numerical solution daes and

7/27/2019 Numerical Solution DAES And

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76 Journal of Marine Science and Technology, Vol. 19, No. 1, pp. 76-88 (2011)

NUMERICAL SOLUTIONS OF

DIFFERENTIAL-ALGEBRAIC EQUATIONSAND ITS APPLICATIONS IN SOLVING TPPC

PROBLEMS

He-Sheng Wang*, Wei-Lun Jhu**, Chee-Fai Yung**, and Ping-Feng Wang***

Key words: differential-algebraic equations, computer algebra, tra-

jectory-prescribed path control.

ABSTRACT

In this paper, we present a numerical method for solving

nonlinear differential algebraic equations (DAEs) based on

the backward differential formulas (BDF) and the Pade series.

Usefulness of the method is then illustrated by a numerical

example, which is concerned with the derivation of the opti-

mal guidance law for spacecraft. This kind of problems is

called trajectory-prescribed path control (TPPC) in the litera-

ture. We reformulate the problem as a Hamiltonian DAE

system (usually with a higher index). After establishing the

system of spacecraft dynamics, we can derive the optimal guid-

ance law of the system by the proposed numerical method.

I. INTRODUCTION

In system theory, a dynamical system is often considered as

a set of ordinary differential or difference equations (ODE);

these equations describe the relations between the system

variables. As pointed out in [4], for the most general purpose

of system analysis, one usually begins by defining the first

order system

( ( ), ( ), ) 0,t t t =F y y (1)

whereF andy are vector-valued functions. Eq. (1) is termed

as differential-algebraic equations (DAE), since it contains

differential equations as well as a set of algebraic constraints.

For control and systems engineers, it is usually assumed that

(1) can be rewritten in an explicit form

( ) ( ( ), ).t t t=y f y (2)

An ODE of the form (2) is called a state-variable (state-

space) description in the systems and control society. Since

then, theorems and design techniques being developed are

largely based on (2). In fact, the state-variable descriptions

have been the predominant tool in systems and control theory.

While the representation (2) will continue to be very important,

there has been an increasing interest in working directly with

(1).

If (1) can, in principle, be rewritten as (2) with the same

state variablesy, then it will be referred to as a system of im-

plicit ODEs. In this paper, we are especially interested in

those problems for which this rewriting is impossible or less

desirable. We consider the general nonlinear DAEs which are

linear in the derivative

( ) ( )( ) ( ) ( ), 0.t t t t + =A y y f y (3)

Suppose that( )

A yy

has a constant rank. Then, in princi-

ple, locally the system (3) can be put in the semi-explicitform

1( ) ( ( ), ( ), ),t t t t =x f x u

20 ( ( ), ( ), ).t t t= f x u (4)

A large class of physical systems can be modeled by this

kind of DAEs. The paper of Newcomb et al. [24] gives many

practical examples, including circuit and system design, ro-

botics, neural network, etc., and presents an excellent review

on nonlinear DAEs. Many other applications of DAEs as well

as numerical treatments can be found in [4]. An existence and

uniqueness theory for nonlinear DAEs has been well devel-

oped in [26] by exploiting their underlying differential geo-

metric structure. Recently, Venkatasubramanian et al. [28]

have extensively studied the bifurcation phenomena of DAEs.

Paper submitted 02/03/09; revised 10/11/09; accepted 01/15/10. Author for

correspondence: He-Sheng Wang (e-mail: [email protected])

*Department of Communications, Navigation and Control Engineering Na-

tional Taiwan Ocean University, Keelung 202, Taiwan.**Department of Electrical Engineering, National Taiwan Ocean University,

Keelung, Taiwan.

***Institute for Information Industry, Taipei, Taiwan.


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H.-S. Wanget al.: Numerical Solutions of Differential-Algebraic Equations and Its Applications in Solving TPPC Problems 77

They have also thoroughly investigated feasibility regions in

differential-algebraic systems. The notion of feasibility regions

provides a natural gateway to the stability theory of DAEs.There are several reasons to consider systems of the form

(4), rather than try to rewrite it as an ODE. Of great impor-

tance, we point out that, when physical problems are simulated,

the model often takes the form of a DAE. DAEs can be used to

depict a collection of relationships between variables of in-

terest and some of their derivatives, which may be treated as

an algebraic constraint between the state variables. These

relationships may even be generated automatically by a mod-

eling or simulation program. In particular, the variables thus

introduced usually have a physical significance. Changing the

model to (2) may produce less meaningful state variables. If

the original DAE can be solved directly, then it becomes easierfor scientists or engineers to explore the effect of modeling

changes and parameter variations. These advantages enable

researchers to focus their attention on the physical problem of

interest. On the other hand, although the state-space models

are very useful, but the state variables thus introduced often

do not provide a physical meaning [8, 27]. Besides, some

physical phenomena, like impulse, hysteresis which are im-

portant in circuit theory, cannot be treated properly in the

state-space models [18, 29]. Differential-algebraic equations

representation provides a suitable way to handle such prob-

lems. It has been proven in the literature that DAE systems

have higher capability in describing a physical system [17, 24,

29]. In fact, DAE system models appear more convenient and

natural than state-space models in large scale systems, eco-

nomics, networks, power, neural systems and elsewhere [17,

20, 24].

In this paper, we investigate some of the control problems

that are well suitable for the DAE system framework. In par-

ticular, we show that the optimal control problem can be re-

formulated as a Hamiltonian DAE. The derived DAE can then

be solved by numerical methods. The numerical method is

mainly based on the backward differential formulas (BDF)

and the Pade series, which can be obtained by using computer

algebra systems (such as MAPLE, Mathematica, etc.). Fur-

thermore, we consider a practical design problem, namely thetrajectory prescribed path control (TPPC) problem. In the

simulation of space vehicles, we often encounter the TPPC

problems. That is, we usually append a set of path constraints

to the equations of motion for describing the shape of the tra-

jectory. Here the optimal guidance law for aircraft dynamics

is derived based on a DAE approach. After establishing the

system of aircraft dynamics, we can derive the optimal guid-

ance law of the system by solving a Hamiltonian DAE.

The rest of the paper is organized as follows: In Section 2,

some of the elementary materials concerning DAEs are in-

vestigated. In Section 3, by using the Lagrange multipliers

method, it is shown that the optimal control problem can berewritten as a Hamiltonian DAE. Simulation results are given

in Section 4. Finally, some concluding remarks are given in

Section 5.

II. ELEMENTS OF DAES

In this section we summarize some basic definitions andpreliminary results that will be needed through-out this paper.

Most of the treatments are purely algebraic and the definitions

are fairly standard. First, the definitions of solvability and

index for the general nonlinear DAEs (1) will be given in this

section. Suppose that the DAE (1) consists of a system of m

equations in the (2m + 1)-dimensional variable ( ( ), ( ), ).t t ty y

A function y(t) is said to be a solution of the DAE (1) on a

interval Tify(t) is continuously differentiable on Tand satis-

fies (1) for all tT. The precise definition is given in the

following.

Definition 1. Let Tbe an open subinterval ofR, a con-

nected open subinterval ofR2m+1, and F a differentiable func-tion from to Rm. Then DAE (1) is solvable on Tin if

there is an r-dimensional family of solutions(t,c) defined ona connected open set , T Rrsuch that:

1. (t,c) is defined on all ofTfor each ,c 2. ( , ), ( , )) for ( , ) ,( , t t tt c c c T 3. If(t) is any other solution with (t, (t, c), (t, c)) ,

then(t) =(t,c) for some ,c 4. The graph ofas a function of (t,c) is an (r+ 1) dimen-

sional manifold.

The above definition means that locally there is an r-dimen-

sional family of solutions. At any time t0T, the initial con-ditions form an r-dimensional manifold (t0, c) and r is in-dependent oft0. The solutions are a continuous function of the

initial conditions on this manifold.

It is known that the property of index [4] plays a key role in

the classification and behavior of DAE systems. It also pro-

vides a measurable scheme of the singularity of a DAE and the

difficulty for numerically solving a DAE. The definition of

index is described in the following.

Definition 2. The minimum number of times that all or part of

DAE (1) must be differentiated with respect to tin order to de-

termine ( )t

y as a continuous function ofy(t) and t, is theindex of the DAE (1).

The definition of the index given above is usually referred

to as the differentiation index of DAEs. According to the

definition, an implicit ODE has index zero. A DAE having

index two or greater is referred to as a higher index DAE.

An alternative, but less general, characterization of the in-

dex is the number of iterations of the following (theoretical)

procedure needed to convert the DAE into an ODE:

1. If / F y is nonsingular, then stop,

2. Suppose that /

F y has constant rank and that nonlinearcoordinate changes make (1) semi-explicit. Differentiate

the constraint equation, letF = 0 denote the new DAE, and

return to 1.


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78 Journal of Marine Science and Technology, Vol. 19, No. 1 (2011)

To motivate the next definition, suppose that we wish to

solve (1) by the implicit Euler method starting at time t0 with

constant step-size h. Let tn = t0 + nh,y0 be the estimate fory(tn).Then the implicit Euler method applied to (1) gives

1, , 0,n nn nth

=

y yF y (5)

which will need to be solved by a nonlinear equation solver.

The Jacobian ofF in (5) with respect toyn is ( )1h +y yF F sothat this matrix pencil will be important. For the linear case,

this was the pencil ( )1 ,h +A B where A, B are constant ma-

trices. Another definition of the index is given in the follow-ing in terms of the matrix pencil.

Definition 3. The local index l of (1) at ( , , )ty y is the index

of the matrix pencil ( , , ) ( , , ).t t +

y yF y y F y y

Proposition 4. Suppose that has constant rank. Then = 1 ifand only ifl 1.

From the above definition and proposition, it can be seen

that the semi-explicit DAE

1 1 1 2( , , )t=x F x x

2 1 20 ( , , )t= F x x (6)

has index one if and only ifF2/x2 is nonsingular.

III. NUMERICAL SOLUTIONS OF DAES

In this section we examine some of the properties of the

numerical methods for solving DAEs. We examine the linear

multistep methods (LMMs) applied to DAEs, especially the

most popular and hence best understood class of LMMs,

namely the backward differentiation formulas (BDF). On theother hand, we also introduce a method that is well suitable for

using computer algebra systems, namely the Pade approxi-

mation method. First we calculate power series of the given

equations system (by using BDF) and then transform it into

Pade series form, which give an arbitrary order algorithm for

solving differential-algebraic equations numerically.

The first general technique for the numerical solution of

DAEs, proposed in 1971 by Gear in a well-known and often

cited paper [11], utilized the BDF. This method was initially

defined for systems of differential equations coupled to alge-

braic equations

( , , ),t=x f x u

0 ( , , ),t= g x u (7)

where u is a vector of the same dimension asg. The algebraic

variables y are treated in the same way as the differential

variables x for BDF, and the method was soon extended toapply to any fully-implicit DAE system

( , , ) 0.t =F y y (8)

The simplest first order BDF method is the implicit Euler

method, which consists of replacing the derivative in (8) by a

backward difference

1 , , ,n n n nth

=

0y y

F y

where h = tn tn1. The resulting system of nonlinear equations

for yn at each time step is then usually solved by Newtons

method. The k-step (constant step-size) BDF consists of re-

placing y by the derivative of the polynomial which inter-

polates the computed solution k + 1 times tn, tn1, , tnk,

evaluated at tn. This yields

, , ,n n nth

=

0

yF y (9)

where0

k

n i i n i = = y y and i, i = 0, 1, , kare the coeffi-

cients of the BDF method. The k-step BDF method is stable

for ODEs for k < 7. The relation 0k

n i i n i = = y y simply

says that the derivative ofy at tn can be represented as apolynomial of order k. We define another type of power series

in the form

2

0 1 2 1 1( ) = + + + + ( + + + ) ,n

n m mf t f f t f t f p e p e t (10)

wherep1, p2, , pm are constants, e1, e2, , em are bases of

vector e, m is the size of vector e. Lety be a vector with m

elements, then every element can be represented by the Power

series in (10). Therefore we can write

2

,0 ,1 ,2 ,n

i i i i iy y y t y t e t= + + + + (11)

whereyi is the ith element ofy. Substitute (11) into (1), we

can get

1

, ,1 1 ,( ... ) ( ),n j n j

i i n i i m mf f p e p e t Q t +

= + + + + (12)

wherefi is the ith element of ( , , )tF y y in (1), and Q(tnj+1) is a

polynomial whose degree is no less than tnj+1. Now suppose

that 0 0 0( , , )ty y is a set of consistent initial conditions for (1),

i.e.

0 0 0( , , ) 0.t =F y y


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Then the solutions of (1) can be assumed to be

20 1 ,t t= + +y y y e (13)

where e is a vector function which is the same size as y0 and

0.y Substitute (13) into (1) and neglect higher order term, we

have the linear equation ofe in the form

Ae =B, (14)

where A and B are constant matrices. Solving Eq. (14), the

coefficient oft2 in (13) can be determined. Repeating above

procedure for higher order terms, we can get the arbitrary

order power series of the solutions for (1).

From (11) and (14), we can determine the linear equation in

(14) as follows:

Ai,j = Pi,j.

Solve this linear equation, we have ei (i = 1, , m). Sub-

stituting ei into (11), we then haveyi (i = 1, , m) which are

polynomials of degree n. Repeating this procedure, we can

get the arbitrary order Power series solution of differential-

algebraic equations in (1). The next step would be to put the

power series into Pade series form and then obtain numerical

solution of it.

Suppose that we are given a power series0

,iii c x=

rep-resenting a functionf(x), so that

0

( ) ,iii

f x c x=

= (16)

where ci = 0, 1, 2, is a given set of coefficients. This ex-

pansion is the fundamental starting point of any analysis using

Pade approximants. A Pade approximant is a rational fraction

[ ]

0 1

0 1

/ ,L

L

MM

a a x a xL M

b b x b x

+ + +=

+ + +

(17)

which has a MacLaurin expansion that is agreed with (16) as

high (in order) as possible. For definiteness, we may take b0 =

1. This choice turns out to be an essential part of the precise

definition and (17) is our conventional notation with this

choice for b0. So there areL + 1 independent numerator co-

efficients andMindependent denominator coefficients, mak-

ing L + M + 1 unknown coefficients in all. This number

suggests that normally the [L/M] ought to fit the power series

(16) through the orders 1, x, x2, , xL+M in the notation of

formal power series,

10 1

0 0 1

( ).L

i L MLi M

i M

a a x a xc x Q x

b b x b x

+ +

=

+ += +

+ + +

(18)

Upon cross-multiplying, the above equation can be re-ar-

ranged as

0 1 0 1( )( )MMb b x b x c c x+ + + + +

1

0 1 ( ).L L M

La a x a x Q x+ +

= + + + + (19)

Equating the coefficients ofxL+1,xL+2,,xL+Myields

1 1 2 0 10,M L M M L M Lb c b c b c + + ++ + + =

2 1 3 0 20,M L M M L M Lb c b c b c + + ++ + + =

1 1 0 0.M L M L L Mb c b c b c + ++ + + =

(20)

Ifj < 0, we define cj = 0 for consistency. Since b0 = 1, Eq.

(20) become a set ofMlinear equations for Munknown de-

nominator coefficients:

1 2 3 1

2 3 4 1 1 2

3 4 5 2 2 3

1 2 1 1

,

L M L M L M L M L

L M L M L M L M L

L M L M L M L M L

L L L L M L M

c c c c b c

c c c c b c

c c c c b c

c c c c b c

+ + + +

+ + + + +

+ + + + +

+ + + + +

=

(21)

from which the bis may be found. The numerator coefficients

a0, a1, , aL follow immediately from (19) by equating the

coefficients 1,x,x2, ,xL:

0 0 ,a c=

1 1 1 0 ,a c b c= +

2 2 1 1 2 0,a c b c b c= + +

min( , )

1

.L M

L L i L i

i

a c b c

=

= + (22)

Thus, Eqs. (21) and (22) normally determine the Pade nu-

merator and denominator and called the Pade equations. Sum-

marizing the previous discussion, the thus constructed [L/M]

Pade approximant is able to agree with0

i

iic x

=

throughorderxL+M.

Before the end of this section, we shall summarize the pro-

cedure for solving DAEs by using the proposed algorithm.

Algorithm Pade Series Solutions for DAEs

Input: The DAEs ( ( ), ( ), ) 0t t t =F y y and set an error toler-

ance .


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Step 1: Calculate a power series solution fory(t) by using Eqs.

(11)-(15). Denote the solution byf(x).

Step 2: Calculate a Pade series solution forf(x). Utilize Eq.(20)-(22) to compute the coefficients for the Pade se-

ries.

Step 3: Put the Pade series solution back into Eq. (1) to see if

the error tolerance is fulfilled. If yes, then outputf(x); if not, then go back to Step 1 and calculate a

solution with higher power.

End.

IV. HAMILTONIAN DAES

In this section, we derive a class of DAEs by the Hamil-

tons principle. We shall call such a system the HamiltonianDAE. The DAE structure allows a model to be obtained from

energy functions, constraint equations, and a virtual work

expression in a systematic manner. Hamiltons principle has

its origin from the classical mechanics. It can, however, also

arise naturally when dealing with the constrained optimization

problem. Here we take the optimization viewpoint.

Consider a nonlinear time-varying dynamical system

( ) ( ( ), ( ), )t t t t =x f x u

together with the associated performance index (cost function)

00

( ) ( ( ), ( ), )d ,T

tJ t L t t t t= x u (24)

Where x(t) Rn is the state, u(t) R

m is the control input,

L(x(t), u(t), t) is the Lagrange function (or Lagrangian). The

LagrangianL can be viewed as a weighting on the state vari-

ables and the control input in order to achieve a prescribed

performance requirement. The main purpose of the control is

to find an optimal feedback law u*[t0, T] such that the per-

formance index (24) is minimized. Usually, in the classical

mechanics, the Lagrangian is a function of the displacement

variables (x, u) as well as theflows ( , )u . However, in thecontrol systems community, the Lagrangian is merely a func-

tion of the displacement. The Hamiltonian and the Lagrangian

can be related by the Legendre transformation:

( ( ), ( ), ) ( ( ), ( ), ) ( ( ), ( ), ),TH t t t L t t t t t t= +x u x u p f x u (25)

wherep is the canonical momentum conjugated to x. Using

the above identity, Eq. (24) can be rewritten as:

0

' ( ( ), ( ), ) ( ) ( ) d .T

T

tJ H t t t t t t = x u p x (26)

By using the Leibnizs rule, the increment inJ' can be ex-

pressed in the increment ofx,p, u, and tas follows:

0

d ' ( ) d .T

T T T

tJ H H t = + x pu p x x p (27)

To eliminate the variation in , integrate by parts to get:

0 0 0

d d .T T

T T

T t t

T T

tt t + + = x p x p xp p (28)

Substituting (28) into (27) yields:

0

d ' d dT TT t

J = +p x p x

0

( ) ( ) d .T

T T T

tH H H t + + + x u p+ p x u x p (29)

According to the Lagrange theory, the constrained mini-

mum ofJis attained at the unconstrained minimum ofJ'. This

is achieved when dJ' = 0 for all independent increments in its

arguments. Setting to zero the coefficients of the independent

increments x, u, and p yields necessary conditions for aminimumJas shown in the following.

, ,T

H H L

= = = +

f

p x xp p

xx (30)

0 .T

H L

= = +f

u u

p

u

(31)

The first two equations are the canonical Hamilton equa-

tions. The third equation can be viewed as an algebraic con-

straint. Therefore we shall call (30)-(31) a Hamiltonian DAE.

The third equation is usually called the stationarity condition

in the optimal control theory. The actual value ofp(t) is by no

means important, but it must evidently be determined as an

intermediate step in finding the optimal control u*(t), which

depends onp(t) through the stationarity condition.

An important point is worth noting. The time derivative of

the Hamiltonian is

TT T

H H H

tH

= + + +

x uu

px

.T T

H H H

t

= + + +

uu p f

x

Ifu(t) is an optimal control, then

.HH

t=

Now, in the time-invariant case, f and L are not explicit

function oft, and so neither isH. In this situation

0.H=


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Hence, for time-invariant systems and cost functions, the

Hamiltonian is a constant on the optimal trajectory.

V. NUMERICAL EXAMPLE

In this section, we first examine a classical trajectory pre-

scribed path control (TPPC) problem which has been thor-

oughly studied in [3]. Then we will find an optimal guidance

law for the TPPC problem by solving a Hamiltonian DAE.

The two results are then compared in the end of the section.

The TPPC problems are often encountered in the simulation

of space vehicle. In this problem, a vehicle is usually modeled

to fly in space when its trajectory is imposed with constraints.

That is, a set of path constraints are usually appended to the

equations of motion for describing the shape of the trajectory.To model the system dynamics, we need a set of state (dif-

ferential) equations

( ) ( ( ), ( ), ),t t t t =x f x u (32)

that described the classical equations of motion together with a

set of algebraic equations

0 ( ( ), ( ), ),t t t= g x u (33)

that corresponds to the path constraints. In general, the x

variables are formed as state variables, while the u variablesare treated as the control (algebraic) variables. The state

variables describe the position, velocity, and possibly the mass

of the vehicle. The control variables are typically described as

angle of attack () and bank angle (), which effectively de-

termine the magnitude and direction of the aerodynamic forces

acting on the vehicle. In a TPPC problem the path constraints

often are functions only of the state variables. In this paper,

the TPPC problems of interest can be written as

( ) ( ( ), ( ), ),t t t t =x f x u (34)

0 ( ( ), ),t t= g x (35)

where the matrix product (g/x)(f/u) is nonsingular for all t

of interest. This condition is merely used to guarantee that the

TPPC problem is solvable. See also [3].

The complete nonlinear kinematics of aircraft is considered

in the TPPC problem. The state variables can be expressed in

an earth centered relative coordinate system (see Fig. 1). In

relative coordinate system, a spherical representation of the

state variables can be utilized. Here we illustrate the represen-

tation of the state variables are altitude h, geocentric latitude ,longitude , magnitude of the relative velocity vector VR,

relative azimuthA, and relative fight path angle .The control variables for TPPC problems are angle of at-tack and bank angle measured in a body coordinate system.

The angle of attack is measured from the relative velocity

North pole

Zr

Greenwich

meridian

Equatorialplane

Yr

Xr

East

h

VR

A

North

Fig. 1. State variables in relative coordinate.

Xn

Zn

Xo

Zo

Yo

Pitch

Body Reference Point

Fig. 2. Rotate about YO through in body coordinate system.

vector of the vehicle to the body XO axis, and corresponds to

pitch about the vehicles latitudinal axis (namely the YO axis)as shown in Fig. 2.

The bank angle is measured from the relative velocity

y-axis to the body YO axis, and corresponds to a roll about the

vehicles longitudinal axis (namely the XO axis) as shown in

Fig. 3.

For most trajectory application problems, position and ve-

locity of the vehicle (namely the rectangular three-component

vectors r

and v

) are often expressed as the differential vari-

ables. The equations of aircraft dynamical system can be

written as

,r v=

1( ) ( ( , , , ) ( , , )),PA Fv g r F r v

mr += +


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Yn

Zn

Xo

Zo

Yo

Body Reference Point

Roll

Fig. 3. Rotate aboutXO throughin Body Coordinate System.

with0 0 0

( ), ( ) andr t v t t

given, and

( ) acceleration vector due to gravity,

mass of the vehicl

, ) aerodynamic force vector,

( , , ) propulsive force vector

e,

.

( , ,A

P

g r

m

F r v

F r

=

=

=

=

In this paper, we limit our discussion to reentry vehicles

with no propulsive forces so that , ) 0,( ,PF r

and sim-

plify the simulation to model only a spherical geopotential and

spherical earth. When state variables are expressed in relative

coordinates, the differential equations of above can be trans-

lated to the following six equations of motion:

sin( ),Rh V =

cos( ) sin( ),

cos( )

RV A

r

=

cos( ) cos( ),RV

Ar =

2sin( ) cos( )R ED

gm

V r =

(sin( ) cos( ) cos( ) cos( ) sin( )),A

2cos( ) cos( ) R

R R

VLg

m V V r

= +

cos(2 )sin( )E A+

2 cos( )(sin( ) cos( ) sin( )

E

R

rAV

+

cos( )cos( )), + (36)

sin(cos(

)) cos( )sin( ) tan( )R

R

VLA

mVA

r

+=

2 (cos( )cos( ) tan( ) sin( ))E A

2 cos( ) sin( ) sin( ),

cos( )

E

R

r A

V

+

where

2gravitational constant,/ , the gravity force,

earth angular rotation rate,

the mass of the vehicle,

the vehicle cross-

the earth radius

sectional reference area,

( ) th

,

e mo

,

at s

e

e

E

a

r H

g r

m

S

a

h

=

=

=

=

=

=

+

=

=

2

2

pheric density,

( ) the aerodynamic lift coefficient,

( ) the aerodynamic drag coefficient,

1, the aerodynamic lift force,

2

1 , the aerodynamic drag force.2

L

D

L R

D R

C

C

L C SV

D C SV

=

=

=

=

Its worth noting that the equations of motion are undefined

if = 90 or = 90. The equation-prescribed path is usually

written as functions of the relative state variables. When the

state (differential) and algebraic equations are both expressed

in relative coordinates, it will be easy to determine the index of

the resulting DAE system. The index of the resulting DAE

system in relative coordinates can be found by a reduction

technique based on repeated differentiations of the algebraic

equations. This procedure can reduce the index of the system

to one. Here, the vehicle is solved by the 5-step backward

differential formulas (BDF). First, let us consider a simple

path constraint so that the vehicle is constrained to fly along a

prescribed azimuth and fight path angle trajectory. It is given

by the state variable constraints as follows:

21 9 ( / 3 ) ,0 00t+ +=

20 45 90 ( /300) .A t= (37)

The above path constraints are also discussed in [4]. The

DAE system (36)-(37) has index two. We have to differentiate

the algebraic Eq. (37) once and substitute forAand from the

differential Eq. (36). Then the resulting index one DAE sys-

tem related to the original system can be derived in the fol-

lowing,


8/13


sin( ),Rh V =

cos( ) sin( ) ,cos( )

RV Ar

=

cos( ) cos( ),RV

Ar

=

sin( )RD

V gm

=

2 cos( )Er

(sin( ) cos( ) cos( ) cos( ) sin( )),A

2

cos( ) cos( ) R

R R

VL gmV V r

= +

2 cos( )sin( )E A+

2 cos( )E

R

r

V

+

(sin( ) cos( ) sin( ) cos( ) cos( )),A +

sin( )cos( ) sin( ) tan( )

cos( )

R

R

VLA A

mV r

= +

2 (cos( )cos( ) tan( ) sin( ))E A 2 cos( ) sin( ) sin( )

,cos( )

E

R

r A

V

+

2cos( ) cos( )0

5000

R

R R

Vt Lg

mV V r

= + +

2 cos( )sin( )E A+

2 cos( )E

R

r

V

+

(sin( ) cos( )sin( ) cos( ) cos( )),A +

sin( )0

500 cos( )R

t L

mV

= +

cos( ) sin( ) tan( )RV

Ar

+

2 (cos( )cos( ) tan( ) sin( ))E A

2 cos( ) sin( ) sin( ).

cos( )

E

R

r A

V

+

Consistent initial values for the DAE system are determined

by selecting the initial differential and control variables to

satisfy the two algebraic equations in the related index one

system (38). Numerical results are obtained for the following

set of initial values:

0 ,

0 ,

12000 ft./sec,

15

1 ,

45 ,

2.6729 ,

0000 f

0.0522 .

t.,

R

h

V

A

=

=

=

=

=

=

=

=

For simplicity, we consider the problem in a model ofspherical gravitational earth. So we can use the relative state

variables in the differential equations. The related coefficients

of the vehicle in this experiment are given as follows:

m = 2.8905 tons,

S= 550 ft.2

The atmosphere is modeled with simple relation, =0.002378exp(h/23800). The lift and drag coefficients are

chosen as functions of angles of attack,

(0. ,01)LC = 20.04 0.1 .D LC C= +

The rest of the constraints needed in (36) are shown as fol-

lows:

16 3 2

5

1.4077 10

2

ft. /

0902900 ft.

s ,

7.2921 10 rad s.

,

/

e

E

a

=

=

=

In our simulation, a truth model is generated by solving theindex one problem with a tight local relative tolerance (EPS =

10-8) and absolute tolerance (EPS = 10-7). The state variables

of vehicle are plotted as functions of time in Fig. 4 and Fig. 5.

Now we shall impose an optimal criterion on the TPPC prob-

lem so that the problem can be reformulated as a Hamiltonian

DAE. Consider the following performance index:

00

( ) ( ( ), ( ), )d .T

tJ t L t t t t= x u

The Lagrange functionL considered here has the following

quadratic form:

1( ),

2

T TL = +x Qx u Ru


9/13


15 1

0.8

0.6

0.4

0.2

0

10

5

0

0.4

0.3

0.2

0.1

0

0 50 100 150

104

h versus t

0 50 100 150

lambda versus t

0 50 100 150

xi versus t

0 50 100 150

VRversus t

15000

10000

5000

0

Fig. 4. State variables.

where the matrix Q is a 6 6 identity matrix andR is a 2 2

identity matrix, x is the state variable, and u is the control

variable. To solve the optimal control problem is then equiva-

lent to solving the following Hamiltonian DAE:

,

H

=

x p (38)

,H

=

px

(39)

0 .H

=

u(40)

whereHis the Hamiltonian function for the aircraft dynamics,

andp is the generalized momentum vector. As mentioned in

Section 3, the Hamiltonian functionHis equal toL +pTf. Here

fis the equations of the vehicular motion. In our experiment,/H = x p of the Hamiltonian DAE system is equal to the

differential equations of the TPPC problem, namely the equa-

tions of motion in relative coordinates. Here we define six

variables as generalized momentum to yield an appropriate

stationary point with choosing state variables x and control

variables u. The control variables u are also the angle of at-

tack and bank angle . Then /H = p x of the Hamilto-

nian DAE system is given as follows:

21 2

3

2

) cos( )) sin( )

)( 209029

cos(cos(

cos( ( 20902900)00)

RRVV

ph

A pA p

h

+= +

+

5 8 2 510 (1.189 10 4.7561.4 36 10 )5 +

238002

4

h

RV p e

0 1000

800

600

400

200

200

150

100

50

0

-50

0

-20

-40

-60

-80

-100

10

8

6

4

2

0

0 50 100 150

gamma versus t

0 50 100 150

A versus t

0 50 100 150

versus t

0 50 100 150

alpha beta versus t

Fig. 5. State variables and control variables.

16

4

3

2.8153 10

sin( )( 20902900)

p

h

+

9

45.3175 10 cos( )p+

(sin( ) cos( ) cos( ) cos( ) sin( ))A

2380010

51.7283 10 cos( )h

RV p e

+

2 16

5 2 3

2.8153 10cos( ) +

( 20902900) ( 20902900)

R

R

Vp

h V h

+ +

9 5cos( )5.3175 10R

p

V

+

(sin( ) cos( )sin( ) cos( ) cos( ))A +

2380010

6

sin( )1.7283 10

cos( )

hRV e

p

+

6

2

cos( ) sin( ) tan( )

( 20902900)

RV A p

h

+

+

9 6cos( ) sin( ) sin( )

5.3175 10 ,cos( )R

A ph

V

+

2 ,p =

23 2

cos( ) sin( ) sin( )

cos ( )( 20902900)

RV A pph

=

+

9

4(5.3175 10 1.1115)sin( )h p +

(sin( ) cos( ) cos( ) cos( ) sin( ))A


10/13


9

4(5.3175 10 1.1115) cos( )h p

+

(cos( ) cos( ) cos( ) sin( )sin( ))A +

4

51.4584 10 sin( )sin( )A p+

9 5sin( )(5.3175 10 1.1115)R

ph

V

+ +

( )sin( )cos( )sin( ) cos( )cos( )A +

9 5cos( )(5.3175 10 1.1115)R

ph

V

+

(cos( ) cos( ) sin( ) sin( ) cos( ))A + 2

6cos( )sin( )(1 tan ( ))

20902900

RV A p

h

+

+

4

61.4584 10 sin( )cos( ) tan( )A p

4

61.4584 10 cos( )p

29 6sin ( )sin( )(5.3175 10 1.1115)

cos( )R

A ph

V

+ +

29 6cos ( )sin( )

(5.3175 10 1.1115) cos( )R

A ph V

+

,

4 1sin( )p p=

2cos( ) sin( )

cos( )( 20902900)

A p

h

+

3cos( ) cos( )

20902900

A p

h

+

8 26.9191(1.189 10 +

238005

44.756 10 )h

RV p e

+

238006

54.1134 10 cos( )

h

p e

5 5

2

2cos( ) cos( )

20902900 R

p p

h V

+

+

2 16

2

1.4077 10

20902900 ( 20902900)

RV

h h

+ +

9 52cos( )(5.3175 10 1.1115)

R

phV+ +

(sin( ) cos( ) sin( ) cos( ) cos( ))A +

238006 6sin( )

4.1134 10cos( )

hpe

cos( )sin( ) tan( )

20902900

A

h

+

9(5.3175 10 1.1115)h +

6

2

cos( ) sin( ) sin( ),

cos( )R

R

A pV

V

25 1

sin( ) sin( )cos( )

cos( )( 20902900)

RR

V A pp V p

h

= +

+

3sin( ) cos( )

20902900

RV A p

h

+ +

16

4

2

1.4077 10 cos( )

( 20902900)

p

h

+

+

9(5.3175 10 1.1115)h +

4cos( ) (sin( )cos( )sin( ) cos( )cos( ))p A

9(5.3175 10 1.1115)h +

5cos( )

(sin( ) cos( ) cos( )

R

pA

V

cos( ) cos( ))

238006 6

2

sin( )sin( )4.1134 10

cos ( )

hRV p e

6sin( ) sin( ) tan( )

20902900

RV A p

h

+

+

4 2

61.4584 10 cos( )cos( ) (1 tan ( ))A p

+ +

9(5.3175 10 1.1115)h +

6

2

cos( ) sin( ) sin( ) sin( ),

cos ( )R

A p

V

26

cos( ) cos( )

cos( )( 20902900)

RV A pph

=

+

2cos( ) sin( )

20902900

RV A p

h

+

+

9(5.3175 10 1.1115)h +

4cos( ) sin( ) sin( ) cos( )A p

4

51.4584 10 cos( )cos( )A p

9(5.3175 10 1.1115)h+ +


11/13


1.6 1.25

1.2

1.15

1.1

15000

10000

5000

0

1.4

1.2

1.0

0.8

1.35

1.3

1.25

1.15

1.2

0 5 10 2015 0 5 10 2015

0 5 10 2015 0 5 10 2015

105

h versus t

lambola versus t

xi versus t

VRversus t

Fig. 6. State variables.

5cos( ) sin( ) sin( ) sin( )

R

A p

V

6cos( ) cos( ) tan( )

20902900

RV A p

h

+

4

61.4584 10 cos( )sin( ) tan( )A p

9(5.3175 10 1.1115)h +

6cos( ) sin( ) cos( )

.cos( )R

A pA

V

(41)

The stationarity conditions are shown below:

238009 2

4100 8.2269h

RV p e

=

238006

54.1134 10 cos( )

h

RV p e

+

238006 6sin( )

4.1134 10cos( )

hRV p e

+

,+

238006

50 4.1134 10 sin( )h

RV p e

=

238006 6cos( )

4.1134 10cos( )

hRV p e

+

.+ (42)

The three parts of the Hamiltonian DAEs (36) (41) (42) can

be put into the following compact form:

0 46.5

46

45.5

45

0.4

0.2

0

-0.2

-0.4

-50

-100

1

0

-1

-2

-30 5 10 15 20 0 5 10 15 20

0 5 10 15 20 0 5 10 15 20

p(1) versus t p(2) versus t

gamma versus t A versus t

106

Fig. 7. State variables and generalized momentum.

0 0

0 0 ,

0 0 0

H

H

H

=

p

x

u

I

I y (43)

where y is equal to .

p

u

Numerical results are obtained for

the following set of initial values:

h = 150070 ft., = 1.1,

= 1.2, VR = 15000 ft./sec,

= 1, A = 45,

p1 = 2000, p2 = 106,

p3 = 10, p4 = 200,

p5 = 2000, p6 = 8.9,

= 2.6725, = 0.0522.

The numerical results are shown in Figs. 6-9.

For the above two cases, we make the following observa-

tion. According to the above numerical results, we can see

that the control variable in the optimal control problem

decreases along with time in compared with the simple TPPC

problem. Althoughincrease slightly, it still doesnt affect the

value of Lagrange function L, so as to attain the minimum.

From those figures, it can be seen that the values of h, VR,

decrease severely and the values of, , andA increase slowly


12/13


10

5

0

-5

6

4

2

0

-2 -6

-4

-2

0

2

1

0

-1

-2

-30 5 10 15 20 0 5 10 15 20

0 5 10 15 20 0 5 10



15 20

106 107

1010 107

Fig. 8. Generalized momentum.

5 0.5

0

-0.5

-1

0

-5

-10

-150 5 10 15 20 0 5 10

alpha vrsus t beta versus t

15 20

104

Fig. 9. Generalized momentum and control variables.

in order to meet the performance requirement, namely the

state variables must have the minimum energies among all

possible trajectories. For the two cases, it is calculated, in

the same time interval, that the Lagrangian L is equal to

2.906314888887725 1015 for the optimal control problem,

and equal to 3.950072046453987 1015 for the TPPC problem.In other words, the TPPC problem with optimal control design

can not only achieve the purpose of flight control, but also

satisfy the minimum energy requirement.

VI. CONCLUSIONS

We have proposed a potential unified framework for solv-

ing optimal control problem in the present paper. The optimal

control problem is reformulated as a Hamiltonian DAE and

then is solved by the BDF method. To serve as an illustrative

example, we also derive an optimal control law for the TPPC

problem. Currently, we just limited the fight path angle and

the azimuthA so that was gradually decreasing from 1 and

A increasing from 45. It is possible to alter the algebraic

constraint to a more complex equation or a realistic fight route.

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