must enforce such logical consistency.

Continuous architecture relaxations¶

Some architecture-like decisions are continuous: actuator placement coordinates, sensor orientation, material distribution, or communication bandwidth. Binary variables are sometimes relaxed to $[0,1]$ to support gradient-based exploration. The relaxed result is a computational aid, not a fractional physical component, unless the variable has a valid mixture interpretation.

Conditional models¶

An architecture-dependent model may be written

with component cost $C_a(a)$ and logical constraints. Big-$M$, complementarity, convex-hull, enumeration, and generative representations are possible implementation strategies; their numerical treatment is deferred to Chapter 8.

Activity 4.9: enforce architectural consistency¶