2014 | Book

# Optimal Control with Aerospace Applications

Authors: James M Longuski, José J. Guzmán, John E. Prussing

Publisher: Springer New York

Book Series : Space Technology Library

2014 | Book

Authors: James M Longuski, José J. Guzmán, John E. Prussing

Publisher: Springer New York

Book Series : Space Technology Library

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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Abstract

Two major branches of optimization are: parameter optimization and optimal control theory. In parameter optimization (a problem of finite dimensions, that is, where the parameters are not functions of time) we minimize a function of a finite number of parameters.

Abstract

We start this chapter with what might be called the fundamental trajectory optimization problem: launching a satellite into orbit. Before setting up the mathematical model for this problem, let us first discuss in words what the problem entails.

Abstract

The brachistochrone problem posed by Johann Bernoulli was a new type of mathematical problem which required a new mathematical approach. Lagrange developed the calculus of variations in which he considered suboptimal paths nearby the optimal one. He then showed that, for arbitrary but infinitesimal variations from the optimal path, the function sought must obey a differential equation now known as the Euler-Lagrange equation.

Abstract

In this chapter we will look at some applications of the Euler-Lagrange theorem. The theorem transforms the Problem of Bolza into a set of differential equations and attendant boundary conditions. In some cases, simple closed-form solutions are available which completely solve the problem. In other cases, numerical methods are required to solve the “two-point boundary-value problem.” In some instances we find that the Euler-Lagrange theorem does not supply enough conditions to determine the optimal control law. In such cases we appeal to another theorem (the Weierstrass condition or Minimum Principle, discussed in Chap. 5) to solve the problem.

Abstract

In Chap. 4 we noted that there are optimization problems that cannot be resolved by the Euler-Lagrange theorem alone. Pontryagin’s Minimum Principle often provides an additional condition that leads to a specific control law and to the solution of the problem. The most general form of the Minimum Principle is stated in Chap. 6 without proof.

Abstract

The Weierstrass condition, which requires the Hamiltonian to be minimized over the set of all admissible controls, is a powerful tool for solving a class of optimization problems that do not immediately yield to our familiar algorithm with the Euler-Lagrange equations and the transversality condition. However, the Weierstrass condition’s “set of all admissible controls” is limited to continuously differentiable, unbounded functions, which are by no means the only feasible controls in practice or in principle. For example, “bang-bang” or “on-off” control schemes are frequently employed in everyday engineering applications, but these controls do not fall within the Weierstrass condition’s set.

Abstract

Why discuss aircraft performance in a text chiefly about optimization of spacecraft trajectories? Because aerospace engineers sometimes wish to capture the best of both worlds—as in the examples of Pegasus and SpaceShipOne which use aircraft as first stages and rockets as second stages to send vehicles into space.

Abstract

So far, we have discussed four sets of necessary conditions which must be met by \(\boldsymbol{{x}}^{{\ast}}(t)\) and \(\boldsymbol{{u}}^{{\ast}}(t)\), over the class of admissible functions: Necessary Condition I:

Abstract

We recall that the general form of the minimization problem can be stated as

Abstract

In this chapter we develop a general theory of optimal spacecraft trajectories based on two pioneering works: Breakwell [1959] and Lawden [1963]. Lawden introduced the concept of the primer vector, which plays a dominant role in minimum-propellant trajectories and also in other types of optimal trajectories. A more complete discussion of the topics in this chapter, including several example trajectories, is in Prussing [2010].