San Fernando Valley State College
OP'l'IJ··1AL CON'ITOL SYSTE!\S
q
A thesis subr;:j_ttcO in partial ~;<'":.tisfaction of tlw
requirement.s for. thE. c1egree. of nast:er of Science. in
Engineering.
by
Wi.lliar.t Joseph
Bor~~c.;ski
f
Jr.
.... 1
The thesis of Williain Joseph .Borkovvski, Jr. is a})proved:
San Fernando Valley State College
May, 1971
ii
DEDI Cli.l I ON
This effort is dedicated to
~v
wife Maureen for
her understanding and help tllrouqhout ·the 1.·1ri tiny·
of this thesis4
AC.i<N O~JLE DGEHENTS
I thank Dr. rc:1iro Hashimoto for his encourage-·
ment and advise throughout the preparation of
this work.
iii
TABLE Of'
£,pprova l
ii
page
Prefv.ce-Declicai:ion and
'l~able
CO:·T'i.'ETl'~;
P-ci~nouledgett1ents
iii
iv
of contents
List of symbols
viii
Abstract
Chapter-Text by section
1.0
Introduction
.l
2.0
Bacl~ground
Information
6
2.1
OptiPal Control 'L'heo:cy
6
2.1.1
Linear State Regulator Theory
9
2.1.2
Linear State Tracking Theory
1 ..,
.....
2.1.3
Linear State Regulators for 'Time
,
Invariant SysterJs
23
2.2
Observer Theory
25
2.3
r•1anual Tracking
35
Formulation of Problem
43
3.1
Statement of I'roblcn
49
3.2
Solution Approach
50
3.2.1
State Equations
51
3.2.2
Optimal Solution
53
-3.2.2.1
State Regulators
54
3.2.2.2
State Tracking
62
3.2.2.3
Obsever Equations
64
Computer Program
65
3.0
4.0
iv
4 .. 1
lUgori tl1ms
68
4.2
Subroutines
72
Result.s and Conclutions
73
Bibliography
81
Appendix 1-.. -· Program Listings
Btl
List of 'l'ables
131
List of Figures
132
v
LIST OF SYr1BOLS
=
matrix of state variables, a function of time
=
controller coefficient matrix, a function of time
=
input matrix, a functicn of t.ime
=
output matrix! a function of time
=
Quadratic cost function
=
final time
=
error vector, a function of tiine
=
time
i
ft.
iI
Q(t)
,-
Ricatti equation, state coefficient matrix
Ricatti equation, control coefficient matrix
R{t)
;--
I
=
solution to Riccati equation
=
cost function due to controlling input
iL""-·
=
cost function due to controlling states
'H
=
Hamiltonian
K(t)
r
Lu
I
I
!<J>
IY
{T ,t) -
ir
)......
I
:s
I;KT
lx
state transition matrix
iden·tity matrix
=
Laplace operation
=
transpose of Riccati equation solution matrix
=
matrix of estimates of state variables
=
array size n elements by n elements
=
Final K(t) values
=
Hamiltonian costate vector
=
partial of the Hamiltonian with respect to the
control vector
i
I
nxn
!
j
F
~ (t)
I
rah
:au
solution to Riccati equation as time approaches
infinity
vi
-----~--------~-----~-·-
··---·--
..
---
-.
.,
linear transformation of state vector
=
constant linear transform matrix
=
human reaction i.:1.me
=
neuromuscular lag
=
human equalization, lead
=
human equalization, lag
=
linear correlation
=
number of equations
I
I
I
I
I
I
I
L____ ........................ -- - - ------------- - ... -- .. ---- ---- ---vii
...........
-
OP'I'Il-lAL (-:CXJ'l'ROI. SYS'l'Er-1S
USING LitJEl\R (}jJS:CR\/EHS
BY
\·Jillia.h~
J.
Borkm~·s}:i,
,Jr.
Master of Science in Engineering
Hay, 1971
By utilizing the Iuethcds of optimal control theory,
state regulator and tracking problems are solved by
using a unique search r:ethod to minimize the performance
index. The search Qethod uses sensitivity equations to
miniHize the cost function. Linear observer theory is
useG to implement optimal coLtrol when states are
physically unmeasurea.ble. 'I'he technique is applied to a
manual control problem.
viii
1.0
INTRODUCTION
Hethods for compensating linear control sys·t:ems to
insure stability have been well defined by classical Bode and Nyquist procedures.
The resulting
control systems may be stable, but the system
performance is usually not 0ptimum with respect ·to
characteristics
s~ch
as rise time, peak overshoot,
settling time, gain and phase margin and bandwidth.
Thus, the above methods usually lead to trial-anderror procedures and at best, provide only an
acceptable
solution~
not necessarily optimum, to
the problem of compensation of control systems.
In
the early 1960's there was developed a more direct.:.
solution to the control system synthesis problem;
this method is nmv called optimum control theory.
The early work was done by L. S. Pontryagin (1962),
C. W. Merriam (1964)
1
J. T. 'fou
Athans and P. L. Falb (1966).
(1964)
1
and H.
The objective of
optimal control theory is to determine the necessary
control signals that will cause a process to satisfy
given constraints and at the same time ::ninimize some
performance index.
The given contraints could be
the usual control system performance characteristic,
e.g., bandwidth, overshoot, etc.
The performance
index, or cost, could be considered a measure of the
efficiency of which the system meets th~ given constraints.
The performance index is usually taken as
1
quadratic in form because it equally penalizes the
cost of the error at the output, with respect to the
input, and the amount of control required.
The
quadratic form is also useful because it makes the
cost independent of the sign of the variations in
output with respect to input and the required
contrcl.
The general form of the system lf7hich is to be
optimized is of the foilowing type:
(1-1)
x(t)
== f(x(t)
where:
1
u{t), t)
x(t)
=
state vector
u(t)
=
control input vector
t
=
time
Figure 1.0-1 shows a diagram of the type system
described in equation (1-1)
+
~(t~1
-I
s -
--r-->-x(t)
Figure 1.0-1 General system type
It can be shown that the solution of (1-1) for x(t)
will be optimum when the cost function
(1-2)
J =K (X (T ) T ) +
r'E 0 F (X ( t ) ' u (t ) ' t ) d t
is minimized.
The optimal control which minimizes
the performance index is
(1-3)
3
where K(t) is the solution to the matrix Riccati
equation
(1-4)
.
~.(t)
=
-!~(t)A -ATK('t) - Q + K(t)B R"'" 1 B'rK(t)
The A and B matrices are functions of the given
control s~rste:rn an.d A :>.nd R- 1 a:r:e functions defined
in the cost function given in equation
(1~·2).
Figure 1.0-2 shows the form of an optimal control
system as defined by equation (1-3) and (1-4)
r
'l-+r
~ ~Js
-.L_:..J
L-·--·
u
---4!
+
~
-7\
- .·I
X
~
+
___L___.:
r=;=-~}-·--­
__._
Figure 1.0-2 Optimal control system
In roost studies, the. values of Q and R-
1
are
nm::mally. chosen by the designer 1 relying on experience and foresight.
Once the choice of Q and R-
1
have been made, it is a simple mechanical process of
solving the Riccati equations for K(t) and then
using this to solve for u(t).
The above procedure
then realizes the required gains to "optimize" the
system.
Since Q(t) and R- 1 {t) are chosen by the designer,
experience is necessary and justification for
parameters is extremely hard.
For instance, one
"-···---
·----··---
. --..
,
....
··----------
.. -------
--
"'---
-.-
---- ---
~
'
..
one could solve for an optimal system for a choice
of Q(t.) and R- 1 (t) but; the choices would :!:1ot
necessarily lead to a system that would minimize the
cost for all possible
Q (t)
and R-·l (t.:).
It. is time
consuming to try different values of Q (t) and R- 1 (t)
until a minimu."Tl. performance index is
dete~mined.
Therefore, a major part of this study was spent in
determining a method for choosing them.
From the
study of simulation problems, the sensitivity equations of the variation of the state equations, x(t)
1
with respec·t to Q(t) and R- 1 (t), were used to deter-·
mine the best choice of Q(t) and R- 1 (t).
The sensi-
tivity equations are a measure or gauge of variation
of one variable with respect to another, in this
case, the state vectors with respect to the prime
variables Q(t) and R- 1 (t).
The form of the sensi-
tivity equations readily lends itself to solution by
using a digital computer.
An automatic search
method utilizing the sensitivity equations
devised and is explained in Section 3.0.
w~s
The search
method determines Q (t) and R- 1 (t) 'tlhich gives mini-mum cost.
Once the optimal gains are determined, all of the
states may not be physically measurable so that implementation of the op·timum system would be impossible.
A solution to this problem exists in the
t:-•-
l
l
i!
·--.~---·
..
"""""
--·
·-·- '
____
_,
....
--- ---.
~-.-.
-- . - . . --
-·~-
'-- . --.
form of linear observer theory.
Linear observers
are -used for two reasons:
1.
Ease of calculations.
2.
Ease of implementation.
Linear observer theory is explained in Section 2.2.
The above approach has several advantages over
conventional methods:
1.
Obtains a control system that has
minimum cost,
2.
The search method using the sensitivity
equation provides automatic selection of
parameters.
3.
The system can be physically implemented
or simulated directly or indirectly on a
digital, analog, or hybrid computer providing reliable and consistent data for
design considerations.
Therefore, the
method provides reliable and consistent
standards for the design of linear control
systems.
6
:2. 0
BA.CKGROlJI.\iD
INFORMATION
The background information necessary will be
described in three sections.
2.1
These are:
1.
Optimum control theory
2.
Linear observer theory
3~
Human operator theory
Optimum control theory
The linear control system can be represented by the
following equations:
(2-1)
(2-2)
1
X
(t) =A (t)
X
+ B (t) ~ (t)
(t)
y{t)=_£(t)~(t)
where:
x(t) - state vector, a function of time
u(t) - control input vector, a function
of time
I
I
y(t)
I
=
output vector, a function of time
I
l!
I
i
The state vector x (-t) is a column vector and has
components x 11 (t)
1
x2
1
(t) , •••• Xn 1 (t) •
The com-
ponents of the control vector u(t) are u 11 (t},
u 21 (t), •••• ur 1 (t).
The output vector y(t) has
the follmving components: y
Ym 1 (t).
the A(t)
I
. I
11
(t), y 21 (t)
1
••••
Because of the three above definitions,
1
B(t), and C(t) matrices to be conformal
have to be of the following form:
~---------1
M. Athans and P. L. Falb, Optimal Control (New
!York: McGrav-r Hill Book Company 1 196.6) ; D. E. Kirk 1
'optimal Control Theory an Introduction (Englewood Cliffs 1
,N • .J.:
Prentlce-Hall, Inc. 1970), Table 5-l, p. 200.
.
I
L_ .~
~,
J
A (t.) is n x n
B(t) is n x r
C(t) is m x m
A block diagram of the state variable system of
equations is shm·;rn in Figure 2 .1-1.
L~
Figure 2.1-1 State variable representation
.
~
The general, quadratic form cost function is
(2-3)2
J(u) = ~<[£(T)-y(T)], F[£{T)-~(T)]>
+ ~[T { < [r
. to
-
(t) -y (t)], Q {t) [r (t) -y (t)] >
-
+<~(t),
where:
-
R{t)u(t)>}
r(t)
=
desired input
T
=
final time
~
=
initial time
-
dt
Physically, the objective is to control the syste1n
as defined in equations
(2-1) and (2-2) such that
the output vector z(t) is near the input vector r(t)
ing:
,2rhe notation is a short form of writing the followmeans
<[£(T)~x(T)],F[E(T)~z(T)]>
<[i{T)=v(T)]Ti[r(T)-y(T)]>
-
-~
-
:;,o_
Q
"
without expending excessive control.
lead to the definition of
(2-4)
~ (t)
= E. ("t)
~(t)
This could
the error vector.
1
l. {t)
-
Resta·ting· i.:he above 1
\'le
desire to keep e (t) small
without exerting excessive control energy. Equation
(2-3) can be rewritten to include the error vector
as follmvs:
(2-5)
J(u)
= ~<~(T)
F
1
~(t)>
J
+ ~~ [<~(t), Q(t) ~(t)
+<~(t),
where:
R(t)
T
=
F
= constant
i·
~(t)>]dt
terminal time
m x m positive semidefinite,
symmetric matrix
Q(t)
=
m x m positive semidefinite synunetric
matrix
R{t)
=
r x r positive
d~finite
,.
matrix
The first term of the cost function is the terminal
cost.
This term will usually be small, note the
term is a function of
~(t)
square, and can be
neglected if the final error is unimportant.
can be done by setting F
=
0.
This
If F is set to 0, we
are assuming the second term in the cost function
will guarantee
~(t)
being small.
The second term
of the cost function is made up of two terms.
we considered the first term
If
9
{2-6)
L6
= !>2 < ~- ( t)
, Q ( t ) .~ ( t) >
m m
= ~~ .I-_.•• ~ qi -i (t) ei (t)
:1:-l.
J-1
eJ· (-t)
..J
It is obvious by examining the functional that the.
larger
.
-e{t), the greater the cost, because of the
ei(t) and ej(t) factors.
Thus, the system is
penalized much more for larger errors than for
small.
The seccnd functional of the second term
of the cost function can be written as follows:
(2-7)
By the argument offered for Le
1
it is clear that
the system will be penalized more for larger control
than for small.
The advantage of using this particular cost
functio~
i.e., quadratic form, is that it leads to optimum,
linear feedback control systems.
The control is
optimum when the cost function is minimized.
The
cost function will be positive if u(t) is greater
than zero and is optimum -.;,vhen it is not equal to
infinity.
2.1.1
Linear state regulator theory
The state regulator problem can be mathematically
stated by making the following assumptions about
equation (2-2) and (2-3) :
(2-8}
£ (t) =
I
=
0
!:_(t}
10
··-~.
-- -· ----·--··-·-·-··..
. .. -
...
-
•rhis leads to the condi t.ion. that
(2-9)
=
y(t)
x(t)
=
-~(t)
and
+
( 2-10)
~
~{~t<~ (t.)
<~(t),
+
Q (t) ~ (t) >
1
R(t)~(t)>]
dt
It is also assumed that the definitions of
equation (2-5) also hold.
The physical meaning
of this is the state vector,
~(t)
1
will be
forced close to 0 at. all times without excessive
control,
~ (t)
•
We now show that the optimal control is a. linear
function of the state and the K(t) matrix,
solution of the Riccati equation, to be defined;
provides a. unique O.f:Ytimum, linear control.
The
Hamiltonian, H, for the system of equations (2-1)
and (2-2) and the cost function (2-10) is
(2-11)
H
=
~<~(t)
+
1
Q(t)~(t)>
<A(t)~(t)
where:
I
p(t}
+
~<~(t)
1
+ <B(t)'
R(t)~(t)>
~(t)
I
p(t)>
p(t) is the solution to
aH
.:.-o (t) = - ax(t>
Taking this partial of H and simplifying yields
(2-12}
E_(t)
=
-'Q(t)~(t) -AT(t)E_(t)
Along the optimal trajectory an extrerritim is
defined at
(2-13)
ClH
au(tf
=
o
ll
r---!
!(2-111 \
:
-
* J
Il
I
:
I
Rearranging this yields
I<2-15)
I
I
r
where
~(t)
has to be a square, positive defined
mat.rix so that R-l(t) can exist.
optimum control, the Hamiltonian H must be
I
l
minimi2ed.
i
i
by equation (2-13)
Ii
I
(2-16)
1
To obtain
I
It is known that H is at an extremum
I
aH/3~ (t)
--- 0.
'l'he extremum
will be a minimum if the square matrix o~/iu (t)
is positive definite.
Taking the second pc:_rtia1
~(t)
of H with respect to
yields:
Cl 2 H
au2 (t)= R(t)
I
I
I
substituting the results of (2-15) into (2-1)
/
yields
b-17)
~ (t) = A (t) X (t) - B (t) R-1 (t) BT (t) p (t)
!
for simplicity of notation, the following will
I!
I
!(2-18)
i
be defined;
D (t)
=
B (t) R- 1 (t) BT (t.)
-
combining equations
(2-12)
1
(2-17)
1
and (2-18)
yields the following system of equations:
(2-19)
The equations of (2-17) constitute a system of
2n time varying, linear, homogeneous differential
__ ________ _ __
___ E?c.ruatio11s_
~l:l:i~h
_have a uni_que_solutio11 if 2n
12
boundry conditions are known.
conditions for
~(t)
··- -
_.
The initial
at time zero will provide n
of the boundry conditions.
The remaining n
boundry conditions are supplied by the following
equation:
2-2 0)
3
taking partial yields
(2-21)
=
p(T)
F(T)~(T)
The solution to equation (2-17) has the form
(2-22)
[
:<:>_
p (T)
j
=
f(T,t)
~(T,t)
where
is the 2n x 2n state transition
matrix of equation (2-19).
Partitioning (2-22)
yields
1
I
_x ~~)]
[ E_(T)
(2-2 3)
i!
I
I
__! 11
where:
;
_! 12 Jj: 21
,
_! 22
are n x n matrices
with the boundry conditions of equation (2-20) ,
equations
I
~(T)
i(2-24)
!
=
(2-23) can be written as follows:
.£141 (T;t)~(t)
+
sQ_ 12 (T,t)E_(t)
I
l' (2-25)
I
I!{2-26)
F~(T)
=
sQ_ 21 (T,t)~(t)
+ f
22
(T,t)£{t)
substituting (2-24) into (2-25) yields
J
I
I---3
F[! 11
(T,t)~(t)
+ !
_2 21
(T,t)~(t)
+ _! 22 (T,t)p(t)
12
(T,t)p(t)] -
D. E. Kirk, Optimal Control Theory an Introduction
!(Englewood Cliffs, N.J.: Prentice Hall, Inc. 1970), 'rable
:s_-:_lJ_P· 200._________________________________________________________________________ _
,
1.3
(2-27) 4
now solving (2-26)
for £(t) yields
p (t) --
f.<P_ 1 2 (T, t)
[¢
-22
(T; t)
-
r
1
[F¢
__ 11 (T,t) -¢
....,21 (T 1 t)] -x(t)
equation (2-27) can be rewritten in the following form:
(2-28)
=
-K(-t)x(t)
-
where :
!5. (t) =
o(t)
.::-..
~<1..
22
[!~L
(T 1 t)
1
(T It)
E! 1 2
-
(T , t) ] -
1
- !2 1 (T 't) ]
Note that !(t) is an n x n matrix depending on
the terminal time T and the final condi·tions
matrix F and not on initial conditions.
Now
substituting the results of (2-28) into (2-15)
yields
(2-29)
u(t)
=
-
~- 1 ( t ) BT ( t ) :!5_ ( t ) ~ ( t )
Notice that {2-29) indicates the optimal control
lavT and is linear.
Also notice that even if the
plant, A and B are fixed, the feedback matrix
-R-1(t)~
K(t) could be time varying.
The state
representation of equation ( 2-2 9) is shO\vn in
Figure 2al.l-l.
4 R.
E. Kalman, "Contributions to the Theory of
Optimal Control," Bol~ Soc. Mat., Mex. 1960, pp. 102-119.
~,
--
14
Figure 2.1.1-1 Optimal control
state representation of
a linear regulator
Notice that to implement an optimal control, all
states will have to be fedback.
This should be
noted and will be discussed again in the section
on observers.
The determination of the feedback matrix,
-R(t)B(t)K(t), is dependent on the state
·'
transition ¢(T,t) of the system defined by
(2-22).
If the
A,~'
R, and Q matrices are
all time invariant the state transition ma·trix
in LaPlace Transform notation is
(2-30)
If any of the matrices A, B, Q, or R are time
varying or the system is of a higher order, then
'•'
_ __.._
15
the solut.ion for 1_ is time consuming and cannot
be solved without resorting to numerical computation methods.
An alternate approach to obtaining the optimal
control of equation (2-29)
K (t) as shown below.
p(t)
=
~(t)
be to solve for
Suppose equa·t.ion (2-2 8)
1
is valid, then differentiating
yields
{2-28)
(2-31)
K(t)~(t),
wou~d
= i(t)~(t) +
!{t)i(t)
from (2-19) we have
A(t)~(t) - D(t)p(t)
=
(2-3'2)
i(t)
(2-33)
E_(t) = -Q(t)x(t) - ~T(t)_J2(t)
substituting equa·tion (2-28) into (2-32) and
simplifying yields
(2-34)
substituting equation (2-32) into (2-29) yields
(2-35)
E_(t)
=
{K(t) + !(t) [A(t) - D(t)K(t)]} ..?S,(t)
now substituting
(2-36)
(2-37)
2_(t)
=
from
(2-35)
(2-~28)
into (2-33) yields
[-9_(t) - AT(t)K(t)] .,?£(t)
and (2-36)
!S_(t)~(t)
:K<t> +
~(t)D(t)K(t)
+ Q(t)
+ AT(t)K(t) = 0
where·:
~ (T)
= F
AT t = T the transversality condition requires:
Fx(t) and AT t = T, p(t) = ~(T)~('l') so [K(T) - E_]~(T)
0 and K(T) - F.
5
· p(t)
~
5
=
16
equation (2-37) is the Riccati equation which
can be rewritten as follows:
K(t) = -K(t)A(t.)
( 2-3 8)
- AT(t)K(t) ·- Q(t.)
-
-
_
+ .............
K(t)B(t)R- 1 (t)BT(t)K(t)
if K(t) is the solution to equation (2-38)
Riccati equation, and
~(T)
sy:rnmetric for all K(t).
.! (t)
(2-39)
=
=F
1
then K(t) is
Then to prove
15? (t)
the definition of a syiU.t-netric matrix, we take
the transpose of equation (2-38) with D
=
B(t)R- 1 (t.)BT (t) substituted yields
(2-40)
6
-
[! ]S(t~
T
=
-AT
+
(2-41)
(t)~T (t) - !._T (t.)A_(t) - .Q.(t)
K'l' (t)p.(t)_!S..T (t)
d
dt
holds.
K(t)
-.
Using (2-41)
1
and comparing (2-38). and
(2-40) that both K(t) and ~(t) are solutions
to the same differential equations.
K (T)
=
At t
=
T,
F and F was originally assumed syrnmetric~
so
(2-42)
and for uniqueness
(2-43)
K(t)
6
=
KT(t)
The identity (AB)
.Q_(t) and ,£(t) are symrnetric.
(Q.E.D.)
=
B A
was used.
Also,
1
..o..l
~,
The results of equation (2-·41) show that K (t) is
symme·tric; this reduces the number of firs·t order,
time varying, ordinary, differential equations
that have to be solved from n
to n(n + 1}/2.
The Riccati equation (2-36) can be solved by
using numerical methods on a digi·t.al computer.
Section 4.1 will give a detailed computer method
for determining
~ (t)
for op-timal control.
A.
sumrr.ary of equations for optima.l control for the
state regulator problem is showri in Table
2.1.1-1.
2.1.2 Linear state tracking theory
The
linear control system as defined by
equations (2-1) and (2-2) and the quadratic
cost function defined by (2-5) are the fundamental equations of linear state tracking
theory.
We now derive the equations following
the development of the linear state regulator.
The results desired from the tracking system is
that the desired system output
be close to
the system input
~(t)
control energy.
Thus, it is required that
~(t)-z(t)
written
(2-44)
~(t)
be small.
without expending excessive
~(t)
Using (2-2}, £(t) can be
=
'I Q
.J.v
TABLE 2.1.1-·1
SUMMARY OF STATE REGULATOR EQUATIONS
.
Linear
System
~(t)
=
A(t)~(t)
y(t)
=
c(t)x{t)
+
~(t)~(t)
vlhere:
c(t) + I
t--------~-
Optimal
Control
where:
~(t)
is symmetric solution
of Riccati equation
K(t)
=
-~(t)A(t) - AT(t)~(t) - Q{t)
+
Riccati
Equat.ion
~(t)D(t)!5_(t)
where:
B(t)~- 1 (t)BT(t)
£(t)
=
K(T)
= F
-~--------------4-----------------------------
~ (t)
State
Optimal
Differential
Equations
where:
E
Cost
= [~ (t)
J
> 0
MIN
=
[xT(t)K(t)x(t)]
·----·-----·----...1
"1
..L
The state tracking cost function is given by
equation (2-5) with (2-44) substituted
(2-45)
= < (£ (T) -,2 (T) x (T), F (r ('I')
J (u)
-.s (T)lS (T} )>
+ ~~~ [< (E,(t)-E_(t)~(t) ,Q(t) (_;:(t)-£{t)~(t))>
+<~(t)R(t)~(t)>]
dt
Again writing the Hamiltonian
(2-46)
H
= ~<
[~(t) ,_Q(t)~Ct) ]>
+
<A(t)~(t)
,p(t)>
+
<B(t)~(t)
,p(t)>
·where:
+~<_£(t)
,R(t)u(t}>
p (t) is the solution t,o
p(t)
=
dH
~ ~ (t)
taking this partial of H yields
(2-47)
E(t)
= -cT(t)Q(t)£(t)x(t) - AT(t)£(t)
+ cT (t) Q ( t) .E. (t)
now defining the following for simplicity of
notation
= B (t) R- 1 (t),!?_T (t)
(2-48)
D (t)
(2-49)
V(t) = cT(t)Q(t),£(t)
(2-50)
W(t)
= cT(t)Q(t)
rewriting (2-4 7)
(2-51)
p(t)
= -::?:(t)~(t) - AT(t)E_(t) + W(t)E_(t)
again to obtain an extremum 3Hjau(t) = 0 yields
(2-52)
au =
~.
R(t)u(t) +
T
~
(t)p(t)
therefore
(2-53)
}!(t)
= -R- 1 (t)BT (t)p_(t)
·~
~J
20
""-
substituting (2-53) into (2-l) yields
(2-5 4)
x(t)
=
A(t)x(t) - B{t)P:-
1
(t)BT (t)E_(t)
now combining equations (2-51) and (2-5 4)
This is again a 2n linear time varying set of
differential equations.
W(t)~(t)
function
conditions on
conditions.
~(t)
But now the forcing
is present.
The initial
provide n sets of boundary
The remaining n conditions are
obtained at £(T).
The following equation is
analogous to equation (2-21) "
(2-56)
p(T)
d
= aX1Tf
=
[~<e(T)
,Fe(T)>]
=
cT(T)Fc(T)~(T) - £T(T)K£(T)
The solution to equation (2-55) is
f_x~t>lj +~T! (~~
.(2-57)
lE(tl
if
~(T,t)
of the form
,J _~ _J
c~(tlE(tJ
d"
is partitioned and the integral is
replaced by
j
-Q~t~:-_AT~t~ F~t~l
7
Note that if c(t) =I which is the usual case,
equation (2-55) would be:-
r::::J
~A(t)
=
-
I
I
-D(t)
-
:e... (t)
....
+
[- Q(t~:(tJ
21
The equations of (2-58) can be written
(2-59)
X (t
{2-60)
p(t)
)
=
1._1 1 (T 1 t ) X ( t ) +
= 5P_21
{T,t)~(·t)
P._1 2 ( T 1 t
) £ (t ) + F 1 ( t )
+ ~22 (T;t)E_(t) +
!'.z
(t)
substituting the boundary conditions of (2-56)
~(t)
into (2-60) and then
in {2-59) into (2-60)
and solving for £(t) yields
-
p (t) -
(2-61)
--
(¢22
(T,t}
_... (T,t)-cT (T)Fc(T)<hz
+¢ 2
1
(T 1 t)
!_1 2
(T 1 t) ] :
1
}
* [CT (T) Fe (T) _!11
- [1_2 2 (T ,t) -c T
(T ,t) -~11 (T ,t)_! 2 1 (T ,t) ]~ (t)
{T)E.£ (T) ¢12 (T ,t)
+!21 (T,t)!_12 (T,t) ]*
* [-cT (T)Fc(T)F1+CT (T)Fr+f2+tP21 {T,t)Fl)
J
- -- - -
_;
-
-
._
the form of £(t) is
(2-62)
p(t)
=
K(t)~(t)
where s(t)
-
~(t)
is a function of £(t) then substi-
tuting the results of (2-62)
{2-6 3)
into (2-53) yields
u(t) =-R- 1 (t)B (t)[K(t)~(t)- s(t)]
Note as in the state regulator case, equation
(2-63) indicates an optimal control law and is
linear.
The state representation of (2-63) is
shown in Figure 2.1.2-1.
r{t)
s(t)
-~r~,c,Ff~
·x.(t)
c (t) -:::-.?
Figure 2.1.2-1 Optimal control of state
·~~ . tracking representation
~)
')
.r• .c.
The state optimal differential equations can be
writt.en from equations (2-62 and (2-55)
{2-64)
.x(t)
=
[A(t)-D(t)K(t)]x(t} + D(t)£(t)
again the Riccati equation p(t) can be written
(2-65)
.
p{t)
.
= K(t)x(t)
.
.
+ K(t)x(t) - s(t)
from equations (2-49)
{2-50),
1
(2-51)
1
(2-62)
and (2-65}
(2-66)
[-Y._(t)-AT(t)K(t)]~(t) + AT(t)~(t)+W(t)E_(t)
p(t) and
=
;(2-67)
•
K(t}~(t)
+
•
K(t)~(t)
•
- £(t)
then from (2-66) and (2-67) the Riccati equation
becomes
(2-68)
and
(2-6 9)
s(t)
= [K ( t) D (t) -Ar.r ( t) ] s ( t) - W( t) r ( t)
~(t)
is an n x n symmetric matrix as in the state
regulator problem and can be solved on a digital
computer.
A more detailed state representation
is shmvn in Figure 2. 1. 2-2.
I!======1==l
K ( t)
ft=====:::::=:.:=::=:=:::::.J
Figure 2.1.2-2 Complete state
representation of tracking problem
23
A summary of equations for optimal control for
the state ·tracking problem is shown in rrable
2.1.2-1.
2.1.3
Linear state regulator for time invariant
systems •.
We now make the following assumptions to ·the
general linear state
regula~or
problem for this
special case.
1.
A(t), B(t), R(t)
1.
Q(t) as· defined in
equations (2-1) and (2-3) are all
assumed to be constant matrices.
2.
F == 0 at rr
=
.
oo
.
there is no terminal
cost this mdicates K (T) == 0.
Now applying th.ese assumptions to the general
case, the system of equations become linear
time invarian·t.
(2-70)
x(t)
=
Ax(t) + Bu(t)
The cost function is
(2-71)
J
=
0
~!0° L<~(t)
1
Qx(t)> + < _;:(t), .Ru(t)>] d·t
The optimal control exists and is unique and is
given by
(2-71)
u(t)
=
-R- 1 BTKx(t)
A
Where K is a constant n :x n positive definite
matrix which is the solution of the following
Riccati equation.
(2-72)
0
= -KA-~TK
-Q+
1
KBR- BTK
24
TABLE 2.1.2-1
SUJYll1ARY OF STATE TRACKING EQUATIONS
~,-------~------~-------~----------~---~--~
Linear
System
I
~(t)
= A(t)~(t)
y_{t) -- ..9
+ ~(t)~{t)
(t)~ (t)
1---·---------t------·-----··---~-------------~
u (t)
-R- 1 BT (t) [K (t)
-
where:
X
(t)
-s
(t) ]
K(t) is solution to Riccati
equation
Optimal
Control
-s(t)
-s (t)
is the solution to
=
[K(t)D(t)-AT(t)]s(t)-W(t)r(t}
and
_______
,...._
t) = cT
-- --·---------·------------(t) Q (t)
\i] (
K(t) -·...; ~(t)~{t) - AT(t)K(t) -· V(t)
+ K(t)D{t)K(t)
Riccati
Equation
D(t)- B(t)R-1 (t)BT(t)
where:
!5_(T)
=
F
·y(t)
=
cT(t)Q(t)E(t)
-~-------------r-------------------~~~~~------------·--~
•
x(t) = [A(t)-D(t)K(t)Jx(t) + D(t)s( t)
State
Optimal
Differential
Equations
where:
x(to)
E:
s(t) is defined above
=
E:
> 0
J = xT(t)~(t)~(t)-sT(t)~(t) + t(t)
where:
Cost
t(t) =-rT(t)Q(t)E(t)
+sT (t) D (t) s (t)
¢(t) - rT(T)K(T)r(T)
/
25
where
(2-73)
The optimal differential equations are given by
(2-74)
=
x(t)
(A-DK)x(t)
and the cost function is given by
(2-75)
J
=
xr <t)i<x<t>
The system, because of the assumption of constant
A and B matrices, is completely controllable.
'l'hat is the composite matrix ·
( 2-7 6)
( B :
~
• • An-1 B
B : A B : -
is not singular.
Another condition is that the
Eigenvalues of (A-DK) in equation (2-75) must
have negative real parts.
Thus we have a stable
system.
2.2 Observer Theory
The optimal theory explained in Section 2.1
utilized state variable tenchiques.
It was
mentioned that in order to obtain an optimal
control system once the values of the K matrix
were determined, all the system states would
have to be observable.
as follows:
Observable can be defined
"A system is observable if every
initial state x(O)
can be exactly determined from
measurements of the output y(t) over a finite
-time interval 1
~inear
O< t~tf. " 8
Where the terms
X
and
sn.G. Schultz and J .L. Helsa 1 s·tate Functions and
Control SysteT:ls 1 (McGraw-H.:J_ll Bo_o_k Company 1 19 6 7) 1 p. 24.
26
yare as defined in equations
(2-1) and (2-2),
repeated here for simplicity,
=
(2-1)
x(t}
(2-2)
X (t) =
where:
A(t)x(t) + B(t)u(t)
-c(t)x(t)
~(t)
=
state vec·tor
u(t)
=
control input vector
y(t)
=
output vector
mathematically, a system of equations is observable if and only if the composite matrix
.....
:(2-77)
·is non-singular.
9
Some control systems fall into the category of
being observable, but not all the states are
physically available to be fedback.
•.
be called unmeasurable states.
These could
If some method
was available to estimate the unmeasurable
states from the measurable st.ates, then optimal
control would be possible.
according to D. G.
The observer theory,
Luenberge~
(1946), will do
just that and is shown below.
Equations
(2-1) and (2-2) will now be simplified
by the assumption that A, B and C are time
9
A complete proof of this definition is shown in
E. Kreindler and P. E. Sarachik, "On the Concepts of
Controllability and Observability of Linear Systems," IEEE
Trans. Auto Control, Vol. AC-9, No. 2, April, 1964,
pp • 12 7-13 6 •
27
invariant matrices.
Now rewriting (2-1) and
(2-2)
(2-7 8)
x(t) = Ax(t) + ~(t)
(2-79)
y<t>
+s~<t>
where the nth order linear system (2-78) with
mth independent outputs (2-79) can be observed
with an (n-m)th order linear system.
The·
observer equations will be done in two parts to
help make the theory a little clearer.
Consider the following case:
Instead of requiring that the observer estimate
the state vector itself, we will require that it
only estimate a constant linear transform of the
state vector.
Mathematically this can be
written:
(2-80)
z (t)
=
where:
T X (t)
x(t)
T
~(t)
once
~(t)
=
state vector cf linear system
- constant linear transform matrix
=
linear transform of state vector
is determined,
if T is invertable.
~(t)
can be obtained
To make these equations
clearer, consider the simple unforced system
shown in Figure 2.2-1.
23
z (t)
:X: (t)
~------?
D
System 2
· System 1
Figure 2.2-1 Unforced system
forcing another system.
Now, if A and .Q. have no common Eigenvalues, then
there is a linear transforma·tion matrix
:r,
such
that initially
. (2-81)
z(O)
=!
~(0)
and for all time t
(2-82)
~(t)
= !.
~(t)
the general equation for
(2-83)
~(t)
is
~(t) =! ~(t) + ~DT[~(O) - T ~(0)]
The proof of this is as follows:
Suppose equation (2-82) is true, then the
equations for systems 1 and 2 of Figure 2.2-1
are
(2-84)
(2-85)
~(t)
.
z(t)
=
A x(t) + 0
= D z(t) + £
~(t)
By differentiating (2-82) and premultiplying
(2-84) by T and substituting into (2-85) yields
29
•
=
(2-86)
T x(t}
T A x(t)
(2-87)
T x(t) - D T x(t) + C x(t)
equating equals and
{2-88)
simp~ifying
'l'A-DT=C
Because A and D have no conunon Eigenvalues, T
has a unique solution.
Subtracting (2-86) from
(2-85) yields
(2-89)
;(t) - T ~(t)
by
'(2-90)
=
subst~tuting
z(t)
- T x (t)
=
D z(t) + C x(t) - T A x(t)
c from (2-89) int.o {2-90) yields
D [z (t) - T x (t)]
This is a first order differential equations in
("z (t) -Tx (t)) with the s·olution
~(t)
=!
x(t) +
£
DT
[z(O) - _! ~(0)]
(Q.E.D.)
Now, extending the above case into a forced
system 1 will lead to the general case.
Figure
2.2-2 shows the system for this case.
z. (t)
u
System 1
F~gure
System 2
2.2-2
Forced System
30
NO"d
vn:iting the equations for system 1 and
system 2
(2-92)
x<t>
(2-93)
~(t)
•
=
A
=
D ~ (t)
X
(t) + B u(t)
+
E
System 1
c x(t) + G u(t)
System 2
-
Post multiplying (2-92) by T and differentiating
(2-82) and substit.uting into (2-93) ·.yields
(2-94)
T ~(t)
=
T A x(t) + T B u(t)
(2-95)
T
~(t)
=
D T x(t) + E c x(t) +.G u(t)
equating equals and simplifying, yields
(2-96)
D T
T A
=
E c
and
(2-97)
G
=
T B
then from (2-94) and (2-95)
z(t) - T x(t)
.(2-98)
D z(t) - T A ~(t) + E c x
=
+ (G - T
B)~
and if T satisfies (2-96) and
(2-99)
j_(t) -
T
x= Q
(2-98) then
(z (t) - T ~ (t))
\vhich is the same result obtained for the
unforced case and has the same solution, i.e.,
·
~q~ation
(2-91).
We will now consider the case where z(t) is
equal to x(t) and no longer a lin~ar transform
of it.
This is making T equal to the identity
matrix I in. (2-82).
If T were
·r,
then we also
are guaranteed that T is invertable.
This will
show that the observer can be made identical to
31
the observed system.
T
(2-100}
=
~\Then
I
is substituted into (2-96) and (2-97) and
solving for D and G then
(2-101)
D
=
A
(2-102)
G
=
B
EC
Now substituting (2-101) and (2-102) into (2-93}
yields
z(t) ~ (A- EC)~(t) + EC ~{t) + ~ ~(t)
(2-103)
. Figure 2. 2-3, · v1hich is a represent.ation of
equation (2-103), shows that system 1 is equal
to system 2.
u
---1
B
System 1
System 2
Figure 2.2-3 System 1
Identical to System 2
32
The estimation of the state vectors can be
given as follows:
x"' (t) == Estimat·e of x (t)
(2-10 4)
Using equations (2-2) and (2-80), an equation
·for x(t) can be written
(2-105)
X (t)
[
:
] =
~]
Now solving for x(t) and saying this is an
estimate of x
(2-106)
It will now be advantageous to give
a simple
illustration of determining observers.
Assume the closed loop transfer function of a
system is given by
y -
u-
(s
+
3 (2)
2) (s
+ 4)
Figure 2.2-4 is the diagram for this system.
u
3
-1s
Figure 2.2-4
1
s
Block Diagram
z
y
_,..
33
'I' he state equations ca.n be written as follows:
~
:xl == -4x.1 +
.
X2
+ 3u
x2 = -2 x2
=
Xl
2yl
c
The A, B,
A r-4 lJ,
==
l
and E mat.rices are as follows:
B
=
c
(:]
0 -2
=
[2
0]
Using equation (2-93) with the block diagram
shown in Figure 2.2-2 and choosing D
=
can calculate the observer constants.
-6, we
It
should be noted that D is a matter of choice
and can be chosen to conform to the system being
observed.
D,
Now writing (2-93) ,with values for
~and
z=
G substitut.ed:
-6z + 1[2 O]x + G u
Where G is given by (2-97) and T satisfies
(2-97).
TA
"'!"
DT
=
[2 0]
=
+
[ 2 0]
the values of T are determined.
Now the value of G can be determined
G == TB +
[1
~]
Now
[~]
- [i
~]
~]
=
-3
4
34
r:r -
ADJ
DET
Then
~J ~-j
=
[:]
Writing these equations out
=
A
J::
z
= 2y -
x2
4z
A block diagram of this system of equations is
shown in Figure 2.2-5.
'U
3
1
1
s
---+--1-T-~-
z
s +·6
+z
A
X2
Original System
Observer
Figure 2.2-5 Sample Problem Block
· Diagram with Observer~
One conclusion that can be made from this
example is that the observer adds poles to the
system transfer function.
Thus, increasing the
35
number of system equations by m - n.
It should
also be noted that this observer is relatively
simple cicuitry.
x
Figure 2.2-5 shows that if
was not measurable, for some reason, its
estimate is available.
If optimal control theory is used, a set of
feedback constants will be determined for the
given system, assuming all statffiare measurable.
Then the system in Figure 2.1-1 could be made
optimal even if all the states were not measurable.
Thus, optimal control is shown to be
possible for systems without measurable states.
For example, we first calculate the optimal
gains as if all states were measurable, then
estimate the physically unmeasurable states;
finally, using the optimal gains with the
observed states, we implement the optimal
control.
2.3
Manual Tracking
Manual tracking control systems are ones in which
the human operator can be an integral part of the
system.
To be compatible with the normal mathe-
matical model of control systems, one must define
a model of the human operator.
The original work
in this area was performed by A. Tustin (1947).
There has
sine~
been much experimentation and study
36
in this area by J. I. Elkind (1956)
1
D. T. McRuer
and E. S.Krendel (1957) and G. A. Bekey (1962).
The human operator in a manual tracking situation
exhibits the following characteristics:
1.
Sensory device
2.
Computing system
3.
Amplifying system
4.
Mechanical linkage
The human response in closed loop control tasks
cannot be uniquely specified, in general, because
the operator's response depends on the following:
1.
Dynamic characteristics of the
controlled element.
2.
Type of forcing function, input power
spectrum.
3.
The motivation, training and psychological
conditions.
Because of the indefinite analytical quality of
human response, parameters of the transfer function
could be varied until the model was made to conform
to the particular case being investigated.
As a
result of the environmental dependence of the human
operator 1 transfer function
1
and occasional disag·ree-
ment between the model and the actual tracking
records, the term quasilinear is used to qualify
the transfer function.
A quasilinear model in this
37
case is taken to mean a model which represents a
smoothed 1 time-averaged type of linearity.
Mos·t
studies to determine the human transfer function
assume human response to be stationary.
Stationary
can be taken to mean t.hat during any given measurement segment, one assumes the human response is not
time dependent.
Many models of the human operators transfer function
have been suggested; but the one suggested by
McRuer and Krendel (1957 and 1959) has been shown
to be
the closest to experimental data.
was arrived at by the following method:
The model
the com-
pensatory manual control system shown is in
Figure
2. 3-1.
r-
~-----,
Rerrmant
1
n
u
G
0
(jw)
Operator
Linear
Characteristics
!Human operator
L------
I
{t)
I
I
·----....
Gp (jw)
Control
Element
I
I
_ _j
Figure 2.3-1 Compensatory Manual
Control System.
r (t)
~---~?
38
The :Fourier Transform domain is used for the following development, thus the j w notation.
Where w
is frequency and j
l( -l
is the complex number
Table 2 .1. 3-1 is a su:rnmary of the notat.ion used in
this derivation.
TABLE 2.1.3-1
. DEFINITIONS
TIME
DOMAIN
FOURIER
TR2\NSFORM
Forclng function
i(t)
I(jw)
or I
Error signal
E: ( t)
E (jw)
or E
Operator's output
u(t)
U(jw)
or U
System output
r (t)
R(jw)
or R
Remnant
Nu (t)
Nu (jw)
Operator's describing function
G (T)
0
G
Controlled element describing
function
~ (T)
~ (Jw)
Cross correlation between
input and output
~r (T)
Auto correlation function
o£ input
~i (T)
Cross spectral density
<Pir (T)
<Pir (jw) or <Pir
Power spectral density
<Pii ( )
<Pii (jw)
The determination of G0
(2-107)
U
+
%E
+
~
0
is as follows:
(jw)
or N
or G0
or
Cp
or <Pii
The error is
=
(2-108) E
=
I -R
I -GPU =- I ~Gp(G 0 E + Nu}
So
(2-109)
u = G0 (I - Gp U) + Nu
or
(2-110) u
=
Go I + Nc == . Goi
Nu
+
1 + G G
1 + G G
1 + GpGo
p 0
p 0
and
(2-111) E
GpNu + ~~I=-=1 + GpGo
1 + GpGo
= I -
NuGp
1 + GpGo
The cross spectral density functions
. (2-112)
(2-113)
LIM
¢iu = T+
oo
1 (I*U)
T
LIM
cJ>iu = T+
oo
T
=
LIM
T+
oo
1 [I (-jw)U(jw)]
T
[tGoi
+ NuJI*]
1 + GpGo
or
(2-114)
¢iu
=
[1 +1GpG o ][G T-+LIM T1
0
(I*I)
+ LIM 1
T+
00
oo
T
{Nci*~
where
: (2-115)
cl>ii =
LIM 1
T+ oo T
(I*I)
and
'(2-116)
¢iNu=
LIM
T+
oo
1
T
but
(2-117)
c)>inu = 0
because by definition there is no linear coherence
between N u
(2-118)
cJ>i u
and the input I.
Go
1 + GfGo
cJ>ii
Thus
40
and the following procedure
(2-119)
I+
1
G G
p 0
then
(2-120)
G0 ==
Equation (2-120) represent.s an experimental set of
data points which define the amplitude and phase
of the complex G •
The experimental data is obtain-
ed from power spectral density measurements under
the assumptions of time-invariance, stationary
process, and with trained operators.
The trained
operators were used so that response data would be
consistent and a mathematical description would
then be possible.
The quantities ¢iuand ¢ie when plotted as a
function of frequency can then
be made fit an
analytical function which takes the following form
(1 + j
(1 + j
where:
WTr)
WT I)
K == gain
-r· = pure time delay 1 human reaction time
delay
TN
=
neuromuscular lag
TL' T1
=
equalization
The terms T and TN are due to inherrent properties
of the human operator.
The terms K,TL , and TI
can be looked at as gain, lead and lag provided by
the human operator: in an attempt to stal:ilize the
control system.
These could also be viewed as part
of the adaptive nature of
man~
Considerable experi-
ment has shown that more complicated models than
equation (2-121)
offer little added accuracy in
fitting the experimental data.
The range of values
for the variables in (2-121) are shown in Table
2.1.3-2.
10
TABLE 2 .1. 3-2
RANGES OF HDr.JAN OPERATOR
PARAMETERS
K
0.6
+
250
Unitless
1"
0.1
+
0.4
Seconds
1"L
+
5.3
Seconds
1"I
o.o
o.o
+
25
Seconds
1"N
0.1
+
0.7
Secon_::_j ·
Jt has been determined experimentally that the
human operator is more satisfied and controls
better when he is able to act as a simple
amplifier.
Birmingham (1954).
The degree of validity of the quasilinear model
can be related by ratio of the operator output
power to the total outpu·t power.
10 A.
This is called
Skolnick, "Stability and Performance of Manned
Control Systems," IEEE Transactions on Human Factors in
Electronics, Vol. HFE-7, No. 3, September,l966, pp. 115-124 •
.
-
.-
42
the linear correlation, p and is given by
/H/ 2 ¢i·
_
(2-122) p 2
-
_ _ ,_--'!.].1._
<Puu
LIM (U*U) =
<Puu = T-)1 CP
where:
2
IH I Oii
+ <Pnn
LIM (I*I)
<Pii = T-+OJ
I Hl
2
Closed loop
= describing
function
=
G0
11 + GoGp
I
thus,
·(2-123)
p
=
"{¢1.1.
.. <P uu
Elkind (1956) summarized all existing experimental
data and determined that for a p of .95 or greater
the remnant could be neglected.
This point corres-
ponds to a cutoff frequency of 0.64 Hz (4.02 rad/
sec).
Thus, for low
model is
linear.
frequencies~
0.64 Hz, the
If these low frequencies are the
only ones that will be considered, the model can
now be written in the standard LaPlace form as
follows:
(2-124)
Go ( s)
=
--rs
K S
(L'LS + 1)
--;(=-rn-s-:-+""1'"")--'--r.('r="J.S--;-+""1")---'-
With the background material as explained in
Sections 2.1 and 2.2, we will now proceed to a
specific problem.
43
3. 0
FOill-1ULATION OF PROBLEM
Originally, the investigation was solely concerned
with being able to standardize and automate the
design of automatic control systems.
After some
thought and research, the following technique was
formulated:
1.
For a given linear point:
Determine the
necessary compensation to make the control
system optimum.
A.
The assumptions are:
The state equations are linear - not
necessarily time invariant.
B.
The plant is observable and
controJJable.
2.
The optimum control system is them
implemented by:
A.
Determining the compensation
parameters.
B.
Determining the states by linear
observer theory.
With the above technique it should be possible to
design and actually implement optimum linear control
systems.
To make best use of the technique, a
manual control type plant will be used for investigation.
This choice was made because all the states
in this plant are physically measurable.
The state
regulator type problem was used because this would
provide insight into system dynamics.
The more
coro:plicated ·tracking problem was chosen because
this was consistent with the normal specifications
of control systems, i.e., for a step input only so
much overshoot with a specified amount of settling
time, etc.
The details of the problem and approach
to the solution will follmv.
To indicate the advantage of implementing a system
using observers, an example will be given that will
compare the cost of an ideal system and one where
only one state is available for feedback.
The
example uses in part, the theory developed by
z.
V.
Rekasius (1967).
Example:
A
=
B
~ ~
[~]
A
=
[~
B
=
[0
-~
1]
Assumptions:
1.
Time approaches infinity, i.e., K used,
time invariant state regulator.
2.
" at infinity equals zero.
K
3.
4.
For this case, the Riccati equation can be written
{3-1)
o=_
rk ll
lkl2
[6
[~]
1
[0 1]
From (3-·1) the following three independent equations
can be written
(3-2)
( 3-3)
0 -· - (k 1 1- 2k 1 2) + k 1 2 k 2 2 - k
(3-4)
0 == -2
(kl2 -
From (3-2),
2k22)
2
+ k22
1 1
1
(3-3), and (3-4) the
Riccati constants
can be determined.
; ( 3-5)
[~
K ==
i]
The cost equation for the optimum solution is
J
=
xT (t) K x {t)
= 6x~
From equation (2-72)
!(
3-7)
u =-r -1 BT"'
Kx
=
(t) + 4x::.{t) x 2 {t) + x; (t)
for optimum control
-2xJt) - lx 2 (t)
Figure 3.0-1 shows the optimal solution representation.
1
X
5+2
-s1
X
-U
Figure 3.0-1
Optimal Solution
Now, if only x were available for feedback, i.e.,
suboptimal, we would have the fullowing cost function
46
( 3-8)
J(x
.. l
x 2 k 1 ) - J*
;,.1here:
xTKx - xTRRTx
(k. )
1
[~~ ~ ~]
R -
The general equation for this case in terms of J*,
K'
{3-9)
and R is
J*(k)
-
[K(A + Bk) +(ATkTBT)K] - RRT(A + Bk)
(AT + kTBT)RRT + kTk + Q
where:
=
0
k
[~:]
=
The new feedback constants
Q is the same as for the optimal case
K is the optimal case solution to the Riccati
equation
J*{ki) new cost constant as a function of k 1
Expanding (3-9)
•
for this case yields the following
three equations:
(3-10)
4klJ*(k 1
( 3-11)
2[2J*(k 1 )+J*
( 3-12)
-2r 11 r
12
)
-
2k 1 r
2
11
r
12
+ k~ + 4
=
0
(k 1 )k 1 -k 1 r~1 r~ 1 +2r 11 r 12 ]
+ 4r 221 + 1
Now solving (3-10),
=
(3-11), and {3-12)
- 1 + 4
(3-13)
1<;
J*(kl) + k
~4
0
0
and r~ 1 yields
(3-15)
=
- 1 + 1
-4
-k
1
for r~ 1 r
1
:t- 21
47
Nm·J
,..,
applying LaGrange Method for expressing RR.l.
{3-·16)
Using (3··13),
(3-14),
(3-15} and (3-16) the follow-
ing equation in J*(k 1 ) and k 1 can be written
{3-17)
'
J*\k1)
=
3
2
3
2
2
2
k1-3k1+ 10kr8-(k1+k1+6k1+8) +32kl
lOk1
The value k1 that minimizes J*(k1) is 1.25.
the value of J*(kr) is 1.71.
Then
Figure 3.0-2 illus-
trates the suboptimal solution.
1
s + 2
-1 . • 25
Figur8 3.0-2 Suboptimal solution
without observer
With the value of J*(k1) and k 1 , the cost from
(3-8) is
(3-18)
J(x 1 x 2 k 1 )=6.15x~ (t)+4.46~(t)x 2 (t)+l.36x~
(t)
Comparing (3-6) and (3-18) shows the optimal and
suboptimal solutions to the same problem.
Notice
that the cost, by definition, is greater for the
suboptimal than the optimal.
case ca.n be determined.
(1-117) with D
=
The observer for this
If the observer equation
-3 and C equals [1 0] is used, the
values· of T become
'48
(3-19)
=
T
[T
T ] =
[~ -~]
and G from (2-102) is
(3-20)
1
=- 3
G
then by (2-106)
,...
[x] =
(3-21)
[:r
[:]
=
or
,..
(3-22)
xl = y
,..
x2 = y - 3z
(3-23)
Using these estima·tes Figure 3. 0-3 can be drawn
showing how the suboptimal system can be made
optimal.
_':_[1
'l~
+
1
s + 3
+
System with
Optimum K's
Observer_
Figure 3.0-3 Suboptimal system with
observer to make it optimal.
z
49
It has been shown by B. Porter and H. A. \t?oodhead
(1968) that the suboptimal system with only x
1
physically measurable, can be made optimal by using
observers.
The only condition is that the observer,
~(0)
determined by assuming
=
Tx(O), holds for
all time and not only for time equals zero.
3.1
Statement of Problem
The following is a statement of the problem which
will be solved to show the optimization technique.
( 3-24)
=
G
-ST
c1
£
(TLS
(TNs+l)
where:
+
1)
(T Is+l)
from experimental data
K = 1
1"
TN
TL
TI
= 0.2
=
=
=
0.3
3.0
20.0
The fixed portion of the plant is taken as a
constant,
c.
Figure 3.1-1 shows a functional
block of the system.
~
r( t)
u
.
c1 e:-sT (
L
s+l)
~
~·
./
~
(T
N
s+l) (T s+l)
I
I
I
Figure 3.1-1
K
Cz
Y ,...
I
-I
Functional block of control system
50
The double lines being fedback through K are
symbolic of all states
K for each state.
bei~
fedback with a different
Using equation (3-1) and Figure
3.1-1 as references, the problem can be stated as
follows:
Find the values of K which will make the
system shown in Figure 3.1-1 and equation
(3-24) optimum for the following ·two cases.
Case 1
State regulator problem using the
following conditions.
A.
r ( t)
B.
All states at time final < .05
C.
Determine estimates of each phy-
== 0
1
x1
( t== 0
) == 1
sically observable state.
Case 2 - State tracking problem using the
following condition.
=
A.
r(t)
unit step
B.
All states at t
c.
Overshoot < 5%
=
D. Steady state error
0 < .005
~
5% or input
E. Determine estimates of each physically unobservable state.
3.2
Solution Approach
The major steps in solving for Cases 1 and 2 are
shown in the following sections:
51
3.2.1 State Equations
The state equations for the system shown in
Figure 3.1-1 will be written after determining
a representation for s-sT
The McLaurin series for s
-sT
is
....
{3-25)
Using the first order Pade
solve for s
(3-26)
E
-ST
approximation to
we have
1 -
-ST
~
1 +
If the division in (3-26) is performed, equation
(3-25) will be determined.
Figure 3.2.1-1 shows
the open loop transfer function of the system.
V.Jhere C 1 is given in (3-27)
(3-2 7)
c1
=
C
1"
1
N
'LL
1"
I
- c1
.1 - ls
1
+
2
ls
2
Time
Delay
1
s +
s +
-1
1"N
s
+
1
-1" L
y
-1
1"
Neuromuscular Compensation
Lag
Fixed
Plant
Figure 3.2.1-1 Open loop transfer
function of the system.
'•
.;...
52
Writing the state equations in physical form from
Figure 3.2.1-1, remembering the input zero has to
be handled specially to avoid any u terms, yields
( 3-2 8)
-
xl
*2
*3
=
1
l
'TI
'TL
1
0
'TN
0
0
1
TN
rx
l
t:
1
-'T2
-c
+
-c
u
~
'T 1
The open loop transfer function with the states
is shown in Figure 3.2.1-2.
1
4/'T
s + 2/'T
Figure 3.2.1-2 Open loop transfer
function with states shown.
The matrices utilizing the same notation as that
in Section 2.0 are
( 3-2 9)
A ==
--'T1
I
-'T1
1
-'T
0
N
L
1
'TN
0
( 3-30)
B
=
1
0
1
--'T2.
[ -c. ]
-c1
4/TC 1
( 3- 31)
0
0
Now checking the controllability equa·tion (2-76)
53
and observability equation (2-70)
S.+
1"
( 3-32)
-c2
N
N
iQ, i _l{~c,( 1,-1 l +4c
;,
c2 <!
"(
-r
L
I
4
-e
1" 2
'T'n
+<-1-1-,
N
-r r
N L
T
j
S+4C, -8K}.
I 1"L -rN_. 1N T 1" 2
!S.
s.+
1"
-rN
-c2
-r
[[s+
1_1
i
1"N TN
4cj
_1"2
scj
T
8
- :rc2
Equation (3-32) indicates the system is controllable because the determinant is non zero.
I
c1 1
-c1
I
I
0
o
I
I
I
1
I
·cl
1
~17f- -:r
-
'T
T
N.
c
I -·
c1
1
1"N +C 1
c
1
2{1
(
TL
)-1
L
N
1
TN) -
2c1
:r
Equation (3-33) indicates the system is observable because the determinant is non zero.
Thus,
the control system being considered is completely
controllable and
states, x 1 , x 2
,
observable even though all the
and x
3
,
are not physically
observable.
3.2.2 Optimal Solution
The optimal solution will be examined in two
sections.
One being for state regulator type
problems and the other for state tracking
54
problems.
3.2.2.1 State Regulators
The equations :Dr an optimal solution to the
state regulator type problem are summarized
in Table 2.1.1-1.
The calculation can be
separated into three parts:
1.
Solution of Riccati equations for K
2.
Solution of state optimal differential
equations for x(t)
3.
Calculation of cost function
The solution of the Riccati equations is of
prime importance because, once this is known
the solution for x(t) can be made and then
the cost J.
The Riccati equation for the
general system shown in Figure 3.1-1, with
r(t)
=
0, and for the specific open loop
transfer function shown in Figure 3.2.1-2, is
( 3-34)
All terms of (3-34) are defined, equations
(3-29),
(3-30), and (3-31), except
and K.
If the cost function is to be mini.
mized, the values of
Q
and R-
l
Q,
R
1
will have to
be chosen such that the value of K will indeed
minimize the cost.
The approach taken here
was to determine the sensitivity of K(t) with
respect to Q and R- 1
•
Then knowing this, the
55
sensitivity of
~{t)
with respect to K(t)
~vi th
could be determined.
the knowledge from
the sensitivity equations, the values of Q
and R- 1 could be chosen to reduce the cost.
The equation for x(t) for the optimal system
of Figure 3.2.1-2 is
(3-35)
or
(3-36)
=
x(t)
-
[A-D K(t)] x(t)
Taking the partial differential of
~(t)
with
respect to Q and K(t) with respect to Q in
(3-36) and (3-34)
( 3-3 7)
( 3-38)
d
dt
d
dt
1
respectively, yields
~X(t)l=(A-DK(T))
Laa J - --
(JX(t) -D £..!Ut1·x(t)
aa
-
ao
-
[dK (t )] ~- aK (t) A-_!!TdK (t) -dQ+dK (t) DK (t)
aa
aQ
aa
ao ao
+ K(t)D(JK(t)
-
-ao
Where because of inital conditions of x(t)
and final conditions of
( 3-39)
we can say
ax(t)
ao
and
( 3-40)
~(t),
dK (t)
ao
I
t=T=
o
The sensitivity equations for
respect to R and
are as follows:
~(t)
~(t)
with
with respect to R are
-r
.30
(3-41)
.
. j= .
d ~ClX(t}
-·- --8
dt
R
[A-DK (t}]
-
--
ClX(t)
~-
oR
- rBClR- 1 BTK (t) +DClK]
L3R
x(t)
""TR
-: -
and
( 3-42)
d
dt
[a~~t)l: -ClK(t) A-ATClK+ClK(t)DK(t)
]
ClR
- - aR dR
-
[
+K(t)DClK(t)+KBCl(R- 1 )BTK
-
-
dR
C3
R
--
again with the following two conditions
(3-43)
ax(t)
dR
I
=
0
I
=
0
t=O
and
( 3-4 4)
t==T
For completeness, the sensitivity equations
of x(t) with respect to F(T), K(T), and K(t)
with respect to K(T) are as follows:
(3-45)
__9_[ax (t)l
dt
ClF
J
=
[A-DK(~]oX(t)
- __
D .. 3FClK X(t)
-ClF
and
0-46}
+ KD ClK
-
where also,
( 3-4 7)
Clx(t)
ClF
and
( 3-4 8)
ClK(t)
ClF
I
t=O
= 0
ClF
5'7
The number of differential equations to be
solved for if F, Q 1 and R where taken as
general n x n for F and Q, and n x m for R,
and an m
(3-49)
N
=
X 1
X{t) WOUld be
[2n x n + n x m] n(n+l) + n
2
+n
If we were considering F, Q and R as diagonal
matrices, plus n x 1 x(t) equations, we would
have
(3-50)
N
=
[2n + m]
[n(~+l)
+ n] +n
Since this a fantastic number of time varying
differential equations, even in our simple
example where n ~ 3, m
=
1, and F neglected
we get 111 for the non-diagonal and 39 for
the diagonal.
If the time invariant case, K at T+oo, is
considered, then
so~simplification
on the
number of differential equations that have
to be solved is made.
(3-51)
A
K
=
LIM
t+OO
In the limit
K(t)
The Riccati equation, equation {2-73), becomes
(3-52)
0
=
A
K
A - ATA
K - Q + KA B R-l BT KA
and equation (2-75) is
(3-53)
•
~(t)
=
(A - D
A
K)~(t)
=
E x(t)
The sensitivity equations become
58
(3-54}
l
where
( 3-55)
Q
=
[
Ql
0-------0
0I
Q2-
0I
-......
I
-
1
. 0- - - - - - - - .: .:".=.Qn
o-------o
I
( 3-56)
R
=
0
R 2,
I
I
I
.,.
I
........
I
I
.... ...,
I
0 - - - - - .... -
I
-~Rn
-
0-------q
I
( 3-5 7)
F
=
F2
0
I
I
I
.,.
'
I
'....
I
6---
(3-58)
=
E
I
. I
I
..._ '
-
-
-
-
I
~Fn
-
[A - D K]
A
(3-59)
K D]
and
(3-60)
d
dt
[A -
D K]ax(t)
-
-
8Qi
and
(3-61)
= K: 1?. a (R- 1 )
8Ri
(3-62)
BTK
-
!2 ~~_x(t)
0 >q,
59
'A
A
A
~·
. ~
~R·~
r
The K equations and the 3K and 3K
equations
can be solved as a set of linear equations not
differential equations.
equations are the
thus
The only differential
ax, ax,
dOi dR·J.
and x(t) equations,
N ; n x n + m x m + n
( 3-63)
In our case n ;
3 and m
= 1,
thus thirteen
differential equations.
One method for determining Q and
ax
setting the maximum value of
aai
~
is by
and
ax
at
3Ri
some predetermined value for all Oij and Ri.
This can be shown with the help of Figure
3.2.2.1-1.
ax
ClQ •
- - - - - .- -
- -
- -
-
-
-
- - -
+ TOL
~
time
- - -
- -
-
-
-
-
-
-
-
-
-
-
- TOL
Figure 3.2.2.1-1 Optimum
Q(ij) search method.
,_.
_.P-
60
At the end of an
iteration cycle, when all
the states are close to zero, state regulator
problem, the maximum partial of all the
partials for the whole
be determined.
iteration cycle can
This is then used to adjust
Q(ij) as follows:
( 3-6 4)
Q (i.)
J new
Q(i j )
new
=
Q(ij)old 1.2
=
Q ( i j ) ld 0 • 5
o
ax
(dQ) max <TOL
ax
( 'd<5) max > TOL
The new value of R(ij) because we are considering R and lt
( 3-65)
1
is as follows:
R ( i j ) new
=
R ( i j ) 0 ld 1. 2
aX max <TOL
(.dR:J
R(ij)new
=
R(ij)old 0 • 5
(~~max >TOL
Only the maximum a~ of all Clx,s and likewise
aQ
ClQ
for the R are changed at the end of each
iteration.
The value of Q(ij) and R(ij) will not be
changed any more after all the maximums of
the partials fall within a tolerance of the
TOL in Figure 3.2.2.1-1.
Figure 3.2.2.1-2
will show the conditions where optimum Q(ij)
will have been determined.
61
TOL
ax
TOL 2
-ao.
l.
Time
TOL 2
TOL
Figure 3.2.2.1-2 Optimum (Q(ij) condition
The complete approach to finding minimum cost
can be summarized as follows:
1.
Pick initial Q and R.
Both diagonal
matrices any positive value will do.
2.
Solve the Riccati equations at T
to get K.
3.
Solve the partials
ax , ax,
CIQi
4.
oR
CIK,
CIQ
If the limits of Figure 3.2.2.1-2
have not been met, change Q(ij) of
the maximum partial
equation (3-64).
ax
aoi nax
as per
Do the same for
R(ij) using (3-65).
5.
Reiterate until conditions of Figure
3.2.2.1-2 are met.
6.
For the time varying case use the
62
values of Q(ij), R(ij), and K for a
starting point.
4, and 5.
Perform steps 2, 3,
The optimum cost should
be reached rapidly.
3.2.2.2
State Tracking
The equations for an optimal
solution~
the
state tracking type problem are summarized in
Table 2.1.2-1.
The general system block
diagram for the state tracking system is shown
in Figure 3.1-1.
The added variable
~(t}
will
have to be considered in the development of
the sensitivity equations.
The state optimum
differential equation is:
(3-66)
=
x(t)
[A-D Kt ~(t} + D S(t)
where
(3-67)
S(t}
= [K
D- AT] S(t) - ~ E(t}
Following the approach used for the state
regulator problems in 2.2.2.1, the sensitivity
equations are as follows:
(3-68)
d
dt
. (3-69)
d
dt
( 3-70}
d
dt
["~cit)]=
E
-
ax (t}
ao
["~cit_L] = ClK(t}
8Q
[-~~t)~
:
E
-
D
3K ( t}
X ( t)
ClQ
ax(t) aR
aR
d
dt
<t>]
~R
[as
D
ClS ( t)
-··
ClQ
.Q §(t)-CTClQf+(KQ-b7)~(t)
ao
aQ
D aK x(t)
- aR -
-B aR- 1 BTK ~(t) + D
(3-71}
+
as (t)
aR
3K(t) D S(t} + K D C!S(t) - ATas
ClR
C!R
---aif-· - -
6.3
(3-72)
~r?x{t)]
=E
axjt) -
L aF J
_E_ fa s <t>l = ~IS.
dt
( 3-7 3)
dt [ 3 F
]
D
aF
aK(t) x(t} +
~-
--
.Q
,qs<tl
aF
n s (t)
3F
Comparing these with similar equations for the
state regulator, it can be shown that added
calculations are necessary to account for
~(t~
The number of differential equations to be
solved are then:
( 3-7 4)
N
=
(4n + 2m) [
h(~+l) + n] + 2n
This number of equations can be changed by
again making the assumption that time
approaches infinity.
The equation which
involves s(T), s, is
( 3-75)
s
=
[~-
This is then substituted for
~(t)
in equations
(2-68) through (2-71) with the K(t)
functions
becoming K, i.e.,
( 3-7 6}
d f<Lx (t) 1- E ax (t)
-·
L aQ J-- aQ
aS : [K D - AT ] rCI' 3 Q
:rQ
-L- aci
dt
(3-77}
(3-78}
d
dt
rax(t)l=
E 3X(t)
L aR J - aR
-
n aK x (t) + n as
- dQ
r ( t)
-
-
-
aK
i
B3 Ir
1
-aR
-:Jsl
D 3K x{t)
-
aR
-
A
-
D
3Q -
ao
BT K _x ( t)
-
A
+
D Cl S
- aR
A
( 3-79)
~
= -
(~
D -AT)- 1
!?_
S
The total number of differential equations are
now the same as the state regulator case.
64
Therefore, the same search routine can be
used.
3.2.3
Observer Equations
The observer will be chosen such that the effects
of the observer on the system will be minimum.
This means that any system effects due to the
observer decay rapidly; i.e., the pole due to
the observer should be large compared to the
system poles.
Assume D of observer to be at
least ten times greater than that of the system.
4.0
COMPUTER PROGRAM
A flow chart for the solution of the problem as
formulated in Section 3.0 is shown in Figure 3.0-1.
Read in
l:t B, }2, .Bt
X3
-,
£, lS.
R-1
A
Solve for K using algorithm
=
ATK + K A + Q. + L RL
-K-K -K-K
-K-K1 BTK
Rwhere: L =
- -K-1
-K
A
= A BL
-
-
--K
A
l<
=
-
-i(
- - -BR- -BTK
1
A
l<-~
"'
Solve for ClK using
dQ
3K
~.
l.J
=
-
dQ
ClR
x 1 {o), x 2 (o)
(o) , time, increment, TOL, TOL 2
A
0
ClR
ao _,
, _,
ao
ao l.J.
..
1
- -
2[A
D!<]
-
B
1
66
:iR
Solve for
....
aK
aR
""--
aR
A
A
K (B R- 1 BT) K
DR) 2 (A
-
-
Solve for
x<t>
x(t)
-
= [A
where:
DK] ?E. (t) + D S (t)
B R-lBT
-D = --
s is
solution of
•
=
-
s (t)
(ax
)
ClQ ..
where:
~J.
D
E
AT] S(t)
-
£T9:_ (t)
}6
Solve for
d
dt
-- - - -
[KD
ax
=.E.
- aK
D
~~~
X
aQ- +
BR- 1 BT
==A
-
-
nt<
Solve for ClX
3R
c
as
- 3Q
D
67
No
<
ax
&
TOL
> TOL2
aR
Do if time
solution is
required.
Change
Q,.
~~
Q(ii)
= Q(ii) 1.2 :Dr(ClX) <TOL
nevl
"'Q
0 1d
0
max
= Q(ii)
.5 for(
old
ax )
Solve for K(t)
..
~(t)=-AT~(t)-K(t)A-Q
+~(t)D
>TOL
30max
K(t)
Change R
Solve for S(t)
R
=R
1. 2 for (ClX) > TOL
new old
a~ax
!(t)=(K(t)D-AT)S(t)
=R
old
• 5 for
(__!,)
R
-cTQr
(t)
< TOL
max
Solve for x(t}
~(t}=[~-DK(t}]~(t}
A
+
£
£_(t}
(End
Figure 4.0-1
Computer Program Flow Chart
)
68
The first portion of the flow chart is a search
method to reduce the total time necessary to obta.in
the time varying solution for K(t) and x(t).
4.1
Algorithms
Riccati Equation Solution
4.1.1
T~is
algorithm was developed by D. L. Kleinman
(1968) to solve the Riccati equation for time
.approaching infinity.
Much of Kleinman's
development is related to work done by N. N.
Puri and w. A. Gruver (1967). The basic
equation is solved
(4-1)
+
Q
+ r:r' R L
L
== R- BT V
K
A
""'A·-BL
AT
= AT
0 == AT V
-.K 1<
where:
-K
-K
-K
Because
~K
+
V A
-:K-:K
1
-
-K-1
-K- - K
=
11 2 • • •
- -K
+ LTBT
-Kis a symmetric matrix, i.e.
and K is symmetric
(4-2)
V
-K
= -K
V T
and
(4-3)
~
VK
=
(AT + ~KBT) VK
and
( 4-4)
so
(4-5)
0 = !JkBTYrz
+
bRB f!K
- YKB ~ + f:~~{ + YK8K
+
Q
As K approaches infinity, we have
1
Voo
+
K
69
{ 4-6)
which is the Ricca·ti equation.
4.1.2
Integration
The integra·t.ion method which was written into
a subroutine is the Gill method developed by
s.
Gill {1951).
The method which is used for
first order differential equations solutions of
equations of the form:
{4-7)
The algorithm can be given as:
Step 2:
Y1
=
Yo +
q
=
g
1
eg 1
1
g2 = F(y1, xo +e)'
2
Step 3:
Y2
=
q
-·
2
{2
-
g3 = F (y 2
V2 ) g 2
I
Xc
+
-
(
2 -
3V7;
2
q 3
g
=
4
F(y
3 ,
x 0 +e)
-
1/3 eq
3
Where the approximate value of y is y
0
+ e.
1
~)
y 4 = y 3 + 1/6 eg 4
x
)q
= Y2 + (l +~ )e(g3-q2)
= <2 +V > g 3 - <2 + ~) q2
Y3
Step 4:
Y1 + (l-..rr;- )e (g2 - q1)
4
at x
=
This method is a modification of the
Runga-Kutta method and maintains the accuracies
of that method.
A flow chart of the Gill method
70
is shown in Figure 4.1.2-1.
Compute
GK (1)
GK (3)
=
=
1.0
=
1,8
GK(2)
=
1.0
GK (4)
=
-1.0/3.0
- {2:0
GK (6)
=
-2.0 + 3.oVO:S
\[2:D
GK ( 8)
=
-2.0
-VO:S
1.0/6.0
GK (5) == 2.0
GK (7)
I
GK (I)
2.0 +
Read
E
X
---
Xp :
step size
initial value
of x
final value of x
Set M
M
= step
counter
Compute
Gx
=
F(x,y)
+~
-
3.0
1
vo:s
71
I f
M
=
3 4
2
1
Calculate
Calculate
Q(I) = G(I}
y (I) =y (I) +xGK (I) [ G (I) -Q (I) ]
y(I) =.y (I) +xEG (I)
+ 0. SE
== X
X
Q
(I) =GK .( 5). G (I) +GK ( 6) Q (I)
M= 3
M= 2
®
Calculate
Calculate
y (I) =y (I) +x GK ( 2 ) [ G (I )
-Q(I)]
y(I)=y(I)+XGK(3)G(I)
+ XGK '4) Q (I )
Q
J
(I) =GK ( 7) G (I) +GK (8)Q (I )
X
=
X
M
=
4
+ 0. SE
Print X
is
X< XF
Yes
No
A
Stop
Figure 4.1.2-1 Flow chart of Gill method
subroutine.
,_.
.-
72
The Gill method of integration was written as
a subroutine.
4.2
Subroutines
The subroutines used in addition to the Gill
subroutine explained in the previous section
were taken from the IBM Scientific Subroutine
Package.
The subroutines are all concerned
with matrix operations:
GMADD - General matrix addition
GMSUB - General matrix subtraction
GMTRA - General matrix transpole
GrilPRD
SIMQ
General matrix products
- Solution to N linear equations
The listings for these are in Appendix A.
The
calling procedure is described in the IBM manual.
73
5.0
RESULTS AND CONCLUSIONS
The program as explained in Section 4.0 was written.
By running a test program, the computed results
sho\-m in Appendix A indicated that
th~
maximum
threshold levels varied wi·th the minimum cost
function.
Figure 3. 2. 2 .1-2 shm·ls the original
scheme for determining Q and R.
If the value of
TOL was made too small, either the cost increased
or the resulting values of
.K
could be made large
enough to make the system have poles in the right
hand plane, i.e., unstable.
If TOL was made too
large, the cost was always too large.
Thus, the
choice of TOL is important in reducing the cost.
The search method that was finally developed had
to find not only the minimum cost, but also the
best value of TOL.A value of TOL was chosen and the
values of Q and R were adjusted until the peak variation of the sensitivity was less than TOL.
Then
the value of TOL was reduced
by a fixed increment
and the process was repeated.
This was done until
a minimum cost was determined.
As an example, the sample problem shown in Appendix
A had a minimum cost equal to 1.915 with a TOL equal
to 0.01 for the Q's and 0.10 for the R's.
These
were determined from examining the results of making
both TOL's larger and smaller.
The effects of
74
making the TOL for the R's smaller was to make the
cost increase while making the TOL for the Q's
larger also increased the cost.
The settling time
for the minimum cost Q's and R's was 12.8 seconds.
Table 5.0-1 shows a list of values of TOL's, Q's
and R's for the test problem.
TABLE 5. 0-1
SAMPLE RESULTS
.----
TOL
FOR R
TOL
FOR Q
Qll
.10
.10
.09
R
COST
Q22
Q33
91.72
1. 00
1.00
1.12
.01
0.6
1.44
1.44
6.19
2.359
.100
.009
1.0
7.43
1.20
4.30
3.525
.10
.01
0.5
1. 20
1.20
4.30
1.915
89.8
--
Figure 5.0-1 is a graph of the cost and settling
time for the runs used in determining R and Q for
TOL R equal to 0.1 and TOL Q equal to 0.01.
The
values of Q and R are tabulated in Table 5.0-2.
Notice that minimum cost leads to nearly maximum
settling time.
This could be significant if termi-
nal settling time was a prominent factor in designing the state regulator.
75
TABLE 5.0-2
SAMPLE PROBLEH RUNS
FOR FIGURE 5.0-1
RUN
R
Qll
Q2 2
Q33
COS 'I'
1
1. 00
1.00
1.00
1. 00
2.58
4
4.30
0.50
1.20
1.20
1.92
8
5.55
0.60"
1. 7 3
1.73
2.33
12
5.75
0.86
2.07
2.07
3.19
16
4.97
1.04
2.49
2.49
3.74
20
5.15
1.24
2.99
2.99
4.26
24
4.78
1.49
3.58
3.58
4.90
28
4.96
1.79
4.33
4.33
5.75
-
-
The observer for this problem could be found using
the method of Section 2.0 and is shown in Appendix
A.
The manual tracking state regulator was attempted
next.
The results of this using the same program
were harder to obtain.
The reason being, the
integration interval for the solution to the differential equations has to be drastically reduced from
0.20 to 0.01 seconds.
If this was not done, the
state regulator solution for x diverges because of
errors in integration.
Once this was done, a
solution was obtained.
The amount of computation
T
COST & SETTLING TIME
vs.
COST
s
RUN
25,0
I
6.0
20.0
I
5.0
I
I
/cosT
4.0
15.0
I
3.0
I
10.0
5.0
I
<t
I
/"----------
SETTLING TIME
I
I
2.0
I
1.0
1
4
Figure 5.0-1
8
12
16
20
24
28
RUN
- Cost and Sttt1ing Time vs. Run
-.....!
0'.
77
time was increased considerably.
Thus a short-
coming of this method is the iteration interval
required to solve the differen·tial equations.
With
this in mind, it was simpler to first try different
values of Q and R in the algorithm for solving just
the Riccati equation because it converged rapidly.
Then the· values of Q and R which yielded a minimum
cost could then be used to search using the sensitivity equations.
Using the above procedure and TOL R equal to 0.1,
and ROL Q equal to 0.01, the minimum cost and
settling time was 0.9958 and 45.75 seconds.
Figure
5.0-2 is a graph of cost and settling time versus
the run.
The run values are shown in Table 5. 0-3.
TABLE 5.0-3
Q AND R FOR RUNS PLOTTED IN FIGURE 5.0-2
RUN
R
Qll
Q22
Q33
COST
1
1.00
1~00
1. 00
1.00
8.21
4
2.00
1.00
2.00
4.00
9.38
0.50
1. 50
4.40
4.99
0.50
1. 00
2.00
4.78
8
12
15.0
5.00
16
10.0
0.25
1.50
3.00
2.47
20
20.0
0.10
2.00
2.00
0.996
24
28.8
0.10
2.00
2.00
0.992
28
34.6
0.10
2.00
2.00
0.998
1
COST & SETTLING TIME {1" s)
COST
Ts
vs.
RUN
I
\
60.0
12.0
50.0
10.0
40.0
8.0
30.0
6.0
20.0
4.0
10.0
2.0
SET'rLING TIME
~
I
COST·
"
1
-4
Figure 5.0-2
8
12
16
Cost & -r 8
20
24
28
RUN
vs. Run for Manual Tracker
......)
CXl
79
Again, notice that the minimum cost has the largest
settling time.
The observer for this manual state
regulator is shown in Figure 5.0-3.
The state tracking problem could not be solved
utilizing the method of reducing the sensitivity
equation maximums below a fixed value because the
sensitivity equations, as shown in Section 3.2.2.2,
contain values which are a function of s.
equations (3-69) and (3-71).
dependent on the input.
See
The value of s i s
This is an area which
could be investigated more fully in another study.
Another area which deserves more investigation is a
comparison of this method of minimizing the performance index with other methods.
Most of the other
methods minimize cost, but only after the values of
Q and R have been picked.
One conclusion that can be made from this study is
that the method of determining minimum cost utilizing sensitivity equations is not the most efficient
in the use of computer time, but does provide a
so~ution
to the problem.
to this type
of problem.
It is a direct approach
t··
....
-.061
-
20
-.J '
' 1
S+. 33
"7'
8+10
~
Xs
p+3.3 3 X2 8+.05
•
.:7~+ z,
+
z2
·y+-
-.055
r-+
-...-l:.
(+
1
1.±.0
X~
-~
+
s +l"I
~
.
:-1-...
'1
27.4
t
,/,
D290
~
i~
I
1-800
+
1138
I
,+
+.
+.
1
s + 15
-t----
:-~
3800
l_____.l
-t7200
Xs
+
Z2
X
4.71Xl0- 3 ~
[ 9. 35X10- 2
2. 6 4Xl0-
3
Fig'l,:tre 5 .0_":':3
l;:,
I
....
x2
~
Optimum system for manual state regulator.
O::t
c
,·
81
BIBLIOGRJI..PHY
Athans, M., and Falb, P. L. Optimal Control.
McGraw Hill Book Company, 1966.
~
New York:
Birmingham, H. P., and Taylor, F. v.
"A Design Philosophy
for Man Machine Control Systems." Proceedings of
the IRE, Vol. 42, 1954.
Carnahan, B., Luther, H. A. 1 and Wilkes, J. o. Applied
Numerical lilethods. New York: John Wiley and Sons,
Inc., 1969.
Costello, R. G., and Higgins, T. J.
"An Inclusive Classified Bibliography Pertaining to Ivlodeling the Human
Operator as an Element in an Automatic Control
System." IEEE. Transactions of Human Factors,
Vol. HFE 7, No. 4, ~~~-----------------------1966.
Craik, K. J. W.
"Theory of Human Operator in Control
Systems." Part I British Journal of Psychology,
Vol. 38, Part II British Journal of Psychology,
Vol. 38.
Davidson, E. J. 1 and .Han 1 F. T.
"The Numerical Solution of
ATQ == QA = -C~"
IEEE Transactions on Automatic
Control, Vol. A C - 13-;-1968.
Derusso, P.M., Roy, R. J., and Close, C. M.
State
Variables for Engineers. New York: John 1\filey
and Sons, Inc.
Elkind, J. I.
"Characteristics of Simple Hanual Control
Systems." Technical Report 111, MIT Lincoln
Laboratories, 1956.
Gill, S.
"A Process for Step-By-Step Integration of
Differential Equations in an Automatic Computing
Machine." Proc. Cambridge Phil. Soc., 1951.
Kalman, R. E., and Englar, T. s.
"A User's Manual For the
Automatic Synthesis Program." NASA CR.475, 1966.
Kirk, D. E.
Optimal Control Theory An Introduction.
Englewood Cliffs, N.J.:
Prentice-Hall, Inc., 1970.
Kleinman, D. L.
"On An Iterative Technique for Riccati
Equation Computations."
IEEE Transactions on
Control, Vol. AC-13, 1968.
82
Luenberger, D. G.
"Observing the State ct a Linear System."
IEEE Transactions on Military Electronics, MIL-8,
1964. 11 0bservers For .Multivarlable Systems."
IEEE
Transactions on Automatic Control. Vol. AC-11 1 1966.
McRuer, D. T., and Kendel, E. s.
"The Human Operator As
A Servo System Element." Journal of the Franklin
Institute, Vol. 267, 1959.
McRuer, D. T., and Kendel, E. S.
"Dynamic Response of
Human Operators." · WADC Technical Report 56-52 4.
Astia Document NR. AD-1106~3.
Merriam, C. w.
Optimization Theory and the Design of
Feedback Control-Systems. New York: McGraw-Hill
Book Company, 1964.
Newman, M. M.
"Optimal and Sub-Optimal Control Using An
Observer When Some of the State Variables Are Not
Measurable."
Int. J. Control, Vol. 9, No. 3, 1969.
Pontryagin, L. s.
The Mathematical Theory of Optimal
Processes. New York:
John Wiley & Sons, Ind.,
1962.
Porter, B., and Woodhead, M.A.
"Performance of Optimal
Control Systems When Some of the State Variables
Are Not Measurable."
Int. J. Control, Vol. 8,
No. 2, 1968.
Puri, N. N., and Gruveri W. A.
"Optimal Control Design Via
Successive Approximations." Joint Automatic Control
Conference, Philadelphia, Pa., June 1967.
Rekasius, z. V.
"Optimal Linear Regulators With Incomplete
State Feedback."
IEEE Transactions on Automatic
Control, Vol. AC-12, 1967.
Sage, A. P.
Optimum Systems Control. Englewood Cliffs,
N. J.: Prentice-Hall, Inc., 1968.
Sarma, V. v. s., and Deekshatula, B. L.
"Optimal Control
When Some of the State Variables Are Not IJ.leasurable."
Int. J. Control, Vol. 7, No. 3, 1968.
Schultz, D. G., and Melsa, J. L.
State Functions and Linear
Control Systems. New York: HcGraw-Hill Book
Company, 1967.
Skolnick, A.
"Stability and Performande of Manned Control
Systems."
IEEE Transactions on Human Factors In
Electronics, Vol. HFE-7, No. 3 1 1966.
83
"System/360 Scientific Subroutine Package (360 A-CH-03X) 1
Version III." IBM Programer's Manual, 4th Edition,
196 8.
Tou, J. T. Modern Control Theory.
Book Company, 1964.
New York:
McGraw-Hill
Tustin, A.
"The Nature of the Operators Response in Manual
Control and Its Application for Controller Design."
Journal of the Institute of Electrical Engineers,
Vol. 94 (IIA).
APPENDIX A
PROGRAM LISTINGS AND SAMPLE DATA
85
l~
OOill
!t_N oor,;:>
\_N 0001
t.N 0004
.LN oons
t.N
nOn"-
{.N 1)007
l.N nons,;
:t_N 0009
r-~ 1'1010
,LN o•JI1
JVJ.
llf•l?
!LN 0013
ll.N 0014
~N 0015
t'~ (101 ..
.r
c
SOLUTIO~ 0~ K RY KLEI~MAN ~(THOO
C 1 ~·[ ~~ 5 I OI-l A 1 3, 3) , i1 C1, 1 1 , Q c 1, 3 1 , T ( 1, 3 I , V ( 3, 3 l , R I ( 3, 3l , "T I 3, 3l , F' ( 3, 1
0 1 t 0 ( 3 t 3 l t A"( ( 3 t 3 I tAT ( 3, 3 l • Y ( 1 t 3 l • Z ( 3 • 3 l • 'rl I l t 3 l t OV ( 3 t 3 I , E ( 3, 3) t H (;,, -,,
0 I t C I n , l l t U Ci. t 1 l • 0 0 (j I ) • 1 I t '"Hl ! 3 • 3 l , 0 K rJ ( .1 t 3 l , UK H ( 3 , 3 l t X ( 3 I • (; K ( 'l l , G X C:
*3) oCi0(01 ( 3) t (;lJX<"l2 ( <) t r;•l~Q) ( ]) 1 f)(Ql (3 I t 0:(1]2 (Jl t Oxrn ( 31 ,GlJXI-> ( 3), Q(Q 1
* 31 • Z •' ! 31 1 G ( t-1 t S ( 6 l t S l< ( 31 t Y()A P (31 t CO ( 3 t 3) t COT I 3 t 31 t R IN ( 3 t 11 t U 1 ( 3, l I
*oU?(3,lloiJ1!3tllotJ4!3.1) tlJ<;(3olloCHK(3)
·
LLL=?
Fl-'=1
1.3=3
I·
II 3:3
K=O
2 FiEAD ui. ( ( (A!ItJ) •J=l•3l IT=lt3l. ( !rl!ItJl •J=1o3) tl=1•3l I ( P:HioJ) •J
o; 1 t 3 l t I= 1 • 3l • I IT I I • J l • J= 1 • 1 l t I= 1 • 3 l , I IV I I • J l t J= 1 • 3 l • I= l 1 3l , ( I R I ( 1 ,'
"J l • J= 1 '3) • t 1 • 3) • ( (I) ll'l (I • J I ' J : 1 • 3) I I:: 1 I 3) • ( (ORR (I ,·J l 'J = 1 '11 'I= 1 • 3)
•• I tCO!I•Jl oJ=lt3l t!=l•3l l
=
. N OOl'T
il.N 001~
Ill
FlJP"~T!<tFiJ,4)
g:;:
13
FQQ1 4 AT!BF'l.4l
3~3
FQQ~AT!3F8,4l
N ·OO?l
N llO?;:>
N 00?.3
LN 00?.4
ILN 00?.5
·L""
[LN
'L"I
oo 26
00=!7
002A
L"l OO:?q
,LN 1)()30
;L"<~ 0031
1LN 0012
~LN 1)031
!LN 0014
fLN Q03<;
•t.N oOV•
IL~ 0037
:LN 001'\
LN 003g
Ct.N 004!1
LN 1)041
LN 1)04:>
LN 0043
:LN 0044
LN 0045
LN 004f..
:t.N 1)047
r
1LN
OO~A
1)01.'1
llO'>'l
LN 0051
N OOS2
LN nos:~
;LN 0054
Fii:.AD !3oCX,TlME,TOLoT0L?.t t:JC(ll tl=lt3loCY
REAO 323o(QJIJ( 1 oil ol=ltJ)
1\=3
fol=3
L=1
CALL GNlR~!Rt~T,N,M)
CALL G~TRA!C0tCOT,~.MI
I'Rl~T l4t((C<JT!toJlhi::J,3loi=lt3l
CALL ANP~~rT•CO,AT,~.~,Ll
CALL AMPHO(CUToaT,ToH.~.Ll
I'RtNT 11.o ( (TiltJl ,.1::1.31 tl=lt3l
S CALL ~~~POtRloRTtFo~t~tLI
CALL AMI-'H0(8tFt0tNoVtLl
27 CALL AMPRO(OtVtDV,~,M,L)
CALL ~~•SU~(Ao[)V,E,>i,P)
CALL A~PRO(ftVoAK,~,M,L)
CALL nMTRA(AK,Ay,N,~)
CALL GNPH0!ATtRtYtNtNoLl
CALL GHPHn(YtAKoZo~oMtLl
CALL AMROO!TtZtHtNt~l
r!loll=2,0<>Eilt1l
rl1•2l=?,Q<>EL~•Il
rlto31=2.0<>fl3tll
1-(114):0,0
1-!lsSl=O.O
... (1,6):0,0
rt2tl>=Ecl.2l
r I? • 21 =F I l , ll + E I 2, ::> l
rt?,3l=E!.3,2l
.. (?. 4 l ::f
1l
.. (?.o5) ::E' (3,1)
r(?of.)::O,l)
rc3,1>=E!l,3l
.. (3t2) =t 12.31
.. ,3,3J:Etl,ll•E!3,3!
'?.
86
LN OO')')
LN OO'if>
Llll 00'57
Llll OO'>M
LN ooc;~
llll OOhO
Llll OOfll
k (:loS I=~- ( 2 tll
"'11t6!=f C3tll
kl4tll=O.•l
t<(4ti'!=?..··)i>f(lt?.l
t-(4t31=0o1
k(4,4!=2,noEC2t21
Ltl 0062
t<(4t'il=?..~i>f(3•21
LN t\0113
LN 0064
LN 001.'-5
LN 0066
LN 00117
LN OOI1'l
LN 006Q
LN 0070
LN non
LN 0072
LN 007~
LN 0074
LN 1)07')
LN 0076
LN OCJ77
LN 007'l
LN 007'1
LN 001\1)
LN 00111
LN OOil?
LN OOi'l3
LN 0084
LN OOil">
LN OQR')
LN OO'l7
Llll OORil
LN OOR'l
LN OO'l'l
LN OO'll
LN OO'l?
LN 00'13
LN 00'14
lN 009<;
LN OO'lf>
LN 00'17
l~J
009R
Llll 009'1
LN o}no
LN 01(11
LN 0102
LN 0101
LN 0104
LN 010'5
LN OlOh
LN 0107
LN 01011
t-(1,41=~.1).
t-(4 thl =0 .o
1-(5.1!=0.0
t- 15•21 :F I 1-•31
t< ( <; • ~I =t I 1, 21
t-(5t41:E.(2,31
1-(5,51 :1' 12,21 +E (3, 11
... ('it 61 =F. ( 3. 21
1-(6.11=0.'1
l-(6.2!:0,1)
t<(6t31:2,0*Eilo3l
.. ,,,41=0.0
"'16t'>I=2,0~EC2•11
k(l1•61=?o1*EI3•1l
Clltli=VIltll
Cl?tl!=Vno?l
Cl3oll=VC1t31
C(4tli=VI2o21
C!5ti):V(2,31
Clt'.tll=V(J,3l
t.. I l• II=-" C1 • 1 l
l; (2tll =-'<' ( lo2)
l:C3tll=-11(1o3)
l ( 4 tll =-1! (?.. 2)
l ( 5 t 1) :-'N ( i' 0 31
I..C6t!l=- .. c1,31
t.=6
lt5=0
CALL c;p1Q CYoUoNtKSl
II ( 1 t 1 I :U ( 1, 11
llllt2l=ltC2tll
II 11 t 3 I :U 13, 1 I
lll2o2):1J(4tll
II l?t3) ::U ('),}I
" (3. 31 =lJ ( 6. 11
V I;>, 11 =U C2, ll
11Doli=UC3.tl
VC3o21=-U15tll
FR ltlT 14 • C CV CI • J I 'J= 1 t3 I • I= 1 • 31
F'HtNT 17tKS
17 FURI-'AT cJH , I 2)
1\=1
11=3
L=3
F=o
1=1
?b
IFtl-1>1 20.20t21
.. ..
87
• lL"' 0109
?.0 ZA!Il=CI!tll-Uilo1l
lF (t>.'lS CZACll)-0,POOn5l 2?t22t23
22 F=P+l
.
FRIIIT 17t P
IFIP-"'l 24,25,25
· 2" I=t+1
au TO 26
23 I=t
FHINT 14• ((CIIollt!=1•6lt(UIItll.I=l•6ll
14 FURMATIIH tbE12,4l
GO TO 27
21 COrJTI•:UE
25 FHINT 14• ((C(loll_.X=t•f.>lt(UIX.ll.I=ltbll
LN ottn
LN 0111
LN 0112
LN 011'
1
l"' 0114
LN OltS
LN 011'>
LN 0117
LN 0111'1
LN
LN
.LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
n1t<l
Ol2fl
0121
FI':PP+1
1112?
0123
0124
Ol?S
Ol<''-
c
tao
"
012il
0129
CHECK=RII1tlJO(V(1oll+TT2~VI2t3l•TT2~TT3°V(3t3l)
o13n
0131
0132
"'=3
FRI~lT
AMPRQ(DRRtRT,ATtNo~tLl
CALL.GMPRO(AtATtAKo~tMtLl
OD~
0}3q
GOXflliii=IJ,O
014"
lll41
0142
014:1
GOX43 Cl I =0, 0
[:KQ lloJl "0,0
0}4~
0149
VI 0150
LN 0151
L~'< 015=?
l.~l 01'51
LN 1)}54
LN 0155
LN
800tCHECK
CALL G~PROtAKtVtZt~tMtLI
CU 90 I=lt3
CO 90 J=lo3
C.t.t"lllll=O,n
OXQ2fli=O,o
0XQ3 ft) :O,Il
1)145
014"
0147
(1}<;<,
Q
CALL
GDX02Cli=O,O
QO
CKP!IoJI='l,O
CO 30 I=lt3
CKP I I , ll =I vI I • 1! ~ 2 c l
3d
0KflCI.Il=-~QO(loll/2,n/EII,ll
Lfl 1'1144
LN
LN
LN
LN
LN
~/R
1\.=3
lT2=1,0
ll1=2.o
0 1;>7
0133
0134
VI 013';
Ltl 0136
LN 0137
LN
LN
LN
LN
LN
lN
IF (P"'-21 2t loOtl"lO
FOR PARTIAL OF K W/R R AND PARTIAL OF K
SOLUTIO~
c
t
t) + V Ct
• 2 l ~ Z I 2 • 1 )· • VI I t3 l 0 l ( 3 • II I/ 2, 0 IE I I • Il
PH tNT t 4, DKR ( 1 t 1 I t I'll< R ( 2, 2 l 1 OKR C 3, 1) t OK Q ( 1 t 1) t OKQ:?. t 2 I , OKO ( 3 t 3 I
SOLVE FOR X
.
•
XOAP:n,O
)IAP=O,O
XO~Pl:O,O
.IO~P2:Q,O
liOt>.F'3::0, 0
CALL
CALL
AMP~O(OtVoOVoNoHoLl
G~SUg(AtOVtEt~•Ml
"::I)
LN 0157
L:o:n
LN
01<;'1
LN 0159
11.=1
LN Olt>!l
LN 0161
LN 01f>2
GKS2=50RT (2,0)
GKill=l·O-AKSl
Gr<(21:::1,0•GKSl
GKSl=<;Q~T
(0o51.
RO
U.J
!.N 0217
ll'!E=Tl~E-TOL •
100 CALL GlLL(~,UX~2tGOXU?oTI~~,TOLoM,L,GK)
IFtL-ll lOJtlOl,lOI
lRl CO 10;> I=I,N.
lll2 GU~Q2(ll=E(ltll~DX~?Ill+E(t•2J08X~2(2l+E(Io3)00X02(3J-(0(ltllOX(ll
o • D (1 t? l "X (;? l • D (I • 3! ., l\ I 3 l l <> nK Q ( 2, 2 l • U2 tIt 1 l
GO TO 100
lll3 ~OP2=hHAil(AAS (0XD2tllloAAS (0XQ2l2lltARS (0X02t3lll
IFIA"lS (XO~P2l-AAS (:0.1P2ll 104tl04tl05
104 X0AP2:XOP2
lnS "=O
.L~
021>!
;LN 021'1
jLN (1220
:LN 0221
!L~ 02??
;LN 02?.1
ill'l 02::>4
· tLN o2;>:;
:LN 022n
'L"-l 02:>7
'LN 027.11
LN
Ll'l
I.N
LN
LN
LN
LN
LN
LN
LN
LN
LS
L.N
LN
LN
L"'
LN
LN
LN
·LN
LN
LN
Ll'l
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
.LN
LN
LN
LN
LN
LN
LN
LN
L=O
T lME=TP~E-TOL
lOb C•LL GILL(~,UXOJ•G~XQJ•Tl"E•TOLoMoLtGK!
IFIL-ll lO<!,l07ol07
107 CO lO'l l=l,N
106 GUX03til=E!l•ll 0 0X0311l+EII•2l 0 0Xv3(2l+EIIt3l"OXQ)I3l-10(Itll"XIl\
ll2~Q
02Vl
o23J
023?
023_1
0214
0235
••Dilt~l
GO TO
109
C23~
0237
023':l
023'1
!17.40
0241
0242
024:1
1)244
n245
ll24h
0247
024i\
1]24q
1}25'1
o251
o252
0253
0254
0255
0256
02<;7
025"1
02'>'1
o2&o
o2n!
o262
0263
02b4
(121'>5
I'J2nA
0267
o26!l
026'l
0270
0
XI?.l•OII•3!<>l\(3)l"OKU!3o3J+U21Itll
11.1~
~OPJ=h~AX!(ABS
c
c
110
!DX03C2lloABS IOX0313lll
llutl10o91
(OX~J(JlloftqS
lf!ARS (l\tJAP3l-ABS !XOP3ll
Jt0~P3:XOP3
CHECK TO SEE IF X HAS SETTLE
91 P=o
1=1
54 lfll-1) 4~,48,49
Tt:MPOClM<Y CARD
48 IFCXII!-O,ll 50oSO,~l
50 P=P+\
IF (P-3) 0:,2o53t53
S2 I.=I+1
GO TO 54
c; 1 I= 1
PRINT 55t(TlMEo(X!tloi=!o3ltiOX'lllloi=lt3l,(DXQ11Il•I=1•3ll
55 FUR~AT(1rl ,Fh,3o'ltl?o4l
F ~ I NT b 5 o I I fl XCJ? I I l • I I , 3 l , 1DXrJ 3 I I l o I 1 , 3 l , t S I ! l , I 1 , 3 l l
65 FOP~AT(1H ,6Xo'lE12.4l
... =0
=
L=O
P,:)
GO TO 31'>
4'l CO~!T l'Jll!':
53 XOAP:ftMAXl(XUAPloXOIP?oXOAP3l
YOAPill=XUAPl
'fUAP(;>):X<juP2
YOAP ( ~ l =X<Hf'3
PRINT )OO,xn•P,XAP
100 FOPMAT(lrl 0 hXt2Fl2,4l
PRINT ElOOtXO~<Pl
FH!NT HOU, ~O•IP2
PRINT !iCO,x04f'3
IFtLLL-11 8~0,850,~6&
A6& 1F!LLL-2l 857tRS7oA~4
8"10 K:4
=
=
90
027\
lN
lN
ll?7:J
~N 11271
.LN 02'74
LN Oi>7<;
027fo
0277
0271\
0279
~~~~ n2AIJ
o2Al
02A2
~N 02A)
~~
[~
;t..N n2A4
l.N 11285
LN
oze,
~N
t!28A
\.N
~~
~2'17
o2w1
1129•1
0291
(12'le!
t.N Q2G3
~~~~ 02'l4
~-N 11295
~N 02'lh
l.N 02Q7
llll
02~A
~N n29'l
;t.N n3n,
:t..N ll301
~~
~N
iLN
lN
~·4
;LN
4'LN
!t..N
t~
~~
lN
!LN
t:~
113:!2
030l
03fl4
030S
1130'>
1)307
031\R
1)309
n3tn
0311
0312
0313
'1314
1)31'1
o31n
0317
n3\o~
031'1
~~ OJ?.D
~N 0321
t~
OJ22
03?3
0321>
K=K-1
Fkt"!T 1.7' K
1-EPE=\o,O
Pt-ltNT BOO, HERE
IF CYO~P (Kl-CI.l R52t<:52tA53
11'53 T (KtKl:O,Sn<>T CKti<l
LLL=l
ADJ=TCKtKl
~~1
t-~~>E=20,0
PH I ~IT HOflt HERE
FRPJi !:IOOtT (KtKl
Go TO bb5
AS2 CHY.tKJ:T(r<,Kl
T (K,KJ:TCK,K)<>l,2
/IOJ:TtKtKl
LLL=l
t-E<>F=1o.o
PI-liNT aoo, HERE
PRINT AOOtT(KtK)
GU TO 865
850 lFCYO~PtKl-CXl 85bt~56t854
!\"i4 T (I<,Kp:AOJ<>O,S
1-EP£=40,0
PRt"'T BOOt HERE
PI-liNT 'lOOt T (K tKl
GO TO 865
R56 IF (K-ll t37o,B70,8S!
R'57 IF ( ~~.P-CY l 8~8t858of!5Q
13<;9 ~C1tl!:O,Sn~R(!tll
'"I::~'~E.=so.o
Pt-lP1T ROOt ~tERE
Fl-ltNT ROO,q Cl.ll
~ltl•ll=l.f)/~(1tl)
t=o
LLL=2
'"I::PF::.;Q,O
FRtr-..T !:IQIJ, HERE
GO TO !165
BSR ChK'4=.tJIP
I< ( 1 tll =R ( 1 , 11 "1 , 2
PRINT !IOOoQ!ltll
ADJ:R t 1, 11
LLL=3
·Rll1ttl=1.~/Rtlt11
1-EIIf=70,Il
FI-IPIT aoo. HI::RE
ROO FU~~·ATt1ti .El5,4l
GU TO 8f:o5
1\(,4 IF P.AP-CY) f<blt86ltllb2
Rl-2 R ( l tll :t.DJ<>IJ ,5
Rl11•1l=1oO/RCltll
1-<E'lE='~O,O
FRir.T PO o, HERE
GO TO 81:-5
91
l
I
I
!
,,
.,
~
Rbl
1FIA8<;(l<.AP-CY)-TOL?J 1180tilROt8513
FI<INT H63oT(ltlJtTC~•?ltTI3•3ltRiltlltRIIltll
R63 FUR~AT(1H ,6Xo5El5.4)
1-:
03?5
G3?.1J
N 0327
N 03;>.:\
N iJ32'l
N o:no
N 0331
R71l
!<E.~<E=90.tl
FtlJNT !lOOt HE.RE
LLt_:?.
K:O
L.N 033?.
C.<:CX-0,01
CY:CY-0,1
TOL?=TOL2-r.illl
lF!CX-0,02) ll6t116o8n8
Rl\8 T11tll=l·O
TC:2o2J:l,O
Tl3o3J:l,O
Fi(loll=l.O
RI!ltll=l.Q
GO TO 865
N 0331
N 0334
N 0335
N 0336
N 0337
N
N
N
N
N
03'1>!
0339
0340
0341
0342
"'
034>
N 0344
1\j
N
N
N
N
N
N
0345
(1346
0347
(!34ii
034Q
03')0
035\
0352
N
N 0353
N l'l354
~J
035':>
1\1 0356
llll
1)3')7
"' 035A
Ill 1)35<1
N 1)36'1
N 03'>1
N n3n?.
~j
0363
N 03f>4
N 0365
N 036'>
L'l !!3·C.7
N
1)3f,~
!II
N
036Q
037'1
!1371
037?.
0373
0374
N
N
N
1\1
ill 0375
N 037.,
N; 0377
N 03711
131\5 1\=3
"=3
L=3
FiE-I'liTIALIZE THE VALUES OF Xlll (X(l) FOR TTHIS PROBLEM ONLY
ll(ll=\,0
X(2)=11,0
)1(31=0.0
Tl"E=n,o
Gu TO 5
C
TII.IE SOLUTION FOR K AND X
116 Slll=VIltll
Sl?l=VIlt21
5(3l=VIlt31
SI41=VI2t21
5(5l=V12o3)
s 161 =v 13.:11
CO 1911 1=1,3
GII1="•0
GXttl=O.Of)
X(IJ:'I,O
190 SJ\!11:1),0
C
THE SX111 VALUE IS F04 THIS PROBLEM D~LY
SX!l1:1,0
TOL=-0.15
LL=O
I=\
C
F=o
"=O
L=ll
2\1) CALL ~ILLI~oSoG•TI~EtTDLt~.LtGKI
IF IL-l I 21t,cl2o21?
212 GI11=-!Sil1"A(l.ll+SI?1<>A(~tl1+'5(3l<>At3tl;1°2,0- Tl1ol1 + 1Sill°C
° C I 1 t 1 I * S I 1 I • U I 1 • 2 1 ° <; I;:> I + 0 I 1 • 3 l "S I 3 ! ) • S I 2 1 I D I 2 t 1 ) 0 S I 11 • 0 I 2 • 21 "S C,
*21•0Po3)"St311 + <;t31<>(0('1tll"Sil1+lJC3o2J 0 512J+D(3t3i"'SI3111
Gl~l=-ISilJ<>A(lo21+'il?.1<>llc;>o2J+SI31<>A(3t2J1 ISIC'l"4(ltl!+SC41<>A(
*2tll•SI':>1"llt3tl11 - T!lt?l • tSill 0 !Diltl1"5121+Dilt21°~141•Dilt3l
°
•~SI51l+SI21°(012tll"SC21+D(?o2J<>St41+lJI2t31"SCSII•SI31°10C3tl1oS!?
I
, :LN r,37'}
1)3ilf)
, LN 03'\l
! LN o3•1?.
f)3il3
, LN 03>14
, iLN 1)3~5
!v~ 03'lh
'LN 0387
LN 03il'l
LN 03fl'l
lN 039,
L~ 0391
LN 039?.
03'H
. LN 0394
: LN 03<l5
, N 03::)(,
03'H
', N 039~
03'19
l' N
N 1)40•1
ILN 1)401
.v~ 0402
.LN 0401
·l~ 041}4
-LN 04()5
°
*l +I) 13, 21 S I 4 I.+ D I 3, 1 l "S I 51 I l
Gl3l=-(Sill<>A(lo31+S(?J<>A(::>,3l+SI31<>A(3o311 - ($(3) 0 A(ltli+S(G)<>At
*:ttll•Si61°A(3tlll- T(l,31 + (5<11"10tl.li<>S(3l+0(1t2!<>SI51•')(lt3l
*" S I 61 1 • S I 2 l 0 I u I 2 • I l ":,I 31 +I) 1 2 • 21 ° S I 5 l + u I 2 • 31 SIb I l + S I 31" Ill 13 t 11 <> S I 3
*I +ll I 3, 21 ° S I !:• I + 0 t 3, :>I<> S t 61 I I
!
G(41=-ISI2J<>A(l,2J+SI41<>il(?•?l+St51<>At3o211"2,0- T(2,2i + 15121<>(!
•c 11 • 11 .. s 1 21 • u t 1• 21 "S < 4 1 •!) < 1 oJl "s <':i 1 1 • s t" 1 " 10 12. 11 .. s t 21 • o <2, c 1 "s 1411
•• D ( 2 I 31 .. s ( <>) ) + s ( 5) .. ( (! ( 3 I 1 ) ., s ( 2) + 0 ( 3. 21 °5 ( 41 • 0 ( 3. 31 .. s ("I I l
I
G(C:,J=-tSI21"'~'(1,3J•SI41<>At?o3l+SI':ii<>A(3o311 - ISI3!"Atl12)+5(51<>Ati
"2•?l•'>l61°11(3o2ll -~1?•31 +<SI2l"IOiltli"SI31+Dilo2l 0 S15l+Dilt3l"~
.. I b l I • '5
I 0 (D.< 2. 1) <> <; ( 3) + 0 ( 2.;::) .. 5 ( 5) + 0 '2. 31 ° 5 ( 6) I + 5 (51 .. ( 0 (3 I 11 .. s ( 3) •'
*C(Jo?.)0$(5) +!l(3,3)<>'5(61) I
I
Glni=-<SI3l<>AI1t31+SISI<>A(?.•3l•SI6l<>A(3o3)1*2,0 -TI3t31 •tSI31<>tOd
0
*ltli '-;(31+Dilo21"5t'il.Oil•~I 0 SI61l+SI5l 0 (012t1l"5131+1)(2o2l"SI5l+d
*12t31<>S(611+St6I"I0!3oli"S!3)+Dt3o2J"SI"ii+OD13l"Sibl)l
:
GO TO 210
lN
°
lN
.
r
lN
- 'LN o4nh
r
. ILN 04(17
040<;
;
'
i
!
1)40'l
N 041n
N 0411
N 04P
0411
11414
0415
N 0411-
tN
:r
!
i
'
N 0417
N 041'1
N 0419
f
N n4;>n
'
(N
; N
!
04~;>
: LN 04?4
04?5
0426
N 0427
N f)4;>>l
N 042'l
N 0430
N OctJl
N 1)432
i
L
<'11
lo':Q
.
L=O
IF ITt HE I
191 TOL=O ,05
191 t 191 , 1 '-l2
lH~E='l,O
LL=1
GO TO 210
192 IFtLLl 210o2l0o230
230 ll~E=TI~E-TOL
193 CALL ~lLL(~,SX,GX,TIHEoTOL,MoLtG~l
lFIL-11 200,220,220
220 GA.Ili:(AI1oll-lfll1•\l<>5(li+flllt21°St21+DI1t31<>S(31ll 0 SXI1)+(t(}o2l
* • In I 1 , 1 l *SIC' l + 0 ( 1,? l "<:I 4 I + n I l, 31 <> S I 51 I l 0 S X I 21 + I A I 1 t 31 - ( ll I 1 t 11 o S ( 3 l
• • on.:> 1 ., s 1 s, •D c1t 1, .. s cf,, , 1 o s x r :n
·
G1.1?l:(II(2,11-CDI?tll<>S(li+Dii!tcl 0 SC2l•DI2t31°S!3!11°SXI11+!~12t21
+rl t 2 , J l "S ( 5) ) I<> S X ( Z I + (A ( 2 t 31- ( 0 ( 2 t 11 <> S ( 3)
*- 10 I?, l l 0 5 ( ;:> l + D ( 2,? l 0 S ( 4)
*•01?•21"5(5) •Dt:'11l"SI61l l<>SXt3l
GX(3l:!ill3oll-(1)(1o\l<>'5!1l+0(3o2loSt21+011oJloSt3111°SX(ll+({1(3o2l
*-1013oli 0 S!?l+O(],?l 0 S(4l•nt3,Jl"SI5lii"SXI2l+IA(3t31-10111ll<>SI3l
*•013t">l 0 S('il+D!it_1) <>St6l ll<>SXI3l
GU TO l'-i3
C
CMFC:K TO SEE IF X IT l TS AT ZE.W)
200 IF 11-11 194olCJ4,l'l<;
194 lF!SX!II-u,O~) 19oo196t197
196 F=P+l
I<.Ki<K:c
FKT~T
04?1
i \.N 0423
:t~
'4
900
CJOOtKKKKtl
FUP~AT(}rl
lr <P-11
o215l
l.'lil,l'l5.1CJS
l'lR l=t•l
GO TO 200
197 l=t
FHINT 20lt TlME,I<:!IIoi=l•F.I
201 FO~~AT(1H oF6.3,6El2,41
PkPlT i:'021(S>-tli•I=1•3l
202 FOR~AT(lrl ,6Xo3El2,41
"=t'l
l=O
93
LN
1)411
L.N 0434
. '
.
Llll
ll435
Go. ro 210
1 QS CUNTl IWE
ENr:l
;
USASI FOPTHAN DIAG~OSTiC RESULTS FOR FTN.MAIN
NO EkRCRS
94
LN
LN
11001
001l2
LN 000.1
LN 0004
LN
OOIIS
L~
ooo'>
LN 0007
LN 000"!
SU~Rn~T!NE
GHAOO(A,RoPoNt~l
Cl'·lEf">lSIO•'I A!lloHilloR!ll
1\.M:N*'-'
co
10 1=1·"~'
10 RIII=atii•H!Il
RETUR•t
END
USASI FOPTf<AN DIAGNOSTIC fiES\JLTS F6R G"AOO
NO ERRCRS
'
~-
95
LN 0001
LN 00(1;>
LN 0003
LN OOIJ4
LN 0005
LN 000'>
LN !!007
LN
oOOil
SUr>ROUTlrJE
GMSUHIA•'~o'loN•-.1
Cl!-'P•SION tdll•>llllo~'<lll
r.l'l:r,<>H
CO }0 I=lorHI
10 l<(!l=Aill-R(ll
I<ETIJih,
ENO
USASI FOPTRAN niftG~OSTIC RESULTS FOR GPSU~
- NO ERRCRS
Lll nO'l1
L~
oon~
LN 0001
LN 0004
LN nooc:;
LN 001\6
Ll'l oon7
lN OOOfl
LN
LN
LN
lN
000'1
OOlll
0011
001?.
USASI
SUP.ROiJTINE
OlNE~SJON
lK:O
10
co
G~ITRA {f.t"lt~·-t'1)
ft(1!tACll
I=I,n
l..J=I-•J
DO 10 ..J=1•'·1
l..J=!..J•N
l~=I'~•l
10 I'IIRI=ACI..Jl
J;ETURN
END
FORTRA~
NO ERRCNS
OIAG~OSTJC
RESULTS FOR cMTHA
97
l~
0001
LN llOO?.
l!ll 000'3
LN 11011<.
LN 00115
LN 0001'>
LN 1'1007
LN 1100"
LN OOC!ll
LN 0010
v~
no 11
LN 0012
LN 1'1013
LN nOl4
LN 001?
LN 0011,
LN 0017
LN 0018
US~Sl
.I
SU<!ROUTINE C,r-IPR0(11o'loPt"'•'-' 1 Ll
ClMENSIO"l 4\!lloR(}),P(l)
li;:O
1~:-M
DO 10 1'-=l•L
lK:IK+I"
co 10 J=l ,Ill
lH=I~hl
.JI=J-N
lt~=!K
li(!R):Q_,')
CO 10 I=l
.J·l=JI •N
t'-1
lo=r"B•l
10 Fi(lRl:R(lR)+A!JI)<Hl(Io'l)
I'ETUR'·I
END
fORTRAN DIAGNOSTIC RESULTS FoR
r,~PRO
LN
LN
0001
SURROIJT
r,Qo?
tr~E
Vl 1)00]
ClM[~lSJOI~
LN 11004
LN 000'5
LN OCIJE'
LN 001)7
LN 1)00F\
LN QOOQ
LN 0010
LN noll
LN 0012
LN 00!3
LN 1\014
LN 0015
LN 0016
LN 0017
LN 001>\
LN nDl'l
LN 00?.0
LN 0021
LN 00?.2
LN 0023
LN 0024
LN ()I) <''1
"'=~I
LN
c
c
USIIS!
+ 1
GO TO llt2o3t4 oSI
,•~
PASS 1
1 l =1
F\E TUR~l
P~<SS ?
~
41
co
41
I= 1 oil
I; ( T I =G I 1 I
SCII=Scll+o.S~EoGcll
X::X+O.S<>E
t::}
I<E TU'-I~J
FASS ::1
c
3
42
co
42 I= loN
SIt I =c; II I •E<>GK ( 1 I* ( G It l -0 CtIl
C(li=GKCSJoG(li+GKc~I~Q(ll
L::l
F\ETUR'-1
PASS 4
c
co 43 I=lolll
Slll=Sill+[OGKC21<>cr;cti-0Ctll
43 C I!<! =GK (71 <>{, (I l +GK (ill <>Q ( IJ
X::(+O.SoE
L=l
F\ETUR•·I
FASS c:;
5 co 44 I= lo M
44 5 CT I = c; (I I +10:* ( r;K C31 <>'3 (I I + GK c 4 I oq rl I I
L=O
F\t. TUR ·J
4
1)021'>
LN 1)0;>7
LN 00?.<.1
LN no;>c;
LN 1)03fJ
LN n•l::l1
LN 003?
LN ljO:n
LN 0034
GILL!N,S.G,XoEtMtLtGKI
<;(b) tG(f.l ,Cl(61 tGKI8) tGl (61 tG2(6) ,G3C61 ,51 (61
c
E~~<D
For.T~AN
DIAGNOSTIC RESIJL TS FOR r;ILL
.~
99
L~ 0001
Llll 0002
LN 0003
LN 0004
t.N 000'>
LN 1\0116
LN 0007
LN OOOP
LN OOO'l
LN 0010
LNOOll
·t.N 0012
.t.l\l 0013
LN 0014
LN 0015
l"4 OOlf,
LN 0017
LN 001'!
LN OOlo
LN 00?11
Llll 0021
L~l 002?
l,N ()02.3
LN 0024
LN 002'>
LN 00?.1',
LN oon
LN 002'1
LN oo?.q
LN 0031
LN 003l
LN
•
003~
lN ()033
L"' oo 34
LN 1103<;
L~'~ 1101<'LN :)037
1.111 (101~
LN 00Jq
1.111 0041)
Ll\l 0041
t.N
004~
v~
oo41
LN 1!044
~.~- 004<;
l,N 004"
11047
L"'
LN
LN
LN
004'<
0040
OOS1
t.N 0051
LN 00">2
LN 1!053
LN 0054
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
e
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
·························································-~········
PU'!POSE
OBTAIN
SOLUTIO~
OF A SET OF
SIHULTA~EOUS
LlrlEA~
EQUATlO~S.
Al.=d
USAGE
CALL SlMOCAtAoNtKS~
OESCHIPTION OF PARAMETEQS
N BY t~ •
OESTAOYEO I~ TrlE C0~ 0 UTATION,
THE SIZE OF MATRIX A IS
A - MATRIX OF COEFFiciE~lS STOR~O COLUHN~ISEo
THESE A~E
8
VECTOR OF"OP!GI~AL CO~STANTS CLENGTH N), TrlESE ARE
REPLACED RY F!~AL SOLUTION VALUES• VECTOR x.
N
NUMBER OF [QUATI0~S A~D VAklABLES
0 FO~ A hO~MAL ~OLUTIO~
liS • OUTPUT OIGIT
1 FOR A SlNGULAP SET OF EQUATIONS
RE•-tt.R!';S
MATRIX A MUST A~ GE~~PALo
lF MAThiX IS SI•;GULAQ , SOLUTION VALUES ARE MEA~INGLESS.
AN ALT~RNATIVE SOLUTION HAY BE QqTAINED BY USING MATRIX
SUAPOUTINES AND
NONE
FU~CT!C~
SUBPROG'!A~S
REQUIRED
METHOD
MET~no OF SOL~TIO~ I~ SY ELI~INAT!O~ USING LARGEST PIVOTAL
OIVISOM. EAC~ STAGE nF ELIMINuTIO~ CONSIST~ OF JNTE~CrlANG(N~
EL£vEJcTS,
RO~S ~HEM NfrES~A~Y TO AVO(O oiVIS!ON BY ZERO OR SMaLL
THE FORw.l.~O SCL"TJO"- TO 0-iTII!'J VI\>1!MlLE N lS OO'lE !;~
N STAG~S. THF ~ftCK SnLUTION FOR TH~ OTHER VARIARLES IS
CALClJLl.TEO flV SUCCES~IVE SU-JSTITUTlONS, FINAL 50LUTION
VALUES AAE DEVELOPEO IN VECTO~ Bt ~ITH VARIA~LE l IN dC1l•
VAqlAHLE 2 t~ RC2l•••••••••• VAN!ftPLE N IN RCN),
IF ,,,, 1--'IVOT C"A'.J HE Fnu~'IJ E.<CH.OI,·l(• A TOLERA•<Ct: nF O.Oo
THE MA ri·<l.l. IS Cn"-'ilUEHEfl S{ ..Jt;Uu,_, A'•tl KS 15 St. T TO 1• T'iiS
TOLENANCE CA~ &F MODtFIED BY REPLACING THE FIRST ~TATEME~T.
• • • • • • • • • ' • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • c- • • • • • • • • • • •
SURROUTlNE S!MO(AtQtNoKSl
DI~•E~SIO<~ 1\ClloBCll
FO'lwARI)
TUL=O.O
SOLUTIO~
100
v:
nose;
KS::C)
l~
005f>
JJ=-~1
LN
LN
LN
LN
LN
LN
Ll>l
LN
LN
L"'
LN
LN
LN
LN
LN
LN
LN
LN
ooc;1
CO 65 J::l tN
005A
ooc;q
Jl'=J+l
no~>o
61Gt\=n
J.J::JJ+N+l
0061
IT=JJ-J
l=JtN
006?
00~>3
0064
006'>
0061>
fJ067
CO 3'l
c
C
c
00611
001','1
0070
0071
0072
007~
c
L"' 007"
LN OORO
LN
LN
LN
LN
LN
LN
• LN
LN
LN
C
INTERCHANGE
40
IF NECESSARY
Il=J•N<>(J-?.1
12=Il+IT
5AVE=~Clll
~1Ill::A(!2)
Alt2l=SAVE
c
C
c
l
OtutGE fQUbTIO'II qy
LEADt~G
COEFFICIENT
r;n A(lli:A(lll/tiiGA
SAVf=R I I•~A l( l
B I I'·1A X l :i3 ( J l
L"' 0095
c
OO<lo
L"'.Qll97
LN Oo}Q"
c
BCJl=StiVE/RIGA
C
EL!MIN~TE
NEXT VARIABLE
l~CJ-~l 55o7U•SS
55 Ius=;,:<> !J-ll
CU 65 lX=Jy,N
oo·o:;q
LN 1110'1
LN 0101
lA,l=It:lS+l~
LN
1110?
IT=J-T~
LN
LN
LN
LN
LN
LN
01113
Cll 60 Jll=JYtN
OltlB
RO~S
IT=P1~X-J
OilA7
0104
0105
0106
0107
TOLERANCE (SINGULAR MATRIX!
00 50 K=JtN
IJ=Il•N
0086
LN
THA~
c
OOR<;
Oll'~
20t30t30
li£TUR~J
c
00.113
0084
LN 009;:>
LN no<u
LN 0094
COLUMN
35 K5=l
001<9
L"~
TEST FOR PIVOT LESS
C
L'll IIOCJ,l
LN
lfCA8SC~lGh)-A8S(AC1Jlll
00111
0013?
Q()A'I
I~
20 E'li';A::A ClJl
li'IAX=t
30 CuNTI'IIUE
c
007'•
LN 0075
LN 0071>
LN 0077
lN 007P
SEARCrl FOR MAXIMUM COEFFICIENT
tJ'"IT+I
l~JX=~<>(JX-ll+II
JJX=IXJX+IT
~0
c
f-5
AIIXJ~l=AIIXJXl-IA!IXJ)<>A(JJXll
8CIXl=8(Iq-(BCJH>~(lXJll
101
LN
LN
LN
LN
LN
LN
I.~
OlQQ
!1111\
Olll
1)112
0113
0114
oll'i
OllF>
LN
LN 0117
LN tlllll
LN !1119
I.N Ol2'l
LN 0121
L~
01?.2
USASI
c
c
FlACK SOLUTION
70
~Y=fl-1
:Ni>~J
ll
co
80 J=ltNY
IA:IT-J
lH=ti-J
lC=N
COROK=loJ
EllRI=Bil~I-AilAI"RIICI
lA:!A-N
flO IC=IC-1
l'f.TUR'-1
EN!)
FO~TAAN
NO ERI-lCflS
OlAGNOSliC RESULTS FOR
~I~Q
102
.. __
.
.... ..,
c
c
c
c
c
0
.;
c
c
c
0
o=N
oN-
~
0
0
~~~
,.. __
"'
0
c
. ..,
....
.........
~
ooc
-oo
c oc
www
www
. ....,_ .........
......
....
coo
ceo
.....
.., ..
··- .. ~~~
~
~
<t<>4
-~
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oco
NN
000
0 cc
CCC
~
•
NN
000
0
oo
.......
........
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..,...,_
0 ~ 0
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~~~
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.;
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0"0
~
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124
SAMPLE PROBLEM -
INPUT DATA
STATE REGULATOR
A=
B =
R =
-0.1
1.0
o.o
o.o
-1.0
1.0
0.0
o.o
-2.0
o.o
o.o
0.0
0.0
1.0
o.o
o.o
o.o
o.o
1.0
o.o
o.o
o.o
0.0
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
1.0
o.o
0.0
o.o
o.o
1.0
o.o
o.o
1.0
1.0
. 1. 0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
R-1 =
T =
v =
DQQ = aQ = DRR = aR
aR
a
a
=
0.1
o.o
o.o
o.o
0.1
o.o
o.o
o.o
0.1
125
1.0
o.o
co = o.o
o.o
1.0
o.o
o.o
o.o
1.0
ex
·- 0.01
Time
= o.o
- initial
TOL
=
0.2
-
TOL2
0.1
- second
X (1)
=
=
X (2)
=
X (3)
=
CY
= 0.1
sensitivity maximum for
o.o
= o.o
= o.o
x (o)
=
RIN (3)
sensitivity limit
X
(l)
X
{2)
X
(3)
- sensitivity
RIN (1) =
RIN (2)
time
integration stepping interval
1.0
o.o
o.o
Q
r
r (t)
=
1
maximum for R
{t)
Non zero
r 2 (t)
for state
rJ {t)
tracking
OUTPUT DATA
2 Rows before
0
**
**
**
**
**
~*
is the K matrix written
"
126
in this case
K =
2.575
1.851
0.6965
1. 851
2.053
0.7727
0.6965
0.7727
0.5584
The line after 0.2586E + 01 contains the following
partials:
The next output is the DS terms for the state tracking problem which in this case are forced to all
zeros.
The next output is the results of one iteration
through the
X
I
, ax , ax , ax ,
aQl aQ2 aQ3
s
Equations:
From the first two lines of this set we have in the
terminology of the text
T
=
0.200 seconds
-. 6758Xl(J' 4
0.9794
X=
-0.01098
-0.1073
=
o.o
o.o
0. 0
. -. 9849Xl(J 3
o.o
o.o
0.0 -.9657Xl0"2
128
ax
aQ
ax
aQ
ax
aQ
o.5842Xl0- 4
o.o
o.o
o.o
0.8514Xl0-: 3
o.o
o.o
o.o
0.5842Xlo-s
o.o
o.o
o.o
0.8514Xl0- 4
o.o
o.o
o.o
0.2284XlO-s
o.o
o.o
o.o
0.3328XlO-'+
o.o
o.o
o.o
=
=
*
s =
0. 9340Xl0- 2
0.8349Xl0- 3
0.3263XlQ-3
p.o
This
o.o
for the state tracking problem
would only have a
v~lue
o.o
The two numbers following last print outs of the
partials are:
ax
= o. 485lxlo-l
aQmax
ax
a~max =
The next three numbers are:
ax
aQ
max
ax
acr-
max
ax
~
max
=
=
0.4851Xl0-2
=
O.l896Xl0- 2
O.l949Xl0-1
129
The following number is the R(l,lhew= 1.2 (R(l,l6ld
= 1,200.
This is repeated until the iterat{ons on
Q(3,3) take place.
This is afbr time equals 10.0
and a 3 is printed signifying Q(3,3) iterations.
Likewise, with Q(2,2) and Q (1, 1) the final printout
is the optimum values of Q(l,l), Q(2,2), Q(3.3)
R(l,l) and R- 1 ( 1 ·, 1) •
Q = T =
These are
0.5
o.o
o.o
1.2
o.o
o.o
0.0
0.0
1.2
R- 1 (1,1) = 0.2326
R ( 1 1 1) =· 4. 3 0
The K matrix for this case is
K
=
1
1.915
1. 607
0.7092
1. 607
2.115
0.8918
.,
!·,
0.7092 0.8918 0.7161
Cost J
=
1.915
The observer for this example
TA - DT
b'
T4
=
= c
T2
T,]
Ts
TG
-.1
1
0
0
-1
1
0
-2
0.
[~
is:
0
0
[: 1
0][T
T2
D2
Ts
T:
T~
TG
:]
~..
_
......
130
=
Assume D
D
-:10
=
-15
The T matrix is
T
=
[099
.067
=
T B
The X"'
=
G
x"' 2
=
=
J
-.011
+.00137
-00475
+.000364
J
[00137
.000364
_l
r:J [:J
is
-36.6x 1 + 242z 1 + 3160z 2
with optimum control U
u = - [ .16 4
.206
'.166]
Xz
Complete optimum system with observers is similar
to that shown in Figure 5.0-3.
131
LIST OF TABLES
Table 2.1.1-1
Summary of State Regulator Equations
Table 2. 1. 2-2
Summary of State Tracking Equations
Table 2. 3-1
Definitions
Table 2.3-2
Range of Human Operator Parameters
Table 5.0-1
Costs as a Function of Q and R
Table 5.0-2
Q and R For Runs Plotted in Figure
5.0-1
Table 5.0-3
Q and R For Runs Plotted in Figure
5.0-2
132
LIST OF FIGURES
Figure 1.0-1
Figure 1.0-2
General system type
·optimal control system
Figure 2.1-1
State variable representation
Figure 2.1.1-1
Optimal control state representation of
a linear regulator.
Figure 2 .1. 2-1
Optimal control of state tracking
representation
Figure 2 .1. 2-2
Complete
state representation of
tracking problem
Figure 2.2-1
Unforced system forcing another system ·
Figure 2.2-2
Forced system
Figure 2.2-3
System 1 identical to System 2
Figure 2.2-5
Sample problem block diagram with
observer
Figure 2.3-1
Compensatory manual control system
Figure 3.0-1
Optimal solution
Figure 3.0-2
Suboptimal solution without observance
Figure 3.0-3
Suboptimal solution with observer to
make it optimal
Figure 3.1-1
Functional block of control system
Figure 3.2.1-1
Open loop transfer function of the
system
Figure 3.2.1-2
Open
loop transfer function with
states shown
133
Figure 3.2.2.1-1
Optimum Q (i 1 j) search method
Figure 3.2.2.1-2
Optimum Q(i,j) conditions
Figure 5.0-1
Cost and settling time for different
Q's and R's
Figure 5.0-2
Cost and 'Is vs. run for manual tracker
Figure 5.0-3
Optimum system for manual state
regulator
'·"
--