Les travaux effectués dans le cadre de cette thèse « Analyse de robustesse par contraintes intégrales quadratiques - Application aux lanceurs spatiaux » ont été menés en collaboration entre le Département Automatique de Supélec, EADS Astrium ST, l’Agence Spatiale Européenne (ESA) et l’université de Stuttgart. Le but était d’adapter et d’utiliser des méthodes analytiques de validation de loi de commande d'un lanceur en phase balistique pour améliorer les résultats obtenus par l’approche probabiliste fondée sur des simulations, technique actuellement majoritaire dans l’industrie. Dans ce cadre, l’utilisation des contraintes intégrales quadratiques (IQC) a permis de caractériser la stabilité et la performance robuste de la loi de commande d’un modèle représentatif du lanceur. Nous avons étudié l’influence de la dynamique non-linéaire des lanceurs sur la stabilité et la performance robuste. Dans ce cadre, nous avons factorisé les équations du mouvement en prenant en compte les incertitudes de la matrice d’inertie ainsi que les couplages gyroscopiques. Le second axe traita de l’influence des actionneurs de type modulateur de largeur impulsions (PWM) sur la stabilité du système par deux études IQC. La conclusion de ces travaux de thèse met l’accent sur l’importance de l’utilisation de méthodes analytiques dans le domaine spatial. Ces méthodes permettent l’obtention de garanties rigoureuses de stabilité et de performance des systèmes. De plus, toutes les méthodes d’analyse possèdent leur extension pour la synthèse de correcteurs robustes. Ainsi on imagine aisément l’immense gain que pourrait produire l’utilisation de ces méthodes pour la synthèse de correcteurs robustes. / The introduction of analytical techniques along the steps of the development of a space launcher will allow significant reductions in terms of costs and manpower, and will enable, by a more systematical way of tuning and assessing control laws, to get flyable designs much faster. In this scope, IQC based tools already present promising result and show that they may be the most appropriate ones for the robustness analysis of large complex systems. They account for the system structure and allow dealing specifically with each subsystems, it means that we can improve the representation contained in the multipliers easily and reuse the set up to assess the improvements. The flexibility of the method is a huge advantage. We experienced it during two phases. The first was dedicated to the analysis of the three-degree-of-freedom uncertain nonlinear equation of motion of a rigid body. Secondly, we studied the influence of the pulse-width modulator behavior of the attitude control system on the launcher stability. IQC-based stability analysis allowed defining estimations of the stability domain with respect to uncertainties and system parameters. Moreover, the results obtained with IQC can go way beyond stability analysis with performance analysis with description of the particular performance criteria of the field with appropriate multipliers. Later on controller synthesis and merging of IQC method with worst-case search algorithms could extend greatly the frame of use of this analytical tool and give it the influence it deserves.

The introduction of analytical techniques along the steps of the development of a space launcher will allow significant reductions in terms of costs and manpower, and will enable, by a more systematical way of tuning and assessing control laws, to get flyable designs much faster. In this scope, IQC based tools already present promising result and show that they may be the most appropriate ones for the robustness analysis of large complex systems. They account for the system structure and allow dealing specifically with each subsystems, it means that we can improve the representation contained in the multipliers easily and reuse the set up to assess the improvements. The flexibility of the method is a huge advantage. We experienced it during two phases. The first was dedicated to the analysis of the three-degree-of-freedom uncertain nonlinear equation of motion of a rigid body. Secondly, we studied the influence of the pulse-width modulator behavior of the attitude control system on the launcher stability. IQC-based stability analysis allowed defining estimations of the stability domain with respect to uncertainties and system parameters. Moreover, the results obtained with IQC can go way beyond stability analysis with performance analysis with description of the particular performance criteria of the field with appropriate multipliers. Later on controller synthesis and merging of IQC method with worst-case search algorithms could extend greatly the frame of use of this analytical tool and give it the influence it deserves.

The study of non-linear input-output maps can be summarized by three concepts: Gain, Positivity and Dissipativity. However, in order to make efficient use of these theorems it is necessary to use loop transformations and weightings, or so called ”multipliers”.The first problem this thesis studies is the feedback interconnection of a Linear Time Invariant system with a memoryless bounded and monotone non-linearity, or so called Absolute Stability problem, for which the test for stability is equivalent to show the existence of a Zames-Falb multiplier. The main advantage of this approach is that Zames–Falb multipliers can be specialized to recover important tools such as Circle criterion and the Popov criterion. Albeit Zames-Falb multipliers are an efficient way of describing non-linearities in frequency domain, the Fourier transform of the multiplier does not preserve the L1 norm. This problem has been addressed by two paradigms: mathematically complex multipliers with exact L1 norm and multipliers with mathematically tractable frequency domain properties but approximate L1 norm. However, this thesis exposes a third factor that leads to conservative results: causality of Zames-Falb multipliers. This thesis exposes the consequences of narrowing the search Zames-Falb multipliers to causal multipliers, and motivated by this argument, introduces an anticausal complementary method for the causal multiplier synthesis in [1].The second subject of this thesis is the feedback interconnection of two bounded systems. The interconnection of two arbitrary systems has been a well understood problem from the point of view of Dissipativity and Passivity. Nonetheless, frequency domain analysis is largely restricted for passive systems by the need of canonically factorizable multipliers, while Dissipativity mostly exploits constant multipliers. This thesis uses IQC to show the stability of the feedback interconnection of two non-linear systems by introducing an equivalent representation of the IQC Theorem, and then studies formally the conditions that the IQC multipliers need. The result of this analysis is then compared with Passivity and Dissipativity by a series of corollaries.