Lesson 6

Read the text: Engineering Specialist

The Aerospace Corporation Trajectory Prescribed Path Control Problem

In a trajectory optimal control or prescribed path control problem, a trajectory is determined which satisfies simultaneously both the equations of motion for the vehicle and additional mission constraints. In general, the equations of motion form a nonlinear system of ordinary differential equations (ODEs). In a prescribed path control problem, additional path constraints are imposed to dictate the shape of the trajectory. In an optimal control problem, a performance index (or cost functional) is minimized or maximized subject to the satisfaction of the differential equations, path constraints, and associated boundary conditions. In both cases, the control profile along the trajectory, as well as the state profile, must be determined.

The first-order necessary conditions for a solution to the optimal control problem generate the Euler-Lagrange system of equations. The Euler-Lagrange system is most generally a boundary value problem for a system of differential-algebraic equations (DAEs). Typically, these DAEs have the semi-explicit form of a set of differential equations coupled with some algebraic equations. The differential equations include the equations of motion and are dependent on both the state and control variables in general. The algebraic equations arise from explicit path constraints specified in the problem, or more subtly from the necessary conditions for an optimal solution. The algebraic constraints may or may not involve the control variables. Hence, there is the potential that the underlying DAE system has an index greater than one.

Over the past 35 years, many methods have been developed to solve these trajectory problems. One such technique, known as the direct transcription method, has been implemented in the trajectory optimization and sizing code FONSIZE. In the direct transcription method, a discretization based on a collocation formula is applied to the differential equations and mission constraints to obtain a parameter optimization problem.

In theory, any nonlinear programming (NLP) algorithm can be used to solve the parameter optimization problem resulting from the direct transcription of an optimal control problem. However, it is critical to the efficiency and ultimate success of this approach to employ an NLP algorithm designed for {\em sparse, large-scale} parameter optimization problems. The sparsity has two origins: the collocation method and the inherent sparse character of the trajectoryproblems (i.e., each variable is involved in relatively few constraints). It is important to exploit the sparsity properties in order to reduce storage requirements and to increase efficiency of the solution of linear systems required by the NLP algorithm.

FONSIZE is used to design new space vehicles, specifically the size of fuel tanks ("sizing") in conjunction with optimally flying the trajectory. For example, in a launch vehicle, one might want to minimize the ascent fuel required, satisfy some trajectory constraints along with the equations of motion, and design the fuel tank lengths.

After text activity

I. Reading Exercises:

Exercise 1. Read and memorize using a dictionary:

trajectory, simultaneously, to impose, constraint, boundary conditions, differential-algebraic equations, semi-explicit, mission constraints, sparsity properties

Exercise 2. Answer the questions:

How is a trajectory optimal control or prescribed path control problem determined?

What was made to solve the problem of a trajectory?

What is the Euler-Lagrange system of equations?

What can be used to solve the parameter optimization problem?

Which two origins does the sparsity have?

Exercise 3. Match the left part with the right:

1) In a prescribed path control problem, additional path constraints	a)generate the Euler-Lagrange system of equations
2)One such technique, known as the direct transcription method,.	b)subtly from the necessary conditions for an optimal solution
3)The first-order necessary conditions for a solution to the optimal control problem	c) has been implemented in the trajectory optimization and sizing code FONSIZE
4)The algebraic equations arise from explicit path constraints specified in the problem, or more	d) are imposed to dictate the shape of the trajectory

II. Speaking Exercises:

Exercise 1. Describe Trajectory Prescribed Path Control Problem, The Euler-Lagrange system, code FONSIZE, any nonlinear programming (NLP) algorithm using the suggested words and expressions as in example:

Trajectory Prescribed Path Control Problem is determined, satisfies simultaneously, motion, vehicle ,additional mission, prescribed path, to dictate the shape, the trajectory Example In a trajectory optimal control or prescribed path control problem, a trajectory is determined which satisfies simultaneously both the equations of motion for the vehicle and additional mission constraints.

The Euler-Lagrange system a boundary, differential-algebraic equations, differential equations, are dependend on, control variables, include

Code FONSIZE mission constraints, to obtain, collocation formula, direct transcription method

Nonlinear programming (NLP) algorithm can be used, parameter optimization problem, the direct transcription, the efficiency, ultimate success, approach, to employ

Exercise 2. Ask questions to the given answers:

1)Question:____________________________________________________?

Answer: The equations of motion form a nonlinear system of ordinary differential equations 2)Question:______________________________________________________?

Answer: Typically, these DAEs have the semi-explicit form of a set of differential equations coupled with some algebraic equations 3)Question______________________________________________________?

Answer: The algebraic constraints may or may not involve the control variables.

4)Question____________________________________________________?

Answer: There is the potential that the underlying DAE system has an index greater than one.

III. Writing exercises:

Exercise 1. Complete the sentences with the suggested words: to exploit, to reduce, to increase, trajectory, linear, collocation

The sparsity has two origins: ______________method and the inherent sparse character of the______________ problems (i.e., each variable is involved in relatively few constraints). It is important______________ the sparsity properties in order______________ storage requirements and______________ efficiency of the____________ solution of systems required by the NLP algorithm.

Exercise 2. Fill in the table with words and expressions from the text:

	Trajectory	parts	processes
Example In an optimal control problem, a performance index is minimized or maximized subject to the satisfaction of the	-	differential equations, path constraints, and associated boundary conditions	-
Typically, these DAEs have the semi-explicit form of a set of
The algebraic constraints may or may not involve
Over the past 35 years, many methods have been developed

Exercise 3. Compose a story on one of the topics (up to 100 words):

“The Euler-Lagrange system”,

“Nonlinear programming (NLP) algorithm.