A new suboptimal guidance law design approach for aerospace vehicles is proposed in this thesis, followed by an advanced control design for supersonic air-breathing engines. The guidance law is designed using the newly developed Generalized Model Predictive Static Programming (G-MPSP), which is based on the continuous time nonlinear optimal control framework. The key feature of this technique is one-time backward propagation of a small-dimensional weighting matrix dynamics, which is used to update the entire control history. This key feature, as well as the fact that it leads to a static optimization problem, lead to its computational efficiency. It has also been shown that the existing model predictive static programming (MPSP), which is based on the discrete time framework, is a special case of G-MPSP. The G-MPSP technique is further extended to incorporate ‘input inequality constraints’ in a limited sense using the penalty function philosophy. Next, this technique has been developed also further in a ‘flexible final time’ framework to converge rapidly to meet very stringent final conditions with limited number of iterations. Using the G-MPSP technique in a flexible final time and input inequality constrained formulation, a suboptimal guidance law for a solid motor propelled carrier launch vehicle is successfully designed for a hypersonic mission. This guidance law assures very stringent final conditions at the injection point at the end of the guidance phase for successful beginning of the hypersonic vehicle operation. It also ensures that the angle of attack and structural load bounds are not violated throughout the trajectory. A second-order autopilot has been incorporated in the simulation studies to mimic the effect of the inner-loops on the guidance performance. Simulation studies with perturbations in the thrust-time behaviour, drag coefficient and mass demonstrate that the proposed guidance can meet the stringent requirements of the hypersonic mission. The G-MPSP technique in a fixed final time and input inequality constrained formulation has also been used for optimal guidance of an aerospace vehicle propelled by supersonic air-breathing engine, where the resulting thrust can be manipulated by managing the fuel flow and nozzle area (which is not possible in solid motors). However, operation of supersonic air-breathing engines is quite complex as the thrust produced by the engine is a result of very complex nonlinear combustion dynamics inside the engine. Hence, to generate the desired thrust, accounting for a fairly detailed engine model, a dynamic inversion based nonlinear state feedback control design has been carried out. The objective of this controller is to ensure that the engine dynamically produces the thrust that tracks the commanded value of thrust generated from the guidance loop as closely as possible by regulating the fuel flow rate. Simultaneously, by manipulating throat area of the nozzle, it also manages the shock wave location in the intake for maximum pressure recovery with sufficient margin for robustness. To filter out the sensor and process noises and to estimate the states for making the control design operate based on output feedback, an extended Kalman filter (EKF) based state estimation design has also been carried out and the controller has been made to operate based on estimated states. Moreover, independent control designs have also been carried out for the actuators so that their response can be faster. In addition, this control design becomes more challenging to satisfy the imposed practical constraints like fuel-air ratio and peak combustion temperature limits. Simulation results clearly indicate that the proposed design is quite successful in assuring the desired performance of the air-breathing engine throughout the flight trajectory, i.e., both during the climb and cruise phases, while assuring adequate pressure margin for shock wave management.

Cette thèse est une contribution au calcul des variations et à la théorie du contrôle optimal dans les cadres discret, plus généralement time scale, et fractionnaire. Ces deux domaines ont récemment connu un développement considérable dû pour l’un à son application en informatique et pour l’autre à son essor dans des problèmes physiques de diffusion anormale. Que ce soit dans le cadre time scale ou dans le cadre fractionnaire, nos objectifs sont de : a) développer un calcul des variations et étendre quelques résultats classiques (voir plus bas); b) établir un principe du maximum de Pontryagin (PMP en abrégé) pour des problèmes de contrôle optimal. Dans ce but, nous généralisons plusieurs méthodes variationnelles usuelles, allant du simple calcul des variations au principe variationnel d’Ekeland (couplé avec la technique des variations-aiguilles), en passant par l’étude d’invariances variationnelles par des groupes de transformations. Les démonstrations des PMPs nous amènent également à employer des théorèmes de point fixe et à prendre en considération la technique des multiplicateurs de Lagrange ou encore une méthode basée sur un théorème d’inversion locale conique. Ce manuscrit est donc composé de deux parties : la Partie 1 traite de problèmes variationnels posés sur time scale et la Partie 2 est consacrée à leurs pendants fractionnaires. Dans chacune de ces deux parties, nous suivons l’organisation suivante : 1. détermination de l’équation d’Euler-Lagrange caractérisant les points critiques d’une fonctionnelle Lagrangienne ; 2. énoncé d’un théorème de type Noether assurant l’existence d’une constante de mouvement pour les équations d’Euler-Lagrange admettant une symétrie ; 3. énoncé d’un théorème de type Tonelli assurant l’existence d’un minimiseur pour une fonctionnelle Lagrangienne et donc, par la même occasion, d’une solution pour l’équation d’Euler-Lagrange associée (uniquement en Partie 2) ; 4. énoncé d’un PMP (version forte en Partie 1, version faible en Partie 2) donnant une condition nécessaire pour les trajectoires qui sont solutions de problèmes de contrôle optimal généraux non-linéaires ; 5. détermination d’une condition de type Helmholtz caractérisant les équations provenant d’un calcul des variations (uniquement en Partie 1 et uniquement dans les cas purement continu et purement discret). Des théorèmes de type Cauchy-Lipschitz nécessaires à l’étude de problèmes de contrôle optimal sont démontrés en Annexe. / This dissertation deals with the mathematical fields called calculus of variations and optimal control theory. More precisely, we develop some aspects of these two domains in discrete, more generally time scale, and fractional frameworks. Indeed, these two settings have recently experience a significant development due to its applications in computing for the first one and to its emergence in physical contexts of anomalous diffusion for the second one. In both frameworks, our goals are: a) to develop a calculus of variations and extend some classical results (see below); b) to state a Pontryagin maximum principle (denoted in short PMP) for optimal control problems. Towards these purposes, we generalize several classical variational methods, including the Ekeland’s variational principle (combined with needle-like variations) as well as variational invariances via the action of groups of transformations. Furthermore, the investigations for PMPs lead us to use fixed point theorems and to consider the Lagrange multiplier technique and a method based on a conic implicit function theorem. This manuscript is made up of two parts : Part A deals with variational problems on time scale and Part B is devoted to their fractional analogues. In each of these parts, we follow (with minor differences) the following organization: 1. obtaining of an Euler-Lagrange equation characterizing the critical points of a Lagrangian functional; 2. statement of a Noether-type theorem ensuring the existence of a constant of motion for Euler-Lagrange equations admitting a symmetry;3. statement of a Tonelli-type theorem ensuring the existence of a minimizer for a Lagrangian functional and, consequently, of a solution for the corresponding Euler-Lagrange equation (only in Part B); 4. statement of a PMP (strong version in Part A and weak version in Part B) giving a necessary condition for the solutions of general nonlinear optimal control problems; 5. obtaining of a Helmholtz condition characterizing the equations deriving from a calculus of variations (only in Part A and only in the purely continuous and purely discrete cases). Some Picard-Lindelöf type theorems necessary for the analysis of optimal control problems are obtained in Appendices.