Sufficient Conditions For Optimal Control Problems With Terminal Constraints And Free Terminal Times With Applications To Aerospace

Sufficient Conditions for Optimal Control Problems with Terminal Constraints and Free Terminal Times with Applications to Aerospace

Motivated by the flight control problem of designing control laws for a Ground Collision Avoidance System (GCAS), this thesis formulates sufficient conditions for a strong local minimum for a terminally constrained optimal control problem with a free-terminal time. The conditions develop within the framework of a construction of a field of extremals by means of the method of characteristics, a procedure for the solution of first-order linear partial differential equations, but modified to apply to the Hamilton-Jacobi-Bellman equation of optimal control. Additionally, the thesis constructs these sufficient conditions for optimality with a mathematically rigorous development. The proof uses an approach which generalizes and differs significantly from procedures outlined in the classical literature on control engineering, where similar formulas are derived, but only in a cursory, formal and sometimes incomplete way. Additionally, the thesis gives new arrangements of the relevant expressions arising in the formulation of sufficient conditions for optimality that lead to more concise formulas for the resulting perturbation feedback control schemes.These results are applied to an emergency perturbation-feedback guidance scheme which recovers an aircraft from a dangerous flight-path angle to a safe one. Discussion of required background material contrasts nonlinear and linear optimal control theory are contrasted in the context of aerospace applications. A simplified version of the classical model for an F-16 fighter aircraft is used in numerical computation to very, by example, that the sufficient conditions for optimality developed in this thesis can be used off-line to detect possible failures in perturbation feedback control schemes, which arise if such methods are applied along extremal controlled trajectories and which only satisfy the necessary conditions for optimality without being locally optimal. The sufficient conditions for optimality developed in this thesis, on the other hand, guarantee the local validity of such perturbation feedback control schemes. This thesis presents various graphs that compare the neighboring extremals which were derived from the perturbation feedback control scheme with optimal ones that start from the same initial condition.Future directions for this work include extending the perturbation feedback control schemes to optimization problems that are further constrained, possibly through control constraints, state-space constraints or mixed state-control constraints.

Optimal Control with Aerospace Applications

Want to know not just what makes rockets go up but how to do it optimally? Optimal control theory has become such an important field in aerospace engineering that no graduate student or practicing engineer can afford to be without a working knowledge of it. This is the first book that begins from scratch to teach the reader the basic principles of the calculus of variations, develop the necessary conditions step-by-step, and introduce the elementary computational techniques of optimal control. This book, with problems and an online solution manual, provides the graduate-level reader with enough introductory knowledge so that he or she can not only read the literature and study the next level textbook but can also apply the theory to find optimal solutions in practice. No more is needed than the usual background of an undergraduate engineering, science, or mathematics program: namely calculus, differential equations, and numerical integration. Although finding optimal solutions for these problems is a complex process involving the calculus of variations, the authors carefully lay out step-by-step the most important theorems and concepts. Numerous examples are worked to demonstrate how to apply the theories to everything from classical problems (e.g., crossing a river in minimum time) to engineering problems (e.g., minimum-fuel launch of a satellite). Throughout the book use is made of the time-optimal launch of a satellite into orbit as an important case study with detailed analysis of two examples: launch from the Moon and launch from Earth. For launching into the field of optimal solutions, look no further!

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