Seeing what can’t be measured
Modern engineering systems are increasingly expected to operate autonomously, adapt to uncertainty, and make decisions in real time. From rehabilitation robots assisting patients to intelligent infrastructure monitoring structural health, a common requirement underpins these technologies: the ability to infer hidden information from measurable data. This process, known as estimation, is central to how machines interpret and interact with the world.
At its core, estimation involves reconstructing quantities that can’t be directly measured – such as internal system states, faults, or environmental variables – based on available measurements. In many real-world systems, however, this task is far from straightforward. The relationships between what can be measured and what must be inferred are often highly nonlinear, meaning they don’t follow simple, predictable patterns.
When mathematics gets in the way
To extract useful information, these nonlinear relationships must be mathematically inverted, a process that can become fragile under certain conditions. One of the most significant challenges comes from the presence of mathematical singularities. These are points where the underlying equations behave irregularly, causing standard estimation algorithms to become unreliable or fail altogether.
Near such singularities, small measurement errors can lead to large estimation inaccuracies, undermining both performance and safety. In more complex cases, the problem is made more difficult by the structure of the solution space, meaning all the possible solutions and how they are related to each other. Instead of forming a single, continuous region, the set of possible solutions may be fragmented into disconnected components. Traditional algorithms, which rely on smooth and continuous search processes, may become trapped in the wrong region and fail to identify the correct solution.
A geometric way forward
Addressing these challenges requires a shift in perspective. Rather than treating singularities and disconnected solution spaces as mere obstacles, Papageorgiou explores them as geometric features that can be understood and navigated. By incorporating tools from topology – the study of shapes and spatial structures – alongside control theory and machine learning, the project aims to develop new estimation frameworks that remain reliable even in the presence of these complexities.
The proposed approach focuses on ensuring that estimation processes are safe, continuous, and robust. This involves designing algorithms capable of detecting and handling singularities without abrupt failure, as well as developing strategies to move between disconnected solution regions when necessary.
Building reliable intelligence
Machine learning techniques can assist in identifying patterns or guiding the estimation process, while control-theoretic principles ensure stability and real-time performance. In settings, where multiple systems interact and share information, these methods could enable coordinated estimation across distributed networks.
Increasingly relevant technologies, such as autonomous vehicles, medical robots, industrial monitoring systems, depend critically on accurate and reliable estimation. Failures in these systems can lead not only to reduced efficiency but also to safety risks and loss of trust.
Toward a new paradigm
Looking ahead, the project aims to establish a new theoretical foundation for estimation in complex systems. By demonstrating that singularities and fragmented solution spaces can be systematically managed, it opens the door to a broader class of problems that were previously considered borderline impossible to solve.
This interdisciplinary effort, bridging control theory, topology, machine learning, and multi-agent systems, has the potential to reshape how estimation and decision-making are approached in engineering. Ultimately, the goal is not just to make algorithms that work under ideal conditions, but to ensure they continue to perform when reality becomes messy, as it inevitably does.
In doing so, the research moves one step closer to enabling intelligent systems that are not only capable, but also dependable in the face of complexity.
