hp–pseudospectral method for solving continuous …

HP–PSEUDOSPECTRAL METHOD FOR SOLVING CONTINUOUS-TIME NONLINEAROPTIMAL CONTROL PROBLEMS

By

CHRISTOPHER L. DARBY

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2011

c© 2011 Christopher L. Darby

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ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Anil V. Rao for guiding me through the journey

of obtaining a PhD. He has taught me to pursue excellence in all that I do. I thank him for

his guidance. I thank him for never holding me to the standard, but instead, holding me

to a higher standard.

I would also like to thank the members of my committee: Dr. William Hager,

Dr. Warren Dixon, and Dr. Norman Fitz-Coy. Dr. Hager’s experience and wisdom

has given me new perspectives on how to continue to grow as a researcher and my

conversations with Dr. Dixon and Dr. Fitz-Coy have always given me motivation to

continue my work.

I would like to also thank my fellow members of VDOL. I will never forget the times

we have shared while I have been a member of this lab. While each member of our

laboratory has enriched my life, I would specifically like to thank Divya Garg for her

ever willingness to look into and answer or ponder questions I may have regarding our

research. No matter how easy or far fetched my questions may be, she is always willing

to ponder them with me.

Finally, I would like to thank my friends, family and Christie for being there to

support and encourage me not only during my pursuit for a PhD, but, in always being

there for me for all of my pursuits in life. I know that whatever I do and wherever I go,

they will always be my constant.

3

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 MATHEMATICAL BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Finite-Dimensional Optimization . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Unconstrained Minimization of a Function . . . . . . . . . . . . . . 252.2.2 Equality Constrained Minimization of a Function . . . . . . . . . . . 282.2.3 Inequality Constrained Minimization of a Function . . . . . . . . . . 302.2.4 Inequality and Equality Constrained Minimization of a Function . . 312.2.5 Finite-Dimensional Optimization As Applied to Optimal Control . . 32

2.3 Numerical Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.1 Low-Order Numerical Integrators . . . . . . . . . . . . . . . . . . . 342.3.2 Gaussian Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.3 Multiple-Interval Gaussian Quadrature . . . . . . . . . . . . . . . . 412.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.4 Lagrange Polynomial Approximation . . . . . . . . . . . . . . . . . . . . . 442.4.1 Single-Interval Lagrange Polynomial Approximations . . . . . . . . 442.4.2 Multiple-Interval Lagrange Polynomial Approximations . . . . . . . 482.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 MOTIVATION FOR AN HP–PSEUDOSPECTRAL METHOD . . . . . . . . . . . 52

3.1 Motivating Example 1: Problem with a Smooth Solution . . . . . . . . . . 523.2 Motivating Example 2: Problem with a Nonsmooth Solution . . . . . . . . 533.3 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 HP–PSEUDOSPECTRAL METHOD . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1 Multiple-Interval Continuous-Time Optimal Control Problem Formulation . 584.2 hp-Legendre-Gauss Transcription . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.1 The hp–LG Pseudospectral Transformed Adjoint System . . . . . . 654.2.2 Sparsity of the hp–LG Pseudospectral Method . . . . . . . . . . . . 70

4.3 hp–Legendre-Gauss-Radau Transcription . . . . . . . . . . . . . . . . . . 734.3.1 The hp-LGR Pseudospectral Transformed Adjoint System . . . . . 78

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4.3.2 Sparsity of the hp–LGR Pseudospectral Method . . . . . . . . . . . 824.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.1 Example 1 - Problem with a Smooth Solution . . . . . . . . . . . . 874.4.2 Example 2 - Bryson-Denham Problem . . . . . . . . . . . . . . . . 90

4.4.2.1 Case 1: Location of Discontinuity Unknown . . . . . . . . 934.4.2.2 Case 2: Location of Discontinuity Known . . . . . . . . . 93

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 HP–MESH REFINEMENT ALGORITHMS . . . . . . . . . . . . . . . . . . . . . 100

5.1 Approximation of Errors in a Mesh Interval . . . . . . . . . . . . . . . . . . 1015.2 hp–Mesh Refinement Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.1 Increasing Degree of Approximation or Subdividing . . . . . . . . . 1035.2.2 Location of Intervals or Increase of Collocation Points . . . . . . . 1055.2.3 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.2.4 hp–Mesh Refinement Algorithm 1 . . . . . . . . . . . . . . . . . . . 106

5.3 hp–Mesh Refinement Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . 1075.3.1 Increasing Degree of Approximation or Subdividing . . . . . . . . . 1075.3.2 Increase in Polynomial Degree in Interval k . . . . . . . . . . . . . 1085.3.3 Determination of Number and Placement of New Mesh Points . . . 108

5.4 hp–Mesh Refinement Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . 109

6 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.1 Example 1 - Motivating Example 1 . . . . . . . . . . . . . . . . . . . . . . 1136.2 Example 2 - Reusable Launch Vehicle Re-entry Trajectory . . . . . . . . . 1166.3 Example 3: Hyper-Sensitive Problem . . . . . . . . . . . . . . . . . . . . . 120

6.3.1 Solutions for tf = 50 . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.3.2 Solutions for tf = 1000 . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3.3 Solutions for tf = 5000 . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3.4 Summary of Example 6.3 . . . . . . . . . . . . . . . . . . . . . . . 124

6.4 Example 4: Motivating Example 2 . . . . . . . . . . . . . . . . . . . . . . 1286.5 Example 5: Minimum Time-to-Climb of a Supersonic Aircraft . . . . . . . 1306.6 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7 AEROASSISTED ORBITAL INCLINATION CHANGE MANEUVER . . . . . . . 138

7.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.1.1 Constraints On the Motion of the Vehicle . . . . . . . . . . . . . . . 140

7.1.1.1 Initial and final conditions . . . . . . . . . . . . . . . . . . 1417.1.1.2 Interior point constraints . . . . . . . . . . . . . . . . . . . 1417.1.1.3 Vehicle path constraints . . . . . . . . . . . . . . . . . . . 142

7.1.2 Trajectory Event Sequence . . . . . . . . . . . . . . . . . . . . . . 1427.1.3 Performance Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.2 Re-Formulation of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.3 Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.4 All-Propulsive Inclination Change Maneuvers . . . . . . . . . . . . . . . . 147

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7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.5.1 Unconstrained Heating Rate . . . . . . . . . . . . . . . . . . . . . . 1497.5.2 Constrained Heating Rate . . . . . . . . . . . . . . . . . . . . . . . 1537.5.3 Optimal Trajectory Structure . . . . . . . . . . . . . . . . . . . . . . 153

7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.1 Dissertation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.2.1 Continued Work on Mesh Refinement . . . . . . . . . . . . . . . . 1648.2.2 Discontinuous Costate . . . . . . . . . . . . . . . . . . . . . . . . . 165

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6

LIST OF TABLES

Table page

3-1 Matrix Density and CPU Time for p– and h–LGR Methods for Example 2. . . . 57

6-1 Initial Grid used in Example 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6-2 Summary of Accuracy and Speed for Example 6.1 . . . . . . . . . . . . . . . . 115



6-5 Initial Grid used in Example 6.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6-6 Summary of Accuracy and Speed for Example 6.3.1 . . . . . . . . . . . . . . . 121









7-1 Physical and Aerodynamic Constants. . . . . . . . . . . . . . . . . . . . . . . . 140

7

LIST OF FIGURES

Figure page

2-1 An Extremal Curve x∗(t) and a Comparison Curve x(t). . . . . . . . . . . . . . 23

2-2 eτ vs. τ and a Three Interval Trapezoid Rule Approximation. . . . . . . . . . . . 34

2-3 Error vs. Number of Intervals for Trapezoid Rule Approximation of Eq. (2-66). . 35

2-4 Legendre-Gauss Points Distribution. . . . . . . . . . . . . . . . . . . . . . . . . 39

2-5 Legendre-Gauss-Radau Points Distribution. . . . . . . . . . . . . . . . . . . . . 40

2-6 Error vs. Number of LG Points for Approximation of Eq. (2-66). . . . . . . . . . 41

2-7 Error vs. Number of LG or LGR Points for Approximation of Eq. (2-66) . . . . . 43

2-8 Approximation of f (τ) = 1/(1 + 25τ 2) Using Uniformly Spaced Support Points. 46

2-9 Approximation of f (τ) = 1/(1 + 25τ 2) Using LG Support Points. . . . . . . . . 47

2-10 Error vs. Number of Linear Approximating Intervals of Eq. (2-86) . . . . . . . . 49

2-11 Error vs. Number of Support Points for Approximating Eq. (2-86) . . . . . . . . 50

2-12 Error vs. Number of LG or LGR Intervals for Approximation of Eq. (2-86) . . . . 51

3-1 Error for p and h–LGR Approximation to Eqs. (3-1)–(3-3). . . . . . . . . . . . . 54

3-2 Error vs. Collocation Pts for p– and h–LGR Approximation to Eqs. (3-5)–(3-8) . 56

4-1 Collocation and Discretization Points by LG Methods. . . . . . . . . . . . . . . 65

4-2 Qualitative Representation of constraint Jacobian for LG points with Large Nk . 74

4-3 Qualitative Representation of constraint Jacobian for LG points with Small Nk . . 75

4-4 Collocation and Discretization Points by LGR Methods. . . . . . . . . . . . . . 79

4-5 Qualitative Representation of constraint Jacobian for LGR points with Large Nk . 85

4-6 Qualitative Representation of constraint Jacobian for LGR points with Small Nk . 86

4-7 Solution to Example 4.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4-8 Convergence Rates for LG Methods for Example 4.4.1. . . . . . . . . . . . . . 90

4-9 Convergence Rates for LGR Methods for Example 4.4.1. . . . . . . . . . . . . 91

4-10 Case 1: Convergence for LG methods for Example 4.4.2. . . . . . . . . . . . . 94

4-11 Case 1: Convergence for LGR methods for Example 4.4.2. . . . . . . . . . . . 95

8

4-12 Case 1: Solution by a Uniform h–2-Method. . . . . . . . . . . . . . . . . . . . . 96

4-13 Case 2: Convergence for p and h–3–LG and LGR Methods for Example 4.4.2 . 98

5-1 Location of Intervals if Subdivision is Chosen. . . . . . . . . . . . . . . . . . . . 105

5-2 Location of Intervals based on Curvature. . . . . . . . . . . . . . . . . . . . . . 110

6-1 State vs. Time on the Final Mesh for Example 6.2 . . . . . . . . . . . . . . . . 118

6-2 State and Control vs. Time on the Final Mesh for Example 6.3.1 . . . . . . . . 122

6-3 State and Control vs. Time on the Final Mesh for Example 6.3.3. . . . . . . . . 125

6-4 State vs. Time on the Final Mesh for Example 6.3.3. . . . . . . . . . . . . . . . 126

6-5 Control vs. Time on the Final Mesh for Example 6.4. . . . . . . . . . . . . . . . 131

6-6 State vs. Time on the Final Mesh for Example 6.5 for hp(2)–3 and ǫ = 10−4 . . . 133

7-1 Schematic of Trajectory Design for Aeroassisted Orbital Transfer Problem. . . . 144

7-2 Schematic of All-Propulsive Single Impulse and Bi-Parabolic Transfers. . . . . . 148

7-3 ∆Vmin, vs. if , and m(tf )/m0 vs. n. . . . . . . . . . . . . . . . . . . . . . . . . . 150

7-4 ∆V1, ∆V2, ∆V3, and ∆VD Impulses vs. n. . . . . . . . . . . . . . . . . . . . . . . 151

7-5 Change in i by Atmosphere, ∆iaero, and transfer time, T , vs. n for Q =∞. . . . 152

7-6 ∆Vmin, vs. n for Various Max Heating Rates and Final Inclinations. . . . . . . . 154

7-7 m(tf )/m0, vs. n for Various Qmax, and if = (20, 40, 60, 80) deg. . . . . . . . . . 155

7-8 Smax, vs. n, for Various Qmax and if = (20, 40, 60, 80) deg. . . . . . . . . . . . . 156

7-9 h vs. v , Q vs. t, and h vs. γ for if = 40-deg. . . . . . . . . . . . . . . . . . . . . 157

7-10 α and σ vs. t for n = 1 and if = 40 deg. . . . . . . . . . . . . . . . . . . . . . . 158

7-11 h vs. v for n = (1, 2, 3, 4) and if = 40 deg and Qmax = 400 (W/cm2). . . . . . . . 159

7-12 Q, vs. t for n = (1, 2, 3, 4), if = 40 deg and Qmax = 400 (W/cm2). . . . . . . . . 160

9

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

HP–PSEUDOSPECTRAL METHOD FOR SOLVING CONTINUOUS-TIME NONLINEAROPTIMAL CONTROL PROBLEMS

By

Christopher L. Darby

May 2011

Chair: Anil V. RaoMajor: Mechanical Engineering

In this dissertation, a direct hp–pseudospectral method for approximating the

solution to nonlinear optimal control problems is proposed. The hp–pseudospectral

method utilizes a variable number of approximating intervals and variable-degree

polynomial approximations of the state within each interval. Using the hp–discretization,

the continuous-time optimal control problem is transcribed to a finite-dimensional

nonlinear programming problem (NLP). The differential-algebraic constraints of the

optimal control problem are enforced at a finite set of collocation points, where the

collocation points are either the Legendre-Gauss or Legendre-Gauss-Radau quadrature

points. These sets of points are chosen because they correspond to high-accuracy

Gaussian quadrature rules for approximating the integral of a function. Moreover,

Runge phenomenon for high-degree Lagrange polynomial approximations to the state is

avoided by using these points. The key features of the hp–method include computational

sparsity associated with low-order polynomial approximations and rapid convergence

rates associated with higher-degree polynomials approximations. Consequently, the

hp–method is both highly accurate and computationally efficient. Two hp–adaptive

algorithms are developed that demonstrate the utility of the hp–approach. The

algorithms are shown to accurately approximate the solution to general continuous-time

optimal control problems in a computationally efficient manner without a priori knowledge

of the solution structure. The hp–algorithms are compared empirically against local (h)

10

and global (p) collocation methods over a wide range of problems and are found to be

more efficient and more accurate.

The hp–pseudospectral approach developed in this research not only provides a

high-accuracy approximation to the state and control of an optimal control problem,

but also provides high-accuracy approximations to the costate of the optimal control

problem. The costate is approximated by mapping the Karush-Kuhn-Tucker (KKT)

multipliers of the NLP to the Lagrange multipliers of the continuous-time first-order

necessary conditions. It is found that if the costate is continuous, the hp–pseudospectral

method is a discrete representation of the continuous-time first-order necessary

conditions of the optimal control problem. If the costate is discontinuous, however,

and a mesh point is at the location of a discontinuity, the hp–method is an inexact

discrete representation of the continuous-time first-order necessary conditions.

The computational efficiency and accuracy of the proposed hp–method is

demonstrated on several examples ranging from problems whose solutions are smooth

to problems whose solutions are not smooth. Furthermore, a particular application

of a multiple-pass aeroassisted orbital maneuver is studied. In this application, the

minimum-fuel transfer of a small maneuverable spacecraft between two low-Earth

orbits (LEO) with an inclination change is analyzed. In the aeroassisted maneuvers,

the vehicle deorbits into the atmosphere and uses lift and drag to change its inclination.

It is found that the aeroassisted maneuvers are more fuel efficient than all-propulsive

maneuvers over a wide range of inclination changes. The examples studied in this

dissertation demonstrate the effectiveness of the hp–method.

11

CHAPTER 1INTRODUCTION

The objective of an optimal control problem is to determine the state and control

that optimize a performance index subject to dynamic constraints, boundary conditions,

and path constraints. The performance index generally represents a desired metric or

combination of metrics (e.g., fuel consumption, time, energy). Solving optimal control

problems fundamentally has its roots in the calculus of variations. Variational calculus

is used to formulate a set of necessary conditions that define extremal solutions to

an optimal control problem [1–3]. Using the calculus of variations, analytic solutions

can be found to a narrow range of simple optimal control problems. Unfortunately, in

practice, analytic solutions to general optimal control problems are difficult or impossible

to determine. Instead, such problems must be solved numerically.

Numerical methods for optimal control fall into two categories: indirect and direct

methods [4, 5]. In an indirect method, the necessary conditions from the calculus

of variations are derived and these necessary conditions result in a Hamiltonian

boundary-value problem (HBVP). Examples of indirect methods include shooting,

multiple shooting [6, 7], finite difference [8], and collocation [9, 10]. Indirect methods

are highly accurate and optimality of the continuous-time system is readily verifiable

because these methods compute extremal solutions. It is noted, however, that indirect

methods are impractical. The first disadvantage is that the necessary conditions from

the calculus of variations must be formulated which is difficult for problems of even

simple to moderate complexity. Next, the first-order optimality conditions are most often

extremely difficult to solve. Furthermore, an indirect method can only be used if good

initial guesses to the state and costate (a quantity inherent in the necessary conditions)

are provided. In particular, guesses on the costate are very difficult to generate because

the costate has no physical interpretation. Finally, for problems whose solutions have

12

active path constraints, a priori knowledge of the switching structure of the solution must

be known.

In a direct method, the optimal control problem is transcribed into a finite-dimensional

nonlinear programming problem (NLP). Many well-known and robust algorithms exist

to solve NLPs [11–13]. In a direct method, the necessary conditions from the calculus

of variations are not formulated. Moreover, direct methods do not require guesses on

the costate. In general, direct methods are not as accurate as indirect methods and

in general the first-order necessary conditions from the calculus of variations are not

satisfied.

Direct shooting [14–16] is a method which has been extensively used for launch

vehicle and orbit transfer applications [4]. A well-known computer implementation

of direct shooting is POST [17]. Successful direct shooting methods approximate

the optimal control problem using a few number of NLP variables. The success of

direct shooting methods in optimizing trajectories for launch vehicle and orbit transfer

applications is largely due to the fact that many of these problems are simple enough

that their solutions can be accurately approximated using a small number of optimization

parameters. As the number of variables used in a direct shooting method grows, the

ability to successfully use a direct shooting method rapidly declines. Furthermore,

when using a direct shooting method, the switching structure of inactive and active path

constraints must be known a priori.

Another class of a direct methods are direct collocation methods [5, 14, 18–23].

In a direct collocation method, the state is approximated using a set of trial (basis)

functions and a set of differential-algebraic constraints are enforced at a finite number

of collocation points. In contrast to indirect methods and direct shooting, a direct

collocation method does not require an a priori knowledge of the active and inactive arcs

for problems with inequality path constraints. Furthermore, direct collocation methods

are much less sensitive to initial guess than the aforementioned indirect methods

13

and direct shooting method. Some examples of computer implementations of direct

collocation methods are SOCS [24], OTIS [18], DIRCOL [25], and GPOPS [26].

In the past two decades, direct collocation methods have become more widely used

in approximating the solutions of general nonlinear optimal control problems. Typically,

direct collocation methods for optimal control are developed as h (local) methods. In an

h–method, fixed low-degree approximations are used and the problem is divided into

many subintervals. Most often, the class of Runge-Kutta methods [4, 19, 21, 27–30] are

used for h–discretization and convergence of the numerical discretization is achieved by

increasing the number of subintervals [19, 30, 31]. h–methods are utilized extensively

in approximating the solution to optimal control problems because the resulting NLP

is sparse. For a specified problem size, sparsity in the NLP greatly increases the

computational efficiency. A drawback to h–methods, however, is that convergence is at

a polynomial rate, and therefore, an excessively large number of subintervals may be

required to accurately approximate the solution. For example, in the design of an optimal

low-thrust trajectory to the moon, Ref. [32] required an approximating NLP of 211031

variables and 146285 constraints.

In contrast to h–methods, the class of direct collocation p–methods uses a

small fixed number of approximating intervals (often only a single interval is used).

Convergence to the exact solution by a p–method is achieved by increasing the

degree of the polynomial approximation in each interval. In optimal control, single

interval p–methods have been developed synonymously as pseudospectral methods

[22, 26, 33–52]. In a pseudospectral method, the collocation points are defined as

the points from highly accurate Gaussian quadratures rules. Furthermore, the basis

functions are typically Chebyshev or Lagrange polynomials. For problems whose

solutions are smooth and well-behaved, a pseudospectral method converges at an

exponential rate [53–55].

14

The most well developed pseudospectral methods are the Gauss pseudospec-

tral method (GPM) [22, 35], the Radau pseudospectral method (RPM) [38–40],

and the Lobatto pseudospectral method (LPM) [33]. In these methods, the set of

collocation points are the Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and

the Legendre-Gauss-Lobatto (LGL) points. Furthermore, these three sets of points

are defined on the interval τ ∈ [−1, 1] and differ primarily in how the end points are

incorporated. The LG points do not include either endpoint, the LGR points include only

one end point, and the LGL points include both endpoints. Because optimal control

problems generally have boundary conditions at both the initial and terminal times,

the LGL points are the most obvious of the three sets of Gaussian quadrature points

to use. Although LGL points are the most obvious choice, recent research shows

that they may not be the most appropriate. Ref. [42] shows that the costate estimate

using LGL points tends to be noisy. In particular, it was explained in Ref. [40] that

the LGL discrete costate approximation is rank-deficient and the noisy LGL costate

approximation is due to the oscillations about the exact solution, and these oscillations

have the same behavior as the null space of the LGL approximation. Furthermore, it

was shown in Ref. [40] that not only is the LGL approximation to the costate inaccurate,

but the approximation to the control may also be inaccurate. Finally, it is noted that

when using the LG or LGR methods [22, 40], the resulting discrete approximations are

high-accuracy approximations to the solution of the optimal control problem.

While p–methods converge at an exponential rate for problems whose solutions are

smooth and well-behaved, p–methods have several limitations. First, if the solution of

a problem is not smooth, exponential convergence does not hold and the convergence

rates to the exact solution are significantly lower. Furthermore, if the solution to a

problem is smooth but has very high-order behaviors, an accurate solution is obtained

only using an unreasonably high-degree polynomial approximation. Next, the NLP

arising from a p–method is dense. Because the density of the p–method NLP is

15

high, as the degree of polynomial approximation becomes large, the NLP becomes

computationally intractable.

A third class of direct collocation methods that has not received a significant amount

of attention in approximating the solutions to optimal control problems is the class of

hp–methods. An hp–method is one in which the number of intervals, placement of

intervals, and degree of polynomial approximation in each interval is allowed to vary.

Previously, hp–methods have been used in mechanics and fluid dynamics. In particular,

Refs. [56–60] describe the mathematical properties of h, p, and hp–methods for finite

elements. It should be noted that in the period from 1975 to 1995, more than 37,000

papers related to h–methods in finite elements have been published [57]. In that same

time period, less than 100 published papers related to either p or hp–methods in finite

elements exist [57]. Therefore, historically, the vast majority of research has been in

h–methods. Some commercially available h–finite element software programs are

MSC/NASTRAN, Cosmos/M, Abaqus, Aska, Adina, and Ansys [57]. With respect to p–

and hp–methods in finite elements, some available software programs are MSC/PROBE,

Applied Structure, PHLEX and STRIPE [57].

In order to combine the benefits of h and p–methods for approximating the solution

to general optimal control problems, that is, combine the sparsity of an h–method with

the exponential convergence rate of a p–method, in this research, an hp–approach

is developed using LG and LGR points. Furthermore, because this research utilizes

collocation at points defined by Gaussian quadrature, we call these hp–pseudospectral

methods. In addition to developing hp–pseudospectral methods that provide high

accuracy approximations to the state and control, we develop high accuracy costate

estimation methods using the hp–LG and LGR methods. By providing an approximation

to the costate and not just the state and control, verification of optimality of the

original continuous-time problem may be determined from the direct hp–method. The

mapping of the Karush-Kuhn-Tucker (KKT) multipliers of the NLP to the costate of the

16

continuous-time optimal control problem are shown by first deriving the KKT conditions

of the NLP, then deriving the transformed adjoint system, and then comparing the

transformed adjoint system to the first-order necessary conditions of the continuous-time

optimal control problem. Because of the continuity constraint on the state at the mesh

points, the mapping from the KKT multipliers to the costate have two forms. If the

costate is continuous, the hp–transformed adjoint system is a discrete representation

of the continuous-time first-order necessary conditions of the optimal control problem.

If, however, the costate is discontinuous, the transformed adjoint system is no longer

a discrete representation of the continuous-time first-order necessary conditions. It

is found, however, that high levels of accuracy are still obtained in approximating the

solution to the continuous-time optimal control problem if mesh points are placed at the

location of the costate discontinuities.

To demonstrate the utility of an hp–method, two hp–mesh refinement algorithms

are presented. Previous refinement algorithms have been developed for h–methods

[30, 61] while an example of a p–mesh refinement algorithm is given in Ref. [47].

Refinement algorithms for h–methods are designed to increase the number of low-order

approximating intervals where errors are largest in approximating the solution to

the continuous-time optimal control problem. By only increasing the number of

approximating intervals where errors are large, these refinement methods increase

the computational efficiency over naive h–method approximations (i.e., a uniformly

distributed grid of approximating subintervals). Next, in the p–refinement algorithm of

Ref. [47], the algorithm increased the accuracy of the approximation, but did not address

computational efficiency. Large, dense single-interval p–approximations are used on

every iteration of the algorithm except for the final one. Only on the final iteration of

the algorithm is the approximating mesh divided. By using large, dense single-interval

p–approximations, the NLP is solved in a computationally inefficient manner.

17

In the hp–algorithms of this research, the number of intervals, width of intervals, and

polynomial degree in each interval are allowed to vary on every iteration. A two-tiered

strategy is employed to determine the locations of intervals and degree of polynomial

within each interval to achieve a specified solution accuracy. If the solution being

approximated by an interval is smooth, then the degree of approximating polynomial is

increased. Otherwise, if the solution being approximated by an interval is non-smooth,

the interval is subdivided into more low-order subintervals. The two hp–algorithms are

different because the first algorithm is more p–biased, that is, increasing the degree

of an approximating polynomial is chosen more often, and the second algorithm is

more h–biased, that is, subdivision of approximating subintervals into more low-degree

subintervals is more often chosen. Furthermore, the hp–algorithms of this research are

significantly different from the aforementioned h and p–algorithms. The hp–algorithms

of this research are different from h–algorithms because the degree of polynomial

is allowed to vary in each approximating interval. Finally, the hp–algorithms of this

research are different from the p–algorithm of Ref. [47] because the number of intervals

and location of intervals are allowed to vary on each iteration.

The contributions of this dissertation are as follows. First, hp–direct collocation

methods are described using Legendre-Gauss and Legendre-Gauss-Radau points.

If the costate is continuous, the transformed adjoint systems of the hp–LG and LGR

methods are demonstrated to be discrete representations of the continuous-time

first-order optimality conditions of an optimal control problem. As a result, high-accuracy

approximations for the state, control and costate of the solution to an optimal control

problem are obtained. Next, the hp–methods of this research combine the computational

sparsity of h–methods and the exponential convergence rates of p–methods on intervals

whose solution is sufficiently smooth. Therefore, the hp–methods result in highly efficient

and highly accurate approximations.

18

Two hp–adaptive algorithms are presented. No a priori knowledge of the solution

structure of an optimal control problem is required in order for these algorithms to

efficiently obtain highly accurate solutions. The algorithms iteratively refine the

approximating mesh using a two-tiered strategy to either increase the degree of

polynomial approximation in an interval or to subdivide the interval into more low-degree

approximating subintervals. The first algorithm presented is a p–biased algorithm,

while the second algorithm presented is an h–biased algorithm. The two algorithms

are used to solve several examples and the utility of each algorithm is demonstrated.

Furthermore, the hp–algorithms of this research are compared against each other and

with p and h–methods. It is demonstrated that the flexibility of an hp–approach adds

considerable robustness to efficiently approximating optimal control problems to a

high level of accuracy regardless of the structure of the solution of an optimal control

problem.

This dissertation is divided into the following chapters. Chapter 2 defines the

mathematical background necessary to understand the hp–methods presented. A

nonlinear optimal control problem is stated and the continuous-time first-order necessary

conditions for this problem are derived. Next, finite-dimensional optimization of a

function subject to equality and inequality constraints is examined and the KKT system

is derived. Finally, numerical integration and polynomial interpolation are examined.

Chapter 3 provides motivation for an hp–method. It is demonstrated that the method that

best approximates a general nonlinear optimal control problem is problem dependent.

Therefore, strict h or p–methods may be computationally expensive for a specific

problem type. Chapter 4 defines the hp–LG and hp–LGR discrete approximation to

the continuous-time optimal control problem. For the hp–LG and hp–LGR methods,

the KKT conditions are derived and are related to the first-order necessary conditions

of the optimal control problem. Finally, the sparsity pattern of the NLP defined by the

hp–methods of this research are examined. It is shown that the sparsity of the hp–LG

19

and LGR methods is dependent upon the degree of approximating polynomial in each

interval. Chapter 5 describes the two hp–adaptive algorithms of this dissertation. First, a

description is given on how errors are assessed in any approximating interval. Second,

the two hp–algorithms are described in detail. The first algorithm refines the mesh by

examining the relative distribution of error within each approximating interval and the

second algorithm refines the mesh by analyzing the curvature of the state. Chapter 6

presents several examples that are solved by the two hp–algorithms of this dissertation

as well as h and p–methods. It is demonstrated that the hp–algorithms are robust to

problem type and result in computationally efficient, high accuracy solutions to a wide

range of problems. In Chapter 7, a particular application of a multiple-pass aeroassisted

orbital maneuver is studied. The optimal minimum-fuel transfer of a small maneuverable

spacecraft between two low-Earth orbits with an inclination change is analyzed. Optimal

aeroglide maneuvers are compared with all-propulsive maneuvers over a wide range

of inclination change. Furthermore, a variable heating rate constraint is enforced for

the aeroglide maneuvers. It is demonstrated that an optimal aeroglide maneuver is

more fuel efficient than an all-propulsive maneuver. Finally, Chapter 8 summarizes the

contributions of this dissertation and suggests future research directions.

20

CHAPTER 2MATHEMATICAL BACKGROUND

This chapter describes the mathematical background necessary to understand the

hp–method presented in this research. The advancements of this research are founded

on optimal control theory and generating highly accurate, computationally efficient NLP

approximations to a continuous-time optimal control problem. Numerical approximation,

function approximation, and quadrature approximation theory are also foundations of

this research and are discussed in detail.

2.1 Optimal Control

The objective of an optimal control problem is to determine the state and control

that optimize a performance index subject to dynamic constraints, boundary conditions,

and path constraints. The optimal control problem studied in this research is given in

Bolza form as follows. Minimize the cost functional

J = Φ(x(t0), t0, x(tf ), tf ) +

∫ tf

t0

g(x(t),u(t))dt, (2–1)

subject to the dynamic constraints

x(t) ≡ dxdt= f(x(t),u(t)), (2–2)

the inequality path constraints

C(x(t),u(t)) ≤ 0, (2–3)

and the boundary conditions

φ(x(t0), t0, x(tf ), tf ) = 0, (2–4)

where x(t) ∈ Rn is the state, u(t) ∈ R

m is control, f : Rn×Rm −→ R

n, C : Rn×Rm −→ R

s ,

φ : Rn × R × Rn × R −→ R

q and t is time. Furthermore, Φ : Rn × R × Rn × R −→ R

is the Mayer cost and g : Rn × Rm −→ R is the Lagrangian. To determine an extremal

solution, the calculus of variations is used. The calculus of variations is used to generate

21

a set of first-order necessary conditions that must be satisfied for a candidate optimal

solution (also known as an extremal solution). An extremal solution may be a minimum,

maximum, or saddle. Second-order sufficiency conditions must be checked to verify that

an extremal solution is a minimum. The continuous-time first-order necessary conditions

for the optimal control problem defined by Eqs. (2–1)–(2–4) are now derived.

First, the constraints are adjoined to the cost functional to form an augmented cost

functional as

Ja = Φ(x(t0), t0, x(tf ), tf )−ψT

φ(x(t0), t0, x(tf ), tf ) (2–5)

+

∫ tf

t0

[g(x(t),u(t))− λT(t)(x(t)− f(x(t),u(t)))− γT(t)C(x(t),u(t))]dt,

where λ(t) is the costate and γ(t) and ψ are the Lagrange multipliers corresponding

to Eqs. (2–3), and (2–4), respectively. The first-order variation with respect to all free

variables is given as

δJa =∂Φ

∂x(t0)δx0 +

∂Φ

∂t0δt0 +

∂Φ

∂x(tf )δxf +

∂Φ

∂tfδtf (2–6)

−δψTφ−ψT ∂φ

∂x(t0)δx0 −ψ

T ∂φ

∂t0δt0 −ψ

T ∂φ

∂x(tf )δxf −ψ

T ∂φ

∂tfδtf

+((g − λT(x− f)− γTC)|t=tf )δtf − ((g − λT

(x− f)− γTC)|t=t0)δt0

+

∫ tf

t0

[

∂g

∂xδx+

∂g

∂uδu− δλ

T

(x− f) + λT ∂f∂xδx+ λ

T ∂f

∂uδu

−λTδx− δγT

C− γT ∂C∂x

δx− γT ∂C∂u

δu

]

dt.

Next, integrating by parts the term containing δx such that it can be expressed in terms

of δx(t0), δx(tf ), and δx gives

∫ tf

t0

−λTδxdt = −λT(tf )δx(tf ) + λT

(t0)δx(t0) +

∫ tf

t0

λT

δxdt. (2–7)

22

Two relationships relating δxf to δx(tf ) and δx0 to δx(t0) are given as

δx0 = δx(t0) + x(t0)δt0 (2–8)

δxf = δx(tf ) + x(tf )δtf . (2–9)

The relationships given in Eq. (2–8) and (2–9) are interpreted by examining Fig. 2-1.

In Fig. 2-1, x∗(t) represents an extremal curve and x(t) represents a neighboring

comparison curve where t0 and x0 are fixed. δx(tf ) is the difference between x∗(t) and

x(t) evaluated at tf . δxf is the difference between x∗(t) and x(t) evaluated at the end of

each curve, respectively. As can be seen in Fig. 2-1, Eq. (2–9) represents a first-order

approximation to δxf . The same result holds for free initial conditions on t0 and x0, giving

the relationship of Eq. (2–8). Substituting Eqs. (2–7), (2–8) and (2–9) into Eq. (2–6) and

Figure 2-1. An Extremal Curve x∗(t) and a Comparison Curve x(t).

23

grouping terms, the variation of Ja is given as

δJa =

(

∂Φ

∂x(t0)−ψT ∂φ

∂x(t0)+ λ

T

(t0)

)

δx0 (2–10)

+

(

∂Φ

∂x(tf )−ψT ∂φ

∂x(tf )− λT(tf )

)

δxf

−δψTφ

+

(

∂Φ

∂t0−ψT ∂φ

∂t0− g(t0)− λ

T

(t0)f(t0) + γT

(t0)C(t0)

)

δt0

+

(

∂Φ

∂tf−ψT ∂φ

∂tf+ g(tf ) + λ

T

(tf )f(tf )− γT

(tf )C(tf )

)

δtf

+

∫ tf

t0

[

(∂g

∂x+ λ

T ∂f

∂x− γT ∂C

∂x+ λ)δx+ (

∂g

∂u+ λ

T ∂f

∂u− γT ∂C

∂u)δu

−δλT(x− f)− δγT

C]

dt.

The first-order optimality conditions are formed by setting the variation of Ja equal to

zero with respect to each free variable. In order to write these conditions in a convenient

form, first, the augmented Hamiltonian is defined. Defining the augmented Hamiltonian

as

H(x(t),u(t),λ(t),γ(t)) = g(x(t),u(t))+λT

(t)f(x(t),u(t))−γT(t)C(x(t),u(t)), (2–11)

the first-order optimality conditions are then expressed as

xT

(t) =∂H

∂λ(2–12)

λT

(t) = −∂H∂x

(2–13)

0 =∂H

∂u(2–14)

λT

(t0) = − ∂Φ

∂x(t0)+ψ

T ∂φ

∂x(t0)(2–15)

λT

(tf ) =∂Φ

∂x(tf )−ψT ∂φ

∂x(tf )(2–16)

H(t0) = −ψT ∂φ∂t0+∂Φ

∂t0(2–17)

H(tf ) = ψT ∂φ

∂tf− ∂Φ

∂tf. (2–18)

24

Furthermore, using the complementary slackness condition, γ takes the value

γi(t) = 0,when Ci(x(t),u(t)) < 0, i = 1, ... , s (2–19)

γi(t) < 0,when Ci(x(t),u(t)) = 0, i = 1, ... , s. (2–20)

The negativity of γi when Ci = 0 is interpreted such that improving the cost may only

come from violating the constraint [2]. Furthermore, γi(t) = 0 when Ci < 0 states that

this constraint is inactive, and therefore, ignored. The continuous-time first-order

optimality conditions of Eqs. (2–12)–(2–20) define a set of necessary conditions

that must be satisfied for an extremal solution of an optimal control problem. The

second-order sufficiency conditions can be inspected to determine whether the extremal

solution is a minimizing solution.

2.2 Finite-Dimensional Optimization

In a direct collocation approximation of an optimal control problem, the continuous-time

optimal control problem is transcribed into a nonlinear programming problem (NLP).

While the continuous-time optimal control problem is an infinite-dimensional problem of

finding continuous-time functions that minimize a cost functional subject to differential

and algebraic constraints, an NLP is a finite-dimensional problem of finding a vector of

static parameters that minimize a cost function subject to algebraic constraints. In this

section the topics of unconstrained minimization, equality constrained minimization, and

inequality constrained minimization of a function are discussed. Moreover, an iterative

Newton method is shown for determining a minimum of a function.

2.2.1 Unconstrained Minimization of a Function

Consider the following problem of determining the minimum of a function subject to

multiple variables without any constraints. Minimize the objective function

J(x), (2–21)

25

where xT= (x1, ... , xn). For x∗ to be a locally minimizing point, the objective function

must be greater when evaluated at any neighboring point, i.e.,

J(x) > J(x∗). (2–22)

In order to develop a set of sufficient conditions defining a locally minimizing point x∗,

first, a three term Taylor series expansion about some point, x, is used to approximate

the objective function as

J(x) = J(x) + gT

(x)(x− x) + 12(x− x)TH(x)(x− x) + higher order terms (2–23)

where the gradient vector, g(x), is

g(x) ≡ ∇xJ ≡(

∂J

∂x

)T

=

∂J∂x1

∂J∂x2

...

∂J∂xn

(2–24)

and the symmetric n × n Hessian matrix is

H(x) ≡ ∇xxJ ≡∂2J

∂x2=

∂2J∂x21

∂2J∂x1∂x2

... ∂2J∂x1∂xn

∂2J∂x2∂x1

∂2J∂x22

... ∂2J∂x2∂xn

...

∂2J∂xn∂x1

∂2J∂xn∂x2

... ∂2J∂x2n

. (2–25)

For x∗ to be a local minimizing point, two conditions must be satisfied. First, a necessary

condition is that g(x∗) must be zero, i.e.,

g(x∗) = 0. (2–26)

The necessary condition by itself only defines an extremal point which can be a local

minimum, local maximum, or saddle point. In order to ensure x∗ is a local minimum, then

26

an additional condition that must be satisfied is

(x− x∗)TH(x∗)(x− x∗) > 0. (2–27)

Eqs. (2–26) and (2–27) together define sufficient conditions for a local minimum.

In order to iteratively find a minimum of the objective function, first, consider again

the approximation of Eq. (2–23) using a Taylor series expansion about some point x(i),

such that an approximation at a new point x(i+1) is given as

J(x(i+1)) ≈ J(x(i)) + g(i)T(x(i+1) − x(i)) + 12(x(i+1) − x(i))TH(i)(x(i+1) − x(i)) (2–28)

where g(i) = g(x(i)) and H(i) = H(x(i)). Next, defining the search direction, p(i+1) =

x(i+1) − x(i), the approximation to J(x(i+1)) can then be written as

J(x(i+1)) ≈ J(x(i)) + g(i)Tp(i+1) + 12p(i+1)

T

H(i)p(i+1). (2–29)

In order to find a minimum of J(x), the Taylor series expansion in Eq. (2–29) is

differentiated with respect to x(i+1) and this derivative is set to zero such that

g(i+1) = 0 = g(i) +H(i)p(i+1). (2–30)

Rearranging Eq. (2–30), the Newton search direction is given as

p(i+1) = −H(i)−1g(i). (2–31)

Finally, to ensure that the search is for a minimum and not a stationary point or a

maximum, the descent condition is enforced,

g(i)T

p(i+1) < 0. (2–32)

Eq. (2–31) is then applied iteratively from some initial guess x(i) such that a new point

x(i+1) is found. If,

J(x(i+1)) < J(x(i)), (2–33)

27

for each iteration, i , then Eq. (2–31) is applied iteratively until a minimum is found.

2.2.2 Equality Constrained Minimization of a Function

Consider finding the minimum of the objective function

J(x) (2–34)

subject to a set of m ≤ n constraints

f(x) = 0. (2–35)

Finding the minimum of an objective function subject to equality constraints uses an

approach similar to the calculus of variations approach for determining the extremal of

functionals. Define the Lagrangian as

L(x,λ) = J(x)− λTf(x) (2–36)

where λT

= (λ1, ... ,λm) are the Lagrange multipliers. Then, a necessary condition for

the minimum of the Lagrangian is that the point (x∗,λ∗) satisfies

∇xL(x∗,λ∗) = 0 (2–37)

∇λL(x∗,λ∗) = 0. (2–38)

Furthermore, the gradient of L with respect to x and λ is

∇xL = g(x)− GT(x)λ (2–39)

∇λL = −f(x) (2–40)

28

where the Jacobian matrix, G(x), is defined as

G(x) ≡ ∂f

∂x=

∂f1∂x1

∂f1∂x2

... ∂f1∂xn

∂f2∂x1

∂f2∂x2

... ∂f2∂xn

...

∂fm∂x1

∂fm∂x2

... ∂fm∂xn

. (2–41)

It is noted that at a minimum of the Lagrangian, the equality constraint of Eq. (2–35) is

satisfied. Next, this necessary condition alone does not specify a minimum, maximum or

saddle point. In order to specify a minimum, first, define the Hessian of the Lagrangian

as

HL = ∇xxL = ∇xxJ −m∑

i=1

λi∇xxfi . (2–42)

Then, a sufficient condition for a minimum is that

vTHLv > 0 (2–43)

for any vector v in the constraint tangent space.

In order to iteratively find a minimum of the Lagrangian, a two-term Taylor series

expansion of Eqs. (2–39) and (2–40) about some point (x(i),λ(i)) is equated to zero

giving

0 = g(i) − G(i)Tλ(i) +H(i)L (x(i+1) − x(i))− G(i)T

(λ(i+1) − λ(i)) (2–44)

0 = −f(i) − G(i)(x(i+1) − x(i)) (2–45)

where G(i) = G(x(i)) and H(i)L = HL(x(i)). After simplification, these equations lead to the

following linear system of equations:

H(i)L G(i)

T

G(i) 0

−p(i+1)

λ(i+1)

=

g(i)

f(i)

. (2–46)

29

It is noted that Eq. (2–46) is called the Karush-Kuhn-Tucker (KKT) system [19] which

is implemented in an NLP such as SNOPT [11]. Eq. (2–46) is solved iteratively until a

minimum to the Lagrangian is found. It is noted that Eq. (2–46) only defines an extremal

point. In order to find a locally minimizing point, Eq. (2–43) must also be satisfied. In

determining a minimum to the Lagrangian, a minimum to J(x) subject to the constraints

f(x) = 0 is found.

2.2.3 Inequality Constrained Minimization of a Function

Consider the problem of minimizing the objective function

J(x) (2–47)

subject to the inequality constraints

c(x) ≤ 0. (2–48)

Inequality constrained problems are solved by dividing the inequality constraints into a

set of active constraints, and a set of inactive constraints. At the solution x∗, some of the

constraints are satisfied as equalities, that is

ci(x∗) = 0, i ∈ A (2–49)

where A is called the active set, and some constraints are strictly satisfied, that is

ci(x∗) < 0, i ∈ A′ (2–50)

where A′ is called the inactive set.

By separating the inequality constraints into an active set and an inactive set, the

active set can be dealt with as equality constraints as described in the previous section,

and the inactive set can be ignored. The added complexity of inequality constrained

problems is in determining which set of constraints are active, and which are inactive. If

the active and inactive sets are known, an inequality constrained problem becomes an

30

equality constrained problem stated as minimize the objective function

J(x) (2–51)

subject to the constraints

ci(x) = 0, i ∈ A (2–52)

and the same methodology is applied as in the previous section to determine a

minimum.

2.2.4 Inequality and Equality Constrained Minimization of a Function

Finally, consider the problem of finding the minimum of the objective function

J(x) (2–53)

subject to the equality constraints

f(x) = 0 (2–54)

and the inequality constraints

c(x) ≤ 0. (2–55)

Define the Lagrangian as

L(x,λ,ψ(A)) = J(x)− λTf(x)−ψ(A)Tc(A)(x) (2–56)

where λ are the Lagrange multipliers with respect to the equality constraints and ψ(A)

are the Lagrange multipliers associated with the active set of inequality constraints.

Furthermore, note that ψ(A′) = 0, and therefore, the inactive set of of constraints are

ignored. A necessary condition for the minimum of the Lagrangian is that the point

(x∗,λ∗,ψ(A)∗) satisfies

∇xL(x∗,λ∗,ψ(A)∗) = 0 (2–57)

∇λL(x∗,λ∗,ψ(A)∗) = 0 (2–58)

∇ψ(A)L(x∗,λ∗,ψ(A)∗) = 0. (2–59)

31

Next, applying a two-term Taylor series approximation to the necessary conditions of

Eqs. (2–57)–(2–59) gives

0 = g(i) − G(i)Tf λ(i) − G(i)Tc(A)ψ(i)(A) +H

(i)L (x

(i+1) − x(i)) (2–60)

−G(i)Tf (λ(i+1) − λ(i))− G(i)T

c(A)(ψ(i+1)(A) −ψ(i)(A))

0 = −f(i) − G(i)f (x(i+1) − x(i)) (2–61)

0 = −c(i)(A) − G(i)c(A)(x(i+1) − x(i)) (2–62)

where Gf(x) is the Jacobian with respect to the equality constraints and Gc(A)(x) is the

Jacobian with respect to the active inequality constraints. Furthermore, HL in this case is

defined as

HL = ∇2xxL = ∇2xxJ −m∑

i=1

λi∇2xxfi −q

∑

i=1

ψ(A)i ∇2xxc (A)i (2–63)

where q is the number of active inequality constraints. Therefore, the resulting KKT

system is given as

H(i)L G

(i)T

f G(i)T

c(A)

G(i)f 0 0

G(i)

c(A)0 0

−p(i+1)

λ(i+1)

ψ(i+1)(A)

=

g(i)

f(i)

c(i)(A)

. (2–64)

For the problem of minimizing an objective function subject to equality and inequality

constraints, the KKT system of Eq. (2–64) is solved iteratively such that a minimizing

points (x∗,λ∗,ψ(A)∗) is found. Again it is noted that for (x∗,λ∗,ψ(A)∗) to be a minimizing

point, Eq. (2–43) must be satisfied as well.

2.2.5 Finite-Dimensional Optimization As Applied to Optim al Control

NLPs are used to approximate a continuous-time optimal control problem by a direct

collocation method such that the finite-dimensional optimization techniques of Section

2.2 can be used and the difficult problem of solving the continuous-time first-order

necessary conditions is avoided. The KKT system of Eq. (2–46) or Eq. (2–64) is solved

iteratively until a minimum is found for the objective function. There are two primary

32

considerations in constructing NLPs that approximate continuous-time optimal control

problems. The first and most important consideration is if the discrete approximation will

provide an accurate approximation to the solution of a continuous-time optimal control

problem. The second consideration is that if the discrete approximation is accurately

approximating the solution of a continuous-time optimal control problem, is it also a

computationally efficient problem to solve. If the KKT system of Eq. (2–46) or Eq. (2–64)

grows large, that is, the number of variables and / or number of constraints is large,

iteratively solving Eq. (2–46) or Eq. (2–64) will become excessively computationally

expensive. Furthermore, if the Hessian or Jacobian matrices are very dense, that

is, a large number of their entries are non-zero, then again, the problem will become

excessively computationally expensive. In order to maintain computational tractability

of the NLP, the KKT system should be constructed to be simultaneously as small and

sparse as possible while providing an accurate approximation of the continuous-time

optimal control problem.

2.3 Numerical Quadrature

Two concepts are important in constructing the discretized finite-dimensional

optimization problem of this research from the continuous-time optimal control problem.

These two concepts are numerical quadrature (numerical integration) and polynomial

interpolation. Numerical quadrature can generally be described as sampling an

integrand at a finite number of points and using a weighted sum of these points to

approximate the integral. In this section, first, low-order numerical integrators are briefly

discussed. Next, Gaussian quadrature rules are presented and it is shown that the

quadrature rules associated with the LG points are the most accurate quadrature rules.

Moreover, the quadrature rules associated with the LGR points are the second most

accurate quadrature rules.

33

2.3.1 Low-Order Numerical Integrators

A common technique used to approximate the integral of a function is to use

many low-degree approximating subintervals that are each easily integrated. The

approximation to the integral of a function is then the sum of the integral of each

approximating subinterval. A well-known low-order method is the composite trapezoidal

rule [62]. The composite trapezoid rule divides a function into many uniformly distributed

subintervals and each subinterval is approximated by a straight line approximation that

passes through the function at the ends of the subinterval. Fig. 2-2 shows exp(τ), τ ∈

[−1, 1], and a three interval composite trapezoid rule approximation of the function. For

f (τ) = exp(τ)

f(τ)

Approximation

τ

00

1.5

2

2.5

3

−1 −0.5

0.5

0.5

1

1

Figure 2-2. eτ vs. τ and a Three Interval Trapezoid Rule Approximation.

N approximating intervals, the composite trapezoid rule is given as [62]

∫ tf

t0

f (t)dt ≈ tf − t02N

(f (t0) + 2f (t1) + 2f (t2) + · · ·+ 2f (tN−1) + f (tN)) (2–65)

34

where tT= (t0, ... , tN) are the grid points such that points (ti−1, ti) define the beginning

and end of the i th interval. Consider using the composite trapezoid rule to approximate

∫ 1

−1

exp(τ)dτ . (2–66)

Fig. 2-3 shows the log10 error from the trapezoid rule for approximating Eq. (2–66) as a

function of the log10 number of approximating intervals. As seen in Fig. 2-3, in order to

achieve high levels of accuracy in the approximation, many approximating intervals are

required. For one hundred thousand approximating intervals, the error is O(10−10).

2 2.5 3 4 5

−4

−6

−8

−10

−12

log10

Err

or

log10 Number of Approximating Intervals

n

3.5 4.5

Figure 2-3. Error vs. Number of Intervals for Trapezoid Rule Approximation ofEq. (2-66).

2.3.2 Gaussian Quadrature

In contrast to low-order piecewise approximating methods such as the composite

trapezoid rule, another quadrature method is to use highly accurate Gaussian quadratures.

35

In using an n point Gaussian quadrature, two quantities must be specified for each

point, namely, the location of the point, and, a weighting factor multiplying the value

of the function evaluated at this point. The three sets of points defined by Gaussian

quadrature are the Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and the

Legendre-Gauss-Lobatto (LGL) points. For the LG points, the points are free to be

placed anywhere on the domain τ ∈ [−1, 1], for the LGR, one end point is fixed at

τ1 = −1 and for the LGL, both end points are fixed at (τ1, τn) = (−1, 1). The goal of a

Gaussian quadrature is to construct an n point approximation

n∑

i=1

wi fi (2–67)

where wi is the quadrature weight associated with point τi and fi = f (τi), such that

the approximation in Eq. (2–67) is exact for polynomials f (τ) of as large a degree

as possible [62]. Let En(f ) be the error between the integral of f (τ) and an n point

quadrature approximation to the integral of f (τ), that is,

En(f ) =

∫ 1

−1

f (τ)dτ −n

∑

i=1

wi fi . (2–68)

First, it is noted that for

En(a0 + a1τ + · · ·+ amτm) = a0En(1) + a1En(τ) + · · ·+ amEn(τm) (2–69)

that En(f ) = 0 for every polynomial of degree ≤ m if and only if

En(τi) = 0, (i = 0, 1, ... ,m). (2–70)

Next, consider the construction of a quadrature to be exact for a polynomial of as

large a degree as possible while requiring τ1 = −1 (that is, consider the construction

of the LGR points). Two examples are provided to demonstrate how this quadrature is

constructed.

36

Example 1: n = 1: For n = 1, there is one free parameter, w1, and τ1 = −1 is fixed.

Because there is one free parameter, the goal is to have zero error for a zeroth degree

polynomial, i.e.,

En(1) = 0. (2–71)

From Eq. (2–71), there is one equation to determine the unknown weight w1 given as

∫ 1

−1

1dτ − w1 = 0. (2–72)

In order to satisfy this equation, w1 = 2.

Example 2: n = 2: For n = 2, there are three free parameters, w1,w2, and t2 and

again t1 = −1 is fixed. Because there are three free parameters, it is desired to have

zero error for a second degree polynomial, i.e.,

En(1) = 0, (2–73)

En(τ) = 0, (2–74)

En(τ2) = 0. (2–75)

This leads to the set of equations

w1 + w2 = 2, (2–76)

−w1 + w2t2 = 0, (2–77)

w1 + w2t22 =

2

3. (2–78)

In solving Eqs. (2–76) and (2–78) for w1, w2, and t2 it is found that w1 = 0.5,w2 = 1.5,

and τ2 = 1/3. As n is increased, a similar approach is used; 2n − 1 free parameters

are determined such that the quadrature is exact for polynomials of degree 2n − 2. In

constructing sets of points in this fashion, the LGR quadrature points and weights are

constructed. Similarly, for the LG method, because t1 is not fixed, for n points, the LG

quadrature rule has 2n free parameters to determine in order to construct quadrature

37

approximations accurate for functions of degree up to and including 2n − 1. As a

result, for a given number of points, the LG method provides the highest accuracy

quadrature approximation of any quadrature rules, while the LGR method provides the

next highest precision. The LGL quadrature rule, for n points, has 2n− 2 free parameters

and are accurate for polynomials of degree up to and including 2n − 3. It should be

noted once again that the LG and LGR points are studied in this research because the

LGL points have previously been demonstrated to result in inaccurate pseudospectral

approximations to continuous-time optimal control problem [40, 42]. As n becomes

large, determining the LG, LGR, or LGL points and weights becomes increasingly

more difficult by the method exemplified in the aforementioned two examples. The

resulting set of equations are nonlinear and their solutions are not obvious. Fortunately,

it has also been shown that the LG, LGR, and LGL points are also defined by a linear

combination of a Legendre polynomial and / or its derivatives [63]. The n LG points are

defined as the roots of an nth–degree Legendre polynomial

Pn(τ) =1

2nn!

dn

dτ n[

(τ 2 − 1)n]

(2–79)

and the corresponding n weights are given as

wi =2

1− τ 2i[

Pn(τi)]2, (i = 1, ... , n) (2–80)

where Pn is the derivative of the nth–degree Legendre polynomial. Furthermore, the n

LGR points are defined as the roots of the sum of the nth and (n− 1)th–degree Legendre

polynomials, i.e., Pn(τ) + Pn−1(τ), and the corresponding quadrature weights for the

LGR points are

w1 =2

n2, (2–81)

wi =1

(1− τi)[

Pn−1(τi)]2, (i = 2, ... , n). (2–82)

38

Figs. 2-4 and 2-5 show the LG and LGR points for various values of n. First, it is

noted that for the LG points the end points, τ = (−1,+1), are not included and for

the LGR points the initial point, τ = −1, is included. Furthermore, both sets of points

demonstrate a similar profile of having more points near the ends of the interval. Finally,

it is noted that the LG points are symmetric about the origin and the LGR points are

asymmetric.

00−1 −0.5 0.5 1

5

10

τ

Num

ber

ofLG

Poi

nts

15

20

25

30

Figure 2-4. Legendre-Gauss Points Distribution.

The accuracy of the LG and LGR points are now demonstrated on approximating

the integral given in Eq. (2–66). Fig. 2-6 shows the log10 error for the approximation

of Eq. (2–66) versus the number of LG and LGR points. It is seen that as the number

of LG or LGR points is increased, the accuracy in approximation rapidly increases.

Furthermore, the rate of convergence by the LG and LGR quadratures to the exact

solution is exponential. For eight LG or LGR points, the approximation is exact with

39

00−1 −0.5 0.5 1

5

10

15

20

25

30

τ

Num

ber

ofLG

RP

oint

s

f

Figure 2-5. Legendre-Gauss-Radau Points Distribution.

respect to numerical round off. Whereas the trapezoid rule required one hundred

thousand approximating intervals (i.e., one hundred thousand and one evaluations of

exp(τ)) to approximate the integral of exp(τ) to O(10−10), only six points are required to

achieve this same level of accuracy using LG or LGR points (i.e., only six evaluations

of exp(τ)). Therefore, for approximating the integral of Eq. (2–66), the LG or LGR

quadratures are significantly more accurate than a low-order method such as the

composite trapezoid rule. Finally, it is noted that the LG points result in a quadrature

approximation that is slightly more accurate than the LGR points for approximating

Eq. (2–66). This is consistent with the fact that the LG quadrature is exact for a

polynomial one degree higher than the LGR points for any given number of quadrature

points.

40

0

2 4

−2

−4

−6

−8

−10

−12

−14

−166 8 10 12 14 16

Number of LG Points

log10

Err

orLG Pts

LGR Pts

Figure 2-6. Error vs. Number of LG Points for Approximation of Eq. (2-66).

2.3.3 Multiple-Interval Gaussian Quadrature

Gaussian quadrature may also be implemented in a multiple-interval sense such

that a highly accurate Gaussian quadrature is applied to each approximating interval.

While the points defined by Gaussian quadrature are on the domain τ ∈ [−1, 1], it is

noted that the integral of a function over an arbitrary time domain t ∈ [ta, tb] may be

approximated as

∫ tb

ta

f (t)dt =tb − ta2

∫ 1

−1

f (tb − ta2

τ +ta + tb2)dτ ≈ tb − ta

2

n∑

i=1

wi f (tb − ta2

τi +ta + tb2).

(2–83)

Next, consider approximating Eq. (2–66) again using uniformly distributed multiple-interval

LG or LGR quadratures. The approximation to exp(τ), τ ∈ [−1, 1] using K uniformly

41

distributed intervals is

∫ +1

−1

exp(τ)dτ ≈K∑

k=1

tk − tk−12

n∑

i=1

wi f (tk − tk−12

τi +tk + tk−12

) (2–84)

where tk−1 and tk define the beginning and end of the k th approximating interval,

respectively, and n are the number of LG or LGR quadrature points in each interval.

Figs. 2-7A and 2-7B show the log10 error versus the total number of LG or LGR points,

Kn, for LG and LGR quadrature approximations using n = (2, 3, 4). In Figs. 2-7A and

2-7B, it is seen that high levels of accuracy are obtained by using a multiple-interval

Gaussian quadrature. Furthermore, for n = 2, two hundred LG points result in an

approximation of accuracy O(10−10) as opposed to the one hundred thousand points

necessary to obtain this level of accuracy using the trapezoid rule. Furthermore, as n is

increased, fewer total points are required to obtain high levels of accuracy. For n = 3,

the approximation is exact for 150 LG points, and for n = 4, the approximation is exact

for 40 LG points. In Fig. 2-7B it is seen that the multiple-interval LGR quadrature is

less accurate than the multiple-interval LG quadrature. Because the LG quadrature is

accurate for a polynomial of one degree higher, for low and mid-degree approximations,

the difference in accuracy is quite noticeable. For n = 4, the multiple-interval LG

quadrature is exact for 40 LG points, whereas the LGR quadrature requires 100 points

before providing an exact approximation with respect to numerical round off.

2.3.4 Summary

Numerical integration methods have been discussed. First, the composite trapezoid

rule is given as an example of a low-order approximation method. Furthermore, this

rule is used to approximate Eq. (2–66) and it is seen that many approximating intervals

are required to achieve high levels of accuracy. Next, Gaussian quadrature rules

are discussed. Gaussian quadrature rules are constructed such that polynomials

of as high a degree as possible are approximated using a set of n points and n

weighting factors. It is shown that very high levels of accuracy are obtained for very

42

0

0

−10

10050 150 200

−5

−15

Number of Multi-Interval LG Points

log10

Err

or

n = 2

n = 3

n = 4

A LG Points.

0

0

−10

10050 150 200

−5

−15

Maxim

n = 2

n = 3

n = 4

Number of Multi-Interval LGR Points

log10

Err

or

B LGR Points.

Figure 2-7. Error vs. Number of Multiple-Interval LG or LGR Points for Approximation ofEq. (2–66).

43

few points by using LG or LGR points to approximate the integral of Eq. (2–66).

Finally, multiple-interval Gaussian quadrature are discussed. High levels of accuracy

in quadrature approximation are obtained again for this case for a few number of

approximating points when compared with the composite trapezoid rule. Furthermore,

the LG quadrature is more accurate than the LGR quadrature. It is noted that while

more points are required to accurately approximate Eq. (2–66) using a multiple-interval

Gaussian quadrature than a single interval, a thorough motivation is given as to why a

multiple-interval Gaussian quadrature is desired in constructing the hp–pseudospectral

method of this research in Chapter 3 and Chapter 4.

2.4 Lagrange Polynomial Approximation

The next important concept used in constructing the discretized finite-dimensional

optimization problem of this research is polynomial interpolation. In this research,

Lagrange polynomials are used as a continuous-time approximation to the state

equations of the continuous-time optimal control problem. In a Lagrange polynomial

approximation, a set of n support points, (t1, ... , tn), are used and a polynomial of

degree n− 1 is fitted to these points. A Lagrange polynomial approximation to a function,

f (t), is given by

f (t) =

n∑

j=1

fjLj(t), Lj(t) =

n∏

l=1l 6=j

t − tltj − tl

, (2–85)

where fj = f (tj). It is noted that at the support points tj for j = (1, ... , n), fi = fi .

2.4.1 Single-Interval Lagrange Polynomial Approximations

It is well known that Runge phenomenon exists for Lagrange polynomials for an

arbitrary set of support points. Runge phenomenon is the occurrence of oscillations in

the approximating function, f , between support points. As the number of support points

grows, the magnitude of the oscillations between support points also grows. Runge

phenomenon can be demonstrated on the function

f (τ) =1

1 + 25τ 2, τ ∈ [−1,+1]. (2–86)

44

Fig. 2-8 shows the Lagrange polynomial approximation to this function utilizing n =

(8, 32) uniformly distributed support points. It is seen that as the number of points are

increased, the approximation near the ends of the interval become increasingly worse.

Next, consider a Lagrange polynomial approximation to the same function utilizing

n = (8, 32) LG support points. This approximation is shown in Fig. 2-9 and it is seen

that by using LG support points, Runge phenomenon is avoided, and the approximations

become better as the number of support points are increased. Insight into the behavior

of the Lagrange polynomial approximations is exemplified by the following theorem [62].

Theorem: Let t0, t1, ... , tn be distinct real numbers, and let f be a given real valued

function with n + 1 continuous derivatives on the interval It = H(t, t0, t1, ... , tn), with t

some given real number. Then there exists ζ ∈ It such that

f (t)− f (t) = (t − t0) · · · (t − tn)(n + 1)!

f n+1(ζ). (2–87)

Eq. (2–87) defines a limit on the error between the function being approximated and a

Lagrange polynomial approximation of the function. It should be noted at any support

point, the error is zero. In designing an n point Lagrange polynomial approximation,

the only design freedom is in placement of the support points, i.e., changing the

behavior of the numerator term in Eq. (2–87). For uniformly distributed points, the

numerator term becomes excessively large at the ends of the intervals. This results

in the Runge phenomenon that is seen in Fig. 2-8. Furthermore, points defined by

Gaussian quadrature have a structure that they are more densely placed near the ends

of the interval (see again Figs. 2-4 and 2-5). This distribution reduces the magnitude

of the numerator term in Eq. (2–87) at the ends of the intervals, therefore, reducing

the effects of Runge phenomenon. Fig. 2-11 shows the log10 maximum error as a

function of degree of approximating Lagrange polynomial for uniformly distributed

support points and support points defined by LG and LGR points for approximating

Eq. (2–86). It is seen that as the degree of polynomial increases, the Lagrange

45

00−1 −0.5 0.5

1

1

0.2

0.4

0.6

0.8

f (τ)

f (τ)f(τ)

τ

A 8 Uniform Support Points.

0

0−1 −0.5 0.5 1

200

400

600

800

−200

Maxim

f (τ)

f (τ)

f(τ)

τ

B 32 Uniform Support Points.

Figure 2-8. Approximation of f (τ) = 1/(1 + 25τ 2) Using Uniformly Spaced SupportPoints.

46

1.2

−0.2

0

0−1 −0.5 0.5

1

1

0.2

0.4

0.6

0.8

f (τ)

f(τ)

τ

f (τ)

A 8 LG Support Points.

00−1 −0.5 0.5

1

1

0.2

0.4

0.6

0.8

Each

f (τ)

f(τ)

τ

f (τ)

B 32 LG Support Points.

Figure 2-9. Approximation of f (τ) = 1/(1 + 25τ 2) Using LG Support Points.

47

polynomial approximation using a uniformly distributed set of support points diverges.

For a 100th degree polynomial defined with uniformly distributed support points, errors

are O(1030). For a Lagrange polynomial defined by either LG or LGR support points, the

approximation converges to the function in Eq. (2–86). For a 100th degree polynomial,

the errors in approximating this function are O(10−10). Furthermore, unlike the case

of using LG and LGR points for the approximation to integrals, it is noted that when

used as the support points to approximate the function in Eq. (2–86) by a Lagrange

polynomial, there is virtually no difference in approximation error by LG or LGR points.

2.4.2 Multiple-Interval Lagrange Polynomial Approximati ons

Consider approximating Eq. (2–86) using multiple-interval approximations of low

or medium degree. Without using LG or LGR points (or any other set of points that

minimizes Runge phenomenon in a Lagrange polynomial approximation), commonly,

Runge phenomenon is avoided by simply limiting the degree of approximation to be

small. Consider a uniformly distributed set of polynomial approximations such that the

degree of approximation is one, i.e., linear, in each interval. Furthermore, define the

support points for each interval to be the beginning and end of the interval. Fig. 2-11

shows the log10 maximum error in approximating Eq. (2–86) as a function of log10

number of approximating intervals. It is seen that for even one hundred thousand

approximating intervals, errors are still O(10−8).

Next, consider a uniformly distributed multiple-interval approximation utilizing LG

and LGR points as the support points in each interval to approximate Eq. (2–86). It is

noted that the LG and LGR points can be transformed from the time domain τ ∈ [−1, 1]

to an arbitrary time domain t ∈ [ta, tb] by the transformation

t =tb − ta2

τ +ta + tb2. (2–88)

Using n = (4, 6, 8) LG and LGR support points in each interval, Fig. 2-12B shows

the log10 maximum error in approximating Eq. (2–86) versus number of approximating

48

2 2.5 3 4 5

−2

−4

−6

−8

−10

f

3.5 4.5

log10

Max

imum

Err

or

log10 Number of Approximating Interval

Figure 2-10. Error vs. Number of Linear Approximating Intervals of Eq. (2–86).

intervals using a Lagrange polynomial defined by n = (4, 6, 8) LG or LGR support points

in each interval. It is seen that high levels of accuracy are obtained for a relatively few

number of intervals. Furthermore, as the number of support points are increased in

each interval, the approximation becomes increasingly accurate for a given number of

intervals. Finally, it is again noted that when using either LG or LGR points as support

points there is virtually no difference in the error of approximation.

2.4.3 Summary

A discussion on Lagrange polynomial approximation to a function is given.

Runge phenomenon is demonstrated to exist when using high-degree polynomial

approximations to a function using a set of uniformly distributed support points.

Furthermore, it is demonstrated that Runge phenomenon does not exist when using LG

49

0

0−10

10

40 60 80 100

20

20

30

f

Number of Support Points

log10

Max

imum

Err

or

Uniform Pts

LGR Pts

LG Pts

Figure 2-11. Error vs. Number of Support Points for Approximating Eq. (2–86).

or LGR support points and high-accuracy approximations are obtained for high-degree

Lagrange polynomials approximations. Next, in approximating a function using multiple

intervals, it is shown that a very low-order approximation requires a tremendous number

of approximating intervals to achieve high levels of accuracy. By using moderately

sized multiple-interval Lagrange polynomial approximations using LG or LGR points,

it is demonstrated that high levels of accuracy in approximating a function can be

obtained for many fewer approximating intervals than when using very low-degree

approximations.

50

0

0

−10

10050 150 200

−5

−15Number of Intervals with n Points In Each Interval

log10

Max

imum

Err

orn = 4

n = 6

n = 8

A LG Points.

0

0

−10

10050 150 200

−5

−15Number of Intervals with n Points In Each Interval

log10

Max

imum

Err

or

n = 4

n = 6

n = 8

B LGR Points.

Figure 2-12. Error vs. Number of LG or LGR Intervals for Approximation of Eq. (2–86).

51

CHAPTER 3MOTIVATION FOR AN HP–PSEUDOSPECTRAL METHOD

A motivation is given for developing an hp–pseudospectral method. The motivation

is to combine the computational sparsity of h–methods and the rapid rates of convergence

of p–methods. In order to motivate the desire for an hp–approach, two examples

with distinctly different solutions are provided. In the first example, a problem with a

smooth solution is studied. In the second example, a problem whose solution has

a discontinuous control is analyzed. Because of the discontinuity in the control, the

solution to this second problem is non-smooth.

3.1 Motivating Example 1: Problem with a Smooth Solution

Consider the following optimal control problem [64]. Minimize the cost functional

J = tf (3–1)


x = cos(θ) , y = sin(θ) , θ = u, (3–2)


x(0) = 0 , y(0) = 0 , θ(0) = −π

x(tf ) = 0 , y(tf ) = 0 , θ(tf ) = π.(3–3)

The solution to the optimal control problem of Eqs. (3–1)–(3–3) is

(x∗(t), y ∗(t), θ∗(t), u∗(t)) = (− sin(t),−1 + cos(t), t − π, 1) (3–4)

where t∗f = 2π. This problem is now solved with a single-interval p–LGR method

[40] and a uniformly distributed multiple-interval h–LGR method that is described in

Chapter 4. The h–LGR approximation uses two collocation points in each interval,

i.e., a second degree polynomial approximation to the state. Because this problem has

a smooth solution, it is expected that a p–method will perform better when compared

52

to an h–method. Fig. 3-1A shows the log10 maximum error at the collocation points

versus the total number of collocation points for the single-interval p–LGR method. It

is seen that for 20 collocation points, the approximation is essentially exact (relative to

roundoff error). Next, Fig 3-1B shows the log10 maximum error at the collocation points

versus the log2 number of approximating intervals for a uniformly distributed h–LGR

approximation. Using the h–LGR method to approximate the solution, the convergence

rates are significantly slower than using a p–method. For an accuracy of O(10−8), the

h–LGR method requires 1024 approximating intervals and 2048 collocation points

while the p–method uses a single approximating interval with only 13 collocation points.

Strictly low-order methods are unable to obtain high levels of accuracy for this problem

in a computationally efficient manner. For this problem, high levels of accuracy in the

approximation are most easily achieved by using a p–method.

3.2 Motivating Example 2: Problem with a Nonsmooth Solution

Consider the following optimal control problem of a soft lunar landing as given in

Ref. [65]. Minimize the cost functional

J =

∫ tf

0

udt (3–5)


h = v , v = −g + u, (3–6)

the boundary conditions

(h(0), v(0), h(tf ), v(tf )) = (h0, v0, hf , vf ) = (10,−2, 0, 0), (3–7)

and the control inequality constraint

umin ≤ u ≤ umax, (3–8)

53

0

0

5

5

−10

10

−5

−1515 20 25

log10

Max

imum

Err

or

Number of Collocation Points

A p–LGR Approximation.

0

0 2 4

−2

−4

−6

−8

−106 8 10

Number of

log10

Max

imum

Err

or

log2 Number of Collocation Points

B h–LGR Approximation.

Figure 3-1. Error vs. Number of Collocation Points for p and h–LGR Approximation toEqs. (3–1)–(3–3).

54

where umin = 0, umax = 3, g = 1.5, tf is free. The optimal solution to this example is

(h∗(t), v ∗(t), u∗(t)) =

(−34t2 + v0t + h0,−32t + v0, 0), t < t∗s

(34t2 + (−3t∗s + v0)t + 3

2t∗s2 + h0,

32t + (−3t∗s + v0), 3), t > t∗s ,

(3–9)

where t∗s is

t∗s =t∗f2+v0

3(3–10)

with

t∗f =2

3v0 +

4

3

√

1

2v 20 +

3

2h0. (3–11)

For the boundary conditions given in Eq. (3–7), t∗s = 1.4154 and t∗f = 4.1641. First,

it is noted in Eq. (3–9) that the control is bang-bang, i.e., the control is at its minimum

value for t ∈ [0, t∗s ] and is at its maximum value for t ∈ [t∗s , t∗f ]. Because of the

discontinuity in the control, the exact solution for the state is non-smooth. Fig. 3-2 shows

the log10 maximum error at the collocation points versus the number of collocation points

for approximating this problem by p and uniformly distributed h–LGR methods. It is

seen that because the solution to the problem is non-smooth, the p–method does not

converge at an exponential rate. Furthermore, the rates of convergence of the p and

h–methods are essentially the same. Therefore, when comparing p and h–methods

with respect to number of collocation points, no significant difference exists between

a p or h–method approximation. The sparsity of the approximating NLPs using p and

h–methods are, however, quite different. A p–method results in a much more dense

NLP as compared to an h–method. Table 3-1 shows the density of the approximating

NLP by p and h–methods for a given number of collocation points. Furthermore, the

CPU time required to solve the given problem is also shown. It is seen in Table 3-1 that

for an equivalent number of collocation points, the p–method results in a problem that

is significantly more computationally expensive because of the density of the NLP. As

the number of collocation points grow for the p–method approximation, the CPU time

55

0

0

−1

1

−2

−410050 150 200

Error

−3

log10

Max

imum

Err

or


p

h–2

Figure 3-2. Error vs. Collocation Pts for p– and h–LGR Approximation toEqs. (3–5)–(3–8).

required to solve the NLP rapidly grows as well. For the h–method, because of the

sparsity of the NLP, the time required to solve the problem does not grow as rapidly

as with the p–method. Furthermore, it is noted that for 200 collocation points, the

h–method solution time is nearly equivalent to the solution time of the 40 collocation

point p–method. Therefore, if the exponential convergence of a p–method cannot be

guaranteed, a much more computationally efficient NLP is obtained using an h–method

approximation.

3.3 Discussion of Results

The two motivating examples demonstrate a key point in approximating the

solution to general optimal control problems by direct collocation methods: the best

approximation method is problem dependent. Furthermore, in some cases that are

56

Collocation Points p–Matrix Density h–Matrix Density p–CPU Time (s) h–CPU Time (s)20 0.37 0.10 0.07 0.0640 0.35 0.05 0.22 0.1160 0.35 0.037 0.38 0.1280 0.34 0.028 0.54 0.14100 0.34 0.023 0.88 0.16200 0.34 0.012 2.97 0.19

Table 3-1. Matrix Density and CPU Time for p– and h–LGR Methods for Example 2.

simple and smooth, a p–method approximation will provide the best approximating NLP.

In the case of a non-smooth problem without higher order behaviors in the solution, an

h–method approximation will provide the best approximating NLP. In the general case,

a solution to an optimal control problem will have mixed behaviors where some regions

of the trajectory will be smooth, and other regions will be non-smooth. Therefore, a

method that allows for the combination of the strength of either h or p–methods will

be capable of providing highly accurate solutions at a low computational cost. An

hp–pseudospectral method has the flexibility of using the computational sparsity of

low-order h–methods and the high convergence rates of p–methods. The hp–LG and

LGR methods are now described in Chapter 4.

57

CHAPTER 4HP–PSEUDOSPECTRAL METHOD

The hp–LG and LGR pseudospectral methods of this research are now described.

First, the continuous-time optimal control problem of Section 2.1 is reformulated in a

multiple-interval form and the continuous-time first-order necessary conditions are stated

for this multiple-interval formulation. The hp–LG and LGR pseudospectral discretizations

of the multiple-interval optimal control problem are then posed. The KKT conditions

are derived for the hp–LG and LGR methods. Then, the transformed adjoint system for

each method is derived from the KKT conditions and a mapping from the transformed

adjoint system of each method to the continuous-time first-order necessary conditions is

derived. Finally, the sparsity of the NLP defined by the hp–LG and LGR pseudospectral

methods is studied.

4.1 Multiple-Interval Continuous-Time Optimal Control Pro blem Formulation

Consider again the continuous-time optimal control problem of Section 2.1. That is,

minimize the cost functional

J = Φ(x(t0), t0, x(tf ), tf ) +

∫ tf

t0

g(x(t),u(t)) dt, (4–1)


x(t) = f(x(t),u(t)), (4–2)


C(x(t),u(t)) ≤ 0, (4–3)


φ(x(t0), t0, x(tf ), tf ) = 0. (4–4)

Next, consider dividing the problem defined in Eqs. (4–1)–(4–4) into K mesh intervals. A

mesh interval k is defined to begin at the mesh point tk−1 and end at the mesh point tk .

58

Furthermore, let t0 < t1 < t2 < · · · < tK where tK = tf . The time domain in each interval

k , t ∈ [tk−1, tk ], is changed to τ ∈ [−1, 1] by the transformation

τ =2t − (tk−1 + tk)tk − tk−1

. (4–5)

Furthermore, the inverse transformation is given as

t =tk − tk−12

τ +tk−1 + tk2

, (4–6)

and it is noted thatdt

dτ=tk − tk−12

. (4–7)

Next, let each mesh point tk be expressed as a function of the initial time, t0, final time,

tK , and a constant ak , k = (0, ... ,K), as

tk = t0 + ak(tK − t0), (k = 0, ... ,K). (4–8)

It is noted that for t0, a0 = 0 and for tK , aK = 1. Furthermore, because tk−1 < tk ,

ak−1 < ak for k = 1, ... ,K . By expressing each mesh point as a function of t0 and tK ,

the number of free variables corresponding to time is reduced from K + 1 free variables

corresponding to each mesh point to two free variables corresponding to the initial and

final times. This is important for constructing a concise set of necessary conditions for

the optimal control problem and for reducing the number of variables in the discretized

NLP. Let x(k)(τ) and u(k)(τ) be the state and control in the k th mesh interval. Finally,

it is noted that the continuity in the state is enforced at the interior mesh points and

therefore, x(k)(+1) = x(k+1)(−1), (k = 1, ... ,K − 1).

The continuous-time optimal control problem defined by Eqs. (4–1)–(4–4)

expressed as a K interval problem is then to minimize the cost

J = Φ(x(1)(−1), t0, x(K)(+1), tK) +K∑

k=1

tk − tk−12

∫ +1

−1

g(x(k)(τ),u(k)(τ)) dτ , (4–9)

59


dx(k)(τ)

dτ=tk − tk−12

f(x(k)(τ),u(k)(τ)), (k = 1, ... ,K) (4–10)


tk − tk−12

C(x(k)(τ),u(k)(τ)) ≤ 0, (k = 1, ... ,K) (4–11)


φ(x(1)(−1), t0, x(K)(+1), tK) = 0, (4–12)

and the interior point constraints

x(k)(+1)− x(k+1)(−1) = 0, (k = 1, ... ,K − 1). (4–13)

It is noted thattk − tk−12

(4–14)

is added to Eq. (4–11) without affecting the constraint such that the optimality conditions

can be posed in a succinct manner.

The first-order optimality conditions of the multiple-interval continuous-time optimal

control problem of Eqs. (4–9)–(4–13) are now stated. Define the augmented Hamiltonian

in each interval k as

H(k)(x(k),λ(k),u(k),γ(k)) = g(k) + 〈λ(k), f(k)〉 − 〈γ(k),C(k)〉. (4–15)

where 〈·, ·〉 is used to denote the standard inner product between two vectors. Using this

definition of the augmented Hamiltonian in each interval, the continuous-time first-order

60

optimality conditions of the problem defined by Eqs. (4–9)–(4–13) are

dx(k)

dτ=tk − tk−12

∇λ(k)H(k), (k = 1, ... ,K), (4–16)

dλ(k)

dτ= −tk − tk−1

2∇x(k)H(k), (k = 1, ... ,K), (4–17)

0 = ∇u(k)H(k), (k = 1, ... ,K), (4–18)

λ(1)(−1) = −∇x(1)(−1) (Φ− 〈ψ,φ〉) , (4–19)

λ(k)(1) = λ(k+1)(−1) + ρ(k), (k = 1, ... ,K − 1) (4–20)

λ(K)(+1) = ∇x(K)(+1) (Φ− 〈ψ,φ〉) , (4–21)

0 =

K∑

i=1

ak−1 − ak2

∫ 1

−1

H(k)dτ +∇t0 (Φ− 〈ψ,φ〉) , (4–22)

0 =

K∑

i=1

ak − ak−12

∫ 1

−1

H(k)dτ +∇tf (Φ− 〈ψ,φ〉) . (4–23)

where ρ(k) is the difference in λ(k)(1) and λ(k+1)(−1) if the costate is discontinuous at a

mesh point k [2].

4.2 hp-Legendre-Gauss Transcription

The hp–LG pseudospectral method is now described. The hp–LG pseudospectral

method is constructed by discretizing the multiple-interval continuous-time optimal

control problem described by Eqs. (4–9)–(4–13). First, the state in each interval k of

the continuous-time optimal control problem is approximated by a Lagrange polynomial

using the initial point, τ (k)0 = −1, and the LG points, (τ (k)1 , ... , τ(k)Nk), where Nk is the

number of LG points used in interval k . The state approximation is then given as

x(k)(τ) ≈ X(k)(τ) =Nk∑

j=0

X(k)j L

(k)j (τ), (4–24)

where X(k)j = X(k)(τ (k)j ). Furthermore, it is noted that in this Chapter, all vector functions

of time are row vectors, i.e., X(k)(τ) = [X (k)1 (τ), ... ,X(k)n (τ)]. Next, an approximation to

the derivative of the state is given by differentiating the approximation of Eq. (4–24) with

61

respect to τ . Therefore, the derivative of the state approximation is given as

dX(k)(τ)

dτ=

Nk∑

j=0

X(k)j L

(k)j (τ). (4–25)

The collocation conditions for interval k are formed by collocating the derivative of the

state approximation with the right hand side of the dynamic constraints of Eq. (4–10) at

the Nk LG points as

Nk∑

j=0

X(k)j D

(k)ij −

(

tk − tk−12

)

f(k)i = 0, (i = 1, ... ,Nk), (4–26)

where f(k)i = f(X(k)i ,U

(k)i ), U(k)i = U(k)(τ (k)i ) is the discretized control approximation, and

D(k)ij = L

(k)j (τ

(k)i ), (i = 1, ... ,Nk , j = 1, ... ,Nk + 1) (4–27)

is the Nk × (Nk + 1) Gauss pseudospectral differentiation matrix [22, 26, 35, 36] in the

k th mesh interval. It should be noted that the state approximation is a continuous-time

Lagrange polynomial approximation with Nk + 1 support points while the control is

calculated discretely at the Nk collocation points. The inequality path constraints of

Eq. (4–11) are enforced at the Nk collocation points in each interval k as

tk − tk−12

C(k)(X(k)i ,U

(k)i ) ≤ 0, (i = 1, ... ,Nk). (4–28)

Next, it is noted that thus far the state approximation only uses the initial point, τ0 = −1,

and the Nk LG points in each interval. No discrete approximation as of yet has been

made for the state at the final point in each interval, τ (k)Nk+1 = +1. In order to approximate

the state at the final point in each interval, the following LG quadrature is used

X(k)Nk+1

= X(k)0 +

tk − tk−12

Nk∑

i=1

w(k)i f

(k)i . (4–29)

62

where w (k)j are the Legendre-Gauss weights in interval k . Next, the state is enforced to

be continuous between mesh intervals, i.e.,

X(k)Nk+1

= X(k+1)0 , (k = 1, ... ,K − 1). (4–30)

In order to increase the computational efficiency of the approximating NLP, a single

variable is used for X(k)Nk+1 and X(k+1)0 for k = (1, ... ,K − 1). Therefore, Eq. (4–29) is then

expressed as

X(k+1)0 = X

(k)0 +

tk − tk−12

Nk∑

i=1

w(k)i f

(k)i , (k = 1, ... ,K − 1) (4–31)

X(K)NK+1

= X(K)0 +

tK − tK−12

NK∑

i=1

w(K)i f

(K)i

and Eq. (4–30) can be ignored as it is being explicitly taken into account. Finally, the

cost functional of Eq. (4–9) is approximated using a multiple-interval LG quadrature as

J ≈ Φ(X(1)0 , t0,X(K)NK+1, tK) +K∑

k=1

Nk∑

j=1

(

tk − tk−12

)

w(k)j g

(k)j , (4–32)

where g(k)j = g(X(k)j ,U

(k)j ).

The discretized hp–LG pseudospectral approximation to the continuous-time

optimal control problem of Eqs. (4–9)–(4–13) is now stated. The hp–LG pseudospectral

approximation is to minimize the cost

J = Φ(X(1)0 , t0,X

(K)NK+1, tK) +

K∑

k=1

Nk∑

j=1

(

tk − tk−12

)

w(k)j g

(k)j , (4–33)

subject to the collocation conditions

Nk∑

j=0

X(k)j D

(k)ij −

(

tk − tk−12

)

f(k)i = 0, (i = 1, ... ,Nk , k = 1, ... ,K − 1), (4–34)


tk − tk−12

C(k)(X(k)i ,U

(k)i ) ≤ 0, (i = 1, ... ,Nk , k = 1, ... ,K − 1), (4–35)

63

the quadrature constraints

X(k+1)0 = X

(k)0 +

tk − tk−12

Nk∑

i=1

w(k)i f

(k)i , (k = 1, ... ,K − 1) (4–36)

X(K)NK+1

= X(K)0 +

tK − tK−12

NK∑

i=1

w(K)i f

(K)i ,


φ(X(1)0 , t0,X

(K)NK+1, tK) = 0. (4–37)

The problem defined by Eqs. (4–33)–(4–37) is the discrete hp–LG pseudospectral

approximation to the continuous-time optimal control problem defined by Eqs. (4–9)–(4–13).

It is noted that the formulation presented allows for an arbitrary number of intervals, with

arbitrary placement, and with an arbitrary number of LG collocation points in each

interval. That is, the hp–LG method presented has the flexibility of approximating a

continuous-time optimal control problem in any manner desired by using LG points.

A few key properties of the hp–LG pseudospectral method are now stated. First, the

discretization points are defined as the points at which the state is approximated. The

discretization points in an interval k , therefore, are the Nk + 2 points, (τ (k)0 , ... , τ(k)Nk+1).

It is noted that control is approximated at the Nk collocation points, (τ (k)1 , ... , τ(k)Nk).

Next, a well-defined Nthk –degree polynomial approximation to the state is given by

Eq. (4–24). However, no well-defined approximating polynomial is given for the control.

It is interesting to note how the discretization points and collocation points differ for

the hp–LG pseudospectral method. When using a single approximating interval as

shown in Fig. 4-1A, the discretization points differ from the collocation points in that the

discretization points are the collocation points plus the end points. Furthermore, for a

multiple-interval discretization as is seen in Fig. 4-1B, the discretization points differ from

the collocation points in that the discretization points include the collocation points and

64

7

9

03−1 −0.5 0.5 1

4

5

6

8

10

Pol

ynom

ialD

egre

e Collocation Points

Discretization Points

t

A Single-Interval Collocation Points and DiscretizationPoints.

7

02

3

−1 −0.5 0.5 1

4

5

6

8

Collocation Points


t

Num

ber

ofIn

terv

als

B Multiple-Interval Collocation Points and DiscretizationPoints.

Figure 4-1. Collocation and Discretization Points by LG Methods.

all the mesh points. It is interesting to note that the collocation points do not include any

mesh points for the hp–LG pseudospectral method.

4.2.1 The hp–LG Pseudospectral Transformed Adjoint System

The hp–LG transformed adjoint system is now derived. It is shown that the hp–LG

transformed adjoint system is a discrete representation of the continuous-time first-order

65

necessary conditions of Eqs. (4–16)–(4–23) if the costate of the optimal control problem

is continuous. First, the Lagrangian of the continuous-time optimal control problem

defined by the hp–LG pseudospectral method is given as

L = Φ(X(1)0 ,X

(K)NK+1)− 〈Ψ,φ(X(1)0 ,X(K)NK+1〉 (4–38)

+

K∑

k=1

Nk∑

j=1

(

tk − tk−12

(w(k)j g

(k)j − 〈Γ(k)j ,C

(k)j 〉)

−〈Λ(k)j ,D(k)j ,1:NkX(k)1:Nk+D

(k)j ,0 X

(k)0 − tk − tk−1

2f(k)j 〉

)

−K−1∑

k=1

〈π(k),X(k+1)0 − X(k)0 − tk − tk−12

Nk∑

j=1

w(k)j f

(k)j 〉

−〈π(K),X(K)Nk+1 − X(K)0 − tK − tK−1

2

NK∑

j=1

w(K)j f

(K)j 〉,

where Λ(k), Γ(k), π(k), and Ψ are the multipliers associated with Eqs. (4–34), (4–35),

(4–36), and (4–37). Furthermore, D(k)j ,1:Nk is the j th row and first through Nthk columns

of D(k) and X(k)1:Nk are the first through Nthk rows of X(k). Next, the KKT conditions are

given by differentiating the Lagrangian with respect to X(k)j , j = 1, ... ,NK + 1 and

U(k)j , j = 1, ... ,Nk in each interval k as well as t0 and tK and setting these derivatives

66

equal to zero. The KKT conditions are

0 = ∇U

(

w(k)j g

(k)j + 〈Λ(k)j + w

(k)j π(k), f

(k)j 〉 (4–39)

−〈Γ(k)j ,C(k)j 〉

)

, (k = 1, ... ,K , j = 1, ... ,Nk)

D(k)T

j Λ(k) =tk − tk−12

∇X(

w(k)j g

(k)j + 〈Λ(k)j + w

(k)j π(k), f

(k)j 〉 (4–40)

−〈Γ(k)j ,C(k)j 〉

)

, (k = 1, ... ,K , j = 1, ... ,Nk)

D(1)T

0 Λ(1) − π(1) = ∇

X(1)0(Φ− 〈Ψ,φ〉), (4–41)

D(k)T

0 Λ(k) − π(k) = −π(k−1), (k = 2, ... ,K), (4–42)

π(K) = ∇X(K)NK+1

(Φ− 〈Ψ,φ〉), (4–43)

0 = ∇t0(Φ− 〈Ψ,φ〉) (4–44)

+

K∑

k=1

Nk∑

j=1

(

ak−1 − ak2

(w(k)j g

(k)j + 〈Λ(k)j , f

(k)j 〉 − 〈Γ(k)j ,C

(k)j 〉)

)

K−1∑

k=1

ak−1 − ak2

〈π(k),Nk∑

j=1

w(k)j f

(k)j 〉+ aK−1 − aK

2〈πK ,

NK∑

j=1

w(K)j f

(K)j 〉

(k = 1, ... ,K , j = 1, ... ,Nk)

0 = ∇tf (Φ− < Ψ,φ >) (4–45)

+

K∑

k=1

Nk∑

j=1

(

ak − ak−12

(w(k)j g

(k)j + 〈Λ(k)j , f

(k)j 〉 − 〈Γ(k)j ,C

(k)j 〉)

)

+

K−1∑

k=1

ak − ak−12

〈π(k),Nk∑

j=1

w(k)j f

(k)j 〉+ aK − aK−1

2〈πK ,

NK∑

j=1

w(K)j f

(K)j 〉

(k = 1, ... ,K , j = 1, ... ,Nk)

where DT

j is the j th row of DT. In comparing Eq. (4–41) with Eq. (4–19), the costate

approximation at t = t0 is given as

λ(1)0 = −D(1)T0 Λ(1) + π(1) (4–46)

Next, in comparing Eq. (4–43) with Eq. (4–21), the costate approximation at t = tf is

given as

λ(K)NK+1

= π(K). (4–47)

67

Consider the change of variables

ψ = Ψ, (4–48)

γ(k)j =

Γ(k)j

w(k)j

, (k = 1, ... ,K , j = 1, ... ,Nk) (4–49)

λ(k)j =

Λ(k)

w(k)j

+ π(k), (k = 1, ... ,K , j = 1, ... ,Nk), (4–50)

and define the matrix D(k)† [40] as

D(k)†ij = −

w(k)j

w(k)i

D(k)ji , (k = 1, ... ,K , j = 1, ... ,Nk) (4–51)

D(k)†i ,Nk+1

= −Nk∑

j=1

D(k)†ij , (i = 1, ... ,Nk). (4–52)

D(k)† defines the differentiation matrix associated with a Lagrange polynomial using the

LG points, (τ (k)1 , ... , τ(k)Nk) and the final point τ (k)Nk+1 in an interval k [40]. Substituting the

change of variables given in Eqs. (4–46)–(4–52) into (4–39)–(4–43), the LG transformed

68

adjoint system is given as

(D(k)†j ,1:Nk

λ(k) +D(k)†j ,Nk+1

π(k)) = −tk − tk−12

∇XH(X(k)j ,U(k)j ,γ

(k)j ,λ

(k)j ), (4–53)

(k = 1, ... ,K , j = 1, ... ,Nk)

0 = ∇UH(X(k)j ,U

(k)j ,γ

(k)j ,λ

(k)j ), (4–54)

(k = 1, ... ,K , j = 1, ... ,Nk)

λ(1)0 = −∇

X(1)0(Φ− 〈ψ,φ〉), (4–55)

λ(K)NK+1

= ∇X(K)NK+1

(Φ− 〈ψ,φ〉), (4–56)

π(k) − π(k−1) = −tk − tk−12

Nk∑

j=1

w(k)j ∇XH(X(k)j ,U

(k)j ,γ

(k)j ,λ

(k)j ),(4–57)

(k = 2, ... ,K),

0 =

K∑

k=1

ak−1 − ak2

Nk∑

j=1

w(k)j H(X

(k)j ,U

(k)j ,γ

(k)j ,λ

(k)j ) (4–58)

+∇t0(Φ− 〈γ,ψ〉),

0 =

K∑

k=1

ak − ak−12

Nk∑

j=1

w(k)j H(X

(k)j ,U

(k)j ,γ

(k)j ,λ

(k)j ) (4–59)

+∇tf (Φ− 〈γ,ψ〉),

where it is noted that Eq. (4–57) is formed by comparing Eqs. (4–40) and (4–42) and the

identity [40]

D(k)i ,0 = −

Nk∑

j=1

D(k)ij , (i = 1, ... ,Nk). (4–60)

From Eq. (4–57), noting that λ(K)NK+1 = π(K) ,

λ(K)0 = π(K−1). (4–61)

Suppose the costate is continuous at mesh point K − 1. This then implies λ(K)0 = λ(K−1)Nk+1

and using Eq. (4–57) again, we obtain

λ(K−1)0 = π(K−2). (4–62)

69

Next, suppose the costate is continuous at all the mesh points k = 1, ... ,K − 1. Then the

costate at the mesh points are given as

λ(k+1)0 = λ

(k)Nk+1

= π(k), (k = 1, ... ,K − 1). (4–63)

In the case where the costate is continuous at every mesh point, the transformed

adjoint system is a discrete representation of the continuous-time first-order necessary

conditions.

Next, suppose that the costate is discontinuous at a mesh point M ∈ [1, ... ,K − 1].

Then

π(M) = λ(M+1)0 = λ

(M)NM+1

− ρ(M) (4–64)

where ρ(M) is the difference between λ(M)NM+1 and λ(M+1)0 . If the costate is discontinuous

at mesh point M, π(M) 6= λ(M)NM+1 in Eq. (4–53). Therefore, Eq. (4–53) is not a discrete

approximation to the continuous-time costate dynamics. As a result, the transformed

adjoint system given in Eqs. (4–53)–(4–59) is not a discrete representation of the

first-order optimality conditions given in Eqs. (4–16)–(4–23).

4.2.2 Sparsity of the hp–LG Pseudospectral Method

The sparsity of the hp–LG pseudospectral method is now described. It is noted that

the only dense blocks of non-zero numbers that occur in the NLP defined by the hp–LG

pseudospectral method are a result of the Lagrange polynomial approximation to the

state in each interval in Eq. (4–34) and also the quadrature constraints of Eq. (4–36).

The size of these dense blocks are dependent upon the degree of approximating

polynomial, Nk , in each interval. The constraint Jacobian for the hp–LG pseudospectral

method is now described in detail. First, define the total number of collocation points as

N =

K∑

i=1

Ni . (4–65)

Next, the total vector of NLP variables corresponding to the discretized state, control,

and initial and final time is constructed. Define the (N + K + 1)n × 1 vector zx as the

70

vector containing each component of the state at all the discretization points, i.e.,

zx =

χ1...

χn

(4–66)

where χi is the (N + K + 1) × 1 vector corresponding to the i th component of the state

evaluated at the discretization points. Furthermore, define the Nm × 1 vector zu as the

vector containing each component of the control evaluated at the collocation points, i.e.,

zu =

σ1

...

σm

(4–67)

where σi is the N × 1 vector corresponding to the i th component of the control at all of

the collocation points. Therefore, the total vector of NLP variables is

z =

zx

zu

t0

tf

(4–68)

and the total number of NLP variables is (N + K + 1)n + Nm + 2.

Next, the total vector of NLP constraints are constructed. Define the Nn × 1 vector

yD as the vector containing the collocation conditions for each component of the state,

i.e.,

yD =

ζ1...

ζn

(4–69)

where ζ i is the N × 1 vector corresponding to the collocation conditions for the i th

component of the state. Next, define the Ns × 1 vector yC as the vector containing the

71

inequality path constraints:

yC =

η1...

ηs

(4–70)

where ηi is the N×1 vector corresponding to the i th inequality path constraint. Define the

Kn × 1 vector yQ to be the vector of quadrature constraints in Eq. (4–36). Finally, define

the q × 1 vector yE to be the vector of boundary conditions in Eq. (4–37). Therefore, the

total vector of constraints for the hp–LG method is

y =

yD

yC

yQ

yE

(4–71)

and the total number of NLP constraints is Nn + Ns + Kn + q. It is interesting to note

for the hp–LG pseudospectral method, the size of the NLP problem (i.e., the number

of variables and constraints) are dependent upon the number of collocation points, N,

and the number of approximating intervals, K . As the number of collocation points or

number of approximating intervals grows large for the hp–LG pseudospectral method, so

do the number of variables and constraints in the NLP. Finally, the constraint Jacobian

for the hp–LG pseudospectral method is given as

G(z) =∂y

∂z. (4–72)

The density of the constraint Jacobian is now examined. For the constraints

associated with yC , each row corresponds to an inequality path constraint evaluated

at a collocation point. By inspection of Eq. (4–35), it is seen that at each collocation

point, the inequality path constraints are only a function of the state and control at that

collocation point and the initial and final time. Therefore, the rows of yC are sparse (i.e., ,

only n + m + 2 elements in any row are non-zero; the remaining (N + K)n + (N − 1)m

72

elements are zero). Furthermore, yE is only a function of the state at the initial and final

time and the initial and final time. Therefore, the rows of yE are sparse as well.

yD and yQ introduce the density of the hp–LG pseudospectral method. It is seen in

Eq. (4–34) that the collocation condition for any component of the state evaluated at any

collocation point is a function of the state and control at that collocation point, the initial

and final time, and Nk additional variables corresponding to the Lagrange polynomial

approximation to the state. Therefore, as Nk grows large in any approximating interval,

the number of non-zero elements in any row of yD grows large as well. Finally, it is

seen in Eq. (4–36) that for each quadrature constraint, the constraint is a function of

Nkn variables corresponding to the state, Nkm variables corresponding to the control,

the initial and final time, and two additional variables corresponding to the initial state

and final state in interval k . Therefore, again, as Nk grows large in any approximating

intervals, so does the number of non-zero entries in any row of yQ .

Figs. 4-2 and 4-3 shows a qualitative representation of the constraint Jacobian

for the hp–LG pseudospectral method. Fig. 4-2 represents an hp–LG pseudospectral

approximation where Nk is large in each approximating interval and Fig.4-3 represents

an hp–LG pseudospectral approximation where Nk is small in each approximating

interval. It is seen that for large Nk , large, dense blocks arise in the constraint Jacobian

corresponding to the rows of yD and yQ . For small values of Nk , the dense blocks are

much smaller. Therefore, the sparsity of an hp–LG pseudospectral method is dependent

upon Nk , or, the degree of polynomial approximation within any interval. Low-degree

approximations result in a sparse NLP while high-degree approximations result in a

dense NLP.

4.3 hp–Legendre-Gauss-Radau Transcription

The hp–LGR pseudospectral method is now described. The hp–LGR pseudospectral

method is constructed by discretizing the continuous-time optimal control problem

described by Eqs. (4–9)–(4–13). First, the state in each interval k of the continuous-time

73

Figure 4-2. Qualitative Representation of constraint Jacobian for LG points with LargeNk .

optimal control problem is approximated by a Lagrange polynomial using the Nk LGR

points, (τ (k)1 , ... , τ(k)Nk), and the final point, τ (k)Nk+1 = +1. It is noted that τ (k)1 = −1. The

state approximation is then given as

x(k)(τ) ≈ X(k)(τ) =Nk+1∑

j=1

X(k)j L

(k)j (τ). (4–73)

74

Figure 4-3. Qualitative Representation of constraint Jacobian for LG points with SmallNk .

Next, the derivative of the state approximation is given by differentiating the approximation

of Eq. (4–73) with respect to τ . The derivative of the state approximation is then given as

dX(k)(τ)

dτ=

Nk+1∑

j=1

X(k)j L

(k)j (τ). (4–74)

75

The collocation conditions are then enforced at the Nk LGR points as

Nk+1∑

j=1

X(k)j D

(k)ij − tk − tk−1

2f(k)i = 0, (i = 1, ... ,Nk , k = 1, ... ,K), (4–75)

where

D(k)ij = L

(k)j (τ

(k)i ), (i = 1, ... ,Nk , j = 1, ... ,Nk + 1), (4–76)

is the Nk × (Nk + 1) Radau pseudospectral differentiation matrix [39] in the k th mesh

interval. The inequality path constraints of Eq. (4–11) are enforced at the Nk collocation

points in each interval k as

tk − tk−12

C(k)(X(k)i ,U

(k)i ) ≤ 0, (i = 1, ... ,Nk , k = 1, ... ,K). (4–77)

Next, the state is enforced to be continuous between mesh intervals, i.e.,

X(k)Nk+1

= X(k+1)1 . (4–78)

In order to increase the computational efficiency of the approximating NLP, a single

variable is used for X(k)Nk+1 and X(k+1)1 for k = (1, ... ,K − 1). Therefore, Eq. (4–75) is then

written as

D(k)j ,1:NkX(k)1:Nk+D

(k)j ,Nk+1

X(k+1)1 − tk − tk−1

2f(k)j = 0 (k = 1, ... ,K − 1), (4–79)

D(K)j ,1:NK

X(K)1:NK+D

(K)j ,NK+1

X(K)NK+1

− tK − tK−12

f(K)j = 0

and Eq. (4–78) can be ignored as it is being explicitly taken into account. Finally,

the cost functional of Eq. (4–9) is then approximated using a multiple-interval LGR

quadrature as

J ≈ Φ(X(1)1 , t0,X(K)NK+1, tK) +K∑

k=1

Nk∑

j=1

(

tk − tk−12

)

w(k)j g

(k)j , (4–80)

where w (k)j are the Legendre-Gauss-Radau weights [63] in interval k .

76

The discretized hp–LGR pseudospectral approximation to the continuous-time

optimal control problem is now stated. The hp–LGR pseudospectral approximation is to

minimize the cost

J = Φ(X(1)1 , t0,X

(K)NK+1, tK) +

K∑

k=1

Nk∑

j=1

(

tk − tk−12

)

w(k)j g

(k)j , (4–81)

subject to the collocation conditions

D(k)j ,1:NkX(k)1:Nk+D

(k)j ,Nk+1

X(k+1)1 − tk − tk−1

2f(k)j = 0 (k = 1, ... ,K − 1), (4–82)

D(K)j ,1:NK

X(K)1:NK+D

(K)j ,NK+1

X(K)NK+1

− tK − tK−12

f(K)j = 0,


tk − tk−12

C(k)(X(k)i ,U

(k)i ) ≤ 0, (i = 1, ... ,Nk , k = 1, ... ,K). (4–83)


φ(X(1)1 , t0,X

(K)NK+1, tK) = 0. (4–84)

The problem defined by Eqs. (4–81)–(4–84) is the discrete hp–LGR pseudospectral

approximation to the continuous-time optimal control problem defined by Eqs. (4–9)–(4–13).

It is noted that the formulation presented allows for an arbitrary number of intervals, with

arbitrary placement, and an arbitrary number of LGR collocation points in each interval.

Therefore, this method has the flexibility of approximating a continuous-time optimal

control problem in any manner desired by using LGR points.

A few key properties of the hp–LGR pseudospectral method are now stated. First,

the discretization points are defined as the points at which the state is approximated.

The discretization points in an interval k , therefore, are (τ (k)1 , ... , τ(k)Nk+1). It is noted that

the control is approximated at the collocation points, (τ (k)1 , ... , τ(k)Nk). Next, an Nthk degree

polynomial approximation to the state is given by Eq. (4–73). The control on the other

hand, is discretely approximated at the Nk collocation points within each interval. The

77

discretization points and collocation points are now compared for the hp–LGR method.

When using a single approximating interval as is shown in Fig. 4-4A, the discretization

points differ from the collocation points only at the final mesh point, tK . Furthermore,

for a multiple-interval discretization as seen in Fig. 4-4B, again, the discretization

points differ from the collocation points only at the final mesh point. While the hp–LG

pseudospectral method did not approximate the control at any mesh points, the hp–LGR

pseudospectral method approximates the control at every mesh point except the final

mesh point.

4.3.1 The hp-LGR Pseudospectral Transformed Adjoint System

The KKT conditions of the NLP defined by the hp–LGR pseudospectral method are

now derived. The Lagrangian of the hp-LGR pseudospectral method is given as

L = Φ(X(1)1 ,X

(K)NK+1)− 〈Ψ,φ(X(1)1 ,X(K)NK+1〉 (4–85)

+

K∑

k=1

Nk∑

j=1

tk − tk−12

(w(k)j g

(k)j − 〈Γ(k)j ,C

(k)j 〉)

−K−1∑

k=1

Nk∑

j=1

〈Λ(k)j ,D(k)j ,1:NkX(k)1:Nk+D

(k)j ,Nk+1

X(k+1)1 − tk − tk−1

2f(k)j 〉

−NK∑

j=1

〈Λ(K)j ,D(K)j ,1:NK

X(K)1:NK+D

(K)j ,NK+1

X(K)NK+1

− tK − tK−12

f(K)j 〉

where Λ(k), Γ(K), and Ψ are the multipliers associated with Eq. (4–82), Eq. (4–83), and

Eq. (4–84) in mesh interval k , respectively. The KKT conditions of the multiple-interval

Legendre-Gauss-Radau discretization are then given as

D(1)T

j Λ(1) =t1 − t02

∇X(

w(1)j g

(1)j + 〈Λ(1)j , f

(1)j 〉 (4–86)

−〈Γ(1)j ,C(1)j 〉

)

− δ1j

(

−∇X(1)1Φ+∇

X(1)1〈Ψ,φ〉

)

, (j = 1, ... ,N1),

D(k)T

j Λ(k) =tk − tk−12

∇X(

w(k)j g

(k)j + 〈Λ(k)j , f

(k)j 〉 (4–87)

−〈Γ(k)j ,C(k)j 〉

)

− δ1jD(k−1)TNk−1+1

Λ(k−1), (k = 2, ... ,K , j = 1, ... ,Nk).

D(K)T

NK+1Λ(K) = ∇

X(K)NK+1

(Φ− 〈Ψ,φ〉), (4–88)

78

7

9

02

3

−1 −0.5 0.5 1

4

5

6

8

10


Collocation Points

Pol

ynom

ialD

egre

e

t

A Single-Interval Collocation Points and Discretization Points.

7

02

3

−1 −0.5 0.5 1

4

5

6

8


Collocation Points

t

Num

ber

ofIn

terv

als

B Multiple-Interval Collocation Points and Discretization Points.

Figure 4-4. Collocation and Discretization Points by LGR Methods.

0 = ∇U

(

w(k)j g

(k)j + 〈Λ(k)j , f

(k)j 〉 − 〈Γ(k)j ,C

(k)j 〉

)

, (4–89)

(k = 1, ... ,K , j = 1, ... ,Nk),

0 = ∇t0(Φ− < Ψ,φ >) (4–90)

+

K∑

k=1

Nk∑

j=1

(

ak − ak−12

(w(k)j g

(k)j + 〈Λ(k)j , f

(k)j 〉 − 〈Γ(k)j ,C

(k)j 〉)

)

0 = ∇tf (Φ− < Ψ,φ >) (4–91)

+

K∑

k=1

Nk∑

j=1

(

ak − ak−12

(w(k)j g

(k)j + 〈Λ(k)j , f

(k)j 〉 − 〈Γ(k)j ,C

(k)j 〉)

)

79

Next, consider the change of variables

ψ = Ψ, (4–92)

γ(k)j =

Γ(k)j

w(k)j

, (k = 1, ... ,K , j = 1, ... ,Nk), (4–93)

λ(k)j =

Λ(k)j

w(k)j

, (k = 1, ... ,K , j = 1, ... ,Nk), (4–94)

π(k) = D(k)TNk+1Λ(k), (k = 1, ... ,K). (4–95)

Furthermore, define the matrix D(k)† [40] as

D(k)†11 = −D(k)11 − 1

w(k)1

, (4–96)

D(k)†ij = −

w(k)j

w(k)i

D(k)ji otherwise. (4–97)

D(k)† is the corresponding differentiation matrix for a Lagrange polynomial approximation

using the Nk LGR points as support points in interval k . Substituting Eqs. (4–92)–(4–97)

into Eqs. (4–86)–(4–91), the transformed adjoint system is given as

D(1)†j λ(1) = −t1 − t0

2∇XH(X(1)j ,U

(1)j ,λ

(1)j ,γ

(1)j ) (4–98)

+δ1j

w(1)1

(−∇X(1)1(Φ− 〈ψ,φ〉)− λ(1)1 ),

D(k)†j λ(k) = −tk − tk−1

2∇XH(X(k)j ,U

(k)j ,λ

(k)j ,γ

(k)j ) (4–99)

+δ1j

w(k)1

(π(k−1) − λ(k)1 ), (k = 2, ... ,K , j = 1, ... ,Nk)

0 = ∇UH(X(k)j ,U

(k)j ,λ

(k)j ,γ

(k)j ), (k = 1, ... ,K , j = 1, ... ,Nk) (4–100)

π(K) = ∇X(K)NK+1

(Φ− 〈ψ,φ〉) (4–101)

0 =

K∑

k=1

ak−1 − ak2

Nk∑

j=1

w(k)j H(X

(1)j ,U

(1)j ,λ

(1)j ,γ

(1)j ) (4–102)

+∇t0(Φ− 〈γ,ψ〉),

0 =

K∑

k=1

ak − ak−12

Nk∑

j=1

w(k)j H(X

(1)j ,U

(1)j ,λ

(1)j ,γ

(1)j ) (4–103)

+∇t0(Φ− 〈γ,ψ〉),

80

where D(k)†j is the j th row of D(k)†. From the results of Ref. [39], we expand Eq. (4–95)

using Eqs. (4–86) and (4–87) to obtain the following relationships:

−∇X(1)1(Φ + 〈ψ,φ〉) = π(1) + t1 − t0

2

N1∑

j=1

w(1)j ∇xH(X(1)j ,U

(1)j ,λ

(1)j ,γ

(1)j ). (4–104)

π(k−1) = π(k) +tk − tk−12

Nk∑

j=1

w(k)j ∇XH(X(k)j ,U

(k)j ,λ

(k)j ,γ

(k)j ). (k = 2, ... ,K) (4–105)

By comparing Eq. (4–21) and Eq. (4–101), the costate approximation at t = tf is given

as

λ(K)NK+1

= π(K). (4–106)

For k = K , the right-hand side of Eq. (4–105) is then an approximation to λ(K)1 .

Therefore,

λ(K)1 = π(K−1). (4–107)

Furthermore, using Eq. (4–107), the final term in Eq. (4–99) disappears for k = K . Next,

let the costate be continuous at mesh point k = K − 1. Then,

λ(K−1)NK−1+1

= λ(K)1 = π(K−1), (4–108)

and for k = K − 1, the right hand side of Eq. (4–105) becomes an approximation to

λ(K−1)1 . Therefore,

λ(K−1)1 = π(K−2), (4–109)

and the final term in Eq. (4–99) disappears for k = K − 1. If the costate is continuous at

every mesh point, then

λ(k)Nk+1

= λ(k+1)1 = π(k), 1 ≤ k ≤ K − 1 (4–110)

and the final term in Eq. (4–99) disappears for k = 2, ... ,K . Furthermore, the left hand

side of Eq. (4–104), then, is an approximation to the costate at t0 and therefore,

λ(1)1 = −∇

X(1)1(Φ + 〈ψ,φ〉) (4–111)

81

and the final term in Eq. (4–98) disappears. Therefore, for the case when the costate

is continuous at every mesh point, the final term in Eqs. (4–98) and (4–99) disappears

and the transformed adjoint system is a discrete representation of the continuous-time

first-order optimality conditions.

Next, suppose that the costate is discontinuous at a particular mesh point M ∈

[1, ... ,K − 1]. Then

π(M) = λ(M+1)1 = λ

(M)NM+1

− ρ(M) (4–112)

where ρ(M) is the difference between λ(M)NM+1 and λ(M+1)1 . Therefore, if the costate

is discontinuous at mesh point M, π(M−1) 6= λ(M)1 in Eq. (4–99). As a result, the

transformed adjoint system given in Eqs. (4–98)–(4–103) is not a discrete representation

of the first-order optimality conditions given in Eqs. (4–16)–(4–23).

4.3.2 Sparsity of the hp–LGR Pseudospectral Method

The sparsity of the hp–LGR pseudospectral method is now described. It is noted

that the only dense blocks of non-zero numbers that occur in the NLP defined by the

hp–LGR pseudospectral method are a result of the Lagrange polynomial approximation

to the state in each interval in Eq. (4–82). The size of these dense blocks are dependent

upon the degree of approximating polynomial, Nk , in each interval. The constraint

Jacobian for the hp–LGR pseudospectral method is now described in detail. The total

vector of NLP variables corresponding to the discretized state, control and initial and

final time is constructed. Define the (N + 1)n × 1 vector zx as the vector containing each

component of the state at all the discretization points, i.e.,

zx =

χ1...

χn

(4–113)

where χn is the (N + 1)× 1 vector corresponding to the i th component of the state at the

discretization points. Furthermore, define the Nm × 1 vector zu as the vector containing

82

each component of the control at all the collocation points, i.e.,

zu =

σ1

...

σm

(4–114)

where σi is the N × 1 vector corresponding to the i th component of the control at the

collocation points. Therefore, the total vector of NLP variables is

z =

zx

zu

t0

tf

(4–115)

and the total number of NLP variables is (N + 1)n + Nm + 2. Next, the total vector of

NLP constraints are constructed. Define the Nn × 1 vector yD as the vector containing

the collocation conditions for each component of the state, i.e.,

yD =

ζ1...

ζn

(4–116)

where ζ i is the N × 1 vector corresponding to the collocation conditions for the i th

component of the state. Next, define the Ns × 1 vector yC as the vector containing the

the inequality path constraints:

yC =

η1...

ηs

(4–117)

where ηi is the N × 1 vector corresponding to the i th inequality path constraint. Finally,

define the q × 1 vector yE to be the vector of boundary conditions in Eq. (4–84).

83

Therefore, the total vector of constraints for the hp–LGR method is

y =

yD

yC

yE

(4–118)

and the total number of NLP constraints is Nn + Ns + q. It is noted that unlike the

hp–LG pseudospectral method, there is no dependence on the number of variables or

number of constraints on the total number of mesh intervals. Therefore, as the number

of approximating mesh intervals grows large, it does not affect the size of the NLP

defined by the hp–LGR pseudospectral method. Finally, the constraint Jacobian for the

hp–LGR pseudospectral method is given as

G(z) =∂y

∂z. (4–119)

The density of the constraint Jacobian is now examined. For the constraints

associated with yC , each row corresponds to an inequality path constraint evaluated at a

collocation point. Examining Eq. (4–83), at each collocation point, yC is only a function

of the state and control at that collocation point and the initial and final time. Therefore,

the rows of yC are sparse. Furthermore, yE is only a function of the state at the initial

and final time, and the initial and final time. Therefore, the rows of yE are sparse as well.

The density of an LGR pseudospectral method is a result of the rows of the

constraint Jacobian corresponding to yD . It is seen in Eq. (4–82) that for any collocation

point, Eq. (4–82) is a function of the state and control at that collocation point, the initial

and final time, and Nk additional variables corresponding to the Lagrange polynomial

approximation to the state. Therefore, as Nk grows large in any approximating interval,

the number of non-zero elements in any row of yD grows large as well.

Figs. 4-5 and 4-6 shows a qualititive representation of the constraint Jacobian for

the hp–LGR pseudospectral method. Fig. 4-5 represents an hp–LGR pseudospectral

approximation where Nk is large in each approximating interval and Fig. 4-6 represents

84

an hp–LGR pseudospectral approximation where Nk is small in each approximating

interval. It is seen that for large Nk , large, dense blocks arise in the constraint Jacobian.

For small values of Nk , these dense blocks are much smaller. Therefore, the sparsity of

an hp–LGR pseudospectral method is dependent upon Nk , or, the degree of polynomial

approximation within any interval. For low-degree approximations, a sparse NLP is

obtained. For high-degree approximations, the NLP is much more dense.

Points

Figure 4-5. Qualitative Representation of constraint Jacobian for LGR points with LargeNk .

85

Time

Figure 4-6. Qualitative Representation of constraint Jacobian for LGR points with SmallNk .

4.4 Examples

The hp–LG and LGR pseudospectral methods are now applied to two optimal

control problems with analytic solutions. In the first example, the optimal state, control

and costate are smooth, while in the second example, the state, control and the second

component of the costate are non-smooth and continuous, and the first component of

the costate is discontinuous. The purpose of the examples is to assess the accuracy of

the multiple-interval LG and LGR methods for approximating state, control and costate.

86

The problems are solved by either p or h–methods. For p–method problems, the number

of intervals is fixed and the degree of approximation within each interval is allowed to

vary. For h–method problems, the degree of approximation is fixed within each interval

and the number of intervals is allowed to vary. Finally, the terminology “h–x” denotes an

h–method with x collocation points in every mesh interval.

4.4.1 Example 1 - Problem with a Smooth Solution

Consider the following optimal control problem [39]. Minimize the cost functional

J =1

2

∫ tf

0

(y + u2)dt (4–120)

subject to the dynamic constraint

y = 2y + 2u√y (4–121)


y(0) = 2 (4–122)

y(tf ) = 1 (4–123)

where tf = 5. The exact solution to the optimal control problem is given as

y ∗(t) = x2(t) (4–124)

λ∗y(t) =λx2√y

(4–125)

u∗(t) = −2λ∗(t)√

y ∗(t) (4–126)

where x(t) and λx(t) are given as

x(t)

λx(t)

= exp(At)

x0

λx0

(4–127)

87

where

A =

1 −1

−1 −1

(4–128)

and

x0 =√2 (4–129)

xf = 1 (4–130)

λx0 =xf − B11x0B12

(4–131)

B =

B11 B12

B21 B22

= exp(Atf ). (4–132)

The convergence rates for p and uniformly distributed h–2, h–3, and h–4–LG

and LGR methods are analyzed. Furthermore, the p–method uses only a single

approximating interval. Because the problem has a smooth solution, it is expected that a

p–method will demonstrate the fastest convergence rates for this problem. Furthermore,

because the problem has a continuous costate, it is expected that the multiple-interval

LG and LGR methods will provide accurate approximations to the continuous-time

optimal control problem.

Fig. 4-7 shows the state, costate, and control solution for this problem. As is seen

in Fig. 4-7, the state, costate, and control are all smooth. Figs. 4-8A, 4-8B, and 4-8C

show the maximum state, control, and costate error at the collocation points as a

function of the number of collocation points, respectively, for both the p–LG method

and h–x–LG methods. Furthermore, the maximum state, control, and costate error at

the collocation points as a function of the number of collocation points for p–LGR and

h–x–LGR methods are shown in Figs 4-9A, 4-9B, and 4-9C. Because the solution to

this problem is smooth, for both LG and LGR methods, the p–method demonstrates the

most rapid rate of convergence to the exact solution. Furthermore, it is noted that the

p–methods converge exponentially. Next, it is seen that the h–methods converge

88

0

0

2

2 3

−1

1

1 4 5

−2

−4

−3

Sol

utio

n

Time

y∗(t)

λ∗y (t)

u∗(t)

Error

Figure 4-7. Solution to Example 4.4.1.

at a significantly slower rate as compared with the p–methods. In this example,

by either LG or LGR method, convergence is achieved much more rapidly using a

p–method. Moreover, as the degree of polynomial approximation is increased utilizing

the h-methods, the rate of convergence is increased as well. It is also noted that there

is virtually no difference between the convergence rates of the LG and LGR methods

for the state and control. Furthermore, because the multiple-interval LGR method is

a more computationally efficient NLP for a given number of collocation points, if only

an accurate state and control are desired, this result implies that the LGR method

should be used. Interestingly, the error in the costate for the h–LG methods is noticeably

smaller than the h–LGR methods for this problem. This result implies that if an accurate

costate approximation is required by a multiple-interval LG or LGR method, the LG

method should be used.

89

0

0

2

−2

−4

−6

−810050 150 200

log10

Max

imum

Err

or


p

h–2h–3h–4

A State Error.

0

0

2

−2

−4

−6

−810050 150 200

log10

Max

imum

Err

or


p

h–2h–3h–4

B Control Error.

0

0

2

−2

−4

−6

−810050 150 200

log10

Max

imum

Err

or


p

h–2h–3h–4

C Costate Error.

Figure 4-8. Convergence Rates Utilizing p, h–2, h–3, h–4–LG Methods for Example4.4.1.

4.4.2 Example 2 - Bryson-Denham Problem

Consider the problem to minimize the cost functional [2]

J =1

2

∫ 1

0

a2dt (4–133)

subject to the dynamic equations

x = v (4–134)

v = a (4–135)

90

0

0

2

−2

−4

−6

−810050 150 200

log10

Max

imum

Err

or


ph–2h–3h–4

A State Error.

0

0

2

−2

−4

−6

−810050 150 200

log10

Max

imum

Err

or


ph–2h–3h–4

B Control Error.

0

0

2

−2

−4

−6

−810050 150 200

log10

Max

imum

Err

or


ph–2h–3h–4

C Costate Error.

Figure 4-9. Convergence Rates Utilizing p, h–2, h–3, h–4–LGR Methods for Example4.4.1.


x(0) = x(1) = 0 (4–136)

v(0) = −v(1) = 1 (4–137)

and the constraint x(t) ≤ l .

91

The solution to this problem is given as

x =

l [1− (1− t3l)3] : 0 ≤ t ≤ 3l

l : 3l ≤ t ≤ 1− 3l

l [1− (1− 1−t3l)3] : 1− 3l ≤ t ≤ 1

(4–138)

v =

(1− t3l)2 : 0 ≤ t ≤ 3l

0 : 3l ≤ t ≤ 1− 3l

−(1− 1−t3l)2 : 1− 3l ≤ t ≤ 1

(4–139)

a =

− 23l(1− t

3l) : 0 ≤ t ≤ 3l

0 : 3l ≤ t ≤ 1− 3l

− 23l(1− 1−t

3l) : 1− 3l ≤ t ≤ 1

(4–140)

λx =

29l2

: 0 ≤ t ≤ 3l

0 : 3l ≤ t ≤ 1− 3l

− 29l2: 1− 3l ≤ t ≤ 1

(4–141)

λv =

23l(1− t

3l) : 0 ≤ t ≤ 3l

0 : 3l ≤ t ≤ 1− 3l23l(1− 1−t

3l) : 1− 3l ≤ t ≤ 1

(4–142)

Unlike the previous example that had a smooth solution and a continuous costate,

the solution to this problem is only piecewise smooth and has a discontinuous costate

solution for λx . Within any piecewise interval, the highest order behavior in the exact

solution is order three. Therefore, the only difficulty in accurately approximating the

solution to this problem is in approximating the nonsmooth features at t = 3l and

92

t = 1− 3l and the fact that λx is discontinuous at t = 3l and t = 1− 3l . In this example,

l = 1/12.

4.4.2.1 Case 1: Location of Discontinuity Unknown

p–LG and LGR collocation methods and uniformly distributed h–2–LG and LGR

collocation methods are used to approximate the solution to this problem in the case

that the location of discontinuity in λx is assumed not to be known. In this case, mesh

points will, in general, not be at the location of the discontinuity in λx . Figs. 4-10A–4-10D

and Figs. 4-11A–4-11D show the convergence rates for state, control, and costate for

this problem as a function of collocation points for p and h–2–LG methods and p and

h–2–LGR methods, respectively. First, it is seen that there is essentially no difference

in the convergence rates of the p and h–2–methods. Because this problem has a

non-smooth solution, exponential convergence rates from a p–method do not exist.

Furthermore, it is seen that there is no substantial difference in accuracy between

LG and LGR methods. It is also noted that the state, control, and λv do converge

to the exact solution as the number of collocation points are increased. It is also

noted that λx does not converge as the number of collocation points are increased.

Figs. 4-12A–4-12E show the approximation for the h–2–method utilizing 150 collocation

points. Visually, the problem is reasonably approximated except for λx . The difficulty

in approximating λx is isolated to the regions near the discontinuities (i.e., the point at

approximately t = 0.25).

4.4.2.2 Case 2: Location of Discontinuity Known

Next, the solution to the problem is approximated knowing the exact locations of

discontinuities in λx . p and h–3–LG and LGR methods are used to approximate the

solution to this problem. First, define the time domains T1 ∈ [0, 3l ], T2 ∈ [3l , 1 − 3l ],

and T3 ∈ [1 − 3l , 1]. For the p–method approximations, a single approximating

interval is used for each of T1, T2, and T3 with a variable degree polynomial in each

interval. Furthermore, for the h–3–method, a variable number of uniformly distributed

93

0

0

−1

−2

−410050 150

−3

log10

Max

imum

Err

or


p

h–2

t)

A Maximum Error in the State.

−1.5

0

0

−1

−0.5

−210050 150

log10

Max

imum

Err

or


p

h–2

B Maximum Error in the Control.

0

1.5

0.5

1

10050 150

plog10

Max

imum

Err

or


p

h–2

C Maximum Error in λx .

−1.5

0

0

−1

−0.5

0.5

1

−210050 150

log10

Max

imum

Err

or


p

h–2

D Maximum Error in λv .

Figure 4-10. Case 1: Convergence for p– and Uniformly Spaced h–2–LG methods forExample 4.4.2.

approximating intervals of third degree are used in T1 and T3, while T2 is approximated

using a single interval of third degree. Figs. 4-13A–4-13F show the maximum errors at

the collocation points when the approximation to the solution of the problem is divided

in the manner described. In Figs. 4-13A, 4-13C, and 4-13E, the x-axis denotes the

number of intervals in T1 and T3. In Fig. 4-13B, 4-13D, and 4-13F, the x-axis denotes

the degree of approximating polynomial in T1, T2, and T3. First, it is noted that the

maximum degree of the exact solution is three. Therefore, the p and h–3–methods

should exactly approximate the solution of this problem. Any deviation from an exact

94

0

0

−1

−2

−410050 150

−3

log10

Max

imum

Err

or


p

h–2

t)

A Maximum Error in the State.

−1.5

0

0

−1

−0.5

0.5

1

−210050 150

log10

Max

imum

Err

or


p

h–2

B Maximum Error in the Control.

1.4

1.6

1.2

0

1

10050 1500.8

plog10

Max

imum

Err

or


p

h–2

C Maximum Error in λx .

−1.5

0

0

−1

−0.5

0.5

1

−210050 150

log10

Max

imum

Err

or


p

h–2

D Maximum Error in λv .

Figure 4-11. Case 1: Convergence for p– and Uniformly Spaced h–2–LGR Methods forExample 4.4.2.

approximation is due to the fact that the transformed adjoint systems are an inexact

discrete representation of the continuous-time first-order optimality conditions if mesh

points are at the locations of discontinuity in the costate.

It is seen that by either p or h–3–LG method approximations, the error for the

state, control and costate are all near machine precision for either a low number of

intervals or degree of approximating polynomial. As the number of intervals or degree of

polynomial is increased, the approximation accuracy becomes worse. Furthermore, by

p or h–3–LGR methods, the error as a function of number of approximating intervals or

95

00 10.2 0.4 0.6 0.8

Sol

utio

n

Time

x∗(t)

x(t)

0.02

0.04

0.06

0.08

0.1

A x∗(t) and x(t).

0

0−1

−0.5

0.5

1

10.2 0.4 0.6 0.8

Sol

utio

n

Time

v∗(t)v(t)

B v∗(t) and v(t).

0

0

2

1

−2

−4

−6

−8

−100.2 0.4 0.6 0.8

Sol

utio

n

Time

t)

a∗(t)

a(t)

C a∗(t) and a(t).

Time

0

0 1

40

60

0.2 0.4 0.6 0.8

20

λ∗x(t)λx(t)

λx(t)

−20

−40

D λ∗x(t) and λx(t).

0

0

2

1

4

−2

6

8

10

0.2 0.4 0.6 0.8

Sol

utio

n

Time

λ∗v (t)λv (t)

E λ∗v (t) and λv (t).

Figure 4-12. Case 1: Solution by a Uniform h–2-Method Using 150 Collocation PointsCompared with the Exact Solution.

96

degree of approximating polynomial neither converges nor diverges. Rather, errors are

banded between O(10−4) and O(10−16). While high levels of accuracy are demonstrated

by p and h–3–LG and LGR methods, the behaviors of these methods with respect to

this case are erratic because the transformed adjoint system is an inexact discrete

representation of the continuous-time first-order optimality conditions.

4.5 Summary

The hp–LG and LGR pseudospectral methods for approximating a nonlinear

continuous-time optimal control problem have been posed. LG and LGR points are

used because they provide highly accurate quadrature approximations and avoid Runge

phenomenon in the Lagrange polynomial approximation of the state. Furthermore, the

hp–LG and LGR methods have been posed such that an arbitrary number of intervals,

with arbitrary location, and an arbitrary number of collocation points per interval may be

used. Over regions where the solution is smooth, increasing the number of collocation

points in approximating intervals will result in efficient high-accuracy approximations.

For regions of the trajectory where the solution is non-smooth, increasing the number of

low-order approximating intervals will result in computationally efficient approximations

of these non-smooth regions. Next, the KKT conditions of the hp–LG and LGR methods

are derived. The transformed adjoint systems are derived from the KKT conditions

and are shown to be discrete representations of the continuous-time first-order

necessary conditions if the costate is continuous. If the costate is discontinuous, then

the transformed adjoint systems are not discrete representations of the continuous-time

first-order necessary conditions. Two examples are used to demonstrate the accuracy

of approximation of the hp–LG and LGR methods. The solution to the first example

is smooth and has a continuous costate while the solution to the second example is

nonsmooth and has a discontinuous costate. It is seen in the first example that as

the number of approximating intervals or degree of approximation in any interval is

increased, the accuracy in approximating the state, control, and costate is increased as

97

−18

0

−6

−8

−10

−12

−14

−16

10 40 5020 30

LG Ptslog10

Max

imum

Err

ort)

Number of Intervals

LGR Pts

A State Error by LG and LGR h–3–Methods.

−18

0

−6

−8

−10

−12

−14

−16

10 40 5020 30

LG Pts

log10

Max

imum

Err

or


LGR Pts

B State Error by LG and LGR p–Methods.

0

−4

−6

−8

−10

−12

−14

−1610 40 5020 30

LG Pts

plog10

Max

imum

Err

or

Number of Intervals

LGR Pts

C Control Error by LG and LGR h–3–Methods.

0

−4

−6

−8

−10

−12

−14

−1610 40 5020 30

LG Ptslog10

Max

imum

Err

or


LGR Pts

D Control Error by LG and LGR p–Methods.

0

−4

−6

−8

−10

−12

−1410 40 5020 30

LG Pts

ointslog10

Max

imum

Err

or

Number of Intervals

LGR Pts

E Costate Error by LG and LGR h–3–Methods.

0

−2

−4

−6

−8

−10

−12

−1410 40 5020 30

LG Pts

log10

Max

imum

Err

or


LGR Pts

F Costate Error by LG and LGR p–Methods.

Figure 4-13. Case 2: Convergence for p and h–3–LG and LGR Methods for Example4.4.2.

98

well. Furthermore, there is little difference in the approximation accuracy of the state

and control using LG or LGR points, but, the costate approximation is significantly more

accurate using LG points. Next, the second example studies a problem whose solution

has a discontinuous costate. It is demonstrated that if mesh points are not located at

the location of discontinuity in the costate, significant errors exist in approximating the

discontinuity. Furthermore, if mesh points are located at the locations of discontinuity

in the costate, high levels of accuracy are obtained in the approximation although the

trends are quite erratic. For the LG methods, a few number of approximating intervals

or low-degree approximations best approximated the solution to the problem. For the

LGR methods, the errors are banded within a large range from O(10−4) to O(10−16).

Therefore, if the costate is discontinuous, it is best to put mesh points at the exact

location of discontinuity, but, guaranteeing high-accuracy by the proposed hp–LG or

LGR methods is difficult for the case of discontinuous costate solutions. Finally, the

sparsity of the hp–LG and LGR methods is studied. It is seen that as the number of

collocation points within an interval grows, so does the density of the approximating NLP.

Therefore, there is a trade off that exists between increasing the number of collocation

points in an interval in order to obtain higher accuracy solutions that also increases the

density of the NLP. If an interval is approximating a smooth region of a solution, the

rapid rates of convergence by increasing the number of LG or LGR points far outweigh

the increase in NLP density. If an interval is approximating a non-smooth region of a

solution, then it is more efficient to increase the accuracy of the approximating NLP

by dividing the problem into more low-order, sparse, approximating intervals. Finally,

it is seen that the size of the hp–LG method is dependent upon both the number of

collocation points and the number of mesh intervals whereas the size of the hp–LGR

method is only dependent upon the number of collocation points. Therefore, if a large

number of mesh intervals are to be used, the hp–LGR method results in a more efficient

NLP.

99

CHAPTER 5HP–MESH REFINEMENT ALGORITHMS

In this chapter, two hp-mesh refinement algorithms are presented using the

hp–LG and LGR methods of Chapter 4. In each algorithm, a naive initial mesh is

selected. Because it is assumed that no a priori knowledge of the solution structure

of a continuous-time optimal control problem being approximated is known, the naive

initial meshes are either a single-interval of moderate degree or relatively few uniformly

distributed intervals of low-degree. From this naive approximation, both algorithms

assess the approximation errors in each interval. If the accuracy of the approximation

in an interval needs to be improved, either the degree of approximation in the interval

is increased or the interval is subdivided into more approximating subintervals. These

strategies are then repeated iteratively until an accurate approximation is obtained.

Because a known exact solution is not generally available, solution errors in an

interval are approximated by examining how closely the differential-algebraic constraint

equations are satisfied between collocation points.

The first mesh refinement algorithm is more heavily weighted as a p–biased

hp–mesh refinement algorithm. An interval is subdivided only if it is believed that

exponential convergence does not hold in that interval. The second algorithm is an

h–biased hp–mesh refinement algorithm. In this second algorithm, subdivision of an

interval is chosen much more frequently and an increase in degree of polynomial

approximation is chosen only if the solution in an interval is considered smooth. First,

a discussion on assessment of the errors in an approximating grid are discussed.

Next, hp–mesh refinement algorithm 1 and 2 are described. It is noted that hp–mesh

refinement algorithm 1 and 2 are adaptations of the algorithms presented in Refs. [66]

and [67].

100

5.1 Approximation of Errors in a Mesh Interval

Consider an hp–LG or LGR approximation to the solution of a continuous-time

optimal control problem as defined in Chapter 4 with K mesh intervals. Furthermore,

let t0 < t1 < ... < tK be the mesh points such that tk−1 defines the beginning of

the k th mesh and tk defines the end of the k th mesh. Next, let Nk be the number of

collocation points in mesh interval k . In designing the hp–mesh refinement algorithms

an assessment of the approximation accuracy of each approximating interval k must

be made without knowledge of the exact solution to a continuous-time optimal control

problem.

In order to provide an assessment on the accuracy of the approximation without

knowledge of the exact solution, the satisfaction of the differential-algebraic constraints

between collocation points are examined. At the collocation points, the differential-algebraic

equations are enforced. However, if an approximating hp–pseudospectral approximation

is accurately approximating the continuous-time optimal control problem, then the

differential-algebraic constraints should be nearly satisfied at an arbitrary point. In

order to evaluate the differential-algebraic constraints between collocation points, the

state and control must be interpolated to an arbitrary point. The state is defined at an

arbitrary point in an interval k by the Lagrange polynomial of Eqs. (4–24) or (4–73)

for the hp–LG or LGR methods, respectively. The control on the other hand, is only

calculated at the collocation points and does not have a unique underlying polynomial

approximation. Therefore, any control approximation that matches the value of the

control at the collocation points is a valid approximation (e.g., linear, cubic, spline,

Lagrange polynomial). In this research, a Lagrange polynomial approximation of

the control is used. If the state is being approximated by a high-accuracy Lagrange

polynomial, then it is counter-productive to use a low-order, less accurate approximation

to the control. Therefore, in either the hp–LG or LGR method, the Lagrange polynomial

approximation of the control with support points at the Nk collocation points in interval k

101

is given as

U(k)(τ) =Nk∑

i=1

U(k)i L(k)i (τ). (5–1)

Using the aforementioned state and control approximations, the differential-algebraic

constraints can now be evaluated at an arbitrary point in a mesh interval k . In order to

sample the differential-algebraic constraints between the collocation points, define

L points in interval k as t(k) = (t(k)1 , ... , t(k)L ) ∈ [tk−1, tk ]. The differential-algebraic

constraints can then be evaluated at these L points as

∣

∣

∣X(k)i (t

(k)l )− f

(k)i (X

(k)l ,U

(k)l )

∣

∣

∣= a

(k)li , (5–2)

C(k)j (X

(k)l ,U

(k)l ) = b

(k)lj , (5–3)

where i = 1, ... , n, j = 1, ... , s , and l = 1, ... ,L. Next, define e(k)max as the maximum

element of a(k)li or b(k)lj . If e(k)max is less than a specified error tolerance, ǫ, then the solution

in the interval is considered accurate. It is noted that Eq. (5–2) is the magnitude of

the violation in the collocation equations while Eq. (5–3) is simply an evaluation of the

inequality path constraints at L points. If the inequality path constraints are inactive,

the left hand side of Eq. (5–3) will be negative, and therefore, be less than ǫ. When the

path constraints are active, between collocation points the left hand side of Eq. (5–3)

may become positive thus violating the path constraint. If any element a(k)li or b(k)lj is

greater than ǫ, then the interval is considered inaccurate and must be refined such

that a more accurate approximation is made. The two hp–mesh refinement algorithms

are now presented that determine how to refine inaccurate approximating intervals. In

hp–mesh refinement algorithm 1, the errors estimated in Eqs. (5–2)–(5–3) are used

directly to determine if an inaccurate interval should be subdivided or if the degree of

approximation should be increased. In hp–mesh refinement algorithm 2, an h–biased

strategy is used. The curvature of the state in an interval is used to determine the

placement of the mesh points and whether to subdivide an interval or increase the

degree of the approximating polynomial within an interval. Even when it is chosen to

102

increase the degree of the approximating polynomial, the degree of the approximation

remains small.

5.2 hp–Mesh Refinement Algorithm 1

A description of hp–mesh refinement algorithm 1 is now given. It is noted that

hp–mesh refinement algorithm 1 is similar to the algorithm presented in Ref. [66] except

for the following differences. In Ref. [66], a cubic interpolation to the control was used.

In this dissertation, a Lagrange polynomial approximation to the control is used within

each interval. Furthermore, in Ref. [66], errors are assessed at the midpoints between

collocation points, whereas in this dissertation, errors are assessed at the arbitrary

points t(k) within each interval. Furthermore, Ref. [66] only modified the mesh based

upon analysis of the collocation conditions. In this dissertation, the collocation conditions

and path constraints are used to modify the mesh.

5.2.1 Increasing Degree of Approximation or Subdividing

The fundamental and distinct property of an hp–mesh refinement algorithm is the

step that determines whether to increase the degree of polynomial approximation in an

inaccurate interval or to subdivide the interval. Consider again the matrices defining the

violation of the differential-algebraic constraint equations in Eqs. (5–2) and (5–3)

A(k) = a(k)li (5–4)

B(k) = b(k)lj , (5–5)

where the column i of A(k) and column j of B(k) defines the magnitude of violation

in the collocation equations for the i th component of the state and violation of the j th

path constraint equation, respectively, at t(k). Let the state X (k)m (t), m ∈ [1, n] be the

component of the state approximation with the maximum value of a(k)li . Furthermore,

let the constraint C (k)p p ∈ [1, s] be the pth inequality path constraint with the maximum

value of b(k)lj . The component of the state or inequality path constraint with maximum

error is always used to determine how to proceed with the mesh refinement. If e(k)max is

103

associated with the inequality path constraints, the interval is subdivided. It is assumed

that, if a path constraint becomes active in an interval, the trajectory is non-smooth

and, therefore, it is necessary to subdivide the interval. If the maximum error occurs

in the collocation equations, the following procedure is used to determine whether to

increase the degree of polynomial approximation or subdivide the interval. First, define

the column of A(k) that contains e(k)max as

e(k) =

e(k)1

...

e(k)L

=

a(k)1i

...

a(k)Li

. (5–6)

Next, define the arithmetic mean of e(k) as

e(k) =

∑L

i=1 ei

L. (5–7)

Finally, define β as

β =

β(k)1...

β(k)L

=

e(k)1 /e(k)

...

e(k)L /e(k)

. (5–8)

β then defines the ratio of the error at a point e(k)1 compared to the average error

e(k). Next, define a parameter ρ, where ρ is used to determine whether to increase

the degree of the approximation in the interval or to subdivide the interval. If every

element of β is less than ρ, then no particular point dominates the errors in interval k .

In this case, the degree of the polynomial approximation in the interval is increased.

Furthermore, if values of β are greater than ρ, then errors in the interval are dominated

at particular points. In this case, the problem is subdivided at these points of maximum

error.

104

5.2.2 Location of Intervals or Increase of Collocation Poin ts

Now that a determination has been made as to whether an inaccurate interval k is

to be modified by increasing the degree of the polynomial or subdivision of the interval,

the details of each choice are now discussed. Consider the case where e(k)max occurs in

the collocation equations and every element of β is less than ρ. In this case, the number

of collocation points in an interval is increased by a specified amount N(+), that is,

N(i+1)k = N

(i)k + N

(+) (5–9)

where N(i)k is the number of collocation points in interval k on grid iteration i . Next,

consider the case where some elements of β are greater than ρ or e(k)max occurs in

the path constraint equations. If e(k)max is in the collocation equations, the interval is

subdivided at every local maximum of β that is also greater than ρ. Fig. 5-1 shows how

to divide an interval for this case. Furthermore, the number of collocation points in each

new interval is set to N(0). If e(k)max occurs in the path constraint equations, the interval

is subdivided at the location of maximum violation of the path constraint and each new

interval has N(0) collocation points.

-

Figure 5-1. Location of Intervals if Subdivision is Chosen.

105

5.2.3 Stopping Criteria

The algorithm stops when the dynamic constraints and inequality path constraints

are satisfied to the tolerance ǫ in every interval at L points between the collocation

points.

5.2.4 hp–Mesh Refinement Algorithm 1

hp–Mesh Refinement Algorithm 1 is now stated.

1. If this is the first iteration, initialize the problem choosing N(0) collocation points andK = 1 number of intervals, where N(0) is chosen by the user. Otherwise, solve theNLP using the current mesh.Begin: For k = 1, ... ,K ,

2. If e(k)max ≤ ǫ, then do not modify interval k . Otherwise, continue to step 3.

3. If the maximum error occurs in the collocation equations and every elementof β(k) < ρ, then increase the degree of approximation of mesh interval k byN(+).

4. Otherwise, if errors are largest in the collocation equations, refine the intervalat the points defining the maximum errors in β. If errors are largest in thepath constraint equations, refine the interval at the point of maximum error.Furthermore, set the number of collocation points to N(0) = 5 in each newinterval.

End: For k = 1, ... ,K .

5. Return to Step 1.

Several important aspects of hp–mesh refinement algorithm 1 are now clarified. It

is noted that ρ serves as a tuning parameter that varies the amount of weighting from

h to p–biases. For very small values of ρ, the h–bias is increased. For large values of

ρ, the algorithm is more p–biased. ρ should be selected such that detection of large

isolated errors is made and the approximation can be subdivided about these points.

Next, for the problems analyzed in this dissertation, N(+) = 10 because increasing

by more than ten collocation points may make the NLP unnecessarily dense. Next, all

newly created intervals are initialized with five collocation points for two reasons. First,

if exponential convergence in a newly created interval holds, five collocation points may

106

provide high-accuracy solutions. Second, initializing a newly created interval with five

collocation points will keep the total number of collocation points small.


hp–mesh refinement algorithm 2 is now defined. Different from hp–mesh refinement

algorithm 1, hp–mesh refinement algorithm 2 is designed to be a much more h–biased

algorithm. It is noted that hp–mesh refinement algorithm 2 is similar to the algorithm

presented in Ref. [67] with the only modification in this dissertation being how the control

is interpolated. In this dissertation, the control in interval k ∈ 1, ... ,K is interpolated by

a Lagrange polynomial using support points at the Nk collocation points. In Ref. [67],

the control is interpolated with a Lagrange polynomial with support points at the Nk

collocation points and one additional non-collocated point for k ∈ 1, ... ,K − 1 and by a

Lagrange polynomial with support points at the NK collocation points in the final interval

K .

5.3.1 Increasing Degree of Approximation or Subdividing

If the solution in a mesh interval k is inaccurate, the first determination is whether

the degree of approximation in the interval should be increased or whether the interval

should be subdivided. Suppose that X (k)m (τ) is the component of the state approximation

in the k th mesh interval that corresponds to the maximum value of a(k)li . Furthermore, let

the curvature of the mth component of the state in mesh interval k be given by

κ(k)(τ) =|X (k)m (τ)|

∣

∣

∣

∣

[

1 + X(k)m (τ)2

]3/2∣

∣

∣

∣

. (5–10)

Next, define κ(k)max and κ(k) be the maximum and mean value of κ(k)(τ), respectively.

Furthermore, let rk be the ratio of the maximum to the mean curvature:

rk =κ(k)maxκ(k). (5–11)

107

If rk < rmax, where rmax > 0 is a user-defined parameter, then the curvature of mesh

interval k is considered uniform and the degree of approximation is increased within the

interval. If rk > rmax, then a large curvature relative to the remainder of the interval exists

and the interval is subdivided.

5.3.2 Increase in Polynomial Degree in Interval k

If rk < rmax, then the polynomial degree in mesh interval k needs to be increased.

Let M be the current degree of the approximating polynomial in the interval. The degree

Nk of the polynomial for the next grid iteration in mesh interval k is given by the formula

Nk = M + ceil(log10(e(k)max/ǫ)) + A (5–12)

where A > 0 is an arbitrary integer constant and ceil is the operator that rounds to the

next highest integer.

5.3.3 Determination of Number and Placement of New Mesh Point s

If rk > rmax, then mesh interval k is subdivided. Two quantities are now defined: The

number of subdivisions and the location of each subdivision. First, the new number of

mesh intervals, denoted nk , is computed as

nk = ceil(B log10(e(k)max/ǫ)), (5–13)

where B is an arbitrary integer constant. The locations of the new mesh points are

determined using the integral of a curvature density function in a manner similar to that

given in Ref. [30]. Specifically, let ρ(τ) be the density function [30] given by

ρ(τ) = cκ(τ)1/3, (5–14)

where c is a constant chosen so that

∫ +1

−1

ρ(ζ)dζ = 1.

108

Let F (τ) be the cumulative distribution function given by

F (τ) =

∫ τ

−1

ρ(ζ)dζ. (5–15)

The nk new mesh points are chosen so that

F (τi) =i − 1nk, 1 ≤ i ≤ nk + 1.

It is shown in Ref. [30] that placing the mesh points in this manner is results in the

smallest L1 norm error when a piecewise linear approximation of a curve is used. Finally,

it is noted that if nk = 1, then no subintervals are created. Therefore, the minimum value

for nk should be at least two. Figs. 5-2A and 5-2B show qualitatively how to divide an

interval into more intervals for a constant curvature in an interval and an interval with a

large curvature relative to the remainder of the interval, respectively.


hp–mesh refinement algorithm 2 is now defined. The algorithm is initialized for a

coarse uniformly distributed mesh with a fixed, low-degree polynomial in each interval.

This approximation is solved, and then each mesh interval is analyzed. If errors are

less than ǫ, then the interval is not modified. Otherwise, if errors are greater than ǫ,

then the interval is modified according to the discussion of Sections 5.1 through 5.3.3.

The refinement process continues iteratively until a suitable approximating mesh is

constructed. hp–mesh refinement algorithm 2 is now given:

1. Solve the NLP using the current mesh.Begin: For k = 1, ... ,K ,

2. If e(k)max ≤ ǫ, do not modify interval k .

3. If either rk ≥ rmax or Nk > M, then refine the k th mesh interval into nksubintervals where nk is given by (5–13). Set the degree of the polynomialson each of the subintervals to be M and proceed to the next k .

4. Otherwise, set the degree of the polynomials on the k th subinterval to be Nk ,which is given in (5–12).

End: For k = 1, ... ,K .

109

A Constant Curvature.

B Variable Curvature.

Figure 5-2. Location of Intervals based on Curvature.

5. Return to Step 1.

Several important aspects of hp–mesh refinement algorithm 2 are now clarified.

hp–mesh refinement algorithm 2 is a h–biased algorithm. It is noted that unlike hp–mesh

refinement algorithm 1, an increase in the degree of polynomial approximation is

only allowed once within an interval. If the errors in approximation are still large after

increasing the degree of approximation, the interval is forced to subdivide as opposed

to increasing the degree of approximation again. Furthermore, Eq. (5–12) defines the

number of collocation points that are added to an interval based on a log10 difference in

110

the current error and desired error plus an arbitrary constant A. If the continuous-time

optimal control problem is known to have a smooth solution or if computational efficiency

on each mesh iteration is not a large concern, A can be chosen to be large. Next,

Eq. (5–13) defines the number of intervals created if subdivision is chosen. Eq. (5–13)

is a function of the log10 difference in the current error and the desired error and an

arbitrary multiplier B. A trade-off exists for large and small values of B. For small values

of B, fewer intervals will be created on each mesh iteration, but the number of mesh

iterations may grow large. For large values of B, more intervals will be created on each

mesh iteration, and the total number of mesh iterations may be reduced, but the number

of mesh intervals may grow large.

111

CHAPTER 6EXAMPLES

The hp–mesh refinement algorithms posed in Chapter 5 are now examined on

a variety of optimal control problems. The first example is the motivating example

of Section 3.1 whose solution, as we recall, is smooth. The second example is a

reusable launch vehicle entry problem whose solution appears to be smooth. The third

example is a problem where the timescale of the dynamics become fast as the final time

increases. The fourth example is the motivating example of Section 3.2 whose solution,

as we recall, is not smooth. Finally, the fifth example is a minimum time-to-climb of

a supersonic aircraft. This final example has two active path constraints along the

trajectory. The objective of this Chapter is to assess how well the hp–mesh refinement

algorithms perform relative to p and h–methods.

The examples were solved with modifications of the open-source optimal control

software GPOPS [37] using the NLP solver SNOPT [68]. All computations were

performed using a 2.5 GHZ Core 2 Duo Macbook Pro running Mac OS-X 10.5.8 with

MATLAB R2009b. Furthermore, analytic derivatives were used in the NLP solver. The

p–method discretizations were solved utilizing a single approximating interval with a

variable number of collocation points in the interval. The p–method simply assesses

the violations of Eqs. (5–2)–(5–3) at L uniformly distributed points between collocation

points where L = 1000. If errors are greater than ǫ, the number of collocation points

is increased by ten. Next, hp–mesh refinement algorithm 1 (hp(1)) uses ρ = 3.5 and

evaluates in each interval the violations of Eqs. (5–2)–(5–3) at L = 100 uniformly

distributed points between collocation points. hp–mesh refinement algorithm 2 (hp(2))

uses rmax = 2, A = 1, B = 2, and L = 30 uniformly distributed points between collocation

points. Finally, the h–method is an adaptation of hp–mesh refinement algorithm 2. For

the h–method, hp–mesh refinement algorithm 2 is used with rmax < 1, permitting only

segment division and preventing growth in the degree of the approximating polynomial.

112

The terminology “h–x–” denotes an h–method with x number of collocation points

in each interval while “hp(2)–x–” denotes the hp(2)–method with a minimum of x

collocation points in each interval. All examples were solved utilizing LGR points.

Finally, a maximum limit of 200 collocation points is given for the p–method. If a suitable

approximation is not obtained for 200 collocation points, then it is decided that an

accurate approximation cannot be obtained. Furthermore, for the hp(1), hp(2), and

h–methods, a mesh iteration limit of 40 is given. If a suitable approximation is not found

within 40 mesh iterations, then it is decided that an accurate approximation cannot be

obtained.

6.1 Example 1 - Motivating Example 1

Consider again the motivating example given by Eqs. (3–1)–(3–3). The problem is

now restated. Minimize the cost functional

J = tf (6–1)


x = cos(θ) , y = sin(θ) , θ = u, (6–2)


x(0) = 0 , y(0) = 0 , θ(0) = −π

x(tf ) = 0 , y(tf ) = 0 , θ(tf ) = π.(6–3)

The solution to the optimal control problem of Eqs. (6–1)–(6–3) is

(x∗(t), y ∗(t), θ∗(t), u∗(t)) = (− sin(t),−1 + cos(t), t − π, 1) (6–4)

where t∗f = 2π. As was demonstrated in Section 3.1, a p–method results in most

efficient approximation to this problem. Therefore, it is expected that the p and hp(1)

strategies will perform the best. Table 6-1 shows the initial uniformly distributed grid for

each strategy studied and the number of collocation points in each interval. It is noted

113

Strategy Nk Number of Intervalsp 10 1hp(1) 10 1hp(2)–2 2 10h–2 2 10

Table 6-1. Initial Grid used in Example 6.1 for Each Strategy Using Uniformly DistributedIntervals with Nk Collocation Points.

that the initial grid for all of the methods results in small NLPs that are solved efficiently.

Next, Table 6-2 shows the maximum absolute error evaluated at the collocation points

and the total CPU time required by the NLP to solve the problem for each value of ǫ.

Furthermore, the number of collocation points, mesh intervals, and NLP density on the

final grid are shown. First, it is noted for any value of ǫ, the maximum absolute error

using any strategy is less than ǫ. Next, it is noted that as ǫ is decreased, the maximum

absolute error by any method decreases. Examining the p and hp(1)–methods, it is

seen that the p and hp(1)–methods produce the exact same approximation grids for this

problem. Because the solution to this problem is smooth, the hp(1)–method chooses

to increase the degree of the approximating polynomial as opposed to subdividing the

initial interval. Moreover, when examining the results for hp(2)–2, again, it is noted that

an increase in the degree of the approximating polynomial was chosen in each interval

as opposed to subdividing the intervals. This is consistent with the fact that the solution

to this problem is smooth. Next, in comparing the CPU times for the hp(2)–2–method

with the hp(1) and p–methods, it is seen that the solutions times are similar for any value

of ǫ. Even though the hp(2)–2–method results in more collocation points than the hp(1)

or p–methods for any accuracy level, the increased NLP sparsity of the hp(2)–2–method

makes up for the increase in total number of collocation points.

Results are drastically different for the h–2–method. For ǫ = (10−2, 10−4), the CPU

time for the h–2–method is comparable to the CPU time of the other three methods. As

the accuracy tolerance is tightened, however, the h–2–method is seen to be significantly

less computationally efficient when compared with the other three methods. For ǫ =

114

ǫ Max Abs. Strategy CPU Colloc Mesh Mesh NLPError Time (s) Pts Interval Iterations Density

10−2 1.51× 10−5 p 0.0084 10 1 1 33.471.51× 10−5 hp(1) 0.0084 10 1 1 33.475.96× 10−3 hp(2)–2 0.0093 20 10 1 9.15.96× 10−3 h–2 0.0093 20 10 1 9.1

10−4 1.51× 10−5 p 0.0084 10 1 1 33.471.51× 10−5 hp(1) 0.0084 10 1 1 33.471.21× 10−7 hp(2)–2 0.025 50 10 2 5.31.39× 10−5 h–2 0.087 160 80 3 1.2

10−6 4.89× 10−15 p 0.027 20 1 2 29.644.89× 10−15 hp(1) 0.027 20 1 2 29.644.88× 10−11 hp(2)–2 0.034 70 10 2 4.51.98× 10−8 h–2 5.36 1568 784 5 0.13

10−7 4.89× 10−15 p 0.027 20 1 2 29.644.89× 10−15 hp(1) 0.027 20 1 2 29.648.51× 10−13 hp(2)–2 0.037 80 10 2 4.263.09× 10−10 h–2 104.60 5760 2880 5 0.035

Table 6-2. Summary of Accuracy and Speed for Example 6.1 Using Various CollocationStrategies and Accuracy Tolerances.

10−7, the p, hp(1), and hp(2)–2–methods all solve the problem in approximately 0.03

seconds. By comparison, the h–2–method requires 104.60 seconds to solve the problem

and 5760 collocation points are used.. Even though the NLP is only 0.035% dense, the

increased size makes the h–2–discretized NLP computationally inefficient.

It is demonstrated on this problem with a simple, smooth solution that a p–method

provides the most computationally efficient approximation. Furthermore, hp(1) behaves

exactly like the p–method. Next, it is seen that the performance of the h–biased

hp(2)–2–method is comparable to the p–method. By modestly increasing the degree

of polynomial within each approximating interval, efficient, sparse approximations are

obtained using relatively few collocation points for hp(2). Finally, for tight accuracy

tolerances, the h–2–method becomes much less computationally efficient when

compared with the other methods. The h–2–method for ǫ = 10−7 requires more than

3000 times the CPU time of the other methods presented in order to obtain an accurate

approximation.

115

6.2 Example 2 - Reusable Launch Vehicle Re-entry Trajectory

Consider the following optimal control problem [69] of maximizing the crossrange

during the atmospheric entry of a reusable launch vehicle. Minimize the cost functional

J = −φ(tf ) (6–5)


r = v sin γ,

θ =v cos γ sinψ

r cosφ,

φ =v cos γ cosψ

r,

v = −Dm

− g sin γ,

γ =L cos σ

mv−

(g

v− vr

)

cos γ,

ψ =L sinσ

mv cos γ+v cos γ sinψ tanφ

r,

(6–6)


r(0) = 79248 + Re m , r(tf ) = 24384 + Re m,

θ(0) = 0 deg , θ(tf ) = Free,

φ(0) = 0 deg , φ(tf ) = Free,

v(0) = 7803 m/s , v(tf ) = 762 m/s,

γ(0) = −1 deg , γ(tf ) = −5 deg,

ψ(0) = 90 deg , ψ(tf ) = Free,

(6–7)

where r is the geocentric radius, θ is the longitude, φ is the latitude, v is the speed, γ

is the flight path angle, ψ is the azimuth angle, α is the angle of attack, σ is the bank

angle, and Re = 6371203.92 m is the radius of the Earth. More details for this problem,

including the aerodynamic model, can be found in Ref. [69]. Similar to Example 6.1,

this problem has a smooth solution. Figs. 6-1A–6-1F shows the state approximation for

ǫ = 10−6 for the hp(2)–3–method. It is seen that all six components of the state appear

116

to be smooth. Therefore, it is expected that a p–biased method will provide the most

computationally efficient approximations to this problem.

Table 6-3 shows the initial uniform grid and number of collocation points in each

interval for each of the methods studied. Next, Table 6-4 shows the results for each

method for various values of ǫ. For ǫ = 10−4, all methods perform comparably with a

slight advantage to the hp(2)–x– and h–2–methods. It is noted for this problem, unlike

the previous problem, the hp(1)–method does not increase the degree of approximation

from the initial interval, but instead, actually subdivides the initial interval into three

subintervals of five collocation points each that satisfy the accuracy criterion ǫ = 10−4.

For ǫ = 10−6, the hp(1) method, interestingly, performs the worst with respect to CPU

time. It is noted, however, that hp(1) also requires the most mesh iterations to obtain

a solution. Furthermore, it is noted that although the p–method results in the fewest

number of collocation points, it is not the most computationally efficient method for

ǫ = 10−6. The reduction in computational efficiency of the p–method is due to the fact

that the p–method NLP density for this problem is significantly greater than either the

hp(2)–x– or h–2–method NLP densities. Interestingly, the hp(2)–x and h–2–methods are

the most computationally efficient for ǫ = 10−6.

For ǫ = (10−7, 10−8), the h–2–method is significantly slower than the other methods.

Because of the large problem size, the resulting NLP becomes computationally

intractable. In comparing the p, hp(1) and hp(2)–x–methods, it is seen again that the hp(1)

method requires the greatest computational time because it requires the most number

of mesh iterations. For ǫ = 10−8, the p–method provides the fastest computational

speeds. For this problem, for very high levels of accuracy, the p–method is the most

computationally efficient. Next, in comparing the hp(2)–x–methods, it is seen that the

hp(2)–3 and hp(2)–4–methods perform twice as fast as the hp(2)–2–method. An increase

in one or two collocation points for high levels of accuracy, therefore, strongly affect the

CPU times by reducing the overall size of the problem for ǫ = 10−8. Finally, it is noted for

117

0

40

60

80

50

20

30

1000 2000500

70

1500 2500t (sec)

h(k

m)

A h (km) vs. Time.

00

40

60

80

20

1000 2000500 1500 2500t (sec)

θ(d

eg)

B θ (deg) vs. Time.

00

10

40

20

30

1000 2000

4985

500 1500 2500t (sec)

φ(d

eg)

C φ (deg) vs. Time.

00

2

4

6

8

1000 2000500 1500 2500t (sec)

v(k

m/s

)

D v (km/s) vs. Time.

0

0

2

−2

−4

−61000 2000500 1500 2500t (sec)

γ(d

eg)

E γ (deg) vs. Time.

00

40

60

80

100

20

1000 2000500 1500 2500t (sec)

ψ(d

eg)

F ψ (deg) vs. Time.

Figure 6-1. State vs. Time on the Final Mesh for Example 6.2.

118

Strategy Nk Number of Intervalsp 20 1hp(1) 20 1hp(2)–4 4 20hp(2)–3 3 15hp(2)–2 2 20h–2 2 20

Table 6-3. Initial Grid used in Example 6.2 for Each Strategy Using Uniformly DistributedIntervals with Nk Collocation Points.

ǫ Strategy CPU Time (s) Colloc Pts Mesh Interval Mesh Iterations NLP Density10−4 p 7.03 40 1 3 15.12

hp(1) 6.04 15 3 2 11.48hp(2)–4 1.58 44 11 2 3.86hp(2)–3 1.93 30 10 1 3.51hp(2)–2 1.16 42 20 2 3.52h–2 1.10 42 21 2 3.46

10−6 p 13.90 70 1 6 14.02hp(1) 32.05 145 11 10 2.31hp(2)–4 4.22 136 32 4 1.31hp(2)–3 3.91 119 30 3 1.49hp(2)–2 4.18 132 34 4 1.32h–2 5.34 256 128 4 0.58

10−7 p 45.37 80 1 7 13.82hp(1) 102.10 165 15 13 1.92hp(2)–4 30.33 254 61 3 0.70hp(2)–3 36.41 224 53 4 0.81hp(2)–2 56.01 272 64 4 0.66h–2 424.05 774 387 5 0.19

10−8 p 64.97 90 1 8 13.69hp(1) 201.78 210 18 13 1.56hp(2)–4 105.3 366 84 5 0.48hp(2)–3 111.2 479 112 5 0.38hp(2)–2 217.2 578 114 4 0.33h–2 1815.6 2480 1240 6 0.06


ǫ = 10−8 that the CPU times using hp(2)–3 and hp(2)–4 are comparable to the CPU times

using the p–method.

119

6.3 Example 3: Hyper-Sensitive Problem

Consider the following hyper-sensitive [70–73] optimal control problem adapted from

Ref. [71]. Minimize the cost functional

J =1

2

∫ tf

0

(x2 + u2)dt (6–8)

subject to the dynamic constraint

x = −x3 + u (6–9)


x(0) = 1 , x(tf ) = 1 (6–10)

where tf is fixed. This problem is the final example problem considered in this Chapter

whose solution is smooth. This problem has a “take-off’,” “cruise,” and “landing” structure

where all of the interesting behavior occurs in the “take-off” and “landing” segments

(i.e., the first ten and last ten time units of the solution). For further details, Ref. [71]

contains a more in-depth discussion of this problem. The solution to this problem is now

approximated for three different values of tf .

6.3.1 Solutions for tf = 50

The problem defined by Eqs. (6–8)–(6–10) is now studied for tf = 50. The initial

approximating grids are defined in Table 6-5 and results for the various approximation

methods are given in Table 6-6. The approximation to the state and control of this

problem for tf = 50 is shown for the hp(2)–2–approximation with ǫ = 10−4 in Figs. 6-2A

and 6-2B. It is seen that in Figs. 6-2A and 6-2B, the initial mesh during the “cruise”

segment was not modified. The only refinement that occurred was to more accurately

approximate the “take-off” and “landing” segments. Next, it is interesting to note, that

while the solution to this problem is smooth, the p–method was not able to obtain a

solution for ǫ = 10−6. Furthermore, each method has a similar CPU time for ǫ =

(10−2, 10−4). As with the previous two examples, for the tightest accuracy tolerance,

120


Table 6-5. Initial Grid used in Example 6.3.1 for Each Strategy Using UniformlyDistributed Intervals with Nk Collocation Points.


hp(1) 0.25 15 3 3 24.42hp(2)–4 0.12 56 14 3 6.75hp(2)–3 0.14 35 11 4 9.52hp(2)–2 0.15 30 14 4 9.43h–2 0.24 34 17 6 8.06

10−4 p 0.33 60 1 6 50.38hp(1) 1.01 55 7 7 12.37hp(2)–4 0.25 86 20 3 4.75hp(2)–3 0.31 74 20 4 5.19hp(2)–2 0.24 65 21 4 5.71h–2 0.69 162 81 6 1.82

10−6 p 7.74∗ 200∗ 1∗ 20∗ 50.12∗

hp(1) 2.15 100 10 9 7.98hp(2)–4 0.96 168 36 6 2.61hp(2)–3 1.26 147 31 6 3.08hp(2)–2 0.80 129 28 5 3.58h–2 74.15 1328 664 7 0.22

Table 6-6. Summary of Accuracy and Speed for Example 6.3.1 Using VariousCollocation Strategies and Accuracy Tolerances.

the h–2–method requires significantly more computational time in order to obtain an

accurate approximation when compared with the hp–methods. Furthermore, unlike

the previous two examples, the p–method is unable to obtain a solution for the tightest

accuracy tolerance. For this example, in order to obtain an efficient high-accuracy

solution, an hp–method is required.


The problem defined by Eqs. (6–8)–(6–10) is now examined for tf = 1000 with the

initial approximating grids defined in Table 6-7 and results for the various approximation

methods given in Table 6-8. First, the p–method is the most computationally inefficient

121

0

0

1

10 40 5020 30−0.2

0.2

0.4

0.6

0.8

1.2

x

t

A x vs. t.

0

0

1.5

2

2.5

−0.5

0.5

1

10 40 5020 30t

u

B u vs. t.

Figure 6-2. State and Control vs. Time on the Final Mesh for Example 6.3.1 for hp(2)–2and ǫ = 10−4.

122



approximation for the lowest accuracy tolerance. The number of collocation points

for the p–method approximation is larger than the hp(1), hp(2)–x , and h–2–methods.

Furthermore, the NLP density of the p–method is much greater than any of the other

methods. Therefore, the p–method is the most computationally inefficient method.

Next, it is seen that the p–method is unable to obtain an approximation for the accuracy

tolerances of ǫ = (10−4, 10−6).

Finally, in studying the h–2 approximations, for ǫ = (10−2, 10−4), the h–2–method

performs similarly when compared with the hp–methods. Again it is noted for the

tightest accuracy tolerance, the h–2–method is significantly more inefficient, requiring

approximately 170 seconds to solve the problem using 9 mesh iterations and 664

approximating intervals on the final mesh. The hp–methods, however, required

approximately five seconds to solve the problem for ǫ = 10−6. For tf = 1000, again,

high-accuracy efficient solutions are only obtained by the hp–methods.


Finally, the problem is analyzed for tf = 5000 with the initial approximating grids

given by Table 6-9 and results given in Table 6-10. It is noted that for the p–method

approximation, L = 10000 in order to increase the resolution of the error assessment.

For this problem, the p–method is unable to obtain a solution for any value of ǫ. It is

again seen that for ǫ = (10−2, 10−4), the hp and h–methods perform comparably, and

for the tightest accuracy tolerance, the h–2–method again is much less computationally

efficient.

123


hp(1) 0.71 30 6 5 13.49hp(2)–4 0.12 56 14 3 6.75hp(2)–3 0.27 56 18 6 6.06hp(2)–2 0.27 38 18 6 7.53h–2 0.37 46 23 8 6.10

10−4 p 9.53∗ 200∗ 1∗ 20∗ 50.12∗

hp(1) 1.97 70 10 8 9.05hp(2)–4 0.25 86 20 3 4.75hp(2)–3 1.17 141 43 7 2.57hp(2)–2 0.66 104 37 8 3.42h–2 1.03 184 92 8 1.60

10−6 p – – – – –hp(1) 4.05 125 15 9 5.84hp(2)–4 4.71 214 46 6 2.06hp(2)–3 5.93 246 59 7 1.73hp(2)–2 2.97 209 53 8 2.08h–2 169.55 1328 664 9 0.22




Figs. 6-3A and 6-3B show the state and control approximation for ǫ = 10−4 using

the hp(2)–2–method and Figs. 6-4A and 6-4B show the state approximation for the first

15 and last 15 time units. As expected, it is seen that refinement occurs only at the start

and terminus of the solution.

6.3.4 Summary of Example 6.3

In Example 6.3, several interesting features are demonstrated for approximating

the solution of this problem by p, hp, and h–methods. As the final time increases,

the timescale of this problem defining the “take-off” and “landing” segments become

increasingly fast with respect to the total trajectory time. Consequently, as tf is

124

0

0

1x

t

−0.2

0.2

0.4

0.6

0.8

1.2

1000 2000 3000 4000 5000

A x vs. t.

0

0

1.5

2

2.5

−0.5

0.5

1

t

u

1000 2000 3000 4000 5000

B u vs. t.

Figure 6-3. State and Control vs. Time on the Final Mesh for Example 6.3.3 for hp(2)–2and ǫ = 10−4.

125

0

0

1

5 10 15

x

t

0.2

0.4

0.6

0.8

A x vs. t.

0

1

x

t

0.2

0.4

0.6

0.8

5000 5005 5010 50154985 4990 4995

B x vs. t.

Figure 6-4. State vs. Time on the Final Mesh for Example 6.3.3 for hp(2)–2 and ǫ = 10−4.

126

ǫ Strategy CPU Time (s) Colloc Pts Mesh Interval Mesh Iterations NLP Density10−2 p 13.11∗ 200∗ 1∗ 20∗ 50.12∗

hp(1) 0.84 45 7 6 12.74hp(2)–4 0.70 76 19 7 5.05hp(2)–3 0.42 68 22 8 5.02hp(2)–2 0.81 46 22 8 6.27h–2 0.38 54 27 8 5.25

10−4 p – – – – –hp(1) 1.41 70 10 9 9.05hp(2)–4 1.14 161 39 6 2.50hp(2)–3 1.21 151 46 8 2.42hp(2)–2 2.74 126 45 10 2.82h–2 1.12 200 100 9 1.48

10−6 p – – – – –hp(1) 3.73 110 12 9 7.78hp(2)–4 8.42 258 59 7 1.64hp(2)–3 5.69 217 55 8 1.93hp(2)–2 4.13 188 53 8 2.24h–2 107.91 1368 684 10 0.22


increased, the p–method becomes increasingly less capable of accurately approximating

the solution to this problem. For tf = 50, the p–method was able to approximate the

solution for ǫ = (10−2, 10−4) and for tf = 1000 the p–method was able to approximate the

solution for ǫ = 10−2. For tf = 5000, the p–method was unable to approximate solution to

the problem for any of the values of ǫ.

The h–2–method is much more insensitive than the p–method to the value of tf for

approximating the solution of this problem. Using a p–method, it is not possible to refine

the mesh specifically in regions where the error may be large. Because refinement

is necessary at the start and terminus of the solution, the h–2–method is much more

successful than the p–method in obtaining high-accuracy approximations to the solution

of this problem. However, because the h–2–method only uses low-order approximating

polynomials, when high accuracy is required, the number of approximating polynomials

is large. Even though the NLP is very sparse for the h–2–method, the size of the NLP

problem for high levels of accuracy results in a computationally inefficient NLP that

needs to be solved.

127

Finally, both hp–methods were able to obtain high accuracy solutions for this

problem for all the values of tf and all the values of ǫ studied. First, while being a more

p–biased method, the hp(1)–algorithm allows for subdivision of approximating intervals.

Because of the ability to subdivide the initial single-interval approximation, hp(1) is

able to densely collect intervals only at the ends of the trajectory regardless of final

time. Furthermore, by allowing higher degree collocation when compared with the

h–2–method, the problem size is much smaller than the h–2–approximation for high

levels of accuracy. Therefore, hp(1) results in an efficient approximation for any of the

accuracy requirements presented. hp(2) also demonstrates a similar capability. First,

the mesh is refined only at the ends of the trajectory. Second, because increases in

the degree of approximating polynomial are allowed within intervals, high accuracy

approximations are made with a much smaller NLP than the h–2–method. For this

problem, it is demonstrated that by allowing the number and location of mesh intervals

to vary and allowing the degree of approximating polynomial to vary, computationally

efficient approximations are obtained for a wide range of accuracy requirements with an

insensitivity to the duration of the trajectory.

6.4 Example 4: Motivating Example 2

Now we will revisit the optimal control problem of Section 3.2. To restate the

problem, this problem is to minimize

J =

∫ tf

t0

udt (6–11)

subject to

h = v

v = −g + u,(6–12)


h(0) = 10 , v(0) = −2

h(tf ) = 0 , v(tf ) = 0,(6–13)

128

and the control path constraint

0 ≤ u ≤ 3 (6–14)

where g = 1.5, and tf is free. The optimal solution to the optimal control problem given

in Eqs. (6–11)–(6–14) is

(h∗(t), v ∗(t), u∗(t)) =

(−34t2 + v0t + h0,−32t + v0, 0) , t ≤ s∗

(34t2 + (−3s∗ + v0)t + 3

2(s∗)2 + h0,

32t + (−3s∗ + v0), 3) , t ≥ s∗

(6–15)

where s∗ is given as

s∗ =t∗f2+v0

3(6–16)

with

t∗f =2

3v0 +

4

3

√

1

2v 20 +

3

2h0 (6–17)

For the boundary conditions given in Eq. (6–13), we have (s∗, t∗f ) = (1.4154, 4.1641).

It is seen that the optimal control for this problem is “bang-bang” in that it is at its

minimum-value for t < s∗ and at its maximum value for t > s∗. For this problem, the

solution is represented by two piecewise low-degree polynomials. The only difficulty,

therefore, in accurately approximating the solution of this problem is the discontinuity

in the control and the non-smoothness of the state at s∗. Table 6-11 shows the initial

grid for each approximation strategy and Table 6-12 summarizes the results from each

approximation method for a variety of values of ǫ. Because the solution to this problem is

not smooth, the p–method is unable to obtain an adequate approximation for any value

of ǫ. Furthermore, the hp(1) method performs significantly worse than either the hp(2)–2–

or h–2– methods. Moreover, for ǫ = 10−6, no solution was found by the hp(1) method.

Therefore, for this problem, the p–biased methods underperform.

Next, for each value of ǫ, the hp(2)–2– and h–2– methods performed comparably.

Furthermore, for each accuracy requirement, these methods provided highly efficient

approximations to the solution. For any accuracy level, the h–biased strategies provide

129

Strategy Nk Number of Intervalsp 10 1hp(1) 10 1hp(2)–2 10 2h–2 10 2

Table 6-11. Initial Grid used in Example 6.4 for Each Strategy Using UniformlyDistributed Intervals with Nk Collocation Points .

ǫ Max Abs. Strategy CPU Colloc Mesh Mesh NLPError Time (s) Pts Interval Iterations Density

10−2 1.76× 10−2 p 4.68∗ 200∗ 1∗ 20∗ 33.78∗

3.02× 10−3 hp(1) 0.88 110 22 22 2.972.74× 10−2 hp(2)–2 0.037 20 10 1 10.502.74× 10−2 h–2 0.037 20 10 1 10.50

10−4 p – – – – –2.64× 10−3 hp(1) 1.18 155 31 25 2.125.20× 10−3 hp(2)–2 0.15 44 21 8 5.184.01× 10−3 h–2 0.094 32 16 5 6.82

10−6 p – – – – –3.57× 10−4 hp(1) 26.57∗ 580∗ 80∗ 40∗ 0.92∗

3.88× 10−5 hp(2)–2 0.72 60 30 19 3.758.15× 10−5 h–2 0.30 54 27 8 4.15


the most efficient approximations. Fig. 6-5 shows the control approximation and the

exact solution for ǫ = 10−4 for the h–2–approximation. Because the only difficult

feature in this problem is in approximating the discontinuity in the control, it is seen that

significant refinement occurs near the control discontinuity.

6.5 Example 5: Minimum Time-to-Climb of a Supersonic Aircraft

Consider the following optimal control problem that is a variation of the minimum

time-to-climb of a supersonic aircraft [72, 74, 75] . Minimize the cost functional

J = tf (6–18)


h = v sin γ , v =T −Dm

− g sin γ , γ =g

v(n − cos γ), (6–19)

130

00

1.5

2

2

2.5

3

3

0.5

1

1 4 5

u

t

u∗(t)

u(t)

Figure 6-5. Control vs. Time on the Final Mesh for Example 6.4 Along with the ExactSolution.


h(0) = 0 , h(tf ) = 19995 m

v(0) = 129.31 m/s , v(tf ) = 295.09 m/s

γ(0) = 0 deg , γ(tf ) = 0 deg,

(6–20)

and the inequality path constraints

γ ≤ 45 deg, (6–21)

h ≥ 0 m,

where h is the altitude, v is the speed, γ is the flight path angle, g is the local acceleration

due to gravity, and n is the load factor (and is the control for this example). Further

131

details of the vehicle model and the numerical values of the constants for this model

can be found in Ref. [72] and [75]. In this problem, during some segments of the

trajectory, non-smooth features are encountered by the active path constraints of

Eq. (6–21). Furthermore, in other segments of the trajectory, the trajectory is smooth.

Figs. 6-6A–6-6C show the state approximation to the solution of this problem for

ǫ = 10−4 using the hp(2)–3–method. First, it is seen by examination of Fig. 6-6A and

6-6C that significant refinement occurs near the location of the active path constraints

in altitude and flight path angle. Furthermore, it is noted in the smooth regions of the

trajectory that a much more sparse grid of points is used. Most of the computational

effort is in approximating the non-smooth features (i.e., the switch between activity and

inactivity of the path constraints).

Table 6-13 shows the initial grids for each method and Table 6-14 shows the results

for approximating the solution of this problem by each strategy. For ǫ = 10−2, each

method performs comparably. For ǫ = 10−4, the p–method was unable to obtain a

solution. Furthermore, it is noted that for 200 collocation points, the p–method required

approximately 12 minutes to not obtain an accurate approximation.

Next, in comparing the h–2, hp(1), and hp(2)–x–methods, it is demonstrated that

for the highest accuracy requirement of ǫ = 10−6, the hp(2)–x–methods are the most

successful in efficiently approximating the solution of this problem. The hp(2)–x–methods

highly refine the mesh around the non-smooth regions of the trajectory. Furthermore,

in the smooth regions of the trajectory, higher degree collocation is used. Therefore,

the hp(2)–x–methods are able to combine the benefits of subdividing intervals and

increasing the degree of approximating polynomial. hp(1) is slower than hp(2) because of

the density of the NLP and the number of mesh iterations are greater when using hp(1)

as opposed to using hp(2). Finally, for high levels of accuracy, the h–2–method is slower

than hp(2) because of the excessive size of the NLP.

132

00

5

10

10050 150 200

15

20

h(k

m)

t (sec)

400

A h vs. t.

0100

10050 150

200

200t (sec)

v(m

/s)

300

400

500

B v vs. t.

0

0−10

10

40

100

50

50 150 200

20

30

t (sec)

γ(d

eg)

C γ vs. t.

Figure 6-6. State vs. Time on the Final Mesh for Example 6.5 for hp(2)–3 and ǫ = 10−4.


Table 6-13. Initial Grid used in Example 6.5 for Each Strategy Using UniformlyDistributed Intervals with Nk Collocation Points.

133


hp(1) 0.60 35 7 4 7.46hp(2)–4 0.37 48 12 2 5.01hp(2)–3 0.44 36 12 4 5.94hp(2)–2 0.25 24 12 3 7.73h–2 0.21 24 12 2 7.73

10−4 p 707.67∗ 200∗ 1∗ 20∗ 25.51∗

hp(1) 4.20 125 15 9 4.30hp(2)–4 1.74 86 21 6 2.89hp(2)–3 0.28 69 20 6 3.42hp(2)–2 0.95 78 29 6 2.85h–2 0.87 110 55 4 1.79

10−6 p – – – – –hp(1) 69.70 245 17 18 7.56hp(2)–4 14.75 217 50 8 1.20hp(2)–3 19.84 241 62 10 1.06hp(2)–2 19.09 359 86 7 0.73h–2 115.01 934 467 6 0.21


6.6 Discussion of Results

The results of the five examples demonstrate some of the key characteristics of

approximating problems by hp–mesh refinement algorithm 1 and hp–mesh refinement

algorithm 2. In Example 6.1, the solution is smooth and the p and hp(1) methods perform

exactly the same. As expected, the most efficient approximation to this problem is by

a p–method. Furthermore, as expected, for tight accuracy tolerances, the h–2–method

is computationally inefficient. hp–mesh refinement algorithm 2 had comparable CPU

times to the p–method for all values of ǫ. By a relatively modest increase in degree of

approximation coupled with the increased sparsity of the approximation in hp–mesh

refinement algorithm 2, efficient approximations to the solution of the first example

problem were obtained.

In Example 2, the solution to the problem appears to be smooth, but is more

complicated than Example 6.1. For this example, as the accuracy tolerance was

tightened, the p–method provided the best approximation. Furthermore, as the

accuracy tolerance was tightened, the h–2–method became computationally inefficient.

134

Surprisingly for this example, hp–mesh refinement algorithm 1 was less efficient than

hp–mesh refinement algorithm 2. The inefficiency of hp–mesh refinement algorithm 1

is largely due to the excessive number of mesh iterations required to find a solution.

Furthermore, while the p–method provides the most efficient solution approximation

to this problem for a high accuracy requirement, it is interesting to note that hp–mesh

refinement algorithm 2 does not significantly underperform. While a single high-degree

approximating interval provides the most efficient approximation to the solution of this

problem, it is interesting to note that many intervals of moderate degree also accurately

and efficiently approximate the solution of this problem. The combined affects of using

moderate degree intervals and a sparse NLP provides comparable efficiency to the

p–method.

Example 6.3 studies a problem for three different values of tf where the chosen

values of tf change the inherent timescale of the problem. In this problem, a p–method

has an increasingly difficult time finding a suitable approximation as the final time is

increased. Furthermore, for very high accuracy requirements, the h–2-method is again

seen to provide inefficient results. Finally, both hp–algorithms performed well for any

value of tf . The combined ability to refine the location of mesh intervals and the use of

a variable degree polynomial in each interval make the hp–algorithms successful for

approximating the solution of this problem.

The solution structure to Example 6.4 is significantly different from the previous

three examples. The solution to this problem is represented by two piecewise low-degree

polynomials. Because the control is discontinuous the connection between these two

piecewise polynomials is non-smooth. In this problem, a p–method was unable to find

an accurate approximation for any value of ǫ because of the non-smooth nature of the

solution to this problem. hp–mesh refinement algorithm 1 performed significantly worse

when compared with hp–mesh refinement algorithm 2 or the h–2–method and was not

135

able to find a solution for the tightest accuracy tolerance. For any value of ǫ, hp–mesh

refinement algorithm 2 or the h–2–method performed well.

The final example is a modified version of the minimum time-to-climb of a

supersonic aircraft. This example has non-smooth regions of the trajectory and smooth

regions of the trajectory. Because of the non-smoothness that exists, the p–method

was not able to find approximations for tight accuracy tolerances. Furthermore,

hp–mesh refinement algorithm 2 outperformed hp–mesh refinement algorithm 1 and

the h–2–method. hp–mesh refinement algorithm 2 more successfully simultaneously

handled non-smooth and smooth features than hp–mesh refinement algorithm 1 or the

h–method.

In summary, the method that will best approximates the solution of an optimal

control problem is problem dependent. If a problem is simple and smooth, a p–method

performs the best. In any other case, a p–method does not perform well. hp–mesh

refinement algorithm 1 improves upon the robustness of a strictly p–method for

approximating the solution to problems that may be non-smooth. For problems with

nonsmooth features, hp–mesh refinement algorithm 1 requires more computational time

than hp–mesh refinement algorithm 2 or the h–2–method.

By an h–method, low-accuracy solutions can be obtained to a wide variety of

problems in a computationally efficient manner. On the other hand, high-accuracy

solutions require a tremendous computational burden because of the large problem

size of the h–methods. hp–mesh refinement algorithm 2 performed much better.

Furthermore, for every example problem of this chapter, hp–mesh refinement algorithm

2 performed either better than the other methods or was comparable to the best

method for each problem (i.e., p for problems with simple smooth solutions or h for

problems with nonsmooth solutions with no high-order behaviors). It is demonstrated by

hp–mesh refinement algorithm 2 that high accuracy solutions to problems do not require

high-degree polynomials. The highest degree polynomial used by hp–mesh refinement

136

algorithm 2 in any of the examples is eighth order. If exponential convergence is

expected in a particular interval, a high-degree polynomial approximation should

not be required. Furthermore, because low and mid-degree polynomials are used,

hp–mesh refinement algorithm 2 results in an NLP that is sparse. Moreover, by

specifically refining only parts of an approximation with the largest errors, the distribution

of the mesh focuses only on those locations that need more mesh intervals or

mid-degree approximating polynomials. hp–mesh refinement algorithm 2 demonstrates

significant robustness irregardless of accuracy requirement or problem type on the

examples in this Chapter. As a starting point, hp–mesh refinement algorithm 2 utilizes

low-degree polynomials that ensure a sparse NLP. Refinement of the mesh occurs in

regions of large error, and mid-degree polynomials are used on intervals with smooth

solutions. Therefore, the computational efficiency of an h–method is combined with the

high-accuracy of a p–method.

137

CHAPTER 7AEROASSISTED ORBITAL INCLINATION CHANGE MANEUVER

In this Chapter, an aeroassisted orbital inclination change maneuver for a small

maneuverable spacecraft in low-Earth orbit (LEO) is studied. It is known that using

atmospheric force (namely, lift and drag) may significantly reduce the amount of fuel

required to change the inclination of the vehicle as compared to an all-propulsive

maneuver [76]. The survey papers of Ref. [77] and [78] summarize early work on

aeroassisted orbital transfers. Ref. [79] studied multiple-pass aeroassisted gliding

maneuvers with a large inclination change of a large spacecraft between geostationary

orbit (GEO) and LEO. A key result of Ref. [79] is that the aeroassisted maneuver was

much more fuel efficient than an all-propulsive maneuver. Furthermore, in Ref. [79],

it was demonstrated for highly constrained heating rates on the vehicle that the fuel

required to accomplish the maneuver decreased as the number of atmospheric passes

increased.

While previous research on aeroassisted orbital inclination change maneuvers has

been for large spacecraft, in this research, we consider a LEO to LEO aeroassisted

orbital transfer of a small maneuverable spacecraft. The aeroassisted maneuver is

compared against all-propulsive maneuvers. The multiple-pass aeroassisted glide

maneuvers are analyzed as functions of the number of atmospheric passes, the required

inclination change, and the maximum allowable heating rate on the vehicle. The results

of this Chapter are similar to the results found in Ref. [80].

138

7.1 Equations of Motion

The equations of motion over a spherical non-rotating Earth are given in spherical

coordinates as [19, 79, 81]

r = v sin γ,

θ =v cos γ cosψ

r cosφ,

φ =v cos γ sinψ

r,

v = −Dm

− µ

r 2sin γ,

γ =1

v

[

−L cos σm

−(

µ

r 2− v

2

r

)

cos γ

]

,

ψ =1

v

[

− L sinσm cos γ

− v2

rcos γ cosψ tanφ

]

,

(7–1)

where r is the geocentric radius, θ is the longitude, φ is the latitude, v is the speed, γ is

the flight path angle, ψ is the heading angle, D is the drag force, L is the lift force, µ is

the Earth gravitational parameter, and σ is the bank angle. Next, it is noted that drag and

lift are modeled, respectively, as [19]

D =1

2ρv 2ACD (7–2)

L =1

2ρv 2ACL (7–3)

where ρ is the atmospheric density, A is the vehicle reference area, CD is the coefficient

of drag, and CL is the coefficient of lift. The atmospheric density is modeled as an

exponential of the form

ρ = ρ0 exp(−βh), (7–4)

where ρ0 is the atmospheric density at sea level, β is the inverse of density scale height,

and h = r − Re is the altitude where Re is the radius of the Earth. The aerodynamic

139

Table 7-1. Physical and Aerodynamic Constants.Quantity Value

ρ0 1.225 kg/m3

1/β 7200 mA 0.32 mCD0 0.032

K 1.4

CLα 0.5699

m0 818 kgIsp 310 s˙Q 19987W/cm2

hatm 110 kmRe 6378145 mµ 3.986× 1014 m3/s2

g0 9.80665 m/s2

coefficients are modeled using a standard drag polar of the form [81]

CL = CLαα,

CD = CD0 + KC2L ,

(7–5)

where K is the drag polar constant, CD0 is the zero-lift drag coefficient, and CLα is the lift

slope. In this research we choose a generic high lift-to-drag ratio vehicle whose model

is taken from Ref. [82]. The aerodynamic data for the vehicle, along with the physical

constants, are given in Table 7-1.

7.1.1 Constraints On the Motion of the Vehicle

The following constraints are enforced on the motion of the vehicle during flight.

The initial and final conditions correspond to a circular low-Earth orbit of altitude h0.

The initial inclination is zero while the final inclination is if . During atmospheric flight,

constraints are enforced at the entrance and exit of the atmosphere on the flight path

angle such that upon atmospheric entry the vehicle must descend into the atmosphere

while upon atmospheric exit the vehicle must ascend. Furthermore, the angle of attack,

stagnation point heating rate, and altitude are all constrained during atmospheric flight.

140

7.1.1.1 Initial and final conditions

The initial conditions for this study correspond to a circular equatorial orbit of

altitude h0. The initial orbital conditions for this problem are given in terms of orbital

elements [83] as

a(t0) = Re + h0 , e(t0) = 0

i(t0) = 0 , Ω(t0) = 0

ω(t0) = 0 , ν(t0) = 0,

(7–6)

where a is the semi-major axis, i is the inclination, ω is the argument of perigee, e is the

eccentricity, Ω is the longitude of ascending node, and ν is the true anomaly. The values

for Ω(t0),ω(t0), and ν(t0) are arbitrarily set to zero. Furthermore, the final conditions for

this problem are for a circular orbit of altitude hf = h0 with an inclination if and are given

as

a(tf ) = Re + hf , e(tf ) = 0 , i(tf ) = if , (7–7)

Because the final orbit is circular and the location of the spacecraft in the terminal orbit

is not constrained, ω(tf ) is undefined and ν(tf ) and Ω(tf ) are free.

7.1.1.2 Interior point constraints

Along with the initial and final conditions on the vehicle, the following interior-point

constraints are enforced at every atmospheric entrance:

r (tatm0 ) = hatm + Re,

γ (tatm0 ) ≤ 0,

(7–8)

where tatm0 is the beginning of any atmospheric flight segment and hatm is the edge

of the sensible atmosphere. These constraints ensure that at the entrance of any

atmospheric flight segment the vehicle is descending into the atmosphere. Furthermore,

at the exit of any atmospheric flight segment, the following constraints are imposed such

141

that the vehicle is ascending upon atmospheric exit:

r (tatmf ) = hatm + Re,

γ (tatmf ) ≥ 0,

(7–9)

where t(atm)f is the time at the end of any atmospheric flight segment.

7.1.1.3 Vehicle path constraints

The following inequality path constraints are enforced during atmospheric flight. The

angle of attack is constrained as

0 ≤ α ≤ αmax. (7–10)

Next, the stagnation point heating rate is constrained. Using the Chapman equation [84],

the stagnation point heating rate, Q, is given as

Q = ˙Q(ρ/ρ0)0.5(v/vc)

3.15, (7–11)

where vc =√

µ/Re . Therefore, during atmospheric flight, the following constraint on

heating rate is imposed:

0 ≤ Q ≤ Qmax. (7–12)

Finally, during atmospheric flight, the altitude is constrained as:

0 ≤ h ≤ hatm (7–13)

or, alternatively,

Re ≤ r ≤ hatm + Re. (7–14)

7.1.2 Trajectory Event Sequence

The trajectory event sequence for a maneuver with n atmospheric passes is now

described. From the initial conditions of Eq. (7–6), a deorbit velocity impulse, ∆V1, is

applied to send the vehicle into the atmosphere. The vehicle then glides in and out of

the atmosphere n times. Upon the final atmospheric exit, a boost velocity impulse, ∆V2,

142

is applied to give the vehicle enough energy to reach the final orbit condition. Upon

reaching the final orbital altitude, a final recircularization velocity impulse, ∆V3 is applied

to recircularize the orbit. The trajectory event sequence is now described in detail (see

Figs. 7-1A–7-1C):

(i) Event: ∆V1 is applied to the initial orbital conditions in Eq. (7–6);

(ii) Phase: an exo-atmospheric glide that terminates at the edge of the sensibleatmosphere (hatm);

(iii) Phase: an atmospheric glide that terminates at the edge of the sensible atmosphere(hatm);

The following two phases are then repeated for the remaining n − 1 atmospheric passes:

(iv) Phase: an exo-atmospheric glide that terminates at the edge of the sensibleatmosphere (hatm);

(v) Phase: an atmospheric glide that terminates at the edge of the sensible atmosphere(hatm);

Upon the final atmospheric exit, the following events and phase occur:

(vii) Event: ∆V2 is applied at the maximum altitude of the sensible atmosphere;

(viii) Phase: an exo-atmospheric glide that terminates at the final orbital altitude;

(ix) Event: ∆V3 is applied at the final orbital altitude to re-circularize the orbit.

7.1.3 Performance Index

The cost functional for the multiple-pass aeroglide maneuver is to minimize the fuel

consumed during the maneuver. This corresponds to minimizing the negative of the final

mass, i.e.,

J = −m(tf ). (7–15)

Assuming impulsive thrust, ∆V ≡ ‖∆V‖, the mass after each impulse is then computed

from the Goddard rocket equation as

m+ = m− exp(−∆Vg0Isp

), (7–16)

143

A Deorbit and First Atmospheric Flight Segment.

B Intermediate Exo-Atmospheric and AtmosphericFlight Segments.

C Final Exo-Atmospheric Flight Segment.

Figure 7-1. Schematic of Trajectory Design for Aeroassisted Orbital Transfer Problem.

144

where m− and m+ are the values of the mass immediately before and after application

of the impulse. Therefore, the final mass after three velocity impulses is given as

m(tf ) = m(t0)exp(−(∆V1 + ∆V2 + ∆V3))

g0Isp. (7–17)

7.2 Re-Formulation of Problem

The manner in which the multiple-pass aeroassisted orbital transfer is solved

differs slightly from the formulation given in Section 7.1. Instead of using α and σ as the

controls, the quantities u1 and u2 are used as controls, where u1 and u2 are defined as

u1 = −CL sinσ,

u2 = −CL cos σ.(7–18)

In terms of u1 and u2, CL and σ are given as

CL =

√

u21 + u22 (7–19)

σ = tan−1(u1/u2). (7–20)

It is noted that u1 and u2 are used instead of the more obvious choices of coefficient

of lift and bank angle because u1 and u2 avoid the wrapping problem that arises

when using angles as controls [19]. In terms of u1 and u2, the differential equations

of Eq. (7–1) are given as

r = v sin γ,

θ =v cos γ cosψ

r cosφ,

φ =v cos γ sinψ

r,

v = −ρv2ACD

2m− µ

r 2sin γ,

γ =1

v

[

−ρv2A

2mu2 −

(

µ

r 2− v

2

r

)

cos γ

]

,

ψ =1

v

[

− ρv 2A

2m cos γu1 −

v 2

rcos γ cosψ tanφ

]

.

(7–21)

145

Next, because the coefficient of lift is linear in α, the constraint on α in Eq. (7–10) is

equivalently written as

0 ≤ CL ≤ CL, (7–22)

where CL denotes the maximum allowable value of CL. Therefore, the controls u1 and u2

are constrained as

− CL ≤ (u1, u2) ≤ CL, (7–23)

0 ≤ u21 + u22 ≤ C 2L . (7–24)

In this study the maximum allowable angle of attack is αmax ≈ 40 deg which corresponds

to CL = 0.4.

Next, the heating rate constraint of Eq. (7–11) is rewritten to be better behaved

computationally. Specifically, instead of constraining Q, we constrain the natural

logarithm of Q, log Q, as

−∞ ≤ log Q ≤ log Qmax, (7–25)

Using the above formulations, the optimal control problem corresponding to the

aeroassisted orbital transfer problem was solved in canonical units where length is

measured in Earth radii, time is measured in units of√

R3e /µ, speed is measured in units

of√

µ/Re , and mass is measured in units of m0, the initial mass of the vehicle. This

system of units is used because it is well-scaled.

7.3 Optimal Control Problem

The optimal control problem corresponding to the minimum-fuel aeroassisted orbital

transfer problem is now stated. Minimize the cost functional of Eq. (7–15) subject to the

dynamic constraints of Eqs. (7–21), the path constraints of Eqs. (7–14), (7–23), (7–24),

and (7–25), and the interior point constraints of Eqs. (7–8) and (7–9). The optimal

control problem is solved using the hp–mesh refinement algorithm of Ref. [67] which is

implemented in the software GPOPS [26, 67] and uses the NLP solver SNOPT [11, 68].

146

All NLP derivatives were computed using the Matlab automatic differentiator INTLAB

[85].

7.4 All-Propulsive Inclination Change Maneuvers

The following two all-propulsive transfers are studied in order to compared with the

aeroassisted orbital inclination change maneuver: (1) a single impulse direct change in

inclination via rotation of the initial velocity and (2) a two impulse bi-parabolic transfer.

Figs. 7-2A and 7-2B show the single impulse and bi-parabolic transfers, respectively. It

is seen that the single impulse transfer rotates the orbital velocity vector by an angle ∆i .

Next, the first impulse of the bi-parabolic transfer sends the orbit to an orbit of infinite

radius where the change in inclination is free. Upon return to the initial orbit, a second

impulse is applied to recircularize the orbit. For initial and terminal circular orbits of the

same size but with different inclinations, the direct impulse and total bi-parabolic impulse

required to change inclination by an amount ∆i are given, respectively, as [86]

∆V = 2v0 sin∆i2,

∆V = 2(√2− 1)v0,

(7–26)

where v0 is the initial orbital speed. By inspection, it is seen that the bi-parabolic

transfer is more efficient for ∆i > 49 degrees. Therefore, when comparing the optimal

aeroassisted orbital transfer fuel consumption with all-propulsive maneuvers, the

aeroassisted orbital transfer fuel consumption is compared with the single impulse

maneuver when ∆i ≤ 49 deg and is compared with the bi-parabolic transfer when

∆i > 49 deg.

7.5 Results

The multiple-pass aeroassisted orbital transfer optimal control problem summarized

in Section 7.3 was solved for n = (1, 2, 3, 4) atmospheric passes. Furthermore,

for each value of n, a change of inclination of if = (20, 40, 60, 80) deg subject to

Qmax = (400, 800, 1200,∞) W/cm2 is studied.

147

A Single Impulse Transfer.

B Bi-parabolic Transfer.

Figure 7-2. Schematic of All-Propulsive Single Impulse and Bi-Parabolic Transfers.

148

7.5.1 Unconstrained Heating Rate

The cost of the multiple-pass aeroassisted orbital transfer is first shown with no

constraint on heating rate. Fig. 7-3A shows the minimum impulse, ∆Vmin, required

to change the inclination of the orbital vehicle for one, two, three and four pass

aeroassisted maneuvers and the all-propulsive maneuvers. First, it is interesting to

note that because the bi-parabolic maneuver has a constant cost, the cost for the

60 deg and 80 deg all-propulsive inclination changes are equivalent. Next, it is seen

for very small inclination change (i.e., 1 deg), there is virtually no difference in cost

between aeroassisted maneuvers and all-propulsive maneuvers. For if = (20, 40, 60, 80)

deg, it is seen that the aeroassisted maneuvers can dramatically reduce costs when

compared with the all-propulsive maneuvers. The cost for a required inclination change

is insensitive to the number of atmospheric passes. For n = (1, 2, 3, 4), the costs are

nearly identical for any given value if . This is seen in Fig. 7-3B. In this figure, the final

mass fraction after completion of the aeroassisted maneuver is shown. It is seen that

for the range of inclination changes, a large fraction of the final mass is retained for the

vehicle after the maneuver is completed. For a 20 degree inclination change, more than

60% of the final mass is retained. Furthermore, for an 80 degree inclination change

maneuver, approximately 30% of the final mass is retained after completion of the

maneuver.

Next, Figs. 7-4A–7-4D show the impulses for ∆V1, ∆V2, and ∆V3 along with the total

loss of velocity from drag, ∆VD . First, it is noted that the deorbit and recircularization

impulses, ∆V1 and ∆V3, are very small. Regardless of the amount of inclination change

that is desired, these impulses are much smaller than the boost impulse, ∆V2. When

comparing Figs. 7-4B and 7-4D, it is seen that the majority of the boost impulse is used

to recover velocity lost due to atmospheric drag. Therefore, as expected, the cost of the

aeroassisted maneuver is largely due to recovering energy lost due to drag.

149

Tota

l∆V

(m/s

)

Inclination Change (deg)

One-Pass

Two-Pass

Three-Pass

Four-Pass

All-Propulsive

00

2000

4000

40 60 8020

6000

8000

max

A Minimum impulse, ∆Vmin vs. final inclination, if .

1.2

0

1

1 32 4

0.8

0.6

0.4

0.2

i(tf ) = 20 deg

i(tf ) = 40 deg

i(tf ) = 60 deg

i(tf ) = 80 deg

Number of Atmospheric Passes

m(tf)/m0

1.4

B Final mass fraction, m(tf )/m0, vs. number of atmospheric passes, n.

Figure 7-3. ∆Vmin, vs. if , and m(tf )/m0 vs. n.

150

1 328

32

34

36

38

30

2 4

i(tf ) = 20 deg

i(tf ) = 40 deg

i(tf ) = 60 deg

i(tf ) = 80 deg


∆V1

(m/s

)

A ∆V1, vs. n.

1 31000

2000

3000

2 4

4000

5000

6000i(tf ) = 20 deg

i(tf ) = 40 deg

i(tf ) = 60 deg

i(tf ) = 80 deg


∆V2

(m/s

)

B ∆V2, vs. n.

01 3

30

50

2 4

10

40

60

20

i(tf ) = 20 deg

i(tf ) = 40 deg

i(tf ) = 60 deg

i(tf ) = 80 deg


∆V3

(m/s

)

C ∆V3, vs. n.

1 31000

2000

3000

2 4

4000

5000

6000i(tf ) = 20 deg

i(tf ) = 40 deg

i(tf ) = 60 deg

i(tf ) = 80 deg


∆VD

(m/s

)

D ∆VD , vs. n.

Figure 7-4. ∆V1, ∆V2, ∆V3, and ∆VD Impulses vs. n.

Figs. 7-5A shows the total inclination change achieved by atmospheric forces as a

function of number of passes and required inclination change. It is seen that for each

required inclination change, the majority of the inclination change is performed by the

atmosphere. It is interesting to note for if = 80 deg that the four-pass maneuver has

less inclination change from the atmosphere as compared with the one, two or three

pass maneuvers. Nevertheless, the cost of the if = 80 deg maneuver is nearly the same

as the cost of the one, two or three pass maneuvers. Next, Fig. 7-5B shows the total

transfer time for n = (1, 2, 3, 4) for if = (20, 40, 60, 80). As expected, the total transfer

151

01 32 4

40

60

80

100

120

20

i(tf ) = 20 deg

i(tf ) = 40 deg

i(tf ) = 60 deg

i(tf ) = 80 deg


∆i aero

(deg

)

A ∆iaero vs. n

01 3 3.51.5 2.52 4

200

400

100

300

i(tf ) = 20 deg

i(tf ) = 40 deg

i(tf ) = 60 deg

i(tf ) = 80 deg


Traj

ecto

ryD

urat

ion

(min

utes

)

B T vs. n

Figure 7-5. Change in i by Atmosphere, ∆iaero, and transfer time, T , vs. n for Q =∞.

time increases as the number of atmospheric passes increases. Finally, it is noted for

a one-pass maneuver, the aeroassisted transfer requires approximately 100 minutes

regardless of desired inclination change.

152

7.5.2 Constrained Heating Rate

The effect of constraining the stagnation point heating rate on the small spacecraft

is now examined. Figs. 7-6A–7-6D shows the total required impulse, ∆Vmin as a

function of if and Qmax for if = (20, 40, 60, 80), Qmax = (∞, 1200, 800, 400) W/cm2,

and n = (1, 2, 3, 4). It is seen that, for any particular value of if , ∆Vmin increases as

Qmax decreases. Furthermore, it is noted again that the total cost is insensitive to the

number of atmospheric passes. Next, for even a highly constrained heating rate, the

aeroassisted maneuver is more fuel efficient than an all-propulsive maneuver. The only

case for which the aeroassisted maneuver has a greater cost than the all-propulsive

maneuver is for if = 80 degrees and Q = 400W/cm2. Figs. 7-7A–7-7D show the final

mass fraction after completion of the maneuvers for if = (20, 40, 60, 80) deg. Again, it

is seen that as Qmax decreases, more fuel is required to achieve a desired inclination

change.

Finally, Figs. 7-8A–7-8D show the maximum value of the sea level gravity-normalized

specific force, Smax, experienced by the vehicle during atmospheric flight, where S is

defined as

S =ρv 2A

2mg0

√

C 2D + C2L . (7–27)

For an unconstrained heating rate, Smax ranges from approximately three to twenty-five.

As the heating rate constraint is tightened, the magnitude of Smax decreases significantly

for all values of inclination change. For Q = 400W/cm2, Smax ≈ 1 for any desired

inclination change.

7.5.3 Optimal Trajectory Structure

The trajectory of the aeroassisted orbital inclination change maneuver is now

examined in detail. A 40 degree inclination change maneuver is used to examine the

structure of the optimal trajectory. Figs. 7-9A–7-9C show the altitude versus velocity,

heating rate versus time, and altitude versus flight path angle for Qmax = (∞, 800, 400)

W/cm2 for if = 40 deg for a one pass maneuver. It is seen for an unconstrained heating

153

All-Propulsive

1 31000

2000

3000

2 4

4000

5000


∆Vmin

(m/s

)

Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

A ∆Vmin vs. n for if = 20 deg.

All-Propulsive

1 32000

2 4

4000

6000

8000


∆Vmin

(m/s

)

10000Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

B ∆Vmin vs. n for if = 40 deg.

All-Propulsive

1 3

12000

2 4

4000

6000

8000


∆Vmin

(m/s

) 10000

Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

C ∆Vmin vs. n for if = 60 deg.

All-Propulsive

1 3

12000

14000

2 4

4000

6000

8000


∆Vmin

(m/s

)

10000

Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

D ∆Vmin vs. n for if = 80 deg.

Figure 7-6. ∆Vmin, vs. n for Various Max Heating Rates and Final Inclinations.

rate, the vehicle dives deeply into the atmosphere once and then ascends back out of

the atmosphere. For the cases of Qmax = (800, 400)W/cm2, it is seen that the vehicle

descends until reaching the maximum value of the heating rate constraint. The vehicle

then glides at the maximum value of the heating rate constraint for a short duration

before re-ascending to near the edge of the atmosphere. The vehicle then descends

again to reach the heating rate constraint before reascending out of the atmosphere.

Therefore, for a one-pass maneuver with an active constraint on the heating rate, the

trajectory essentially becomes a two-pass maneuver. Figs. 7-10A and 7-10B show the

154

All-Propulsive

0

1

1 3

0.5

1.5

2 4Number of Atmospheric Passes

m(tf)/m0

Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

A m(tf )/m0 vs. n for if = 20 deg.

1.2

All-Propulsive

0

1

1 32 4

0.8

0.6

0.4

0.2


m(tf)/m0

1.4Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

B m(tf )/m0 vs. n for if = 40 deg.

All-Propulsive

0

1

1 32 4

0.8

0.6

0.4

0.2


m(tf)/m0

Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

C m(tf )/m0 vs. n for if = 60 deg.

All-Propulsive

0

1

1 32 4

0.8

0.6

0.4

0.2


m(tf)/m0

Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

D m(tf )/m0 vs. n for if = 80 deg.

Figure 7-7. m(tf )/m0, vs. n for Various Qmax, and if = (20, 40, 60, 80) deg.

angle of attack and bank angle for the vehicle for the one-pass maneuver. It is seen that

for each value of heating rate constraint, the vehicle enters and exits the atmosphere

with a near maximum value of angle of attack of approximately 40 deg. Furthermore, it

is seen that the vehicle enters at a nearly 90 deg bank angle, rotates 180 degrees, and

exits at a nearly -90 deg bank angle.

Figs. 7-11A–7-11D show the altitude versus velocity for one, two, three and four

pass maneuvers for if = 40 degrees and Qmax = 400W/cm2. It is interesting to note that

the profiles are nearly identical regardless of number of passes. For the one-pass

155

01 3

8

2

2

4

4

6


Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

Smax

(Dim

ensi

onle

ss)

A Smax vs. n for if = 20 deg.

01 3

5

15

2 4

10


Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

Smax

(Dim

ensi

onle

ss)

B Smax vs. n for if = 40 deg.

01 3

5

25

35

30

15

2 4

10

20


Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

Smax

(Dim

ensi

onle

ss)

C Smax vs. n for if = 60 deg.

01 3

30

50

2 4

10

40

20


Qmax =∞ (W/cm2)

Qmax = 1200 (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

Smax

(Dim

ensi

onle

ss)

D Smax vs. n for if = 80 deg.

Figure 7-8. Smax, vs. n, for Various Qmax and if = (20, 40, 60, 80) deg.

maneuver, the vehicle descends and ascends to nearly the edge of the sensible

atmosphere. At this point, the vehicle descends again before exiting the atmosphere.

For the two-pass maneuver, the vehicle simply splits the one-pass maneuver between

each pass. For the three and four-pass maneuvers, these maneuvers collapse into a

two-pass maneuver. The second atmospheric pass for the three-pass maneuver and

the second and third atmospheric passes for the four pass maneuver never go deeply

into the atmosphere. Regardless of number of passes, the same general two-pass

profile is exhibited. In examining Figs. 7-12A–7-12D, the similarity of the maneuvers

156

12

Alti

tude

(km

)

Velocity (km/s)5 7 118 96 10

40

60

80

100

120

20

Qmax =∞ (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

A h, vs. v .

Time (sec)

Q(W

/cm2)

00

1000

1000

1500

1500

2000

2000

2500

2500

3000

3000 3500

500

500

Qmax =∞ (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

B Q, vs. t for if = 40-deg.

Alti

tude

(km

)

γ (deg)0 5−5 10

40

60

80

100

120

20

Qmax =∞ (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

C h vs. γ.

Figure 7-9. h vs. v , Q vs. t, and h vs. γ for if = 40-deg.

regardless of number of passes is seen again in examining the heating rate versus time

in the atmosphere. Regardless of number of passes, the same heating rate profile is

seen. The first time the maximum value of the heating rate is met, the vehicle rides

along the maximum value of the heating rate and then reascends. The second time the

vehicle achieves the maximum value of the heating rate constraint, it only touches it

before reascending. Again, it is seen that the interior passes for the three and four-pass

maneuvers collapse into passes that do not descend far into the atmosphere.

157

Time (sec)

Ang

leof

Atta

ck(D

eg)

00

30

50

1000 1500 2000 2500 3000 3500

10

500

40

20

Qmax =∞ (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

A α vs. t.

Time (sec)

Ban

kA

ngle

(Deg

)

0

0

−100

−2001000 1500 2000 2500 3000 3500

200

500

100

Qmax =∞ (W/cm2)

Qmax = 800 (W/cm2)

Qmax = 400 (W/cm2)

B σ vs. t.

Figure 7-10. α and σ vs. t for n = 1 and if = 40 deg.

158

Alti

tude

(km

)

7 7.5 8 8.5 9 9.540

60

80

100

120

Velocity (km/s)A h vs. v for n = 1 and if = 40 deg.

Alti

tude

(km

)

7 7.5 8 8.5 9 9.540

60

80

100

120

Velocity (km/s)

First Pass

Second Pass

B h vs. v for n = 2 and if = 40 deg.

Alti

tude

(km

)

7 7.5 8 8.5 9 9.540

60

80

100

120

Velocity (km/s)

First Pass

Second Pass

Third Pass

C h vs. v for n = 3 and if = 40 deg.

Alti

tude

(km

)

7 7.5 8 8.5 9 9.540

60

80

100

120

Velocity (km/s)

First Pass

Second Pass

Third Pass

Fourth Pass

D h vs. v for n = 2 and if = 40 deg.

Figure 7-11. h vs. v for n = (1, 2, 3, 4) and if = 40 deg and Qmax = 400 (W/cm2).

7.6 Discussion

The results of this study demonstrate several interesting features of using

aeroassisted glide maneuvers to change the inclination of a small spacecraft in LEO. For

an unconstrained heating rate, it is demonstrated that the total fuel required to change

the inclination of the small spacecraft is much less than for an all-propulsive maneuver.

As the heating rate is tightened, the cost of the aeroassisted maneuvers increase, but,

are still more efficient than the all-propulsive maneuvers except for the single case of

if = 80 degrees and Qmax = 400W/cm2. For an unconstrained heating rate, regardless

159

Q(W

/cm2)

00 1000 1500 2000 2500 3000

200

400

500

500

100

300

Time Since Atmospheric Entry (sec)A Q vs. t for n = 1 and if = 40 deg.

Q(W

/cm2)

00 1000 1500 2000 2500 3000

200

400

500

500

100

300

First Pass

Second Pass

Time Since Atmospheric Entry (sec)B Q vs. t for n = 2 and if = 40 deg.

Q(W

/cm2)

00 1000 1500 2000 2500 3000

200

400

500

500

100

300

First Pass

Second Pass

Third Pass

Time Since Atmospheric Entry (sec)C Q vs. t for n = 3 and if = 40 deg.

Q(W

/cm2)

00 1000 2000 3000

200

400

4000 5000

500

100

300

First Pass

Second Pass

Third Pass

Fourth Pass

Time Since Atmospheric Entry (sec)D Q vs. t for n = 4 and if = 40 deg.

Figure 7-12. Q, vs. t for n = (1, 2, 3, 4), if = 40 deg and Qmax = 400 (W/cm2).

of number of atmospheric passes, the vehicle dives deeply into the atmosphere only

once and affects all of the inclination change with this single atmospheric dive. For

the constrained heating rate cases, it is demonstrated that regardless of number of

atmospheric passes, the vehicle essentially conducts a two-pass maneuver. In the

first pass, the vehicle dives into the atmosphere and rides the heating rate constraint

for a finite duration. In the second pass, the vehicle merely touches the heating rate

constraint before reascending.

160

7.7 Conclusions

A study has been given on changing the inclination of a small high lift-to-drag ratio

spacecraft in low-Earth orbit using aeroassisted maneuvers. Three velocity impulses

(a de-orbit, boost, and re-circularization) are used along with lift and drag forces in the

atmosphere to change the inclination of the vehicle. It is found that the optimal solution

is insensitive to the number of atmospheric passes. For an unconstrained heating rate,

the solution reduces to a one-pass maneuver regardless of number of passes. For

highly constrained heating rates, the solution reduces to a two-pass maneuver. Finally,

it was found that for all but the highest inclination changes with the tightest heating rate

constraints that an aeroassisted maneuver was more fuel efficient than an all-propulsive

maneuver.

161

CHAPTER 8CONCLUSIONS

8.1 Dissertation Summary

Numerical methods for solving optimal control problems fall into two categories:

indirect and direct methods. An indirect method solves the continuous-time first-order

necessary conditions from the calculus of variations. While indirect methods in general

provide high-accuracy approximations, it is often hard to find a solution to these

necessary conditions. Direct collocation methods, on the other hand, are a more

robust numerical approximation technique. Direct collocation methods discretize the

original continuous-time optimal control problem such that the infinite-dimensional

continuous-time problem is transcribed into a finite-dimensional discrete problem, or

nonlinear programming problem (NLP). Direct methods have most commonly been

applied as h–methods. An h–method uses many low-degree approximating subintervals

and guarantees that the NLP is sparse. In contrast to h–methods, p–methods use a few

fixed number of intervals with a variable degree polynomial in each interval. A p–method

results in a dense NLP, but, if the solution to a problem is simple and smooth, high

accuracy approximations to the continuous-time optimal control problem are obtained

with small NLPs.

In this dissertation, an hp–direct collocation method has been presented. The

hp–method allows for a variable number of approximating intervals with a variable

degree approximation within each interval. The result is a method that is robust with

respect to the solution structure of a problem (that is, some optimal control problems are

better approximated by an h–biased method, while other optimal control problems are

better approximated by a p–biased method). The discrete approximation in each interval

is defined by the set of Legendre-Gauss or Legendre-Gauss-Radau points. These

points are used because they are defined by highly accurate quadrature rules and when

used as the support points in a Lagrange polynomial, do not exhibit Runge phenomenon

162

for high-degree approximations. Next, in analyzing the KKT conditions of the NLP, it

is found that the hp–method of this research results in a discrete representation of the

continuous-time first-order optimality conditions if the costate is continuous.

Two hp–mesh refinement algorithms have been presented. In each algorithm, the

accuracy of the approximation in each mesh interval is assessed by evaluating the

differential-algebraic constraints between collocation points. If an approximating interval

is inaccurate, the interval is modified by either increasing the degree of approximation

within the interval, or by subdividing the interval. The hp–mesh refinement algorithms

iteratively update the approximating grid until an approximation is accurate in every

approximating interval. The hp–mesh refinement algorithms are capable of utilizing

the best approximation strategy for different regions of a solution by modifying the

mesh either by increasing the degree of the approximating polynomial or increasing

the number of mesh intervals. The two hp–mesh refinement algorithms presented

in this research are distinctly different from one another because the first algorithm

is a p–biased algorithm while the second algorithm is an h–biased algorithm. Both

algorithms are compared with one another along with comparisons to p and h–methods

over several examples. It is demonstrated that the hp–methods are robust to problem

type and accurately solve the problems presented in a computationally efficient manner.

Furthermore, hp–mesh refinement algorithm 2 is the most successful strategy when

compared with p, h, and hp–mesh refinement algorithm 1 over the range of problems.

Utilizing the computational sparsity of an h–method as a starting point, hp–mesh

refinement algorithm 2 results in a sparse NLP. Furthermore, by using mid-degree

collocation only on smooth intervals, high accuracy approximations are obtained in a

computationally efficient manner. hp–mesh refinement algorithm 2 provides accurate

approximations to continuous-time optimal control problems with simultaneously small

and sparse NLPs.

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Finally, an optimal aeroassisted orbital inclination change maneuver is studied.

In this study, a small maneuverable spacecraft in low-Earth orbit (LEO) uses the

aerodynamic forces of lift and drag to change the orbital inclination. The maneuver

is studied for a multiple number of atmospheric passes and variable heating rate

constraints. The aeroassisted maneuvers is demonstrated to be more fuel efficient

than all-propulsive inclination change maneuvers. Furthermore, it is found that the

aeroassisted maneuvers are insensitive to number of atmospheric passes.

8.2 Future Work

8.2.1 Continued Work on Mesh Refinement

Two hp–mesh refinement algorithms have been presented in this research. It is

believed that automatic mesh generation is a rich subject and many more advances

may be made. Typically, mesh refinement strategies are studied for h–methods. In

an h–method, there are only two degrees of freedom in refining a mesh, namely, the

number of mesh intervals and the placement of the mesh points. hp–mesh refinement

algorithms are significantly more complex, however, because there are three degrees

of freedom for refining a mesh, namely, the number of mesh intervals, the placement of

the mesh points, and the degree of the approximating polynomial within each interval.

Because of this extra degree of freedom in generating an approximating mesh, the

choice of how to refine a mesh is more complicated, but also more interesting and

fruitful. Presented in this research are two static refinement algorithms that only

increase the size of the NLP on every iteration. Two interesting future concepts to

explore are now stated. The first concept is a refinement process to decrease the size

of the NLP if any intervals are more accurate than the desired accuracy level, further

increasing the computational efficiency of the NLP. The second concept is to add a

dynamic mesh refinement component that can be used to exactly locate the switch in

activity between active and inactive path constraints. One of the strengths of a direct

collocation method is that the location of active and inactive constraint arcs does not

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need to be known in order to approximate the solution of the problem. If the locations of

these arcs can be determined, however, the approximations can be made significantly

more accurate.

8.2.2 Discontinuous Costate

It has been demonstrated in this research that high-accuracy approximations are

achieved by the proposed hp–method if the costate is continuous. If mesh points are

at the location of discontinuity in the costate, the transformed adjoint system is an

inexact discrete representation of the continuous-time first-order necessary conditions.

For a dynamic refinement algorithm that will exactly locate the switch in activity of

inequality path constraints, it is likely that the costate may be discontinuous at mesh

points. It is necessary to determine how to make the transformed adjoint system a

discrete representation of the continuous-time first-order necessary conditions for a

discontinuous costate solution if mesh points are at the location of discontinuity in the

costate.

165

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BIOGRAPHICAL SKETCH

Christopher Darby was born in 1983 in Gainesville, Florida. He received his

Bachelor of Science degree in mechanical engineering in December 2006, his Master of

Science degree in mechanical engineering in May 2010, and his Doctor of Philosophy

in mechanical engineering in May 2011. Each degree was earned at the University

of Florida. His research interests include numerical approximations to differential

equations, optimal control theory, and numerical approximations to the solution of

optimal control problems.

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hp–pseudospectral method for solving continuous …

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