5.4 Simultaneous or All-at-Once CCD

One sparse nonlinear program¶

Simultaneous CCD exposes physical variables, controller parameters, states, and controls to one optimizer:

min_{p,θ,X,U} J_{sys}(p,θ,X,U), R_i(p,θ,X,U)=0, g_j(p,θ,X,U) ≤ 0.

After direct transcription, $R_i=0$ are dynamic defects. The nonlinear program is large but sparse because each defect depends mainly on neighboring time nodes and the shared design variables.

Interpretation¶

There is no inner control optimizer. Plant and control decisions move together according to the KKT system of the joint problem. This can improve derivative consistency and remove inner-solve noise, but it requires careful scaling, sparse derivatives, feasible initialization, and mesh verification.

Simultaneous CCD and nested CCD may represent the same mathematical problem. Their numerical behavior, memory use, initialization sensitivity, and derivative implementation can nevertheless differ substantially.

Activity 5.4: size and verify a simultaneous CCD problem¶

Activity 5.4: size and verify a simultaneous CCD problem

A transcription uses 4 plant variables, 6 controller parameters, 8 states, 2 controls, and 101 time nodes. States and controls are decision variables at every node.

Compute the total number of decision variables.
Trapezoidal defects provide 8 equations on each of 100 intervals. Compute the number of defect constraints.
There are 3 path constraints at every node and 12 boundary constraints. Compute the total constraint count.
If each defect row depends on 4 shared design/controller variables and 20 local state/control entries, estimate the defect-Jacobian nonzeros.
With final normalized defect $3 × 10^{-7}$ , maximum inequality -0.012, and KKT residual $6 × 10^{-6}$ , determine whether tolerances of 10^-6, 0, and 10^-5 are met.