10.11 Choosing a Transcription Method

Core idea¶

Choosing a Transcription Method 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 continuous problem, 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 choose transcription. 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 10.11: select a transcription using evidence¶

Activity 10.11: select a transcription using evidence

Five methods are tested on the same constrained suspension OLOC problem using ten randomized initial guesses.

Method	Successful starts	Variables	CPU (s)	$J$	Validation defect	Path violation
Single shooting	4/10	80	6	1.012	$2.0\times10^{-5}$	0
Multiple shooting	9/10	260	18	1.009	$6.0\times10^{-5}$	0
Trapezoidal	10/10	620	22	1.018	$9.0\times10^{-5}$	0
Hermite–Simpson	10/10	360	28	1.008	$2.0\times10^{-5}$	0
hp-LGR	8/10	210	45	1.007	$8.0\times10^{-6}$	$1.0\times10^{-5}$

Requirements are at least 8 successful starts, validation defect $\leq10^{-4}$ , path violation $\leq2\times10^{-5}$ , and CPU $\leq30$ s.

Apply all four requirements and identify the admissible methods.
For each admissible method compute $S=J+0.002t_{\mathrm{CPU}}+0.01(n_v/1000)$ . Rank the methods and recommend one.
For a 100-case CCD parameter study, estimate serial CPU hours for every method. How much time does the recommended method save relative to the most accurate reported method?
The solution is later found to contain a control discontinuity. Predicted CPU multipliers for single shooting, multiple shooting, trapezoidal, Hermite–Simpson, and hp-LGR are respectively $[1.2,1.3,1.1,1.4,2.0]$ ; defect multipliers are $[4,1.5,1.2,3,5]$ . Reapply the requirements and score using the predicted values.
State which additional evidence—derivative checks, mesh convergence, independent integration, constraint sampling, or initialization trials—could change the selection, and assign a quantitative acceptance threshold to each test.

Chapter summary¶

The chapter connected continuous problem, time mesh, state and control samples, defect equations, sparse NLP through one system formulation. Engineering conclusions require aligned models, information, numerical accuracy, and validation.

Common mistakes¶

changing assumptions while comparing alternatives;
reporting objective improvement without verified feasibility;
hiding information, architecture, or uncertainty;
treating solver convergence as validation; and
reporting runtime without accuracy, derivatives, and tolerances.

Exercises¶

Recreate the workflow for minimum-time mass–spring motion, a suspension road event, and a two-phase energy converter.
State every variable, unit, dependency, and constraint.
Construct a common sequential or nominal baseline.
Identify active constraints and the physical bottleneck.
Design a test that could falsify the claimed benefit.

Principal sources¶

Patterson and Rao on GPOPS-II; Garg and coauthors on LG, LGR, and LGL pseudospectral methods.

Open research question¶

Can mesh selection use physical events and optimization sensitivity, not state error alone?