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OPTIMIZATION OF DISCRETE STATE LINEAR

IMULTHUUHABLEEWSEDMS

1962

MICHIGAN STATE UNIVERSITY

TJFS’I'S

0-169

This is to certify that the

thesis entitled

OPTIMIZATION OF DISCRETE STATE LINEAR

MULTIVARIABLE SYSTEMS

presented by

Chan Kyu Kim

has been accepted towards fulfillment
'of the requirements for

Ph. D degree in Electrical Engineering

 

 

Date November 20, 1962

LIBRARY

Michigan State
University

 

 

 

OPTIMIZATION OE DISCRETE STATE LINEAR
HULTIVARIABLE SYSTEMS

03

Chan Kyu Kin

E
1...

ABSTRACT OF A THISIS

fiubnittcd to the
School of Advanced Graduate Studies of
ichi; n State UniVJrfiity
in Dartlal fulfillment of the chuircncntc
fer the degree of

DOCT 3 OF PHILOSOPHY

.

ucpcrtncnt of ElectriCil Engineerin

(T
u.‘)

Afierovcd

 

OPTIMIZATION OF DISCRETE STATE LIInAR MULTIVARIABLE
SYS TBIJIS

by Chan Kyu Kim

The object of this dissertation is to study
techniques for Optimizing the response characteristics
of multivariable systems with respect to input signals
having discrete levels.

To characterize the system behavior, a normal-form,
ordinary differential equation model is assumed to be
given. The output signals obtains as a solution to
this model is compared to the desired or preferred
output signal to produce an error functions. The
objective of the Optimization procedure is to minimize
this error functions in the mean-square sense.

In Section II, a discrete state system model is
deveIOped from the given normal-form model for discrete-
level pulses of T second durations. From this model
minimalization for the error is realized by differen-
tiating the mean-squared error function with respect
to the (nxm) discrete level input pulses, where n
represents the number of inputs and m the number of

intervals or stages. In Section III, the problem is

Abstract Chan Kyu Kim

1".)

viewed as an m stage process and the dynamic pregramming

technique introduced by Bellman is used as a mean of

Optimization. I In Section IV, the discrete-state

control for the multivariable systems is analyzed.

 

OPTIMIZATIOH OF DISCRETE STATE 1132i?
MULTIVARIABLE S STEMS
by
Chan Kyu Kim

A THESIS

Submitted to the
School of Advanced Graduate Studies of
Michigan State UnIVersity
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1962

PLEASE NOTE: Thesis is not an original
coyy. Indistinct type on several pages.
Filmed as received.

UNIVERSITY MICROFILMS, INC.

 

CTZJSSYT.
(,hli‘bl

ACKNOWLEDGEMENT

The author wishes to eXpress his sincere
gratitude to Dr. H. E. Koenig for his guidance in
preparing this thesis. Much gratitude is also due
to Dr. L. W. Von Tersch, Dr. M. B. Reed, Dr. G. P.

Weeg and Dr. Y. Tokad for their encouragements.

*-3€-%i$****ié*%****%

- 11 -

TABLE O? CONTENT

pages

80013101.“.
1

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-, Hodfls of Yul . _ .
A

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.11: ...Q...0......00.0............... 2‘)
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J'ri'"‘:'-;ji‘flr..............0.9.00.0..O..... (L3
.0.. :2

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.............

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-117“-

 

OPTIn ZATION or DISCRB 3 STATE LINEAR MULTIVARIABLE

SYSTEI..

I. Introduction.

Control of the multivariable systers through the
use of the system configuration shown in Fig. 1.1 Ias
9
been discussed by Kavanaugh. Compensation of the
main plant G is attempted through diagonalisation

of the transfer matrix to ramove the interaction

between the system input and output variables.

 

Fig. 1.1 Synthesis of the multivariable
system by compensating networks
Horowitz:3 presents a method of synthesis based on
the frequency response characteristics of the
multivariable feedback system. In this method, the
desired transfer matr‘x is achieved without diago-

nalization.

The stability problem in multivariable system
is discussed in general by Bohn4 with a discussion
on the application of root locus plots in adjusting
the parameters of compensating networks to achieve
stability.

snares discusses control or svnthesis problems
in multivariable system with the non-deterministic
signals along with the problem of noise reduction.

A compensation network (or filter) is synthesized
using the criterion of least-square. Similar system

6 for the

was synthesized by Reich and Leondes
multivariable control system in terms of correlation
function between signals and noise in Wiener sense.

7 analysed the system in the inter-

Mesarovic
actions of the input and output variables in binary
form and established the existence and uniqueness
of the optimum system along with a synthesis
procedure.

The objective of this thesis is to show a
procedure for establishing the Optimum values of
discrete-level input signals to a multivariable
system using the mean-square error of the output

signals as a basis. In this develOpment a mathematical
model of the system in the form Of a set of first-order

\JJ

°. - n L' 3‘- ”Urn.
linear countless l; uncrnu'.

-, J-n..‘ .3. "-0-" ..- an". .\ “halved-
L- 'i.)‘.':/~ ‘. {10.IVI-LOKIJ “bio. C(Jm-.L)JL(/ilu

U

“.5 1 ‘ 'I 4‘. _' ‘I w __o ‘_"_ M‘
LuiCA sucn a nOdCl oils U an'
O
' . . ,.,‘ '1 ~. v.1 ,, .‘ k) (‘1‘ r‘ ‘J.
tinge” seu.:¢, ‘liud, .Ll; Opti
formulated as a; n-stafe docis
too n4 e; (m (infifillc progra

;nnin3 in

chariots-my tics under
has a solution is

Fixation problem is

ion problem and uses the

solution.

ll. Discrete-State Models of Multivariable Control

Systems.

Let the mathematical model of a linear system
be written in the form of set of ordinary differen-

tial equation in normal form

F s P s

 

cm F, (Ht). cm. R t). ids)
d _._._ (2.1)
* .
at 1 V(t) F2 ( V(t), C(t), R(t), sit)
h- L.

 

 

 

where C(t)r is a vector of cross and through variables
representing the output signals and Bit) is a vector
representing the input signals. The vector V(t)
represents a set of variables internal to the system.
The derivative vector in Eq. 2.1 is sometimes referred
to as the state vector since the magnitude and the
derivative of this vector uniquely describes the system
at any time t. In general, the vector functions F1 and

F2 may be nonlinear. However, throughout this thesis

‘ _.__n_

# Capital letters, such as C(t), are used for the
notation of vector, small case letters, such as
r(t), are used for the sea er, and capital letter

t-rith bar, such as mt), are used for the matrix.

- 5 -

only linear functions are considered with Eq. 2.1

 

 

 

written as
P -'1 P- _ ‘1 "" .1 :- - T P’
d C(t) A11 A12 C(t) B11 B12 R(t)
—-——- = - - + - __ .‘. (2.2)
dt V(t) A21 A22 V(t) 321 322 ntt)
d h- ..d L ... h- “ '4

 

 

 

 

 

 

 

where the submatrices Xi: and E13 contain real,
constant coefficients and the initial conditions

on the state vector are given by

0(0)
V(0)

Let P(t) be a vector representing the preferred
or desired output signals with the same dimension
as C(t) and let the criterion of performance be
taken as the integral

2

cf q (P(t) - C(t)) dt

t

t2

1

H

r (yct). 0(t)) dt (2.3)

where q and f are scalar functions of the vectors
P(t) and C(t). For example, q might represent the

square of the error

n
_ . 2
q [as - C(12)] .. X [we eke) ]
k=1
The basic problem is to determine the vector Kit)
such that
132
0’ = r [C(t), P(t)] dt
t

is minimum.

*
The input vector R(t) is considered as a set

of dis rete levels, as shown in Fig. 2.1
R t

A

 

 

 

 

 

 

 

 

 

 

 

 

 

‘V

Fig. 2.1 Actuating input signals for the
system input

- 7 -

Such an input signal might, for example, be obtained
as the output of a digital controller. For continuous
input signals, a discrete level approximation of the
above form is assumed. In general, a finite interval
(0, T) is divided into uniformly Spaced sections as
indicated with (5 = T/m representing the width of
each discrete-level interval. The magnitude of the
n-inputs over interval 3 is represented by the vector
12:16 ). The set of numbers representing the n input
vectors over the entire interval (0, T) is represented
by the nxm rectangular matrix.

5'.

ii" = [32%) 3’56 ) . . . RE:.(m-1)é )] (2.4)

The output signals, over the j~th interval, in
general, are a function of time as well as the magnitudes

of the input signals and are represented by the time-

varying vector 0(3 (5 + T ) where
0 4 T g 6 :: T/m

The set of m time-varying vectors representing the
output signals over the m discrete time intervals

is represented by the n x m rectangular matrix

O

- K.) -

5(7’) = [C(T) C((S+T) . . . C((m-1)5+T)] (2.5)
To establish the relationship between the output

or r spouse vector C(3(5+/T) cnd.the input or reference

vector Hal.) 6), cor:-:ider tic e-lon'LiE t."!.“-.SfOI‘ relation

obt lied by taking the Laplace transform of Eq. 2.2.

 

. =- _. 1 _*
’J(f‘) 1 “L1? 0(8) 1.11 D12 .2 (C) 0(0)]
8 —- _ - + _ _ 0.2:, + .
V(s) a, p? 7(s 3.1 so” u (s) v(o)
nol .1 For C(e gives hc form

)
0(3) as) 12*(3) + 5(3) c'(o) 2.6)

TfllTi“

and

( C(O)
c'(o> .. 3*(0)

1. v<o) J

 

 

For the first interval, 1. 0., 0 -¢ t :2 (5

'2. fl»
R (t) : R (O) : constant

Therefore

0(a) == 12(0) + {1(8) Me) (2.7)

 

cw) :2 63(7) R*<o> + mr) c’(o> (2.8)

T-rhere O '3 7" g 6

For the second interval, consider the response
C(t) to a step function of magnitude R (0) applied at

t :o- O, i. 0.,

C(t) at) 12*(0) + mt) c’(o) (2.9)

I!

To this response add the reSponse to a step function
e
of magnitude - R (O) a_plied at t :(5 and a step
1‘ O 1 fl 0
function. or magnitude R ((5) applied also at t z: 6 .

Using the shifting theorem, the result is
- '3’: Ills-r '3' *
C(65-z-T) = G((5+T) R (O) 4- GU) [11(6) - R (0)]
- I
+H((S+T) C(O)

where O < T g (5

In a similar manner, it can be shown that the
response for interval (3 + 1) is given by
. _ e:- 7 .3. fi- '
c(;c§+7’) :..- G(J(5+T) R (0) + -r((J-1)5+T)[R ((5) - R (0)]

+ . . . +‘c:(r)[n*(jd) - a““(<3-1)6>]

.. / .
+Hu6+rwcm> (am)
tfiiere
5(36+T) : O for 0’7
fhase resultina solutions can be shosn effectively

in matrix form as‘

 

 

 

 

 

 

-c( T) q FEE-'(T) . o — —R(O)
0.((mr1)(5+ T) Late-1 )6 e7") 5(7)! Elihu-1) (5:
L J 73(7) 1 [010)]
: (2.11)
+
L__:E-I"((I:1---1)6+T)__)

 

 

2u6)=¥ed)-$urud>

Using the n r m matrices defined in

2.5, the above model can be written as

[C(T) . . . C((n-1)6+T)J

1 r-

: [3(0) . . . R((:n-1)CS)J C—(T)

 

O

 

L.

4- 0(0) [5(T) . . . H((m-—1)6+ T)]

Or

6UiT>=md)m7)+nmiun

where

Te. 2.4 and Eq.

5(7)
(L012)
(2.13)

§(T) : [H(T) . . . H (m-1)()+T)]

Eq.2.11 and Eq. 2.12 represent the system models

to be used as a basis of Optimizing the input

vector.

To establish a measure of the error let the

preferred or desired response vector P(t) be

approximated by a set of m vector functions of T'and

 

 

6: 10 0.,
3375.7’) =[P(T) . . . P((m-1)6+T)] (2,14)

where O 4: T 3 6
In general, 5(t) may or may not be continuous at the
points t z j (5. In a similar manner, let the error

function defined as
Mt) == P(t) - C(13) (2.15)

be represented by an n x m matrix of functions on the

m discrete intervals
7(697):§(62T) “ES-(627') (2016)

micro C((j, T) is related to EMS, T) by Eq. 2.12.
Thus, the error function is relate; He the discrete

levels of the input vector by

5(6,T) .2 1316.7) - 2nd) 516.7) ~§<5: “010)
(2.17)

In the following develOpment the error criteria

for each of the n signals, say the k-th, is taken as

T T

0'k = [ [ek(t)] 2m: :f [pk(t) - oxen] 4 dt (2.18)

O O

 

- 13 -

When the error functions ek(t), where k = 1, 2. o . . n,

for the n output signals are represented as a vector

231(5)”
62%)

 

 

en(tfl

b

the error criterion in Eq. 2.19 can be written as

T

T
d = [ [E(t)] . [ E(t)] (it (2.20)
0
Substituting
E(t)=E(T) Octc-(S
zE(6+T) 6 «st-s26
. . (2.21)

:: E((m-1)5+T) (J-Hé-zt .g 36

into Eq. 2.20 gives

 

- 14 -

m-1 ‘5
T d?
0’ =2] [2(36+T)] -[E(36+T)] dT (2.22)

3:0 0

where the 3(364- T) represents the (J+1)-th column
of the error matrix §( (5, T ). Substituting the
expression for 3(6, T) as given in Eq. 2.17 into

Eq. 2.20 gives
m 1 (5 n -1

0’= if Z{pk<36+r> -j Gkuén).
O O _

j: kz1 .0

F0

men-M6) - hkmSn) c130) M (2.23)“

Minimization of the integro-error function
(76?)

with respect to the n x m discrete input levels in

ff: [ R(O) . . . R((m-1)6)]

can be realized by taking the partial derivatives of

 

* See Appendix 3.1 for the derivation of Eq. 2.22.

** See Appendix B.2 for the derivation of Eq. 2.23.

- 15 -

as)
with reSpect to the coefficients in E. The result
is the set of n x m simultaneous linear algebraic equa-

tions

amt)

 

 

:: 0
51‘1”)
60’s?) __ O
arnw)

60’s?)
ar1((m-1)6)

 

amt")
arn< (rm-1 ) 6)

 

A solution of these n x m simultaneous equations

gives either a minimization, maximization or

-15..

the saddle point of the integro-error (751-).
The eXplicit form of the n x m simultaneous equations

represented in Eq. 2. 24 is

q .—
C’mR) = 2 pk( (35+T)
a r,(o) o k—

 

3-1
- Z GKWS + T)R((J-1-l)(5) - hk<36+r>cg<o)}

m-1

 

66“” = :2{pk(36+f)
arn((m-1) 6 ) 32.0 0 :1
3-1
- Z Gk(A (5+ T ) R((J-1-A ) C5) - hfliéfl’héW}
i=0
e (T) (11' =0 (2.25)

kn

- 17 -

Rearranging Eq. 2.25 in matrix form, we have a set
of n x m linear algebraic equation with n x m unknown,

namely,

(5 n
j Z RT) dT
O k=1

where 3(7) and B(T) are

(Sn

2] EMT) dT (2.26)

0 1:30

2”

 

 

 

 

 

m-I
ng1(7'): Gk(36+7') . . . 5k1(T) Gk(T) -
J=0
l"( T) = °
a 13-1 0 o
Skim" )6+ T )Z Gk<x5+r>. . amps-1 @me
L 3:0 _
(2.27)
m-1
"ek,(1’){>: [12k(36+r) - hk(;15+7’)c1;(0)]}1
3:0
1': . . - ‘
B(T) (2.28)
m-1
L_€;]m((m«-1)6+‘l"){§_:[1’k(36+7’) - hk(35+7') 012(0ii
3:0
‘I

See Appendix B.3 for the derivation.

 

~18-

The solution to Eq. 2.26

<5 n -1 n
i = j Z RT) (17' [6 2 MT) dT (2.29)
O k=O O k=O

gives either a maximum, minimum or saddle point.
To establish a minimum, differentiate Eq. 2.23 a
second time with respect to the entries of

WI

The result is

626d) M 6n . .2 '
' .-.-.. Z I 2 [sk1(T)] . (IT
0

ar1(0)2 3:0

 

 

 

2 __ m-1 n
a 6(R) Z [15: 2
2 = 2 g (7') dT
a rn(0) 3:0 0 k=1 [ kn ] (2.30)
2 m-1
6 6(3) :-.. Z [6211: 2 2
a I'1((m--1)(5)2 [gk1((m")6+7')] dT
3:0 0 k=1

m-1 n
82 6(5) 2
WE) gum-H6? 3 Z 21 2 [Skn((m-1)6+T)] ‘17

3:0 0

 

- 19 -

Since the integrands to the right of Eq. 2.30
are positive real functions, the solution to Eq.

2.29 represents a minimum.

Example; 1

Let the s-domain relation between the input and

output function for a system be given as

C(S) :

 

 

e
R (s) + .. + 0(0) (2.31)

C
.m

s + 1

and let

p1t) = t (2.32)
be the preferred response. Determine the magnitudes
of two discrete levels each of one second duration
that gives minimum integrc-error over the two
periods with the initial condition

0(0) 2 0.25 (2.33)
The output signal during the interval of (0, 1) is

-‘r -1'
0(7') = (1 - e ) r(0) + 6(0) (1 - e ) (2°34)

 

- 20 ~ '

For the interval of (1, 2) second, the output signal
is

-<6 7') -T
c(6+T) =(1-e + )r(0)+(1-e >145)

46.7)
+ 0(0) (1 - e ) (2.35)

The integro-error function of the two intervals are

e(T)=T-(1-e )r(O)-(1-e )C(O)

-( T) «-
e((S+T)=(C5+T)-(1-e 6* )r(O)-(1~eT)r(5)

- T
{(1 - e‘ (6+ )) cm) (2.36)

The system output functions for this multi-stage

process is shown in Fig. 2.1.

cm ‘

 

 

Fig. 2.1 System output variables as a function of time.

The error criteria 0’ [r(0), r(6 )] is
‘ 2

0' '7’ ' -T
z] {7-(1-e )r(o)-(1-e )0(0)}d7’

O

1 ~(6+T) '7'
{05,714. - . )r(o)-(1- e Md)
0

-((5+T) 2
.. (1 - e ) (40)} dT (2.37)

In order to determine the discrete input levels, take

the partial or 0' with respect to r(0) and r(1') to

obtain

1

-T -T . _
—g)-9(:5-)—= 2 (1 -' e ){T " U ' e )1‘(0)-(1-e 730(0)}{1’
r

O
1

_ ~<6 T)
+f2 (1... (5*7’){(6+r) - (1 - e + MO)

0

_ ~(6+T)
~(1-eT)r(6)-(1-e )c(o)}dT

 

 

 

 

1

*Ma 6 =[2 (1 - e-T) {(6+T) - (1 - e-(6+T)) NO)
and)

 

O
-(1- e-T)r((5)-(1- e-(6+T)) 0(0)}dT
(2.38)
or
r , ., . ~ r- 1 ..

 

 

 

 

 

r(0)
] 1(7) (:1 . [ RT) dT (2.39)
o J N6) 0

where UT) and B(T) are

 

"‘(1-e )+(1-e"‘5*T)) (1-e'-‘5*T’)<‘-9-W

 

 

 

 

(1-9 )(1 - e + ) (1 " e-T)2 _J
L. *
(2J0)
' (1-e"T>[ T-o(o)(1-e'T)] +(1~e- 6+”) [(6*T7"°(°m'°{ri
MT»: -‘r “‘6‘“
__ (1-9 )[(5+T) - c(O)(1-e )] * __1
(2.41)

 

* See Appendix 3.4

- 23 _

By taking the inverse of coefficient matrix of Eq. 2.39
we have as the Optimum values of r(O) and M6)

r10) 1 -1 1 1.109

=[ I(T) dT][ 3(1) aT] .-.-

r115) 0.982
0

Example‘ 2

Let the s-domain system equations-for a two input

and two output port system be given as

 

F- "1 r- 2 2 ‘1 P ‘1
I'
1 . s + 1 s + 1 r1( )

 

 

 

 

 

 

 

= 3 5 *
; __~.___ .__.___. r (s)
r ‘ 1 P 1
e +11 0 01(0)
* (2.42)
1
0 ------ c (O)
L. B * ‘4 .. 2 .4

 

 

 

 

with preferred functions and intitial condition

P‘(t) - -105
p2”) . _1.o

01(0) _ —o.2
02(0) ~ L“0.1

 

The entries of at) are

an): (1-e-t) [2
3

(2.43)

(2.44)

2
]
5

The "discrete functions as given in Eq. 2.11 are

”0‘17') q
02(7)
31(6 +T)

 

ll

 

L°2(‘6 4' TL

”211-64) "211%?“

30-67) 5(1-."T)

2(1-6.(6 + T?) . 2(1-‘9‘7'641

 

 

F1310) “ Fem) (1-67)
ram) 02(0) (1-e"T)
me) + mo) u-o"

-r‘.(6)_J __°2(0) ("94

 

 

O O
O 0

) 2(1-84) 2(1-e
_3(1-e'(6+T)) 5(1-e-( +7") 311-84.) 5(1'9“.TU

q

6.1), (2.46)
(541)”

 

The matrix of discrete error functions is,

therefore,

q

.T)

 

-25..

1(7) 1(7) [0:T)
E‘t) :: 1 2
115.11 H617) o(6+r)
F2 17') 1 2(1-eT 2(1~0 T, 0 01 .-
p (T) 3"(1-117 ) 511-11; ) o o
= p (617’) - 20-11 “6”") 20-11 m) 2(1-e‘7') 20-67)

 

 

Lei-(6.11") 30-1741547) 5(1-1"‘Sm)-3<1-e‘7) stt-e’T)

 

 

"rum-1 1rc110) (1-e‘T) .1
13(0) . «12(0) (1-1" T)

rfld) " c‘(0) (1-e'(6+T))
.1; (6) Law) (1-1." “5+T’)
; _ r

(2.47)

"H .

 

 

 

 

The integro-error function is

6:];{[e1(r)] 2 1 [e2<T)]2}
.f; {[e‘((§+T):} 2 +[e2(6+T)]2} M (2.48)

where 81(T). e2(T), e1(6+T) and c2(6+T) are
given in Eq. 2.47. Differentiating the integra-
error function shown in Eq. 2.48 with respect to the

entries of the vector it. gives

 

 

 

 

 

1 r.111”) I12”)- ill'1‘“)

dT r2“),
' r15)
0_ 5'12 (T) 122(T)J _12~;1'((5)

 

 

 

 

1 [23 (TH-BB 2(T)] (1-e-T)+[2B (6:T)+339 26,11] (1.9 3464:”) “

._., [231171632171] (1-. 734213,(6117')+5132(6+T)](1-1-1“5"“)
/ [231(6+T)+332 (6+T)] (1-6 7')
° [21.«6111.11:<51r1] (1.... T )

aT

 

 

bi: (2049) J

where 111(7’), I12 (Th: 21(T)9 122(T)o §1(T)9 32(T)c

31(611') and 32‘6”) are

13 19
- 2 - r
..I,,(T) =1 [11-11 T) 10-11“5+ ’)2][ J

19 29

13 19
I‘2(T)=I21(T)-..1'22(T)=(1.e ”1-8 (5+ ))

 

 

19 29

31171 = mm - 11(0) (1-1177)

3.11) = 112(1) - cam) (1-e'T)

B,(5+T) = p116 1ST) - 0,10) (1-e'(6+T))

 

- 27 -

-(6+T)

 

 

 

 

 

 

32(6+ T) 3132(6+T) "' 02(0) (1‘3 ) (2050)fl
The solution to Eq. 2.49 gives
"1' (0) " .. _. " --1
H! 1 A11(T) A12(T)
r (O)
: dT
r (6) "' (T) K (T)
O A 0’)
LI‘ (6)_ L 2‘ M- J
r' .. ' "
1 1[231(‘rh-3132C7') (1-e T)+ 231 (6911035 44") (he-(6+n)
r
j [231 (TM-5132 (7')] (1 ~0-T)+ [2131 (C547) +SB,)((S+T)] (1 -e"(6+T))
a . _%r - d7'
0 - [2316+T) 4- 3B2((5+T)] (1-0 )
[231(6+T> + 5132(5+T)] (1-67)
“.304 '
~1.211
= -0.482
L007}? 4

 

 

 

* 'See Appendix 3.5 for the derivation.

 

 

 

 

 

 

 

 

 

III. Control of Hultivorieblo System by Dynamic

Programming.

..-, — 4--.v -,- --§ o-fi'. -" 1 .- '.‘ t. r-\-\ ..
Ego ogoiiuo maglloudes of lupus LLUhClS own

fl '!

also be realized by a process re;

s
as dynamic bro: amning. To show 0135 proeeflure and

O
’3'
I-)

J
’D

to by Bellman

its apolieetion to the uypo 0
tion here, consider the solution $0 the svston model

GiVSIlle 11. 2?.11 enfl.1iréiorroz'cn1liorvga

To lininiuo Cf(fi)l'rith respect to E, let the

minimization of 3g. 3.1 be written:es
Ill
,1
2:21,; o’ (if) Min E(t) '“ [ 3m dt
'1‘? E

 

* Bellman, R., "Dynamic Pregramming," Priaeeton

UnLVorsltJ Press, 1957

svseo: unfier considere-

 

=1§3){L6[ []E(T) T[E(T)]dT}

m
+ Min , {f[(E2(t)2+. . . +Em(t)2]at}
3(6) . . .n((m-1)6

=§(i)n{[[é 111(k)2 at + (341%) £64; 32(t)2 dt + . . .
m6
at” J} (3.2)

«911111 E (1-,)2
:11
Applying the change of variable

RHm-UCS) .-
(m'-1)6
t =: (3-1) (3 + T

for the j-th interval, EQ. 3.2 becomes

6
Min OWE): Min{fE(T)2 aT
.. 3(0) ‘
R o
6 6 2
+(lgifné) I; 32(5+T)2 dT+ . +R.Mi|(.§-1)6)fEm ((m-1)6+T)d7)}

(3.3)

 

- 30 -

Note that the minimization at each stage is with
respect to the levels of the n input signals. The
basic procedure in dynamic programming_is to
minimize each stage, starting from the last and working
forward. At each stage the minimum is expressed as
an eXplicit function of the signal levels of the
previous,interval.

To effect the last minimizatian with respect
to R((m-1)6 ), let the response function

C((m—1)6+T)

as given in Eq. 2.11 (or Eq. 2.12) be written as

c<<m-1>c5+ T) = Qm_1(T)+5(T)R((m-1)6) (3.4).

where Qm-1(7') is

m~1 '
’ Qm-1(T)=z 5(36+T)R((j-1)6)+H((m-1)6+T)C(O) (3.5)
3:1

Note that Qm_1(7’) is independent of the signal

levels in the last stage. Assuming that Qm-1(1 ) will

be Optimized later, the Optimum values of the signals

for the last stage are obtained by differentiating
dm(§) with respect to R((m-1)6). The result is

a set of n linear simultaneous equation in the n

variables of the vector R.

 

- 31 -

Error functions in the last interval is therefore

3m((m-1)6+T)=P((m-1)(5+T)-5(T)R((m-1)6)-Qm_,(T) (3.5).

and for the last stage the error criteria is

6 n
drums] Zip ((m-1)6+T)-61(T)R((m-1)5)-Qm_,(7’)}2 (W

O

(3.7)

'Minimizing with rGSpeot to_the set of discrete-level
input functions gives

aO’m (if)
61‘ (Cm-16) =[6 : 2{p1((m-1)6+ T) "' 311T)R((m-1)(S)

-q(m-1)l(T)} 811(T) (1T 3 O

 

 

66,, (if) zfé n

arn ((m-1)§) 2{p1((m")(5+T>-Gi(T)R((m-1)(5)

0 1:1

-1-i(m_,)1<r)} any) a7: 0 (3.8)

 

-32..

Rearranging Eq. 3.8 in matrix form gives

(5 .
[[ Am(T) dT]R((m-1)(5)

0

where Xm(T), Bm(T) and Dm(T) are

--

G1(T) {311(7)

11 o
Xm(T) = Z . (3.10)

T)

r: I
Uin\ ‘

 

 

LG1(T)

1— 91((m-1)6+T) 811(7')

[V].

B (T)
m

1'“
fl.

 

 

L p1((m-1)5+T)s1n(T)J

 

- 33 -

r‘q(m_1)1(7, gum)“

12mm

[\v/]

(3.12)
1:1

 

 

bq(m-1 )11T) 51n1T)

.4

Taking the inverse of Am(7'), assuming that it exists,
the Optimum value of the input signals in the last
stage with the assumption of all signals up to

that last stage are Optimum, is

(5 -1
R((m-1)5) =[[ Am(T) d7]

0
5 6
. + _ ‘T
[L Bm(T) dT £13m [Qm,(T)]d J
= gbm [Qm_,(T)] (3.13)

Where gb [Q (7')] is referred to as a
m 111-1

functional since it is a function of a function Qm_1(T').

 

-34-

Consider next the second last stage. The

function to be minimized is

6
{ [Em_12((m-2)6+T) + E:((m-1)6+T)}d1'
’0

1113111336)
R m-1)6 )

[[62 T
2M1 -2

6
4- Min (5) [OE§((m-1)C5+T)}dT (3.14)

R( (In-1 )

But the minimum of the last term is an erplicit
function of Qm_1('r) and hence as a function of
R((m-?)(5), R((m-3)6), . . . , 3(0) and 0(0). This
minimum has already been established in Eq. 3.13.

To use this result in Eq. 3.14, let

Qm_,(T> = 9314(7) + 5(7) R((m-2)6) (3.15)
where
m-1
Qm 2(T) =2: ‘é’((a-1)6+T)R((m-3-1)6) + 0(0) (3.16)
i=2

Note that Qm_2(7') is a function only of the signal

 

 

- 35 -

levels up to the second last stage.

The error for the second last stage is

E _1 (1.1-2) 6n )=1>((m-2)6 + T)-E( T mum-2) 6 )-qm_2(r) (3.17)

where Q (7') has already been defined in Eq. 16.

The function to be minimized is therefore

(im-=[6:[p1((m-2)6+T)-Gi(T)R((m-2)6)

0 i=1

m-2 1

2
-Q (T) ] (N (3.18)

The minimum of the function is established by differen-
tiating with respect to R((m-2)6 ). The result is
a set of n linear algebraic equations in the n signal

levels for the second last stage and are of the form

-1

[féh_1(T) 111’] R((m-2) 6)
o
= f Am_1(T) dT *me-1 [Qm_2(T)]dT (3.18)
o

O

 

-35..

Again, assuming the invo so of A

Ti-

1e signal for the second last

exists, the

Optimum value of t

stage is

Repeating the above process gives a set of m

functionals of the form

Rum-H6) = (tn [Qm_1(T)]

R((m-:2)6) = (by, [ 0(7)]
. (3.20)

 

 

- 37 -

where now) = 21(T) c’(o).

The functions qb “(at each stage are determined
d

by the solution of n simultaneous equations to
establish the Optimum magnitude of the input signals
at that state as an explicit function of the

arbitrary function Q'j 1(7"). All Q3( T) except

Q0(7') are known only in terms of another arbitrary
function yet unknown. Consequently, the Optimum
magnitudes of the signals at each stage is not actual-

ly evaluated until.all.n1 qu have been established.

Once R(O) is him-1n, Q1 (T) and EH6) are established.
These, in turn, establish 11(26) and 12-22(7’), etc.
The application of this procedure is demonstrated

in the following two examples.

 

- 3a -

Examnle; 1

Let the s-domain relation between the input

and output function for a system be given as

 

 

0(8) = R (S) + 0(0) (3.21)
. 1

.12
+
m
+

p(t) = t (3.22)

be the preferred response. Determine the magnitudes
of two discrete levels each of one second duration
that gives minimum integro-error over the two

periods with the initial condition

0(0) = 0.25 (3.23)

The output signal during the interval of (O, 1) is
~ uT'
0(7) = (M: T) r(0) + 0(0) (1-e ) (3.24)

For the interval of (1, 2) second, the output

signal is ‘
c(6+T) =z (1-e-(6+T))r(0)+(1-e-T )r(6)
+o(0)(1-e'(6”7)) (3.25)

 

 

-39..

The integro-error function of the two intervals are

an: -<1-e‘7)r<o)-ccom-e“ >

f

e<6+r )=(6+1>-(1-e“6” ’)r(0)-<1-e‘7 )r(5)

 

 

 

-C(O)(1-9-(6+T)) (3.26)
or
”arm 7 " T ) "(M-2'7) 0 ’m) "
Le(6+T) a L6+T ‘ L-(1.,__‘.3"(64-7fl)) (he-T) brusu
Pc(0)(1-e-T) W
" (3.27)
Lc(o)(1_e-(6+T ))
The error functicn is
e<6 + T)=(6+T)-o-e"’ yum-cam (3.2a)

 

 

 

 

 

 

 

where Q2(T) is

Q2( T)=(1-e'(6+ T))r(0)’rc(0)(1--e'u5 +fl) (3.29)

For the last stage, the error criteria is

. 1 T 2
62 m5) .1] [(6+T>-<1-e‘ )r(6)-Q2(T)] 4T

0
(3.30)

Taking the partial derivatives with respect to M6)
gives as the optimum for r( 6)

n “ l

,rt6)=[£2<r>ar] [Ingmar] 0.31)

' o
where A2(T) and 32(7') are

‘2‘“ = (1-5432 (3.32)

32m =- [(6.7) - 02m] (i-e'T) (3.33)

01"
1

5(5) = 3.59 - 5.95], QE'(T)(1-e'T) N (3.34)
o

= ¢2m

 

-41..

The error criteria' Cf [ r(0) ] in the first

stage is

5 --T 2
firm] z.- [pnmu-e )r(o)-o,(r)] ar
0
(3.35)
Where Q1(T') is
Q1(T) = o<o)(1~e'T) (3.36)

Taking the partial derivative with respect to r(0)

gives as the Optimum for r(0)

6

-1 <5
r(0) =[fI‘(T) dT] [f B‘(T) dTJ (3.37)
O

0

where K1(T) and B1(T) are
If

,(T)=(1—e”T)2

and
B1(T)==[p(T)-<:z,(7')J (i-e‘T)

or r(0) = 1.005
Substituting this do) into Eq.3.34 . 11(5) is obtained

r((5 ) = 0.993

 

Exam 6:

Let the s-domain system equations for a two

- 42 -

2

input and two output port system be given as

“1(a)-1

c2(é)

b— 0-1

 

 

with the preferred function and initial condition

and

The discrete state functions as given in Eq. 2.11

are

 

 

 

We‘ve)-1

p (t)

 

 

1.2 ..

r“01(0)?

 

 

cfi(0)
u—c

~

 

 

 

 

 

r‘ 2 ‘ F
r (s)
s+1 s+1 ‘
3
rq(s)

 

1.5

 

 

-

1

 

 

(3.38)

(3.39)

(5.40)

n;

 

L,<o)q

 

 

co(0)
L’ ._.J

 

 

+w)) )

 

p-

 

L.

c1(T)
62(1’)
c1(6+T)
c2<6+n

The matrix

01(7') W
02(1)

e1( +7.)
e2( +1)

 

 

0—1

 

(1-(2-T) A” K12
(1 n'(6*T))K’m (137)”

 

 

.'A1 I), J‘LI') 1‘1 ALGA l
c-. ¢_(.
2 0
A21 : «(3') 7 L‘
" J .2

of discrete error function is t1

"mm 2 F(1-O‘T)r
p2(T)

171(64-7') (1_n’(6+T))"
L.z){j(6 +TZJ s... V A21

F0 (0)(1-c-T) 1
o (om-e”)
C (O)(1-C-(6+T))

c(O)(1-e-(6+T))
*- J

11

 

 

 

 

 

1",,(0)

r1(5)

 

 

2:2 1.“ ( (5 )
b (2.

(3:41 )

lerefore

'5- ‘1
"1:2
I -7, "'
\1-e )A22

(3.42)

.1 P1303) '1

 

 

r—r1(O)-!
r210)
r1(6 )

 

r215)
.J

 

 

-44..

The error function in the second stage is
01164.1) 2 men) ~(1-e'7) 2 2 r,(6)
e2(6'+T) we) 3 5 ram)

3”") (3 23)
.. ‘- a].
Q (T)

For the second stage, the integro-error is

1

62 [11(6)] 2: [{[e1(6+T)] 2 +[32(6+T)] 2}dT (3.45)

0

Taking; the partial derivative with respect to R( 6)

gives as the optimum for R(CS )

1 -1 1
2(6) = [f Ian m] U 32m" NT] (3.47)
O O

 

 

“21‘5-

.where 332(7) and 132(7') are

am

and

32(7)

11

ll

01‘

r(CS)
1

 

p

 

 

L515)

+[p2(6+7) - Q

.. T .
(1~0 ) 5 - (1-e ) 2Q21+BQ22

61(7") s1,(7’) f G2(7') 821(7’)

21(7 ) 5,2(7') + 62(7 ) g22(7 )

'7 2 13 19
19 29

T
[a(5+r)- Qm(T)] e(T)

22

-7 r"

 

8 2Q21+5Q22

L.

(3.49)

 

fi

 

_

(3.48)

T
(7)] G(T)
2

(3.50)

1

 

1.5 "jf (he-7'> 7.42Q2‘+2.93Q22
o

~4.44021+2.98Q22

47'

 

 

-46—

The error criteria 0’ [R(O)]in the first stage is
2 2
U[R(O)] :f{[e1(T)] +[eO(T)] )dT’
O

(3.51)

where the error functions e1(T) and e217) are
3(7) 21.1(7) - ((-e'T) [2r1(0)+3r(o~)] -Q,,(7)

12(7) 7-” P2(T) " ('l‘C-T) [3r1(0)+5r2(0)] “Q12(T)

Taking the partial derivative with respect to R(O) gives

as the Optimum for R( O )

(S -1 5
R(O) {I 1117') 017' ] [f B1(T)dT] (3.52)
0

where K1( T) and 381(7) are

 

-47-

__ G1(T)SH(T)+%(T)S (T)
A1(T)= T T " 2‘7.
G1( ) s12( ) + G217') 322( )

 

.. 13 19
z (1-e T)"2 [ ] (3.53)
19 29
and
T T
32(7) .. [pun-Qflm] a, (7) +[p2m 212(7)] 92(7)
-7 -7
~ (1-0 i b - (1-e ) 2Q 3Q
[ J [ “+ ‘2] (3.54)
8 20 +5Q
11 12
or 1.300
R(C) :
bf1.211-J

 

Substituting the Optimum value of step input found

at t = O in Eq. 3.50 gives

P -O.48 ‘-

R((5) =

0.73

 

 

- 48 -

IV. Discrete State Control of the Multivariable

:System.

In applications where the preferred function
is specified only at discrete points, such as the
end of each stage, the values of the input signal
vector at the respective stages is uniquely determined
by the solution of n simultaneous linear algebraic

equations. Specifically, let the values of

c10).c(6), . . . , cum-1x5)

be specified. These values are Obtained from the
system model of Eq. 2.11 when we set T': 0.

Therefore, the required values of

R10). R15). . . . . 11((m-1)<5)

as obtained from this models are

-1 i

7 "(3(0) . . . ()1 $10) - 1110mm)

13(0)

 

 

 

 

 

 

_R(..(..m_-;1)(S)_J LEarn-4)c5)...."(§(o) gum-1M) -H((m-1)6)0101J

(4.1)
where the inverse of matrix G(0) is assumed to exist and

 

 

-49..

where the constant values
2(0), N6), . . . .P((m-1)6)

are the specified (or preferred) values at beginning

of each steses.

"‘k)
‘ns an example of application, consider the

problem of establishin; the input signals required

)

to hold an aircraft on a specified path. The “ositiOn

is specified at regular intervals of tine in terms of

the three coordinate Variables as indicated in Fig. ‘.1.

 

”:5
A
{J

 

 

 

 

 

3(t)

 

- 50 -

The s-domain model of the system, representing the

inter-relationship for the system input and output

variables is given as

” xgs):T

3'18)

 

 

uz(s) .

 

L.

 

 

 

 

 

 

 

 

 

 

 

F g11(8) 312(5) 813(3) 1 P 11(8) q
3 5521(5) {522(8) eds) fls)
_s3116) g32(s) $3318) . _ 1-r(s)_
+ "'hi11s) O O 1 ' ox(o)'1
0 h2216) b ey(o) (4.2)
L O O h33(8’a L.°z(o)_.
with the preferred function and initial candition
"1 1' 1
9x“) I3:136)
123,10) ° ‘ 12,136) (4.3)
p (0) 10,136)
Z - — .1
3(0).“ cxm)‘
1r1o) = cyw) (21.4)
Z(O)_1 CZ(O)_J

 

 

 

 

 

 

- 51 -

The discrete level input values R(0), . . .,R(3
are obtained from Eq. 4.1 by substituting the
given values of Eq. 4.2, Eq. 4.3 and Eq. 4.4.

The result is

 

 

 

 

 

—-u(o) '1 - .1 ‘1 rp-(0) -
V10) . 5(0) ’ ° ° 0 1311(0) -
w(0) Pi(o) -
u(3(5) __ __ . - 'P.£:(36) -
v06); Law) ' ' ' Gm) J _ 1 2.1.136) -

(.1496); t .- 1.9;(36) -

(4.5)

where 5(T) is
r- '1
8,1(T) 312(T) 313(T)
EXT) 2: s2117) 5522(7) 1323(7)

 

 

)

cx(0)h11(0) ‘
cy(0)h22(o)

cz(O)h33(0)

oxen“ (3(5)
cy(0)h22(36)

oz(O)h33(30).1

 

 

1‘.)

V. Conclus one.

The dynamic programming method for evaluating
the Optinun‘coniand signals for discrete level control
of the nultivariable systems is very effective
for the intepro-error criteria used in this thesis.
By means of this process the Optimum is found by

- o A '_I H. ' Q' . ‘_ _I_ _‘ 1 . _ o J. _'_
solvisfl s1cceJSiJely n linear 31;?091LC equatIOns

- --4 —!0 PO --' . .0
11;.(1. 0.1-3 SOL) Of 01111:.t..'_011‘3 O).

O
1...]
’)
{3
’11
1...:

'h s procedure is most effective
involving a relatively few control
variables and a large number of stages, 1. e.,
a lar C iuuhtr of discrete control intervals. It
is particularly effective, for examplf, for s~
involving one control variable ahl, s1" tr.nty

staces.
is for the discrete State cantrol of the
multivariable systems, this technique can he

auulled only in the case of systems where the

f
V
d-
,
ed
,..... A
J
r. O
1.4
L—Jc
c:-
I
r.
O
1
’ )
OJ
(.9

w,sure incc the output values are

ctr time re ardlcss of the

-ch disc

*1

R

O
’3
(+-
F.
J
i...-
1
b;
{.1
C"
L?
}J

M
-)

functional behavior of the intervals.

 

Appendix A Definitions

Hatri: of n I n order is

O
O
("I
4
hr
,4
A
d-
v

{311(t) .

) t (7.1)

\
r--
.- .

0
g.
:3

CF

gn1(t) °

 

 

I-Ihere‘: each entry 5313(13) is inverse of Laplace

transform of

-1
813(t) = L s13(S) / s (e.2)

of the system transfer function G(s)

F .
811(3) 0 o o I (S)

("N

A
C)
V
o

o

o

g D
A
U)
v

 

 

n1
J

 

 

 

R)

Ros matrix of 1 x n order is

33.. (J5) (3-04)

 

 

A"?
I-Iatri:: of n 2: 11 order
R(O)
___ R( 6)
W = (3.5)
O
R((m-1) (5)
i t:; eonhin tion of the discrete level innut functions.
A.4

Vector or column matrix

Frflsé)

was) = ' 3.6)

I‘D-(3(5)
.. .J

-

 

 

 

 

 

o ;‘ T < 1
I a__ of” ..._J“. - ‘
of th* nutri“ 1U dei god .

 

dt

 

a
C")

 

Capital letters, such as 0(t), ere used for
the notation of v:etor, small case letters, such
es r('¥, nre used for the serl r, w-l eawitel

\ I

letter with bar, such as R(t/, are used for matrix.

The trans ose of the vector A(t) is

Transposc of A(t) : A(t) (“.9)

 

 

Appendix B Derivations

The error criterion of Eq. 2.20 is

T 1
[MM] [ 33(15)J dt (b.1)
(m-H5

Thu result of the translation of time is, then,

,6
0’ g; [:(T)]i[13(7)]d7' + . . .

J

F.

J

n~1

6
Z f['3(36+7 )]T[ ..(3 5+T)] d7 (b.3)

4] ==0 0

 

F5
R)

The error finetion of the (3-1)-th stage is

geé+r>2reé+T>-§s6+r>
: P(36+T) -Z' €<A5+T)R(<j-1-A)6)
i=0
~ H<36+T> c’(o) (m4)

m
«L r
TIL-”x _')i‘('u_7.:L(;t of {"136 + T) ] [14.113 + T)] is: than:

L
m n 2
[TL—36“) Ii[n<aé+7)]=§: [est-164) ]
. J 33:21

it

3-:
2: [vbménfi- Z Gk(ic5+r)n<<n-1-i)c5>-hk(36+T)okm]
_ i=0

kzi

(b.5)

'1 hrs--

n‘: establishes the result shown in we. soc).

1'51
{.1
}_..

2

 

 

.342
By rearrnging Eq. 2.25, we obtain
111-1 (5 11
cr 4- ' 1 '
3:1 0 1:21

= fz[pk(36+T)-clé(o)hk(jé m] 5mm at

3:1 0 11".“:1

m-1 6 n
E: i, E[Skn((tfl-1IU+T) [ak((J-1)6+T) . . Gk,T)] R] (17'
3:1 J 0 11:1

111-1 6 n
:1 j Z [pkmé +T)—ck’(0)hk(jé +73] {ERNIE-1) +T)dT

3:1 0 k:1

(b.6)

 

Or, the m

We

 

f\/1

0 k=1

strix form of Eq. b.6 i

m-1

(T) 2:3 G,,.(36+T)
J: ,-

m-1

Ls‘:,m( (In-1 {5+T)Z Gk(J(S+T) .

 

5332(T)Gk(T) ‘

dTR

 

. 531a“ (21-1 )(S+T) Gk (7)4

 

3:0
m-1
r_C‘::1(T)z PRU +7)”°1:(O)hk(36+7“) “
3:0 '
dT
m-1
Ewe-”5%) Z pk<36+7>-cg<o)%<36+71
3:0 .

(b.7)

 

O "
T l
71’1“;

 

 

 

 

 

 

 

o , b 1 Lil 10rd 0L 1 h 3 is
r. -7 n «(5+7 -<6+n 4 - -
1 (. ) 3 ( ~o ) He)
J‘ 0”"
.. «<6 T) -T r» M5)
0 _<1-~ Tun-e J” > o-n > J ~
_ _ «Sm -( T) i
1 (1—‘uT‘; [T—(1—e 730(0)]4-(1-«3 )[iéJ-T)'(1'e )}3(0)
2 aT
] .. -( T)
O UNBT) (5+?W'Ufi3 6+ )Cm) .
L J '
(b.9)
1.3;;

From 3g. 2.48, we obtain

..1 ' 2
{)1CT)-[?(1-:'T) 2(1-e”T)] ~(1-e'T)c (0)} dT
r2(0) ‘
1
_ .T __ F (0)1 _er' 2
+f<7)g(T)-[3(1-0 > 5(1-e WP -<1-<> ‘ >c2<0)> dT
OK -.

' «s I EM 5 2
prMG-fl’ [ (1-3" m) 2(1-0'6mfl -(1-e"( mm (0) (IT
q 13(5) 1

r 46m __ r (o)
+f€g<6m¢e~s ) 5(1-o <5+T>>] ‘ ~(1-e (6mm ”0%
O .

 

 

 

 

{-

 

 

 

+
M.
f)

 

‘2: error eriterie o? )1. b.10

r o
J. A r. ‘1 W .. , _,_ '
‘4‘) Q('~;) un‘lfl L( (V ) )1n‘f if)

 

 

 

ex
3

0. .l.‘ , ,. .|-- - - A ,_ _. 9"
in tie ne-ii. lone J -l

l
h. --0

give

 

i
1
O

3.

4

I ’1
-J._ .31 -J,!\/.J.,)

Kocnig, H. D. d 31: knell, I. A., "Jlectro-
mechanical System Theory," NeGrsr-Hill Co.,

1961.

"The Multivarieble Control

J
J
1-3

ranssctions, Part II,

Horonits, I. M., "Synthesis of Linear Multi-
varieble Feedback Control System," I. R. E.,
P. G. A. 0., Transactions, June, 1960, p.94.
Bohn, E. V., "Stabilization of Linear Multi-
veriable Feedback Control Systems, I. R. n.,
P. G. A. C., Transactions, September, 1960,
p. 321.

Amara, R. C., "The Least Squares Synthesis of
Multivarieble Control Systems," A. I. B. m.,
Transactions, Part II, May 1950.

Hsieh end Leondcs, "On the Optimum Synthesis
of Multipole Control Systems in the Wiener
Sense," I. R. 1. Proceeding for the International

Convention, Part IV., 1959, n. 95.

A.

 

10.

.'.". . _ J... . ~- 1.:- H. 0 '1 N 4., .. m - J-
b e Cytimal nultiv.riaele system JJJUHQS s,

‘— ,... - .-.- . "A- -x.‘ A ”an “4- . l- ,. ”.0 .
_0.-:',.‘ cit, LR ‘. ...J.‘.!J".ilCC’ “(hi Uniqueness Of

'1

G. t. C., iranssetions, August, 1960.

firth J. I Vfime Domain noeels of Physical

4,
Systems and Existence of Solutions," Technical
Regert Lo. 1, National Science Foundation, NSF G-
Cn‘n‘ Hichiran State University, 7. Laising, Mich.

‘1
./ -,

Bellman, R.,"Dynamic Programming," Princeton,

1?. C“. , 1359.

 

 

 

 

 

 

 

 

mum

3062 25

1293 0