Hans Josef Pesch

Numerical treatment of delay differential equations by Hermite Interpolation

SummaryA 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.

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

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 users guide to solve inequality constrained real-life optimal control problems by multiple shooting.

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

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 users 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.

New general guidance method in constrained optimal control, part 1: numerical method

A very fast numerical method is developed for the computation of neighboring optimum feedback controls. This method is applicable to a general class of optimal control problems (for example, problems including inequality constraints and discontinuities) and needs no on-line computation, except for one matrix-vector multiplication. The method is based on the so-called accessory minimum problem. The necessary conditions for this auxiliary optimal control problem form a linear multipoint boundary-value problem with linear jump conditions, which is especially well suited for numerical treatment. In the second part of this paper, the performance of the guidance scheme is shown for the heating-constrained cross-range maximization problem of a space-shuttle-orbiter-type vehicle.

Solution differentiability for parametric nonlinear control problems with control-state constraints

This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameterp of a Banach space. Using recent second-order sufficient conditions (SSC), it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parameteric boundary-value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. This is achieved by building a bridge between the variational system corresponding to the boundary-value problem, solutions of the associated Riccati ODE, and SSC.Solution differentiability provides a theoretical basis for performing a numerical sensitivity analysis of first order. Two numerical examples are worked out in detail that aim at reducing the considerable deficit of numerical examples in this area of research.

Complex differential games of pursuit-evasion type with state constraint, Part 1: necessary conditions for optimal open-loop strategies

Complex pursuit-evasion games with state variable inequality constraints are investigated. Necessary conditions of the first and the second order for optimal trajectories are developed, which enable the calculation of optimal open-loop strategies. The necessary conditions on singular surfaces induced by state constraints and non-smooth data are discussed in detail. These conditions lead to multi-point boundary-value problems which can be solved very efficiently and very accurately by the multiple shooting method. A realistically modelled pursuit-evasion problem for one air-to-air missile versus one high performance aircraft in a vertical plane serves as an example. For this pursuit-evasion game, the barrier surface is investigated, which determines the firing range of the missile. The numerical method for solving this problem and extensive numerical results will be presented and discussed in Part 2 of this paper; see Ref. 1.

Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting

SummaryThe numerical solution of two-point boundary value problems and problems of optimal control by shooting techniques requires integration routines. By solving 15 real-life problems four well-known intergrators are compared relative to reliability, fastness and precision. Hints are given, which routines could be used for a problem.

Combining Direct and Indirect Methods in Optimal Control: Range Maximization of a Hang Glider

When solving optimal control problems, indirect methods such as multiple shooting suffer from difficulties in finding an appropriate initial guess for the adjoint variables. For, this initial estimate must be provided for the iterative solution of the multipoint boundary-value problems arising from the necessary conditions of optimal control theory. Direct methods such as direct collocation do not suffer from this problem, but they generally yield results of lower accuracy and their iteration may even terminate with a non-optimal solution. Therefore, both methods are combined in such a way that the direct collocation method is at first applied to a simplified optimal control problem where all inequality constraints are neglected as long as the resulting problem is still well-defined. Because of the larger domain of convergence of the direct method, an approximation of the optimal solution of this problem can be obtained easier. The fusion between direct and indirect methods is then based on a relationship between the Lagrange multipliers of the underlying nonlinear programming problem to be solved by the direct method and the adjoint variables appearing in the necessary conditions which form the boundary-value problem to be solved by the indirect method. Hence, the adjoint variables, too, can be estimated from the approximation obtained by the direct method. This first step then facilitates the subsequent extension and competition of the model by homotopy techniques and the solution of the arising boundary-value problems by the indirect multiple shooting method. Proceeding in this way, the high accuracy and reliability of the multiple shooting method, especially the precise computation of the switching structure and the possibility to verify many necessary conditions, is preserved while disadvantages caused by the sensitive dependence on an appropriate estimate of the solution are considerably cut down. This procedure is described in detail for the numerical solution of the maximum-range trajectory optimization problem of a hang glider in an upwind which provides an example for a control problem where appropriate initial estimates for the adjoint variables are hard to find.

A modified continuation method for the numerical solution of nonlinear two-point boundary value problems by shooting techniques

SummaryA modification of the well-known continuation (or homotopy) method for actual computation is worked out. Compared with the classical method, the modification seems to be a more reliable device for supplying useful initial data for shooting techniques. It is shown that computing time may be significantly reduced in the numerical solution of sensitive realistic two-point boundary value problems.

Solution Differentiability for Nonlinear Parametric Control Problems

Perturbed nonlinear control problems with data depending on a vector parameter are considered. Using second-order sufficient optimality conditions, it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof exploits the close connections between solutions of a Riccati differential equation and shooting methods for solving the associated boundary value problem. Solution differentiability provides a firm theoretical basis for numerical feedback schemes that have been developed for computing neighbouring extremals. The results are illustrated by an example that admits two extremal solutions. Second-order sufficient conditions single out one optimal solution for which a sensitivity analysis is carried out.