Distributional Uncertainty and Adaptive Decision-Making in System Co-design

Yujun Huang; Gioele Zardini

math.OCMar 14, 2026

Distributional Uncertainty and Adaptive Decision-Making in System Co-design

Authors: Yujun Huang, Gioele Zardini

Organizations: Laboratory for Information & Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139

Abstract

Complex engineered systems require coordinated design choices across heterogeneous components under conflicting objectives and uncertain specifications. Monotone co-design provides a compositional framework for such problems. Performance of each subsystem is modeled with a design problem: a relation specifying what resources suffice to provide each functionality. 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 compositions remain compositional under this richer uncertainty, and introduce queries and observations extracting probabilistic trade-offs, including feasibility probabilities, confidence bounds, and distributions of minimal required resources. A task-driven unmanned aerial vehicle case study shows how the framework captures risk-sensitive and information-dependent design choices that interval models cannot express.

Figures & tables

Panel of Figure 1 — Fig. 1 : Markov kernel diagrams $b\mathbin{\circ}\mu$ and $b\mathbin{\circ}a$ in two orientations.

Panel of Figure 2 — Fig. 2 : DPs can be composed in different ways.

Fig. 3 : Diagram of $a\mathbin{\mathbin{\circ}_{\mathcal{P}}}\langle{\mathrm{obs}}_{T},\pi_{T}\rangle\mathbin{\mathbin{\circ}_{\mathcal{P}}}\cdots\mathbin{\mathbin{\circ}_{\mathcal{P}}}\langle{\mathrm{obs}}_{1},\pi_{1}\rangle\mathbin{\mathbin{\circ}_{\mathcal{P}}}\langle{\mathrm{obs}}_{0},\pi_{0}\rangle\colon{\color[rgb]{0.258,0.527,0.957}S}_{0}\times{\color[rgb]{1,0.58,0.063}I}_{0}\Rightarrow\mathsf{DP}_{\text{A}}\{{{\color[rgb]{0,0.5,0}F}},{{\color[rgb]{0.7,0,0}R}}\}$ , a $T$ -stage adaptive decision process. The right-most kernel, $a\colon{\color[rgb]{0.258,0.527,0.957}S}_{T+1}\times{\color[rgb]{1,0.58,0.063}I}_{T+1}\Rightarrow\mathsf{DP}_{\text{A}}\{{{\color[rgb]{0,0.5,0}F}},{{\color[rgb]{0.7,0,0}R}}\}$ , is illustrated in a similar way to those in Figure 8 .

Fig. 4 : Co-design diagram for a task-driven UAV , decomposed into functional components; designs carry implementation color. Red and green lines leaving the lower diagram are the four system-level functionalities / resources of the interface above.

Actuator	Mass [ $\mathrm{g}$ ]	Cost [ $$$ ]	Velocity [ $\mathrm{m}\mathrm{/}\mathrm{s}$ ]	$p_{0}$ [ $\mathrm{W}$ ]	$p_{1}$ [ $\mathrm{W}\mathrm{/}\mathrm{N}^{2}$ ]
${a_{1}}$	50.0	50.0	3.0	1.0	2.0
${a_{2}}$	100.0	100.0	3.0	2.0	1.5
${a_{3}}$	150.0	150.0	3.0	3.0	1.5

TABLE I : Deterministic parameters of component choices [ 9 ] .

Fig. 5 : Required payload versus minimal lifetime cost for a fixed task profile , deterministic and interval-uncertain. Pairs above the pessimistic curve (green) are robustly feasible, below the optimistic one (orange) infeasible even optimistically; between them, feasible for some realizations only. Labels give optimal implementations at some payloads; for large payloads the pessimistic model is infeasible, with minimal cost

+\infty

Panel of Figure 7 — Fig. 6 : Uncertain design results for the task-driven UAV and selected components, maximally adaptive benchmark. Upper and lower curves are inner confidence bounds; violin plots are distributions of minimal resource via Example 4 . Some distributions are annotated with the probability of each implementation being optimal and share of samples inside/outside the bound.

Panel of Figure 8 — Fig. 7 : Design processes for the UAV with different adaptivity levels.

Fig. 8 : Partly adaptive decision process combined with the co-design decomposition of the task-driven UAV .

Fig. 9 : Design results for three adaptivity levels: distributions of minimal lifetime cost induced by each design process at the given payloads .

Decision-Making under Uncertainty

Decision-making under uncertainty studies how to choose actions when outcomes, states, or models are imperfectly known.

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Task-Driven Co-Design of Heterogeneous Multi-Robot Systems

Maximilian Stralz, Meshal Alharbi, Yujun Huang, Gioele Zardini

Laboratory for Information & Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139

Multi-Robot Systems

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Designing multi-agent robotic systems requires reasoning across tightly coupled decisions spanning heterogeneous domains, including robot design, fleet composition, and planning. Much effort has been devoted to isolated improvements in these domains, whereas system-level co-design considering trade-offs and task requirements remains underexplored. In this work, we present a formal and compositional framework for the task-driven co-design of heterogeneous multi-robot systems. Building on a monotone co-design theory, we introduce general abstractions of robots, fleets, planners, executors, and evaluators as interconnected design problems with well-defined interfaces that are agnostic to both implementations and tasks. This structure enables efficient joint optimization of robot design, fleet composition, and planning under task-specific performance constraints. A series of case studies demonstrates the capabilities of the framework. Various component models can be seamlessly incorporated, including new robot types, task profiles, and probabilistic sensing objectives, while non-obvious design alternatives are systematically uncovered with optimality guarantees. The results highlight the flexibility, scalability, and interpretability of the proposed approach, and illustrate how formal co-design enables principled reasoning about complex heterogeneous multi-robot systems.

Abstract

Decision-Making under Uncertainty

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