The report deals with theoretical and computational aspects or optimal control problems with state variable inequality constraints. The computational methods developed here solve bounded state variable problems by searching the constraint surface (or boundary) for the optimal junction point between the subarc on the boundary and the unconstrained subarc. The results obtained for the bounded state variables problem are extended to problems with discontinuous state variable and/or discontinuous differential equations. (Author).
The topic for this thesis is the state-variable inequality constrainted optimal control problem. The problem is formulated as a standard optimal control problem with one additional constraint of the form S(x(t)) or = 0. This constraint is assumed to be of p-th order where p is an integer and p or = 1. In particular, the p-th time derivative of the constraint is the first to contain the control variable explicity.
This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control problems with state variable inequality constraints in the framework of continuous linear programming. The duality theory in this framework makes it possible to relate the adjoint variables arising in different formulations of a problem; these relationships are illustrated by the use of a simple example. This framework also allows more general problems and admits a simplex-like algorithm to solve these problems.
While optimality conditions for optimal control problems with state constraints have been extensively investigated in the literature the results pertaining to numerical methods are relatively scarce. This book fills the gap by providing a family of new methods. Among others, a novel convergence analysis of optimal control algorithms is introduced. The analysis refers to the topology of relaxed controls only to a limited degree and makes little use of Lagrange multipliers corresponding to state constraints. This approach enables the author to provide global convergence analysis of first order and superlinearly convergent second order methods. Further, the implementation aspects of the methods developed in the book are presented and discussed. The results concerning ordinary differential equations are then extended to control problems described by differential-algebraic equations in a comprehensive way for the first time in the literature.
Computational Optimal Control: Tools and Practice provides a detailed guide to informed use of computational optimal control in advanced engineering practice, addressing the need for a better understanding of the practical application of optimal control using computational techniques. Throughout the text the authors employ an advanced aeronautical case study to provide a practical, real-life setting for optimal control theory. This case study focuses on an advanced, real-world problem known as the “terminal bunt manoeuvre” or special trajectory shaping of a cruise missile. Representing the many problems involved in flight dynamics, practical control and flight path constraints, this case study offers an excellent illustration of advanced engineering practice using optimal solutions. The book describes in practical detail the real and tested optimal control software, examining the advantages and limitations of the technology. Featuring tutorial insights into computational optimal formulations and an advanced case-study approach to the topic, Computational Optimal Control: Tools and Practice provides an essential handbook for practising engineers and academics interested in practical optimal solutions in engineering. Focuses on an advanced, real-world aeronautical case study examining optimisation of the bunt manoeuvre Covers DIRCOL, NUDOCCCS, PROMIS and SOCS (under the GESOP environment), and BNDSCO Explains how to configure and optimize software to solve complex real-world computational optimal control problems Presents a tutorial three-stage hybrid approach to solving optimal control problem formulations
A slack variable is used to transform an optimal control problem with a scalar control and a scalar inequality constraint on the state variables into an unconstrained problem of higher dimension. It is shown that, for a p-th order constraint, the p-th time derivative of the slack variable becomes the new control variable. The usual Pontryagin Principle of Lagrange multiplier rule gives necessary conditions of optimality. There are no discontinuities in the adjoint variables. A feature of the transformed problem is that any nominal control function produces a feasible trajectory. The optimal trajectory of the transformed problem exhibits singular arcs which correspond, in the original constrained problem, to arcs which lie along the constraint boundary; this suggests a duality between singular and state-constrained problems, which should be explored. Generalizations of the approach to cases where the constraint and the control are vectors of equal dimension, as well as to problems involving multiple constraints and a single control variable, are considered. Owing to the appearance of singular arcs in the solution of the transformed problem, a direct application of second-order or second-variation algorithms is not possible. However, gradient or conjugate gradient methods are applicable and computations, using the conjugate gradient method, are presented to illustrate the usefulness of the transformation technique. (Author).
"It is the purpose of this text to provide in introduction to the development and utilization of techniques applicable to the solution of optimal control problems. Such problems are within the domain of system optimization theory. It is felt that the text is a suitable beginning point for the engineering reader interested in the fields of optimal control and system optimization. No prerequisites in control theory are required and use of the text is not limited to any one special field of engineering. Several methods of formulating and solving deterministic optimal control problems are presented." --Preface.
A review of the computational method of conjugate gradients for linear and nonlinear operator equations is given with emphasis in applying this technique to state variable constraint control problems. The first and second Frechet derivatives of the performance functional are derived. The search directions generated in the iteration process for the optimal control are locally conjugate with respect to the second Frechet derivative. The convergence is along the expanding sequence of sets, the itersection of the linear spaces spanned by the search directions and the set of admissible controls. The computational technique is applied to two state variable constraint problems, in one of which a penalty function is employed to convert the constraint problem to an unconstrained one in addition to the approach considering the constraints directly. For this same problem the method of steepest descent also is studied, and comparison of the results obtained is made and discussed. (author).
