Distributional Uncertainty and Adaptive Decision-Making in System

Abstract

Complex engineered systems require coordinated design choices across heterogeneous components under multiple conflicting objectives and uncertain specifications. Monotone co-design provides a compositional framework for such problems by modeling each subsystem as a design problem: a feasible relation between provided functionalities and required resources in partially ordered sets. Existing uncertain co-design models rely on interval bounds, which support worst-case reasoning but cannot represent probabilistic risk or multi-stage adaptive decisions. We develop a distributional extension of co-design that models uncertain design outcomes as distributions over design problems and supports adaptive decision processes through Markov-kernel re-parameterizations. Using quasi-measurable and quasi-universal spaces, we show that the standard co-design interconnection operations remain compositional under this richer notion of uncertainty. We further introduce queries and observations that extract probabilistic design trade-offs, including feasibility probabilities, confidence bounds, and distributions of minimal required resources. A task-driven unmanned aerial vehicle case study illustrates how the framework captures risk-sensitive and information-dependent design choices that interval-based models cannot express.

Figures (9)

• Figure 1: Diagrams for Markov kernels $b \mathbin{\circ} \mu$ and $b \mathbin{\circ} a$ in two orientations.
• Figure 2: can be composed in different ways.
• Figure 3: Diagram of $a \mathbin{\mathbin{\circ}_\mathcal{P}} \langle {\mathrm{obs}}_{T}, \pi_{T} \rangle \mathbin{\mathbin{\circ}_\mathcal{P}} \cdots \mathbin{\mathbin{\circ}_\mathcal{P}} \langle {\mathrm{obs}}_{1}, \pi_{1} \rangle \mathbin{\mathbin{\circ}_\mathcal{P}} \langle {\mathrm{obs}}_{0}, \pi_{0} \rangle \colon \textcolor{specificationcolor}{S}_0 \times \textcolor{imporange}{I}_0 \Rightarrow \mathsf{DP}_{\text{A}}\{{\textcolor{dpgreen}{F}}, {\textcolor{dpred}{R}}\}$, a $T$-stage adaptive decision process. The right-most kernel, $a \colon \textcolor{specificationcolor}{S}_{T+1} \times \textcolor{imporange}{I}_{T+1} \Rightarrow \mathsf{DP}_{\text{A}}\{{\textcolor{dpgreen}{F}}, {\textcolor{dpred}{R}}\}$, is illustrated in a similar way to those in \ref{['fig:uav-adaptive-dp-battery-tech-after-actuation']}.
• Figure 4: Co-design diagram for a task-driven , showing a decomposition into functional components. Designs are marked with implementation color.
• Figure 5: Trade-off between required payload and minimal lifetime cost for a fixed task profile under deterministic and interval-uncertain specifications. Pairs above the pessimistic curve (green) are robustly feasible, while pairs below the optimistic curve (orange) are infeasible even optimistically; the region between them is feasible for some realizations only. Labels indicate optimal implementations at the same payload on the three curves. For large payloads, the pessimistic model is infeasible and the minimal cost is $+\infty$.

Theorems & Definitions (83)

• Lemma 1
• Definition 1: Poset
• Definition 2: Opposite poset
• Definition 3: Product poset
• Definition 4: Upper closure
• Definition 5: Upper set
• Definition 6: Monotone map
• Definition 7: Measurable spaces & measurable maps
• Definition 8: Probability distribution
• Definition 9: Completion of a sigma algebra