Survey of Numerical Methods for Trajectory Optimization

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

This paper gives a survey of the various forms of Pontryagin’s maximum principle for optimal control problems with state variable inequality constraints. The relations between the different sets of optimality conditions arising in these forms are shown. Furthermore, the application of these maximum principle conditions is demonstrated by solving some illustrative examples.

A survey of the maximum principles for optimal control problems with state constraints

From the Publisher:
Introduction to Numerical Continuation Methods continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra is adequate preparation for reading this book; some knowledge from a first course in numerical analysis may also be helpful.

/pdf/introduction-to-numerical-continuation-methods-ily9kpadk5.pdf

Introduction to Numerical Continuation Methods

생체의 모델링과 Simulation

A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically.

Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.

http://www.ingmath.uni-bayreuth.de/LEHRE/ps-files/Oberle-Pesch_81.pdf

Numerical treatment of delay differential equations by Hermite Interpolation

In Part 1 of the paper (Ref. 2), we have shown that the necessary conditions for the optimal control problem of the abort landing of a passenger aircraft in the presence of windshear result in a multipoint boundary-value problem. This boundary-value problem is especially well suited for numerical treatment by the multiple shooting method. Since this method is basically a Newton iteration, initial guesses of all variables are needed and assumptions about the switching structure have to be made. These are big obstacles, but both can be overcome by a so-called homotopy strategy where the problem is imbedded into a one-parameter family of subproblems in such a way that (at least) the first problem is simple to solve. The solution data to the first problem may serve as an initial guess for the next problem, thus resulting in a whole chain of problems. This process is to be continued until the objective problem is reached. Techniques are presented here on how to handle the various changes of the switching structure during the homotopy run. The windshear problem, of great interest for safety in aviation, also serves as an excellent benchmark problem: Nearly all features that can arise in optimal control appear when solving this problem. For example, the candidate for an optimal trajectory of the minimax optimal control problem shows subarcs with both bang-bang and singular control functions, boundary arcs and touch points of two state constraints, one being of first order and the other being of third order, etc. Therefore, the results of this paper may also serve as some sort of user's guide for the solution of complicated real-life optimal control problems by multiple shooting. The candidate found for an optimal trajectory is discussed and compared with an approximate solution already known (Refs. 3–4). Besides the known necessary conditions, additional sharp necessary conditions based on sign conditions of certain multipliers are also checked. This is not possible when using direct methods.

Abort landing in the presence of windshear as a minimax optimal control problem, part 2: multiple shooting and homotopy

A Practical Guide to the Solution of Real-Life Optimal Control Problems

The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.

http://www.ingmath.uni-bayreuth.de/LEHRE/ps-files/Bulirsch-Montrone-Pesch_91a.pdf

Abort landing in the presence of windshear as a minimax optimal control problem, part 1: necessary conditions

A numerical method is developed for the real-time computation of neighbouring optimal feedback controls for constrained optimal control problems. The first part of this paper presents the theory of neighbouring extremals. Besides a survey of the theory of neighbouring extremals, special emphasis is laid on the inclusion of complex constraints, e.g. state and control variable inequality constraints and discontinuities of the system equations at interior points. The numerical treatment of these constraints is particularly emphasized. The linearization of all necessary conditions of optimal control theory leads to a linear, mulitpoint, boundary value problem with linear jump conditions that is especially well suited for numerical treatment.

Hans Josef Pesch

Papers

Numerical treatment of delay differential equations by Hermite Interpolation

Abort landing in the presence of windshear as a minimax optimal control problem, part 2: multiple shooting and homotopy

A Practical Guide to the Solution of Real-Life Optimal Control Problems

Abort landing in the presence of windshear as a minimax optimal control problem, part 1: necessary conditions

Real‐time computation of feedback controls for constrained optimal control problems. part 1: Neighbouring extremals