arXiv subjects

Luca Furieri

Publications and source records attributed to Luca Furieri.

36 records · Page 2Linked to original sources

arXiv2022

Neural System Level Synthesis: Learning over All Stabilizing Policies for Nonlinear Systems

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.

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arXiv2022

Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Design

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.

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arXiv2022

Safe Control with Minimal Regret

As we move towards safety-critical cyber-physical systems that operate in non-stationary and uncertain environments, it becomes crucial to close the gap between classical optimal control algorithms and adaptive learning-based methods. In this paper, we present an efficient optimization-based approach for computing a finite-horizon robustly safe control policy that minimizes dynamic regret, in the sense of the loss relative to the optimal sequence of control actions selected in hindsight by a clairvoyant controller. By leveraging the system level synthesis framework (SLS), our method extends recent results on regret minimization for the linear quadratic regulator to optimal control subject to hard safety constraints, and allows competing against a safety-aware clairvoyant policy with minor modifications. Numerical experiments confirm superior performance with respect to finite-horizon constrained $\mathcal{H}_2$ and $\mathcal{H}_\infty$ control laws when the disturbance realizations poorly fit classical assumptions.

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arXiv2022

Follow the Clairvoyant: an Imitation Learning Approach to Optimal Control

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.

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arXiv2022

Near-Optimal Design of Safe Output Feedback Controllers from Noisy Data

As we transition towards the deployment of data-driven controllers for black-box cyberphysical systems, complying with hard safety constraints becomes a primary concern. Two key aspects should be addressed when input-output data are corrupted by noise: how much uncertainty can one tolerate without compromising safety, and to what extent is the control performance affected? By focusing on finite-horizon constrained linear-quadratic problems, we provide an answer to these questions in terms of the model mismatch incurred during a preliminary identification phase. We propose a control design procedure based on a quasiconvex relaxation of the original robust problem and we prove that, if the uncertainty is sufficiently small, the synthesized controller is safe and near-optimal, in the sense that the suboptimality gap increases linearly with the model mismatch level. Since the proposed method is independent of the specific identification procedure, our analysis holds in combination with state-of-the-art behavioral estimators beyond standard least-squares. The main theoretical results are validated by numerical experiments.

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arXiv2022

System-level, Input-output and New Parameterizations of Stabilizing Controllers, and Their Numerical Computation

It is known that the set of internally stabilizing controller $\mathcal{C}_{\text{stab}}$ is non-convex, but it admits convex characterizations using certain closed-loop maps: a classical result is the Youla parameterization, and two recent notions are the system-level parameterization (SLP) and the input-output parameterization (IOP). In this paper, we address the existence of new convex parameterizations and discuss potential tradeoffs of each parametrization in different scenarios. Our main contributions are: 1) We reveal that only four groups of stable closed-loop transfer matrices are equivalent to internal stability: one of them is used in the SLP, another one is used in the IOP, and the other two are new, leading to two new convex parameterizations of $\mathcal{C}_{\text{stab}}$. 2) We investigate the properties of these parameterizations after imposing the finite impulse response (FIR) approximation, revealing that the IOP has the best ability of approximating $\mathcal{C}_{\text{stab}}$ given FIR constraints. 3) These four parameterizations require no \emph{a priori} doubly-coprime factorization of the plant, but impose a set of equality constraints. However, these equality constraints will never be satisfied exactly in floating-point arithmetic computation and/or implementation. We prove that the IOP is numerically robust for open-loop stable plants, in the sense that small mismatches in the equality constraints do not compromise the closed-loop stability; but a direct IOP implementation will fail to stabilize open-loop unstable systems in practice. The SLP is known to enjoy numerical robustness in the state feedback case; here, we show that numerical robustness of the four-block SLP controller requires case-by-case analysis even the plant is open-loop stable.

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arXiv2021

Distributed neural network control with dependability guarantees: a compositional port-Hamiltonian approach

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.

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arXiv2021

Sample Complexity of Linear Quadratic Gaussian (LQG) Control for Output Feedback Systems

This paper studies a class of partially observed Linear Quadratic Gaussian (LQG) problems with unknown dynamics. We establish an end-to-end sample complexity bound on learning a robust LQG controller for open-loop stable plants. This is achieved using a robust synthesis procedure, where we first estimate a model from a single input-output trajectory of finite length, identify an H-infinity bound on the estimation error, and then design a robust controller using the estimated model and its quantified uncertainty. Our synthesis procedure leverages a recent control tool called Input-Output Parameterization (IOP) that enables robust controller design using convex optimization. For open-loop stable systems, we prove that the LQG performance degrades linearly with respect to the model estimation error using the proposed synthesis procedure. Despite the hidden states in the LQG problem, the achieved scaling matches previous results on learning Linear Quadratic Regulator (LQR) controllers with full state observations.

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arXiv2021

A Behavioral Input-Output Parametrization of Control Policies with Suboptimality Guarantees

Recent work in data-driven control has revived behavioral theory to perform a variety of complex control tasks, by directly plugging libraries of past input-output trajectories into optimal control problems. Despite recent advances, a key aspect remains unclear: how and to what extent do noise-corrupted data impact control performance? In this work, we provide a quantitative answer to this question. We formulate a Behavioral version of the Input-Output Parametrization (BIOP) for the optimal predictive control of unknown systems using output-feedback dynamic control policies. The main advantages of the proposed framework are that 1) the state-space parameters and the initial state need not be specified for controller synthesis, 2) it can be used in combination with state-of-the-art impulse response estimators, and 3) it allows to recover suboptimality results on learning the Linear Quadratic Gaussian (LQG) controller, therefore revealing, in a quantitative way, how the level of noise in the data affects the performance of behavioral methods. Specifically, it is shown that the performance degrades linearly with the prediction error of the behavioral model. We conclude the paper with numerical experiments to validate our results.

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arXiv2020

An Input-Output Parametrization of Stabilizing Controllers: amidst Youla and System Level Synthesis

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.

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arXiv2020

Sparsity Invariance for Convex Design of Distributed Controllers

We address the problem of designing optimal linear time-invariant (LTI) sparse controllers for LTI systems, which corresponds to minimizing a norm of the closed-loop system subject to sparsity constraints on the controller structure. This problem is NP-hard in general and motivates the development of tractable approximations. We characterize a class of convex restrictions based on a new notion of Sparsity Invariance (SI). The underlying idea of SI is to design sparsity patterns for transfer matrices Y(s) and X(s) such that any corresponding controller K(s)=Y(s)X(s)^-1 exhibits the desired sparsity pattern. For sparsity constraints, the approach of SI goes beyond the notion of Quadratic Invariance (QI): 1) the SI approach always yields a convex restriction; 2) the solution via the SI approach is guaranteed to be globally optimal when QI holds and performs at least as well as considering a nearest QI subset. Moreover, the notion of SI naturally applies to designing structured static controllers, while QI is not utilizable. Numerical examples show that even for non-QI cases, SI can recover solutions that are 1) globally optimal and 2) strictly more performing than previous methods.

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arXiv2020

Learning the Globally Optimal Distributed LQ Regulator

We study model-free learning methods for the output-feedback Linear Quadratic (LQ) control problem in finite-horizon subject to subspace constraints on the control policy. Subspace constraints naturally arise in the field of distributed control and present a significant challenge in the sense that standard model-based optimization and learning leads to intractable numerical programs in general. Building upon recent results in zeroth-order optimization, we establish model-free sample-complexity bounds for the class of distributed LQ problems where a local gradient dominance constant exists on any sublevel set of the cost function. %which admit a local gradient dominance constant valid on the sublevel set of the cost function. We prove that a fundamental class of distributed control problems - commonly referred to as Quadratically Invariant (QI) problems - as well as others possess this property. To the best of our knowledge, our result is the first sample-complexity bound guarantee on learning globally optimal distributed output-feedback control policies.

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arXiv2019

First Order Methods For Globally Optimal Distributed Controllers Beyond Quadratic Invariance

We study the distributed Linear Quadratic Gaussian (LQG) control problem in discrete-time and finite-horizon, where the controller depends linearly on the history of the outputs and it is required to lie in a given subspace, e.g. to possess a certain sparsity pattern. It is well-known that this problem can be solved with convex programming within the Youla domain if and only if a condition known as Quadratic Invariance (QI) holds. In this paper, we first show that given QI sparsity constraints, one can directly descend the gradient of the cost function within the domain of output-feedback controllers and converge to a global optimum. Note that convergence is guaranteed despite non-convexity of the cost function. Second, we characterize a class of Uniquely Stationary (US) problems, for which first-order methods are guaranteed to converge to a global optimum. We show that the class of US problems is strictly larger than that of strongly QI problems and that it is not included in that of QI problems. We refer to Figure 1 for details. Finally, we propose a tractable test for the US property.

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arXiv2019

On the Equivalence of Youla, System-level and Input-output Parameterizations

A convex parameterization of internally stabilizing controllers is fundamental for many controller synthesis procedures. The celebrated Youla parameterization relies on a doubly-coprime factorization of the system, while the recent system-level and input-output characterizations require no doubly-coprime factorization but a set of equality constraints for achievable closed-loop responses. In this paper, we present explicit affine mappings among Youla, system-level and input-output parameterizations. Two direct implications of the affine mappings are 1) any convex problem in Youla, system level, or input-output parameters can be equivalently and convexly formulated in any other one of these frameworks, including the convex system-level synthesis (SLS); 2) the condition of quadratic invariance (QI) is sufficient and necessary for the classical distributed control problem to admit an equivalent convex reformulation in terms of Youla, system-level, or input-output parameters.

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arXiv2019

On Separable Quadratic Lyapunov Functions for Convex Design of Distributed Controllers

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.

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arXiv2019

Robust Distributed Control Beyond Quadratic Invariance

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.

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arXiv2019

Unified Approach to Convex Robust Distributed Control given Arbitrary Information Structures

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.

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arXiv2017

Gone with the Wind: Nonlinear Guidance for Small Fixed-Wing Aircrafts in Arbitrarily Strong Windfields

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.

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