4.8 Deterministic and Uncertain Formulations

Deterministic means one specified problem realization¶

In a deterministic formulation, parameters, models, initial conditions, and disturbances are treated as known. The positioner may use one payload mass, nominal motor resistance, a fixed disturbance trajectory, and exact sensor values. Deterministic studies are essential for understanding coupling and debugging a formulation, but nominal feasibility does not imply reliable operation.

Introduce uncertain quantities with $\theta$. They may include payload, friction, motor constants, manufacturing variation, sensor noise, disturbance timing, environmental inputs, or model discrepancy:

Uncertainty can change the objective, dynamics, equalities, inequalities, and even the effective design variables.

Common uncertain problem statements¶

An expectation formulation minimizes average performance:

A chance constraint limits violation probability:

A worst-case formulation protects against every realization in an uncertainty set $\Theta$:

These formulations answer different engineering questions. Expected performance does not guarantee a low failure probability, and worst-case protection can be unnecessarily conservative if the uncertainty set is poorly chosen.

Information and uncertainty are related but distinct¶

A quantity can be uncertain during design yet measurable during operation. Adaptive or feedback control can respond to revealed information. Conversely, a deterministic future disturbance may still be unavailable to a causal controller. Specify what is uncertain, when it becomes known, and to whom.

Activity 4.8: choose an uncertainty statement¶