Robert R. Bless

Weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamiltons weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamiltons principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamiltons weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Finite element method for optimal guidance of an advanced launch vehicle

A temporal finite element based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables that are expanded in terms of nodal values and simple shape functions. Unlike other variational approaches to optimal control problems, however, time derivatives of the states and costates do not appear in the governing variational equation. Instead, the only quantities whose time derivatives appear therein are virtual states and virtual costates. Numerical results are presented herein for an elementary trajectory optimization problem which show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The goal of this work is to evaluate the feasibility of this approach for real-time guidance applications. A simplified model for an advanced launch vehicle application is presented which is suitable for finite element solution. Numerical results for this model will be presented in a later paper.

Finite Element Solution of Optimal Control Problems with State-Control Inequality Constraints

NEW method is presented for solving optimal control problems with inequality constraints that are functions of both control and state variables. The method is based on the weak Hamiltonian formulation derived in Ref. 1 and uses finite elements in time. The extension of the weak formulation is already outlined in Ref. 2 where the methodology is applied to a two-stage rocket trajectory optimization problem. This Note is intended to fill in some of the details omitted in Ref. 2 concerning the solution of problems with state-control inequality constraints. Herein it is shown that the formulation does not require element quadrature, and it produces a sparse system of nonlinear algebraic equations. Since these algebraic equations may be derived before specifying the problem to be solved, the formulation is conducive to development of a general purpose computational environment for the solution of a large class of optimal control problems. After the derivation is given, a simple example problem is presented. The numerical results are compared with the exact solution. Of particular interest is the performance in terms of execution time and accuracy vs the number of elements used to represent the time span of the problem.

Finite Element Method for the Solution of State-Constrained Optimal Control Problems

This paper presents an extension of a finite element formulation based on a weak form of the necessary conditions to solve optimal control problems. First, a general formulation for handling internal boundary conditions and discontinuities in the state equations is presented. Then, the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions with touch points and solutions with stateconstrained arcs are considered. After the formulations are developed, suitable shape and test functions are chosen for a finite element discretization. It is shown that all element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form, yielding a set of algebraic equations. To demonstrate and analyze the accuracy of the finite element method, a simple state-constrained problem is solved. Then, for a more practical application of the use of this method, a launch vehicle ascent problem subject to a dynamic pressure constraint is solved. The paper also demonstrates that the finite element results can be used to determine switching structures and initial guesses for a shooting code.

Axial instability of rotating rods revisited

Abstract For strain sufficiently small such that Hookes Law is valid, it is shown that only a linear model for axial deformation of rotating rods can be derived. As discussed in the literature, this linear model exhibits an instability when the angular speed reaches a certain critical value. However, unless this linear model is valid for large strain, it is impossible to determine whether this instability really exists; because, as the angular speed is increased, the strain becomes large well short of the critical speed. Next, axial deformation of rotating rods is analyzed using two strain energy functions to model non-linear elastic behavior. The first of these functions is the usual quadratic strain energy function augmented with a cubic term. With this model it is shown that no instability exists if the non-linearity is stiffening (i.e. if the coefficient of the cubic term is positive), although the strain can become large. If the non-linearity is of the softening variety, then the critical angular speed drops as the degree of softening increases. Still, the strains are large enough that, except for rubber-like materials, a non-linear elastic model is not likely to be appropriate. The second strain energy function is based on the square of the logarithmic strain and yields a softening model. It quite accurately models the behavior of certain rubber rods which exhibit the instability within the validated range of elongation.

Analysis of beam contact problems via optimal control theory

A class of large-deflection contact problems for beams is treated within the framework of optimal control theory, using an automated software system developed for variational optimization (e.g., trajectory optimization). Analysis of both Euler-Bernoulli and Timoshenko-type models is presented. Essentially exact numerical solutions are obtained by shooting, and the accuracy of a very efficient mixed finite element approach on which the automated software is based is demonstrated. Advantages of posing the problem within the optimal control framework are discussed. These include 1) being able to establish from the Euler-Bernoulli equations the existence of a physically inappropriate touch-point solution and 2) the ease by which an integral of the governing equations can be found to independently check the accuracy of a numerical solution

State-constrained booster trajectory solutions via finite elements and shooting

This paper presents an extension of a FEM formulation based on variational principles. A general formulation for handling internal boundary conditions and discontinuities in the state equations is presented, and the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions which only touch the state constraint and solutions which have a boundary arc of finite length are considered. Suitable shape and test functions are chosen for a FEM discretization. All element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form. The final form of the algebraic equations is then derived. A simple state-constrained problem is solved. Then, for a practical application of the use of the FEM formulation, a launch vehicle subject to a dynamic pressure constraint (a first-order state inequality constraint) is solved. The results presented for the launch-vehicle trajectory have some interesting features, including a touch-point solution.

A Hybrid Symbolic/Finite-Element Algorithm for Solving Nonlinear Optimal Control Problems

The general code described is capable of solving difficult nonlinear optimal control problems by using finite elements and a symbolic manipulator. Quick and accurate solutions are obtained with a minimum for user interaction. Since no user programming is required for most problems, there are tremendous savings to be gained in terms of time and money.