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Dynamical models identified from data are frequently employed in control system design. However, decoupling system identification from controller synthesis can result in situations where no suitable controller exists after a model has been identified. In this work, we introduce a novel control-oriented regularization in the identification procedure to ensure the existence of a controller that can enforce constraints on system variables robustly. The combined identification algorithm includes: (i) the concurrent learning of an uncertain model and a nominal model using an observer; (ii) a regularization term on the model parameters defined as the size of the largest robust control invariant set for the uncertain model. To make the learning problem tractable, we consider nonlinear models in quasi Linear Parameter-Varying (qLPV) form, utilizing a novel scheduling function parameterization that facilitates the derivation of an associated uncertain linear model. The robust control invariant set is represented as a polytope, and we adopt novel results from polytope geometry to derive the regularization function as the optimal value of a convex quadratic program. Additionally, we present new model-reduction approaches that exploit the chosen model structure. Numerical examples on classical identification benchmarks demonstrate the efficacy of our approach. A simple control scheme is also derived to provide an example of data-driven control of a constrained nonlinear system., Comment: 12 Pages, Implementation found on https://github.com/samku/Concur-qLPV-RCI

We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an $\epsilon$-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

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., Comment: 14 pages, paper presented to the L-CSS/CDC combined submission

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.

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., Comment: 27 pages, 9 figures, 2 tables

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., Comment: 23 pages, 4 figures

One of the most important objectives of solar physics is the physical understanding of the solar atmosphere, the structure of which is also described in terms of the density (N) and temperature (T) distributions of the atmospheric matter. Several multi-frequency analyses show that the characteristics of these distributions are still debated, especially for the outer coronal emission. We aim to constrain the T and N distributions of the solar atmosphere through observations in the centimetric radio domain. We employ single-dish observations from two of the INAF radio telescopes at the K-band frequencies (18 - 26 GHz). We investigate the origin of the significant brightness temperature ($T_B$) level that we detected up to the upper corona ($\sim 800$ Mm of altitude with respect to the photospheric solar surface). To probe the physical origin of the atmospheric emission and to constrain instrumental biases, we reproduced the solar signal by convolving specific 2D antenna beam models. The analysis of the solar atmosphere is performed by adopting a physical model that assumes the thermal bremsstrahlung as the emission mechanism, with specific T and N distributions. The modelled $T_B$ profiles are compared with those observed by averaging solar maps obtained during the minimum of solar activity (2018 - 2020). The T and N distributions are compatible (within $25\%$ of uncertainty) with the model up to $\sim 60$ Mm and $\sim 100$ Mm of altitude, respectively. The analysis of the role of the antenna beam pattern on our solar maps proves the physical nature of the atmospheric emission in our images up to the coronal tails seen in our $T_B$ profiles. The challenging analysis of the coronal radio emission at higher altitudes, together with the data from satellite instruments will require further multi-frequency measurements., Comment: 12 pages, 8 figures, 5 tables, accepted by A&A; v1

The Sun is an extraordinary workbench, from which several fundamental astronomical parameters can be measured with high precision. Among these parameters, the solar radius $R_{\odot}$ plays an important role in several aspects, such as in evolutionary models. Despite the efforts in obtaining accurate measurements of $R_{\odot}$, the subject is still debated and measurements are puzzling and/or lacking in many frequency ranges. We aimed to determine the mean, equatorial, and polar radii of the Sun ($R_c$, $R_{eq}$, and $R_{pol}$) in the frequency range 18.1 - 26.1 GHz. We employed single-dish observations from the newly-appointed Medicina "Gavril Grueff" Radio Telescope and the Sardinia Radio Telescope (SRT) throughout 5 years, from 2018 to mid-2023, in the framework of the SunDish project for solar monitoring. Two methods to calculate the radius at radio frequencies are considered and compared. To assess the quality of our radius determinations, we also analysed the possible degrading effects of the antenna beam pattern on our solar maps, using two 2D-models. We carried out a correlation analysis with the evolution of the solar cycle through the calculation of Pearson's correlation coefficient $\rho$. We obtained several values for the solar radius - ranging between 959 and 994 arcsec - and $\rho$, with typical errors of a few arcsec. Our $R_{\odot}$ measurements, consistent with values reported in literature, suggest a weak prolatness of the solar limb ($R_{eq}$ > $R_{pol}$), although $R_{eq}$ and $R_{pol}$ are statistically compatible within 3$\sigma$ errors. The correlation analysis using the solar images from Grueff shows (1) a positive correlation between the solar activity and the temporal variation of $R_c$ (and $R_{eq}$) at all observing frequencies, and (2) a weak anti-correlation between the temporal variation of $R_{pol}$ and the solar activity at 25.8 GHz., Comment: 18 pages, 12 figures, 6 tables, accepted by A&A; v1

This work presents a nonlinear MPC framework that guarantees asymptotic offset-free tracking of generic reference trajectories by learning a nonlinear disturbance model, which compensates for input disturbances and model-plant mismatch. Our approach generalizes the well-established method of using an observer to estimate a constant disturbance to allow tracking constant setpoints with zero steady-state error. In this paper, the disturbance model is generalized to a nonlinear static function of the plant's state and command input, learned online, so as to perfectly track time-varying reference trajectories under certain assumptions on the model and provided that future reference samples are available. We compare our approach with the classical constant disturbance model in numerical simulations, showing its superiority., Comment: (8 pages, 4 figures)

Metis, the coronagraph on board Solar Orbiter, provides for the first time coronagraphic imaging in the ultraviolet HI Ly-alpha line and, simultaneously, in polarized visible light, thus providing a host of information on the properties of CMEs and solar eruptions like their overall dynamics, time evolution, mass content, and outflow propagation velocity in the expanding corona. We analyzed in this work six CMEs observed by Metis between April and October 2021, which are characterized by a very strong HI Ly-alpha emission. We studied in particular the morphology, kinematics, and the temporal and radial evolution of the emission of such events, focusing on the brightest UV features. The kinematics of the eruptive events under consideration were studied by determining the height-time profiles of the brightest parts on the Metis plane of the sky. Furthermore, the 3D position in the heliosphere of the CMEs were determined by employing co-temporal images from two other coronagraphs: LASCO/C2 onboard SOHO, and COR2 onboard STEREO-A. Finally, the radiometrically calibrated Metis images of the bright UV features were analyzed to provide estimates of their volume and density. From the kinematics and radiometric analyses, we obtained indications of the temperatures of the bright UV cores of these events. The analysis of these strong UV-emitting features associated with coronal mass ejections demonstrates the capabilities of the current constellation of space coronagraphs, Metis, LASCO/C2, and COR2, in providing a complete characterization of the structure and dynamics of eruptive events in their propagation phase from their inception up to several solar radii. Furthermore, we show how the unique capabilities of the Metis instrument to observe these events in both HI Ly-alpha line and polarized VL radiation allow plasma diagnostics on the thermal state of these events., Comment: 26 pages, 26 figures, 2 appendices

Harmonic model predictive control (HMPC) is a recent model predictive control (MPC) formulation for tracking piece-wise constant references that includes a parameterized artificial harmonic reference as a decision variable, resulting in an increased performance and domain of attraction with respect to other MPC formulations. This article presents an extension of the HMPC formulation to track periodic harmonic references and discusses its use to track arbitrary references. The proposed formulation inherits the benefits of its predecessor, namely its good performance and large domain of attraction when using small prediction horizons, and that the complexity of its optimization problem does not depend on the period of the periodic reference. We show closed-loop results discussing its performance and comparing it to other MPC formulations., Comment: (11 pages, 14 figures)

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., Comment: 7 pages, 4 figures, 2 tables

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., Comment: 16 pages, 6 figures

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., Comment: 8 pages, 6 figures

We introduce a notion of self-concordant smoothing for minimizing the sum of two convex functions, one of which is smooth and the other may be nonsmooth. The key highlight of our approach is in a natural property of the resulting problem's structure which provides us with a variable-metric selection method and a step-length selection rule particularly suitable for proximal Newton-type algorithms. In addition, we efficiently handle specific structures promoted by the nonsmooth function, such as $\ell_1$-regularization and group-lasso penalties. We prove the convergence of two resulting algorithms: Prox-N-SCORE, a proximal Newton algorithm and Prox-GGN-SCORE, a proximal generalized Gauss-Newton algorithm. The Prox-GGN-SCORE algorithm highlights an important approximation procedure which helps to significantly reduce most of the computational overhead associated with the inverse Hessian. This approximation is essentially useful for overparameterized machine learning models and in the mini-batch settings. Numerical examples on both synthetic and real datasets demonstrate the efficiency of our approach and its superiority over existing approaches. A Julia package implementing the proposed algorithms is available at https://github.com/adeyemiadeoye/SelfConcordantSmoothOptimization.jl., Comment: 39 pages, 7 figures, 3 tables

Coronal Mass Ejections (CMEs) correspond to dramatic expulsions of plasma and magnetic field from the solar corona into the heliosphere. CMEs are scientifically relevant because they are involved in the physical mechanisms characterizing the active Sun. However, more recently CMEs have attracted attention for their impact on space weather, as they are correlated to geomagnetic storms and may induce the generation of Solar Energetic Particles streams. In this space weather framework, the present paper introduces a physics-driven artificial intelligence (AI) approach to the prediction of CMEs travel time, in which the deterministic drag-based model is exploited to improve the training phase of a cascade of two neural networks fed with both remote sensing and in-situ data. This study shows that the use of physical information in the AI architecture significantly improves both the accuracy and the robustness of the travel time prediction.

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., Comment: 8 pages

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.

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.

The coronal magnetic field, despite its overwhelming importance to the physics and dynamics of the corona, has only rarely been measured. Here, the electron density maps derived from images acquired during the total solar eclipse of August 21st, 2017 are employed to demonstrate a new technique to measure the coronal magnetic fields. The strength of the coronal magnetic fields is derived with a semiempirical formula relating the plasma magnetic energy density to the gravitational potential energy. The resulting values are compared with those provided by more advanced coronal field reconstruction methods based on MHD simulations of the whole corona starting from photospheric field measurements, finding a very good agreement. Other parameters such as the plasma-$\beta$ and Alfv\'en velocity are also derived and compared with those of MHD simulations. Moreover, the plane-of-sky (POS) orientation of the coronal magnetic fields is derived from the observed inclination of the coronal features in the filtered images, also finding a close agreement with magnetic field reconstructions. Hence, this work demonstrates for the first time that the 2D distribution of coronal electron densities measured during total solar eclipses is sufficient to provide the coronal magnetic field strengths and inclinations projected on the POS. These are among the main missing pieces of information that limited so far our understanding of physical phenomena going on in the solar corona., Comment: 20 pages, 11 figures

Optimization problems involving mixed variables (i.e., variables of numerical and categorical nature) can be challenging to solve, especially in the presence of mixed-variable constraints. Moreover, when the objective function is the result of a complicated simulation or experiment, it may be expensive-to-evaluate. This paper proposes a novel surrogate-based global optimization algorithm to solve linearly constrained mixed-variable problems up to medium size (around 100 variables after encoding). The proposed approach is based on constructing a piecewise affine surrogate of the objective function over feasible samples. We assume the objective function is black-box and expensive-to-evaluate, while the linear constraints are quantifiable, unrelaxable, a priori known, and are cheap to evaluate. We introduce two types of exploration functions to efficiently search the feasible domain via mixed-integer linear programming solvers. We also provide a preference-based version of the algorithm designed for situations where only pairwise comparisons between samples can be acquired, while the underlying objective function to minimize remains unquantified. The two algorithms are evaluated on several unconstrained and constrained mixed-variable benchmark problems. The results show that, within a small number of required experiments/simulations, the proposed algorithms can often achieve better or comparable results than other existing methods., Comment: code available at https://github.com/mjzhu-p/PWAS

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., Comment: Python package available at https://github.com/bemporad/gnep-learn

Type U radio bursts are impulsive coherent radio emissions produced by the Sun that indicate the presence of subrelativistic electron beams propagating along magnetic loops in the solar corona. In this work, we present the analysis of a type U radio burst that was exceptionally imaged on 2011 March 22 by the Nan\c{c}ay Radioheliograph (NRH) at three different frequencies (298.7, 327.0, and 360.8 MHz). Using a novel modelling approach, we show for the first time that the use of high-resolution radio heliograph images of type U radio bursts can be sufficient to both accurately reconstruct the 3D morphology of coronal loops (without recurring to triangulation techniques) and to fully constrain their physical parameters. At the same time, we can obtain unique information on the dynamics of the accelerated electron beams, which provides important clues as to the plasma mechanisms involved in their acceleration and as to why type U radio bursts are not observed as frequently as type III radio bursts. We finally present plausible explanations for a problematic aspect related to the apparent lack of association between the modeled loop as inferred from radio images and the extreme-ultraviolet (EUV) structures observed from space in the same coronal region, Comment: A&A, in press (8 pages, 6 figures)

Stealthy Coronal Mass Ejections (CMEs), lacking low coronal signatures, may result in significant geomagnetic storms. However, the mechanism of stealthy CMEs is still highly debated. In this work, we investigate whether there are differences between the stealthy and standard CMEs in terms of their dynamic behaviors. Seven stealthy and eight standard CMEs with slow speeds are selected. We calculate two-dimensional speed distributions of CMEs based on the cross-correlation method, rather than the unidimensional speed, and further obtain more accurate distributions and evolution of CME mechanical energies. Then we derive the CME driving powers and correlate them with CME parameters (total mass, average speed, and acceleration) for standard and stealthy CMEs. Besides, we study the forces that drive CMEs, namely, the Lorentz force, gravitational force, and drag force due to the ambient solar wind near the Sun. The results reveal that both the standard and stealthy CMEs are propelled by the combined action of those forces in the inner corona. The drag force and gravitational force are comparable with the Lorentz force. However, the impact of the drag and Lorentz forces on the global evolution of the stealthy CMEs is significantly weaker than that of the standard CMEs., Comment: 14 pages, 7 figures, 1 table. Accepted by ApJ

The determination of the chemical composition of interplanetary coronal mass ejection (ICME) plasma is an open issue. More specifically, it is not yet fully understood how remote sensing observations of the solar corona plasma during solar disturbances evolve into plasma properties measured in situ away from the Sun. The ambient conditions of the background interplanetary plasma are important for space weather because they influence the evolutions, arrival times, and geo-effectiveness of the disturbances. The Reverse In situ and MHD APproach (RIMAP) is a technique to reconstruct the heliosphere on the ecliptic plane (including the magnetic Parker spiral) directly from in situ measurements acquired at 1 AU. It combines analytical and numerical approaches, preserving the small-scale longitudinal variability of the wind flow lines. In this work, we use RIMAP to test the interaction of an ICME with the interplanetary medium. We model the propagation of a homogeneous non-magnetised (i.e. with no internal flux rope) cloud starting at 800 km s-1 at 0.1 AU out to 1.1 AU. Our 3D magnetohydrodynamics (MHD) simulation made with the PLUTO MHD code shows the formation of a compression front ahead of the ICME, continuously driven by the cloud expansion. Using a passive tracer, we find that the initial ICME material does not fragment behind the front during its propagation, and we quantify the mixing of the propagating plasma cloud with the ambient solar wind plasma, which can be detected at 1 AU., Comment: Movie available at https://www.aanda.org/articles/aa/olm/2021/10/aa41892-21/aa41892-21.html

As a key feature, NASA's Parker Solar Probe (PSP) and ESA-NASA's Solar Orbiter (SO) missions cooperate to trace solar wind and transients from their sources on the Sun to the inner interplanetary space. The goal of this work is to accurately reconstruct the interplanetary Parker spiral and the connection between coronal features observed remotely by the Metis coronagraph on-board SO and those detected in situ by PSP at the time of the first PSP-SO quadrature of January 2021. We use the Reverse In-situ and MHD Approach (RIMAP), a hybrid analytical-numerical method performing data-driven reconstructions of the Parker spiral. RIMAP solves the MHD equations on the equatorial plane with the PLUTO code, using the measurements collected by PSP between 0.1 and 0.2 AU as boundary conditions. Our reconstruction connects density and wind speed measurements provided by Metis (3-6 solar radii) to those acquired by PSP (21.5 solar radii) along a single streamline. The capability of our MHD model to connect the inner corona observed by Metis and the super Alfv\'enic wind measured by PSP, not only confirms the research pathways provided by multi-spacecraft observations, but also the validity and accuracy of RIMAP reconstructions as a possible test bench to verify models of transient phenomena propagating across the heliosphere, such as coronal mass ejections, solar energetic particles and solar wind switchbacks., Comment: Astronomy & Astrophysics, Solar Orbiter First Results (Nominal Mission Phase), (in press) DOI: 10.1051/0004-6361/202244535

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.

This paper proposes a construction-free algorithm for solving linear MPC problems based on autoregressive with exogenous terms (ARX) input-output models. The solution algorithm relies on a coordinate-descent augmented Lagrangian (CDAL) method previously proposed by the authors, which we adapt here to exploit the special structure of ARX-based MPC. The CDAL-ARX algorithm enjoys the construction-free feature, in that it avoids explicitly constructing the quadratic programming (QP) problem associated with MPC, which would eliminate construction cost when the ARX model changes/adapts online. For example, the ARX model parameters are dependent on linear parameter-varying (LPV) scheduling signals, or recursively adapted from streaming input-output data with cheap computation cost, which make the ARX model widely used in adaptive control. Moreover, the implementation of the resulting CDAL-ARX algorithm is matrix-free and library-free, and hence amenable for deployment in industrial embedded platforms. We show the efficiency of CDAL-ARX in two numerical examples, also in comparison with MPC implementations based on other general-purpose quadratic programming solvers., Comment: 6 pages, 1 figure

In this work we performed a polarimetric study of a fast and wide coronal mass ejection (CME) observed on 12 July 2012 by the COR1 and COR2 instruments onboard Solar TErrestrial RElations Observatory (STEREO) mission. The CME source region was an X1.4 flare located at approximately S15W01 on the solar disk as observed from the Earth's perspective. The position of the CME as derived from the 3D Graduated Cylindrical Shell (GCS) reconstruction method was at around S18W00 at 2.5 solar radii and S07W00 at 5.7 solar radii, meaning that the CME was deflected towards the Equator while propagating outward in the corona. The projected speed of the leading edge of the CME also evolved from around 200 km s$^{-1}$ in the lower corona to around 1000 km s$^{-1}$ in the COR2 field of view. The degree of polarisation of the CME is around 65 % but it can go as high as 80 % in some CME regions. The CME showed deviation of the polarisation angle from the tangential in the range of 10$^\circ$ - 15$^\circ$ (or more). Our analysis showed that this is mostly due to the fact that the sequence of three polarised images from where the polarised parameters are derived is not taken simultaneously, but at a difference of few seconds in time. In this interval of time, the CME is moving by at least two pixels in the FOV of the instruments and this displacement results in uncertainties in the polarisation parameters (degree of polarisation, polarisation angle, etc.). We propose some steps forward to improve the derivation of the polarisation. This study is important for analysing the future data from instruments with polarisation capabilities.

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., Comment: 26 pages, 11 figures. Submitted for publication

On February 12, 2021 two subsequent eruptions occurred above the West limb, as seen along the Sun-Earth line. The first event was a typical slow Coronal Mass Ejection (CME), followed $\sim 7$ hours later by a smaller and collimated prominence eruption, originating Southward with respect to the CME, followed by a plasma blob. These events were observed not only by SOHO and STEREO-A missions, but also by the suite of remote sensing instruments on-board Solar Orbiter (SolO). This work shows how data acquired by the Full Sun Imager (FSI), Metis coronagraph, and Heliospheric Imager (SoloHI) from the SolO perspective can be combined to study the eruptions and the different source regions. Moreover, we show how Metis data can be analyzed to provide new information about solar eruptions. Different 3D reconstruction methods were applied to the data acquired by different spacecraft including remote sensing instruments on-board SolO. Images acquired by both Metis channels in the Visible Light (VL) and H I Lyman$-\alpha$ line (UV) were combined to derive physical information on the expanding plasma. The polarization ratio technique was also applied for the first time to the Metis images acquired in the VL channel. The two eruptions were followed in 3D from their source region to their expansion in the intermediate corona. Thanks to the combination of VL and UV Metis data, the formation of a post-CME Current Sheet (CS) was followed for the first time in the intermediate corona. The plasma temperature gradient across a post-CME blob propagating along the CS was also measured for the first time. Application of the polarization ratio technique to Metis data shows that, thanks to the combination of four different polarization measurements, the errors are reduced by $\sim 5-7$\%, thus better constraining the 3D distribution of plasma., Comment: 15 pages, 14 figures, accepted for publication on A&A

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., Comment: Accepted version of the article published in IEEE Transactions on Automatic Control (8 pages, 5 figures)

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., Comment: 23 pages, 4 figures. Submitted for publication

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., Comment: published in Computational Optimization and Applications 2023

Ultraviolet spectra of Comet 96/P Machholz were obtained during its 2002 perihelion with the UVCS instrument aboard the SOHO satellite. Emission from H I, C II, C III, and O I, is detected near the nucleus. The outgassing rate is in line with the value extrapolated from rates at larger distances from the Sun, and abundances of C and O are estimated. Reconstructed images show a nearly spherical cloud in H I Ly$\alpha$ emission and an ion tail seen in C III. Radiation pressure on the hydrogen atoms produces a modest distortion of the shape of the Ly$\alpha$ cloud as seen from SOHO, and it produces Doppler shifts up to 30 km s$^{-1}$ in the outer parts of the cloud. We estimate a ratio of C to H$_2$O similar to what is observed in other comets, so low carbon abundance does not account for the anomalously low C$_2$ and C$_3$ ratios to NH$_2$ observed at optical wavelengths.

This paper presents a method to identify an uncertain linear time-invariant (LTI) prediction model for tube-based Robust Model Predictive Control (RMPC). The uncertain model is determined from a given state-input dataset by formulating and solving a Semidefinite Programming problem (SDP), that also determines a static linear feedback gain and corresponding invariant sets satisfying the inclusions required to guarantee recursive feasibility and stability of the RMPC scheme, while minimizing an identification criterion. As demonstrated through an example, the proposed concurrent approach provides less conservative invariant sets than a sequential approach.

Linear models with additive unknown-but-bounded input disturbances are extensively used to model uncertainty in robust control systems design. Typically, the disturbance set is either assumed to be known a priori or estimated from data through set-membership identification. However, the problem of computing a suitable input disturbance set in case the set of possible output values is assigned a priori has received relatively little attention. This problem arises in many contexts, such as in supervisory control, actuator design, decentralized control, and others. In this paper, we propose a method to compute input disturbance sets (and the corresponding set of states) such that the resulting set of outputs matches as closely as possible a given set of outputs, while additionally satisfying strict (inner or outer) inclusion constraints. We formulate the problem as an optimization problem by relying on the concept of robust invariance. The effectiveness of the approach is demonstrated in numerical examples that illustrate how to solve safe reference set and input-constraint set computation problems.

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., Comment: 21 pages, 3 figures, submitted for publication

This Letter addresses the first Solar Orbiter (SO) -- Parker Solar Probe (PSP) quadrature, occurring on January 18, 2021, to investigate the evolution of solar wind from the extended corona to the inner heliosphere. Assuming ballistic propagation, the same plasma volume observed remotely in corona at altitudes between 3.5 and 6.3 solar radii above the solar limb with the Metis coronagraph on SO can be tracked to PSP, orbiting at 0.1 au, thus allowing the local properties of the solar wind to be linked to the coronal source region from where it originated. Thanks to the close approach of PSP to the Sun and the simultaneous Metis observation of the solar corona, the flow-aligned magnetic field and the bulk kinetic energy flux density can be empirically inferred along the coronal current sheet with an unprecedented accuracy, allowing in particular estimation of the Alfv\'en radius at 8.7 solar radii during the time of this event. This is thus the very first study of the same solar wind plasma as it expands from the sub-Alfv\'enic solar corona to just above the Alfv\'en surface., Comment: 10 pages, 4 figures

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., Comment: 8 pages, 2 figures

The determination of solar wind outflow velocity is fundamental in order to probe the mechanisms of wind acceleration in the corona. We aim to study, via the Doppler dimming technique, the effects that the chromospheric Ly{\alpha} line profile shape causes on the determination of the outflow speed of coronal HI atoms. The Doppler dimming technique takes into account the decrease of coronal Ly{\alpha} radiation in regions where HI atoms flow out in the solar wind. Starting from UV observations (UVCS/SOHO) of the coronal Ly{\alpha} line and simultaneous measurements of pB (LASCO/SOHO and Mk3/MLSO), we studied the effect of the pumping chromospheric Ly{\alpha} line profile through measurements from SOHO/SUMER, UVSP/SMM and LPSP/OSO-8, taken from representative on-disk regions and as a function of time during the solar activity cycle. In particular, we considered the effect of four chromospheric line parameters: line width, depth of the central reversal, asymmetry and distance of the peaks. We find that the range of variability of these parameters is of about 50% for the width, 69% for the depth of the central reversal, 35% for the asymmetry, and 50% for the distance of the peaks. Then, we find that the variability of the pumping Ly{\alpha} profile affects the estimates of the coronal HI velocity by about 9-12%. Therefore, this uncertainty is smaller than other physical quantities uncertainties, and a constant in time and unique shape of the Ly{\alpha} profile over the solar disk can be adopted in order to estimate the solar wind outflow velocity., Comment: 12 pages, 10 figures, 1 table, it will appear in "Astronomy & Astrophysics", accepted for "ESPM-16" and "SWICo 2021", the uploaded abstract is abridged

This work focuses on the combination of White Light (WL) and UV (Ly-alpha) coronagraphic images to demonstrate the capability to measure the solar wind speed in the inner corona directly with the ratio between these two images (a technique called "quick inversion method"), thus avoiding to account for the line-of-sight (LOS) integration effects in the inversion of data. After a derivation of the theoretical basis and illustration of the main hypotheses in the "quick inversion method", the data inversion technique is tested first with 1D radial analytic profiles, and then with 3D numerical MHD simulations, in order to show the effects of variabilities related with different phases of solar activity cycle and complex LOS distribution of plasma parameters. The same technique is also applied to average WL and UV images obtained from real data acquired by SOHO UVCS and LASCO instruments around the minimum and maximum of the solar activity cycle. Comparisons between input and output velocities show overall a good agreement, demonstrating that this method that allows to infer the solar wind speed with WL-UV image ratio can be complementary to more complex techniques requiring the full LOS integration. The analysis described here also allowed us to quantify the possible errors in the outflow speed, and to identify the coronal regions where the "quick inversion method" performs at the best. The "quick inversion" applied to real UVCS and LASCO data allowed also to reconstruct the typical bimodal distribution of fast and slow wind at solar minimum, and to derive a more complex picture around solar maximum. The application of the technique shown here will be very important for the future analyses of data acquired with multichannel WL and UV (Ly-alpha) coronagraphs, such as Metis on-board Solar Orbiter, LST on-board ASO-S, and any other future WL and UV Ly-alpha multi-channel coronagraph., Comment: 14 pages, 8 figures, accepted for Astronomy & Astrophysics

We report the first unambiguous observational evidence in the radio range of the reflection of a coronal shock wave at the boundary of a coronal hole. The event occurred above an active region located at the northwest limb of the Sun and was characterized by an eruptive prominence and an extreme-ultraviolet (EUV) wave steepening into a shock. The EUV observations acquired by the Atmospheric Imaging Assembly (AIA) instrument on board the Solar Dynamics Observatory(SDO) and the Extreme Ultraviolet Imager (EUVI) instrument on board the Solar TErrestrial RElations Observatory(STEREO-A) were used to track the development of the EUV front in the inner corona. Metric type II radio emission, a distinguishing feature of shock waves propagating in the inner corona, was simultaneously recorded by ground-based radio spectrometers. The radio dynamic spectra displayed an unusual reversal of the type II emission lanes, together with type III-like herringbone emission, indicating shock-accelerated electron beams. Combined analysis of imaging data from the two space-based EUV instruments and the Nancay Radioheliograph (NRH) evidences that the reverse-drifting typeiiemission was produced at the intersection of the shock front, reflected at a coronal hole boundary, with an intervening low-Alfv\'en-speed region characterized by an open field configuration. We also provide an outstanding data-driven reconstruction of the spatiotemporal evolution in the inner corona of the shock-accelerated electron beams produced by the reflected shock., Comment: Accepted by Astronomy & Astrophysics, 6 pages, 7 figures

Scenario-based stochastic optimal control problems suffer from the curse of dimensionality as they can easily grow to six and seven figure sizes. First-order methods are suitable as they can deal with such large-scale problems, but may fail to achieve accurate solutions within a reasonable number of iterations. To achieve solutions of higher accuracy and high speed, in this paper we propose two proximal quasi-Newtonian limited-memory algorithms - MinFBE applied to the dual problem and the Newton-type alternating minimization algorithm (NAMA) - which can be massively parallelized on lockstep hardware such as graphics processing units (GPUs). We demonstrate the performance of these methods, in terms of convergence speed and parallelizability, on large-scale problems involving millions of variables.

The investigation of the wind in the solar corona initiated with the observations of the resonantly scattered UV emission of the coronal plasma obtained with UVCS-SOHO, designed to measure the wind outflow speed by applying the Doppler dimming diagnostics. Metis on Solar Orbiter complements the UVCS spectroscopic observations, performed during solar activity cycle 23, by simultaneously imaging the polarized visible light and the HI Ly-alpha corona in order to obtain high-spatial and temporal resolution maps of the outward velocity of the continuously expanding solar atmosphere. The Metis observations, on May 15, 2020, provide the first HI Ly-alpha images of the extended corona and the first instantaneous map of the speed of the coronal plasma outflows during the minimum of solar activity and allow us to identify the layer where the slow wind flow is observed. The polarized visible light (580-640 nm), and the UV HI Ly-alpha (121.6 nm) coronal emissions, obtained with the two Metis channels, are combined in order to measure the dimming of the UV emission relative to a static corona. This effect is caused by the outward motion of the coronal plasma along the direction of incidence of the chromospheric photons on the coronal neutral hydrogen. The plasma outflow velocity is then derived as a function of the measured Doppler dimming. The static corona UV emission is simulated on the basis of the plasma electron density inferred from the polarized visible light. This study leads to the identification, in the velocity maps of the solar corona, of the high-density layer about +/-10 deg wide, centered on the extension of a quiet equatorial streamer present at the East limb where the slowest wind flows at about (160 +/- 18) km/s from 4 Rs to 6 Rs. Beyond the boundaries of the high-density layer, the wind velocity rapidly increases, marking the transition between slow and fast wind in the corona.

Preference-based global optimization algorithms minimize an unknown objective function only based on whether the function is better, worse, or similar for given pairs of candidate optimization vectors. Such optimization problems arise in many real-life examples, such as finding the optimal calibration of the parameters of a control law. The calibrator can judge whether a particular combination of parameters leads to a better, worse, or similar closed-loop performance. Often, the search for the optimal parameters is also subject to unknown constraints. For example, the vector of calibration parameters must not lead to closed-loop instability. This paper extends an active preference learning algorithm introduced recently by the authors to handle unknown constraints. The proposed method, called C-GLISp, looks for an optimizer of the problem only based on preferences expressed on pairs of candidate vectors, and on whether a given vector is reported feasible and/or satisfactory. C-GLISp learns a surrogate of the underlying objective function based on the expressed preferences, and a surrogate of the probability that a sample is feasible and/or satisfactory based on whether each of the tested vectors was judged as such. The surrogate functions are used iteratively to propose a new candidate vector to test and judge. Numerical benchmarks and a semi-automated control calibration task demonstrate the effectiveness of C-GLISp, showing that it can reach near-optimal solutions within a small number of iterations., Comment: A MATLAB and a Python implementation of C-GLISp is available at http://cse.lab.imtlucca.it/~bemporad/glis

Re-planning in legged locomotion is crucial to track the desired user velocity while adapting to the terrain and rejecting external disturbances. In this work, we propose and test in experiments a real-time Nonlinear Model Predictive Control (NMPC) tailored to a legged robot for achieving dynamic locomotion on a variety of terrains. We introduce a mobility-based criterion to define an NMPC cost that enhances the locomotion of quadruped robots while maximizing leg mobility and improves adaptation to the terrain features. Our NMPC is based on the real-time iteration scheme that allows us to re-plan online at $25\,\mathrm{Hz}$ with a prediction horizon of $2$ seconds. We use the single rigid body dynamic model defined in the center of mass frame in order to increase the computational efficiency. In simulations, the NMPC is tested to traverse a set of pallets of different sizes, to walk into a V-shaped chimney,and to locomote over rough terrain. In real experiments, we demonstrate the effectiveness of our NMPC with the mobility feature that allowed IIT's $87\, \mathrm{kg}$ quadruped robot HyQ to achieve an omni-directional walk on flat terrain, to traverse a static pallet, and to adapt to a repositioned pallet during a walk., Comment: Video available on: https://youtu.be/r0-KIiw0eWM

The Metis coronagraph is one of the remote sensing instruments hosted on board the ESA/NASA Solar Orbiter mission. Metis is devoted to carry out the first simultaneous imaging of the solar corona in both visible light (VL) and ultraviolet (UV). High-energy particles penetrate spacecraft materials and may limit the performance of on-board instruments. A study of galactic cosmic-ray (GCR) tracks observed in the first VL images gathered by Metis during the commissioning phase for a total of 60 seconds of exposure time is presented here. A similar analysis is planned for the UV channel. A prediction of the GCR flux up to hundreds of GeV is made here for the first part of the Solar Orbiter mission to study the Metis coronagraph performance. GCR model predictions are compared to observations gathered on board Solar Orbiter by the EPD/HET experiment in the range 10 MeV-100 MeV in the summer 2020 and with previous measurements. Estimated cosmic-ray fluxes above 70 MeV n$^{-1}$ have been also parameterized and used for Monte Carlo simulations aiming at reproducing the cosmic-ray track observations in the Metis coronagraph VL images. The same parameterizations can also be used to study the performance of other detectors. By comparing observations of cosmic-ray tracks in the Metis VL images with FLUKA Monte Carlo simulations of cosmic-ray interactions in the VL detector, it is found that cosmic rays fire a fraction of the order of 10$^{-4}$ of the whole image pixel sample. Therefore, cosmic rays do not affect sensibly the quality of Metis VL images. It is also found that the overall efficiency for cosmic-ray identification in the Metis VL images is approximately equal to the contribution of Z$>$2 particles. As a result, the Metis coronagraph may play the role of a proton monitor for long-term GCR variations during the overall mission duration., Comment: 11 pages, 7 figures, accepted for publication on Astronomy & Astrophysics

In this paper we present a dual active-set solver for quadratic programming which has properties suitable for use in embedded model predictive control applications. In particular, the solver is efficient, can easily be warm-started, and is simple to code. Moreover, the exact worst-case computational complexity of the solver can be determined offline and, by using outer proximal-point iterations, ill-conditioned problems can be handled in a robust manner.

This paper introduces a novel model-free approach to synthesize virtual sensors for the estimation of dynamical quantities that are unmeasurable at runtime but are available for design purposes on test benches. After collecting a dataset of measurements of such quantities, together with other variables that are also available during on-line operations, the virtual sensor is obtained using machine learning techniques by training a predictor whose inputs are the measured variables and the features extracted by a bank of linear observers fed with the same measures. The approach is applicable to infer the value of quantities such as physical states and other time-varying parameters that affect the dynamics of the system. The proposed virtual sensor architecture - whose structure can be related to the Multiple Model Adaptive Estimation framework - is conceived to keep computational and memory requirements as low as possible, so that it can be efficiently implemented in embedded hardware platforms. The effectiveness of the approach is shown in different numerical examples, involving the estimation of the scheduling parameter of a nonlinear parameter-varying system, the reconstruction of the mode of a switching linear system, and the estimation of the state of charge (SoC) of a lithium-ion battery.