followed, for example, in [61 and [7]. ..... The closed-loop migration due to various
f in (3-7) and (3-13) ap- pear in Fig. 3-1 and 3-2, ..... 0.676, al = 0,0625, and a0 = -
0.0338 from (3-32)-(3-43), yields the pole-zero ...... X111 J = W11-1 ). Y11J -r11 ...
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(NASA-C6-1b3992) SIMULATED LU11YED-PANAMETEM STSTLM hFDULr.D-O-T)dh ALAeIIVL Lob'rRUL .iTU1.j1;: (V1r jiaid Polytaca nlc duu Stdt fa Uuiv . ) USLL 098 11b P HL AJb/MF A01
s
081-20760
lust.
Unclas
5 R's1
I
t ^ 1"A Rev
Virginia Polytechnic Institute q
nd nia
te Uni'versitv
Electrical Engineering BLACKSBURG, VIRGINIA 24061
SIMULATED LUMPED-PARAMETER SYSTEM REDUCED-ORDFR ADAPTIVE CONTROL STUDIES
by C. Richard .Johnson, Jr., D. A. Lawrence, T. Taylor and M. V. Malakooti
March 1981
1
Technical Report No. EE-8119 Department of Electrical Engineering Virginia Polytechnic Institute and State University Blacksburg, VA 24061
.
i.
i
This work supported by NASA Grant NAG-I-7. NASA Technical Officer: Dr. R. C. Montgomery, Flight Dynamics and Control Division, NASA Langley Research Center, Hampton, VA 23665.
i
I. INTRODUCTION Very little conclusive insult- is currently available for predicting l
the behavior of reduced-order adaptive controllers (ROAC), i.e. an adaptive controller based on a plant model of lower dimension than that of the actual plant. In [11 the ROAC problem is seen to be an unI
avoidable consequence of the application of any finite-dimensional, lumped-parameter system (LPS) adaptive control strategy to an infinite
C dimenHtona1.distr'buted-parameter system (UPS). In [2] and [31 the relative orders of the plat:;,, model, and controller are noted to be critical. In
[4] the problem
of
adaptive parameterization of a C order controller for
a M order m,.,,lc 1 of a N orderHystCm it, addressed when N - M > C. However, j
aH notL-d 111 121 and 131, the practical, poorly understood ense Is when
1
N ^ M
t:. This conditirn, i.e. the system order exceeds that of the plant
fmodel used for adaptive controller parameterization, will be considered to constitute the ROAC problem in this report. Though numerous strategies have been espoused [5] for extracting a low order model from a too complex system description, none currently seem 1
fully applicable to the real-time, recursive requirements of on-line adaptive control algorithms. These reduced-order modaling techniques seem to fall into two broad categories: (1) Extract tbuse "modes" (or componentsubsystems) from the full ;system description that are most tnfluentlil in the performance of the model in its subsequent use. This strategy is followed, for example, in [61 and [7]. (ii) Parameterize the reducedorder structure• to provide the best prediction of the desired output. This latter strategy will produce a model the "modes" of which need not currespon:: to any of those of the full system as noted in [3]. This latter strategy includes the model reference approach prominent in adaptive systems [8j. Two methods of interpreting the misbehavior of ROAC, one
P
1
L
1.2
based on system input - output description
[
3) [91 and one on state-
variable description [11 1101, have emerged recently from attempts to !
1
develop adaptive controllers for flexible spacecraft. Consider the implementation of the proper, single-input, singleoutput (STSO), autoregroo live, moving-average (ARMA) system N
^
y(k) _
[ai y(k-i) + b i u(k-i)) (N even),
(1-1)
i=1
where u is the system input and y the output, in parallel, i.e. partial fraction expanded, form as
I
Y^
(k) -
t
[exi t
y t (k-i) +
pit u(k-i))
(1-2)
i =1
N/2 y (k) - X Y;( k ).
(1-3)
^=1
where y y is the output of the R-th mode. As interpreted in 131 the first strategy in the preceding paragraph when used for identification would lead to approximation of (1-3) as M/2 of [1,N/21 Y s (k) -
YJt(k)
(1-4)
t with (1-2) describing the dynamics of each y,(k). The M/2 modes chosen in i.
(1 +) may by selected as those M/2 from the N /2 in (1-3) that provide the "best" fit of y s to y. Note that the order M of (1-4) refers to the order of the underlying reduced-order ARMA model. If the dimension of (1-4) (and (1-1)) were based on the number of quadratic "modes" then M (and N) would be replaced by M = 2 (and N = 2). This quadratic "'modal" designation of order is common in the flexible spacecraft literature [3). As noted in [11, [3), and [111 the extraction of the y f chosen in (1-4) from the y in
1.3
(1-3) remains an open que= stion. However, these y, would be required
for identification of the appropriate (1-2) parameters a iR and Bit. Alternatively, the second strategy would approximate tl-3) with
N M/2 of [1, Y1 9(k)
Y$(k)
(1-5)
in, e.g., a le=ast-squares sense by selection of the j,M M. These estimated "moual" outputs need not match the corresponding values in ( 1-2) and, therefore, need not lead to identification of the corresponding a il and
Oil
in (1 - 2) as noted in [3]. The successive y l chosen to fit (1-5) to (1-3) may not even obey a time-invariant difference equation of the form of (1-2)
With y t replaced by y C The problem with closing, the adaptive control loots via a simultaneous ide= ntification and control strategy could be intensified
by
feeding hack the ^^^ instead
of the unavailable y, to meet .1 modal
control objective. The alte= rn ate interpretation of the ill-effects of ROAC, derived from the flexible spacecraft control problem [1) [101, begins with the separation of a SISO, state-space model for, e.g., (1-1) into the reduced-order and unmodeled (or residual) segments as
xN(k+l) xR(k+1)
AN ANR
xN(k)
ARN A R
xR(k)I
+
bN
u(k)
(1-6)
b
XN(k) (1-7)
y (k) - [ c N cR1 xR(k)
As outlined in [1] the derivation of (1-6) - (1-7) can be viewed as a
I.4
Iprojection operation on the full system. From (1-7)
the "spillover"
of the residual modes into the observation of y(k) via c g x R (k) and from
(1-6) the ":spillover" of tl• e control designed for the reduced model into
residual
the
possible
modes x R (k+l) via b
u(k) are clearly displayed. Also the
coupling of reduced model modes and residual modes via
I
ANR
and
ARN
is
immediately apparent. Shown in [1] and [10; is the predictable fact that j
if
ANR
0 and c R
0 then,
assuming the residual modes remai.a stable,
any full-order (N), stable adaptive controller identifying AN , bN , and cN explicitly or implicitly from y and u alone would retain its stability. If ANR = 0 but
C
and b
are nonzero then the degradation of RUAC has two
sources. Unmodeled components in y via c
will generate an error that
Is Indistinguishable from parameter error thereby causing .ada.ption. 11io .application of control u to the residual states x
via b
will contribute
further Lo this unmodeled component of y. Not only will the parameter e:stim,aLo!; be :ncorrvt , t from use of y and a to identify only AN , b N , and c but also ti l t , s g ait, cast im.aters provided by an adaptive observer will be incorrect, whiell if fed back could lead to an unpredictable "controlled" resp,,nse. These two problems of inappropriate parameter and state estimation are the same ones noted in the first interpretation with y approximation of y in (1-5). The ability to extract the y t and obtain y $ corresponds to an effective zeroing of c R . This report will, in part, attempt to implement, compare, and contrast these two strategies embodied in (1-4) and (1-5). The next seetion details the specific objectives of this study. Section III presents the example autoregressive, moving-average plants that are to be used in
the simulations.
Section IV presents the adaptive
t control algorithms to bs used and their sources in the literature. Section V outlines the formats for the simulated tests including that
i 1
description of numerical figures of merit to be tabulated in Section VI.
I
Section VII offers interpretations of the test results and Section VIII draws conclusions relevant to the RnAC problem. The last sections of this report include the referenced literature and the appendices Including computer program listings.
1
11.1
11. OBJE CTIVES
}
The principal objective of this study is to test the usefulness of the folklore of reduced -order modeling with respect to adaptive control. In particular four "facts" will be tested: (i)
heavily damped modes may be neglected relative to more lightly
damped nx)dcs in reduced- order -model derivation. (ii)
Finite bandwidth actuatoraz limit the number of moles necessary to be
models-".. (iii) An "optinwal" reduced-order controller neglects the modes contributice measure. ing the least degradation in the control system perforwr (iv)
Indirect and direct adaptive control are essentially equivalent and
intvirchangeable. The iml"1 icat irna to the BOAC. problem of each of these statements will be developed in the following paragraphs. The .9imulations of the following sections will be chosen to test the veracity of these implications. The conclu:aion of this report will summarize the useful "facts" that either escape unscathed or emerge from these tests. Since each of these facts has
been accepted into the reduced-order
and/or adaptive control folklore it is difficult to pinpoint particular references suocinctly stating these points. However, several classical control texts contain the source of point (i) in the concept of dominant roots or poles. For example: "The complex conjugate roots near the origin of the s-plane relative to the other roots of the closed-loop system are labeled the dominant roots of the system since they represent or dominate the transient response. The relative dominance of the roots is determined by the ratio of the real parts of the complex roots and
{
11.2
will result in reasonable dossinance for ratios exceeding five. ... Dominance .. also depends upon the relative magnitudes ... of the residues evaluated at the complex roots, [which) depend upon the I,ocation of the zeros ict the s-plane" [121. Or: "The relative dominance of closed-loop poles is determined by the ratio of the real parts of the closed-loop poles, ns well as by the relative magnitudes of the residues evaluated at the closed-loop poles. The magnitudes of the residues depend tipon both the closed-loop poles and zeros. If the ratios of the
real parts exceed five, and there are no zeros nearby, then the
closed-loop poles nearest the jW axis will dominate in the transientresponse behavior because these poles correspond ro trans ient-response terms which
viol IV
SI„wIN" I I i, p. ." )1 1.
'his t,vllarat I«n
cc,ssovpt has hvvii
forma-
lized via singular perturbation theory [141. Despite the concomitant warning in both 1121 and [13] for caution in the use of this rule and the explication of its applicatIon to the closed-loop system, point (i) is commonly (though admittedly inappropriately) used for model reduction prior to control design. The (mis)implication for discrete systems is that if the open-loop singulafities are separable into a group in the z-plane outside a radius of r (rR CKI + F: 0 tK1 RESULTING CHAR. FUN. s Pkx: -Ik tA1 + A?+F« (B + EI ) + A1wA2 +F>4 t8wt1?*1 xAll PARAMETERS: At-0.9
A4=0.1
810.1
E=-0.01
Fig. 3-3: Closed-loop Root Migration due to Output Feedback for Example 2.
I1I.9
10.00
._...POL i_ S _?[HO PLAN, t Y (x) - (Ai + API r (K-11 - (A1%A21 T (K-4`1 + (B+Ft oW (K 11 - (A1 «E+BkA;' xl1 tK- 2) MESULTING CHAR. EON.t 7Ak2-1x(A1+A2+Fk5j+RIxA2+BmA2xF i
PRRAMETERSt AI : 0.9
R2=0.1
0=0.1
E=-0.01
Fig. 3-4: Closed-loop Root Migration due to Partial State Feedback for Example 2.
r
III.1O
9
Z
• Ir.
—POLES ._...ZERO "L ANT: T IKl x (Ai+A21 T lK-1J - (AINA21 T tK
CONTROL = U tK1 = FINK W +F, I %T
-21 + t8+E1 xUIK-11 - IA1xE+IiKA2j KIIIK-21
IK1
MESULT;NG CHAR, FUN.: ?x+.2 ix i n 1 + A2 + Fx ($+EI ) +A1KA2+Fx tBx p 2*FxP; )
PARAMETERS: Al=0.9
A2-('.R
B=0.08
E=0.02
Fig. 3-5: Closed-loop Rost,: Migration due to Output Feedback for Example 3.
ou
C
71 LW
ja -
I. -
- - -
- -, -
n
a
4
- 1
v
V.
0z) -_ - - , b, 00' 0 - 0
VA4.
A
-,
- T -13 U
IV. GO
r C Tci
— Pot
t5
- __ 71 Flo N ANT ,
1
y K
ME 5UL T I N5 C PWIAMt I t R )-
k
I - :A1
I 11:
0. 9
P,:
Fig. 3-6: Closed-loop Root Migration due to Partial State Feedback for Example 3.
a
111
.12
important mode for partial state feedback by either swapping the elements
of they state feedback gain vector in (3-14) or swapping the designation of
a i , b and a 2 , t results in the cl o sed -loop roo t s shown in Fig. 3-7. Note in Fig.
3-5 for f ^ -
the faster
mode
3.8 the near-c a ncellation of the slower mode while
matches that desired. The degree of closeness of this near
cancellation is not maatLhed in either Fig. 3-6 or 3-7. Mote that in Fig.
3-6 if f is chosen as -5 tee cause q l - 0.5, use of the same f in Fig. 3-5 WOUld result in overcoanpcnsation. Neither in the open-loop plant nor for any f placing one pole near the desired location of d - 0.5 are the
singu,arities separable as in cattg;ory (iii).
?:x 0
(4-83)
u 2 > 0
(4-84)
P 1 > 0
(4-85)
P 2 > 0
(4-86)
1V.9
z -1 + q 2 x-2 1 -1 a 11 z - a12x -z
1+ q x^ 1 -
> o
q l could be chosen as -0.98 and q.,
v
as
(Z(K 1
(4-137)
zero for the rapidly sampled
examples 5 and h and (4-87) would be satisfied.
I ; la
(4-88)
li is nominally chosen as one unlotisa such a value would c:a yse e 2 (k)
0 or
.4 1 (k)a I (k) + h' 2 (k) - a,(k) ti l` (k)/.^,(k) = 0. Then h is chuNen within the range of (4-88) such that the absolute value of both terms
c = 0+.
Each of thr last three algorithms in (4-50)-(4-8H) provides a
( fx.,(k), t, M
r2(k) for
user
I s l (k-1) - 7 1 - a i (k)
vl(k-1)
l (k),
in (refer to (3-21)-(3-30)).
(4-89)
l(k)--^ 2 (k)al( W) Pi I (k-1) a 2 (k)+(n l (k-IYx l (k)-a 2 (k)+ti l ) t eax
_._ ... ^. ___ a 2 (W (k)-a (k)/61(k)l I (k)+e 2 1 ( y l (k)
2(k))l
(4- 9n) n,(k-1) ,(k-1)
x l (k)si l (k-1) + '2 -• t z l (k)v 1 (k-1)
(k)
n.,(k-1) ^2(k),'RZ(k)
(4-91) (4-92)
for parameterization of u(k-1)
a 6r(k-1) + 1 1 1 (k-1)u(k-2) + n2(k-1)u(k-3) + V (k-1)y(k-2) + %-,,(k-1)y(k-1)
(4-93)
tor application to the difference equation repres ont.at ion 1 1 (3-17) y ( k)
21 + tx ll )^ (k-1) + ( x L2 - .x 11 "` 21 + "1L)y(k-2) - (xll`a2^, + ^`12"21)y(k-3) - a12`a22y(k-s) +(6 11 + ^, 1 )u(k-1) - (;^ ll : x V1 612 + 21 (t 11 - Bz.y)u(k-2) - 6 11x 22 + 6 12`x 21 + ti, 1 `a 12 + F22"11)u(k-3) t2 - (tz12 x "''' + r 4 :'x )u(k-4)
(4-94)
^.
IV. 10
!
generating y(k). Mote that all of the preceding schows in (4-4)-(4-93) use the current information as efficiently as possible and therefore I result
in causal rather than strictly causal adaptive controllers (111.
(See Appendix F for programs of each of these controllers.)
•R
v.1
V. TEST FORMATS Having specified the non-adaptive &ad adaptive control strategies and objectives the remaining items to be selected for the testing of the guidelines in the second section are the reference inputs, adaption stepsize constants, parameter estimate initializations, and performance measures. Selection of the reference signal is important for two reasons: richness and magnitude. The magnitude of the reference signal is important due to the nunlinearity of the adaptive control system and therefore the anticipated transient (at least) differences due to different signal levels. The richness is Important because it is well-recognized [381 that a sufficient "input" richness in necessary to perturb all of the modus of the system for their "identification." This "input" differs for the indirect and direct approaches and the equation and output error formulation.. For the equation error indirect approach the plant input and output must be sufficiently rich. Since the plant is linear this translates to
it
sufficient frequency content of the plant input (or control)
signal, of which the reference is only one component. The difficulty of assuming adaptive control input richness based on reference signal richness is addressed in 1301. For the output error indirect approach the plant input and identifier output must 1-e sufficiently rich. In an openloop output error identification task the identifier output richness can be translated [391 solely to input richness necessity under the assumption that the identifier output asymptotically converges to the plant output, which requires sufficient identifier order absent from ROAC. For direct adaptive control the control paraw-ter identification problem must be
r
i
v. 2
recast as an open-loop identification task as in 119
1
and [371. In the
equation error case the richness rrquiresents are converted principally to conditions on the model output to be tracked or, if generated by a 1
linear, timer - invariant model, to the forcing or reference signal 139].
Once regain this assumes sufficient controller order. The requirements for output error based direct aaiaptive control, though as yet unspecified, are assumed to be similar.
All of the preceding richness requirements are for complete identification and assume non-reduced adaptive model order. As is well-recognized
[221 (341 1371,for a control objective complete identification may nut be necessary. Consider the control of any order system with n single constant
i a
output feedbat-k gain and a ref erenve.= signal gain. Tea achieve ARymptotic convergence to a particular DC level set-point any feedback grain stabilizg
T S
in
the system in can jun: tion with the appropriate reference gain will be
adequate. This applies readily to examples 1-4 in section three. As noted in [401 for open-loop output error (and by implication equation error) identification the richness of the input signal for stable reducedorder identification its dependent on the reduced-order model dimension, not that of the higher order plant. From a frequency response point of view, the viewpoint promote) in [411 for reduced-order modelling, the minimum number of sinusoids providing this richness matchets the number of points at which the controller can specify the steady-state frequency response. Therefore, if this reduced-order controller stabilizes the system, asymptotically convergent steady-state control seems possible up to the richness level of the reduced-order model. When the signal to be tracked (or equivalently the reference signal) has a higher frequency content,
^I T
reduced-order tracking will onlv be approximate at best. The reference
G
t
v. 3
signal should be chosen to test this boundary condition, with the principal case of interest being reference signal over-richness. Therefore the following reference signals (all zero prior to k - 0) will be considered: INPUT 1 (unit step input): r
1
M - 1
(5-1)
INPUT 2 (single low-frequency sinusoid): r1 (k) - 2
Kin(kw 2T) (5-2)
Note that the sample period T Ilia f requenev :. r 2 ex.t111111
`x 1-4, i.e.
resspvt , t iveiy, and 5 and
iss
0,
In assumed
to be one second for examples 1-4.
chosen an 201 below the desired cutoff frequency for
0.17851, 0.34443. 0.5502, and 0.40866 radianK per second, 2U'; below
the dominant natural fregt-eney for examples
i.e. 0.40 radian-: per
isc • comd,
with T - 0,5 second$,
INPUT 3 (;single high-frequency minussoid): r (k) - 2 sink, 1)
(5-3)
ABaiss the sample period is assumed to be one second for examplvs 1-4. The frvgtjvnev . 3
1.4 double the dt • sirea cutoff for examples 1-4, I.e. respective-
ly 0.44tQ9, 0.86157,
1.38629, .snd 1.02170, and double thc= dominant natural
frequency for examples 5 and 6, i.e. 1.00, with T - 0.5 second;., INPUT 4 (non-zero mean, uniform, white noise): r 4 (k,)
t
1-0-5,1- S ]
(5-4)
This is clearly the over-rich input of most interest. Since the rate of convergence and parameter estimate variance near convergence are commonly acknowledged as being affected by the adaptive step-size (SS) coefficients pi and p i , two cases will be considered. t
TOO
v.4
SS - 1.
ui -Pi -
1
(5-.S',
SS 2 - 0.1: (5-6)
u i - p 3 - 0.1.
Given the approximate nature of the
gradient
formulas for GOI embodied in the
required u and p smallness noted in (4-20)-(4-21) and (4-71)-(4-14) the u acrd p of (5-5) may lead to unstable behavior for C;01 while (5-6) :should prove more satisfacLory. (Note that h in the normalizing term of all of the algorithms is selected as unity). For output error type algorithms such as SOI and SOD the error smoothing coefficients q in (4-24), (4-47), and (4-77) are to be selected in satisfaction of (4-32), (4-48), and (4-b7), respectively. As noted in [42] selection of these q to make (4-32), (4-48), and (4-87) equal unity does not offer tht, most rapid convergence. Furthermore, in a reducedorder application these q can influence the mean convergence paint. Such factors may be used to advantage in (4-48) where f is known a priori, but In (4-32) and (4-87) the strictly positive real (SPR) condition is dependent on unknown p1nnt parameters. 'Therefore several values will be tested. They are sunurzarized in Table 5-1. Since only Local convergence is currently anticipated three settings of the initial plant (and therefore controller) parameter estimates (EST) will be considered corresponding to the nominal values from control design based on neglecting the second mode (i.e.
e =
0 for examples 1-4 and a 2 - 0
for examples 5 and 6) and ± 20% of these val , aes. These values are summarizer) in Table 5-2. The controller parameter initializations for the direct schemes are based on the corresponding plant parameter initializations and
V.5
Lx^^raplc ( s)
Algorithm
Coefficients)
1.r)l^el
-0.9
SC-
-0.95
SCO
-0.97
SC+
-0.7
SG-
-0.8
SCO
-0.9
SG+
So
-0.9
SCO
St)1
-0.9
SCO
Sol)
-0.5
SC
-0.8
Sc:—
--0.9
SCO
-0.95
SC-+-
-0.4
Sc-
-0.6
SCO
-0.g
SC+
Sol
Sul)
-- -
Error S mo othing
1
Su 1
SOD
-.
5&6
`Sul
q1--0.9,g2-0
SG-
g 1 =-1.846,g 2 -0.9515
SGO
g 1 4 -0.96,g2-0
SC+
TABLE 5-1; Error Smoothing Coefficients
i
I^ I'
1
~
- -
_--__~~^
V.6
Example
Parameter
SST -f20%
EST
20%
0.052 -
0.76 0.08
0.72 -------~---0.064 0.72 0.4 0.72 --------'--0.046 14 12
0.139
0.093
0.116
ti&6
l.477 1.086
-0.905
-0.?-)4 ^ -'-'--------/
TABLE 5-2: Paromo,r Estimate Initial loztIons.
f
I
V.7
1 (3-8)
and
( 3-9)
for examples 1-4 and
(3-27)-(3-30)
for examples 5 and 6.
Nnte the instability of the EST - + 20% initializations. Since the control objective of all of the examples in section 3 is model-following, the tracking error is the principal performance measure. For the expectation of asymptotic tracking, the figures of merit should be divisible into short and long term quantities. The mean and variance of they tracking errors should be normalized and tabulated for comparison among the examples and between the fixed and adaptive contro' schemes. Similarly the input cost and controller (and plant, for indirect schemes) parameter estimates should be observed. Therefore the following quantities will be tabulated at k
- 1, 2, 5, 10, 20, 50, 100, 200, 500, and 1000
iterations. Instantaneous desired output: IDO (k) = s (k)
(5-7)
Instantaneous controlled output: `
IO(k) - y(k)
(5-8)
Normalized instantaneous tracking error:
NITE(k) - (s(k) - y(k))
(5-9)
s (k)
Normalized average segmented tracking error:
(
l ki
NASTE - I
1
l k
i
- k
(5-10)
NITE( ,j )
J k1-1+1 i-1 Jj
Normalized segmented tracking error variance
NSTEV
(
^n (ri 1) j n l
ki NITE2 (j ) j=k
i-1+1
r k
L
i
j^ki_1+1
2l NITE (j )] r 1
(5-11)
r f
V.8
(Note that n is k
- k 1-1 minus the number of times s ( k) - 10 -3 , which
are excluded due to (5-9).
If NSTL•'V is negative due to round-off, it
is print ed as zero.) Seymentvd average squared input ki 1
ki - k
Segmented average
l-1
i-ki-l4.1
controller parameter c:st im.-atC for f
i
._.k
1
k
^'
- J .3'kt-1+1 i-1
(Similarly for g, a, b, ,t, r,
f (k)
(5-13)
v, and n.)
Segmented controller parameter estimate variance for g ki (k i "1kl-1) (k
i- k
i - 1- 11
j ( k i - ki-1) `
Y
f*1(.1)
i-1
ki
ink 1-1*1 (5-14)
(Similarly for g, a, b, i t e, v, and n.) These quantities are tabulated for each of the combinations of example, adaptive controt algorithm, reference input, step-:size weights, error Smoothing coefficients (if necessary), and parameter estimate initialization.
(Sve Appendix F for listings of the programs ooml•ilinyt these
figures tit mor i t .) Table S-3 summarizeas all the various combinations that were simulated.
a
V. 9
Example
control Alg.
nonadaptive
1
--
S ize
Smooth
Coeff -
Actuator Conf1guration -
Total
FST - 0
zero
all 3
both
-
-
Gc)1 (2)
all 4
all 'I
both
-
-24
Sol (3)
all 4
all i
both
all 3
-
72
GED/SFD (4)
all 4
all 3
but11
-
-
24
SUD (5)
call 4
all 3
both
all 3
-
72
all 4 all 4
EST = 0 EST - 0
zero SSl
-
-
4 4
GUI (2)
all 4
EST = 0
SS1
-
-
4
Sol (3)
all 4
EST - 0
SS1
SCO
-
4
GED/SED (4)
all 4
EST - 0
SS l
-
-
4
SOD (5)
all 4
EST - 0
SS 1
S CO
-
4
all 4
EST - 0
zero
-
-
4
all 4
I:ST - 0
SS1
-
-
4
(DUI (2)
all 4
EST = 0
SSl
-
-
4
Sl)1
(3)
'111
4
FS -
0
I's
SCO
-
4
GI:1)/sr:11 (^^
al
4
I:ST ^
n
ss l
-
-
4
Sol) (5)
all
4
EST - 0
SS1
SCU
-
(1)
nonadnpt ive _ (1111/SEI (1)
-
24
-
-
24
-
72
-
24
xo-aro " hoth
t:lI (2)
all 4
' ill 3
both
(3)
all 4
all 3
both
all
GED/SFD (4)
all 4
all 3
both
-
Sol) (5)
all 4
all 3
both
all 4 all 4
all 3 all 3
zero both
all 3 _ ^ __
GUI (2)
all 4
all 3
SUI (3)
all 4
nonadaptive (;1;I/SF1 (1)
4
4
p' EST x111 3
! 1
4 24
: -
1111 4 ill 4
Sol
3
.^^_
72
-
-
12 24
both
-
-
24
all 3
both
All 3
-
72
all 4 all 4
all :3 all 3
zero both
-
ig.12a&b all 3
24 72
GUI (2)
all 4
all 3
both
-
all 3
72
SO1 (3)
all 4
all 3
both
all 3
all 3
216
nonadaptive (=E1/SE1 (1) b
snit.
error
all 4
GEE /SEl
5
Step
all 4
nonadapt ive
3
Par. Fmt.
GEI/SFl (1)
nonadaptive GEI/SFI (1)
2
Inputs
1008
TABLE 5-3; Tested Combinational.
V1.1
Vl. RESULTS Due
to the number (1008) of printouts for the examples shown in
Table 5-3 the tabulated results are under separate cover. (Sec s Appendix F for pert iuent program listings.) The next section provides an evaluation of thus a numc• r ic.al results.
V11.1
VII. TEST RESULTS INTERPRETATIONS
Refer to section 2 for the statement of the ob_jertives of this study. The four questicxas raised in than section will be addressed In this section with respect to the simulations of Section 6. QUESTION (1) : Are heavily damped open-loop nacsdes negIvet.able relative to more lightly damped mudrs? In particular: a)
Are the accepted criteria for neglecting modes, baaPsvd tin relative damping, realistic for the examples conesidere • d here in a fixed (non- adaptive) reduced-order controller?
b)
For these examples, haw dt. the vartoues adaptive control eachrmvs l area under they reduced order modeling "rulvs" e concerning relativ damping? More specific questions might be: Do the reduced-order adaptive control systemab remain stable? Can the adaptive mechanism compensate for mismodeling to yield improved performance over non-adaptive (modal) controllers of the same order?
In example 1, the open loop and closed loop modes are separable based tin the accepted criteria for relative damping, so the good closedloop (non-adaapeive) reduced-order performance was to be expected (despite the fact that in this example and all the others the reduced-order modal model was based strictly can the partial fraction expansion term without DC gain correction). Fxnmple 2 represents a marginally separable system in closed loop, yet the (nun-adaptive) reduced-order controller traacRing errors are only slightly worse than in example I. 'this Indicates
that, for these examples, this boundary between separable and inseparable
VII.2
modes is reasonable but fuzzy (as expected). Example 3 possesses neither closed loop nor open
loop
separability,
but sa pole-zero near-c-incellation should leave one mode dominating, the system bahavior. As shtwn in the stmulat ion, t l, e reduced-order lnatdeling cause•sa significant errors. This is presumably due to the relative closeness of the "cancelled" pole to the unit rircle< compared to the "domillant"
pule.
Smill errorn in cancellation In this erase may have
large effects t »t system behavior and the performance of reduced-circler t ont ro I. i'he approximat:lon of a pair of complex pules by a simple real axis pole ias studied in example 4. Here, no first order aasodel can be found to
clost > l y represt -tit
this
second order :system, and considerable deagrad-
:at lon in portoinwat-v by re.luced order coo;,trol should be expected. Tile sltrul.at ions show this to
lit-
true; the ssto ady state tracking errors are
sst gni f ic.aut . tit
the five .adaptive control algorithms, all but the second one,
tile gradIt'llt --otitptit -vrror- ludiroot (Gol) scht'mv. were s-table.
Ill v%ample tint',
whit'll has the most accurate reduced order model, a larger slumber of test combilnations
tut
input:t, step sizes, and Initial parameter estimatt'ss
produce stabl y responses than the other examples. Even at), the only consistent results were for zero Initial parameter error (with respect to the extracted modal model) and the step and random inputs. In ticese cases, the step response is improved over the non-adaptive case to essentially zero :steady state error, while the .random input causes a stable yet more widely ranging response. The other four adaptive algorithms produced essentially equivalent responses for most inputs, step sizes and smoothing coefficients. Parameter convergence occurred
(t
{Fj{ 2
between 50 and 500 iterations for
t VI1.3 I most run g , the random input ca g e being consistently faster. Also, t1,e error between the desired and actual outputs alw+aym resultNd in improvement over the non-adaptive output error. Some minor differences in responses were: • Example 3 usually responded f:.:.ter, due to the dominant pole being, more highly damped. • Algorithm 5 (stahiaity-output-P , ,or-ciirect, SOD) prod red consistently smaller steady sta f r output tracking errors. • Smaller step sizes improved the steady state tracking error variance, but lengthened convergence times. • The smmpomition of linear svat ems with applications to modal roduct ion," Ittt . ,1. Control, to appear.
The Model Refer -eAl,^roach, Now York: 181 Y. 11. I.indau, Adccwivc Control: _ _.. Kircel-Dokker, 190.
. and R. C. Montgomery, "A distributed sytstem adaptive [91 C. R. Johnson, Jr. control strateg y ," IEE E Tra ns. on Ae raaspac e an d I:lo1 tr ^^., vol. AFS-15, no. 5, pp. 601-612, Scptember^1979.
(101 M. .1. Iialas and C: R. Johnson, Jr., "Townrd adaptive control cif large space structures," In Ali.Elicat i onts of Adapti ve Control, eds. K. S. Narcndrea and R. V. Motiollol 1. New York: Academic Pres.-4, pp. 113-344, 1980. [ 1 1 1 C. R. Johnson, Jr. and M.
T. R alrals, "Distributed parameter system coupled ARMA expansion Went If teat ion and adaptive parallel I IR fI Ito rIngl* A uuit it , d problem :statement," Pror. lath AsIIomnr Conf. on ('IrcuIts Svti -• and Computers, , 1"aril it t,ruvc , CA, p p 21N - '".S November
1979,
[12] R. C. Dort, Modern Control Sy -st1-ms. Reading, MA: Addis " in-IN . •Av, p.
1131 K. t [
'i
F 4
e
t)9AL 1 %,
11.111,
M,,dorn Control l_ at;itll^^rin^,. _
1()70.
Fnglowood C1ifis, N.1. Pronttee-.
l
IX.2
[ 14
j
B. F. Gardner, Jr. and
.l.
B. Cruz, Jr., "Lower order control for
systems with fast and slow modes," Autoamatica, vol. 16, no.2, pp. 211-213, March 1980.
1151 P. C. Hughes And T. M. Abdel-R.ahmman, "Stability of proportionalxible spacecra ft," plus-derivative-plus-integral control of Journal of Guidance and Control, vol. 2, pp. 499-503, NovemberDecember 1979. [161 K. S. Narendra and L. S. Valavani, "Direct and indirect model reference adaptive control," Automatica, vol. 15, no.6, pp. 653-664, November 1979. 1171 C. R. Johnson, Jr., "Input matching, error augmentation, self-tuning, and output error identification: Algorithmic similarities in discrete adaptive model following," IEEE Trans. or -A uto. Control, vol. AC-25,
:eo.4, pp.697 -703,
f
Augut't 1980.
118 1 C. R. .tolanaaon, Jr., "Explication of a simple output error based model
reference adaptive controller," VI'I&SU Dept. of Elec. MR. Tech. Report No. EE8013, June 1980. 119] C. R. Johnson, Jr., "An output error identification interpretation of
modei reterviii-v adaptive control," Autonuiticaa, vol. I6, no.4, pp. 419421,
1980,
[201 K. S. Narendraa and Y-H. Lin, "Stable discrete adaptive control," IEEE
Trans. on Auto. Control, rol. AC-25, no.3, pp. 456-461, ,tune 1980.M
Jr.,
(.' 1 1 B. Widrow, .1. M. McCool, M. G. Larimore, and C. R. Johns on , "Stationar y and nonstrationary learning characteristics of the LMS .adaptive filter," Proc. IEEE,vo1. 64, no.8, pp. 1151-1162, August 1976. [ 2 2] C. R. Johnson, Jr., . "Adaptive parameter matrix and output vector estimation via an equation error formulation," I EEE Tra ns . _En .+mss. , M. an^^a nd C yt+ernetic s, vol. SMC-9, no.7, pp. 392-397. July 1979.
1231 1). Parikh and N. Ahmed, "the an adaptive algorithm for 11R filters," Prue. IEEE, vol.66, no.5, pp. 585-587, May 1978. [241 C. R. Johnson, Jr., "A convergence proof for a hyperstaable adaptive recursive filter," IEEE Trans. on Info. Ttay., vol. IT-25, no.6, -pp. 745-749, November 1979. [251 C. R. Johnson, Jr., "A stable family of adaptive IIR filters," Proc. 1980 IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Denver, CO, pp. 997-1000, April 1980. 126] C. R. Johnson, Jr. and M. J. Bralaas, "Reduced order adaptive controller studies," Proc. 1980 .It. Auto. Control Conf., San Frnncisco, CA, Paper WP24, August 1980.
f
[271 C. P. Newman and C. S. Bfaraadello, "Digital transfer functions for microcomputer control," IEEE Trans.
on Sys., Mean, sand Cybern etics,
no. 12, pp. 856-860, Uecomher 1979.
vol. SMC - 9,
--
T
..
I
1X.3
I
[28] C. R. Johnson, Jr., "On direct adaptive control of nonminimum-phase plants," Information Sciences, vol. 20, pp. 165-179, April 1980. [29] G. F. Franklin and J. D. Powell, Digital Control of Dynamic Systems. Reading, MA: Addison-Wesley, p. 64, 1980. [30] G. Kreisselmeier, "Adaptive control via adaptive observation and asymptotic feedback matrix synthesis," IEEE Trans. on Auto. Control, vol. AC-25, no.4, pp. 717-722, August 1980. [31] G. C. Goodwin and K. S. Sin, "Adaptive control of nonminimum phase systems," Univ. of Newcastle Dept. of Elec. Eng. Tech. Report No. EE 7918, August 1979. [32] A. L. Hamm and C. R. .Johnson, Jr., "Model reference ada p tive control using an adjustable model," Proc. 12th Southeastern Symp. on Sys. Thy., Virginia Beach, VA, pp. 269-273, May 1980. [33] C. R. Johnson, .Jr., "An adjustable model reference adaptive control reconfiguration of self-tuning pole replacement," Proc. IEEE, vol. 68, no.2, pp. 295-296, February 1980. [34] J. M. Mendel, Discrete Techniques of Parameter Estimation: The Equation Error Formulation. New York: Marcel Dekker, 1973. [35] K. J. Astrom, U. Borisson, L. Ljung, and B. Wittenmark, "Theory and application of self-tuning regulators;'Automatica, vol. 13, no.5, pp. 507-517, September 1977. [36] G. C. Goodwin, P. J. Ramadge, and P. E. Caines," Discrete-time multivariable adaptive control," IEEE Trans. on Auto. Control, vol. AC-25, no.3, pp. 449-456, June 1980. (37] C. R. Johnson, Jr. and E. Tse, "Adaptive implementation of one-stepahead optimal control via input matching," IEEE Trans. on Auto. Control, vol. AG-23, no.5, pp. 865-872, October 1978.
[38] J. S.-c;.
Yuan and W. M. Wonliam, "Probing signals for model reference identification," IEEE Trans. on Auto. Control, vol. AC-22, no.4, pp. 530-538, August 1977.
[39] B. D. 0. Anderson and C. R. Johnson, Jr., "Exponential convergence of adaptive identification and control algorithms," University of Newcastle Dept. of Elee. Eng. Tech. Report, April 1980. [40] C. R. Johnson, Jr. and B. D. 0. Anderson, "On reduced-order adaptive output error identification and adaptive IIR filtering," May 1980.
i
[41] S. A. Marshall, "The design of reduced-order systems," Int. J. Control, vol. 31, no.4, pp. 677-690, April 1980. [42] C. R. Johnson, Jr. and T. Taylor, "CHARY convergence studies," Proc. 13th Asilomar Conf. on Circuits, Sys., and Computers, Pacific Grove, CA, pp. 403-407, November 1979.
W
i
t
Appendix A: Hoot Locus Program Listings and Tabular outputs
Contents: p. A-1: A-2: A-3: A-4: A-5: A-6:
A-7:
Program determining
p
and p 2 in ( 3-12) for Examples 1-4
p l , p 29 h, and f for Example 1 (Fig. 3-1) pit
p2, h, and f for Example 2 (Fig. 3-3)
p l , p 2 , h, and f for Example 3 (Fig. 3-5) P
1' P2' h, and f for Example 4 (Fig. 3-8)
Program determining q for Examples 1-4 ql ,
and q 2 in (3-15)
q2, h, and f for Example 1 (Fig. 3-2)
A-8:
q l , q 2 , h, and f for Example 2 (Fig. 3-4)
A-9:
q l , q 2 , h, and f for Example 3 (Fig. 3-6)
A-10:
q l , q 2 , ;,, and f for Example 4 (Fig. 3-9)
A-11:
Program determining q l - a l , q 2 = a 2 + cf,
i.e.[0 f]
in (3-14), for Example 3
A-12:
q l , q 2 , h, and f for Example 3 (Fig. 3-7)
A-13:
Sample Plot Program
(Fig. 3-1)
A-1
IJola C C C C
4AiFIV91(,P-^49PALEx1009IIMtalOn k7L1: ROOTLOCUS IN7LkPKE7A71L'N OF TMi- UNACAPTif PLANT, T.F.:Y(Z )/F ( Z)=( (B+E )*Z-(E3*A2+E*Al IM Z**2- (A! A20 *( +E ) ) *Z-tAI*A F +F*(8*AZ+L *A1 ) ^ L DIMENSION t( 4) XX1(50 0) t XX2(500) YYl(5U0 )9YY21`.U(lvA Z(^u0)9Fi(50) UIMENSION 9 A2(4) 9 6(4) 9YM1l50U ► 9YMA2l!UO) COM P LEX GC } FCrZC(2I
Al(41
DATA E/U.U19-U.U1U.U19-0.3/
DATA Al/C.95g0.')96.Yr0.9./ rA7A A2/U.^ ii 0.1 O.tIr0.7/ L`ATA B /0.()67 9 0. 90.0E9005/ UU 3UO J=1 94 2U l.I
40.)
Wk1TE(4 20U)A1(J) A2 WvB(J) FORMAT( ;Xr'Ain' F13.t)95X99AZ='9FI3.695Xr'ii¢'rF13.E,) WKIICl(,94UU)E(Jj FORMAT(// ' E.= '9F13.b)
DC 100 K=l950 1F(J.EC.4) VU Tv 13
13 1
F2(K)=-0.2*(K-1) G(i r1) 1
F?lK)=0.?*1K-11 G=(AlIJ1+A2(J?*r^IK)*lblJl+t:l ))) F = (A1(J)*A2(J)+F2(K)*(B(J1*AZ J)+E(J)*A1(J)))
GL=CMPLX(Ut0.0) FC=cc iP X ((F J.D) ZC (1 ,)=^GCtZS6;k1 (GC#*^-4.0*F 1 1/2.0 ZC (2)=(GC-C:((;C-T (`'2C**?- 4.0*F^) 1/x.0 1G1=Q EALILC(?11 Z11=AI M r 1C(111 Z I : A I M A ZCC-6
XX1(KI=:.R1 XX70 ) =ZR? YYZ00=Z 14: YY1(K)=ZI1
YMA ; ^ cC. tT tZK!**?+CI?**r 1 Z
Y'M1(K)=YMAG1
YM2(K) =YMAu2
V=R(J)+E(J) W=(t3(J)*A?(J)+i(J)*A1(J)) AZ (K)=w /V 10 11) ("U
!^9 UU .Q UO
C^NT1NUL w^II^(L'9U,) FLkMAT(/95X9'REAL'•EX9'IMAUIN'98XrlMA;;l'9^-Xr'I:EAL'9GX9'IMAGIN'93X +•'MAG2)'•£'X'ZLk!)'•8X9'FELDBACK'1
DO 500 1=^ 950 WRITE(e-9::-))XX11I1rYYl(I)rYMl(I)9XX2(1)9YY2lI)9YM:;(1)•AI(I)rFz(1) FORMAT (P F13.6) CiINTINUk CLINT IVUE
SKIP
END
A-2
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I
AND ZEkU't20t10.0990.0tYSTART9DY
CALL NLWPEN(3 LALL PLOlIXXxl11/UT9YYl(l)/VYq3) DO t0 I=2 N CALL PLUTlXXXfIJ/UTqYY1(I)/DYv2) Lau PLUTIXXX(l) / DT9YYZ111/DY0) DG 30 I*2 N LALL PLUT^XXX(I)/UT9YY2(I)/D"92) GALL PLOT - U.br-C.5t3j LALL PLOT( - 0929-b.9tZ CALL NLWPEN(I) CALL PLOT(-0.59-7.0:31 CALL PLOT(-092t-7.Or2) ALL PLU110.0 000931 L ALL SYMBJLI-a.lbr-b.5r0.15t • PULLS • O.Oc51 CALL SYMbUL(-U.15r-7.0.0.1!`v'ZERC'ra.0 4)
20 ,^G
CALL SYMbUL(-0.5 9 -7.5r'J.15 PLAN12 YIK) (A l +A2)YIK-1l-IA1*A21Y(K+(b+L)*U(K-1)-(A1*E+b*A2 ) (K-21' 9 0.0 71) ALL SYMbULI — U.79 — bee tU.l5 9 LUN UL: U(KlsF1*KtK)+F2 *Y(KI9t09092V
t^
CALL SYMBUL ( - 19.5 -b.5 0 15 :'RLSULTINI. CHAR. L4N.: Z**2 - Z *(Al+A2 +F *(q+L )+A1*AZ+F*(A*AZ+t^*All .0.0.64) +^ShLL SYMBULt-0.59"9.090.15 9 ' PARAMETERS: A1*0.95 A2=092 B*0.0 L*0.01' 9 0.0 v) CALL PLUTIlb.O 0.09-3) LALL PLOT! O.V t6?3 r+ yq 9 )
l
+
STOP L NE;
99:99
C
Appendix B: Supporting Alebra for (3-31) - ( 3-43) and Simulated Check
}
Contents:
p. B-1-3: Supporting; ++lgebra B-4-5: Step response test program
e B-6
I f
R
t
Step response watch
B-1
From (3-31)
6z2^
Y(z)
(Bllz+612)(x2 - a21z -0 22) + (^1z + 622)(z2 2 R(z) _ (z 2 _ n l z - n 2 )(z - a 11 z - a 12 )( z 2 - a 21 - a22)
- (1 22 ) + (6 21 z + 622) ( x2 - a 11 z - a12))
- (v l z + v2 ) 1(811z + 612)(z2 - C121
= b 5 z 5 + b 4 z4 + b 3z 3 +
a z 6 + a z5 + u z 6 5 4
4
b2
Z
2
+ b z + b0
N z D(z)
I
+ a z3+a x2+a 3
2
a ll z - ai 2)^
1
z+a
(B-1)
0
Evaluating the numerator N(z) _ Sz 2 [a lz3 - 01 12 `122 0 + 22 z2
11121z 2 - Bll t22z +
0 21 z ^
5
'z 2 -
1.
6 ()L,)Iz
21 (16210112z (1
+ [,5(-[;
11 11 22 B 12 a 21
+[6(-61
B
^.'.)°1 1 z - d22U121
_ [n( o 11 + V21) 1z +[6(
22 - 0 a 12
11 `121
0 21 x 12
+ 12 - 621"`11 + 622)1
z4
`22a11))z3
z2
(B-2)
yields (3-32) - (3-36). Define the denominator as D(z)
(B-3)
D 1 (z) - D 2 (z)
where D 1 (z) _ (z 2 - n z - rj 2 )(z 2 - a ll z - a 12 )( z 2 - a 21 z - a 22 )
(B-4)
1
and
D2 (z) _
0
z + v2)
N(z) 6z
(B-5)
W-2 F
t
Expanding D 1 (z) yields D 1 (z) - (z2 - n 1 z - n 2 )(z 4 - a21z3 - a22z2 - a11z3 + a11a21z2 1 + 11 a22 z2 a 12 z2 + u12`121z + a12a22)
!
_ (z2 - n 1 r. - 1 1 2 )[ z
+(Yll`A22 -4 a = z('- (u,
'.
i t
4
- ( `x 21 + a 11 ) z 3 - ( a 22 - (111a21 + a12)z2
12' x 21 )z
+ 112a22I - a ^x
+ On 00000000
0.000C(w
2• s>28,4
O .U000UO 00 100000 0.0L►U(100
2.423[1411
Po 3b5542 2.349549, 2.3255 8V
n•()C► no (,*n
090("OCOO
:. 3( 1 743&1
O.OR nnno 000 0u ( ) nu
7.241431,
O•uODUUU 00000c(u
1'.
2&12944
&9c06 080 ko 207924
0.00OOGO
U.OL10000
00000000
0.0C1uuoJ
o•ou()0 00
e 2 U3118 y
c. q 00uuo
2.209402 2.206744
U•ODUUUO 0.00OCOO
?•198537 2.20549'u 1:.199988 :.1056313
2.2"'0881 2.221515
2.22064? a 2.222 2. 22 049 6
OOOVOOM
do 4' 8698
4.41557" 2.4172 F+b
2.2:0891 1516 cc4 15 4 c.2.2^ 2 2210 69 29223914 2.2??711 2.: &1 17 9 2.22 3778 2.212480 2.222UoU
t rc-rul**2
000UQODU () . nt)no ()O O,OL► 0t,U0
2.ti47(i1 Z 2125?3 2.41ts133
4.201040
-0.4712t^4
Z.3'J51 t11 7.35 b 45 1)
2.214733 2.2125 5 2o •.16134
2o219666 2.219666 2.217241 2.272535
NUZ
^• 7 tt^
C► •4490506'
402zbIO2
?:203193
NUI
U. 1:._'94911
LoU951aS
2.2[ullO
1.1G6086 2 20 7929
-1.523571
095!cbt3 O.00OCK)O 0.0o0C((^ 0• '700000
SY S YU: UVLKALL 1 .F u• uuuu uu
2.2'1303U :..:u 1j39
950
0.116359
u. rib? 111
O.u70 0 16
-0.1590bb
A21
L22
621
2.2u3e4!, 2.441444
.
A14
() (10()GO 0.000(IUO U00GOU00 0.000C.00
00000000 CF.nonnt►t) O.00JCUO
7. r y bht, 2.217240 ?.??2535
00000000 0.0 (,UU00 G•OUOCCO
1.214151
0.0UUUU0
202"'3912 2.272713 . f-,41736 2.2&13774
U.OLOCOO O.1t► (1U(10
1.221067
2.121037 2.212470
2.212U!7 2.2208 o 2.2: &134. 2. 2.: r► 4t t
00000000 (1.0oC)(i('o
('40001,00
().000('0()
0.1)(.0('(10 U40uU000 Va0tl()(IC10
U•3OU(K'0 C.0('OL00
00±)000!)l
0• () L I CN OO( ►
. . ??OSrN? 2o2ZU979 2• ^: 1 05. 3 0979 e.1109b I .2r G'+R2
000000100 9*000000 t),ov n ( •n0 O.JUJUuO 0.000000 oeuoouuu
2•220 y 7 40
o.oGa(f)O
Appendix C: Factorization of i for BxsWle 5 Contents: p. C-1-2: Factorization program C-3-7: IMSL routine - ZRPOLY C-8:
Destabilizing A 2 effect
C-4:
Plot routine for Fig. 3-10
C-10:
Plot routine for Fig. 3-11
...._.,r .,
^ — C-1
1f;1^ t'f^:,CKAt° , ► IIRMIALL YLANI NAb.. AN., ilht',,. L/ fl, tt.;/Ltv LIi. ).F1 1•Wl?i+ALIe ud.l/i•F.1.! U m -, LtA t -^l
L
•talc I/t •:`/• /1/I.^ rl.^/rt{i
L,u lc. 1=1rG
/:•irl.• ^t/rt./l,.`>r:. ►
/•/t
Ik /t
•t/
^l l l ls•.+Exi•1`^ t) 11 ► ^..{ ; ► +1 ► t +. ► : Iwl 1 i^'1+...r.11 1 — t.t t t l /^"• GI 1
1f i 1•E:.•I Ai ir4.II1
+
IFII.Lt 0 . I i1: I li=-t At- ( +tt]li 1' ^rlllxll lF 1 1 .tL'9 IA1. _^.. I1 1 It t I -)A^^ », *r 1(li =I^L111/>^1,1**.'/'+tl — thF^ — colt
III1aJJ +Il11
+j)
1F1^.t l..l«ia -i :I1) 1h11.I ^....lt.al- ► 1 t.. ) F11I1=INl +^Italil*1*s;t.it;`.t i111 # +:
1f • f I •LL.I)L>1 lc
i
1n 111+^1 ► xlt.^:tw.11l*i:..hill—.:
4 t I I1 '..) - I
il^tttlll
F. Ir,r1))^► i*'.'. ► 1t1—..;'1:1 ► ^. 1)
it. It1+11ItAir tr{11x11-1:. l!'.:t.711 — .}1111++. J1^'•1'vtkll/°'I^
=f . I l 1
1h 110Et,01.11L..
tl,hII'tit.t :"+1 Z :4tXYi — Lltt ri
L—t^
:—LXY( —: a lt tr'^4♦111 ^1 I h3gc:I1+.:.1 L r ;` — t 1 1'+ M.. l +t' l a — t L l^hi l ^^ Gl t , lr—t l 1*e^. —t la * t.^l—< < 1*'D1.:.—^.^.'^► ^ 1 I
A4z- 1 .^ F_ —a.l—rtl
F4= ► !i't's..:.
t (r tl,
wom^Iltlt,t.t,ls
t.Ar t
r . ` :f jt.y .L, ,",ti,, r t,l Sl gF't.t,.f., ) IIF"s,,. i- iJ1%,'"T1111Aq'L,._19F'. t Wl',iI ` (ti. 1ti,,._ ,A.,t-ItA L ^ i -A M A I1 ^Ar r A-,.af , l y .L, hr'A_= r ,f 1.r,,. A t r A 4 9.6 9 F ;.r ..: Xr'Alurri .j'.t. f•. f LI1 A f 4Ul=
.
tt11* A 1C+t : i ;A . t — sL
+t, M. )*la.l — t► lt k t.11/L I. I ► /l.
il r rl
L T« = A 1 1 A r I 1 }:.I 1'. — x 1 .1"r '.l I —A 2c I :*A1-.41 i. w H.liclt •.;1,)cil•'1fr`v.^t•% +uf •r'1•^"'c r9 1-V L r• R , 'NQ19 F r] {iA}r^1,-'}r4. t ,.,A,ft 9
t F,
^.
.,i,., 1r+I041lLt.rrI,LAI t f'I L k4 t It FIIf- Ai (1/ 1 r I't Lt hhi It It ,-,.,1 F —,I+,.11/rr
t.^r
r.L"z, Lt ttiu: f )
I•i.Lf,yti,rl't:arr`+r,rhtttr•it,^1"'ti.•1
^ t I =1,« 4AL1ic.iI)) .^rLI..I t I i
L
S
i
l.• I t t,
a. IL t t Ii It • v/^- ^f._1 M f I M i. 1 t l x r t t, _, . t ) - T I. L L,aa.%
}
t^ltltULtd AC. pAQrZ'ri^' I9
POUR
QUALM
9
:
+c :.
—'i !
,r't.t
C-2
{
INI LUck
NLJ iu,ItK
C,,hPA^c LL l4
N. t(yay
F11)a ,4 FI 4
FI
zi j
.1=AL
r j r.,% t h a^
..+1,..al`txt`t R
-
EaL WIN ►
t (tit- s^ k t 1 1.
Et„
.•a7
`^ti.,
t so,iL
l.::
r:
',t a,t t,^ • t l
C-3
IMSL ROUTINE NAME
C
- ZRPOLY
N!!Nl M^!!-i!N!f!i ll----li^!!r!/!!^
4 C
C
-Ia
COMPUTER
- IBM/DOUBLE
LATEST REVISION
- JANUARY 1 9 1973
PURPOSE
- dE
USAGE
- CALL ZRPOLY IArNDEG,Ztl[RI
ARGUMENTS
a
WITH COEFFICIENTS Y IJENK1N L AUD)
INPUT -
NDEG
ORDER OF NGEP"RERSEUFITMLENTS I N - INPUT INTEGEk 01: RE RF POLYNOMIAL. NMEG MUUS
BE
tHAN 0 AND LESS
GPEATE
Z ^CONTAINI NGtTHEFC G OM PUT ED S OF POLYNOMIAL. L NOTE - THE ROUTINE TREATS Z AS A REAL VECTOR OF LENGTH 2*NQM AN APPROPRIATE G EQUIVALENCE SiTATEM ENT MAY BE REQUIRED.
C
`
C
IER
G`
IER ; 130
C
N
INDICATES THAT THE LEADING
IER=1319 INDICATES THAT ZRPOLY FOUND FEWER THAN NDEG ZEROS. IF ONLY M ZEROS ARE
`
L L C
TEAM
COIFFICIENT IS ZERO.
C
G
AL ERROR
POLYNOMIAL IS GREATER THAN 100 OR LESS
G
C
IE,si6Rr INDICATES THAT THE DEGREE OF THE
C
C
RM OR EXIOUTPUT PARAMETTR I
-
POSITfVEIMAaHINE1 NF INIAYG ARE SET TO PRECISION / HARDWAkE - SINGLE AND DOUBLE/H32 - SINGLE/H369H48,H60 REQO. IMSL ROUTINES -- UERTST UGETIU,ZRPQLE^ ZRPQLC ZRPQLD,ZRPQLt, ZRPQtF , ZRPQLGrZRPQCHr1RPQtI NOTATION - INFORMATION ON SPECIAL NOTATION AND CONVENTIONS IS AVAILABLE IN THE MANUAL INTRODUCTION OR THROUGH IMSL ROUTINE UHELP COPYRIGHT - 1978 BY IMSLr INC, ALL RIGHTS RESERVED.
G
WARRANTY
- IMSL WARRANTS ONLY THAT IMSL TESTING HAS BEEN
APPLIED TO THIS CODE. NU OTHER WARRANTY,
C
EXPRESSED OR IMPLIED, IS APPLICABLE.
!!!i!!lN^M!!!l Nri!!!!!l^lY^!-^^!!!!!!!!!^, !A!!-^CM!!i!! M^•!!M!!^!l^l^r^f!
C C
C
SUBROUTINE ZRPOLY (A,NDEG , Z91LR) SPECIFICATIONS FOR ARGUMENTS
INIEGER DOUBLE PREC ISION
NDE691ER
INTEGER
N NN J JJ I,NMI ICNT^N2 L,NtrNPI EfA ^IM E , I , FPr^EPSP 9 RAdIX Rl0 XX,YY,SINR
I RE AL
REAL 1 DOUBLE PRECISION
1 UOUBLE PRECISION 2LOi^ICAL
I COMMON /ZRPQLJ/
A11),t(l)
SPECIFICATIONS FOR LOCAL VARIABLES
COS aR1MAXRINrXrSCrXM,FFttxgR rBNOrXXX,AI^E PTIl TEMPI101)r"(101)rQP1101)rRK1101)rQK(101)r SVK(101) SR,SI,UrVrRA,RB CvD A1rA2rA3r A6 A7 ErF G H ssOR , SL1 RLZR RLZI
ZETROKaSICar^A^TORrREP R1rZ^P.0r0NErFN P iQP,RK QK,SVK,SR SI U V,RA RB,C:D A1rA2rA39A6r A7, E, F , a ,H,SZR,SZ^A ZR,RLZ^,ETAA^tE,RMRE,N,NN
O-4
V^ LOW4 S, AET S
I
I
$ON PANT SV1, t E " Rfi " N 4 S PA S' " Me AN t0 14"Uh I HE ROG I AMI 0 1 9 OF T h k FOUR C ST NT AR — REM k xj;Uwl RlAlIVE R E kpk A ON OR WHI A N Q ,TIAS '" I t .1 AMLEST L 05 111% L DA ING P POINT NUMBER SUCH THAT le+REPSR1 IS GREATER T HAN I RINFP 111! LARGEST FLOATING-PUINT
tqkv l
ERR
NUMB Vt
i ST RLPtS 14E SRA ff
F A IN G —PUN 19 t I TI Y l: TH E XPONENT RANGE UYFFERS R I IN SINGLE N i ND UOUBV PRECISION THEN REPSP ANU kINF $1 HOUL NDICAIL THE SMALk[K RANGE
RADIX HE 6AJkUFlUE FLOATING—POINT
NUMBER SYS EM USED E k IN V P/L 7 V F F f-f- kF,*
DATA
REP SP / RADI X/ 600.4 REPSN1/13410000000000000/ ZERO/000UO/AONE/ld / Z POLY S S SINGLE PRECISIUN Cj%CULArlUNS FOR SCALING 9 BOUNDS ERROR CALCULONSe A
DA TA DATA
DATA DATA
EXECUTABLE STA71 ATLMEN7
TER IF (NDE6 . Tale 100 9 0R * NDEG * LT * 11 60 IU 165 LIA a RLP.11*1 ARkEm,EIA ETA RM
RLO
E
RLPSP/LIA INII JAVIJAIIUN Or CONSTANTS FOR H
XX a 01071068
0 AlION
YY s -XX
NR x . 9975641 U SR r. -,06975647 GO
N
C c
NN
NDEG
N+l
IF (A(I).NE.ZEKO) GO 70 5
IERTO a130 GO 19 000
E
5 If- (AfNN)oNEoZER0) GO 10 10 J NDEG-N+1 ii J+NDE6 ztj) a ZERO I(JJI ZLKIJ NN NN-1 N
C C C C
ALGORITHM,FAILS IF THE LEADING COL FIC LN I S ZERO* REMOVE THE ZLROS AT THE URIGIN IF ANY
N-1
IF (NN.kQol) GO 10 9005 GO TO 5 10 DO 15 1- 1 9NN r. A Pi )-M 15 CONTINUE
TV30 20 IF (N.GI * 2) GO TO IF IN.Llol) GU
MAKE A COPY OF THE COEFFICIENTS
STAR) THE ALGORITHM FUR ONE ZERO
9005 CALCULATE THE FINAL ZERO UR PAIR OF Ztfaus
IF (N * EQ o Z) GO TO 25 ZINDEG) a -P(2)/P(l) Z NOEG + NDEG) z ZERO GO TU 145 25 CALL ZRP^LI4PI119PI2)vP(3)vZ(NDEG-I)tZ(NDk(i*NUEG-I)tZINDEG)o E) I Z(NDE + N EG))
GO 10 145
C C
30 RMAX a Go RMIN a RINFF
DO 35 1=1 NN X = Ab9( S NGLQ(I))) IF IX.,G] , RMAX ) RMAX v X
FIND LARGEST AND SMALLEST MODULI OF CULFFICIENTS*
IF (XoNEo0ooAND.XoL`T9RMIN) R14IN a X
35 CONTINUE
C-5 SC ALL
THERE ARE LARGE OR VERY FCC ii X FA TpF RC TMNM^TIP LV T MEA MEN OF THE ROLVNIAL.
CE O THE FAQ.
AND I TO DA ID UADE:T CTED LO INTERFERR NG WITH HE NGEN ` E CRITERION.
rOR 1S A POWER OF THE BASE
SC : RLOAMIN
*10.l OGOO1^055 IF JRSCMA- Xe[1 .O. S +^ REPSP *RADIX*RADIX lf• St.E p
GQ TO 45
40 IF I RINFP / C.LT.RMAX) GO TO 55 ALOGIRADIXI*.5 : ALOGISCO),/
45
DbLE **l FACTaR ov1RADIX! %a W/IIOc1FAtNInk*P1I1
COMPUTE LOWER BOUND ON MUDULI OF
ZcKus.
cc
=$ASS
{SNGL(PII ► G) 60 PT 110! PT INN! ; °P11NN!
COMPUTE UPPER ESTIMATE OF BOUND X = EXP(^ALUG(-PT` NNI!-ALUGIPTI111) /N)
C
IF IPTIN .k4 . 0.) GU TU 65
C
i
C
BETTER S USE IT.
XM a -P11 NN 1/P11 N!
IF IXM . LI.XI X v XM
CHOP THE INTERVAL 10 * X) UNTIL rF.LC.0
65 XM : X*01 FF a P1 1 I) Go 70 1=29NN 70 FF s FF*XM*PT11! IF tFF . LE-0o) GO TU 15 X = XM GO TU 65 75 DX = X
G
If NEWTON STEP AT THE ORIGIN IS
80
IF fF
OF
/ Xl.LE..005)
s A FT^II c FF
DO NEWTON ITERATION UNTIL X CUNVERGLS TO TWU UECIMRL PLACES GrOTO
90
UU FF s F^*X+P1111 DO- s DF*X+FF 85 CONTINUE FF s FF*X+PT(NN) DX = FP /DP
X = X-DX
GO TO 80
90 BND = X
COMPUTE THE DERIVATIVE AS THE 1NTIAL
C C
K POLYNOMIAL AND 00 5 STEPS WITH NO SHIFT
C
NMI = N-1 FN = ONE/N
a
95 Kt1 ) s
AA :
N- I) * Pll)*FN
P(NN) Be : PIN) ZEROK = RK(N).LQ.ZERo
DO CG5 J RK IN1 IF 1ZLkUK) 60 TO 105 C
C
T = -AA/CC DO 100 I=1tNMl 100
K
USE SCALED FORM OF RECURRENCE IF VALUE OF K AT 0 IS NUNZERO
J = NN -* RKIJI = T*RKIJ-11+PIJ1 CONTINUE P(1)
ZERUK s GABS(RK(N ) l.LE.DAI$)S(88) G O 1 0 115
* ETA*10.
C
0-A
105 110
C
vu
ll q Iw1M M1
KKIJl N +^ f kKtJ^-ll
kKI1 1 0E ZERO x w RKIN
) 115 Co NT ROK
. EQ*Zk RO
SAVE K FUR RESTARTS WITH NEW SHIFTS
DU 120 I a l N
120 TE MPI l)
_
AK(I)
LOON T O SELECT T HE QU ADRA TIC
CORRESPONDING TO EACH NEW SHIFT
DO 14G ICNT x l 9 2V
C
USE UN SC ALED FORM OF RECURRENCE
aUA^1F)I^IlCARNUNI-^A6 P^IIAV OANn l 1 S OMPLLX CONJUGA E. THE PUNT HA MUUl1US RNU AND AMPITUDE ROTATED
C
XXX
CUSR * XX-SINR*YY
^k6RkES FkOM
HE
PREVILIUS
SINk* XX+COSR*YY XXX
YY XX
SK Sl U V
SHIFT
bNU*XX 8NU*YY -SR-SR 6NU*HNU
G
SECOND STAGE CALCULAT IUNt FIXED
C
CALL ZkPQLb 120*ICNT NI) IF
(NZ.E0.0)
QUADRATIC
GO TO 1 .30
C
1NE. SECOND STAGE JUMPS DIRECTLY TO
C
ONE OF THL THIRD STAGE ITERATIONS
L
AND R( TURNS HLkL IF SUCCE SSPYL.
L C
G
J s NDE V-N^!, I JJ J +NDL'G
DtF^LATE THE POLYNUMIAL STORE 1`HE ZERO Ok ZER[^ S ANU RE,tURN TU HE MAIN ALGLIRITHM.
ZIJ) x S1k l(JJ) a SZi
NN = NN-NZ N NN-1 $ Utz 125 I
125
& 1 NN
P(l) * QP(11 IF
G>UI
.
J
J r t J .1 . • 1' I ?
li it ^U
U
.CS •d
4
.1 J
tt J1 l Ul
1 a to to 7'A
11;
'1 J •
It) J`
-n
.'11'i .',
ON
i
v-n
x
y^ t^n•,w N cc
N
(f
t
4001®"00 3c 0000. •-+4f^t y t^ c.1 i
n
111.41 r) c Ol(')*OC Jio.Q. -P- .t,*1in44 .4 J.v•-•- mocnec
GL w V-4,0 J, NIOLfN►c1G J` T'r-,' i ' Or)t... • •7J 0c) 1 ^^^^11OC ill it 1-•>
Q
1Q soU,in NN
c?^OCaOA 1
N WNP-F-00
(To P.- --4P.-.W J ..lOOU1nU1^10 o Qenl•inil,ao" 0. W••'^"'^•.t^tNN 4c inr. o •o aoao
OO,afjr'1t^ 111
^l.1QUICa
Q H
s
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tlniGiNAL PAGE V 4
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1
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C-9
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4
L
SYSILM Its 110 U1MEN` lUN X11 hLGli X.ibi)(11 X3(00011 X44 bout I Yl(6t^0lltlikt1-001)tY*k I +4d(j I19Y-4 80L ) XP1 ,91 YP44 X101 0 YZ^3ItAXZ,44 AV
X111)eX^?I1•X.11^rX11^/t.0«^► •LtU,4.^^^.t^/ DATA Y iI) t Y^ t119Y 1I V4111 / 4 .o U.t t -4.C; G.^ #C UAIA
uAlA — 0.7t +1 N`>t-t► . lta 'Ikl / !,t-00!)1966 UAIA Y) I /L.,33Cr(.Ab. t -U«4'(J uts ro UAIA Xt/ — L.1441^ t — c .1^,Ai^a --v.4► a y t 8/ 01 UATA N*6000
YMAXXI.0 YMly=—l.c LrJ 45 1 = s r vtrl
X/ 1)=X. I1-1 )U.UUO -° Y.:^ I )--fiUS( s4;Kl 11ta.( — X/1 I )**Z) 1 J -L^ tUJ-1 X111 J = W11-1 ) Y11J -r11 -1 X.^(J -^x^l 1 — I ) Y.11 J )= Yr (1-1 ) A4 1 1 ) = — X.'( I ) Y41 1 )=-Y;,! I ! WN 1NUL
:.ALL PLL;1' 4 1Q' t C.^ tLV011 LALL PLoIla.Lrc.Ut-J) LALL )-ALTUR(C.6t ) XS1Ahl=-1.t Lt1-1/^t.l
LALL AX15 YI - AW,IYt-,AXl Ii^-1At A .S YN1N i.u19Yf ) Yl=Aba 4YM1N)
Y51'Ak1: -YT UY=Y 1 /w.c
CALL AX1:^ 4L.l^9-- 4 .tor LALL N1 WPkkN( .t)
• I M At>1NARY • 9 9 +43« (,t 9k, . 9 VAlAKI L'Y)
LALL PLL)IIX111)rYlill93) C-u (j 1µ49N aU
LALL CALL
PtUI(XIMvYll1 ► tc)
PLkalX-11)fY2 1 )r:+) LALL PLt1(Xe(1)9Y2tI)91') t,ALL PLO X.11)tYa11)f3) L.1 4U 1=..rN GALL 1'1.011Xi 1)tYJ1I)t.:) LL) Jo 1 = tN
^L
CALL PL1^1( X4 Ui,I ',I, 1--ctN
I) f Y`t
1 1 9 3)
CALL PL(Jl(X tt111fY41I )rwl U Y Y Ll .:; 5
L)O 100 1=1 t4 GALL '"YMbUL1 XP( l WUXtYF'111YMbJLIAXZIJ)•AVZ(J)tJ.6'•11-'-'vL-Ut-11 k. Q 0
CUNT I'iu^
CALL PLUT ( I b -U 90- 6 #--0
LALL FLUI(O-LvU*UV+199 ► .S1 UP
LNU
I^
''I Appendix V: Overall Transfer functions for Example 6 Contents: p. D-1-2: Supporting algebra D-3-4: Simulated stop response check for (3-48) D - 5 - 6: Simulated step response check for (3 - 61)
Note: a's and b's In privitout correspond to p's and q's in (3-48) and (3-61)
Consider the control configuration in Fig. 3-12 a ( or c) with a convergent
identifier, i.e. the time-invariant, asymptotic c an *. For the plant in (3-17) with the controller in (3-46) and the actuator of (3-45) with B A -aT
(D-1)
then, since
z
6 R(z) + (v1 z + v2 U(z) c
z
n2
Mz) 2
z
(D-2)
and
Y- z UU (Z) C
Y r, U(z)
U (Z)
U
C
(Z) 11 1z
(D-3) z -2 )
-1 -
Y W
n2 ( Z—) -2 U (z) 0 C 6 R(z) + (v i z-1 + V 2 Z ) Y(Z)
a
6 11 z + 6 12
I z 2 - a 11
Cross-multiplying
I FS Z)
Z - 'A
a 21
12
I
Z - 01
L=.4 I . z - g
I
(p- 4)
22
in (Uf-^4) and rearranging yields
(%,)I Z - (1
32
it
(z - g)(z 2_ OL
- (vi
021 z + a 22 _ Z2
2 0 ) (z z + ^
R(Z)
+
+ "' ) ( 1
2
Z 11
-
22
z2 _ 1 12)( 0
g)
(1 - g) IN (z)
(z - g) D 1 (z) - (1 - g) U
) a
+ 01.1 z + fA
21
11,z + X 12)
Z - CI
22 )
(z 2 - a
q5 z5 +
21
12
(z
)( z 2 a-
2 _
Z - 01
it V —a
12 1
ts (I - g )1-,
t) z - r'2) I
2_rA Z-C% 11 22 )+( 11 z + a 12) (z IL 2
q4z 4 + q3z 3 + q2z
+ q lz + qo
2 (z) -1p7z 7 + P6z 6 + p 5z 5 + P4z 4 + P3z 3 + p 2 z + P z + Poll 1
2
(D-5)
wliere N(z), D I (z), and D2 (z) are from (B-2), (B-4), and (B-5), respectively.
o-2
Note that the q, of (D-5) are equal to the 0-0 b, of (3-321 ) - (3-36) an shown in (3-49) - 0-53). From (h-6) and (P-7) Ov,
pt
Of (3-54) - (3-6o)
are readily foniied. Similarly for 010 control configuration of Fig. 3-12b after Identifier convergence, from nkwhip, the pickoff point across the actuator Z - 2 ) Ile(z) ^^^^) U W - 6 RW + Ili(V_g
z
2
+ IV V_ + v ') z^~ I Y(r) 1 6
(D-6)
and Y .(.- L— . ( j-7.0 11 c
W
) uv_^
Z-g UW
Therefore
3 I Z - 9z2
(1-0 z - n,, (1-8) 1 YW O-g)
2
z 6 RW + 1%, 1 z+ V, 1 1 Y
F 6 11 z L r-
+ 11 (4
Ji
12
+
11 -
0
z + 6 21
22 -
7 2 - t1ilz - (I
12
22-1
(D-7) Again eross-mullipl}filg and rearranging yields
Y , (z)
Z,
6
0 - g) I
!t (x)
rl
11 + 1 ^12 I
W -
o ') I z
(1
2
(-j
O-Oz - g 2 0 -1,0
z
") 21
11 Z
+ (0 21 Z + (^" )(z U 12)(z
2
0
11
Z - Q
I z - a22)
^i (1 7 21 a v.) 1(1-9) 14. 11 z + i-))(z2 21 22) 4 (02- 1 Z + 11 22 ) (
v 1, +
[ z 3 _ gz 2-
i l z -112(1-8)1(x2
_ tA 11 Z - (1 12)(.-
where N(z) and 1),,(z) are from (11-2) and
yields (3-62) - (3-68).
(B - 7),
12
Kx21 z - (1
it
Z - kx
121
22) -(I-g)D2(z)
respectively. Expansion
p-9
S,N lU (vi( t' 1- 11,!)94LIUitWl.. rAf ik' Irtr1 i11t F11/ 1 .^}.t /rG► /f..5< ZETA/t o6/ g 8IMEN ATA T/4J.^/9A L/1:011 V/ u zbI/U04r^.tar / UAlA
Y1
luto
9YZ9Y. tY4Y: rt r 19.* f pU ,39UNtU / 0 . ► 90.U 4 . t► tU.Catt► . l► t^1.Ur^J.f^ 0*0
UA11. YUlo Let YL .A tY%."rYI, K rYLCsYL // U*Uv0•Ur0 • UtG•1;r 1^.Q909C1r0.0/rK1rR +.' ftt+iwtR`^ftt , tK7r ►t/tCry . l;tt..C^r4 . tr9C^.C^rll . C'rU.lwt/ Ut lt! 1=1r^
r^111l^2*^x ►^i-^^!.T11{*will*11*c.t^.^iwlll^► [*,^^111-l^Tii!**t11 IFIIotnj.1lAll-t,ll l; 1 Fi1.*t7,2 ),t I&C 214 1 t,2IIll a -2 L t^11I ► *^I l ► *1 ! II- I L.v.l..l #Ali =L,241!
Ilw II,r(J.2 )A rC, l& ► lIIImIALII1/ 1%111 *Y*,;Ivt1-Lx614-11.111 *wi 11*11*1G1::^1w11!*M WIRI11-Zh 1^i 4+Ft1t1-41 411** 11 0111*vd!I1+IL^.141 ► /SQKII1-1L1i1!**'1 *5► 1Ni11^1+
+11 1F1 l.t;^i.11k^ll =t°1 111 1F11.LtJ.2)Is2L^^lli: ► _ I *^Iw it! *T*4^1'1ti Z* 1iI1 *tili +t 111 111 %svk ll +4QPII 11-11.1 1 1!**.I1! 1 If I I-kQoIIt;UL+e. III
1 FI .Lt).: )144-tti2 ►
tiMl^^: +^X^1-ZElt► *will*11*t, fJ:±tt^t/i! *T*S^til t Mo*&' m - hXN 1- ,*1 Ll A *W ( l) *1 ►
_ *ZEL 1 Ili :
_ .! 1 I+S1NtWi! i *
I1.(}- L CiA* *111
k T l =UM 1-/.11 ANUI =I c.T Lrnle+ i l l l*A3 1-A lt+uMt l*1 All-A1l*tall/r«l^ l 1/ Ik+l, l ♦ A11+b12-AY f;T: ^AL1*tl l+t^F;::-k:,11*AtvUl-A1^
ANU2-=0: *A1L/t I+' t3 ^ = U* 1 1 . t^ -C^ 1 '^ t t^ 11 + b « 1 1 Lea=l^wl l,t : - 1, ► * 1-1j11't A.. 1+tal c - 1^.: t'tAll+hc::l 13:^=1^*I1.G - 1 1* t - LL1+'o..:. - ^al^*A.:1 - t^21*Al2 - is2c"*Ali ) ^+=L+il l.u - ^l+1 - 1^1.'^► ^.«: - '^^.."^tic1 t< 1 ^ +,+. U +f)=1 1 A7=l.Li
A Li m -t,-A,' I - A At)
1 1 L11
It lPt + I1+1.I1i-A..:+All
I A I i I11*A::I+L11* ►`,11-t 11 ►
OQ1*1.11 A4`1-Cx*1-A,,«+AII A.!I-F1.!+1•II A..14LII*AII-LIi1+14111":4+i L AiI+#,,.z *LII`A1 *1 1f:*A,^1+1,12*AII-ANU1' t°II-II I&I1*II.J-(Y1) ++A41. -L A.3 c 1-0+I0 11 A. +A11*ALI+LI I A,6, -L 11*A11+A41+LIIIAI: +LI.,*A,GI*I 'I: *A1 II*n"..--CII-% I.:*A..14 1..+rc. -kIie I1*A4'1+ F :b+Al4- O'NU *11 *A1,.*1..' - ►.1I +1^ 1 - ,.+11 4, A.1 +t+.:. + ►;aI., - t,: 1+AII 1^Ii. * - GI - ANN # Ib1I+«1!x'11.1: - G)! A IA1. 0 4,1.6 - 1, 11 41'aIt'a il4,! - 1 II*Al2 0 A,!I *LIt! 110 A/ ItII*A21*112*A1 A.:.--kI4+A1I*Ad4-LIL 1vA1«.+A'1 4 ANUL*IF'l1 A&,4LIk*A`1+I'41*A AI +12+ L^/ 4+t,11!*I I ov-:,t1-FNVe. °t l-E II *h.^l*1'42+L12-bdI* t,II I I I C-U ) Afju2*It3II* A .: 21 1 le AII t,.!;o+L_14':*AI "X A«: 1) - LI2 m i+It" A1 = t I L h I 12)*(IOV+11 )*I Io U - L, ► +ANUI*IL' +L, ) 1 AU=
c,J
C` i #A1."IrA .:+A14Uo.'V(01. *Avg+c^.:,:*A1 1 *11.-1 11-5 ,t O t t#3.1W « t b L tt•C• I- KM AIIIXt'Lt) 9 t V o 0 9 C Xt 0 b-4 vF 1Y,t,r,eXt F
w
y *LI
111 16 jCU
+•L^1=•tl'`,'.U^e.Xt•t:t,-• ♦ i-y.f► t2X1
•4X9 b2r•9F`F.ht2Ar
wK11L(o /G!A`lrA tAhtAtir i► . t A. j Al AG tF°"1.tarlXt•A`^-•rF4.Lr1Xr•114=•tFY. t,t^Xr 1^ UkMAlI1Xt•i,7='rF'`1.Lvt I • AL. +• A ^ =• t hy .b• iX t • F 4 =• •1-`i.fa•2x •• A l ^• •h^.bt «'X ♦ • l^G-' t F y .t^ t ^'X 1 WK1TL( bt.;tllt 1 9 LT2rANU1 •ANU2tt•M1 UMd t~uFMACIIX • L11 = • F •y .b t .'X • L11- • .!+l ?',t•t1U1-•tF^1.6r^Xt•NU1=•rl•Cl.b ^n ++2'X • GM1= t t h y obvix9 • uM = jtF- Y.hr1xl 4 R :If , rll► l}) 100 1= CJkh1AT(IX t 'Y1:LLN1KLLLLU SYS. 6 t -3 Xt 9 Y2:L3VLKALl T .F.•r.Xt •IY1- Y:1**'2 /G
OU :)5 J=19100
Y=t(,+Ae I+All)*Y1-( U *(ht!.1+A 11)-A22+Al1*A/1-Al2)* Yk - 1Al1 * Aie2 *A1k*A21 ♦ -U* 1- ►>12 *A1 1 *F,at -w12 i 1*Y,^-1 P1 2^A."_2- G *IA 11*AZi+A 12*A2111*Y4+G*^ 12*
+AZ2*Y:^+11.1.-U6*1111+c^^11*U2+11.U-t^1*i-E111*A21+1x27+1?11-t1+'1*Atli*U3 ♦ +t 1.G-G)*I-L! I1* AtL- ^L2*A21-till*Al2-b21*Aill*U4 + 1 l.C-G ► *1-'1112+A1c-132
+^ *Al21 *U^
U =Ca* i;+E1 1*L,1+k1c*UL+AP4Ul*Y1+,At4U.:* Y2 Y9 = Y4
Y4=Y3 Y ,=YL
Y=Y1 YL=Y
U5=l,-.
U4=U3 U =L1.? K -U1 U I U,
Y( a -AO*YL1-1!J
*.j *t • , y 7=Y,()
'
*YLw #t 1 v t>tti ♦ ea t *1-
-Al;*YLl)-AI*YL( ► -Aeto^Y(.7*U!)*Kt+ 4*K3+f4
l
y (I YLd YLt-=YL4
YL4=YCI YL,zYL;l YG;e-YC1 YL 1tYL K7xKo kbxKS K!)sK4
K4:K3 R3s92 K2's ►t1 K1;tt
uY=IY-Y41wt a y
w R I I E 6 5 V YL U F URMu111A9PIa. s., 9X t • 1:1 .1 5X1 lIi d L OCI INUL
X)
S7LN tNU
kid•
C1.O37 444
(.4-
< 1=
(l.UO(talltl 1.UL.'Ut,u^.)
EAU= ( .t O(Jv()L P)= - U07"lt.t-r.,l
(`.ty"ILL l,#
H1--l).:..»'^i I4
A7^ Ass
► »ht1=
I of o7iQ1
t,*" L t
E
E11--U .It)0561
j
Y1:LUNrKULLLi.. Z)Y;t. VOQUOOQU(:ti
i
u.(Uts^U((oUt, U. L3 1 4 k 44 4 , L • 11.4 01 0 4*0bb5LUL ,
i
ea .t^1'nIt-4#4
u1 r► vcv r► N1
1
a/ cis
14
.r1CJl1
1/ N
J r It IV 4ti l` D 0 0 ,jr,`,n.I ^ Q +^J'Q► +^1^
NMI 4
,Z-
^(^1f1JN'111
1^ 1!N a►1 t JO
^
N^
^^ J ^,^ ,!^ •„^
* .n
ti*^l3
ac^ OVUJ
r O^ux
r* ^D K ^ ^ ,x'11-
1
+i> 111f^AC
M olP4 It It .sJ^tir`f^t^u
u4a 1.a
NN:lft
46 W40
v^► ^a^vna
i1 N Al 3
rq .11f► Y +*11 - 4 u tN • J It .a1.a
nA
j
aD ~ ^c
F^ nA>
•
y
^°dcv^
P4 00
sly •••••
P
.t • 11
N u
x
_., 11
i^
sa
1r
1
r^1
^,.
^.,s
J
^1 (l .h "d d.'1
v
I'-.- A
I NO re
1
>D
r- +l
^a
•
a
•o A a ZO0% +-4
r r 10 ,^^,v + •..• T
./
• •
4
V
qr
s .A
4.4 It
d^v
all J O ^'• • •E
4
xc^.J••l...s nu
..•.•
a^^
I11
4 GJ J • .J a
ysnJ
1
n l%W A n
o ^ +n
4dP4 • dJ:i J tU v
J.t^cn V^V IAA^`^If+a+J► 9aTA^^iOau « zoo Q4 0
1% I 'll
-.' ►
JatoN Y4 J% 40 • J^ O 1it'. ' II f 1 V 'd dF- NIV 4 (1 1 V f„M- I" 04-41-4 u 4 1^,- r % I •••••••
•
.t e a
P4 if • •V
...•...
de
11
a0 I'YA = SS1'PA+ANAT(K-1) S5^' l'E+E=IIAT ( K-1) ** S.aYI ' Is SSCPF+F**2 SaCPG = S SCPG +(i *+2
**i:
300 TIME = FLOAT (N)
k r
it
IIr (N . LT.KD ( M+1)) F2TURN IDO (M) S (K) IO ( M ) = Y (K) RANGE = FLOAT ( KD (M+ 1) —KD (M) ) DIV = (RANGE — FLOAT (KUUNT)) * (RANGF — FLOAT(KOUNT) —1) RNG = RANGE — FLOAT ( KUUNT) NSTEV ( M) = 0.0 IF(DIV . GT.O.0) NSTEV (M) = (hNG*NASTE(M)-5TE ** 2)/DIV IF (NSTEV ( M) .LT.0 . 0D0) NSTlV (M) = O.ODO N ITE (M) = TE DIVSOR = BANGE* (RANGE-1) IF'(DIVSOR . LE.O.0) GOTO 1000 SI)PEVA ( M) : (RANGE * SSPPA —SAPPEA ( M) **2) /DIVSOR IF (SI'PEVA (M) .LT.O . ODO) SYPLVA (M) = 0.01)0 SITEVE (M) _ (RANGE*SSPPB--SAI)PrP(M) **2) /DIVSOR IN (SPPEVD ( M) .LT.O.ODO) SPPEVB (M) r O.ODO SCPF.VF (M) = (RAN(,E*SSC:II F — SA PEF(M) **2) /DIVSOR IF(SCPI VFI?I) .L T .O.ODO) SC:PFVF ( M) = O.ODO SCPEVG (M) = (RANGL *SSCPG —SACPLG ( M) **2) /DIVSOR I F (SCPEVG ( M) .LT.O . ODO) SCPEVG (M) = O.ODO GUTO 2000 l uou SI'PEVA (M) = 0.000 SPPI1VB ( M) =, O.UDO
SCPFSVF (m) s 0.0vo SCPtVG (lI) - O.OItO 000 CONTINUE SASI (M) s SASI (m) /MANGE SAPPEA (M) a SAPPRA (M) /IRANGE SAPPRO (M) s SAPPEL3 (H) /RANGt SACPEr (m) a SAc pzr (M) /MANGE SACPEG (M) s SACPEG (M) /RANGR N ASTE (H) a NASTE (M) /RANGE KOUNT s O STV s O.ODO SSPPA = 0.080
I
SSPPB = 600DO SSCPF = 0.080 SSCPG : 00080 h & M+1 RETURN
a
END
1
F-12
cccccCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCcCCCc cCCCCCCCCCCC4C GEI/SEI ALGORITHM EQUATIONS (4-4) - (4-8) , SUBROUTINE ADAFTI (CvDrKU,RH0dl) IMPLICIT REAL*H (A-H2O -Z) F+EAL*8 KU,8(4),U(4),Y(4),5(4),AliAT(4)oBHAT(4),E(4) REAL*8 THAT (4) ,GHAT(4) COMMON /SIGNAL/R,U,Y,S COMMON /C:ONTItL/F,(a,AHAT,EiHAT,FHATeCliAT DATA K /4/ AHAT (K-2) = AHAT lin-1) AHAT (K-1) = All" f (K) IJHAT (K-2) = HHAT (K-1) HHAT (K-1) Ulf AT (K) E (K-2) = F. (K-1) ,U (K-1) z E (K) FHAT (K-2) s THAT (K-1) FHAT (K-1) = FHAT (K) GHA`!' (K-2) = GIIAT (K-1) G11AT (K-1) = GHAT (K) E (K-1) = Y (K-1) — itHAT (K-1) *Y (K •-2) — GHAT (K-1) *U DE:NOM = H tMU*Y (K-2) **2+RHC)*U (K-2) **2 AHAT (K-1) +MU*Y (K-2) *E (K-1) /DENOM AHAT (K) IsHAT (K) = F HAT (K-1) *RHC)*U (K-1) *E (K-1) /DENOM GHAT (K-1) - C/AHAT (K) FIIAT(K-1) = (D—AHAT(K))/BHAT(K) F = FHAT (K-1) G = GHAT (K-1) RETURN END
(K-2)
F-11
:CCCL(;ccCCCCCCCCCCCCCCLVCC CCCCC:C"CC.CCCCCCCCcccce :CCCCCceccC (
GUT ALGORITHM EQUATIONS (4-12) - (4-19) iUBI',. ilTINE ADA11 T:! (C, D,MU,RHO,H) IMPLICIT RLA ! .*6 (A-H2O-Z) IiF,AL*Ci MU, R(4),U(4),Y(4),S(4),YHAT(4),AHAT(4),DHAT(4) hEAL*H E(4) vFHAT(4),GHAT(4) REAL*8 LAMBDA (4) ,GAMMA (4) COMMON /SIGNAL/BrU,Y,S COMMON /CONTtiL//P',G,AMAT,BHAT,PHAT,GHAT COMMON /MISC1/GAMMA,LAMBDA,YllAT DATA K/4/ LAMBDA (K-2) LAMI.iDA (K-1)
= LAMBDA (K-1) = LAMBDA (K)
GAMMA (K-2)
=
GAMMA(K-1) AHAT (K-2) AI)AT(K-1) WIAT (K-2)
AHAT (K-1) = All AT(K) = LA HAT (K-1)
1311AI (K-1) YHAT (K-2) YHA`!' (K-1)
tillAT (K) WHAT (K--1) r YHAT (K)
FflAT (K-2) FRAT(K-1)
E (K) = FIIAT (K-1) = FHAT(K)
GHAT (K-2) CHAT (K-1)
GHAT (K-1) = GHAT (K)
E (K- 1)
»-
GAMMA(K-1) = GAMMA(K)
=
YHAT (K-1) : AHAT (K-1) *YHAT (K-2) *DHAT (K-1) *U (K-2) LAMBDA ( K -1) = YIiAT (K-2) *AHAT(K-1)*LAMBDA (K-2) = U (K-2) *AHAT(K-1) *GAMMA (K-2) GAMMA (K-1)
E (K-1) = Y (K-/) -YHAT (K-1) »
VENOM = H*MU*LAMBDA (K-1) * *2 *RHO*GAMMA (K-1) * *2 AHAT (K) = AHAT (K-1) *MU*LAMBDA (K-1) *E (K-1) /DENUM AHAT (K) = BtIAT (K-1) *RHO*GAMMA (K-1) *E (K-1) /DENOM GHAT (K-1) = C/IiHAT (K) AIiAT(K-1) = (D-AHAT(K))/aHAT(K) F _ FHAT (K-1) G GMAT (K-1) RETURN END
i
F-14
'CCCCCCCCCCCCCCCCCCccCCCCCCCCCU'CCCCCCCCCCCCCCCCCCCCCCCCCCCC OI ALGORITHM EQUATIONS (4-23) - (4-24) SUbROUTINF. ADAPT3(C,D,MU,RHb,H,C) IMPLICIT HEAL.*H (A-U O-Z) REAL *F MUth(4),U( 4), Y(4 ),S(4),YHAT(4),AHAT(4)vl3HAT(4) HFAT. 0 8 V (u) ,Z (4) ,Ft,AT (u) #(;IIAT(4) COMMON /SIGNAL/k,U,Y,s COMMON /CONTIIL/FoUvAHATtVIIA7°,FHAT,c:IIAT COMMON /MISC2/YHAT,Z,,V DATA K/4/ Z (K-2) = Z (K-1) Z (K-1) = x (K) AIIAT (K-2) = AHAT (K-1) AIIAT(K-1) = AHAT(K) UIiAT (K-2) : BHAT (K-1) Ti IIAT (K-1) = VIIAT (K) VIIAT (K-2) = VHAT (K-1) YIIAT (K) YIIAT (K-1) V (K-2) = V (K-1) V ( K - 1 ) = V (K) FHAT (K-2) = WHAT (K-1) FHAT ( K -1) = PH All (K) GNAT (K-2) = GHAT (K-1) GNAT ( K -1) = GHAT (K)
'
YHAT ( K -1) = ANAT ( K -1) *Z (K-2) +BRAT(K-1) *U (K-2) DENOR = H4MU *2, (K-2) **2*hHO*U (K-2) **2 V (K-1) = (Y (K-1) -Y NAT ( K -1) *Q* (Y (K-2) -Z (K-2) )) /DFNOM ANAT (K) = AIIAT (K-1) +MU*Z (K-2) *V (K-1) 13HAT (K) 13H AT ( K -1) *RHO *U (K-2) *V (K-1) Z ( K -1) v AIIAT (K) *Z (K-2) *GHAT (K) *U (K-a) GIIAT (K-1) = C/BHAT (K) THAT (K-1) _ (D-AHAT(K))/I311AT(K) P = FIIAT(K-1) G = GHAT (K-1) RETURN END
F
^
F•1S
ICCCCCCCCCCCCCCCCCCCecccceccceeceeccccececcccccecccccccccc GtD/SED ALGORITHM EQUATIONS ( 4-14)
- (4-36)
Stf!)ROUTINr ADA1,T4(CeD,,MU*Hll0dl) IMPLICIT HLPL*8 (A—Hof?—Z ► REAL*b MU, R(4)tl)(4)tYI4)o$(4)tV(4)oYHAT(4)t6tfAT(4) HEAL*8 AHAT ( 4) sl31lAT(4) COMMON /SIGNAL/housyss CONNOR , "' O WTI(L /F e V jA HATs bit AT * F HATeGlIAT DATA K/4/ GHAT (K-2) w GliAT(K-1)
GHAT ( K-1) FHAT ( K-2)
GIIAT(K) PRAT (K-1)
FHAT(K-11) FHAT(K) V(K-2) V(K-1) V ( K -1) V (K) V(K-11 a - 11 0 (1. 0*MU*Y (K-2) **2+lkll0*U (K - 2) * *2) GHAT (K - 1) a GHAT (K-2) + RiiO*ii ( K-2)*V(K—I) / DENOII F1IAT(K-1) - PHAT(K-2) 4 mU*V ( K-2)*V ( K—I)/DENON Dh-NUM
G F
GHAT(K-1)
FHAT(K-1)
IR ETU H END
ggG
N
R-16
CrcCCCcCcc CCCCCCCCcCcucc "CCCCCccceCCCK'CCCCCcCCGCCCCCCCGL'CC SOD ALGORITHM VQUATIONS ( 4-40) - (4-44) SUbROUTINS ADA&T5 ( C,D,MU v RHQoH,Q) IMPLICIT EtEAL * 6 (A-HO-Z) ltEAL*f MU , It 1 4 ) . U14 )e Y 1 4 ) ^ S 1 4 ) ^V14 ) , F'HAT ( 4 ► •CIIAT(4) IRLAL*8 GAMMA(4),Hi1'A (4) ,AHA'1'(4) ► WHAT(4i
COMMON /SIGNAL/R,U,Y,S COMMON /CONTRL/F,G,AHAT#bHAT,FHAT,{THAT COMMON /MISC3/D TA,GAMMA,V DATA K/4/
GHAT (K-2) = GHAT (K-1) GHAT(K-1) = GHAT(K) WHAT ( K-2) - THAT (K-1) FIIAT (K-1) = THAT (K) BETA (K-2) = BETA (K-1) bf^'TA ( K-1) : BETA (K)
GAMMA (K-2) = GAMMA (K-1) GAMMA (K-1) = GAMMA (K) V (K-2) V (K--1) V (K-1) : V (K)
BETA ( K-1) = NitO * R (K-2) **2 + MU*Y (K-2) **2 GAMMA ( K-1) _ (D+U) *N * BETA (K-2.) * V (K-2) +D * GAMMA (K-2) DENUM = 1.0+H*bLTA (K-1) V (K-1) _ ( S ( K - 1) - Y ( K -1) *Q* (S(K- 2 )-Y (K-2) )-GAMMA ( K-1) )/DENOM GHAT ( K- 1) = GHAT ( K-2) +RHu+R (K-2) *V (K-1) FHAT ( K-1) FIIAT (K - 2) +MU *X ( K - 2) *V (K - 1) G = GHAT (K-1) F = FHAT(K-1) RETURN 1.ND
A
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t•
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7125 kUFMAIiI1 t
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