Luca Furieri

John Cao, Luca Furieri

We study distributed control of networked systems through reinforcement learning, where neural policies must be simultaneously scalable, expressive and stabilizing. We introduce a policy parameterization that embeds Graph Neural Networks (GNNs) into a Youla-like magnitude-direction parameterization, yielding distributed stochastic controllers that guarantee network-level closed-loop stability by design. The magnitude is implemented as a stable operator consisting of a GNN acting on disturbance feedback, while the direction is a GNN acting on local observations. We prove robustness of the policy to perturbations in both the graph topology and model parameters. Numerical experiments validate the effectiveness of the proposed approach.

Luca Furieri, Thomas Stastny, Lorenzo Marconi et al.

The recent years have witnessed increased development of small, autonomous fixed-wing Unmanned Aerial Vehicles (UAVs). In order to unlock widespread applicability of these platforms, they need to be capable of operating under a variety of environmental conditions. Due to their small size, low weight, and low speeds, they require the capability of coping with wind speeds that are approaching or even faster than the nominal airspeed. In this paper we present a principled nonlinear guidance strategy, addressing this problem. More broadly, we propose a methodology for the high-level control of non-holonomic unicycle-like vehicles in the presence of strong flowfields (e.g. winds, underwater currents) which may outreach the maximum vehicle speed. The proposed strategy guarantees convergence to a safe and stable vehicle configuration with respect to the flowfield, while preserving some tracking performance with respect to the target path. Evaluations in simulations and a challenging real-world flight experiment in very windy conditions confirm the feasibility of the proposed guidance approach.

Luca Furieri, Yang Zheng, Antonis Papachristodoulou et al.

This paper proposes a novel input-output parametrization of the set of internally stabilizing output-feedback controllers for linear time-invariant (LTI) systems. Our underlying idea is to directly treat the closed-loop transfer matrices from disturbances to input and output signals as design parameters and exploit their affine relationships. This input-output perspective is particularly effective when a doubly-coprime factorization is difficult to compute, or an initial stabilizing controller is challenging to find; most previous work requires one of these pre-computation steps. Instead, our approach can bypass such pre-computations, in the sense that a stabilizing controller is computed by directly solving a linear program (LP). Furthermore, we show that the proposed input-output parametrization allows for computing norm-optimal controllers subject to quadratically invariant (QI) constraints using convex programming.

Luca Furieri

We derive a state-space characterization of all dynamic state-feedback controllers that make an equilibrium of a nonlinear input-affine continuous-time system locally exponentially stable. Specifically, any controller obtained as the sum of a linear state-feedback $u=Kx$, with $K$ stabilizing the linearized system, and the output of internal locally exponentially stable controller dynamics is itself locally exponentially stabilizing. Conversely, every dynamic state-feedback controller that locally exponentially stabilizes the equilibrium admits such a decomposition. The result can be viewed as a state-space nonlinear Youla-type parametrization specialized to local, rather than global, and exponential, rather than asymptotic, closed-loop stability. The residual locally exponentially stable controller dynamics can be implemented with stable recurrent neural networks and trained as neural ODEs to achieve high closed-loop performance in nonlinear control tasks.

Andrea Martin, Luca Furieri, Florian Dörfler et al.

We consider control of dynamical systems through the lens of competitive analysis. Most prior work in this area focuses on minimizing regret, that is, the loss relative to an ideal clairvoyant policy that has noncausal access to past, present, and future disturbances. Motivated by the observation that the optimal cost only provides coarse information about the ideal closed-loop behavior, we instead propose directly minimizing the tracking error relative to the optimal trajectories in hindsight, i.e., imitating the clairvoyant policy. By embracing a system level perspective, we present an efficient optimization-based approach for computing follow-the-clairvoyant (FTC) safe controllers. We prove that these attain minimal regret if no constraints are imposed on the noncausal benchmark. In addition, we present numerical experiments to show that our policy retains the hallmark of competitive algorithms of interpolating between classical $\mathcal{H}_2$ and $\mathcal{H}_\infty$ control laws - while consistently outperforming regret minimization methods in constrained scenarios thanks to the superior ability to chase the clairvoyant.

Luca Furieri, Clara Lucía Galimberti, Giancarlo Ferrari-Trecate

We address the problem of designing stabilizing control policies for nonlinear systems in discrete-time, while minimizing an arbitrary cost function. When the system is linear and the cost is convex, the System Level Synthesis (SLS) approach offers an effective solution based on convex programming. Beyond this case, a globally optimal solution cannot be found in a tractable way, in general. In this paper, we develop a parametrization of all and only the control policies stabilizing a given time-varying nonlinear system in terms of the combined effect of 1) a strongly stabilizing base controller and 2) a stable SLS operator to be freely designed. Based on this result, we propose a Neural SLS (Neur-SLS) approach guaranteeing closed-loop stability during and after parameter optimization, without requiring any constraints to be satisfied. We exploit recent Deep Neural Network (DNN) models based on Recurrent Equilibrium Networks (RENs) to learn over a rich class of nonlinear stable operators, and demonstrate the effectiveness of the proposed approach in numerical examples.

Luca Furieri, Yang Zheng, Antonis Papachristodoulou et al.

We consider the problem of designing a stabilizing and optimal static controller with a pre-specified sparsity pattern. Since this problem is NP-hard in general, it is necessary to resort to approximation approaches. In this paper, we characterize a class of convex restrictions of this problem that are based on designing a separable quadratic Lyapunov function for the closed-loop system. This approach generalizes previous results based on optimizing over diagonal Lyapunov functions, thus allowing for improved feasibility and performance. Moreover, we suggest a simple procedure to compute favourable structures for the Lyapunov function yielding high-performance distributed controllers. Numerical examples validate our results.

Daniele Martinelli, Clara Lucía Galimberti, Ian R. Manchester et al.

In this work, we introduce and study a class of Deep Neural Networks (DNNs) in continuous-time. The proposed architecture stems from the combination of Neural Ordinary Differential Equations (Neural ODEs) with the model structure of recently introduced Recurrent Equilibrium Networks (RENs). We show how to endow our proposed NodeRENs with contractivity and dissipativity -- crucial properties for robust learning and control. Most importantly, as for RENs, we derive parametrizations of contractive and dissipative NodeRENs which are unconstrained, hence enabling their learning for a large number of parameters. We validate the properties of NodeRENs, including the possibility of handling irregularly sampled data, in a case study in nonlinear system identification.

Luca Furieri, Maryam Kamgarpour

We consider the problem of computing optimal linear control policies for linear systems in finite-horizon. The states and the inputs are required to remain inside pre-specified safety sets at all times despite unknown disturbances. In this technical note, we focus on the requirement that the control policy is distributed, in the sense that it can only be based on partial information about the history of the outputs. It is well-known that when a condition denoted as Quadratic Invariance (QI) holds, the optimal distributed control policy can be computed in a tractable way. Our goal is to unify and generalize the class of information structures over which quadratic invariance is equivalent to a test over finitely many binary matrices. The test we propose certifies convexity of the output-feedback distributed control problem in finite-horizon given any arbitrarily defined information structure, including the case of time varying communication networks and forgetting mechanisms. Furthermore, the framework we consider allows for including polytopic constraints on the states and the inputs in a natural way, without affecting convexity.

Luca Furieri, Maryam Kamgarpour

The problem of robust distributed control arises in several large-scale systems, such as transportation networks and power grid systems. In many practical scenarios controllers might not have enough information to make globally optimal decisions in a tractable way. We propose a novel class of tractable optimization problems whose solution is a controller complying with any specified information structure. The approach we suggest is based on decomposing intractable information constraints into two subspace constraints in the disturbance feedback domain. We discuss how to perform the decomposition in an optimized way. The resulting control policy is globally optimal when a condition known as Quadratic Invariance (QI) holds, whereas it is feasible and it provides a provable upper bound on the minimum cost when QI does not hold. Finally, we show that our method can lead to improved performance guarantees with respect to previous approaches, by applying the developed techniques to the platooning of autonomous vehicles.

Danilo Saccani, Luca Furieri, Giancarlo Ferrari-Trecate

The growing complexity of modern control tasks calls for controllers that can react online as objectives and disturbances change, while preserving closed-loop stability. Recent approaches for improving the performance of nonlinear systems while preserving closed-loop stability rely on time-invariant recurrent neural-network controllers, but offer no principled way to update the controller during operation. Most importantly, switching from one stabilizing policy to another can itself destabilize the closed-loop. We address this problem by introducing a stability-preserving update mechanism for nonlinear, neural-network-based controllers. Each controller is modeled as a causal operator with bounded $\ell_p$-gain, and we derive gain-based conditions under which the controller may be updated online. These conditions yield two practical update schemes, time-scheduled and state-triggered, that guarantee the closed-loop remains $\ell_p$-stable after any number of updates. Our analysis further shows that stability is decoupled from controller optimality, allowing approximate or early-stopped controller synthesis. We demonstrate the approach on nonlinear systems with time-varying objectives and disturbances, and show consistent performance improvements over static and naive online baselines while guaranteeing stability.

Danilo Saccani, Haoming Shen, Luca Furieri et al.

We study a control architecture for nonlinear constrained systems that integrates a performance-boosting (PB) controller with a scheduled Predictive Safety Filter (PSF). The PSF acts as a pre-stabilizing base controller that enforces state and input constraints. The PB controller, parameterized as a causal operator, influences the PSF in two ways: it proposes a performance input to be filtered, and it provides a scheduling signal to adjust the filter's Lyapunov-decrease rate. We prove two main results: (i) Stability by design: any controller adhering to this parametrization maintains closed-loop stability of the pre-stabilized system and inherits PSF safety. (ii) Trajectory-set expansion: the architecture strictly expands the set of safe, stable trajectories achievable by controllers combined with conventional PSFs, which rely on a pre-defined Lyapunov decrease rate to ensure stability. This scheduling allows the PB controller to safely execute complex behaviors, such as transient detours, that are provably unattainable by standard PSF formulations. We demonstrate this expanded capability on a constrained inverted pendulum task with a moving obstacle.

Vaibhav Gupta, Daniele Martinelli, Giancarlo Ferrari-Trecate et al.

In this paper, we present a novel method for synthesising an optimal distributed spatial regret controller using experimentally obtained frequency-response data. Spatial regret provides a measure of the performance gap between a structured distributed controller and an oracle with enhanced communication topology. We relax assumptions on the communication topology, allowing the oracle to adopt any enhanced structure. While this generalisation requires an iterative solution in place of a single convex program, we provide a tractable algorithm that synthesises optimal controllers from frequency-response data while preserving stability and the desired communication structure. Through numerical examples, we illustrate the better performance of the spatial regret controller compared to classical H2/Hinf designs, underscoring the effectiveness of the proposed methodology.

Junan Lin, Paul J. Goulart, Luca Furieri

The Alternating Direction Method of Multipliers (ADMM) is a widely used method for structured convex optimization, and its practical performance depends strongly on the choice of penalty and relaxation parameters. Motivated by settings such as Model Predictive Control (MPC), where one repeatedly solves related optimization problems with fixed structure and changing parameter values, we propose learning online updates of the relaxation parameter to improve performance on problem classes of interest. This choice is computationally attractive in OSQP-like architectures, since adapting relaxation does not trigger the matrix refactorizations associated with penalty updates. We establish convergence guarantees for ADMM with time-varying penalty and relaxation parameters under mild assumptions, and show on benchmark quadratic programs that the resulting learned policies improve both iteration count and wall-clock time over baseline OSQP.

Luca Furieri, Clara Lucía Galimberti, Giancarlo Ferrari-Trecate

The growing scale and complexity of safety-critical control systems underscore the need to evolve current control architectures aiming for the unparalleled performances achievable through state-of-the-art optimization and machine learning algorithms. However, maintaining closed-loop stability while boosting the performance of nonlinear control systems using data-driven and deep-learning approaches stands as an important unsolved challenge. In this paper, we tackle the performance-boosting problem with closed-loop stability guarantees. Specifically, we establish a synergy between the Internal Model Control (IMC) principle for nonlinear systems and state-of-the-art unconstrained optimization approaches for learning stable dynamics. Our methods enable learning over arbitrarily deep neural network classes of performance-boosting controllers for stable nonlinear systems; crucially, we guarantee L_p closed-loop stability even if optimization is halted prematurely, and even when the ground-truth dynamics are unknown, with vanishing conservatism in the class of stabilizing policies as the model uncertainty is reduced to zero. We discuss the implementation details of the proposed control schemes, including distributed ones, along with the corresponding optimization procedures, demonstrating the potential of freely shaping the cost functions through several numerical experiments.

Luca Furieri, Sucheth Shenoy, Danilo Saccani et al.

We introduce magnitude and direction (MAD) policies, a policy parameterization for reinforcement learning (RL) that preserves Lp closed-loop stability for nonlinear dynamical systems. Despite their completeness in describing all stabilizing controllers, methods based on nonlinear Youla and system-level synthesis are significantly impacted by the difficulty of parametrizing Lp-stable operators. In contrast, MAD policies introduce explicit feedback on state-dependent features - a key element behind the success of reinforcement learning pipelines - without jeopardizing closed-loop stability. This is achieved by letting the magnitude of the control input be described by a disturbance-feedback Lp-stable operator, while selecting its direction based on state-dependent features through a universal function approximator. We further characterize the robust stability properties of MAD policies under model mismatch. Unlike existing disturbance-feedback policy parametrizations, MAD policies introduce state-feedback components compatible with model-free RL pipelines, ensuring closed-loop stability with no model information beyond assuming open-loop stability. Numerical experiments show that MAD policies trained with deep deterministic policy gradient (DDPG) methods generalize to unseen scenarios - matching the performance of standard neural network policies while guaranteeing closed-loop stability by design.

Andrea Martin, Ian R. Manchester, Luca Furieri

In high-stakes engineering applications, optimization algorithms must come with provable worst-case guarantees over a mathematically defined class of problems. Designing for the worst case, however, inevitably sacrifices performance on the specific problem instances that often occur in practice. We address the problem of augmenting a given linearly convergent algorithm to improve its average-case performance on a restricted set of target problems - for example, tailoring an off-the-shelf solver for model predictive control (MPC) for an application to a specific dynamical system - while preserving its worst-case guarantees across the entire problem class. Toward this goal, we characterize the class of algorithms that achieve linear convergence for classes of nonsmooth composite optimization problems. In particular, starting from a baseline linearly convergent algorithm, we derive all - and only - the modifications to its update rule that maintain its convergence properties. Our results apply to augmenting legacy algorithms such as gradient descent for nonconvex, gradient-dominated functions; Nesterov's accelerated method for strongly convex functions; and projected methods for optimization over polyhedral feasibility sets. We showcase effectiveness of the approach on solving optimization problems with tight iteration budgets in application to ill-conditioned systems of linear equations and MPC for linear systems.

Andrea Martin, Luca Furieri

The increasing reliance on numerical methods for controlling dynamical systems and training machine learning models underscores the need to devise algorithms that dependably and efficiently navigate complex optimization landscapes. Classical gradient descent methods offer strong theoretical guarantees for convex problems; however, they demand meticulous hyperparameter tuning for non-convex ones. The emerging paradigm of learning to optimize (L2O) automates the discovery of algorithms with optimized performance leveraging learning models and data - yet, it lacks a theoretical framework to analyze convergence of the learned algorithms. In this paper, we fill this gap by harnessing nonlinear system theory. Specifically, we propose an unconstrained parametrization of all convergent algorithms for smooth non-convex objective functions. Notably, our framework is directly compatible with automatic differentiation tools, ensuring convergence by design while learning to optimize.

Luca Furieri, Clara Lucía Galimberti, Muhammad Zakwan et al.

Large-scale cyber-physical systems require that control policies are distributed, that is, that they only rely on local real-time measurements and communication with neighboring agents. Optimal Distributed Control (ODC) problems are, however, highly intractable even in seemingly simple cases. Recent work has thus proposed training Neural Network (NN) distributed controllers. A main challenge of NN controllers is that they are not dependable during and after training, that is, the closed-loop system may be unstable, and the training may fail due to vanishing and exploding gradients. In this paper, we address these issues for networks of nonlinear port-Hamiltonian (pH) systems, whose modeling power ranges from energy systems to non-holonomic vehicles and chemical reactions. Specifically, we embrace the compositional properties of pH systems to characterize deep Hamiltonian control policies with built-in closed-loop stability guarantees, irrespective of the interconnection topology and the chosen NN parameters. Furthermore, our setup enables leveraging recent results on well-behaved neural ODEs to prevent the phenomenon of vanishing gradients by design. Numerical experiments corroborate the dependability of the proposed architecture, while matching the performance of general neural network policies.

Clara Lucía Galimberti, Luca Furieri, Liang Xu et al.

Deep Neural Networks (DNNs) training can be difficult due to vanishing and exploding gradients during weight optimization through backpropagation. To address this problem, we propose a general class of Hamiltonian DNNs (H-DNNs) that stem from the discretization of continuous-time Hamiltonian systems and include several existing DNN architectures based on ordinary differential equations. Our main result is that a broad set of H-DNNs ensures non-vanishing gradients by design for an arbitrary network depth. This is obtained by proving that, using a semi-implicit Euler discretization scheme, the backward sensitivity matrices involved in gradient computations are symplectic. We also provide an upper-bound to the magnitude of sensitivity matrices and show that exploding gradients can be controlled through regularization. Finally, we enable distributed implementations of backward and forward propagation algorithms in H-DNNs by characterizing appropriate sparsity constraints on the weight matrices. The good performance of H-DNNs is demonstrated on benchmark classification problems, including image classification with the MNIST dataset.