Adjoint Differential Equations Research Articles

Trajectory Optimization Using Nonsingular Orbital Elements and True Longitude

The system and adjoint differential equations needed to solve low-thrust transfer and rendezvous trajectories are derived in terms of nonsingular orbit elements, with the true longitude selected as the sixth variable that also dee nes the radial distance. The perturbation accelerations are resolved in the rotating polar frame such that the equations of motion are written in polarcoordinates. This formulation is particularly convenient forthe treatment of the J2 perturbation acceleration whose components are easily expressed in terms of the true longitude. The iterations needed for the solution of Kepler’ s equation are also eliminated, and the analytic form of the adjoint equations is provided in their simplest aspect thus far.

Mechanics of Trajectory Optimization Using Nonsingular Variational Equations in Polar Coordinates

Thecompletesetofthestateand adjoint differentialequationsforthenonsingularequinoctialorbitelementsexpressed in polar coordinatesand written in terms of the truelongitudeispresented. Previous formulations adopted the equinoctial frame as the orbital frame for component resolution of the various perturbation accelerations. The consideration of the position dependency of the J2 acceleration in the context of precision integrated orbital transfer trajectories led us to adopt the more convenient polar frame coupled with the use of the true longitude as the accessory variable needed in the description of the variational equations, inasmuch as it is more dife cult to generate the contribution of the J2 perturbation to the adjoint equations if the variational equations are left in terms of the eccentric longitude. I. Introduction T HEconsideration of thenonsingularequinoctialorbitelements has been of great benee t in trajectory propagation and optimization.References1 ‐7 madeuseofthecorresponding variational equations resolved in the equinoctial orbital coordinate system. In Refs. 8‐12, modern numerical methods based on the techniques of collocation and nonlinear programming are applied to the solutionoflow-thrustEarth-boundandinterplanetarytrajectories.These direct methods have introduced robustness characteristics to the optimization process and were e rst used with great success by launch vehicle trajectory optimization and performance analysts. In particular, Betts 9,10 uses a variety of equinoctial element sets within the framework of the direct method. Following the more traditional approach based on the calculus of variations and the indirect shooting method,thispaper developsthe differentialequationsforthe adjoint or multiplier variables to be numerically integrated simultaneously withthestatedifferentialequationstosolvethetwo-point-boundaryvalue orbit transfer problem. As noted by Betts, the elements of the matrix of partial derivatives of the Hamiltonian with respect to the equinoctial elements derived here in analytic form can be used to developacertainanalyticJacobianmatrixinconnectionwithnonlinear programming (NLP) solvers used with direct methods, thereby saving considerable computation time with respect to the adoption ofa numerically determined Jacobian. The variational equations are usually expressed in terms of the eccentric longitude, which can either be left as an accessory variable or be adopted as the fastor sixth orbital element. In Ref. 3, the averaged J2 perturbation effect was accounted for in the design of the optimal transfer trajectory. However, when the exact or rather position-dependent J2 effect must be considered in the context of precision integrated transfer solutions, the J2 perturbation acceleration components become dependent on the eccentric longitude. Because these components are given in the rotating Euler‐Hill or polar frame, it is much simpler to use this frame as the orbital frame instead of the equinoctial framebecause otherwise the transformed expressions for these components become more complicated. These expressions are still quite complicated if expressed in terms of the eccentric longitude even if the polar frame is used. This is important because we must take the partial derivatives of these expressions with respect to the elements to generate the adjoint differential equations. This task is made easier by expressing the variational equations for the J2 perturbation in terms of the true

Adjoint variable solutions via an auxiliary optimization problem

The problem addressed is to obtain an initial guess for the adjoint variables along a trajectory suitable for use in an indirect optimal control method. The proposed method assumes that good estimates of the control and state histories are available. The adjoint estimates are obtained through an auxiliary unconstrained optimization problem of low dimension. The goal of this optimization problem is to find the adjoint values at nodes along the trajectory that cause the control values from the application of the minimum principle to approach the estimated control values as closely as possible in a manner consistent with the adjoint differential equations and transversality conditions. Experience has shown that the method produces adjoint variable estimates sufficiently accurate to cause convergence of a multiple-shooting indirect method. Nomenclature H = Hamiltonian u — p-dimensional control vector x - n -dimensional vector of state variables A = n -dimensional vector of adjoint variables \I> = m -dimensional vector of constraints at terminal time, m < n

Adjoint Differential Equations and Oscillation

The paper offers effective oscillation criteria for the equation [formula] where p i and q are positive continuous functions on an interval [ a, ∞) in the case n ≥ 4 is even and the operator [formula] is not in canonical form. The technique is established on properties of adjoint equations of an equation r n ( t)( r n−1 ( t)...( r 1( t)( r 0( t) y)′)′...)′ + cy = 0 with 0 ≠ c ∈ R and r i ( i = 0,1,..., n) positive continuous functions. The conception of adjoint equations is stated in the paper.

PANSYM: a symbolic equation generator for mathematical modelling, analysis and control of metabolic and pharmacokinetic systems

Software is presented for automatic generation of fist-order ordinary differential equations (ODE) that arise from lumped parameter representations of metabolic and pharmacokinetic systems. The definition of system structures is accomplished by fractional transfer rates between state variables, together with input/output equations and initial conditions of state variables. General non-linear mathematical expressions can be assigned to all structure definition items. The software parses and interprets the system definitions and generates symbolically the mathematical expression of the model's set of ODE. In addition, symbolic derivatives of state equations are determined with respect to model parameters, state variables and external inputs. These derivatives represent the constituents of systems of sensitivity-differential and adjoint-differential equations that arise in identification and optimal control problems. Finally, output routines generate source code that, once compiled and linked to simulation programs, allows efficient numerical integration of the system of ODE. This software has been developed in PROLOG on Macintosh computers and has been extensively used with the programming environment MATLAB ™. Possible applications of this software include model building, sensitivity analysis, identification, optimal experiment design and numerical solution of optimal control problems.

Differential and integral sensitivity analyses for optimal heating of solids by the BEM

Differential and integral formulations of sensitivity analysis of a general boundary-integral functional are presented for a heat-conduction problem. By using the adjoint-variable method of optimization, adjoint differential and integral equations are derived for the two formulations. Some distinguishing differences appear in the boundary-integral equations of the relevant adjoint functions and also in the sensitivity expressions of the objective function. Discretizations af all equations are achieved by the BEM. Analytical expressions are checked with available exact results, and comparisons of the numerical solutions are made. A problem of the optimal cooling of turbine blades is also analyzed by the proposed formulations.

Path-constrained trajectory optimization using sparse sequential quadratic programming

One of the most effective numerical techniques for the solution of trajectory optimization and optimal control problems is the direct transcription method. This approach combines a nonlinear programming algorithm with discretization of the trajectory dynamics. The resulting mathematical programming problem is characterized by matrices that are large and sparse. Constraints on the path of the trajectory are then treated as algebraic inequalities to be satisfied by the nonlinear program. This paper describes a nonlinear programming algorithm that exploits the matrix sparsity produced by the transcription formulation. Numerical experience is reported for trajectories with both state and control variable equality and inequality path constraints. T is well known that the solution of an optimal control or trajectory optimization problem can be posed as the solution of a two-point boundary value problem. This problem requires solving a set of nonlinear ordinary differential equations; the first set defined by the vehicle dynamics and the second set (of adjoint differential equations) by the optimality conditions. Boundary conditions are imposed from the problem requirements as well as the optimality criteria. By discretizing the dynamic variables, this boundary value problem can be reduced to the solution of a set of nonlinear algebraic equations. This approach has been successfully utilized1'5 for applications without path constraints. Since the approach requires adjoint equations, it is subject to a number of difficulties. First, the adjoint equations are often very nonlinear and cumbersome to obtain for complex vehicle dynamics, especially when thrust and aerodynamic forces are given by tabular data. Second, the iterative procedure requires an initial guess for the adjoint variables, and this can be quite difficult because they lack a physical interpretation. Third, convergence of the iterations is often quite sensitive to the accuracy of the adjoint guess. Finally, the adjoint variables may be discontinuous when the solution enters or leaves an inequality path constraint. Difficulties associated with adjoint equations are avoided by the direct transcription or collocation methods.6'10 In this approach, the dynamic equations are discretized, and the optimal control problem is transformed into a nonlinear program, which can be solved directly. The nonlinear programming problem is large and sparse and a method for solving it is presented in Ref. 7. This paper extends the method of Ref. 7 to efficiently handle inequality constraints and presents a nonlinear programming algorithm designed to exploit the properties of the problem that results from direct transcription of the trajectory optimization application.

Symbolic Determination of Parameter Sensitivities and Adjoint Differential Equations: A Fortran Source Code Generator for Their Numerical Integration

This paper describes a software for the automated generation of FORTRAN source code sub-routines for the numerical integration of non-linear differential equations describing dynamical systems. The use of directed graphs for representing the systems greatly simplifies the definition of models and allows an efficient use of symbolic differentiation rules for generating the source code for: i) the sensi-tivity equations for the state/output trajectories with respect to model parameters; ii) the differential equations for computing the adjoint variables of optimal control problems. Possible applications of this software include model building, sensitivity analysis, identification, optimal experiment design and numerical solution of optimal control problems.

A boundary element method based solution procedure for non-linear boundary value problems

This paper describes a technique for the solution of non-linear boundary value problems in one dimension based on the boundary element method. The nonlinearity is handled by a quasilinearization over subintervals or elements within the main domain. The weighting functions used are the solutions to the one dimensional diffusion operator or the solutions to the adjoint differential equation to the original problem. The paper also describes the development of a computer code to handle the numerical aspects of the problem. A number of illustrative problems are solved using this computer code and it is demonstrated that the method is very robust and leads to accurate numerical results for both the dependent variable and its derivative at the nodal points. A method of estimating the accuracy of the solution is also presented. A number of problems for which the boundary element method is likely to be particularly suitable are pointed out.

Diffusion—reaction problem revisited via a new boundary element discretization

This paper provides a new solution method for diffusion—reaction problems. Here the boundary element concepts are used to define a discretized version of the differential equation over each subinterval within the main domain. The nonlinearity in the reaction kinetics is handled by a process of quasi-linearization over each subinterval. The weighting functions are defined separately over each subinterval corresponding to the solutions of the quasi-linearized adjoint differential equation. Thus the discretization is exact for a linear problem and the accuracy for the nonlinear case depends mainly on the validity of the quasi-linearization process which in turn can be easily improved by mesh rearrangement. A banded global matrix is obtained by assembling the resulting equations for each subinterval with both the concentration and its gradient as the primary unknown variables. Application of the method is illustrated for a number of common rate forms. The capability of the method to predict multiple steady states is also illustrated with reference to a substrate-inhibited biochemical reaction.

Boundary conditions optimal control

AbstractA simple rigorous approach is given to finding boundary conditions for the adjoint differential equation in an optimal control problem. The boundary conditions for a time-optimal problem are calculated from the simpler conditions for a fixed-time problem.

THE KRALL POLYNOMIALS: A NEW CLASS OF ORTHOGONAL POLYNOMIALS

Abstract A new set of orthogonal polynomials is found that are solutions to a sixth order formally self adjoint differential equation. These polynomials are shown to generalize the Legendre and Legendre type polynomials. We also show that these polynomials satisfy many properties shared by the classical orthogonal polynomials of Jacobi, Laguerre and Hermite.

Design Sensitivity Analysis of Planar Mechanism and Machine Dynamics

A method of formulating and automatically integrating the equations of motion of quite general constrained dynamic systems is presented. Design sensitivity analysis is carried out using a state space adjoint variable method that has been employed extensively in optimal control and structural design optimization. Both dynamic analysis and design sensitivity analysis formulations are automated and numerical solution of state and adjoint differential equations are carried out using a stiff numerical integration method that treats mixed systems of differential and algebraic equations. A computer code that implements the method is applied to two numerical examples. The first example concerns a relatively simple slider-crank mechanism. The second example treats a more complex agricultural trip plow that undergoes intermittent motion.

Optimal design of dynamically loaded continuous structures

AbstractA computational algorithm is developed and applied for optimization of beam and plate structures, subject to constraints on transient dynamic response. A continuous design formulation is retained, with dynamic response governed by partial differential operator equations. Adjoint equations are employed for sensitivity analysis and a function space gradient projection optimization approach is presented. Finite element analysis methods are applied for solution of the system dynamic and adjoint differential equations. Displacement constrained beam and plate minimum weight examples are solved, with a variety of boundary conditions.

Optimal control problems with non-differentiable isoperimetric constraints

Necessary conditions are derived for optimality of a control in a differential process with an isoperimetric constraint that is not necessarily differentiable. It is found that the gradient of the functional describing the isoperimetric constraint can be replaced by its Gateaux differential. The techniques of mathematical programming and the Dubovitskii-Milyutin theorem are employed. The necessary conditions derived are in the form of an adjoint differential equation and a pointwise maximal condition.

Fixed-Range Optimum Trajectories for Short-Haul Aircraft

An algorithm, based on the energy-state method, is derived for calculating optimum trajectories with a range constraint. The basis of the algorithm is the assumption that optimum trajectories consist of, at most, three segments: an increasing energy segment (climb); a constant energy segment (cruise); and a decreasing energy segment (descent). This assumption allows energy to be used as the independent variable in the increasing and decreasing energy segments, thereby eliminating the integration of a separate adjoint differential equation and simplifying the calculus of variations problem to one requiring only pointwise extremization of algebraic functions. The algorithm is used to compute minimum fuel, minimum time, and minimum direct-operating-cost trajectories, with range as a parameter, for an in-service CTOL aircraft and for an advanced STOL aircraft. For the CTOL aircraft and the minimum-fuel performance function, the optimum controls, consisting of air-speed and engine power setting, are continuous functions of the energy in both climb and descent as well as near the maximum or cruise energy. This is also true for the STOL aircraft except in the descent where at one energy level a nearly constant energy dive segment occurs, yielding a discontinuity in the airspeed at that energy. The reason for this segment appears to be the relatively high fuel flow at idle power of the engines used by this STOL aircraft. Use of a simplified trajectory which eliminates the dive increases the fuel consumption of the total descent trajectory by about 10 percent and the time to fly the descent by about 19 percent compared to the optimum.

A Result on Differential Inequalities and Its Application to Higher Order Trajectory Derivatives

A result on differential inequalities is obtained by considering the adjoint differential equation of the variational equation of the right side of the inequality. The main theorem is proved using basic results on differentiability of solutions with respect to initial conditions. The result is then applied to the problem of determining solution behavior using comparison techniques.

Multi-Parameter Optimization of Damped Linear Continuous Systems

A method is presented for obtaining and optimizing the frequency response of one-dimensional damped linear continuous systems. The systems considered are assumed to contain unknown constant parameters in the boundary conditions and equations of motion which the designer can vary to obtain a minimum resonant response in some selected frequency interval. The unknown parameters need not be strictly dissipative nor unconstrained. No analytic solutions, either exact or approximate, are required for the system response and only initial value numerical integrations of the state and adjoint differential equations are required to obtain the optimal parameter set. The combinations of state variables comprising the response and the response locations are arbitrary.

Aircraft performance optimization.

Using the calculus of variations, the solutions to various fixed end point flight-path optimization problems are developed. These include the minimum fuel-fixed range problem, the minimum time-fixed range problem, and the minimum fuel-fixed range-fixed time problem. Altitude profiles and throttle control laws are presented. A variety of aircraft mathematical models is initially examined, and the existence of a classically optimal controller is verified for a simple model. For this model, the first integral condition is used to eliminate the requirement of integrating the Euler Lagrange adjoint differential equations. The resulting computational algorithms are attractive for both laboratory investigations and airborne implementations.

Optimization of a non-linear distributed parameter system using periodic boundary control†

Conditions for loeal optimality are worked out using simple calculus of variations. To find the optimum control, a two-point boundary value problem in space and time has to be solved, which involves the solution of the adjoint differential equation together with the prooess equation. The method is applied to the optimization of a periodic process, consisting of a tubular reactor where a second-order homogeneous reaction takes place and a periodicity oondition of the state is satisfied everywhere along the reactor. Tho plug flow and the diffusion model are assumed. In the first case an exact solution is carried out. The improvements in yield compared with steady-state conditions are obtained and shown in graphs.