Computing Methods in Optimization Problems: Papers presented at the 2nd International Conference on Computing Methods in Optimization Problems, San Remo, Italy, September 9–13, 1968

199 Pages·1969·8.73 MB·English

by  D., G. Arienti, A. Colonelli Daneri, M. Auslender, E. J. Beltrami, L. F. Buchanan, A. R. Stubberud, Philippe A. Clavier, R. Cosaert, E. Gottzein, A. De Maio, G. Guardabassi, A. Locatelli, S. Rinaldi, Mark Enns, H. O. Fattorini, Jean Fave, F. Caroti Ghelli

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Lectu re Notes in
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Edited by M. BeckmannJ Providence and H. P. KOnzi, ZOrich
14
Computing Methods in
Optimization Problems
Papers presented at the 2nd International Conference
on Computing Methods in Optimization Problems,
San Remo, Italy, September 9-13, 1968
Springer-Verlag Berlin Heidelberg GmbH
ISBN 978-3-540-04637-0 ISBN 978-3-642-85974-8 (eBook)
DOI 10.1007/978-3-642-85974-8
AII rights reserved. No part of this book may be translated or reproduced in any form without written permission from
Springer Verlag. © by Springer-Verlag Berlin Heidelberg 1969
Originally published by Springer-Verlag Berlin Heide\berg New York in 1969
Library of Congress Catalog Card Number 78· 94162 Title No. 3763
PREFACE
This volume is based on papers presented at
the 2nd International Conference on Computing Methods in
Optimization Problems held in San Remo, Italy, September
9-13, 1968. The Conference was sponsored by the Society
of Industrial and Applied Mathematicians (SIAM), with the
cooperation of the University of California and the Univer-
sity of California and the University of Southern Califor-
nia. The Conference focussed on recent advances in com-
putational methods for optimization problems in diverse
areas including:
Computational Aspects of Optimal Control and Trajec-
tory Problems;
Computational Techniques in Mathematical Programming;
Computational Techniques in Optimization Problems in
Economics, Metero1ogy, Biomedicine and Related Areas;
Identification and Inverse Problems;
Computational Aspects of Decoding and Information
Retrieval Problems;
Pattern Recognition Problems.
The Organizing Committee of the Conference consisted
of:
A. V. Balakrishnan (U.S.A.) - Chairman
L. W. Neustadt (U.S.A.) - Co-Chairman
L. A. Zadeh (U.S.A.) - Co-Chairman
G. Debreu (U.S.A.)
E. Gilbert (U.S.A.)
H. Kelley (U.S.A.)
J. Rosen (U.S.A.)
J. Lions (France)
A. Ruberti (Italy)
A. Lepschy (Italy)
E. Biondi (Italy)
G. Marchuk (U.S.S.R.)
N. Moiseev (U.S.S.R.)
B. Pschenichniy (U.S.S.R.)
The Conference was hosted by the Consiglio Naziona1e
Delle Ricerche, Italy.
CONTENTS
G. Arienti and A. Colonelli Daneri: Computation
of the Switching Times in Optimal Control
Problems of Bang-Bang Type •••••••••••••••••••• 1
M. Auslender: Methodes d'optimisation dans la
theorie du controle •••••••••••••••••••.•••.•• • 9
E. J. Beltrami: A Comparison of Some Recent
Iterative Methods for the Numerical Solu-
tion of Nonlinear Programs •••••••..••••••••••• 20
L. F. Buchanan: Problems in Optimal Control of
Macroeconomic Systems ••••••••••••••••••••••••• 30
P. A. Clavier: Economic Optimization by Simula-
tion: The Confidence Level Approach ••.••••••••• 43
R. Cosaert and E. Gottzein: A Pro~ramme for Orbit
Determination Associated with Launching and
Station Keeping of 24 Hour Satellites ••••••••• 55
A. De Maio, G. Guardabassi, A. Locatelli and
S. Rinaldi: Optimal Manpower Training ••••••••• h8
M. Enns: Optimal Control of a Nuclear Reactor
Power Plant...... . . . . . . . . . . . . . . . . . . . . . • . . . . . .. 77
H. o. Fattorini: Control with Bounded Inputs ••••••• 92
J. Fave: Critere de convergence par approxima-
tion de l'optimum pour la methode du
~radient ••••.••..•.•••••.•••..••••••.••••••. •• 101
F. C. Ghelli: Statistical Optimization of Circuit
Design •......................•.....•.......... 114
D. H. Jacobson: New Algorithms for Determining
Optimal Control: A Differential Dynamic
Programming Approach ••.•••••••••••••••.••••••• 129
S. Kau and K. S. P. Kumar: Successive Linearization
and Nonlinear Filtering ••••.•••••••••••••••••• 133
H. J. Kelley and W. F. Denham: Modeling and Adjoints
for Continuous Systems ••••••••••••••••.••••••• 141
A. Miele: Variational Approach to the Gradient
Method: Theory and Numerical Experiments •••••• 143
R. Petrovic: Optimization of a Quasistochastic
Class of Multiperiod Investments •••••••••••••• 158
J. K. Skwirzynski: Optimisation of Electrical
Network Responses ••••••••••••••••••••••••••••• 167
R. G. Stefanek and P. V. Kokotovic: Obtaining Fuel-
Optimal Controls for Linear Time-Varying
Plants by Newton's Method ••••••••••••••••••••• 172
L. E. Weaver and D. G. Schultz: Reactor Control Via
State Variable Feedback ••••••••••••••••••••••• 180
- 1 -
COMPUTATION OF THE SWITCHING TIMES IN OPrIMAL
CONTROL PROBLEHS OF BANG-BANG TYPE
G.Arienti * and A.Colonelli Daneri **
* 29th rvrathematical Division, C.N .R., Nilano, Italy
**A.R.S., Milano and Pavia University, Pavia, Italy
1. INTRODUCTION
This paper describes a numerical iterative method for
computing the optimal control actions in linear or nonlinear
control systems with control appearing linearly (l),(l),(i).
The method we propose belongs to the class of direct methods
but also makes use of the knowledge of the general form of
the optimal control derived by the maximum principle (~).
He will consider the class of optimal control problems where:
1) the state equations and the performance functional J(u) are
linear in the control variables u(t), that is :
x = f(x,t)+B(x,t)u ; x(O) = x ; (1)
t 0
J(u) = f f {,(x,t)+< fo(x,t),u"}dt,
o
where 0 ~t~ T; tf..s; T; .x, ~o and f are n-vector functions
with components xi, x6, f~; u(t) and fo are m-vector func-
tions with components ui(t), fb; B is a nxm matrix function
with elements d~noted by b·j , while. is a.scalar function;
t
the functions f~(x,t), bij x,t), ~x,t), f~(x,t) are defin-
ed for x, Xo E. X (n-dimensional state space) and o~ t -< T.
All these functions are assumed to be continuous in the va-
riables x and t, and continuously differentiable with res-
pect to xi.
2) the set U of the admissible controls is the collection of
piecewise continuous functions of t which lie in the time
invariant unit hypercube in Rm
U ~ { u( t ); Illj (t) I < 1, j = 1,2, ••• , m } • (3)
The case of an arbitrary rectangular parallelepiped is al-
ways reducible to the unit hypercube without upsetting the
form of Eqs.(l) and (2).
3) the final conditions on the state variables are specified
- 2 -
as follows
where g is a convex n-vector function continuously diffe-
reni;;iable with respect to x and t for x€ X and o~ t ~T,
that is we require X(tf)E Stwhere ~is the target set
f
St~{x(t); gi(x(t),t) = 0, i = 1,2, ••• ,n}.
The time tf may be f~xed or unspecified; this includes the ca-
se where ~ :: 1 and f6:: 0 (time optimal pro blem (~)).
According to the maximum principle (l),(i), we can see that
an optimal control u * must satisfy the control law (of bang-
bang type) :
. n ..
u¥(t) = sign{ L: b .. (x,t)pJ.(t)_fJ.(x,t) }, i=l, ••• ,m, (5)
. 1 J.J 0
J=
where p(t)eRn is the adjoint vector.
In the remainder of the paper, the existence of a unique
optimal control with a finite number of switching points is
assumed and the singular case is disregarded.
Relation (5) gives us the qualitative behaviour of the opti-
mal control. To determine the optimal control using the maxi-
mum principle, it is necessary to solve the canonical system
which is a two-point boundary value problem containing the
signum function. The purpose of our method is to avoid the
difficult numerical integration of such systems. Taking into
account the control law (5) to compute the optimal control we
only need to find the number r an~ the values t~;j=~, ••• ,r,
of the switching times in which uJ. = + 1 jumps to uJ. = - I and
vice-versa, for i = l, ••• ,m. To accomplish this, the control
problem is reduced to a minimization of a function of the va-
riables t~ in finite dimensional spaces. Such a method of so-
lution is direct and eliminates the need of solving the two-
point boundary value problem obtained by the application of
the Pontryagin maximum principle':'
2. THE SWITCHING TH'lES Cm'lPUTATION
The penalty method (2), (~), (1) is used in order to
take into account the constraint (4) on the final point x(tf).
The solution of the original problem is then approximated by
the solution of a modified problem in which the constraints
are eliminated by adding penalty terms to the functional J(u):
n .. 2
Io(u) = J(u)+ ~ ~ [~(x(tf),tf)J ( 6)
J.=l
- 3 -
where Ki are constants for a fixed °a nd Ki .... co as ° .... co.
It can ~e shown that the solutions o£ the' Rew problem approxi-
mate as closely as desired the infimum in the original control
problem for sufficiently large K •
It is easily verified that theOperformance functional (6) is
linear in the control variable.
The case of a single control function u(t)eRI will be trea-
ted here (the multicontrol case, u(t)eRm, may be treated in
the same way). Now let us consider the case of the fixed time
problem and .,let u( t) be a bang-bang control defined as follows:
?+l 1 2
u(t) = L 1/I.(t) = u(r;t ,t , ••• , trr), (7)
. I J r r
J=
where 1/I j are step functions :
0 if t rf [t;-l, t; ]
1/1 • (t) = +1 if t E [t j - l t j ] and j odd (8)
J r ' r
-1 if t E [t j - l t j ] and j even
r ' r
and the switching times t~, j=l, ••• ,r, satisfy the constraints
012 r r+l
o = t ~ t ~t ~ ••••.• ~ t ~t
r r r r r
Another definition of the functions 1/1. can be used by inter-
changing odd with even in (8); this later definition will be
called "alternative definition" of (8) and vice-versa.
Let Rr be the Euclidean space of coordinates t~, j=l, ••• ,r,
and let TrC Rr be the set of points tr satisfying (9). If a
bang-bang control u( t) is assigned for 0", t< t f , then x( t) and
g(x(t),t) are completely determined as functions of the varia-
bles t~ and we can put :
(10)
where IQ(r~tr) is a function defined for integer r> 0 and
t~Tr. ~urthermore Io(r,tr ) is continuous in the variables
t¥, j=l, ••• , r. ~
For a fixed r the function Io(r,tr ) has a minimum ~(r) be-
cause I (r1tr ) is restricted to be a £ontinuous mapping on
TrCRr ...o. R • The sequence of minima {I (r)} , r=1,2, ••• is
monotonely non increasing as r increas~s, since one can regard
every tr-l, which is admissible for the (r-l)th minimum pro-
blem, as an admissible value for the rth minimum problem. For,
if a bang-bang control can be obtained by (7) with (r-l) and
tr-l, the same control can be obtained with rand tr having
- 4 -
the first (r-1) oomponents ooinoident with the oomponents of
t r -1 and the last oomponent equal to tf; that is the set of
admissible values for the rth ~rob1em is larger than that for
the (r-1)th problem and henoe Io(r) .~;f (r-1). Furthermore
ihere is a minimum non negative intege~ r..1 suoh that Io(r) =
IQ(r~) for r "t-r~. For, if the solution uO\.t) of the problem
w1th penalty terms has s switohing times t~ T~, ••• 'T~ with
u;C t) =+1 f2I;' 0 ~ 't" T~ and we use ~he ~efini tion (8), then
ro = s and t~iF TJ, ~=1, ••• ,s,,.."l~i1e l:f ~i!(t)= -1 for o~ ~
T~ , then ~ s+l, t r *= 0 and t~* =T~, J=l, ••• ,s. In a simi-
lar way it is easy to obtain the relation between ~ and s
when we use the alternative definition of (8).
The algorithm we propose oan be sketohed in the following
way:
a) se1eot the definition (8) or its alternative and choose
the guess va1~es for r, 0 and tr;
bo) find a point trE Tr whioh minimizes !o(r t r );
) inorease r and repeat from b) until Io(r) reaohes its mini-
mum value. A few words are required hete about the determi-
nation of ~ and o! the oorresponding tlO*' • Use the last
oa1ou1ated veotor tr to define a veotor t r , w!thout coinci-
dent components as follows: if in the vector tr there are h
oollections Ci , i=l, ••• ,h of coincident components, then
the oomponents of the veotor t r , are formed by retaining
only one element of every oollection Ci sqntaining an odd
number of elements and all components t~!._u h Ci •
, l.-I, ••• , ,
Now, if t r
r , 1= t f , then r*0 = r'; on the", " contrary if tr ,=tf ,
then r~ =r'-l. In both cases we have t~!= t~"j=l, ••• ,rt •
d) Put q = r* ; update 0 and repeat from b) starting with r=q
until theOfina1 oondition (4) is satisfied within the re-
quired acouraoy. At the beginning of step d) it is useful
to change the selected definition of theN ~j functions with
their alternative definition if we have t;* = 0; then use
q = r* -1. 0
o
The above algorithm can be easily extended to the class of
time optimal control problems with state equation (1) and
terminal condition (4). In this case the final time is unspe-
cified and oonsequent1y it becomes a further variable t~+l=tf
in 10 • Finally it is necessary to stress the importance of
the ohooce of t numerical method for finding the values of the
r parameters tr , t~, ••• , t r satisfying (9), which minimi ze
Io(r,tr ). Our computationa1r experince has suggested the minimi-
zation algorithm which make use of the penalty method for the
constraints (9) and the Powell method (8) for the minimization
of unconstrained functions. This latter-algorithm does not
require the evaluation of the derivatives which is frequently
laborious or practically impossible.