10.5 Direct Transcription - Control Co-Design

Core idea¶

Direct Transcription must be treated as a system-level decision rather than an isolated technique. For minimum-time mass–spring motion, a suspension road event, and a two-phase energy converter, state what is fixed, what is optimized, what information is available, and what equations define feasibility.

The relevant quantities are discrete $X$ , $U$ , plant $p$ , and possibly final time. The chapter-level formulation is

\min_vJ_h(v)\;\mathrm{s.t.}\;\zeta_h(v)=0,\;g_h(v)\le0.

For this section, trace how the choice changes sparse NLP, the active constraints, and the implementable engineering design. A method is useful only when its assumptions are explicit and its result answers the same system question as the baseline.

Engineering interpretation¶

Ask three questions:

Which physical, informational, computational, or economic resource changed?
Which objective component or active constraint made the change valuable?
Does the conclusion survive model, disturbance, initialization, uncertainty, and implementation checks?

A practical action is to refine and verify. Record units and assumptions before optimization, report component objectives and margins afterward, and verify the result using an independent calculation or higher-fidelity model.

Activity 10.5: assemble a sparse direct transcription¶

Activity 10.5: assemble a sparse direct transcription

Transcribe the fixed-time double-integrator problem $\dot q=v$ , $\dot v=u$ over $T=2$ s using $N=4$ trapezoidal intervals. The boundary conditions are $(q_0,v_0)=(0,0)$ and $(q_4,v_4)=(1,0)$ ; enforce $|u_k|\leq2$ and minimize

J=\frac{h}{2}\sum_{k=0}^{3}(u_k^2+u_{k+1}^2),\qquad h=0.5.

Use decision ordering $w=[q_0,v_0,u_0,q_1,v_1,u_1,\ldots,q_4,v_4,u_4]^T$ .

Write the eight trapezoidal defect equations and four boundary equations explicitly.
Count NLP variables, equality constraints, and control bounds. Determine the degrees of freedom after equality constraints.
Assemble the equality matrix $A_{\mathrm{eq}}w=b_{\mathrm{eq}}$ and quadratic objective matrix. Solve with MATLAB quadprog or an equivalent QP solver.
Report all state and control nodes, objective value, peak control, and bound margin. Independently integrate the piecewise-linear control on a 1 ms grid and report endpoint errors.
Count the nonzeros in $A_{\mathrm{eq}}$ and compare its density with a hypothetical dense $12\times15$ matrix. Explain how local defects create sparse NLP structure.