Alberto Padoan

Alberto Padoan, Fulvio Forni, Rodolphe Sepulchre

The paper introduces notions of robustness margins geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and in engineering, a theory of robustness for behaviors away from equilibria is lacking. The proposed framework addresses this need in the framework of p-dominance theory, which aims at generalizing stability theory for the analysis of systems with low-dimensional attractors. Dominance margins are introduced as natural generalisations of stability margins in the context of p-dominance analysis. In analogy with stability margins, dominance margins are shown to admit simple interpretations in terms of familiar frequency domain tools and to provide quantitative measures of robustness for multistable and oscillatory behaviors in Lure systems. The theory is illustrated by means of an elementary mechanical example.

András Sasfi, Ivan Markovsky, Alberto Padoan et al.

We propose a modeling framework for stochastic systems, termed Gaussian behaviors, that describes finite-length trajectories of a system as a Gaussian process. The proposed model naturally quantifies the uncertainty in the trajectories, yet it is simple enough to allow for tractable formulations. We relate the proposed model to existing descriptions of dynamical systems including deterministic and stochastic behaviors, and linear time-invariant (LTI) state-space models with Gaussian noise. Gaussian behaviors can be estimated directly from observed data as the empirical sample covariance. The distribution of future outputs conditioned on inputs and past outputs provides a predictive model that can be incorporated in predictive control frameworks. We show that subspace predictive control is a certainty-equivalence control formulation with the estimated Gaussian behavior. Furthermore, the regularized data-enabled predictive control (DeePC) method is shown to be a distributionally optimistic formulation that optimistically accounts for uncertainty in the Gaussian behavior. To mitigate the excessive optimism of DeePC, we propose a novel distributionally robust control formulation, and provide a convex reformulation allowing for efficient implementation.

Shreyas Bharadwaj, Bamdev Mishra, Cyrus Mostajeran et al.

The paper studies a geometrically robust least-squares problem that extends classical and norm-based robust formulations. Rather than minimizing residual error for fixed or perturbed data, we interpret least-squares as enforcing approximate subspace inclusion between measured and true data spaces. The uncertainty in this geometric relation is modeled as a metric ball on the Grassmannian manifold, leading to a min-max problem over Euclidean and manifold variables. The inner maximization admits a closed-form solution, enabling an efficient algorithm with a transparent geometric interpretation. Applied to robust finite-horizon linear-quadratic tracking in data-enabled predictive control, the method improves upon existing robust least-squares formulations, achieving stronger robustness and favorable scaling under small uncertainty.

Alessio Rimoldi, Carlo Cenedese, Alberto Padoan et al.

Urban traffic congestion is a key challenge for the development of modern cities, requiring advanced control techniques to optimize existing infrastructures usage. Despite the extensive availability of data, modeling such complex systems remains an expensive and time consuming step when designing model-based control approaches. On the other hand, machine learning approaches require simulations to bootstrap models, or are unable to deal with the sparse nature of traffic data and enforce hard constraints. We propose a novel formulation of traffic dynamics based on behavioral systems theory and apply data-enabled predictive control to steer traffic dynamics via dynamic traffic light control. A high-fidelity simulation of the city of Zürich, the largest closed-loop microscopic simulation of urban traffic in the literature to the best of our knowledge, is used to validate the performance of the proposed method in terms of total travel time and CO2 emissions.

Noa Kaplan, Alberto Padoan, Anastasia Bizyaeva

Understanding how training shapes the geometry of recurrent network dynamics is a central problem in time-series modeling. We study the emergence of low-dimensional dominant manifolds in the training of Reservoir Computing (RC) networks for temporal forecasting tasks. For a simplified linear and continuous-time reservoir model, we link the dimensionality and structure of the dominant modes directly to the intrinsic dimensionality and information content of the training data. In particular, for training data generated by an autonomous dynamical system, we relate the dominant modes of the trained reservoir to approximations of the Koopman eigenfunctions of the original system, illuminating an explicit connection between reservoir computing and the Dynamic Mode Decomposition algorithm. We illustrate the eigenvalue motion that generates the dominant manifolds during training in simulation, and discuss generalization to nonlinear RC via tangent dynamics and differential p-dominance.

Alberto Padoan

The paper extends the Scaled Relative Graph (SRG) framework of Ryu, Hannah, and Yin from Hilbert spaces to normed spaces. Our extension replaces the inner product with a regular pairing, whose asymmetry gives rise to directional angles and, in turn, directional SRGs. Directional SRGs are shown to provide geometric containment tests certifying key operator properties, including contraction and monotonicity. Calculus rules for SRGs under scaling, inversion, addition, and composition are also derived. The theory is illustrated by numerical examples, including a graphical contraction certificate for Bellman operators.

Shreyas Bharadwaj, Bamdev Mishra, Cyrus Mostajeran et al.

This paper discusses robustness guarantees for online tracking of time-varying subspaces from noisy data. Building on recent work in optimization over a Grassmannian manifold, we introduce a new approach for robust subspace tracking by modeling data uncertainty in a Grassmannian ball. The robust subspace tracking problem is cast into a min-max optimization framework, for which we derive a closed-form solution for the worst-case subspace, enabling a geometric robustness adjustment that is both analytically tractable and computationally efficient, unlike iterative convex relaxations. The resulting algorithm, GeRoST (Geometrically Robust Subspace Tracking), is validated on two case studies: tracking a linear time-varying system and online foreground-background separation in video.