GM THE DETERMINATION OF TIME OPTIMAL CONTROLS
FOR LINEAR STATIONARY SYSTEMS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY

Narberfi' Barnard Hamesaih
1964

THESIS

This is to certify that the

thesis entitled
ON THE DETERMINATION OF TIME

OPTIMAL CONTROLS FOR LINEAR

STATIONARY SYSTEMS
presented by

Norbert B . Heme sath

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Electrical
Engineering

Major professor 7

Date_M§\; 15. 1964

0-169

__ u.-_— _ 7 _ 7

LIBRARY

Michigan Sm:
University

 

ABSTRACT

ON THE DETERMINATION OF TIME OPTIMAL
CONTROLS FOR LINEAR STATIONARY SYSTEMS

by Norbert Bernard Hemesath

The last decade has witnessed an intense interest in
time optimal control, i.e., the minimum time transfer of the
nth order system

it =AX + BU(t)
from an arbitrary initial state to an arbitrary terminal
state, subject only to the constraint that the components of
the control vector, U(t), be bounded and measurable, The
optimal control, when it exists, is known to have components
which are piecewise continuous and assume only their extreme
values. Furthermore, when A has real, distinct, non-positive
eigenvalues, each control component has (n-l) or less discon-
tinuous points. The optimal control is uniquely determined
when the (n-l) discontinuous points or ”switching times” and
the initial sign of each of its components are known;

This thesis develops nonlinear equations which the
optimal control satisfies and some techniques for solving
these equations. The special case in which U(t) is a scalar
is analyzed separately, and it is shown that the optimal con-

th

trol is the solution of an n order transcendental set in

the (n-l) switching times and the minimum control time, tn,

ON THE DETERMINATION OF TIME OPTIMAL

CONTROLS FOR LINEAR STATIONARY SYSTEMS

BY

Norbert Bernard Hemesath

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1964

 

ACKNOWLEDGEMENTS

I wish to thank Dr. H. E. Koenig for his help
and guidance during the writing of this thesis and to
acknowledge the support of the National Science Founda-
tion during my years of graduate study. I am grateful
to Suzanne for her patience and encouragement through

some difficult periods.

ii

TABLE OF CONTENTS

Page

LIST OF FIGURES . . . . . . . . . . . . . . . . . . iv

LIST OF APPENDICES . . . . . . . . . . . . . . . . . v
Chapter

I. INTRODUCTION . . . . . . . . . . . . . . . . 1

II. MATHEMATICAL THEORY . . . . . . . . . . . . 7

Statement of the Problem . . . . . . . . 7

Pontryagin' 5 Maximum Principle . . . . . 9

Maximum Principle . . . . . 11

Properties of the Optimal Control . . . 12

III. SCALAR CONTROL PROBLEM . . . . . . . . . . . 19

The Switching Equations . . . . . . . . l9

Bounds on the Control Time . . . . . . . 22

IV. THE VECTOR CONTROL PROBLEM . . . . . . . . . 26

The Switching Equations . . . . . . . . 26

Constrained Minimum Problem . . . . . . 30

Modified Lagrange Technique . . . . . . 32

Bounds on the Control Time . . . . . . . 38

V. EXAMPLES . . . . . . . . . . . . . . . . . . 43

Scalar Control Problems . . . . . . . . 43

Vector Control Problem . . . . . . . . . 51

VI. CONCLUSION . . . . . . . . . . . . . . . . . 56

REFERENCES . . . . . . . . . . . . . . . . . . . . . 99

iii

Figure

.ts

CUU'IUIUI

LIST OF FIGURES

Relay Controller

f(x) = <x-1)3

Simple R-L-C System
Mechanical System

Two Driver R-L-C System

Contour Map of (2: Constant

iv

Page

32
43
48
52

82

LIST OF APPENDICES

Appendix

A. NUMERICAL SOLUTION OF NONLINEAR
ALGEBRAIC EQUATIONS

Algebraic Systems . .
Mixed Algebraic and Differential
Systems . . . . . . . . .
B. NUMERICAL METHODS

Newton's Method
Gradient Method

C. SOME DIGITAL COMPUTER PROGRAMS
Program I--Kurung

Program II--Optima
Program III--Inequa

Page

58
59
62
77

77
81

8O
8O

87
88

I. INTRODUCTION

Any realistic design of a physical system imposes
constraints on some if not all of the system parameters.
Voltages, currents, forces, diSplacements, torques, veloc-
ities are all limited in magnitude. System designs ignor-
ing such restrictions are often trivial and physically mean—
ingless. A realistic design involves choosing components
and parameters from within a given set of constraints such
that "acceptable" performance is achieved as measured by
some predetermined criterion. Optimization theory is con—
cerned with the question: "Given a performance index, what
is the 'best' design within the framework of the constraints
imposed on the parameters?” If the physical system is des-

cribed by the state model*
x = P [x<t),U<t>] (1-1)-

then an important class of optimal problems involves choos-

ing the vector U such that the scalar integral

 

* . . . .
In this thes1s upper case letters indicate vectors
and matrices, lower case letters are scalars.

 
 
 

ti

m = h[X(t),U(t)]dt (1—2)

to

is minimum, where X(t0) and X(ti) are the initial and termi-

th order

nal states, reSpectively, of (1-1), and U is an r
excitation vector constrained to lie within a compact region
R of the r-dimensional Space.

Typical problems in this class are: ”What control
function, subject to magnitude constraints, will transfer a
system from one state to another in minimum time?” ”What
control will transfer a system from one state to another
with minimum energy expenditure?” These and many similar
questions have been investigated in recent years.

Time optimal control, the problem of taking a system
from one state to another in minimum time, represents one of
the most widely discussed optimization problems. It had its
genesis in the efforts of researchers to determine when a
relay controller should be switched to simultaneously reduce
the error and its derivative to zero in minimum time follow—
ing a step input.

The relay controller (also called bang-bang because
of its on-off nature) is a simple, economical approach to
closed loop control and is therefore attractive in many ap-
plications [1],[2]. Figure 1.1 below is a simple position
servo using a DC motor whose armature voltage is applied

discontinuously (off-full on) by the relay.

 

 

INPUT + ERROR OUTPUT

 

 

V

RELAY .__. MOTOR
ei(t) ' E(t) eon)

 

 

 

 

 

 

 

 

Figure 1.1 Relay controller.

The operation may be briefly described as follows: If the
input and output positions differ, the relay applies arma-
ture voltage of the proper polarity to the motor to reduce
the error. When the error changes sign the relay reverses
the armature voltage. The system reSponse finally depends
upon the characteristics of the motor and its load. For no
damping the system oscillates with constant amplitude; a
damped system oscillates with decreasing amplitude and in-
creasing frequency [1].

Reversing the relay before the error reaches zero
reduces the "hunting” and shortens the settling time. Sys-
tem design procedures incorporating linear anticipatory
switching, wherein the relay is reversed as some linear
function of the error and its derivative before the error
reaches zero, have been developed in the phase plane for
second order systems such as that of Figure 1.1 [3],[4].

The concept of optimum performance is a natural out-
growth of the more SOphisticated relay controller with antic-
ipatory switching. Hopkin in 1950 defined Optimum performance

as ”that behavior in which the system returns to rest with

zero error in the shortest time following a step input” [5].
Hopkin and McDonald both demonstrated with heuristic proofs
in the phase plane that a second order system with real
characteristic roots and a bounded forcing function achieves
zero error in minimum time when the forcing function assumes
only its extreme values and reverses sign at a critical
boundary which is a nonlinear function of error and error
rate [5],[6].

In 1954 Bogner and Kazda, considering higher order
systems with real roots, attempted to extend the phase plane
concepts to a phase space [4]. In the case where the number
of relay reversals is one less than the order of the system,
their results indicate that a unique path exists from an
arbitrary initial point in the phase Space to the origin.
Bushaw in 1952 discussed the second order system with com-
plex roots as an abstract mathematical problem [7].

Bellman, Glicksberg and Gross in 1956 ”imbedded”
the Optimum relay controller problem in a more general,
precisely stated mathematical problem, and gave the first
rigorous proof that for a rather general class of systems
there exists an optimal controller and it is "bang—bang" in
character [8]. LaSalle further generalized the theory to
include time varying linear systems [9]. Concurrently sev-
eral Russian authors, working independently, developed sim—
ilar results. The approach of Bellman and LaSalle is topo—

logical; Desoer arrived at many of their conclusions using

variational calculus [10]. Finally, the Russian mathemati-
cian, L. S. Pontryagin, devised a ”maximum principle” which
is applicable to a very broad class of systems and problems
[11]. It too can be used to derive properties of the time
optimal controller.

Thus, a distinguished body of theory relevant to the
bang-bang control problem has been developed with the mathe-
matical form of the control firmly established for systems
of arbitrary order with both real and complex eigenvalues.

Although the form (bang-bang) of the control func-
tion is well known, the problem which has not been satisfac-
torily solved is this: ”Given a system with an initial

state, X for which a time optimal control exists, how is

O,
that control found?” If this question has no reasonable

answer, then optimal control remains a mathematician‘s game.
The object of this dissertation is to derive sets Of

conditions which the time optimal control for linear, con—

stant coefficient systems necessarily satisfies, and to show

 

that the equations representing these conditions can be
solved by numerical techniques.

The body of this thesis contains six sections fol-
lowed by three appendices. Section two is devoted exclusive-
ly to developing the mathematical theory of the time optimal
problem considered in this thesis. Important properties of

the controlled system and of the control itself are discussed.

Section three deals with the scalar control problem and
develops the equations which the scalar control must satisfy.
Section four introduces the vector control concept and devel-
ops the necessary optimization equations. A modified
Lagrange multiplier method is used to obtain a solution to
the resulting equations of Optimization. Two sections on
examples and conclusions complete the main body of the
thesis.

Appendix A includes a new method for solving non—
linear equations by transforming the algebraic equations
into differential equations whose solution at one endpoint
represents the solution to the nonlinear algebraic equations.
Appendix B contains standard material on numerical techniques
and is included primarily for continuity. Appendix C gives
some computer programs used to solve the optimization equa—

tions set down in sections three and four.

II. MATHEMATICAL THEORY

Statement of the Problem

 

The physical systems considered in this thesis can
be described by a system of linear, constant coefficient,

ordinary differential equations

n r
Xi = z aijxj * z bikuk(t) (2—1)
j=1 k=1

1 = l, 2, --—, n

where x , ---, x are the state variables which completely
1 n

define the system, and xi indicates differentiation of Xi
with reSpect to the independent variable, t. Equation
(2-1) may be written in the vector notation

X = AX + BUCt) (2—2)

where X is an n-component column vector, A is an nxn con-
stant matrix, B is an nxr constant matrix, U is an r—compo—
nent column vector.

The trajectory X(t) of (2-2), as a function of t, is
uniquely determined on an interval 0 j t 5 t1 when the con-
trol, U(t), and the initial condition X(O) = X0, are Spec-

ified. The ability to control the system lies in the freedom

to choose U(t), the entries of which are assumed to satisfy

the inequality
luk(t)l _<_ 1 k = 1, —-—, r. (2-3)

Suppose also that the controls, uk(t), are piecewise con-
tinuous, i.e., continuous for all t, 0 j t 5 t1, except at a
finite set of points ti at which the controls may have finite
discontinuities. Any control, U(t), whose components satisfy

these two conditions is called an admissible control.

 

The time optimal control problem may now be stated as
follows:

Given two points X0 and Xlin the state Space,

 

among all admissible controls, U(t), which trans—

fer the state point from XO to X1 (if such con-

 

trols exist), find one which minimizes the time,

Here X(to) = X0 and X(tl) = X1, and X(t) is the solution to
(2-2) correSponding to control U(t).

The maximum principle as given by Pontryagin, can be
used to establish some of the mathematical properties of the

control which is the solution to the problem posed above.*

 

*The development of this chapter is not intended to
present any new material, and therefore theorem proofs,
readily available from such sources as Bellman, LaSalle, and
Pontryagin, are omitted [8],[9],[10],[ll].

Pontryagin's Maximum Principle

The maximum principle developed by Pontryagin and
his associates states a necessary condition which an optimal
control and the associated Optimal trajectory Of a System
must satisfy. The process to be controlled is assumed to

have a state model of the form
x = F (X,U) <2-4)

where: X is an n-component vector (state vector)
U is an r-component vector

F and its partials with respect to x-, i = l,
2,---,n, are continuous on the direct product
of the control Space and the state Space.

The performance is to be measured by the functional
t1
J z 'fTO[X(t),U(t)]dt (2-5)

to
where fo(X,U) together with its partial derivatives is de—

fined and continuous on the direct product of the control
and state Space. Then the fundamental problem of optimal
control is stated as follows:

Given any two points X0 and X1 in the phase Space,

 

select from among all admissible controls, U(t),

which transfer the phase point from XO to X1 (if

 

such controls exist), the one which minimizes the
functional, J,
where X(to) = X0 and X(tl) = X1, and X(t) is the solution

to (2-4) associated with control U(t).

10

Application of the maximum principle requires:

1.

Augmentation of system (2-4) with the equation

i0 = fO(X,U) (2-6)

Introduction of the adjoint set of equations

W: UTE] (2-7)

where II/ = (1,10, l/jl, ---, Mn)

A scalar function H relating the augmented sys-

tem and the adjoint System.

The augmented system is

where

T = P(X,U) (2-8)
R - (f0, f1, -——, fn) = (f0, F)
X = (x0, x1, ---, xn) = (x0, X).

(2-9)

A .

13 afi(X,U)/axj.

Observe thatlflfi F, and X are all (n+l)-component vectors and

that F(X,U) is not a function of x0. The scalar function H

ll

relating (2-7) and (2-8) is defined
_T _ .
H =11” F(X,U). (av-10)

Systems (2-7) and (2-8) are obtainable from (2-10) as

x 0
ll

i 9H/ 3 WI i
WI

Note that for constant values of X andl/T/ the function H de-

II
0
I—‘

I

I

I
:3

“3H/ aXi

H
ll
0
l—‘
I
l
I
:3

pends only upon the vector parameter U. The maximum prin-

ciple may now be stated as follows.

Maximum Principle

 

Let U(t), t0 5 t 5 t1, be an admissible control Such
that the correSponding trajectory, X(t), beginning at XO at
time tO passes through X1 at time t1. If X(t) and U(t) are
optimal it is necessary that:

1. There exist a non-zero continuous vector func-

tionlflkt) = lpg(t),lp&(t), ---,lph(t) corre-
Sponding to U(t) and X(t).
2. For every t, t0 5 t 5 t1, the function H of the

variable U attains its maximum at U = U(t).

The maximum principle stated above is a necessary

 

condition for optimality, but the fact that it has been
satisfied does not assure the existence of an optimal con-

trol. Mathematical questions concerning the existence and

 

12

uniqueness of optimal controls are very important and dif-

 

ficult. The following sections deal with some of these
questions in the particular case of time optimal, linear,

stationary Systems.

Properties of the Optimal Control

 

For the time Optimal problem stated in the first

section of this chapter the functional, J, is

t1
J = .jTO(X,U)dt = tl-tO (2-11)
to
which implies
fO(X,U) = 1. (2-12)

The augmented system (2-7) is

x0 = 1
(2—13)
x = AX + BU
and the adjoint system (2-7) defined in terms of
wle/ltl/jzv "'"9 W11 is
7 = 0
V0 (2-14)

13

The function H can be written as
_ T .T
H(l/f,X,U) 3W0 «Ly! AX +15] BU (2-15)

The maximum principle states that if U(t) is optimal then H,
considered as a function of U alone, assumes its maximum at
U = U(t). Since (2-15) is linear in U this implies that each
component of U assumes its greatest magnitude and the Sign of
its coefficient. Since from (2-3) IuiI: l, the control U(t)

may be written

T
U(t) = Sgnyy B (2-16)
where for r—dimensional vectors A and B, A = sgn B means
that ai = Sign of bi’ i = l, ---, r. This result may be

formally stated as

Theorem 2-1

 

If an Optimal control function for the time optimal

problem exists it is of the form
_ T
U(t) - sgnyy B
wherelflKt) is a non-zero solution of the adjoint* system

11/ = fw-

 

' T . . °
*The system Y = -A Y 15 called the adj01nt to x = AX.

14

Therefore the form of the optimal control, if it
exists, is established as piecewise constant or "bang-bang”,
and each component of the control switches Sign at the zeros
Of the corresponding component oft/1TB.

Perhaps a logical question at this point is: "Under
what conditions does an Optimal control exist, and is it
unique?” A simple example may provide some insight into this

problem. Consider the scalar equation

x = ax + bu (2—17)

By the maximum principle u i l and never switches Sign Since

the adjoint solution, y(t) = e-atyo, has no zeros. Thus the
solution to (2-17) is
at
x(t)=e (x+_b_u -22.
0 a a

and if the desired terminal state is X(t) = O, the optimal

solution, if it exists, must satisfy

1 _ at
axO — e t 3 0 (2—18)

+1
bu

 

Whether or not (2-18) has a solution depends upon the values
of the parameters, a and b, and the sign of u. There are

three cases of interest.

15

Case 1: a < O, b f 0

An optimal solution exists for arbitrary x Since

0

the left hand Side is always less than one if u = sgn (a X0

D,

 

while the right Side approaches zero as t grows large.

Case 2: a > O, b # O

. . a x
NO solution eXists for values of xO such that .7;2

 

> 2. Under this condition the left hand Side is always less

than one while the right hand side exceeds unity.

Case 3: a arbitrary, b = O

. . . at
No solution ex1sts Since X(t) = e x never reaches

0
the origin.
The behavior in case three above is related to the

concept of controllability as introduced by R. E. Kalman [l2].

 

He defines a System to be completely controllable if for

 

arbitrary states X0 and X1, and times t0 and t1, there exists
a control which transfers the system from state XO at time tO
to state Xl at time t1. Kalman has also stated the follow-

ing [13]

Theorem 2-2

 

The system described by (2-2) is completely control-

n‘lB, An-ZB, ---, AB, B]

lable if and only if the matrix [A
has maximum rank.
The system described in case three does not satisfy

this theorem, and is, therefore, not controllable. It is

16

apparent that a system which is to be optimally transferred
from an arbitrary initial State to the origin must be neces-
sarily completely controllable.

However complete controllability is not sufficient

 

for optimal control of a System with arbitrary initial state.
Consider case two; the system is completely controllable by
theorem 2-2, yet for certain initial states there is no
optimal control.

On the other hand the system of case one is complete-
ly controllable and always has an Optimal control. This
leads to the final factor affecting the existence of optimal

controls, stability. System one has a stable characteristic

 

root while system two has an unstable characteristic root,
where a stable root is defined as an eigenvalue Of the matrix
A in (2-2) with non-positive real part. The intuitive evi-
dence might lead one to expect that which the following the-

orem states [11]

Theorem 2-3

 

If the matrix A has stable eigenvalues and if the
system (2-2) is completely controllable, then there exists
an optimal control which transfers an arbitrary initial

phase point, X. to the origin.

0’

17

The previous theorem establishes the existence Of
an optimal control under certain conditions. Another very
important property of the control is its uniqueness which

is assured by

Theorem 2-4

 

Let U1(t) and U2(t) be two Optimal controls (defined
on the intervals t0 5 t 5 t1 and t0 5 t 5 t2 respectively)
which transfer the phase point XO to X1. Then these controls
coincide, i.e., t1 = t2 and Ul(t) = U2(t).

In systems which have real eigenvalues the optimal
control has an especially significant property wherein the
number of switchings of each control component is related

to the order of the System.

Theorem 2-5
If the matrix A of (2-2) has real, non-positive

roots, then each component, u- i = 1, ---, r, of the op—

17
timal control will switch not more than (n-l) times where

n is the order of the system.

The property stated in theorem 2-5 above is exploited
in the following chapters of this thesis to develop sets of
equations which the optimal control must satisfy. Henceforth,
unless otherwise stated, the system under consideration is

that characterized by (2-2) with the additional restrictions

18

that the eigenvalues of A are real, non-positive and simple
(non-repeated), and that the System is completely control-
lable, i.e., all subsequent development is concerned with
linear, stationary systems for which a unique, time optimal

control always exists.

III. SCALAR CONTROL PROBLEM

The time Optimal control of physical systems governed
by a single control variable is perhaps the most Significant
of the class of time optimal problems in applications. In~
deed, all single input, Single output, linear control sys-
tems currently designed with s—domain techniques fall into
this class. For these systems usually (but not necessarily)
the error and its first (n-l) derivatives are reduced to
zero. Because of its importance and its mathematical trac-
tability, the scalar control problem is considered first as

a Special case.

The Switching Equations

 

Many of the early efforts to determine time Optimal
controls were based on the phase Space, a generalization Of
the phase plane which is SO useful for second—order systems
[4],[lO]. The approach is that of establishing switching
surfaces in the phase Space, i.e., surfaces at which the
control function changes Sign. However, as the order of the
system increases the problem of eliminating the time variable
from the system equations becomes very difficult. Recently
several authors have suggested techniques for determining
the control as a function of switching times rather than as

a function of the system state [17],[l8]. This approach will

19

20

be used here.

Consider the completely controllable System
x = AX + Bu(t) (3-1)

where A is a Square matrix with simple, real, non-positive
eigenvalues, B is a column vector and u(t) is a scalar con-
trol function. The theorems of the previous section guar-
antee the existence of a unique control function which will
transfer the solution of (3-1) from an arbitrary initial
state to the origin in minimum time.

The equations of optimal control are simplified if
the system (3-1) is reduced to principal coordinates, i.e.,
A is diagonalized. Defining the nonsingular linear transfor-

mation

X = PY (3-2)
equation (3-1) becomes
9 = (P'lAP)Y + P—lBu(t) (3-3)
where [14],[15]
P—lAP = D = diag(/\1,A2, ---, An) (3-4)

and the ,Kiare the eigenvalues of A. If we let Zi2yi/(P-1B)i
_1 . .th -1
where (P B)i is the 1 component of the vector P B then

(3-3) reduces to

21

z = D2 + Iu(t) (3-5)

where I is a column vector of ones.
The solution to (3-5) is [19]
t

Dt D(t-r)
‘I' e

Z(t) = e Z Iu(r)dr (3—9)

If tn represents the time required for an Optimal control to

transfer ZO to the origin, then the optimal solution is

tn
Z(tn) = O = eDthO + J/geD(tn-r)lu(r)dr. (3-10)
0
Multiplying both sides of (3-10) by (e Dtn)-l = e_Dtn gives
tn
-ZO = J[e_DrIu(r)dr. (3-11)

0

Since u(t) iS piecewise constant and reverses Sign at most
(n-l) times on the interval [0,tn], (3—11) may be written as

the sum of n integrals of alternating Sign

 

 

/ t1 t2 tn \
_ —D n-l —Dr
-ZO = u( e DrIdr- e rIdr+---+(-l)( ) e 1dr? (3—12)
t
x l tn‘1 1
where u = i l and t1, --—, tn—l are the (n-l) switching times

which must satisfy the ordering constraints

22

< t2 < ___ < t -l j t . (3—13)

If none of the Ad.: 0 integration of (3-12) gives n equations

_ ~t - -t - -t - -t
71.2. = u [l—2e A1 1+2e)Ll 2— ——- -(-l)n2e A1 r1'14-(_1)ne 1 n]
l .10
i : 1: —--i n (3‘14)

where zi0 is the ith element of the vector Z0. The modifica—
tion required where some Ad.= O is Obvious.
TO find the unique optimal control it is necessary to

find the ordered set

0 < t < t < ___ < t < t

1 2 n-l - n

with minimum tn which satisfies the n transcendental equa-

tions in (3-14) and to find the correct Sign for u.

Bounds on the Control Time

 

Systems Of transcendental equations, in general,
cannot be solved analytically and the solutions are not
unique. However, any solution of (3-14) which simultaneously
satisfies (3-13) is unique [4]. Since numerical methods are
required to solve (3-14), a ”good" first approximation to the
desired solution is necessary if an iterative procedure is
to prove successful. Initial estimates must be made for t1,

t —--, tn as well as the Sign of u. The switching times,

23

of course, must be positive and must satisfy the ordering
constraints (3—13). However, the ordering constraints con-
tain no information about the magnitudes of these switching
times. Intuitively one might expect that the control time
required to bring a system to the origin is a function of
the System initial state and the eigenvalues. The following
theorem shows that this is, indeed, true, and provides a
useful basis for choosing the initial vector with which to

begin an iterative solution technique.

Theorem 3-1
If the eigenvalues are all distinct from zero, the

minimum time T0 required to transfer the normalized system

z. = -z- + u(t) i (3—15)

1 11

II
[.1
N

l

I

I
D

from an arbitrary initial state, Z to the origin satisfies

0,

the inequality
T > max t.
o — i
where:

ti ='17%;rlog( Lliziol+l) i = l, --—, n (3-16)

Proof:
Since the minimum time solution requires the Simulta-
neous transfer to the origin of all states, the minimum time

solution, To’ must equal or exceed the maximum of the Set

24

t1, t2, ---, tn, where ti is the minimum time required to
transfer the initial state, Zio’ to zero. From (3-14) the
control which transfers 2- to the origin in minimum time is

10

the solution to

alizio = u(l-eflliti) t. > 0. (3-17)

The solution to (3-17) is

_ 1 i 10
ti — - Ai log(l + u ).
. . i io
and Since ti > 0 it follows that (l + u )> 1
and
u = sgn ( Aizio). (3—18)

It follows, therefore, that

t. = 1' lo (1 +IAgz- I) (3-19)

1 IiiI g 1 10

and the theorem is proved.

In case one of theA.i = O, say,l1 = O, the solution

corresponding to (3—17) is
O = ut + 210 (3-20)
and the corresponding ti is

t1 = 'leI . (3-21)

25

Hence for a system with A1 = 0

TO _>_ max ”210' ,ti] (3-22)

where ti = 7171' log (1+lAiziol ).

where:

.lu

IV. THE VECTOR CONTROL PROBLEM
Consider the linear system

x = AX + BU(t) (4-1)

X is the n component state vector

A is an nxn matrix with real, non-positive, Simple
eigenvalues

B is an nxr matrix
U is the r-component control vector

il: l i = l, 2, ---, r 2 j r j n.

From the results of the previous section and other

investigators the control vector, U0, which transfers an

arbitrary initial state, X

is known
only its
not more

lish the

o’ to the origin in minimum time
to be unique. Further, each component of U0 assumes
extreme values, is piecewise constant, and switches

than (n—l) times. The set Of equations which estab-

switching times on the components of U0 is Obtained

by extending the development of the previous section.

The Switching Equations

 

Let the System in (4-1) be transformed to principal

coordinates by the linear transformation X = PY so that

26

 

 

‘IJ

27

I = DY + CU(t) (4-2)
where
_ - - —l
D — diag (Ad-~7Kn) 1 P AP
--1
C = P B

Since the linear, constant coefficient differential

equation (4-2) has the vector solution

t

Y(t) = eDtYO + -[eD(tJT)CUCT)dT’ (4-3)
0

the Optimal control vector, U(t), must satisfy the relation

tn
D D t -
—e tnY0 : je ( n T>CU<T>C1T (4-4)
0
or
tn
-D7'
“Y0 = er CUCT)d7' (4—5)
0
tn tn
-DT' [ -D7’
-Yo — dfe ClulCr)dT + -—- + J e CrurCT)d7’ (4-6)
0 0
where the Ci, i = l, 2, ---, r, represent the columns of C.
The fact that luil = l for all t on the interval

(0, tn) with (n-l) switchings or less, makes the integration

of (4-6) elementary when it is written as n Scalar equations

_A-T .T
-y, = J[; l eiluldi --- + jfe'A“ c. urdT’. (4-7)

Now each of the r integrals in (4-7) can be written as n

integrals SO that (4-7) becomes

 

 

tii t12
_AiT _AiT
-yi0 = Cilu1< fife d7’- JIe d7’+ --—
O tll
\
tn ( trl
_ _l-T A-T
+ (-l)(n l) ’[e l dT'+ -—- + cirur '[e- 1 d7,
tl,n-1 O
tri T tnA T N
_ ' -1 _ '
- -je l d7'+ --- + (-1)(n ) fife l d7’>
tr1 tr,n-l /
i = l, 2, ---, n (4-8)
where the tij’ i = 1, ---, r, j = l, --—, (n-l) are the (n-l)

switching times associated with ui and satisfy the inequal-

ities

O 5 till: ti2 : --- : ti,n-l : tn 1 = l, --- r. (4-9)

29

Assuming ’li # 0 (4-8) reduces to

_ .t — .t — .
Cilul l _ 28 A1 11 + 2e 1 12 _ ___ + (_l)n e Altn

i0 :1

 

 

Ci2u2 l _ 28-Ait21 + ___ + (-l)n e'Aitn

+ .

A}
I A A A
‘ Cir r 1 _ 2e- itri + 2e- itr2 ___ + (_1)n e- 'tn
+

Ii

i = 1, 2, ---, n (4-10)

The result for a particular ’Xi = 0 calls for a trivial mod-
ification.

The optimal control vector U0 is completely specified
by the solution to (4-10) for minimum tn subject to the con-
straints in (4-9). Since ui may be i 1 (4-10) is actually 2r
distinct sets of n equations, and each of these sets involves
r(n-1) + 1 variables. Since 2 j r j n the number of unknowns
exceeds the number of equations. If such a system has one
solution it has an infinity of solutions each of which is
Obtained by arbitrarily Specifying r(n-l) + l - n of the
variables and solving the resulting normal* set. However

the problem of finding the unique time optimal control remains.

 

* . . .
A normal set is one With the same number of equations
as unknowns.

3O

Constrained Minimum Problem

Finding the time optimal control from set (4-10) may
be viewed as a constrained minimization problem [17]. Since

tn appears in each equation of (4-10) only once, in a term

_ -t
of the form, Kie 1 n, it is possible to solve explicitly
for tn in terms of the remaining r(n-l) switching times,

thus

t = f(t

n 11’ t --—, t ). (4—11)

Substitution Of (4—11) into the remaining (n-l) equations of
(4-10) gives a set of n-l equations which is independent Of

tn

gi(tllyt127__-7tr n-l) : O i = 1,27_--$(n-1)° (4-12)

The optimal time solution is now obtained by minimizing

f(t -,t n 1) Subject to the constraint equations of the

11""
form given in (4-12), and the ordering constraints, (4—9).
Using Lagrange multipliers to find the minimum, form

the Scalar function [20],[2l],[25]

H<tii’""’tr,n-iJTGfl'"““TChi):f(tii’""tr,n—i)
(4-13)

*'7581(tii""’tr,n-i)*"'In;—1gn-i(tii’""tr,n—i)

where the TG are constant multipliers. Consider now the

 

u -53?
-u.._ 5

o .

Vi‘h

‘~4...

'r-
, J
"‘-...u

31

problem of locating the extrema of H. A necessary condition

is that all its partial derivatives vanish

3H 2 _ ._
SE-i—j O .‘L - l, ---, r J - l, --- n-l

(4-14)
3H = 0 k - l, ——— n-l

5TH.

Set (4-14) contains (r+1)(n-l) equations in the same
number of unknowns, and its solutions are the stationary
points* of H. Among its solutions are the minima of the
original function, f, subject to the constraints, gi = 0.

Two difficulties arise:

l. The solutions Obtained are minima satisfying the
constraint set, gi = 0, yet they may not satisfy
the inequality constraints, (4-9), which order
the switching times.

2. The desired minimum time solution may be on the
boundary of the closed constraint set defined by

inequalities (4-9), in which case it is not nec-

essarily a stationary point of H, i.e.,

3H
zfifj' To # O for some tij (4—15)

lJ

where TO represents the time optimal solution

vector.

 

*A stationary point of a function is one at which
all its partials vanish.

 

 

.iu.

 

 

32

As a simple example illustrating the second point,
consider the extreme values of f(x) = (x-l)3, O f x j 2.
Figure 4.1 below shows the minimum at x = O and the maximum

at x = 2. The only stationary point is x = 1.

f(x)

 

 

Figure 4.1 f(x) = (x-l)3

The two difficulties discussed above impair the use—
fulness of the classical Lagrange technique; in the first
case undesired solutions are Obtained while in the second
the minimum time solution cannot be found because it is not
a stationary point. A method which permits use of the con—
straints, gi = O, as well as the ordering constraints in

(4-9) is given next.

Modified Lagrange Technique

 

Several authors have extended the Lagrange multiplier
technique so that inequality as well as equality constraints
may be handled [22],[23],[24]. This is done by observing

that an inequality can be transformed into an equality by

33

introducing a variable parameter s such that

f(x) 2 0 implies f(x) = 52. (4-16)

Thus the inequality constraints in (4-9) can be re—

written as

tii : 0
tiz 3 til
: 1 = 1,---,r (4-17)
I
I
tn 3 ti,n-l

and replaced by the following set of equalities

_ 2
til ‘ 511

2

. = +
t12 til 512

t. +5.
n i,n-1 in

Sucessive substitution of each member of (4—18) into the

following one yields each tij defined in terms of the Sij

alone

34

_ 2
1:ii ‘ sil
_ 2 2
ti2 ‘ si1+si2
: i = l,2,—--,r (4-19)
I
I
I
tn = 511*Si2*"'*sin

Now finding the minimum time solution requires minimizing

tn : f(tll""-’tr,n-1)
subject to the constraints
gi(t11,---,tr n-l) = O i = l---,n-l
t _ 2
il ‘ Sii
t, = s? +52 1 = 1,2,---,r (4-20)
12 11 12
I
I
I
I
I
t = 52 +5 +---+s2
n 11 12 in

Observe that by substituting into f and gi the relations
from (4-20) defining the tij the remaining constraints in
(4-20) are r in number. Thus, the problem becomes one of

minimizing

t = f(s-.) i = 1,---,r (4-21)

1,--—,n—l

Q...
II

35

subject to the constraints

gj(sij) = O i = 1,---,r j = 1,---,n-l (4-22)
2 2 2
h = -f S. +5 +5 +---+S = O
k ( 13) kl k2 kn
k = l,---,r (4-23)
1 = l,-—-,r
j : l,---,n-1

Now f(sij) may be minimized by the usual Lagrange technique

of forming

.. . = + . +
m<rr> f Re.

1 ‘ l,—--,r (4-24)
J - 17_--)n
k = l,---,n-1

and setting partials with respect to all variables equal to

ZEIO

3O

n-l r
3H - af TT agk + th': O
s.. + k35.. um 55..
1J 1J 1J 1J
k=1 m=l
i = l,---,r
3H 2 g : O
371: k j = l,-—-,n (4—25)
k - 1,---,n-1
.13 = .=
Bu. hi 0
1
This is a set of nr + n-l + r = (r+l) (n+l)-2 equations in

the same number of variables. However, in practice, r of
these equations may be eliminated immediately. None Of the

functions, f and gi, contain variables S S -—- S

1n? 2n 3 In,
since tn does not appear in them and Sin, ---, Srn do not

appear in the equations defining the tij (see (4-20)).

 

Therefore
n-
2H 2 at , n’k agk, um ahm_0
Esin sin 3sin 2sin
k=1 m=
i = 1,---,r
= O + O + Zuisin = 0
3H : = ' :
35E ZUiSin O 1 1,2,---,r (4-26)
and it is necessary that either ui = O or Sin = O. In the

usual case where each control component switches (n-l)

37

distinct times, all Sij # O. The conclusion is that u1 = u2

- --- = ur = 0, thus reducing the number of equations and the
number of unknowns in set (4-25) by r, and leaving n(r+1)-l
equations in n(r+l)-l variables. The cases in which some or
all of the Sin are zero can also be handled by the ”either-or”

rule. Therefore, the system to be solved iS always the fol-

lowing normal one Of dimension n(r+1)-1

n-l r
h
—>-I;I—=bf+ZTTk agk‘t- u amzo
35.. 55.. 35.. m 35..
ij ij ij lJ
k=1 m=1
i = 1,---,r
j = 1,---,n-1 (4—27)
_ k = 1,--—, -1
2H _ gk : O n
gjht
3H : _
Eui hl - 0

Among the solutions to this Set will be the unique
time optimal solution. Furthermore, the two difficulties
encountered in the classical Lagrange development have been
eliminated:

1. Every solution (only real solutions are con-
sidered) is a realizable control since, by
virtue of (4-20), all the switching times are
positive and ordered.

2. The minimum will always occur at a Stationary

38

. . . 1 ‘.
p01nt of H Since the variab es le,fl&, um are

all unrestricted in range.

Bounds of the Control Time

 

The System of equations in (4-27) is highly nonlinear
in the variables, sij’ and therefore not amenable to analytic
solution. More Often than not the convergence of numerical
methods of solution is dependent critically upon a ”good”
first approximation to the solution. The motivation for the
following two theorems is the establishment of upper and
lower bounds on the optimal control time, tn, as an aid in

choosing an approximate solution vector.

Theorem 4-1

 

The time, t representing the time optimal solution

n,

to (4-1) with the origin as terminal state satisfies

> .
tn _ max Pl

 

 

where
Pi ___ 1. log (-Sgn Xio)(IbilI+”‘+IbirI )+41Xio (448,
IAlI ("sgn Xio)( Ibiil*‘"*IbInI>
i = 1,—--,n

Proof: Completely analogous to the proof of theorem 3-1.

39

If each entry u2 --- ur of the control vector, U,

is chosen to be 1 ul, i.e.

’

ui = 1 ul 1 = 2,—-—,r

then the vector control problem becomes a Scalar control

problem, for a typical equation from the set (4-1) is

Xi = Aixi+bilul 1 b312111 I b13111 I “‘ : birul
OI'
Xi -_— Aixifibil i biz .1 --- _t birDul (4—29)
1 = l,---,n

Evidently the Set (4-29) can be written in 2(r'1)

ways since each of the last (r-l) columns of B may assume

either the plus or minus Sign. Thus (4-29) represents 2(r-l)

different scalar control problems. Let

2(r—l)

7 _T-a

. . . . th
be the Optimal control time assoc1ated With the k control

problem, and consider the minimum of the set Tk’ say T

T0 is the time required to reduce an arbitrary initial

state, X

0'

o’ to the origin under the influence of a particular

 

control vector, U in which each entry is some Specific

0?

choice of 1 ul. Either U0 is the Optimal control vector or

it isn't; in either case

40

and we have:

Theorem 4-2

 

1. Let tn be the Optimal solution to the vector

control problem
X=AX+BU
. (r-1)
2. Define 2 scalar control problems by

ui = 1 ul 1 = 2,---,r

3. Define the set Tk’ k = 1 ___ Z(r-l)

7 7

, where

Tk is the optimal solution, if it exists, to

the kth scalar control problem.
4. Define T0 = min Tk
then
tn 5 TO

Theorems 4—1 and 4—2, taken together, establish
both a lower and an upper bound on the optimal control time,
tn. These bounds are very useful for choosing an approximate
solution vector with which to initiate an iterative numerical
scheme. In most cases the bounds are reasonably ”Sharp”; the
lower bound always exists, while in certain cases the upper

one may not exist. Consider the example

 

 

 

 

41

 

 

 

 

 

 

(4-30)

This is a well-defined second—order vector control problem.

Since r = 2,

there are two derived

 

First Scalar Problem, u2 = ul
r-" j r- -1 7
x1 A1 0 le
x O x
2 .. A2- L. 2-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

scalar problems.

 

 

 

 

 

 

 

 

 

 

I' ‘ F 7
1 1
U1
+
l -1
— _I _uld
I2”
2‘ ul
0
_ 1 17 I u1_
+-
_ l ’1‘ --ul‘
0
1‘ ul
2i

 

 

Neither of the two problems has a time Optimal control Since
neither system is controllable. Therefore, theorem 4—2 can-
not be used to establish an upper bound on the optimal solu-
tion to the vector problem. However, this difficulty arises

if and only if each of the 2(r‘l)

derived Scalar systems is
not controllable.
Most of the effort in this section has been directed

toward deriving a set of equations, (4-27), which the time

optimal solution to the vector control problem necessarily

 

satisfies. A generalization of Lagrange's multiplier tech-
nique was used to handle the inequality constraints on the
switching times. Two theorems bounding the control time,
tn, above and below were stated.

Finally, the results of this section hold also for
the scalar problem (r=l), but, in practice, the nth order
set, (3-14), of the previous section is easier to use Since

its dimension is (n-l) less than that of (4-27).

V. EXAMPLES

The previous sections have been devoted to develop-
ing the theory of time optimal control. Equations for which
the optimal control is a solution have been derived. In
this section application of the theory developed above to

several Simple, physical Systems is considered.

Scalar Control Problems

 

Example 1

 

Consider the Simple R-L-C system Shown in Figure 5_l(a).

 

 

 

 

 

 

 

 

; 2
2 R
4.
e(t) ’91 3 L lI'I V 3
4:-;.-.-c if
(a) (b)

Figure 5.1 Simple R-L—C System.

A state model of the system based on the linear graph of

Figure 5.1(b) is [20]

43

 

,— — _
r— —1 1 I— I— T
V4(t) O E V4(t) 0
d _
a - + (5-1)
, 21 mg , e(t)
13(t) L L 13(t) L
I—. ._ R— .A _ _ _I

 

 

 

 

 

 

 

 

The second-order system in (5-1) has eigenvalues

2
A. _ __ R R - 1
1’2 ‘ *2L:\/_ _ <5—2)

4L2 LC

 

If R, L, and C are positive, and if

R2 1

_>_—

4L2 LC

the theorems of Chapter II establish the existence of a
unique, bang-bang control which Switches once, and reduces
both the initial voltage, v4(O), and the initial current,
13(0), to zero in minimum time. Since the equations which
must be solved to yield the optimal control are derived from
a state model written in prinCipal coordinates, the coeffi-
cient matrix in (5-1) must be diagonalized- The nonsingular

linear transformation which does this is

(5-3)

 

 

 

 

 

 

 

45

and the transformed system is

 

 

 

 

 

 

 

LR.¥-1. 7
2L 4L2 LC 0
Y1 y1
£L =
dt 2
,2 o -E.-JL-i ,2
2
2L 4L Lcj
F e(t)
2L iii - EL
V 4L2 LC
+ (5-4)
- e(t)
2
2L E__ _ JL
4L2 LC j

 

 

d y1 = -l O yl + 1
a? e(t) (5-5)

y2 O -2 y2 -1

One more linear transformation

2 (5-6)

 

 

 

 

 

 

reduces (5-5) to the normalized form of (3-8)

46

Ed? : t e(t) (5-7)

 

 

 

 

 

 

 

 

The nonlinear switching equations corresponding to (3-14)

from which the optimal solution is obtained are

t t
A1210 = e(t)(1-2e l+e 2)
(5-8)
2t1 2t2
712220 = e(t)(l-2e +e )
where 210 and 220 are initial conditions related to v4(O)

and i3(O), respectively, by the product Of the linear trans-

formations (5-6) and (5-3). Since e(t) assumes only the
values +1 and -1, (5-8) may be solved for both cases, and the
results in Chapter III indicate the optimal solution is the
one which satisfies 0 5 t1 5 t2. Indeed, the control is now
uniquely Specified: (1) the sign of e(t) is the Sign of the
control for 0 j t 5 t1, (2) t1 is the time at which the con-
trol switches, (3) t2 is the time at which the control is
removed and at which the System state is zero.

Table 1 below lists the optimal solution of equations

(5-8) for eight different Sets of initial conditions.

47

Table 1. Optimal Solutions for R-L-C System

 

 

 

 

21(0) 22(0) v4(0) i3(0) e(t) t1 t2
2 3 -2 4 —1 .3863 .6094
3 2 2 1 -1 .8477 .1622
-5 9 -28 23 1 .4112 .7909
37 25 24 13 -1 .1650 .5085
-20 —12 -16 -4 1 .7739 .2212
—12 -20 16 -28 l .0445 .3673
5 87 -l64 169 l .7442 .7371
-75 17 —184 109 1 .8667 .2138

The above data were obtained using Program I of

Appendix C. This program solves the second order set (5-8),

using the
braic set
set which
and (3-2)

with which to begin the numerical process.

is solved by a Runge-Kutta method.

iS transformed into an equivalent differential

technique of Appendix A; i.e., the nonlinear alge-

Theorems (3-1)

guide the choice of an approximate solution vector

The results ob-

tained indicate that it is rather easy to choose an initial

approximation which will converge to the desired solution.

Example 2

 

A higher order system Shown schematically in Figure

5.2(a) consists of two masses interconnected with Springs

and dashpots and excited by the force driver, f(t).

Springs and dashpots are

tions.

 

 

 

 

 

 

 

 

 

 

 

4

Mle

B4=B

48

2

K3=K5=4

 

//// ///////////////

(a)

Figure 5.2 Mechanical

=1

=1

System

 

 

The

described by-linear terminal rela-

A state model of this system based on the linear graph of

Figure 5.2(b) is

_

 

"I

 

 

_
-8 4
4 —4
l o
o l
L—

 

 

 

 

 

(5—9)

49

where x1 and x2 are the diSplacements of masses M1 and M2,
respectively, from the equilibrium position while x1 and x2
are the correSponding velocities.

The coefficient matrix in (5-9) has real, simple,

negative eigenvalues, and the linear transformation which

diagonalizes the system is

 

 

 

 

 

 

 

 

 

 

 

 

F'. q, -- -—r r- —
x1 0.1315 0.3103 -0.6319 0.8673 yl
R2 -0.0813 0 5021 -1.o224 -0.5360 y2
= (5-10)
xl —0.5131 —0.9854 0.5209 —0.0849 y3
x2 0.3171 —l.5943 0.8429 0.0525 y4
._ ... _ _J _. A
And (5-9) written in principal coordinates becomes
r- ], ’- _ F— —'
y1 -0.2563 0 0 0 yl
dt y3 o o -l.2l30 0 y3
y4 0 0 0 -10.2159 y4
._ _ _. _J I.— J
F _
-0.0875
0.5054
+ f(t) (5-11)
0.9559
0.5289

 

 

50

The linear transformation

 

 

 

 

 

 

_ 1 F .1 _ _
Y1 -0.0875 0 0 0 21
y 0 0.5054 0 O 22
2 = (5—13)
y3 0 0 0.9559 0 23
y4 0 0 0 0.5289 24
reduces (5-11) to the normalized form
P z '1 F 0 2563 0 0 0 “72 — 51'
1 ° 1
z 0 -0.3l49 0 0 z 1
6%- 2 = 2 + f(t)
23 0 0 -1.2130 0 Z3 1
(5-13)
24 0 0 0 -10.2159 24 1
— .— — —. _I h -.I

 

 

 

 

 

 

 

 

From (3-14) the nonlinear switching equations from which the

optimal solution is obtained are

~A1t1+2€-Ait2_2€-Ait3+e-Adt4)

”A1210 = f(t)(l-2e
- t — t - t _

”K2220 = f(t)(l-ZeA2 1+2eA2 2-2e/\‘2 3+e A2t4)

(5-14)
-A.t - t - t -
3230 = f(t)(l-2e 3 l+2e A3 2-2e 3 3+e A~3t4)

- t 4A t 4A t 4A

- 4240 = f(t)(l-2e 4 1+2e 4 2-2e 4 3+e 4t4)

where the A-

1 are the diagonal entries of (5-13).

Table 2 below lists the optimal time solution of
equations (5-14) for two different sets of initial diSplace-

ments and initial velocities of the masses, M1 and M2.

51

Table 2. Optimal Solutions for Mechanical System

 

 

x1 x2 x1 x2 Sign t1 t2 t3 t4

 

1.533 -2.596 —0.633 -0.722 1 2.6299 5.4552 6.0723 6.1399

1.700 -4.405 0.229 0.971 1 3.1660 5.7931 6.4022 6.4699

 

The column labeled ”Sign” indicates the value of f(t),
0 j t 5 t1, and the ti are the successive times at which f(t)
reverses Sign. Program II of Appendix C was used to solve
(5-14) for the optimal controls of Table 2. The choice of
the initial, approximate solution was based upon theorems
(3-1) and (3-2). As one might expect, the choice of an
initial approximation which converges to the desired solu-
tion is somewhat more critical for the fourth—order system

than it is for second order-Systems.

Vector Control Problem

 

A Simple example of a physical System whose mathe-
matical model fits the structure of the vector control prob-

lem is the R-L—C system of Figure 5.3(a).

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 3 4
+ 4b = A#-
O 5
+ O
- 2
91(t) N O 1 V
: «4
.1 ’
C 2 2
(a) (b)
Figure 5.3 CDMDdriverR-L-C System.
A state model of the system, formulated from the linear
graph 5.3(b) is [26]
_v N _0 ZIP-Vfl F2 0-1-i(tf
d 2 - 2 + 5 (5 15)
HT — I
14 —1 -3 14 o 1 ei(t)
h— ...I ._ _ _. __I ___. _J L— _I
The diagonalized System becomes
Fyl‘ -_1 O“ 7y: 74 7 350:)—
d
a? + (5-16)
YZ O ‘2 Y2 2 €~(t)
— _I _— J». _J h. — 1 —I.

 

 

 

 

 

 

 

 

 

 

53
where

= 1 (5-17)

Application of the extended Lagrange method to (5-15) leads
to the following fifth-order System correSponding to (4-27),
which must be solved to obtain the optimal control

éf__.z§_g_
as11 asii

II
0

 

3521 3521
f-52 ~52 = 0
ll 12

2 2
f_ _ :
s21 S22 0

where

54

 

2 2
5 s
_ + 1_ 11 + 1_. 21
11 21 -4u -2u
1 2
2 2
g(511’521) _ _ y20 ”1(1' e ) u2( — e )

2 2
s11 521
+ 2 +2 -y10+4ul(1-2e )+2u2(l-2e )
( ul ”2)
-4u1-2u2

If t11 and t21 represent, respectively, the times at which the
first and the second controls switch and t0 is the time at
which control is removed, then the following equations relate

the tij and the Si.

J
_ 2
t11 ‘ S11
_ 2
t21 ‘ S21
2 2 (5-19)
= +
to S11 S12
_ 2 2
t0 — 521+522

Program III of Appendix C was used to solve (5-18) for the
variables, Sij and 2. Equation (5-19) was used to determine
the switching times of the optimal control. Table 3 below
lists the solutions for several different sets of initial

conditions.

 

 

 

Table 3. Optimal Solutions for Two Control R-L—C System
V2(O) i4(0) i5 ei t11 t21 to
-14 11.5 -1 1 0.8765 .1616 2.1810
12 6.5 l -l 0 .3013 2 6498
-8 -2.0 -1 l 0 .9729 2.0423
-82 84.5 1 1 1.6468 .2030 2.1943

 

i and e

5

reSpectively.

The columns labeled i5 and e

on the intervals, 0 j t j

1
t

11’ O E t i t21’

indicate the Sign of

VI. CONCLUSION

The time Optimal control of physical systems des-
cribed by a set of first-order linear, constant coefficient
differential equations has been extensively discussed in
the literature during the past few years. Most researchers
have been concerned with establishing the salient mathemat-
ical features of the Optimal control, a.e., existence and
uniqueness, while only a handful have studied techniques for
finding the control. This thesis has developed and extended
techniques for determining the optimal control for that class
of systems which is completely controllable and which has
simple, real eigenvalues.

The introduction traces the history of the time
Optimal problem from its genesis in the relay controller up
through the rigorous analysis in a precise mathematical form.
SectiOn two contains a precise mathematical statement of the
time optimal problem. Pontryagin's maximum principle is
used to show the bang-bang nature of the optimal control,
and a theorem relating the number of switchings of each con-
trol component to the order of the system is stated,

The scalar control problem is discussed extensively,
and a transcendental set of equations in the switching times

which the optimal control must satisfy is developed-

56

57

Theorems bounding the optimal time are stated and proved.

The vector control problem is considered separately
since it is considerably more complex than the scalar case.
The equations which the vector control must satisfy contain
more unknowns than equations, and consequently the set has
infinitely many solutions. The problem is reformulated as
a minimization of the final control time, tn. subject to a
set of equality constraints and a set of inequality con-
straints. Simple extensions of the Lagrange multipliers
permit handling the inequalities, and a normal set which the
optimal vector satisfies is derived. Again, theorems bound-
ing the optimal control time are given.

In Appendix A a numerical technique for solving
nonlinear algebraic equations is developed. This procedure,
based upon some of Wirth's work [26], transforms the alge-
braic set into a differential set whose solution at one end-
point of the interval, [0, 1], is a root of the algebraic
set. Other methods applicable to the solution of nonlinear
equations are included in Appendix B.

Several examples of physical systems are analyzed.
The state models and the transcendental equations in the
switching times are developed. The results Of numerical
solutions carried out on a digital computer are listed in

tabular form for both the Scalar and the vector control.

APPENDIX A

NUMERICAL SOLUTION OF NONLINEAR

ALGEBRAIC EQUATIONS

The analysis of many engineering problems has been
hindered by systems of nonlinear algebraic equations. Very
little is known about the properties of the solutions of
such systems. Indeed, the very question of the existence of
solutions to such a set can be fully answered only for very
special subclasses such as polynomials.

Wirth gives sufficient conditions for the existence

 

of a unique solution to such a set and devised an algorithm

for obtaining the solution [26]. His theorem is stated
below.
Theorem

Let G(T,X) = 0 be an n—dimensional vector function

of the n-dimensional vector, T, and the r—dimensional para~
meter vector, X. For every X such that:

l. The entries of g; exist, are bounded for all T

and satisfy a Lipschitz condition on T for all T.

2.

det §£,: k > O for all T, k a constant.

T

 

 

Then there exists a unique T such that G(T,X) = O.

58

59

The hypotheses of this theorem are necessarily quite
restrictive since a unique solution is required. Thus, for
example, a single polynomial equation of degree two or higher
will not satisfy the hypotheses.

Therefore, while the theorem is extremely useful for
a narrow class of systems, it is not applicable to a broad
class of problems of engineering interest.

In practice, existence alone may be the significant
property of the system of equations, i.e., even though a
set may have many, even infinitely many, solutions one par-

ticular solution may provide an acceptable result.
Algebraic Systems

The theorem stated and proved below is an extension
of Wirth's work. It lists sufficient conditions for the
existence of a solution to a nonlinear equation set, G(T,X)
= O, in some compact region, R. Perhaps more significant
is the fact that an algorithm for obtaining a solution is

contained in the proof.

Theorem A

 

Given:

1. G(T,X) = 0, an n-dimensional vector function of
the n-dimensional vector, T, and the r-dimensional
vector, X.

2. A compact region, R, in the (n+r) Space defined

then:

Proof:

60

by:

IA
C)

“T ’ T0”

”X - XO” < Dl

For all TER, XER the entries of 3% exist, are

bounded and satisfy a Lipschitz condition with

reSpect to T.

det-figg ,3 k > O in R

 

 

 

 

_-l
_ C __ max BG(T X)
|G(TO,X)' < M where M - R ——:;;—-

 

 

and X is a particular X.5 R

There exists T 8.R such that G(T,X) = O

T = T(O)
where
— —l
dT _ 3G(T X) ,
“a? — [WT-2F] G(TO,X) T<l>=TO (A-l)

Consider T as a function of a scalar independent

variable t so that we have
G(T(t),X = 0

Define a function H(T(t),t) = G(T(t),X)-tG(TO,Y)

and differentiate with reSpect to t:

61

 

3% = BGgétLX) . .3; — G(TO,X) (A-2)
NOte that 3% exists and is nonsignular every-
where in R by hypothesis.

Now assume g? = O which implies that

 

8? 3T

dT :(3G(T(t),i)
3G ‘1 . . . .
Each entry in ST satisfies a Lipschitz condi-

t1 —.
I G(Toyx) (A‘3)

tion with reSpect to T. This holds because

gg
det 3T

functions are again Lipschitz functions from

3 k and sums and products of Lipschitz

 

 

Lemma 2.

The right hand side of (A—3) also satisfies

éG—l

max fi G(TOE)

T

<

 

 

-1
3G _ C : _.
ST N “G(TO’X) < M ' M C (A 4)

 

 

 

 

 

 

from the hypothesis.

Thus the differential equation (A-3) satisfies
all the hypotheses of Lemma 1; therefore, it has
a unique solution, T(t), such that T(t) is in-
terior to R for t£[0,l] and T(l) = To“
Substituting (A—3) into (A-2) shows that

alo—

rtm
m
0

Integrating we have

62

O O

'/%% H(T(t),i,t)dt _ O=H(T(t),i,t)

 

Therefore

H(T(O),X,O)-H(T(1),i,i)

H
O

and
G(T(O),i)—G(T(1),i)+c(ro,x) a O (A—O)

8. Since the initial condition for the differential
equation (A-3) is chosen such that T(l) = To,
(A-O) above becomes
G(TCO),X) a O (A-7)
and T(O), the solution of (A-3) evaluated at

t = O, is a solution to the algebraic set,

G(T,X) = 0.

Mixed Algebraic and Differential Systems

 

In formulating nonlinear mathematical models of
physical systems using linear graph techniques, the final

representation is often of the form

x = F(X,Y.E(t)) x<o> = c (A-8)

G(X,Y) = O

63

where E(t) is a known function of time. If G(X,Y) = O
can be solved analytically for Y = H(X), the solution can

be substituted in the differential set to yield
x = F(X,H(X),E(t)) X(O) = C (A—9)

and this equation can then be solved for X(t). Hence the
statement that the system has a unique solution. Wirth, in
the theorem previously mentioned, stated conditions suffi-
cient for the existence of a unique solution to G(X,Y) = O,
and, therefore, also to the whole system (A-8). However,
this result includes a rather limited subclass of the total-
ity of functions, G(X,Y) = O. For example, the polynomial
g(x,y) = yZ-x2 = O has two well-behaved solutions yl(x) =
x, y2(x) = -x. It does not satisfy the hypotheses of Wirth's
theorem because it does not have a unique solution; neverthe—
less in this case arunhunique complete solution to (A-8) can
be obtained.

Therefore, it appears that in some cases sufficient
conditions for the existence of a complete solution to (A-8)

are desirable even though that solution is non-unique. The

following theorem is addressed to this problem.

Theorem B

 

Given:

1. x = F(X,Y,E(t)) X(O) = x0
(A—lO)
G(X,Y) = O

Then there

defined on

where

Proof

0 5 t,: t

64

F continuous in X, Y and E for X, Y £‘R defined

by “x—XO”,: D1, “Y-YO“ 5 c.

G satisfies all hypotheses of theorem A, and in
—1

addition[%$] satisfies a LiSpschitz condition

with reSpect to X as well as Y for all X, Y in R.

E(t) is piecewise continuous.

exists a continuous solution, X(t), to (A-lO)

satisfying X(O) = X

2 o

t = min [t ,t ]

2 1
is first t such that "X(tl) - x0“ - D1
is first t such that "Y(to) - Y0“ = C.

By Theorem A there exists YO such that G(YO,XO)

 

= O and Y0 is interior to R.
Y is determined as the solution of the differen—
tial equation

—1
dY = bG(Y X) = -
_§ T G(X,X) Y(l) X (A 11)

Q.

evaluated at S = O, Y(O). But since the right
hand side of (A-ll) satisfies a Lipschitz condi-
tion with reSpect to both X and Xsby Lemma 4

the solution vector Y(S,X) is a continuous

65

function of the parameter vector, X, so that

HY(O,XO)-Y(O,Xl)“ <5 if onlyHXl-XO”< 0’ (PL—12)

Thus applying a numerical solution technique to
x = F(X,Y,E)

gives

 

X1 = XO+F1(XO,YO,E) where ,Fl(xO,YO,Ejl<<{ (A—l3)

by taking a suitably small step size.
Using X1 and Y0 in (A-ll) a new solution Yl is

determined, which by (A-l2) satisfies

”Y1 - Yo

 

 

Repeating the above procedure a sequence
Y(XO),Y1(X1),---Yn(Xn),---

is generated where

Using this sequence of Yi's a numerical solution

<€for all i

 

 

Yi ‘ Yi-J

to (A—lO) is obtained over range

1

 

 

X. - X l < D
O —

Y. — Y
i o

 

 

 

 

66

Lemma 1
Given:
1. i = f(Z) 2(1) = c (A-l4)
2. A closed region, R, defined by “Z - C” : Cl

3. There exists C such that for all 2

3 Z in R:

1 ’ 2

"We.” : cue-21H

4. C2 = max “f(Z)H , C2 < Cl

Then there exists a unique solution, Z(t), for t 8 [0,1]

such that Z(t) E‘R and 2(1) = C.

Proof (Existence)
1. The following integral equation is completely

equivalent to the initial value problem (A-14):
l

Z(t) = c - J[f[2(t1)]dt1 (A-lS)
t

2. Define a sequence of functions <2n(t) as follows:

20 = C

I
' l
l
l

_ 1 1
Zn+1 — c - ‘/;[Zn<t )]dt
t

Note that we know about the existence of the

above integral only if ZnER for all n.

67

3. Assertion: Zn_E R for all n, all t 6 [0,1]
By induction
a) 20 = C 8.R

b) Assume Zn(t) E R

c) Then: 1

_ 1 1
Zn+1(t) — C - jinnu )]dt

t
l

”Zn'tl-C”: fl'f(2n)”dtl 5 C2(l—t) 5 c2 for t$[0,l]
t

Hence:

 

 

Zn+1-C”: C2 < Cl for t€[0,l], for all n (A-lO)

from hypothesis. Hence we restrict t, O f t j 1.
Note the strict inequality implies Zn is an
interior point of R.

4. Now consider:

1
z1 — 20 = — jf(ZO)dtl
t
1
“21 — ZONE. lif<zojldtl : C2(l-t)
t

l

”22 — 21“: fl'f(zl)-f(zo)”dtl
t

Uniqueness:

 

6.

68

And by hypothesis (Lipschitz condition):

 

1 1
z z < C 2 — 2 dtl < C C (1-tl)dtl
2 ' 1 _ 3 I 1 o” — 2 3
t t
C2C3(l-t)2
= 2

By induction for any n:

n n+1
< C C (l-t) (A-l7)

Z
— 2 3 (n+1)!

n+l-zn

 

 

 

 

Hence it follows that [Zn] converges uniformly

for any interval [0,t0], t j l, to a limit func-

0
tion,

1

Z(t) = C - f[Z(tl)]dt1 (A—18)
t

which satisfies the integral equation and hence

the differential equation (A-l4).

It remains to show that the solution, Z(t),
derived by successive approximations is, under

our hypotheses, the only solution of
2 = f(2) 2(1) = C

on the interval 0 j t j l.

69

Assume there exists a second solution, Y(t),

such that Y(l) = C. Since Y is continuous and

in R at time t = 1, it is in R for O 5 t1 5 t j 1.

Let t2 = max [0,tl], then we have for t2 5 t j l:
l

Y(t)-Z(t) = - [f(Y)—f(Z)] dtl

t

1 1
1 1
”Y(t)-Z(t)” _<_ C3 fllY—ledt < 6+ C3 flY-ledt (A—19)
t t

whereEis constant > 0

By Gronwall's Lemma [28]
”Y(t)-Z(t)” _<_ 2633‘“) (A-ZO)

Hence Y(t) = Z(t), t2 5 t j 1,

since E'is arbitrarily small.

If t2 = O the proof is complete; if t2 = t1 > O
we have a contradiction since tl was chosen so

that Y(tl) was on the boundary of R. But
Y(tl) = Z(tl) (A—Zl)

which implies Z(tl) is on the boundary of R also.
However by equation (A—lO) Z(tl) is an interior

point of R. Hence for 0 j t j l, Y(t) = Z(t).

70

Lemma 2
Given:

1. |f(x)—f(y)

IA

Kilx'yl

 

|g(X>-g<y) < KZIX-yl
2. x, y ER, some compact region

Then:

H

. |f(x)+g<x)-f<y>-g<y>| 5 K3lx-yl

 

2. |f(X)gCX)-f(y)g(y)| : K4IX-y'
l l .
3. 'm‘mstlx-ylif f(Z)_>_k>O, ZER
Proof
1. |f(X)+g(X)-f(y)-g(y)| :If(X)-f(y)|

+|g(x)'g(y)| 5 kllx'yl+ kzlx‘yl
hence:

If(x)+g(x)—f(y)-g(y) : k3'x-yl , k3 = k +k2

l

 

hence assertion l.

2. lf(x)g(x)-f(y)g(y)|

|f(x>g(x)—f(x)g(y)+f(x)g(y)

-f(y)g(y)| 5 f(X)g(X)—f(x)g(y)|+ -f(y)g(y)

 

 

+f(x>g(y)| : f(x)Hg<x)-g(y)|4g(y)Hf(x)-f(y)

9

But since f(x) and g(x) are continuous on a compact

region they ha

in R so that

 

71

ve maxima, M1 and M2, reSpectively

 

 

 

 

 

 

 

 

 

|f(x)g(x)-f(y)g(y)l : M1K2lx—yl+M2Kl'x-y]: K4lx-yl
lf<x>g(x)-f<y)g<y)| : K4IX—yl
hence assertion 2.
1 l _. 1 1
3- ----‘ f -f
f(x) f(y) f(x) f(y) [ (y) (X)]l
l ___;L_ < l | 1 K1 lx—yl
f(x) f(y) - f(x)|f(y)
l 1 K1
___... _ _ =K _
|f(X) f(y) 5 k2 ix VI SIX Vi
hence assertion 3.

 

Lemma 3 (Gronwall's Lemma)

Given:
1. u, V’: 0, Cl’ a positive constant
t
2. u 5 C1 + j’u v dtl (A-22)
0
Then: 1:
IV dtl
u 5 C1 e 0 (A-23)

Proof

72

From (1):

IA
H

 

 

: v (A—24)

Integrate over 0 to t:

t t

 

S
Cl+ .]u(Sl)v(Sl)dSl

O

5 d5
u( )V($) j ‘/;(S)dS (A-25)

note:
S
0
Thus:
S t t
log [C1 + fu(51)v(Sl)d51:| : fv(S)dS
o o o

f t \
t

C + Iu(S )v(S )dS
log< 1 ° 2; 1 1 )5 fv(S)dS (A-26)
l o

 

 

 

73

And:

t
t J[V(S)d5
0

Cl +Ofu(51)v(51)d5l : Cl e

3. Finally using the second hypothesis:
t j;(S)dS
O
u :ICl + .]u(S)v(S)dS : C1 e~
O

t

jv(S)dS

O

u : Cle

Lemma 4
Given:

1. i = F(X,-() X(O) = x (A—27)

O
where X is an n-vector, ‘( an r-vector
2. P(X,o() satisfies a Lipschitz condition with
reSpect to both X and °(in some compact (n+r)
-Space, R.

Then:

the solution, X(t,°(), given by:

t
X(t.°<) = XO + fP[X(t1),°(]dt1

O

is a continuous function of the vector

parameter, 6( .

Proof

74

 

 

 

 

 

 

 

 

 

 

Define:
s -‘ I‘ ‘
F’fi Fxl fig”
X x'n then: in fn(Y)
X: °< = a1 X: 0 = O =G(X) (A-28)
l
I
' O O
a
_ Ii _ 1 _ J
where:
rxl(0)
Fa}; : x
l — ' O
°<= I X(O) = Xn(0) =
. °<
-arl :31
1
_ar 1
Now we have the system:
XO
'1? = G(X) X(O) = a( (A—29)

where G(X) now satisfies a Lipschitz condition
with reSpect to X in R.
Solution to (2) is:
t
X(t) = X(o) + fGGDdtl (A-30)
o

75

Consider two solutions at time t, Xl(t) and

X2(t), with different initial conditions

X2(t)-Xl(t) = 562(0)-R'1(0)

t
+ j’[G(X'2(t))-G(Xl(t))Jdtl
O

t
nun-w: : IIY2<0>-i1<o>ll+ [Mean-«ralldtl
0

And by Lipschitz condition:

t

.Kf

0

 

 

“Y2(t)-Xl(t)” 5 X2(O)-Xl(0)

 

 

 

 

- - 1
X2(t)-X1(t4ldt

Now by Lemma 3, Gronwall's Lemma:

 

 

 

”X2(t)-'X'l(t)”_<_“X2(O)-X1(O) eKt
1f ”X2(O)-X1(O)” < J (A-3l)
Then:
_ Kt
IIX2(t)-Xl(t) <Je = E (A—32)

 

 

Hence

”gen-x1... < J (A-..)

 

|<£if only “22(0)-21<0).

 

and the conclusion that X(t) is a continuous func-

tion of its initial conditions.

76

In particular if Xl(O) = X2(O) then
x2(o) - x1(o) =o(2 -o(1 (A-34)
From the definition of X in (A—28)
“X2(t, 0(2)—Xl(t, Kl)“ :"x2(t)-X1(t)” (A-35)
Using (A-33), (A—34), (A-35)

“X2(t, °<2)—Xl(t, e(1) <J(A—36)

 

<6 if only ”0(2— °<1

 

 

 

But (A—36) is precisely the statement that X(t,°()

is a continuous function of the parameter,fi’.

APPENDIX B

NUMERICAL METHODS

This section briefly describes two of the standard
techniques, Newton's method and the gradient method, which
may be employed to generate numerical solutions to any set
of n equations in n unknowns. In particular, these algo-
rithms are applicable to the time optimal control equations

of Chapter III, (3-14), and Chapter IV, (4-27).

Newton's Method

 

The n equations

fi(yl,y2,---,yn) = O i = l,2,--—,n
in the n unknowns yl,y2,---,yn can be written in vector
form as
F(Y) = 0 (B-1)
If each of the functions, fi(yi) = O, can be expanded in a

Taylor series in n variables about some n-dimensional point,

Y1(O) Y2(O) --- yn(0), we haVe

77

f1<yl,y2,---,yn) f1(yl y2< >__ yn< ))
n
(O) (o)
+ (y _ y (0) 3fi<Y1 —-yn )
- j J BYj
i=1
n n
2
l (O) (0) a (O) (O)
+ 2 (YR-YR )(Yj—yj ) S;33§; fi(y1 ___yn )
k=1 321
+ —'_ i 2 1, 2, ---, n (B-2)

Discarding the second and higher order terms fi(Y) can be

approximated by

n
_ (O) (0) a (O)
fi(Y) - fi(Y )+' E (yj-yj ) 5?; fi(Y ) (B-3)
i=1
1 = l 2 --- n

Rewriting the system (B-3) in vector form we have

(0) (O)

F(Y) = F(Y ) + J(Y )(Y-Y(O)

) (B-4)

where J(Y) is the usual Jacobian matrix defined by

3fi<y>
J(Y) = (A -) A-- = ----
13 i .
J IBYJ
If the vector, Y(O), is an approximate solution to (B-l),

and if the matrix J(Y(O)) is nonsingular, then we would

79

expect the vector

(O)-J'1(Y(O) (0))

Y = Y )F(Y (B—S)

to be a more accurate approximate solution. This concept
leads naturally to the following algorithm by which, hope-
fully, we are able to generate a sequence of successively

Y(n)

closer approximations, , given by

Y(n) : Y(n-l)_J-1(Y(n-l))F(Y(n-l)) (B-O)

(k)

. -l .
prOVided that J (Y ) ex1sts for each k = O, l, 2, ~--.

This is Newton's Method for solution of the system, (B-l)

 

[311,[33].

The sequence Y(O),Y(l),___,y(n)

(O)

,--- beginning with
an arbitrary Y may not always converge to a solution of

(B-l). Sufficient conditions for the convergence of Newton‘s

 

Method to a solution were published by L. V. Kantorovich in

 

1937 [30],[3l]. His theorem gives conditions under which the

l
Y(O),Y( ),--—,Y(n),---, converges to a solution,

sequence,
without assuming the existence of a solution a priori. His

theorem is stated below.

Kantorovich's Theorem

 

Given:

1. The normal set, F(Y) = 0, where for Y = Y(O) the

-1(Y(O)

Jacobian inverse, J ), exists and satisfies

80

-l
I‘J (y(0))” : Bo’ BO a positive constant.

 

 

2. HJ-1(Y(O))F(Y(O)) : D D a positive constant,

0’ O

3. In the region defined by (B-7) below, F(Y) is
twice continuously differentiable with reSpect

to the components of Y and

n
32fi

———————-< K ' s 1, 2, --- n

j,k=l
4. The constants BO, DO, K satisfy
h = B D K < 1/2
0 OO '—

Then:

1. F(Y) = O has a solution Y which is located in

the region

 

 

”Y—Y 5 N(hO)DO — to DO. (B-7)
. . . (n) .
2. The succeSSive approx1mations, Y , defined by

Y(n) - Y(“'l)-J'1(Y(n’l))F(Y(n'l))

exist and converge to Y, and the Speed of con-

vergence may be estimated by the inequality

n-
””0“?” 5- 2011—4) (2%)” 1)DO°

81

Gradient Method

 

The gradient method (also called steepest descent)
. th
generates solutions to the n order normal set, F(Y) = O,

by minimizing the scalar function
C): FTMF (B-8)

where M is a positive definite matrix, usually but not neces-
sarily the unit matrix. Since M is positive definite Q takes
its absolute minimum if and only if F = 0. Thus every solu-
tion of F(Y) = O is an absolute minimum of §) , and every
absolute minimum of {? correSponds to a solution of F(Y) = O.
The technique used to find the minimum Of Qdepends
upon its geometry in hyperSpace [33],[34]. For any partic-

ular value of K

{)2 K (3-9)

is an n-dimensional surface in hyperSpace. For the two—
dimensional case a contour map such as Figure 1 may be found.
Here C is the point at which {)2 O, and the contour lines are

constant values of f}.

82

 

 

 

1%
Figure 8.1 Contour map of ()= constant.
We seek a method of progressing from a vector point
Y0 = (Ylo, YZO’ ---, Yno) to another vector point, Y1 = YO+kZ,

where k is a scalar and Z is an arbitrary vector, such that
()(YO+kZ) < {)(YO). (B-lO)
Thus we minimize the function
Y(k) = Q<v0+k2) (13-11)

of the single variable k by setting its derivative equal to

zero:
d _ T _
d£Y(k) — z §2Y(Yo+k2) - O (B-l2)

Since condition (B-lO) requires k # O we exclude this possi-

bility by further specifying that

T
2 C)Y(YO) # 0 (B-13)

83

where {)Y, commonly called the gradient, is the column vector

whose entries are

 

aQ
(2y. = (B—l4)
.1 ayi
Condition (B-13) states that the line in the 2 direction
through YO may not be orthogonal to the gradient at that

point, and hence not tangent to the surface
§7<YO) = Ko'

Similarly, (B-12) says the line in the Z direction through

Y is tangent to the surface

0
Q(Yl) 2 K1 K1 < K

at Y1 as shown in Figure B.l.
At Yl we choose a new direction, Z, and proceed as
before. In this way a monotonically decreasing sequence,
{2(Yi)’ is obtained which is bounded below by the minimum
of {3(Y) and therefore has a limit which is the minimum.
In the method of steepest descent the vector, 2, is

always chosen as the gradient

2 = Q (13-15)

since (7Yis the direction of most rapid variation of {2.
However, each step involves many calculations, and a variety

of simpler choices of 2 can be used.

84

Since solving (B-12) for the minimizing k may be
exceedingly difficult, an approximate solution which sim-
plifies the calculation is highly desirable. Consider the
Maclaurin series expansion of‘fkk), the function to be

minimized:

Y(t) = Y(O) +\ij’(0)k + 'éfO) k2 + (B—lO)

If we assume the second order approximation and minimize we

have:

'00 = '(O) + "(O)k = O
k : - Y(O)

\J-JII(O)

And from (B-12)

T
z f)y(YO)

(111(0)
3M0)

(B—l8)

T
Z J(YO)Z
where

32
J(YO) = (A..) = ——————-§?(Y)

l . .
J aylay, Y 2 Yo

Therefore, in the steepest descent method where
z =§)Y(YO)

an approximate solution for the scalar, k, at the nth step

of the process is

85

T
—()Y(Yn) §2Y(Yn)
k = (B-19)
n §?$(Yn)J(Yn)§?Y(Yn)

 

and the iterates are defined by

Yn+1 : Yn + knCZY(Yn)° (B-ZO)

The gradient method, by its very nature, always con-
verges, yet there are some difficulties. These stem from the
fact that the method converges to a stationary point of the
surface,§7. However, in general, the surface may have maxima,
minima or saddle points all of which are stationary points.
Therefore each "solution" obtained must be carefully examined
to see if it is a true minimum. When a machine is used for
solution this is readily accomplished by substituting the
"solution” in the set, F(Y) = O, to see if it checks.

In the case where J(Yi), i = l, 2, ---, is positive
definite we see from (B-19) that kn is always negative.
Therefore, the descent direction is that of the negative

gradient and the sequence (B—20) always converges to a min—

imum though it may be a relative minimum.

APPENDIX C

This section contains a brief description of the
digital computer programs used to obtain the numerical solu-
tions tabulated in the examples. A complete listing of the
programs is included after the program descriptions. All

programs are written in the Fortran language.

Program I-—Kurung

 

This program determines the unique time optimal con-

trol for the second order system

 

r- —
. —' " F '- r —
x1 P 0 x1 b1

: + U(t) (C’l)

 

 

 

 

 

 

 

It may be used exactly as listed provided a data deck, as
outlined below, is inserted at the end of the program.

The first data card contains a single integer N,
the number of initial conditions for which optimal solutions
are desired, located on the card by statement 12. The second

card contains the parameters P, Q, b and b2 in the format

1,
of statement 10. The N remaining data cards contain the

initial conditions and the associated approximate switching
times for which optimal solutions are desired. The initial
conditions are x10 and x20 while tl is the estimated switch-

ing time and t2 is the final control time.

86

87

The output consists of ten numbers: x10, x20, P, Q,

U’ 510’ 520’ t1’ t2’

is given by U while t1 and t2 are the switching time and the

ERROR. The value of U(t), t1 5 t 5 t2,

final control time, reSpectively. The ERROR is the norm of
the state vector at time t = t2. The approximate switching
times, S10 and 520, are included for reference.

This program uses the numerical solution technique
of Appendix A, and the differential set is solved by the

fourth-order Runge—Kutta method.

Program II——Optima

 

This program solves the optimization equations which

yield the time Optimal control for the fourth-order system

 

T-" P O O O—‘ 5 _ F1.“
X1 r1 X1
x O r O O x 1
2 = 2 2 + U(t) (c-2)
x3 0 0 r3 0 x3 1
_x4d _ O O O r4J __X4j _l_i

 

 

 

 

 

 

 

It may be used as tabulated with the addition of a data deck.
The first data card, in the format of statement 25,
contains a single integer N, the number of points for which
optimal solutions are desired. The second card contains the
four eigenvalues in order of increasing magnitude in the
format of statement 15. The N remaining cards each contain

the coordinates of an initial point and the approximate

88

switching times associated with that point in the order:
x10, x20, x30, x40, t1, t2, t3, t4 where O < tl < t2 < t3
< t4. Format statement 20 describes the field on the card.
The output consists of the initial conditions x10,
the value of U(t), O f t 5 t1, the switching

X2O’ X30’ X40’
times t1, t2, t3, t4, and the norm of the state vector at
time t = t4, E.

This program also uses the numerical technique of
Appendix A employing the fourth—order Runge-Kutta method for
solving the differential equations. The step size may be
changed by replacing the statement H = -0.005 by one which
reads the desired size. If this is done the 200 in the

statement, DO 75, must be replaced with an integer K such

such that HK = -l.O.

Program III--Inequa

 

This program determines the time optimal control

vector for the system

 

 

 

- - r—

x1 = P 0 x1 + bll b12 Ul(t) (C_3)
x2 0 Q x2 b21 b22 U2(t)

b _J ._

If a data deck is supplied it may be used as listed.
The first data card, in the format of statement 14,
contains an integer N which is the number of initial points

for which solutions are desired. Card two, in the format of

89

statement 12, contains the numbers P, Q, bll, b12, b21’ b22
in that order. The N remaining cards each contain a set of

numbers X10, X20, 811’ $21, $12, $22 in the format Of State-

ment 10. The numbers x10 and x20 are initial conditions
while the Sij are related to the switching times t1, t2 and t0
by
_ 2
t1 S11
_ 2
t2 ‘ S21
2 2 (C—4)
to ‘ S11 + S12
2 2

to 2 S21 + S22

where t1 and t2 are the switching times of the first and
second controls, respectively, and t0 is the final control
time. The initial choices of the Sij are made by assuming
initial values for the switching times and then solving
equations (C-4).

The output consists of U1 and U2, the values of the
controls U1(t) and U2(t) initially and the switching times

t1, t t . The norm of the state vector at t = t0 is given

2’ o
by E while F is a measure of the error in the solution of
the set correSponding to (4-27). A desired solution is one

in which both P and E approach zero. The quantity D is the

determinant of the Jacobian matrix which must be inverted

90

at each iteration. Its value is significant only if it
approaches zero, in which case the solution obtained is of
questionable value.

Newton's method is used to solve the minimization
equations in this program. In the event that the successive
iterates do not converge to the desired solution a message

to that effect is printed out.

91

Program Kurung

 

PROGRAM KURUNG
5 FORMATIIH1934HN B HEMESATH RUNGE KUTTA SOLUTION)
10 FORMAT(4(FIOOZ))
12 FORMAT(I3)
15 FORMAT(IHOQBHXIOQ7X93HXZOQ7X9IHP09X91HOO9X01HU09X93H51007X¢3H5209
17X02HT1OBXQZHTZOBXQSHERROR)
20 FORMAT(1H007(F1002)93(51508)3
PRINT 5
PRINT 15
READ 12 MI
READ IO P90081052
DO 25 I=19M1
35 READ 10 XIOOXZOOSIOOSEO
HB-OOO4
N=25
DO 25 18192
YI'SIO
Y22520
U:(-100)**I
Fl=((X10*P*U)/BI*I.0)+200*EXPF(-P*YlI-EXPF(“P*Y2)
F2=((X20*O*U)/BZ-loO)+2.0*EXPF(-Q*Y1)-EXPF(-Q*Y2)
DO 30 J=XQN
D=2oO*P*Q*EXPF(.P*YI-Q*Y2)*(EXPF((Q-P)*(Y2‘Yl))-loO)
P118¢O*EXPF(“O*Y2))/D
P12=(‘P*EXPF(-P*Y2))/0
P21=(200*O*EXPF(*O*Y1’)/D
P22=(-200*P*EXPF(-P*Yl))/D
GICPI!*F1+912*F2
628P21*Fl+pZZ*F2
PllsHiGI
RIE:H*GZ
T1=Y1+O.5*Rll
T28Y2+005*R12
D2200*P*Q*EXPF(-P*T1~O*T2)*(EXPF((G-P)*(T2-TI))*loO)
R21:H*(Q*EXPF(—Q*T2)*Fl‘PiEXPF(-P*T2)*F2)/D
R22:H*(2.O*Q*EXPF(-Q*T1)*F1‘200*P*EXPF(~P*T13*F21/D
T18Y1+005*R21
T2=Y2+O.55922
D=ZooiP*0*EXPF(~P*T1*G*T2)*(EXPF((Q-P)*(T2—T1))-loO)
R31=H*(Q*EXPF(-Q*T2)*Fl“P*EXPF(-P*T2)*F2)/D
R328H*(200*0*EXPF(-Q*TI)*F1—200*P*EXPF(~P*T1)*F2)/D
T18Y1+R31
722Y2+R32
08200*P*G*EXPF(-P*T1~O*T2)*(EXPF((Q-P)*(T2*T1))~loO)
R412H*(Q*EXPF(-Q*T2)*F1*P*EXPF(-P*T2)*F2)/D
R42=H*(200*O*EXPF(-Q*T1)*F1-2.O*P*EXPF(-P*T1)*F2)/D
YI=Y1+(RI1+200*R21+200*R31+R4l)/600
Y2=Y2+¢Rl2+200*R22+200*R32+R42)/600
IF(Y1*Y1+Y2*Y2-1000**500) 30945045

92

3O CONTINUE

45 CONTINUE
X1=(X10‘IBI/P)*U)*EXPF(P*Y2)+(2.0*EXPF(P*(Y2~YI))*loO)*(Bl/P)*U
X2=(X20“(82/O)*U)*EXPFIOiY2)+(2.0*EXPF(O*(Y2-Yl)3-100)*(82/03*U
ERROR=SQRTFIX1§XI+X2*X2)

25 PRINT 209 XIOOXZOOPQQQUOSIOOSZOOTIOTZQERROR
STOP
END
END

Progam Optima

 

PROGRAM OPTIMA

5 FORMATIlHloésHTIME OPTIMAL SOLUTION FOR FOURTH ORDER SYSTEM WITH 5
ICALAR CONTROL)

10 FORMAT(1H0‘3x93HX109SXOBHXZO95X03HX3005X03HXQ094XO1HU09XQ2HTIOIBXQ
12HT2913X02HT3913X02HT4913X01HE)

II F0RMAT‘1H0967HJACOBIAN DETERMINANT APPROACHING ZERO. TRY NEW APPRO
IXIMATE SOLUTION)

12 FORMATI1H0639HSOME SWITCHING TIME HAS BECOME NEGATIVE)

13 FORMAT(1H0033HPRODUCT OF R4 AND T4 IS TOO LARGE)

15 FORMATI4F1092)

17 FORMAT! IHOI4F8029F501 05E1508)

20 FORMATIBFIOOZ)

25 FORMAT(I3)
PRINT 5
PRINT 10
DIMENSION AI4§4IQR(4)OPI494)OTI4)OY(4)QF(4’96(4)9$(4)
H3-00005
N124
D3100
53000
J13!
READ 25 M1
READ 15 (R(K)0K=lo4)
DO 100 Ifllqu
READ 20 XIOOXZOOXBOGXAOQ (SCK)OK=194)
DO 100 J2102
U=(-100)**IJ+I)
G(I)=U*R(l)*XIO+100-2004EXPF(-RII)*S(l))+200*EXPF(-R(I)*S(2))
l-Z.O*EXPFI“RII)*$(3))+EXPF(-R(I)*SI4))
G(Z)=U*R(2)*XZO+I00*200*EXPFI~R(2)*S(I))+200*EXPE(*RI2)*S(23)
1-2.0*EXPF(-R(2)*S(3))+EXPF(-R(2)*S(4))
G(3)=U*R(3)*X30+loO-ZOO*EXPF(‘R(3)*S(I))+2.O*EXPF(-R(3)*S(2))
1‘290*EXPF(-RI3)*S(3))+EXPF(~R(3)*S(4))
6(4)=U*R(4)*X40+1oO~2¢O*EXPF(-R(4)*S(I))+2.0*EXPF(-R<4)*S(2))

30

35

37
38
40

41

45

50

55

65
70
50
67
68

69
72

75

42

93

1-2.0*EXPF(-R(4)*S(3))+EXPF(-R(4)*S(4))

DO 30 K8194

YIK)=S(K)

DO 75 L=I9ZOO

DO 35 K3194

TIK)3Y(K)

DO 70 M2194

DO 40 K3194

DO 40 Nfil94

IF(N-4) 37938938
AIKON)8(*IOO§*(N‘II)*200*R(K)*EXPF(~RIK)fTIN))

GO TO 40

AIKoN)=(-100**(N“I))*R(K)*EXPF(-R(K)*T¢N))

CONTINUE

CALL INVERT (AoNIODoJI)

IF(D*O*(lOoO**(*40))) 42942941

CONTINUE

DO 45 K3194

FIK)=090

DO 45 N3194

FIK)=F(K)+A(K9N)*GCN)

DO 50 K3194

PIKOM)3H*F(K)

IFIM-B) 55965965

DO 70 K3194

T(K)=Y(K)+P(K9M)/Zoo

GO TO 70

DO 70 N=IQ4

TIN)=Y(N)+p(N9M)

CONTINUE

DO 80 K8194
Y(K)=Y(K)+II.O/600)*IPIK9I)+2.O*P(K92)+ZOO*PIK93)+PIK94))
IFIYI1)) 43967967

IFIYI2)) 43968968

IFIYC3)) 43969969

IFIYI4)’ 43972972

IFI-RI4)*T(4)‘IOOOO) 75944944

CONTINUE
G(I)3EXPFIRII)*Y(4’)*(U*R(I)*XIO+IOO‘EOO*EXPFI‘R(1)*Y(1,’+200*EXPF
II-RII)*Y(2))-200§EXPF(*RII)*Y(3))+EXPF(-R(I)*Y(4)))
G(Z)=EXPF(R(2)*Y(4))*IU*R(2)*X20+I00-200*EXPF(*R(2)*YII))+2oO*EXPF
I(-RI2)*YI2))‘200*EXPFI‘R(2)*YI3I)+EXPFC“RI2)*YI4I))
G(3)=EXPF(R(3)*Y(4)’*IU*R(3)*X30+I00-200*EXPF(‘R(3I*YII))+200*EXPF
l{-R(3)*Y(2))‘2.0*EXPF(*R(3)*Y(3))+EXPF(~R(3)*Y(4)))
GI4)=Epr(RI4)*Y(4))*(U*RI4)*X40+IoO“2.0*EXPF(‘R(4)*YII))+2.0*EXPE
I(-RI4)*YI2))’EOO*EXPF(-RI4)*YI3))+EXPF(*R(4)*YI4)I)
EI=GII)*G(I)+G(2)*G(2)+GI3)*G(3)+G(4)*GI4)
E=SORTFIEI)

PRINT 17 X109X209X309X400U9‘YIN)9N2194)9E

GO TO 100

PRINT II

GO TO 100

94

43 PRINT I2
GO TO 100
44 pRINT I3
100 CONTINUE
STOP
END
SUBROUTINE INVERT (BoNoDEToJ)
THIS SUBROUTINE INVERTS AN ARBITRARY MATRIX BY THE GAUSS
ELIMINATION METHOD
DIMENSION BI494)9A(498)
IF DIVIDE CHECK 26926
26 IF ACCUMULATOR OVERFLOU 25925
25 DETIIoO
N2=N+N
LgN-I
DO 10 I=ION
DO 10 J=I9N
IO AIIoJ)=BII9J)
K=N+I
DO I1 I=IoN
DO I2 J=K9N2
I2 ACIOJ)=OoO
M=N+I
II AIIQM)81.O
DO 6 J=19L
M=J+I
IF(A(J9J)) I929!
2 DO 3 I=M9N
IF(A(I9J)) 49394
3 CONTINUE
24 DET=OoO
IE ACCUMULATOR OVERFLOW 28930
30 IF DIVIDE CHECK 28027
DO 5 K=I9N2
AIJOK)=A(J9K)+A(I9K)
DO 6 I3M9N
IFIAIIoJ))89698
XBAIIOJ)/A(J9J)
DO 9 K=J9N2
9 AIIoK)=A(I9K)‘X*A(J9K)
6 CONTINUE
IFIAINoN)) 23924923
23 DO 13 KX=19L
MzN-KX
J=M+1
DO 13 IthoM
I:M-IX+I
IF(A(I9J)) 15913915
15 XSAII9J)/A(J9J)
DO 14 K=I9N2
I4 A(I9K)=A(I9K)-X*AIJ9K)
13 CONTINUE

”019

cu

95

DO 16 1319N

X=A(Iol)

DET:DET*X

DO 16 J=19N2
I6 A(I9J)=A(19J)/X

DO 18 I819N

DO 18 J=19N

L=J+N
18 BIIoJ)=AII9L1

IF ACCUMULATOR OVERFLOW 28929
29 IF DIVIDE CHECK 28927
27 J81

RETURN
28 J=“I

RETURN

END

END

Program Inequa

 

PROGRAM INEOUA

5 FORMATI1H1947HN B HEMESATH LAGRANGE MULTIPLIER SOLUTION WITH9
123H INEQUALITY CONSTRAINTS)

IO FORMATI6IF10.2))

I2 FORMATI6F1296)

I4 FORMAT(I3)

15 FORMATI1H092F89294E159894E1093)

20 FORMATI 1H095X92HU1 96X92HU297X92HT1913X92HT2913X92HT0914X91HT912X9
11HZ9IOX91HF9IOX91HE9IOX9IHD)

35 FORMAT(IHO925HLOGF ARGUMENT IS NEGATIVE95X93HTO=E1598)
PRINT 5
PRINT 20
DIMENSION AC595)
N25
0:100
J31
READ 14 M1
READ 12 P909811981298219822
DO 50 I=I9M1
READ 10 X109X209T119T219TI29T22
DO 50 K3192
U13(*190)**K
DO 50 L8192
U2=I-I90)**L
M=O
SIIeTII

96

521=T21
5122T12
522:722 -
u=1.O/I-BII*U1«812*UZ)
3:521*U1+822*02
OI:P*X10+811*U1*(190*290*EXPF(‘P*511*511))
1+812*U2*(1.0-2.0*EXPF(~P*521*521))
Bl=4oO*Ul*p*BI1*EXPFK-P*Sl1*5111
82:4.O*u2*9*812*521*EXPF(—P*521*821)
83:4.0*U1*O*821*Sll*EXPF(—O*Sl1*511)
84:4.0*U2*Q*822*521*EXPF(~O*521*S21)
IFIDI*U) 40940930

30 Hl=(~loO/P)*(Bl/Dl)
H2=(41.0/P)*(82/Dl)
Gl:83+(0/P)*B*BI*U*((01*U)**(Q/P~1.0))
GZ=B4+(O/P)*B*82*U*((DI*U)**(Q/P~190))
Z:(*HI*Gl-H2*GZ)/(Gl*GI+G2*62)

25 DI=P*XIO+811*U1*(190-290*EXPF(-P*Sll*511))
1+812*U2*<loo-290*EXPF(~P*SZI*521))
B!=4.O*UI*P*BI1*EXPFt-P*SII*SII1
BE:4.0*U2*P*BI2*521*EXPF(-P*521*521)
83:490*U1*G*B21*SI1*EXPF(«Q*SII*SII)
84:4.0*U2*Q*822*521*EXPF{—O*SZX*521)
IF(01*U> 40940.32

32 H1=(*190/P)*(81/Dl)
H2=I~loO/P)*(Ba/Dl)
GI=BB+(Q/P)*8*BI*U*((DI*U)**(O/P—l.0))
GZ=BQ+(O/P)*B*82*U*((DIiU)**(O/P~190))
HllzteloO/P)*(IDI*(-2.0*P*SIl*Bl+Bl/Sll)-Bl*81)/(DI*D )3
H12=(*loO/P)*((-Bl*82)/(DI*DI))
H22:(-1.O/P)*((Dl*(”290*P*521*82+82/521)~82*82)/(DI*DI))
H21=H12
511=-2.0*o*511983+83/511+(G/P)*B*U*(t(DI*U)**(Q/P-I.O))*<-293*p*
1511*Bl+Bl/Sll)+(Q/P~1.0)*U*81*81*((DI*U)**(Q/P-290)))
612:(O/P)*8*U*(Bl*82*U*(0/P-1oO)*((Dl*U)**(O/P~290)))
621:612
622=~290*0*S21*84+84/521+(O/P)*8*U*(t(DI*U)**(O/P-190))*€«2.O*P*
1521*82+82/521)+(Q/P*1oO)*U*82*82*((01*U)**(Q/P*290)))
A(191)=H!1+Z*GII
AI192)=OOO
A(I93)=H12+Z*GIZ
AI19413090
AII95)=GI
A(291)=H21+Z*621
A(292)=O.O
A(293)=H22+Z*622
AI29413000
At295)=62
A(391)=Gl
AI39213000
AI393)=GZ
AI39QI=OOD

97

AI3QSI:OQO
A(491)=HI*290*SII
AI49213“290*512
A(493)=H2
A(494)=OOO
A(495)=0.0
AI59113HI
A1592)=O.O
AI593)=H2-290*S21
AI594)=-290*S22
AI59513090
FI=H1+Z*GI
F2=H2+Z*62
F3=O*X2O+821*UI*I190‘290*EXPFI-O*SII*SII))+822*U2*(190*290*
1EXPF(‘Q*S2I*S21))+B*((DI*U)**(O/P))
F4=(“1oO/P)*LOGFIDI*U)‘SII*SII“512*512
F5=€*loo/P)*LOGFCD1*U)”S21*SZI~522*S22
M=M+l
CALL INVERT (A9N9D9J)
SII:SII’AIIOII*FI’AII02)*F2“A(I93)*F3“A(I94I*F4“AII951*E5
5122512’A(291)*F1*A(292)*F2“A(293)*F3-A(294)*F4~A(295)*F5
$213521"A(391)*FI~A(392)*F2'A(393)*F3-A(394)*F4~A<395D*F5
822:522“A(491)*FI*A(492)*F2“A(493)*F3-A(494)*F4-A(495D*F5
Z=Z~Af591)*F1~A(592)*F2-A(593)*F3~A(594)*F4~A(595)*F5
T1=SII*SII
T22521*521
TO=TI+SI2*512
T=T2+522*522
IFC~Q*T0~4OOO) 55940940
55 IFIwO*T-40.0) 60940940
60 EleXPFIP*TO)*IXlO-IBI1*Ul/P)*(290*EXPF(*P*TI)“EXPFI-P5TO)-190)
l~{832*U2/P)*(290*EXPFI-P*T2)~EXPF(~P*TO)‘I90)I
E2=EXPF(0*TO)*(X20-(821*U1/Q)*I290*EXPF(“O*TI)-EXPE(-Q*TO)~I9O)
1~(822*U2/O)*I290*EXPF(*Q*T2)~EXPFI~O*TO)~I90))
E=SORTFIEI*EI+E2*E2)
F:F1*F1+F2*F2+F3*F3+F4*F4+F5*F5
PRINT 15 UI9U29TI9T29TO9T9Z9F9E9D
IFIMw30) 45945950
45 IF(F~0.000I) 50950925
40 PRINT 35
5O CONTINUE
STOP
END
SUBROUTINE INVERT (89N9DET9J)
THIS SUBROUTINE INVERTS AN ARBITRARY MATRIX BY THE GAOSS
ELIMINATION METHOD
DIMENSION BI595)9A(59IO)
IF DIVIDE CHECK 26926
26 IF ACCUMULATOR OVERFLOW 25925
25 DET=IOD
N2=N+N
L=N~I

IO

12

11

30

Hmp

CD

23

15

14
13

16

mm
‘1 ‘0

N
CO

98

DO 10 1:19N

DO 10 J=19N
AIIOJI=BII9JI

KzN+l

DO 11 I=laN

DD 12 J=K9N2
AII9J)=OOO

M=N+I

A‘IOM13100

D0 6 J=19L

M2J+1

IFIAIJoJ)) 19291

CO 3 I=M9N

IF(A(19J)) 49394
CONTINUE

DET=OoO

IF ACCUMULATOR OVERFLOW 28930
IF DIVIDE CHECK 28927
DO 5 K=19N2
A(J9K)=A(J9K)+AII9K)
DO 6 I=M.N
IF(A(19J))89698
X=A(I.J)/A(J9J)

DO 9 K=JQN2
AII9K1=AII9K)“X*A(J9K)
CONTINUE

IFIAIN9N)) 23924923
DO 13 KXtI9L

M=NuKX

J=M+1

D0 13 IX:19M

InM—IX+1

IFIAIIin) 15913915
X:A(I9J)/A(J9J)

DO 14 K=I9N2
A(19K)=A(I9K)“X*A(J9K)
CONTINUE

DO 16 I=IoN

X=AII9I)

DET:DET*X

DO 16 J=I9N2
AII.J)=A(I.J)/x

DO 18 I=I9N

DO 18 J=19N

L=J+N

8(I9J)=A(19L)

IF ACCUMULATOR OVERFLOW 28929
IF DIVIDE CHECK 28927
J31

RETURN

J=*1

RETURN

END

END

10.

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Wirth, J. L. ”Time Domain Models of Physical Systems

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Bellman, R. Stability Theory of Differential Equations.

 

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Kantorovich, L. V. Approximate Methods of Higher
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Henrici, P. Discrete Variable Methods in Ordinary
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Zaguskin, V. L. Handbook of Numerical Methods for the
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r ,2. '
I‘d-am 1:33.". 01.1.1,

 

 

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