A Mathematical Theory of Co-Design
This foundational theory addresses the problem of designing complex systems, such as in robotics, for engineers and researchers, though it is incremental in building on existing design frameworks.
The paper tackles the challenge of designing complex systems with optimality guarantees by introducing a theory of co-design that defines design problems and their interconnections, leading to optimization problems for minimal resources, with results enabling rigorous system design.
One of the challenges of modern engineering, and robotics in particular, is designing complex systems, composed of many subsystems, rigorously and with optimality guarantees. This paper introduces a theory of co-design that describes "design problems", defined as tuples of "functionality space", "implementation space", and "resources space", together with a feasibility relation that relates the three spaces. Design problems can be interconnected together to create "co-design problems", which describe possibly recursive co-design constraints among subsystems. A co-design problem induces a family of optimization problems of the type "find the minimal resources needed to implement a given functionality"; the solution is an antichain (Pareto front) of resources. A special class of co-design problems are Monotone Co-Design Problems (MCDPs), for which functionality and resources are complete partial orders and the feasibility relation is monotone and Scott continuous. The induced optimization problems are multi-objective, nonconvex, nondifferentiable, noncontinuous, and not even defined on continuous spaces; yet, there exists a complete solution. The antichain of minimal resources can be characterized as a least fixed point, and it can be computed using Kleene's algorithm. The computation needed to solve a co-design problem can be bounded by a function of a graph property that quantifies the interdependence of the subproblems. These results make us much more optimistic about the problem of designing complex systems in a rigorous way.