arXiv subjects

Alberto Padoan

Publications and source records attributed to Alberto Padoan.

At least 19 recordsLinked to original sources

arXiv2026

Gaussian behaviors and stochastic data-driven control

We propose a stochastic behavioral modeling framework, termed Gaussian behaviors, which augments a deterministic linear time-invariant (LTI) behavior with a Gaussian noise component. We show that this notion is a tractable subclass of stochastic behaviors and encompasses classical parametric stochastic LTI state-space system models as special cases. Analogously to deterministic LTI behaviors, the framework enables simple and tractable stochastic data-driven control methods. To this end, we obtain a method for prediction by conditioning the Gaussian behavior on the known part of the trajectory, which is identified directly from the sample covariance of trajectory data. Building on this method, we develop predictive control formulations that optimize over feedforward or disturbance affine feedback policies. The resulting formulations are shown to be convex. We further derive a finite-sample confidence bound on the prediction accounting for both aleatoric and epistemic uncertainty, and incorporate it into a robust control method, for which a tractable convex upper bound is obtained. Within this framework, subspace predictive control is recovered when only the mean prediction is used, while data-enabled predictive control is shown to account for the prediction uncertainty in an optimistic fashion. Numerical case studies illustrate the benefits of the proposed methods.

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arXiv2026

On Dominant Manifolds in Reservoir Computing Networks

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.

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arXiv2026

Scaled Relative Graphs in Normed Spaces

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.

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arXiv2026

Min-Max Grassmannian Optimization for Online Subspace Tracking

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.

eess.SY↗

arXiv2026

Data-driven generalized perimeter control: Z\"urich case study

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 Z\"urich, 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.

eess.SY↗

arXiv2026

From time series to dissipativity of linear systems with dynamic supply rates

This paper studies the problem of verifying dissipativity of linear time-invariant (LTI) systems using input-output data. We leverage behavioral systems theory to express dissipativity in terms of quadratic difference forms (QDFs), allowing the study of general dynamic quadratic supply rates. We work under the assumptions that the data-generating system is controllable, and an upper bound is given on its lag. As our main results, we provide sufficient conditions for the data to be informative for dissipativity. We also show that for a specific class of static supply rates, these conditions are both necessary and sufficient. For the latter supply rates, it turns out that certification of dissipativity is only possible from data that enable unique system identification. As auxiliary results, we highlight some properties of QDFs, such as upper bounds on the degree of storage functions.

math.OC↗

arXiv2025

Robust Least-Squares Optimization for Data-Driven Predictive Control: A Geometric Approach

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.

math.OC↗

arXiv2025

Geometrically robust least squares through manifold optimization

This paper presents a methodology for solving a geometrically robust least squares problem, which arises in various applications where the model is subject to geometric constraints. The problem is formulated as a minimax optimization problem on a product manifold, where one variable is constrained to a ball describing uncertainty. To handle the constraint, an exact penalty method is applied. A first-order gradient descent ascent algorithm is proposed to solve the problem, and its convergence properties are illustrated by an example. The proposed method offers a robust approach to solving a wide range of problems arising in signal processing and data-driven control.

math.OC↗

arXiv2025

Least Squares Model Reduction: A Two-Stage System-Theoretic Interpretation

Model reduction simplifies complex dynamical systems while preserving essential properties. This paper revisits a recently proposed system-theoretic framework for least squares moment matching. It interprets least squares model reduction in terms of two steps process: constructing a surrogate model to satisfy interpolation constraints, then projecting it onto a reduced-order space. Using tools from output regulation theory and Krylov projections, this approach provides a new view on classical methods. For illustration, we reexamine the least-squares model reduction method by Lucas and Smith, offering new insights into its structure.

math.OC↗

arXiv2025

Split-as-a-Pro: behavioral control via operator splitting and alternating projections

The paper introduces Split-as-a-Pro, a control framework that integrates behavioral systems theory, operator splitting methods, and alternating projection algorithms. The framework reduces dynamic optimization problems - arising in both control and estimation - to efficient projection computations. Split-as-a-Pro builds on a non-parametric formulation that exploits system structure to separate dynamic constraints imposed by individual subsystems from external ones, such as interconnection constraints and input/output constraints. This enables the use of arbitrary system representations, as long as the associated projection is efficiently computable, thereby enhancing scalability and compatibility with gray-box modeling. We demonstrate the effectiveness of Split-as-a-Pro by developing a distributed algorithm for solving finite-horizon linear quadratic control problems and illustrate its use in predictive control. Our numerical case studies show that algorithms obtained using Split-as-a-Pro significantly outperform their centralized counterparts in runtime and scalability across various standard graph topologies, while seamlessly leveraging both model-based and data-driven system representations.

math.OC↗

arXiv2025

Gaussian behaviors: representations and data-driven control

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.

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arXiv2025

Distances between finite-horizon linear behaviors

The paper introduces a class of distances for linear behaviors over finite time horizons. These distances allow for comparisons between finite-horizon linear behaviors represented by matrices of possibly different dimensions. They remain invariant under coordinate changes, rotations, and permutations, ensuring independence from input-output partitions. Moreover, they naturally encode complexity-misfit trade-offs for Linear Time-Invariant (LTI) behaviors, providing a principled solution to a longstanding puzzle in behavioral systems theory. The resulting framework characterizes modeling as a minimum distance problem, identifying the Most Powerful Unfalsified Model (MPUM) as optimal among all systems unfalsified by a given dataset. Finally, we illustrate the value of these metrics in a time series anomaly detection task, where their finer resolution yields superior performance over existing distances.

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arXiv2025

Mixed Small Gain and Phase Theorem: A new view using Scale Relative Graphs

We introduce a novel approach to feedback stability analysis for linear time-invariant (LTI) systems, overcoming the limitations of the sectoriality assumption in the small phase theorem. While phase analysis for single-input single-output (SISO) systems is well-established, multi-input multi-output (MIMO) systems lack a comprehensive phase analysis until recent advances introduced with the small-phase theorem. A limitation of the small-phase theorem is the sectorial condition, which states that an operator's eigenvalues must lie within a specified angle sector of the complex plane. We propose a framework based on Scaled Relative Graphs (SRGs) to remove this assumption. We derive two main results: a graphical set-based stability condition using SRGs and a small-phase theorem with no sectorial assumption. These results broaden the scope of phase analysis and feedback stability for MIMO systems.

math.OC↗

arXiv2025

Stability results for MIMO LTI systems via Scaled Relative Graphs

This paper proposes a frequency-wise approach for stability analysis of multi-input, multi-output (MIMO) Linear Time-Invariant (LTI) feedback systems through Scaled Relative Graphs (SRGs). Unlike traditional methods, such as the Generalized Nyquist Criterion (GNC), which relies on a coupled analysis that requires the multiplication of models, our approach enables the evaluation of system stability in a decoupled fashion, system by system, each of which is represented by its SRG (or an over-approximation thereof), and it provides an intuitive, visual representation of system behavior. Our results provide conditions for certifying the stability of stable and square MIMO LTI systems connected in closed loop.

eess.SY↗

arXiv2024

GREAT: Grassmannian REcursive Algorithm for Tracking & Online System Identification

This paper introduces an online approach for identifying time-varying subspaces defined by linear dynamical systems. The approach of representing linear systems by non-parametric subspace models has received significant interest in the field of data-driven control recently. This system representation enables us to provide rigorous guarantees for linear time-varying systems, which are difficult to obtain for parametric system models. The proposed method leverages optimization on the Grassmann manifold leading to the Grassmannian Recursive Algorithm for Tracking (GREAT). We view subspaces as points on the Grassmann manifold and adapt the estimate based on online data by performing optimization on the manifold. At each time step, a single measurement from the current subspace corrupted by a bounded error is available. The subspace estimate is updated online using Grassmannian gradient descent on a cost function incorporating a window of the most recent data. Under suitable assumptions on the signal-to-noise ratio of the online data and the subspace's rate of change, we establish theoretical guarantees for the resulting algorithm. More specifically, we prove an exponential convergence rate and provide an uncertainty quantification of the estimates in terms of an upper bound on their distance to the true subspace. The applicability of the proposed algorithm is demonstrated by means of numerical examples.

eess.SY↗

arXiv2024

DeePC-Hunt: Data-enabled Predictive Control Hyperparameter Tuning via Differentiable Optimization

This paper introduces Data-enabled Predictive Control Hyperparameter Tuning via Differentiable Optimization (DeePC-Hunt), a backpropagation-based method for automatic hyperparameter tuning of the DeePC algorithm. The necessity for such a method arises from the importance of hyperparameter selection to achieve satisfactory closed-loop DeePC performance. The standard methods for hyperparameter selection are to either optimize the open-loop performance, or use manual guess-and-check. Optimizing the open-loop performance can result in unacceptable closed-loop behavior, while manual guess-and-check can pose safety challenges. DeePC-Hunt provides an alternative method for hyperparameter tuning which uses an approximate model of the system dynamics and backpropagation to directly optimize hyperparameters for the closed-loop DeePC performance. Numerical simulations demonstrate the effectiveness of DeePC in combination with DeePC-Hunt in a complex stabilization task for a nonlinear system and its superiority over model-based control strategies in terms of robustness to model misspecifications.

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arXiv2023

Urban traffic congestion control: a DeePC change

Urban traffic congestion remains a pressing challenge in our rapidly expanding cities, despite the abundance of available data and the efforts of policymakers. By leveraging behavioral system theory and data-driven control, this paper exploits the DeePC algorithm in the context of urban traffic control performed via dynamic traffic lights. To validate our approach, we consider a high-fidelity case study using the state-of-the-art simulation software package Simulation of Urban MObility (SUMO). Preliminary results indicate that DeePC outperforms existing approaches across various key metrics, including travel time and CO$_2$ emissions, demonstrating its potential for effective traffic management

eess.SY↗

arXiv2023

Controller implementability: a data-driven approach

We study the controller implementability problem, which seeks to determine if a controller can make the closed-loop behavior of a given plant match that of a desired reference behavior. We establish necessary and sufficient conditions for controller implementability which only rely on raw data. Subsequently, we consider the problem of constructing controllers directly from data. By leveraging the concept of canonical controller, we provide a formula to directly construct controllers that implement plant-compatible reference behaviors using measurements of both reference and plant behaviors.

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