13.14 Uncertainty Propagation - Control Co-Design

Core idea¶

Uncertainty Propagation must be treated as a system-level decision rather than an isolated technique. For a suspension with uncertain mass, road, damping, measurement noise, and actuator efficiency, state what is fixed, what is optimized, what information is available, and what equations define feasibility.

The relevant quantities are $\theta$ , design $p,c$ , random trajectories, failure events, and risk $\alpha$ . The chapter-level formulation is

\min\mathbb E[J]\;\mathrm{s.t.}\;\mathbb P(g_i\le0)\ge1-\alpha_i.

For this section, trace how the choice changes robust design, 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 optimize risk. 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 13.14: quantify uncertainty propagation¶

Activity 13.14: quantify uncertainty propagation

For an uncertain wave-energy-converter design, compare three formulations for Uncertainty Propagation using 200 validation scenarios.

Formulation	Mean cost	Cost SD	95th-percentile cost	Violations
Nominal	1.140	0.142	1.350	32
Risk-aware	1.190	0.093	1.300	7
Robust	1.260	0.054	1.330	2

Compute empirical violation probabilities and test a 3% limit.
Calculate coefficient of variation and a risk score equal to mean plus 1.645 standard deviations.
Rank only feasible formulations by risk score.
Repeat if every violation count increases by four and report the resulting design decision.
State which uncertainty samples, seeds, and model versions must accompany the result.

Chapter summary¶

The chapter connected uncertainty sources, propagation, risk formulation, robust design, Monte Carlo validation 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 a suspension with uncertain mass, road, damping, measurement noise, and actuator efficiency.
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¶

Azad and Herber’s overview of uncertain CCD formulations.

Open research question¶

How should model-form uncertainty and distribution shift enter dynamic chance constraints?