arXiv subjects

Alberto Bemporad

Publications and source records attributed to Alberto Bemporad.

At least 55 records · Page 3Linked to original sources

arXiv2024

An L-BFGS-B approach for linear and nonlinear system identification under $\ell_1$ and group-Lasso regularization

In this paper, we propose a very efficient numerical method based on the L-BFGS-B algorithm for identifying linear and nonlinear discrete-time state-space models, possibly under $\ell_1$ and group-Lasso regularization for reducing model complexity. For the identification of linear models, we show that, compared to classical linear subspace methods, the approach often provides better results, is much more general in terms of the loss and regularization terms used (such as penalties for enforcing system stability), and is also more stable from a numerical point of view. The proposed method not only enriches the existing set of linear system identification tools but can also be applied to identifying a very broad class of parametric nonlinear state-space models, including recurrent neural networks. We illustrate the approach on synthetic and experimental datasets and apply it to solve a challenging industrial robot benchmark for nonlinear multi-input/multi-output system identification. A Python implementation of the proposed identification method is available in the package jax-sysid, available at https://github.com/bemporad/jax-sysid.

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arXiv2024

An active learning method for solving competitive multi-agent decision-making and control problems

To identify a stationary action profile for a population of competitive agents, each executing private strategies, we introduce a novel active-learning scheme where a centralized external observer (or entity) can probe the agents' reactions and recursively update simple local parametric estimates of the action-reaction mappings. Under very general working assumptions (not even assuming that a stationary profile exists), sufficient conditions are established to assess the asymptotic properties of the proposed active learning methodology so that, if the parameters characterizing the action-reaction mappings converge, a stationary action profile is achieved. Such conditions hence act also as certificates for the existence of such a profile. Extensive numerical simulations involving typical competitive multi-agent control and decision-making problems illustrate the practical effectiveness of the proposed learning-based approach.

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arXiv2024

Parameter Dependent Robust Control Invariant Sets for LPV Systems with Bounded Parameter Variation Rate

Real-time measurements of the scheduling parameter of linear parameter-varying (LPV) systems enables the synthesis of robust control invariant (RCI) sets and parameter dependent controllers inducing invariance. We present a method to synthesize parameter-dependent robust control invariant (PD-RCI) sets for LPV systems with bounded parameter variation, in which invariance is induced using PD-vertex control laws. The PD-RCI sets are parameterized as configuration-constrained polytopes that admit a joint parameterization of their facets and vertices. The proposed sets and associated control laws are computed by solving a single semidefinite programing (SDP) problem. Through numerical examples, we demonstrate that the proposed method outperforms state-of-the-art methods for synthesizing PD-RCI sets, both with respect to conservativeness and computational load.

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arXiv2024

Configuration-Constrained Tube MPC for Tracking

This paper proposes a novel tube-based Model Predictive Control (MPC) framework for tracking varying setpoint references with linear systems subject to additive and multiplicative uncertainties. The MPC controllers designed using this framework exhibit recursively feasible for changing references, and robust asymptotic stability for piecewise constant references. The framework leverages configuration-constrained polytopes to parameterize the tubes, offering flexibility to optimize their shape. The efficacy of the approach is demonstrated through two numerical examples. The first example illustrates the theoretical results, and the second uses the framework to design a lane-change controller for an autonomous vehicle.

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arXiv2024

A Long-Short-Term Mixed-Integer Formulation for Highway Lane Change Planning

This work considers the problem of optimal lane changing in a structured multi-agent road environment. A novel motion planning algorithm that can capture long-horizon dependencies as well as short-horizon dynamics is presented. Pivotal to our approach is a geometric approximation of the long-horizon combinatorial transition problem which we formulate in the continuous time-space domain. Moreover, a discrete-time formulation of a short-horizon optimal motion planning problem is formulated and combined with the long-horizon planner. Both individual problems, as well as their combination, are formulated as MIQP and solved in real-time by using state-of-the-art solvers. We show how the presented algorithm outperforms two other state-of-the-art motion planning algorithms in closed-loop performance and computation time in lane changing problems. Evaluations are performed using the traffic simulator SUMO, a custom low-level tracking model predictive controller, and high-fidelity vehicle models and scenarios, provided by the CommonRoad environment.

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arXiv2024

Regularized Gauss-Newton for Optimizing Overparameterized Neural Networks

The generalized Gauss-Newton (GGN) optimization method incorporates curvature estimates into its solution steps, and provides a good approximation to the Newton method for large-scale optimization problems. GGN has been found particularly interesting for practical training of deep neural networks, not only for its impressive convergence speed, but also for its close relation with neural tangent kernel regression, which is central to recent studies that aim to understand the optimization and generalization properties of neural networks. This work studies a GGN method for optimizing a two-layer neural network with explicit regularization. In particular, we consider a class of generalized self-concordant (GSC) functions that provide smooth approximations to commonly-used penalty terms in the objective function of the optimization problem. This approach provides an adaptive learning rate selection technique that requires little to no tuning for optimal performance. We study the convergence of the two-layer neural network, considered to be overparameterized, in the optimization loop of the resulting GGN method for a given scaling of the network parameters. Our numerical experiments highlight specific aspects of GSC regularization that help to improve generalization of the optimized neural network. The code to reproduce the experimental results is available at https://github.com/adeyemiadeoye/ggn-score-nn.

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arXiv2023

Certification of the proximal gradient method under fixed-point arithmetic for box-constrained QP problems

In safety-critical applications that rely on the solution of an optimization problem, the certification of the optimization algorithm is of vital importance. Certification and suboptimality results are available for a wide range of optimization algorithms. However, a typical underlying assumption is that the operations performed by the algorithm are exact, i.e., that there is no numerical error during the mathematical operations, which is hardly a valid assumption in a real hardware implementation. This is particularly true in the case of fixed-point hardware, where computational inaccuracies are not uncommon. This article presents a certification procedure for the proximal gradient method for box-constrained QP problems implemented in fixed-point arithmetic. The procedure provides a method to select the minimal fractional precision required to obtain a certain suboptimality bound, indicating the maximum number of iterations of the optimization method required to obtain it. The procedure makes use of formal verification methods to provide arbitrarily tight bounds on the suboptimality guarantee. We apply the proposed certification procedure on the implementation of a non-trivial model predictive controller on 32-bit fixed-point hardware.

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arXiv2023

Data-Driven Synthesis of Configuration-Constrained Robust Invariant Sets for Linear Parameter-Varying Systems

We present a data-driven method to synthesize robust control invariant (RCI) sets for linear parameter-varying (LPV) systems subject to unknown but bounded disturbances. A finite-length data set consisting of state, input, and scheduling signal measurements is used to compute an RCI set and invariance-inducing controller, without identifying an LPV model of the system. We parameterize the RCI set as a configuration-constrained polytope whose facets have a fixed orientation and variable offset. This allows us to define the vertices of the polytopic set in terms of its offset. By exploiting this property, an RCI set and associated vertex control inputs are computed by solving a single linear programming (LP) problem, formulated based on a data-based invariance condition and system constraints. We illustrate the effectiveness of our approach via two numerical examples. The proposed method can generate RCI sets that are of comparable size to those obtained by a model-based method in which exact knowledge of the system matrices is assumed. We show that RCI sets can be synthesized even with a relatively small number of data samples, if the gathered data satisfy certain excitation conditions.

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arXiv2023

Computation of safe disturbance sets using implicit RPI sets

Given a stable linear time-invariant (LTI) system subject to output constraints, we present a method to compute a set of disturbances such that the reachable set of outputs matches as closely as possible the output constraint set, while being included in it. This problem finds application in several control design problems, such as the development of hierarchical control loops, decentralized control, supervisory control, robustness-verification, etc. We first characterize the set of disturbance sets satisfying the output constraint inclusion using corresponding minimal robust positive invariant (mRPI) sets, following which we formulate an optimization problem that minimizes the distance between the reachable output set and the output constraint set. We tackle the optimization problem using an implicit RPI set approach that provides a priori approximation error guarantees, and adopt a novel disturbance set parameterization that permits the encoding of the set of feasible disturbance sets as a polyhedron. Through extensive numerical examples, we demonstrate that the proposed approach computes disturbance sets with reduced conservativeness improved computational efficiency than state-of-the-art methods.

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arXiv2023

Learning Critical Scenarios in Feedback Control Systems for Automated Driving

Testing is essential for verifying and validating control designs, especially in safety-critical applications. In particular, the control system governing an automated driving vehicle must be proven reliable enough for its acceptance on the market. Recently, much research has focused on scenario-based methods. However, the number of possible driving scenarios to test is in principle infinite. In this paper, we formalize a learning-based optimization framework to generate corner test-cases, where we take into account the operational design domain. We examine the approach on the case of a feedback control system for automated driving, for which we suggest the design of the objective function expressing the criticality of scenarios. Numerical tests on two logical scenarios of the case study demonstrate that the approach can identify critical scenarios within a limited number of closed-loop experiments.

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arXiv2023

SCORE: Approximating Curvature Information under Self-Concordant Regularization

Optimization problems that include regularization functions in their objectives are regularly solved in many applications. When one seeks second-order methods for such problems, it may be desirable to exploit specific properties of some of these regularization functions when accounting for curvature information in the solution steps to speed up convergence. In this paper, we propose the SCORE (self-concordant regularization) framework for unconstrained minimization problems which incorporates second-order information in the Newton-decrement framework for convex optimization. We propose the generalized Gauss-Newton with Self-Concordant Regularization (GGN-SCORE) algorithm that updates the minimization variables each time it receives a new input batch. The proposed algorithm exploits the structure of the second-order information in the Hessian matrix, thereby reducing computational overhead. GGN-SCORE demonstrates how to speed up convergence while also improving model generalization for problems that involve regularized minimization under the proposed SCORE framework. Numerical experiments show the efficiency of our method and its fast convergence, which compare favorably against baseline first-order and quasi-Newton methods. Additional experiments involving non-convex (overparameterized) neural network training problems show that the proposed method is promising for non-convex optimization.

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arXiv2023

Counter-example guided inductive synthesis of control Lyapunov functions for uncertain systems

We propose a counter-example guided inductive synthesis (CEGIS) scheme for the design of control Lyapunov functions and associated state-feedback controllers for linear systems affected by parametric uncertainty with arbitrary shape. In the CEGIS framework, a learner iteratively proposes a candidate control Lyapunov function and a tailored controller by solving a linear matrix inequality (LMI) feasibility problem, while a verifier either falsifies the current candidate by producing a counter-example to be considered at the next iteration, or it certifies that the tentative control Lyapunov function actually enjoys such feature. We investigate the Lipschitz continuity of the objective function of the global optimization problem solved by the verifier, which is key to establish the convergence of our method in a finite number of iterations. Numerical simulations confirm the effectiveness of the proposed approach.

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arXiv2023

Specification-Guided Critical Scenario Identification for Automated Driving

To test automated driving systems, we present a case study for finding critical scenarios in driving environments guided by formal specifications. To that aim, we devise a framework for critical scenario identification, which we base on open-source libraries that combine scenario specification, testing, formal methods, and optimization.

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arXiv2022

Active Learning for Regression by Inverse Distance Weighting

This paper proposes an active learning (AL) algorithm to solve regression problems based on inverse-distance weighting functions for selecting the feature vectors to query. The algorithm has the following features: (i) supports both pool-based and population-based sampling; (ii) is not tailored to a particular class of predictors; (iii) can handle known and unknown constraints on the queryable feature vectors; and (iv) can run either sequentially, or in batch mode, depending on how often the predictor is retrained. The potentials of the method are shown in numerical tests on illustrative synthetic problems and real-world datasets. An implementation of the algorithm, which we call IDEAL (Inverse-Distance based Exploration for Active Learning), is available at http://cse.lab.imtlucca.it/~bemporad/ideal.

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arXiv2022

Efficiently solving the harmonic model predictive control formulation

Harmonic model predictive control (HMPC) is a model predictive control (MPC) formulation which displays several benefits over other MPC formulations, especially when using a small prediction horizon. These benefits, however, come at the expense of an optimization problem that is no longer the typical quadratic programming problem derived from most linear MPC formulations due to the inclusion of a particular class of second order cone constraints. This article presents a method for efficiently dealing with these constraints in operator splitting methods, leading to a computation time for solving HMPC in line with state of the art solvers for linear MPC. We show how to apply this result to the alternating direction method of multipliers algorithm, presenting a solver which we compare against other solvers from the literature, including solvers for other linear MPC formulations. The results show that the proposed solver, and by extension the HMPC formulation, is suitable for its implementation in embedded systems.

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arXiv2022

Recurrent Neural Network Training with Convex Loss and Regularization Functions by Extended Kalman Filtering

This paper investigates the use of extended Kalman filtering to train recurrent neural networks with rather general convex loss functions and regularization terms on the network parameters, including $\ell_1$-regularization. We show that the learning method is competitive with respect to stochastic gradient descent in a nonlinear system identification benchmark and in training a linear system with binary outputs. We also explore the use of the algorithm in data-driven nonlinear model predictive control and its relation with disturbance models for offset-free closed-loop tracking.

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arXiv2022

Training Recurrent Neural Networks by Sequential Least Squares and the Alternating Direction Method of Multipliers

This paper proposes a novel algorithm for training recurrent neural network models of nonlinear dynamical systems from an input/output training dataset. Arbitrary convex and twice-differentiable loss functions and regularization terms are handled by sequential least squares and either a line-search (LS) or a trust-region method of Levenberg-Marquardt (LM) type for ensuring convergence. In addition, to handle non-smooth regularization terms such as $\ell_1$, $\ell_0$, and group-Lasso regularizers, as well as to impose possibly non-convex constraints such as integer and mixed-integer constraints, we combine sequential least squares with the alternating direction method of multipliers (ADMM). We call the resulting algorithm NAILS (nonconvex ADMM iterations and least squares) in the case line search (LS) is used, or NAILM if a trust-region method (LM) is employed instead. The training method, which is also applicable to feedforward neural networks as a special case, is tested in three nonlinear system identification problems.

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arXiv2022

A Simple and Fast Coordinate-Descent Augmented-Lagrangian Solver for Model Predictive Control

This paper proposes a novel Coordinate-Descent Augmented-Lagrangian (CDAL) solver for linear, possibly parameter-varying, model predictive control (MPC) problems. At each iteration, an augmented Lagrangian (AL) subproblem is solved by coordinate descent (CD), exploiting the structure of the MPC problem. The CDAL solver enjoys three main properties: (i) it is construction-free, in that it avoids explicitly constructing the quadratic programming (QP) problem associated with MPC; (ii) is matrix-free, as it avoids multiplications and factorizations of matrices; and (iii) is library-free, as it can be simply coded without any library dependency, 90-line of C-code in our implementation. To favor convergence speed, CDAL employs a reverse cyclic rule for the CD method, the accelerated Nesterov's scheme for updating the dual variables, a simple diagonal preconditioner, and an efficient coupling scheme between the CD and AL methods. We show that CDAL competes with other state-of-the-art methods, both in case of unstable linear time-invariant and linear parameter-varying prediction models.

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