10.6 Trapezoidal and Hermite–Simpson Methods

Core idea¶

Trapezoidal and Hermite–Simpson Methods 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 10.6: compare trapezoidal and Hermite–Simpson defects¶

Activity 10.6: compare trapezoidal and Hermite–Simpson defects

For $\dot x=f(x,u)=-x^2+u$ , consider one interval with $h=0.4$ , $(x_0,u_0)=(1.00,0.20)$ , and $(x_1,u_1)=(0.78,0.40)$ .

The trapezoidal defect is $c_T=x_1-x_0-\tfrac h2(f_0+f_1)$ . For Hermite–Simpson, use

x_c=\frac{x_0+x_1}{2}+\frac h8(f_0-f_1),\quad u_c=\frac{u_0+u_1}{2},\quad c_{HS}=x_1-x_0-\frac h6(f_0+4f_c+f_1).

Calculate $f_0$ , $f_1$ , $c_T$ , reconstructed midpoint $x_c$ , $f_c$ , and $c_{HS}$ .
Determine which candidate defect is closer to zero. Explain why a higher-order formula does not guarantee a smaller defect for arbitrary, inconsistent endpoint guesses.
A converged study gives the validation errors below. Compute observed order $p=\log(e_h/e_{h/2})/\log2$ for each refinement.

$h$	Trapezoidal error	Hermite–Simpson error
0.20	$1.6\times10^{-3}$	$1.2\times10^{-5}$
0.10	$4.1\times10^{-4}$	$7.6\times10^{-7}$
0.05	$1.0\times10^{-4}$	$4.8\times10^{-8}$

Require error below 10^-5. Predict the largest acceptable $h$ for each method using the measured order and compare the resulting interval counts over a 2 s horizon.
If Hermite–Simpson uses twice as many variables per interval but one quarter as many intervals, quantify the total-variable ratio relative to trapezoidal transcription.