6.10 Why an OLOC-Optimal Plant May Not Be Closed-Loop Optimal

Core idea¶

Why an OLOC-Optimal Plant May Not Be Closed-Loop Optimal must be treated as a system-level decision rather than an isolated technique. For an active suspension with complete road preview and a wave-energy converter with known future waves, state what is fixed, what is optimized, what information is available, and what equations define feasibility.

The relevant quantities are $p$ , $u(t)$ , $x(t)$ , and known $d(t)$ . The chapter-level formulation is

\min_{p,u,x}\Phi+\int L\,dt\;\mathrm{s.t.}\;\dot x=f(t,x,u,p,d).

For this section, trace how the choice changes performance limit, 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 translate insight to feedback. 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 6.10: quantitative design audit¶

Activity 6.10: quantitative design audit

Two candidate plants are optimized first with perfect-preview OLOC and then tested with the same implementable preview controller. Closed-loop operating scenarios have probabilities 0.50, 0.30, and 0.20.

Plant	Mass (kg)	Perfect-preview OLOC cost	CL nominal	CL forecast bias	CL sensor delay	Worst peak force (kN)
A: light, narrow-band	210	0.62	0.75	1.15	1.32	4.30
B: heavier, broad-band	245	0.70	0.88	0.94	1.02	3.80

Use the total design metric $J_{\mathrm{tot}}=\mathbb E[J_{\mathrm{CL}}]+0.002m$ and require worst-case peak force to be no greater than 4.0 kN.

Rank the plants using perfect-preview OLOC cost alone. Compute the apparent percentage advantage of the selected plant.
Compute expected closed-loop cost, mass penalty, $J_{\mathrm{tot}}$ , and force margin for both plants. Apply the feasibility requirement and select the implementable design.
Quantify the closed-loop regret incurred by choosing the OLOC-optimal plant rather than the feasible closed-loop optimum. Report regret both with and without the mass penalty.
Find the maximum probability of the sensor-delay scenario for which Plant A would have lower expected closed-loop cost than Plant B, holding the forecast-bias probability at 0.30 and assigning the remaining probability to nominal operation. Then reapply the force constraint.
Determine how much Plant A’s worst peak force must be reduced, in kN and percent, to become feasible. Explain which plant property or controller capability would need to change before its OLOC advantage could support a hardware decision.

Chapter summary¶

The chapter connected plant variables, known disturbance, optimal control, state response, performance limit 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 an active suspension with complete road preview and a wave-energy converter with known future waves.
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¶

Deshmukh, Herber, and Allison on bridging open- and closed-loop co-design; the active-suspension direct-transcription studies.

Open research question¶

How can noncausal trajectory structure be converted systematically into robust causal control?