Local Lipschitz Constant Computation of ReLU-FNNs: Upper Bound Computation with Exactness Verification

This paper is concerned with the computation of the local Lipschitz constant of feedforward neural networks (FNNs) with activation functions being rectified linear units (ReLUs). The local Lipschitz constant of an FNN for a target input is a reasonable measure for its quantitative evaluation of the reliability. By following a standard procedure using multipliers that capture the behavior of ReLUs,we first reduce the upper bound computation problem of the local Lipschitz constant into a semidefinite programming problem (SDP). Here we newly introduce copositive multipliers to capture the ReLU behavior accurately. Then, by considering the dual of the SDP for the upper bound computation, we second derive a viable test to conclude the exactness of the computed upper bound. However, these SDPs are intractable for practical FNNs with hundreds of ReLUs. To address this issue, we further propose a method to construct a reduced order model whose input-output property is identical to the original FNN over a neighborhood of the target input. We finally illustrate the effectiveness of the model reduction and exactness verification methods with numerical examples of practical FNNs.Comment: 8 pages, 3 figures, Fixed authors' display in the bibliograph

Commande variant dans le temps pour le contrôle d'attitude de satellites

Cette thèse porte sur la commande variant dans le temps avec comme fil directeur l application au contrôle d attitude de satellites. Nous avons étudié trois types de commande: une commande à commutation, une commande LPV et une commande adaptative directe. Pour cette dernière nous avons proposé des résultats théoriques nouveaux portant sur la structuration du gain et de l adaptation. Les résultats ont été validés en simulation et sont testés à bord d un satellite. En partant de la loi à commutation actuellement utilisée sur les satellites Myriade, une première partie de nos travaux est dédiée à la commande LPV. Notre approche, basée sur la spécification des objectifs de commande à travers un modèle de référence LPV, permet d'obtenir de nouveaux algorithmes exprimés dans ce formalisme. Testées en simulation, ces lois de commande répondent à la problématique de notre application. Toutefois, le choix du modèle de référence LPV s'avère délicat. Cette difficulté a été levée en utilisant la commande adaptative. Dans cette approche, les spécifications sur le comportement temps-variant sont traduites par des contraintes au niveau des lois d'adaptation des gains de commande. Nous introduisons ainsi une nouvelle méthode de synthèse de lois adaptatives structurées. Les preuves de stabilité établies s'appuient sur des outils de la théorie de Lyapunov. Les résultats obtenus sur un simulateur complet montrent l'intérêt de tels algorithmes adaptatifs. Ils permettent en particulier de modifier la dynamique du satellite selon les capacités disponibles des actionneurs. Sur la base de ces résultats, une campagne d essai en vol sur le satellite PICARD est actuellement en cours.This manuscript considers time varying control, with a strong emphasis on a satellite attitude control application. Three types of control structures have been studied: a switch-based approach, LPV control and direct adaptive control. In this last field we have introduced new theoretical results which allow structuring the gain and the adaptation law. The results have been validated in simulation and are currently tested on board a satellite. Starting from the switch-based control law currently implemented on the Myriade satellites, a first part of our work isdedicated to LPV control. Based on the specification of the control objectives by using of an LPV reference model, our approach allows obtaining new control algorithms expressed within this framework. The simulations carried out with theLPV algorithms obtained by using this method show that they meet the needs of our application. Nonetheless, the choice of a reference model proves to be difficult. This obstacle has been surpassed by using direct adaptive control. In this approach, specifications regarding the timevarying behaviour are added through constraints on the laws defining the control gains adaptation. We thus introduce anew synthesis method, based on which structured adaptive control laws are obtained. Stability proofs are established based on tools of the Lyapunov theory.The results obtained on a complete simulator show the interest of using such adaptive algorithms, which allow in particular to modify the satellite dynamics depending on the available capacity of the actuators. Based on these positive results, a fight-test campaign on the PICARD satellite is underway.TOULOUSE-ISAE (315552318) / SudocSudocFranceF

Synthèse de correcteurs robustes périodiques à mémoire et application au contrôle d'attitude de satellites par roues à réaction et magnéto-coupleurs

Les travaux présentés dans ce mémoire constituent une contribution à la conception de méthodes systématiques pour l analyse et la commande de systèmes périodiques et incertains. Une partie importante de cette thèse est également consacrée au contrôle d attitude de satellites dont la dynamique se prête naturellement à une représentation sous forme de modèles périodiques soumis à des incertitudes. La première partie propose une présentation unifiée des résultats d analyse et de synthèse de modèles périodiques et incertains à temps-discret via des méthodes basées sur des inégalités linéaires matricielles (LMI) et en s appuyant sur la théorie de Lyapunov. Par la suite, l accent est mis sur une nouvelle classe de correcteurs périodiques à mémoire pour lesquels l entrée de commande est construite en utilisant l historique des états du système conservés en mémoire. Des exemples numériques démontrent que ces nouveaux degrés de liberté permettent de repousser les limites des performances robustes. La seconde partie s intéresse aux aspects de périodicité et de robustesse du contrôle d attitude de satellite rencontrés notamment lors de l utilisation des magnéto-coupleurs. Ces actionneurs s appuient sur le champ géomagnétique variant périodiquement le long de l orbite du satellite. Différentes stratégies de commande sont mises en œuvre et comparées entre elles avec le souci constant de tenir compte des principales limitations des actionneurs. Cette démarche conduit à une nouvelle loi de commande périodique régulant le moment cinétique des roues à réactions sans perturber le contrôle d attitude dont l effort de commande est réparti sur l ensemble des actionneurs.This manuscript reviews contributions to the development of systematic methods for analysis and control of periodic uncertain systems. An important part of this thesis is also dedicated to the design of attitude control systems for satellites whose dynamics is naturally represented as a periodic model subject to uncertainties. The first part is devoted to the developpement of a unifying presentation of the analysis and synthesis results of periodic, uncertain and discrete-time models via methods relying on linear matrix inequalities (LMI) and based on Lyapunov theory. Subsequently, the focus is on a new class of periodic control laws with memory for which the control input is constructed using history of the states of the system kept in memory. Numerical experiments show that these new degrees of freedom can outperformed the existing results. The second part deals with periodic and robustness aspects of attitude control of a satellite using magnetorquers. These actuators use the geomagnetic field that varies periodically along the orbital trajectory. Different control strategies are implemented and compared with one another with the constant concern of taking the main limitations of the actuators into account. This approach leads to a new control law regulating the momentum of the reaction wheels without disturbing attitude control for which the control effort is shared by all actuators.TOULOUSE-ISAE (315552318) / SudocSudocFranceF

Induced Norm Analysis of Linear Systems for Nonnegative Input Signals

This paper is concerned with the analysis of the $L_p\ (p\in[1,\infty), p=\infty)$ induced norms of continuous-time linear systems where input signals are restricted to be nonnegative. This norm is referred to as the $L_{p+}$ induced norm in this paper. It has been shown recently that the $L_{2+}$ induced norm is effective for the stability analysis of nonlinear feedback systems where the nonlinearity returns only nonnegative signals. However, the exact computation of the $L_{2+}$ induced norm is essentially difficult. To get around this difficulty, in the first part of this paper, we provide a copositive-programming-based method for the upper bound computation by capturing the nonnegativity of the input signals by copositive multipliers. Then, in the second part of the paper, we derive uniform lower bounds of the $L_{p+}\ (p\in[1,\infty), p=\infty)$ induced norms with respect to the standard $L_{p}$ induced norms that are valid for all linear systems including infinite-dimensional ones. For each linear system, we finally derive a computation method of the lower bounds of the $L_{2+}$ induced norm that are larger than (or equal to) the uniform one. The effectiveness of the upper/lower bound computation methods are fully illustrated by numerical examples.Comment: 12 pages, 3 figures. A preliminary version of this paper was presented at ECC 2022 (arXiv:2401.03242) and IFAC WC 202

Stability Analysis of Feedback Systems with ReLU Nonlinearities via Semialgebraic Set Representation

This paper is concerned with the stability analysis problem of feedback systems with rectified linear unit (ReLU) nonlinearities. Such feedback systems arise when we model dynamical (recurrent) neural networks (NNs) and NN-driven control systems where all the activation functions of NNs are ReLUs. In this study, we focus on the semialgebraic set representation characterizing the input-output properties of ReLUs. This allows us to employ a novel copositive multiplier in the framework of the integral quadratic constraint and, thus, to derive a linear matrix inequality (LMI) condition for the stability analysis of the feedback systems. However, the infeasibility of this LMI does not allow us to obtain any conclusion on the system's stability due to its conservativeness. This motivates us to consider its dual LMI. By investigating the structure of the dual solution, we derive a rank condition on the dual variable certificating that the system at hand is never stable. In addition, we construct a hierarchy of dual LMIs allowing for improved instability detection. We illustrate the effectiveness of the proposed approach by several numerical examples.Comment: 8 pages, 5 figure

Séparation Intégrale Quadratique et Applications en Commande Robuste

The presented work tackles the theoretical approach called “integral quadratic separation”. The manuscript discusses the origines of this approach, it describes in details some mathematical properties and illustrates the contributions it allows for some robust analysis issues. During the seminar we shall discuss the extension of integral quadratic separation to descriptor forms and the sequences of LMI relaxations with decreasing conservatism it allows to build. Examples will cover: robust analysis with respect to parametric uncertainties, the time-varying uncertainty case, the time-delay case, as well as the analysis of systems with saturations.L’HDR porte sur une approche théorique intitulée “séparation intégrale quadratique”. C’est l’occasion de revenir sur les origines de cette approche, de décrire dans le détail certains aspects techniques et surtout d’illustrer les contributions qu’elle permet pour certaines questions d’analyse robuste. Lors de la soutenance nous parlerons des résultats de séparation intégrale quadratique pour le cas de modèles descripteurs et des suites de relaxations LMI au pessimisme décroissant que cela permet de construire. Des exemples seront abordés traitant de: stabilité robuste vis à vis d’incertitudes paramétriques, le cas d’incertitudes variant dans le temps, les systèmes à retard, ainsi que les systèmes avec saturations

Exploring robust structured static output feedback design

International audienceMany design problems in control can be recast as the search for static structured (diagonal) output-feedback gains stabilizing an augmented and rearranged dynamical plant model. Moreover, in first approximation that plant may be considered as linear, or at least, one will usually request the linearized plant to be stable before considering the more complicated non-linear version. Because of the approximations leading to the linearized version, the plant parameters are usually uncertain and time-varying. In the presentation we discuss the possibility to design such static diagonal output-feedback gains for uncertain linear systems using a recently proposed matrix inequality based formulation. As expected for this hard problem, the methodology does not provide a guaranteed to succeed result, but provides some interesting promising paths for an efficient algorithm. If we have time we also mention an adaptive control strategy for updating (learning) the structured static gains

Formulation générique de problèmes en analyse et commande robuste par les fonctions de Lyapunov dependant des paramètres

The thesis addresses robustness problems in automatic control. Some property of a given system is said to be robust if it is invariant with respect to all admissible model uncertainties. Our aim is to give methods for finding a control law that improves and/or guarantees different robust properties. Linear time-invariant models are considered. The structured parametric uncertainties enter the model by linear fractional transformations while the uncertainty domain may be polytopic or dissipative. The robust properties studied in the thesis are robust stability, disturbance rejection (robust guaranteed cost) and transient response (pole location). First are given some analysis methods and then algorithms for control law synthesis are proposed. The theoretical tools employed are issued from the Lyapunov theory and from topological separation. Parameter dependent Lyapunov functions are considered in order to reduce the conservatism of the quadratic stability framework. This last framework is also described thoroughly for comparison in terms of conservatism and computational complexity. Most mathematical results are given in a unified formulation in order to describe their generality. All the results are based on linear matrix inequalities that can be solved efficiently with nowadays classical techniques. The theoretical results are illustrated by some examples.Cette thèse porte sur la commande robuste des systèmes. La robustesse caractérise l'invariance de propriétés de stabilité et de performance vis à vis des inévitables incertitudes affectant le modèle. Le problème de commande est d'améliorer et/ou de garantir les propriétés robustes. Les modèles considérés sont linéaires à temps invariant. Les incertitudes sont paramétriques réelles structurées et interviennent sous forme rationnelle. Les classes d'incertitudes polytopiques et dissipatives sont plus particulièrement prises en compte. Les propriétés étudiées sont principalement la stabilité robuste, le rejet des perturbations (coût garanti robuste) et le comportement transitoire (localisation des pôles). Pour ces propriétés nous proposons dans un premier temps des méthodes d'analyse puis des méthodes de synthèse de correcteurs. Les outils théoriques utilisés sont issus de la théorie de Lyapunov et de la séparation topologique. De manière à garantir les performances avec le moins de pessimisme possible, nous proposons de faire appel à des fonctions de Lyapunov dépendant des paramètres. Comme on attache une importance à la mise en oeuvre numérique, des méthodes issues du cadre de la stabilité quadratique, plus pessimistes mais moins demandeuses en capacité de calcul sont également proposées. La formulation volontairement unifiée des différents problèmes met en évidence les sources de pessimisme. Toutes les méthodes proposées sont formulées en termes d'Inégalités Matricielles Linéaires (LMI) dont la mise en oeuvre numérique est désormais classique. Les résultats de recherche sont illustrés sur des exemples

Continuous quaternion based almost global attitude tracking

International audienceThis paper considers the attitude control problem of a generic rotating 3 degrees of freedom fully actuated rigid object. The specific studied problem is the deviation control of this object around a theoretically feasible attitude trajectory. The rotation motion has an intrinsic non linear behaviour (trigonometric, 2π-periodicity) that need to build non linear and hybrid controllers to get global stability of the closed loop system. This paper considers the opportunity to use the quaternion framework to build a continuous non linear state feedback that reaches an almost global asymptotical stability. Some perspectives to enhance this result with integrators to cancel out static and drag errors are eventually proposed

An Efficient Numerical Solution for H2 Static Output Feedback Synthesis

This paper addresses the problem of static output feedback synthesis and focuses on § ©¨opt imisation. The bilinear problem of finding a control feedback gain and a Lyapunov matrix is shown to be equivalent to a BMI problem that involves slack variables and a state feedback gain. This BMI condition is a promising theoretical result that links the two state and output feedback questions in a unified formulation. Based on this new expression, an efficient numerical procedure is derived. Some telling examples show that it is quite competitive compared to other algorithms proposed in the literature. Extensions to robust § ¨opt imal synthesis are derived. They show that unlike other BMI approaches, the method is of the same complexity if the system is certain or uncertain