5.3 Nested CCD - Control Co-Design

Joint formulation through a value function¶

Nested CCD is a genuine CCD formulation. For every physical design $p$ , an inner problem finds its best admissible controller and trajectory:

V(p) = min_z J_{sys}(p,z), R(p,z)=0, g(p,z) ≤ 0.

The outer problem then solves

min_{p ∈ P} V(p).

Unlike a sequential handoff, the physical design is judged by the optimized closed-loop system objective. Controller authority, dynamic feasibility, and coupled constraints therefore influence the selected plant.

Inner-solve accuracy¶

The outer optimizer sees only the value and derivatives returned by the inner solve. Loose inner tolerances make $V(p)$ noisy and can corrupt outer gradients. Failed inner solves do not necessarily prove that a plant is physically infeasible; they may indicate poor initialization or numerical conditioning.

For differentiable solutions,

dV/dp = ∂L/∂p.

where $L$ is the inner Lagrangian evaluated at a converged primal–dual solution.

Activity 5.3: account for inner-solve error¶

Activity 5.3: account for inner-solve error

An outer iteration evaluates three spring designs:

$k_s$ (kN/m)	reported $V(p)$	inner optimality residual	max constraint	inner CPU
18	0.742	$8 × 10^{-7}$	0.97	14 s
21	0.711	$4 × 10^{-3}$	0.99	5 s
24	0.724	$9 × 10^{-7}$	1.01	17 s

Accept an inner result only if its residual is at most 10^-5 and its maximum normalized constraint is at most 1.

Identify trustworthy and feasible evaluations.
The 21 kN/m case is re-solved to residual $7 × 10^{-7}$ and returns $V=0.731$ . Compute its apparent objective bias from the loose solve.
Select the best admissible plant after re-solving.
Compute total inner CPU including the additional 16 s re-solve.
Explain how accepting the original 21 kN/m value could misdirect the outer optimizer.