Search

The Loewner framework is an interpolatory approach designed for approximating linear and nonlinear systems. The goal here is to extend this framework to linear parametric systems with an arbitrary number n of parameters. One main innovation established here is the construction of data-based realizations for any number of parameters. Equally importantly, we show how to alleviate the computational burden, by avoiding the explicit construction of large-scale n-dimensional Loewner matrices of size $N \times N$. This reduces the complexity from $O(N^3)$ to about $O(N^{1.4})$, thus taming the curse of dimensionality and making the solution scalable to very large data sets. To achieve this, a new generalized multivariate rational function realization is defined. Then, we introduce the n-dimensional multivariate Loewner matrices and show that they can be computed by solving a coupled set of Sylvester equations. The null space of these Loewner matrices then allows the construction of the multivariate barycentric transfer function. The principal result of this work is to show how the null space of the n-dimensional Loewner matrix can be computed using a sequence of 1-dimensional Loewner matrices, leading to a drastic computational burden reduction. Finally, we suggest two algorithms (one direct and one iterative) to construct, directly from data, multivariate (or parametric) realizations ensuring (approximate) interpolation. Numerical examples highlight the effectiveness and scalability of the method., Comment: 32 pages, 4 figures

This paper presents a systematic methodology for the discretization and reduction of a class of one-dimensional Partial Differential Equations (PDEs) with inputs and outputs collocated at the spatial boundaries. The class of system that we consider is known as Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type of boundary inputs and outputs. This is, for instance, the case of waves and beams with Neumann, Dirichlet, or mixed boundary conditions. Based on a Partitioned Finite Element Method (PFEM), we develop a numerical scheme for the structure-preserving spatial discretization for the class of one-dimensional BC-PHSs. We show that if the initial PDE is passive (or impedance energy preserving), the discretized model also is. In addition and since the discretized model or Full Order Model (FOM) can be of large dimension, we recall the standard Loewner framework for the Model Order Reduction (MOR) using frequency domain interpolation. We recall the main steps to produce a Reduced Order Model (ROM) that approaches the FOM in a given range of frequencies. We summarize the steps to follow in order to obtain a ROM that preserves the passive structure as well. Finally, we provide a constructive way to build a projector that allows to recover the physical meaning of the state variables from the ROM to the FOM. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach.

A structure-preserving Finite Element Method (FEM) for the transport equation in one- and two-dimensional domains is presented. This Distributed Parameter System (DPS) has non-collocated boundary control and observation, and reveals a scattering-energy preserving structure. We show that the discretized model preserves the aforementioned structure from the original infinite-dimensional system. Moreover, we analyse the case of moving meshes for the one-dimensional case. The moving mesh requires less states than the fixed one to produce solutions with a comparable accuracy, and it can also reduce the overshoot and oscillations of Gibbs phenomenon produced when using the FEM. Numerical simulations are provided for the case of a one-dimensional transport equation with fixed and moving meshes.

Flow control aims at modifying a natural flow state to reach an other flow state considered as advantageous. In this paper, active feedback flow separation control is investigated with two different closed-loop control strategies, involving a reference signal tracking architecture. Firstly, a data-driven control law, leading to a linear (integral) controller is employed. Secondly, a phenomenological/model-driven approach, leading to a non-linear positive (integral) control strategy is investigated. While the former benefits of a tuning simplicity, the latter prevents undesirable effects and formally guarantees closed-loop stability. Both control approaches were validated through wind tunnel experiments of flow separation over a movable NACA 4412 plain flap. These control laws were designed with respect to hot film measurements, performed over the flap for different deflection angles. Both control approaches proved efficient in avoiding flow separation. The main contribution of this work is to provide practitioners simple but yet efficient ways to design a flow separation controller. In addition, a complete validation campaign data-set is provided.

In this work, we detail a procedure to construct a reduced order model on the basis of frequency-domain data, that preserves the non-strictly passive property and the port-Hamiltonian structure. The proposed scheme is based on Benner et al. (2020) contribution, which has been adapted (i) to handle non-strictly passive model, and (ii) to handle numerical issues observed when applying the Loewner framework on complex configurations. We validate the proposed scheme on a very complex two-dimensional wave equation, for which the discretized version preserves the port-Hamiltoninan form., Comment: Accepted at European Control Conference 2023

This note describes a constructive heuristic to select frequencies of interest within the context of reduced-order modelling by interpolation. The approach is described here through the Loewner framework. Numerical illustrations highlight the benefit it can bring to decrease the required number of interpolation points which is key when the data come from numerically expensive solvers.

In this contribution, we discuss the modeling and model reduction framework known as the Loewner framework. This is a data-driven approach, applicable to large-scale systems, which was originally developed for applications to linear time-invariant systems. In recent years, this method has been extended to a number of additional more complex scenarios, including linear parametric or nonlinear dynamical systems. We will provide here an overview of the latter two, together with time-domain extensions. Additionally, the application of the Loewner framework is illustrated by a collection of practical test cases. Firstly, for data-driven complexity reduction of the underlying model, and secondly, for dealing with control applications of complex systems (in particular, with feedback controller design)., Comment: 32 pages, 13 figures

The dynamical response of an aircraft subject to gust perturbations is a key element in a preliminary design phase. In particular, the loads induced by gusts along the wing should not exceed some limit values and should even ideally be decreased. Active control is one lever to address this problem. However, evaluating the benefit that active control may bring considering some actuators characteristics or some delay in the loop is a difficult task, especially in the early design phase. This problem is addressed in this paper with an open-loop optimal control framework and more specifically with a direct transcription method resulting in a linear optimisation problem. The approach is illustrated on a realistic aeroelastic aircraft model built with a coupled fluid-structure solver which order is reduced to decrease the number of optimisation variables.

From simple physical systems to full production lines, numerical models could be used to minimize downtime and to optimize performances. In this article, the system of interest is the SPIRAL2 (Syst\`eme de Production d'Ions RAdioactifs en Ligne de 2e g\'en\'eration) particles accelerator cryogenic system. This paper illustrates three totally different applications based on a SPIRAL2 cryostat model: optimal controller synthesis, virtual sensor synthesis and anomaly detection. The tow firsts applications have been deployed on the system. Experimental results are used to illustrate the benefits of such applications., Comment: 11 pages, 8 figures

SPIRAL 2 is a high intensity heavy ions beams accelerator project that has been going on for more than 10 years now. Countless efforts in different disciplines made it what it is today. One of the most important steps after the set up of the different equipments has been the very first full cool down of the superconducting cavities in an accelerator operation type configuration. While this has been a major achievement for the SPIRAL 2 teams, it also hi-lighted new challenges and constraints that would have to be addressed in order to have a high availability rate of the beam from the cryogenics side. This paper retraces this particular episode.

Cryogenic thermoacoustic oscillations is an area of interest of several studies. For superconducting accelerators, it is an unwanted phenomena that we usually want to get rid off. The SPIRAL2 superconducting accelerator had distributed Taconis all over its cryostats. This paper spans the different steps from their first detection to their damping with a highlight on the methods and the instrumentation that has been used. The presented study also sets the ground for a real life experimental investigation of thermo-ascoustics in complex geometries such as superconducting LINACs. With modern Big data analysis and simulation tools, it also sets the ground for the developments of new data linked codes that fit these complex situations.

The paper's main contribution concerns the use of interpolatory methods to solve end to end industrial control problems involving complex linear dynamical systems. More in details, contributions show how the rational data and function interpolation framework is a pivotal tool (i) to construct (frequency-limited) reduced order dynamical models appropriate for model-based control design and (ii) to accurately discretise controllers in view of on-board computer-limited implementation. These contributions are illustrated along the paper through the design of an active feedback gust load alleviation function, applied on an industrial generic business jet aircraft use-case. The closed-loop validation and performances evaluation are assessed through the use of an industrial dedicated simulator and considering certification objectives. Although application is centred on aircraft applications, the method is not restrictive and can be extended to any linear dynamical systems., Comment: 23 pages, 9 figures, submitted to journal

On the basis of input-output time-domain data collected from a complex simulator, this paper proposes a constructive methodology to infer a reduced-order linear, bilinear or quadratic time invariant dynamical model reproducing the underlying phenomena. The approach is essentially based on linear dynamical systems and approximation theory. More specifically, it sequentially involves the interpolatory Pencil and Loewner framework, known to be both very versatile and scalable to large-scale data sets, and a linear least square problem involving the raw data and reduced internal variables. With respect to intrusive methods, no prior knowledge on the operator is needed. In addition, compared to the traditional non-intrusive operator inference ones, the proposed approach alleviates the need of measuring the original full-order model internal variables. It is thus applicable to a wider application range than standard intrusive and non-intrusive methods. The rationale is successfully applied on a large eddy simulation of a pollutants dispersion case over an airport area involving multi-scale and multi-physics dynamical phenomena. Despite the simplicity of the resulting low complexity model, the proposed approach shows satisfactory results to predict the pollutants plume pattern while being significantly faster to simulate., Comment: 25 pages, 10 figures, submitted to journal

The versatility of data-driven approximation by interpolatory methods, originally settled for model approximation purpose, is illustrated in the context of linear controller design and stability analysis of irrational models. To this aim, following an academic driving example described by a linear partial differential equation, it is shown how the Loewner-based interpolation may be an essential ingredient for control design and stability analysis. More specifically, the interpolatory framework is first used to approximate the irrational model by a rational one that can be used for model-based control, and secondly, it is used for direct data-driven control design, showing equivalent results. Finally, this interpolation framework is employed for estimating the stability of the interconnection of the irrational model with a rational controller., Comment: Book chapter, 17 pages, 11 figures

Pulsed fluidic actuators play a central role in the fluid flow experimental control strategy to achieve better performances of aeronautic devices. In this paper, we demonstrate, through an experimental test bench, how the interpolatory-based Loewner Data-Driven Control (L-DDC) framework is an appropriate tool for accurately controlling the outflow velocity of this family of actuators. L-DDC combines the concept of ideal controller with the Loewner framework in a single data-driven rationale, appropriate to experimental users. The contributions of the paper are, first, to emphasise the simplicity and versatility of such a data-driven rationale in a constrained experimental setup, and second, to solve some practical fluid engineers concerns by detailing the complete workflow and key ingredients for successfully implementing a pulsed fluidic actuator controller from the data acquisition to the control implementation and validation stages., Comment: 20 pages, 7 figures

In this paper, we address extensions of the Loewner Data-Driven Control (L-DDC) methodology. First, this approach is extended by incorporating two alternative approximation methods known as Adaptive-Antoulas-Anderson (AAA) and Vector Fitting (VF). These algorithms also include least squares fitting which provides additional flexibility and enables possible adjustments for control tuning. Secondly, the standard model reference data-driven setting is extended to handle noise affecting the data and uncertainty in the closed-loop objective function. These proposed adaptations yield a more robust data-driven control design., Comment: 9 pages, 4 figures

This paper revisits the modal truncation from an optimisation point of view. In particular, the concept of dominant poles is formulated with respect to different systems norms as the solution of the associated optimal modal truncation problem. The latter is reformulated as an equivalent convex integer or mixed-integer program. Numerical examples highlight the concept and optimisation approach.

This report aims at presenting (yet) a(nother) methodology to design and implement a linear controller for linear dynamical systems on practical applications. The specificity of this report is that authors try to cover (obviously in a non exhaustive way) a wide range of control engineering fields. Indeed, the main purpose is to give a quick overview of standard control engineer approaches to non familiar readers. More specifically, using a simple toy example, we discuss the main steps control engineers usually follow. Namely, (i) the excitation signals construction, (ii) the (continuous-time) linear model construction and approximation, (iii) the (continuous-time) control design, and finally, (iv) its time-domain discretisation and control signal modulation in view of practical implementation. This report is clearly user-oriented and thus focuses on practical aspects (using Matlab code) rather than on theoretical ones, let to the reader's curiosity with few but relevant references., Comment: 27 pages, 21 figures, code material

In the framework of a national funded program for nuclear safety, a first prototype of portable gamma camera was built and tested. It integrates a Caliste-HD CdTe-hybrid detector designed for space X-ray astronomy coupled with a new system-on-chip based acquisition system (FPGA and ARM microprocessor) and thermo-electrical coolers for a use at room temperature. The complete gamma part of the camera fits in a volume of 15 x 15 x 40 cm^3 for a mass lower than 1 kg and a power consumption lower than 10 W. Localization and spectro-identification of radionuclides in a contaminated scene were demonstrated during several test campaigns. A new generation of system is under development taking into account feedback experience from in-situ measurements and integrating a new generation of sensor cost-optimized by industrial applications called Caliste-O. Caliste-O holds a 16 x 16 pixel detector of 14 x 14 mm^2 and 2 mm thick with 8 full-custom front-end IDeF-X HD ASICs. Two prototypes were fabricated and tested. The paper will present the results of in-situ measurements with the first gamma camera, the spectroscopic performance of Caliste-O and the design of the second generation of gamma camera which aims for real time imaging and spectro-identification., Comment: 6 pages, 7 figures

This note describes how the Loewner framework can be exploited to create a discrete-time control-law from frequency-data of a continuous-time plant so that their hybrid interconnection matches a given continuous-time reference model up to the Nyquist frequency. The resulting Hybrid Loewner Data Driven Control scheme is illustrated on two numerical examples.

An interpolation method for discretising continuous-time Linear Time Invariant (LTI) models is proposed in this paper. It consists first in using the Loewner interpolation framework on a specific set of frequency data and secondly to project the resulting model onto a stable subspace. The order of the discretised model may be chosen larger than the initial one thus allowing for trading complexity for accuracy if needed. Numerical examples highlight the efficiency of the method at preserving a satisfactory matching both in magnitude and phase in comparison to standard discretisation methods like ZOH or Tustin.

The choice of a reference model in data-driven control techniques is a critical step. Indeed, it should represent the desired closed-loop performances and be achievable by the plant at the same time. In this paper, we propose a method to build such a reference model, both reproducible by the system and having a desired behaviour. It is applicable to Linear Time-Invariant (LTI) monovariable systems and relies on the estimation of the plant's instabilities through a data-driven stability analysis technique. The L-DDC (Loewner Data Driven Control) algorithm is used to illustrate the impact of the choice of the reference model on the control design process. Finally, the proposed choice of specifications allows to use a controller validation technique based on the small-gain theorem.

Traditionally, the delay margin of a looped system is computed by considering both the controller and system representations that evolve in the same space (e.g. either continuous or discrete-time). However, as in practice the system is continuous and the controller is mostly embedded in a computer, the looped - controller / system pair - model is hybrid. As a consequence, the computed delay margin might vary with respect to the continuous (or discrete one). This paper proposes a novel approach to compute the exact delay margin of hybrid systems, and more specifically, when a discrete-time controller is looped with a continuous-time system. The main interest is then to provide the practitioners with a way to select the appropriate discretization technique for maximizing the delay margin and to be able to exactly evaluate the delay margin before implementation on target. The main idea is to approximate the discrete-time controller with an equivalent continuous-time one (often with higher order) and to exploit the classical continuous-time frequency-based analysis strategies., Comment: 17 pages, 4 figures, submitted to the European Control Conference 2019

This paper proposes an simple but yet effective approach to structured parametric controller design in a linear fractional form. The main contribution consists in using structured $\mathcal{H}_\infty$ oriented optimization tools in an original manner to either (i) construct a parametric controller or (ii) a family of controllers with varying performances. Practical and numerical issues are also discussed to provide readers and practitioners a simple way to deploy the proposed process. The overall approach is illustrated through two numerical academical (but still complex) examples illustrating two applications: first, a parametric controller design adapted to a parameter dependent model of a clamped beam and, second, a controller with parameter dependent performance applied on a building model., Comment: Submitted to IEEE CDC 2017

In this paper, the $\mathcal{H}_{2}$ optimal approximation of a $n_{y}\times{n_{u}}$ transfer function $\mathbf{G}(s)$ by a finite dimensional system $\hat{\mathbf{H}}_{d}(s)$ including input/output delays, is addressed. The underlying $\mathcal{H}_{2}$ optimality conditions of the approximation problem are firstly derived and established in the case of a poles/residues decomposition. These latter form an extension of the tangential interpolatory conditions, presented in~\cite{gugercin2008h_2,dooren2007} for the delay-free case, which is the main contribution of this paper. Secondly, a two stage algorithm is proposed in order to practically obtain such an approximation., Comment: 14 pages, 3 figures, submitted to Automatica Journal