FAO-A.90 119 ARM MSILE COMIMAND REDSTONE ARSENAL AL GUIDANCE A-ETC F/B 17/1DIGITAL SIMULATION FOR DESIGN OF A DISTURBANCE ABSORBING CONTRO-ETC(U)MAY G0 W L GICCOWAN, W HOOKER

U7NCLASSIF lEO nRSMI/RS-80-25-T0

-ENOMEEEEE

TECHNICAL REPORT RG-80-25

DIGITAL SIMULATION FOR DESIGN OF A DISTJRBANCEABSORBING CONTROLLER FOR A FOURTH-ORDER PLANT

0WITH SECOND-ORDER DISTURBANCE AT INPUT

Wayne L McCowanWilliam HookerGuidance and Control DirectorateUS Army Missile Laboratory DTIC

SELECTEhSOCT 1 01980 012 May 1980

B

IF __i F?4edt)n4& A~r-eoial, Alkabma~ 35609

Approved for public reloee; distribution unlimited.

CL.

ROM OFW 1021, 1 JUL 79 PfIOWS EDITION IS OBSOLETEo.,.,.,u ..°ou.o,.o.,o... ()R 9 2 2 1 1

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REPORT DOCUMENTATION PAGE BEFORE COPEIGFORM1. REORT HUMMER 12. GOVT ACCESSION NO: 31 ,7S CATALOG NUMIER

TR-RG-80-25AA /1 9L4. TITLE (an~d Subtitle) S. TYPE OF REPORT & PERIOD COVERED

Digital Simulation for Design of a DisturbanAbsorbing Controller for a Fourth-Order Plght Technical Reportwith Second-Order Disturbance at Input S. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(s) S. CONTRACT OR GRANT NUMSER()

Wayne L. McCowan and William Hooker

9. PERFORMING ORGANIZATION NAME ANC ADDRESS I0. PROGRAM ELEMENT PROJECT, TASK [Commander AREA & WORK UNIT NUMBERS

US Army Missile Command, ATTN: DRSMI-RGRedstone Arsenal, AL 35809It. CONTROLLING OPPICE NAME AND ADDRESS 12. REPORT DATECommander, US Army Missile Command 1 A 90 r1ATTN: DRSMI-RPT .NUBROPAERedstone Arsenal, AL 35809 7I&. MONITORINfG -AGENCY NAME A ADDRIESS(It O~ftener from Cameffinum Ott#*.) IS. SECURITY CLASS. (of this. reprt)

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Approved for Public Release, Distribution Unlimited.

17. DISTRIBUTION STATEMENT (of th. abstract 4046m a ISock i INHIfmn fhem ROPort)

1S. SUPPLEMENTARY NOTES

is. KEY WORDS (Ceaiuaae an rev'ee* adsineessar old fdmnIFp by week rmmoas)

Disturbance-Accommodating ControlDisturbance-Absorber ControllerSimulation

246 ADST*AcrmY m~wn ebb Onmm tie 11 FI or~ w u. eek 0010eb4e)

A simulation is presented vhich utilizes user-input plant and stateobserver pole placement data to generate a disturbance-absorbing controlcomponent,. u, v hich will cancel the effects of a disturbance which Isentering the system at the plant Input.

PD W 3 Sns PISVSSWS UNCLASSIFIEDBEJUIV CLASSIICATION OF THIS PA49 (AR~ Data Sneered)

SECUITY CLASSIFICATION OF THIS PAGM(Mb" De. Balm**

SECURIY CLASSIFICATION OF THIS PAGgEWhon Data Etnod)

CONTENTS

Page

I. Introduction .. . . .. ... . ... .. . . . .. . ... 2

II. Plant Model ................................................ 3

III. Disturbance Model .......................................... 7

IV. Disturbance Absorber Control ................................ 7

V. State Reconstructor Design .................................. 7

VI. Digital Simulation ........................................ 12

REFERENCES....................................................... 20

APPENDIX A - DIOITAL SIMULATION LISTING .......................... 21

APPENDIX B - SAMPLE RUNS ........................................ 46

FAccesion

For

NTIS GRA&IDTIC TAR

AvalabilitY CodesiAvail and/or

Dit Special

I. INTRODUCTION

The Disturbance Accommodating Controller (DAC) design method wasdeveloped by Dr. C. D. Johnson (References 1-5) of the University ofAlabama in Huntsville. This method uses a combination of waveform-modedisturbance modeling and state-variable control techniques and permitsthree primary modes of disturbance accommodation: (1) cancellation(absorption) of disturbance effects, (2) minimization of disturbanceeffects and, (3) utilization of disturbance effects as an aid inaccomplishing the primary control task. This report, and the digitalsimulation presented herein, will deal with the methods associated withthe first mode, i.e., absorption.

The plant considered is one which can be described by state equationsof the Form

= Ax+Bu+Fw(1)

where

x is the plant state vector

u is the plant control input vector

w is the vector of external disturbances actingon the plant

is the system output vector, and

A, B, F, C, E, G are appropriate size, knownmatrices which are not necessarily constant.

The disturbances considered are assumed to be described by the followinggeneral set of linear disturbance state equations:

w Hz+Lx(2)

= Dz+Mx+o

where

z is the disturbance "state" vector

a is a sequence of randomly arriving vector impulses, and

D, H, L, H, are known, time- -invarient matrices.

Since neither a complete set of plant state variables nor the variouscomponents of the disturbance are available for direct on-line measurementin most practical applications, the DAC is restricted to operate only oninformation in the available on-line measurements of the system outputs and

3

commands and on any disturbance components which may actually be availablefor direct measurement. Since the plant and disturbance states (x, z) arerequired for a practical DAC implementation, the necessary data, if notavailable, must be generated via use of state reconstructors (observers)operating on real-time system outputs y and control inputs u.

A full-dimensional observer which can be used to generate the plantand disturbance staie estimates (i, _) for the equations of the form (1)and (2) is given in Reference 2 as

A+FL+K (C+GL) [_+K G

M+ K2 (C+ G L) D+K G H

+ B(t) (3)

where

El E2 are gain matrices to be designed, and

A, F, L, C, G, H, D, M! are as previously defined.

For acceptable observer performance, the real-time estimation errors

C XX

(4)

must settle to zero rapidly in comparison to system settling times wherex and £ dynamics are governed by-- --zLA A+FL + K (C + GL) [+ K1 0G H-

_ + , ( 5 )

(:i) +K 2 (C+ GL) D + K2 (c a

I. PLANT HODEL

This simulation will model a fourth-order plant expressed in the formshown in Figure 1.

4

I.

*1I

Li

4.

+

K

'-I.

a~ 4. I

+

0

'-I. 00

N

4.

a 24.

9.13 01.4

* I4.

K

'-'a

24. I

9.a

5

The transfer function across the plant is

y(s) K [S4 + b s3 +bs2+b +b

- - 4 3 2u 1(s) s +b 5s' + bs6' +b s-b8

and this can be diagrammed as shown in Figure 2. As can be seen fromFigure 2,

i = x2 + Kb 1u1b5

- x +4Kbu b2 3 2 1 6

it x +Kbu -by (6)

it= Kb u1 - by

y + Ku

For purposes of DAC design, equations (6) need to be expressed asfunctions of x, u, and w. Therefore, since

u u +w, then

y = x+ K(u + w)

= -b + x + K(u+vw) (b - b)5 1 21 5

b -x + x + K(u +w) (b2 - b6 (7)2 6 1 32 6

i= -bx +xK~ + w) (b b)7 81 4 3 7

In matrix form, Equations (7) can be written as

-b 51 0 0 xIb 1-b 5b Ib5

5 1 5 1 5

-b 0 1 0 X b -b b -b

4 4 8- 4 81

[1 [0o0 x, + [I u+ [K] !E (9)

X4

These correspond to Equations (1).

6

~ --- '

I II I

0

0

I ''I

0r ** I 4.J K~ "ID

I 0+ *

+ +

I q~q II L..J

I I- _-j

2

3

4

4.

2

U

a

2

7

III. DISTURBANCE MODEL

The general set of equations describing the disturbances were givenin Equations (2). In this report, it is assumed that the disturbance isnot dependent on the plant state, i.e., L = M _0 . Therefore, thedisturbance modeled in the subroutine is

w = Hz(10)

= D z + a(t)

and it has been restricted to be a second-order model which can be representedas

H = z = (h1 h2

1 2 ( z l ) 11

= 2 + =Fdl d3]/Z +3d2 d4 z2 +- (12)

IV. DISTURBANCE ABSORBER CONTROL

For the complete absorption mode of DAC design, the object is to obtaina control vector which will completely cancel out the effects of thedisturbance input. First, however, it must be verified that such a controlexists for the particular case being considered.

It has been shown (Reference 1) that such a control vector, uc, will exist

if, and only if,

F B_

for some r. With the disturbance summed at the plant input, as shown inFigure 1, we can see from the representation in Equation (8) that F r B Pfor r [11. Therefore, for this plant-disturbance model, u exists andis U -PW = - w.

Since the disturbance states zI and z2 cannot, ir general, be measured,in order to implement the control u the state reconstructor given byEquation (3) must be used to provide ii and z2 " The DAC control for thisconfiguration will then be given by

u = -w = -hl -h2 (13)

C1 1 2 2

V. STATE RECONSTRUCTOR DESIGN

In order to implement the state reconstructor, it is first necessary

8

to design the gain matrices K and E2. This is done by using Equation (5)with -

= 2= [k 12 1 (14)

k21 LR22Jki

k31

k41

Substituting the appropriate values into the first term on the right-handside of Equation (5) and performing the indicated matrix multiplicationsand additions will result in the relation

(kll - b5 ) 1 0 0 h1K(bI - b5 + kll) h2K(bI - b5 + kll)

(k 21 - b 6 ) 0 0 h 1 K(b 2 - b6 + k2 1) h2K(b2 - b6 + k 21 )

(k 3 1 - b7) 0 0 1 hlK(b 3 - b 7 + k 3 1 ) h2K(b3 - b7 + k3 1 )

__x (k - b8) 0 0 0 hlK(b - b + k1) h2K(b - b + k1)41 8 1 4 8 41 2 4 841 ]

k 12 0 0 0 (d1 + Kh1 k1 2) (d3 + Kh2kl2 ) -

k 2 2 0 0 0 (d2 + Kh1 k22 ) (d4 + Kh2k 22) (15)

For computation simplification, let this be

A- + 0I-= A U _ (16)

and represent A as

e0 1 0 0 e6 e12e 0 1 0 e e

17 13e2 0 0 1 e8 e14 (17)

e3 0 0 0 e9 e1 5

e 4 0 0 0 e1 O el 6

e5 0 0 0 eli e1 7 -

Now, A represents the characteristic matrix of the £ dynamics. As statedearlier, it is desired that E(t)- o "rapidly" for good reconstructor performance.This means that the roots of the characteristic equation,

det[ - = o,

should be "large" negative numbers. The next step, therefore, (and generallythe most tedious),is to calculate

det[ N!].

9

Remember that A has unknown gain components included and is not just anarray of known numbers. Therefore, we have

(e -A ) 1 0 0 e6 el2 1

-A 1 0 e7 e1 3

e2 0 -A 1 e8 el4

det[ X-I1] e 0 0-A e e 0L- J3 9 15

e 0 0 0 el- e410 16

e 0 0 0 ell e17 -A

Evaluating this gives

I 6 15A- AI (e + e 0 + e1 7 A + (e0 e10 + e0e1 7 - e11 e1 6 + e10el7

e1 e4 e6 e5e1 2 ) A + (e0ee1 1e16 e0 e10 e1 7 + e1 e1 0

+ e1e7 - e2 + e4e6e17 e 4e11e12 - e4e7 - e5e6e16 + e5el0e12

3A + (e e -e e e + ee + ee -ee5e 1 3) (ee 1 1 1 6 1 1017 2e10 217 3

+ e e e1 e elle3 e e - e e e1 + e el e3 e e144 717 4 11 13 48a 5 716 5 10 13 5l14 A+ (e2e11el6 2e10e17 + e3e0 + e3e17 + e4e8 - e4el1e14

e ee - eee + eee4 e e +e e ee

4 81 1 4 5 15 3 10 17

+ eeel + e4e9e1 7 - e4ee 5 - e5 e9 e1 6 + eee)3116 491 4111 -5916 5 10 15 =0 (18)

If the desired roots of Equation (18) are A1, A2, A3, A4, A5, A6, thenthe desired characteristic equation is

(A -A 1 (4 X 5 ) (X - X6 ) 0 (19)

Expanding Equation (19) and equating coefficients of like powers of X betweenEquations (18) and (19) and substituting the proper symbols which the eirepresent results in the following:

(a) k + Khlk + Kh k + (d1 - b 5 + d4 ) + + + 411 1 12 2 22 + 1=

+ 5 + A6 A 1

10

... . ......... , , ,, i |

---------------------------------------------

(b) (dI + d4 ) k1 1 + (- Kd 2h2 + Kd4hI - KhIb ) k1 2 + (-Kd3hI + KdIh2

-Kb 1h2 ) k 2 2 - k2 1 + (- b5dI - b5 d4 - d2 d 3 - dld 4 + b6)

5 6

S 1 -A 2i-i .j i+1

(c) (d d - dld 4 ) k + (Khbd - Kh bld - b2 ) k + (- Kh bldd3 2 1 4 k1 1 + 1 14 2 1 2-K 1 2 1 2 C 1 113

21+Kh12bId - Kh2b2 k22 + (dI + d4 ) k 21 - k 31 + (-b 5d3d2

+ b5d Id 4- b6d 1-b6d 4+ b )-5 4 + 1 1 J + I

l-AAl -A3 ;

(d) (Kb2h1d4 22Kb2h2d2 - Kb3hI ) k12 + ( K- 2h1d3 + Kb2h2dI - Kb3h2 ) k22

3 d1d4)2 + (d1 + d k - k + b - + bd d(d3d2 - k21 d4 ) k3 1 k41 b6d3d2 6 4 1

- db d +[ 671 7d4 +b 8) = + 1

= J=i+ ii=J+ I n +- I+i

LijxIxn A4 ;

(e) (d3d2 - d1d4 ) k31 + d 3d2b7 + dld 4b7 - d1 b8 - d4b8) + (-Khb3d3

+ Kh b d + Kh b4 ) k + (Kh b d - Khbd - Khb) k + (d + d k2 31 2 4 2 2 + K 1 b3 d4 - 2 32 1 4 12~ ( 1 +d 4) 4 1

1- [i'2'3'4 ('5 + 6 ) + )' 1 2 5 6 (A 3 + A4 ) + A3 A4 5 6 I + )'2) ]-A ;5

(f) ( d- d 4 + d3d 2 ) k41 + (b8d114 b 8d2d3 ) + (Khlb4d4 - Kh2b4d 2) k12

+ (- Kd3b4 hI + Kd h2b4 ) k2 2 = x1 2 3 A4 A5 6 - A6

For ease of manipulation, let us re-express (a) - (f) as

(a) k11 + mok 12 + mtk 22 + m2 - A1

(b) m3 k 1 1 k21 + m4k12 + m5k22 + m6 A 2

(c) m7k1 1 + m8k21 k 31 + M9k1 2 + m1 0 k 2 2 + m - - A 3

(d) m1 2 k 2 1 +m 1 3k 3 1 -k 41 + m1 4 k 1 2 + m1 5 k 2 2 + 1 6

(e) m 17k31 + m18k4 1 + m1 9k12 + m 20k22 + m21 = -A 5

Mf m22k41 + m23k12 + m24k22 + m25 A 6

11'I

or, in matrix form,

1 0 0 0 rn0 rn1 k 11 A-rnmr-i 0 0 m -kl -A-rn2

m3 -1 0 0 m4 m5 k21 A2 - 6

m7 m -i 0 m m k -A- m7 8 9 10 31 3 11

(20)0 m12 m13 -1 M14 M15 k41 A 4 - 16

0 0 m1 7 mi1 8 m19 m20 k1 2 -A5 -m 21

L0 0 0 m22 m23 M24_ k22- 6 m25I

k[_K /21l [12jwe ['= ,whr K1 =[k31] -_2= 2J ()Therefore, we have X K2 R where k E

' --2 - 4 1 -

Solving for K7 gives 1 = X R where X denotes the inverse of

the matrix X . Since X is composel of known nmumbers when the desired valuesof A to A are substituted in, this matrix can be inverted via use of amatrix inversion subroutine.

Therefore, the components of the gain matrices K and K are determinedas functions of the plant and disturbance parameters and the values of Xthrough A chosen by the designer. It will usually be necessary to go

6through several iterations on values for the A's before the desired observerperformance is obtained.

Having these gains, it is now possible to construct the state observer,Equation (3), as

xI 1 -(k11 - b5 ) 1 0 0 h1K(b1 - b5 + k11 ) h2"X(bI - b5 + k 11 1

x2I (k21 - b6 ) 0 1 0 h1K(b2 - b6 + k21 ) h2K(b2 - b6 + k21 ) 2

3 (k3 1 - b7 ) 0 0 1 hlK(b3 - b7 + k31 ) h2K(b3 - b7 + k31 ) x3

x 4 (k4 1 - b8 ) 0 0 0 h1K(b4 - b8 + k4 1 ) h2K(b4 - b8 + k4 1 ) 4

i 1 k12 0 0 0 (dI + Khk ( Kh2k2) z

z2 k22 0 0 0 (d2 + Khlk22) (d4 + Kh2k22) 2

12

IIk K (b -b +k

111 5 k11)

k K(b -b + k)21 2 6 21

k 31 K(b 3 b 7 + k31

k41 K (b4 8 + k4 1 ) (t) (21)

k Kk12

1212Lk 22- -Kk 22-

and thus obtain the disturbance state estimates, Zl and a2, required forthe DAC control u

.--c

Figure 3 presents the overall block diagram for the composite plant-state reconstructor model. The symbols r, pi,' v, relate to matrix elementsfrom Equation 21 as shown in Table 1.

VI. DIGITAL SIMULATION

This simulation has been assembled, for use on a CDC 6600 digital computer,in order to permit the design of DAC's for systems of the type shown inFigure I without the necessity of having to go through the tedious task ofexpanding determinants by hand. This simulation can be used in a designprocess to determine the necessary gains for a given system and then simulatethat system's operation for various disturbance conditions. Or, the simulationcould be modified and used as a subroutine in a larger program to provide anecessary disturbance control value when called.

As a design tool used by itself, the simulation will perform the followingtasks: (1) calculate the elements of the gain matrices K and K utilizingthe plant and disturbance input parameters and the X's input by The designer;(2) implement the state reconstructor; (3) calculate the DAC control vector;

u = _ h1z -h 2z2P

and (4) close the DAC control loop through the plant to provide output datashowing the overall performance obtained.

As a subroutine, the necessary plant output and other data can be transferredin through a COMMON block; the gains can be updated, if required by changingplant parameters; the value for us can be determined; and then required datacan be transferred out through a COMMON block.

An overall program dictionary is presented in Table 2. Table 3 liststhe NAMELIST inputs for the program, and Table 4 lists the outputs. ASystem Library Line Printer Plot Routine is used to plot the output, Y, andthe disturbance state estimates il, z2 "

13

4WW-

_ + .. .. .. .. ..

A listing of the simulation is given in Appendix A and the results ofseveral disturbance cases for a given plant are shown in Appendix B.

The line-plot and matrix inversion subroutines used in this simulationwere taken from Reference 6.

14

TABLE 1. EQUIVALENCES FOR FIGURE 3 SYMBOLS

r " k1 1 - b5 P7 - Kh2 (bI - b5 + k)

r2 - k21 - b6 P8 = Kh2 (b2 - b6 + k2 1 )

r 3 . k31 - b 7 P9 m Kh2(b3 - b7 + k31)

r4 m k41 - b8 PlO . Kh 2 (b4 - b8 + k 4 1

P1 . Kh1(bl - b5 + k1 1 ) pll 1 d3 + Kh2k1 2

P2 a Khl(b2 b6 + k21 P12 d4 + Kh2k22

P3 a Kh 1(b3 - b 7 + k3 1) v a K(b 1 - b5 + k1 1 )

P4 ft KhI(b4 - b8 + k4 1) v 2 K(b2 - b6 + k21 )

P5 1 d + Khkl2 v 3 K(b3 - b7 + k 31 )

p5 f d2 + Kh1k22 v4 K(b4 - b8 + k4 1 )

15

L-i

C1414

X cc

.

K 0N

Nu

C44x -[to

IN 0

-116

TABLE 2. PROGRAM DICTIONARY

FORTRAN NAME SYMBOL DEFINITION

A A Coefficients of the desired character-istic equation associated with

[A- X] calculated using inputelgenvalues.

AMO-AM25 m1 Coefficients of the characteristicequation associated with[1- XI]calculated using actual plant anddisturbance input parameters,factored according to components ofthe KI and K2 matrices.

B b Coefficients in the plant transferfunction.

CO, Cl Co, C1 Coefficients used in defining w(t).

C K Plant transfer function gain value.

CUl Defined as K'U

CK K K2 Array containing computed values forthe gain matrices. CK(l) - CK(4)correspond to K1, CK(5) and CK(6)correspond to i.

D di Array consisting of the elements of theD matrix associated with the disturbancestate model.

DT At Integration step size.

H hi Array consisting of elements of the Hmatrix associated with the disturbance

state model.

KUTTA Integration loop counter.

KU Integration loop counter.

LM Xi Eigenvalues of 11 - XI I - 0 chosen bydesigner to settle out state reconstructorresponse.

NX Number of derivatives to be integrated.

PGO ur Plant Input Command

17

TABLE 2. (CONCLUDED)

FORTRn NAME SYMBOL DEFINITION

R R Matrix used in solving for K1 and K2. V

STPSZ Used to define integration step sizeAt - I./STPSZ.

T Intermediate terms, comprised of variouscombinations of the X's, defined for use inlater equations.

TIME t Total elapsed time (sec).

TSTOP tstop Run end time (sec).

Il uI Suimmation of u with disturbance magnitude, w.

U u Sumation of plant input command, u , DACcontrol term, uc, and plant output-feedback,

Y.

1IDA u DAC control vector.

W w(t) Disturbance vector.

Xl -X4 xI -x 4 Plant states.

XD- XD4 Xl x4 Plant state derivatives.

XDH1 - XDH4 xI - x4 State reconstructor state derivativescorresponding to Xl - x 4

XH1 -XH4 x - 4 State reconstructor states corresponding toxI1 - x4 .

XM Array of elements of X matrix.-qm

y y Plant output.

Z Intermediate terms, composed of variouscombinations of the 's, defined for use insimplifying later equations.

ZDHl, ZDH2 zl, 2 State reconstructor disturbance statederivatives.

ZH1, ZH2 ill i2 State reconstructor disturbance stateestimates.

18

TABLE 3. NAMELIST INPUT VARIABLES

FORTRAN NAKZ SYMBOL DEFINITION

B b Array consisting of the coefficients, b1 -b 8 9of the plant transfer function y/u1 .

C K Plant transfer function gain value.

CO, Cl Co , c Coefficients used in defining w(t).D d I Array consisting of the elements of the D

matrix associated with the disturbance state

model. The elements are entered accordingto the subscripts shown in Equation (12).

H h Array consisting of elements of the H matrix

i associated with the disturbance model. The

elements are entered according to thesubscripts shown in Equation (11).

LM Xi Array consisting of designer's choice of rootsfor the characteristic equation of

JA - XI{. The array permits input of complexconjugate values for the roots in the forma + Jb. For this reason, the input formatwhich must be used is:

(REIIMI), (RE2, IM2), (RE3 P IM3), (RE4, M4 ),

(RE5, IM5), (RE6, IM6).

NPRT Used to control output print interval.

NUMBR Used to control data storage for plots.

NX Number of derivatives to be integrated by theRunge-Kutta integration subroutine.

PGO ur Plant input command.

STPSZ Used to define integration step size as,DT l./STPSZ (sec).

TSTOP tSTOP Run end time (sec).

19

4V-'

TABLE 4. nIGITAL SIMULATION OUTPUT VARIABLES

FORTRAN NAME SYMBOL DEFINITION

PGO u Plant input command

TIME t Total elapsed time since beginning ofrun (sec)

UDA U DAC control vector

W w Disturbance magnitude as determined fromdifferential equation used to describe it.

Xl - X4 x - x4 Plant states

XDI XD4 1 - 4 Plant state derivatives^ *

XDH1 - XDH4 x - x4 State reconstructor state derivativescorresponding to XDl - XD4.

XHI - XH4 x1 - x4 State reconstructor state estimates

corresponding to Xl - X4.

Y y Plant output.

ZDH1, ZDH2 zip z2 State reconstructor disturbance state

derivatives.

ZHI, ZH2 Zl' z State reconstructor disturbance state

22

20_ _ _ _ _ _ _ _ _ _ _

REFERENCES

1. Johnson, C. D., "Accommodation of Disturbances in Optimal ControlProblems," Int. Journal Control, Vol. 15, No. 2, 1972, pp. 209-231.

2. Johnson, C. D., "On Observers for Systems with Unknown and InaccessibleInputs," Int. Journal Control, Vol. 21, No. 5, 1975, pp. 825-831.

3. Johnson, C. D., "Accommodation of External Disturbances in LinearRegulator and Servomechanism Problems," Institute of Electrical andElectronics Engineers Transactions on Automatic Control, Vol. AC-16,No. 6, December 1971.

4. Johnson, C. D., "Algebraic Solution of the Servomechanism Problem Vwith External Disturbances," Transactions of American Society ofMechanical Engineers, Journal of Dynamics Systems, Measurements andControl, March 1974, pp. 25-35.

5. Johnson, C. D., "Theory of Disturbance-Accommodating Control," Chapterin Book, Control and Dynamic Systems; Advances in Theory and Applications,Vol. 12, Edited by C. T. Leondes, Academic Press, Inc., New York, 1976.

6. Sellers, W. R., Jr., and Gibbs, B. G., "Descriptions, General-PurposeComputer Subroutines," U.S. Army Missile Command, Report No. TR-WS-79,April 1979.

7. McCowan, W. L., "Investigation of Disturbance Accommodating ControllerApplication to a Missile Autopilot," U.S. Army Missile Research andDevelopment Command Technical Report No. T-79-63, May 1979.

I.

(I

21

4 .

APPENDIX A

*DIGITAL SIMULATION LISTING

23

kaiiiCDmG pAua aumI-mOT nib=

PRCGRAM I 74/74

c 7 CCPA'v CUTPUT fT L, M T U

I N P L ' , , . , , , F ,LT .

LCC~'~cN/rr L-LF T T 7') ET, S T

AI C *I IC

15 A ~A, 2410 AN/25

2 C

y F~1 I C', 2

3 C F. F L C, L T,, C R 1 ) " , Y , ^ ) yy

LN' I. LFNC

C 9A C 0 C, )~ 9CC 9 9-A0) IL?~ k' 9( t ),4:

STPS7 (1 ) \. 1 )q TC',C),'i~~

C

CA ' L~ r I p 1jj 1,ET

*F L L

6PT CELv L~~

2C 1-E16DN PAG SAKO Fl

CL" 2 * L' ( 4 )

,(P) L v C P') k' L C (C 7(9) = L ( 2 ) * F' ( &

I' C S ) L 7 L V )L7(11) = Lr ( 3 ) * L F C

( 2) L" ( L ) * L' ( _ )(13) -- LW V 4 ) * Lv C -

4( ) L ( 4 ) L L 6 )S(15) = LW 5 ) * L" !v

T( 1 ) - L' ( 7 ) + L' 4 ) r L ' S ) ' L' ( L )-(2) -- L" ( 4 ) * Lp¢ ( 5 ) * L.W ( ' )

"(3) " L = ) + ' (7L (4) : LV Y 7 + L! m

"(5) L IV ( ) L ' ( f )1T'U,30) T

() T ( )+ T (4 )+ T( )

"(2) 2 + Z 3 + 77"- ,, ? ( 5 ) + 7 6 + 7 ) + Z +

* ? ( € ) + 2 C 10 ) + + ( 11 ) 1 7 12 ) ** 7 (1")4+ C ( I" ) * Z C 1I1

( 7 C ( * * 37 ) 7 2 ) * T( ( ( "" ( " ) * T ( 0 ) + ( 1 ) * L T (- )

7 7(1.) * 2 2 ) + ? 1 ' * T T 3 + &

• Z( P) (1) * C 7 C: 1C ) *7T(11 )47( ?Z4 1 L IV + '7 V F ( 1 1. } + 7 ( 1

* 7 C 13 ) e 2 C 1 l - ) * 2 C 1 ) ) +

* 7 ( 3 ) * 7. ( 1= ) 4 "' ( ) * ( " C 13 ) 4 7 C 14 32 7 ( 14 ) ) 4 7 ( 7 ) * 7 7 1 ) * 2 C 1F ) * + C 15 )

" ) = 7 ( . ) * 7 C 10 ) * T ( " ) 4

7 C 7 C " i ) 1 C. C 1 5'(E) = 7 ( I 3 * 7 1 10 T * C 15 3

',. ITS" (,47C)

"Ivo= C * F- ( 1 )CIv1 = C C ( 4 )

'2(1 )- (F + r

A"vl3 = ti=(

4= C * - 7 ( + * C 2 3 4 C 4 3 * - C 1 1 -

' - C-C2 ( F 3

7 (3)* (2)- C (13*7, ( " )IV= AM17 = AM 7

.' = - .= 5 3 * 4!"' * 5 C €: 3 - 4W''

-2 1 P F A - E C 7 ) * AV7f'IE - " C 6 3 * 4 7 - F ( 7 ) * A --F ( " )IV'1 - - A-_-l = . r ( = ) * / W7 . ," r ) , 4W,! * r (7 )

tlc - C * C C' 1 3 * -S ( 1 ) * rl (4 )'- "

F- C C(- ( C 3 * ( 3 r

• ( 2 ) P ( 1 )*F( 13+ - ( )'F (2))

" * C 26 * " ( I * r, 3 -0 " C " ) * I- C 2 3 * C 2 3 - R C( 3 * I- C ] )

= 26

PR1 51 C * C I- ( I ) , P C ) * ( 'T ) -

* F (2)*(2)*P( )-*P(I )*F( s) )

f-- C * ( ) ( 1 ) * C ) * - ( ) -

120 3=2 2 * ( 3 ) * ( ) ( ) * ( )

( 2 ) * F ) ) L(( ))P 2 ( C * - ( 2 ) * L ) +

125 [C 3" 1 1 I, 2, 5±0 ~1 T-(6C-6C 0 1, T qI T +1,o A(1,1)9 T+29 t(I+2)vT+!, (I!

• 144, M(I.4) A'

: (1) = A C 1 ) - RP2

1 30 (2) A C ) -

(3) =- A ( A ) - AVIlr( ) ( 4 ) -

(5) - A f ) - 21(6) : A C ) - A 2

135 Q ITF6,'-O0) F- CLU - NC. 1 --- FLEEI T c ThPll 6

yV (1,1) : q7

140 v ( 4 l1) : f ( 5,91) IV ( EvI) C. .,

C -------------------- COLUM'. NC. 2 --- FLE"E\Tc 7 TPOU 12v 1, ) : 0.0 *

YV(3,2) = -.

145 ~'4,2 A 11 2xm(5,2) X Y (692) = V.0

c -------------------- CLLMN N. 3 --- rLEYFNTq I! TPL, It-x (l Iv X v(2 ) 0.

vV(3 ) :-1.0

X(5,5) = A 17! 0.0

C--------------------- CCLUMN NC. 4 --- ELP NTS Ic TI-RL '14

Ylv(1,4) XN(2,4) = Y (3,4) = P.O'"I 5 v(49 ) : -1.0

Y IVM94 418 Lx (5,4) : 22I

C----------------------OLLMN C* 5 FLEVPNT 25 TFPL 3fL)( 195) = AO

16C. YP(2,F) = Av4X M( 395) A M 9Y (495) A Av14

5Y p A v1 1

IC C---------------------CCLUFN NC. 6 --- ELEVFNTS 31 T RIu !E×V(It ) = A I .. ..

xk(296) Av5-

P 4i t6) A Y15170 'P(5,e AI2C

PU(f,) 6 A 24

27

C 1 0 L '

CC 2C I =1, 6

17 ~ C ------------------ CALCULATE 1NVERSF-CF' 7V.ATLI' *i V- ..

["ALL SESOMI C Xms 69 69 19 6, CET, RA, E, 6rpKv IHLC, It 19 1IIF ( CET oFZ 0.0 erPe F* sECO 19C ) CU2 TC 2200IF ( E .FG* 2.0 ) W~lTE((-9 9200), C 4 r I = 1, 6

IF9 40 PITF(69200) DT XP(o,~ 19YF9) T ((9

'C 5C T 1, f

CALL PfPY ( X'1, Re CKi 69 6, 1A;ITr(6,1700) CK'C 1X2 0.0%I X', = 3 4 0.0

1% F X C F2 XCrI- '(CH- = ?CI-1 7C1-2 0.CXFI-= VH! = X[ 4 ZF

r T 1.0 / FTrPVZ

TIPE 0.0-

1:TF PF~ I

VSTR =STFSZ /NL!'PPIPLT IF1TR -1

VVX -10000co."

2CC PI 1%CflOO.0

"PINI V'IK2 = irOOrCO.O

T1 TIY'E a.37E. TSTCC7 ' TC iCOC

-C 2 'C 0<KL = 1~ 4

L T TA VL*CC + CI EXP( ALP *TIM~E

2 1 r FC C-c' - y

C 1C * L IC 4 CLi1 * F. 4 )-FC )*v

r (4 + C L. I P 3 ) P ' 7 ) * Y2 1 F r2 ) (3 4 CIJ I E C 2 ) - F F 6 Y

CE I Y'2 + CUl * E ( 1 1 - F C ) * y

*C C (1) 1~(~)*C l )

2 2 1 - ) * w -1 F - 2 ) 12 * L ) v ( C 1 1 * YI

C I - ) * 7>1 +- F 2 )*Z-2 L ' )-CK T - Y. * -

( CK ( 3 ) - C 7 ) *H 4 *e 2 F 3 ) - 4 C

C -C I 1 * ? 1 4 F- 2 1 * Z'- P L I - CI' 7 Y4r-' C XI 4 1 - F ) * I'1

28

PRCCGRt4 VAIN 74/74 CPT=2 FTI' 49+!

7* ( - ( 1 ) ?-1 4 F- 2 Z F 2? # t, re t iy~

23,) 'Ct Fk I ) * )U- 1 ( * C 1 1 C" C ' ) )

*~~ ~ Z?-h( )*Y C *CK( Al)*7tV-2 E ~ D I- 2 + r - CK c )** 7-I + D (4 )*C *1(2 C K 1(

235 * 7F-2 -C'< (7 Y CK )

'~C TO 0009 E0flo, CC, 4COD )3, KU7T5000 £CKTlNUE

T PL T TFPL T 1IF ( IPLT oNF. ISTR ICC TC 2020

2 4 1 IL T =IPTS NPTS + 1xT(KPTS)= Tlm'rY T (NKP 7S Y ~ YJ~

245y YIVIN A M If C I~ IN K s 3

722(NPTS)= 71-e'vlv~i = VAX1 CyvtxI, ZP1I.

y M I N1 = A M IN K C Yv I NI 9 ZHI25 Y AX, = AlvA)1 ( Yt-fl(:) 71< 3

Y"'N2 =AVI!KI ( YvIN29 7H< 32 2 0 " CkT INLF

IF ( IF .11'. T\PRT ) C TC pq30

255 i p

X* X"-39 X n 9 7?CHlI Z C-2, d1 2

26fl 3 UCA Xy-4 -

2, 30 (ChTIA LE3'000 TIMlE 'TItE + G.'4'00 3CNTINUVrE200 CALL RLNCK

26r 2'I30 CCNTIKLJF' C TO I CI 0

I'DO (CTINULE'PTS z - PTS + 1

)(T(N'FTS)z TIr27C YT(NF'TSI)= Y

Y fvAX A N A XI Y'AY YYV~IN AtlINI( Yh'IN, YZ1TV'PTS3 7'-1I?2TeIPIS)= Z-2

~'VIN1 AlvIN' Y v"INl * 2- 1

'vPl2 A INI Y%'IN2* Z -2

*P r( 9 10 Tl'q' v%1, Nr~.

CALL LIKCLT C XT9 VT, CUV/M, CUf'v', %-T Is Vt'I%, YVX Got Cot

II 29

FcpclApd AAIK 74/74 CPT=? FTN 4.(-+4!r,

CALL LINFLT C XT9 ZlT, Z2T9 tCUFulq NPTS, 29 YFINl, YF'AXlYI!1N2* Y!'AX2* 0., 0.)

r C T(' 2 10C2CC kCTINLE

2 1Z^C C h1N LF*cCo'LL EXIT

1c C CR!v'AT(8AiO)

.,c p I 1(! 1- 2( E12 aIX s3 ) /,X'T( I 194 3 1E262Y ,IY 1__X

4 b FCR-' (9X)A:*F~E12,.(l *A()=I2.Et ..-

1 * 9 C 3 9 F 12 6 ,1'Y(, * A ( 4 9 ,Ei 2 6'0- Y

r-50 FCRWT(/v4X,6FTlVE =ofi4o794)(96'-XCI =,F1497,4Yt

* y,*K(=qr:*,12.X6,XC3- *K( e74):r2.qFP3jr 94 C-'.-* =oEt4*794,9(2.P ,1,KI) * Fl4712.(,/)EIo74

5100 W FC! *T(F/f/,28X,(PFXCI =)C4X,4I7T1XI 9/9'R UNI AC*4 % X H 9 1 * 9 y 6 X~-3 = E 4 9 6-C-

*.hSCLUlC 9j*i4Y;TVF7C TFAPX"tY2iSNULRC L

Y 2 =F4744 X- q~~~4 jH?4 =~~~~31 0-P =~j*7/4X6F~e v~4*94(vNCrG

sc.6 =v 14 79 4 -CA =q 14 79 Yq - =~l o7 //

U. M II

c a,~) L14 C%

L)~~14 4 C' U O(,C OC 4C C)0UC C, C l

I" I :, :'. I Z: -,"vk z

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N. 04 14 ) V

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a ,a. r: a j a vm aU

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w w-3D z Z D D Z D:3 :)LI

u U~ ,(U~ .0 '-0 , C 111 uC m.ICC, 4C 4 LLJI6LJ4J.~4LJ44.6*I4 U66 4 6)4 b. ' It u a u.. 4

LC- 0 .-61 w-o- 4) c4~ULa~t U U L ~~. UW~.4-4 ~ .a L..

e'66LJ4~l 4.6 .4

L. Z I I, Z Ll. W, jLLC L ZWL4)w.. wdw4L . w w w.w6W L.

u - 4 '1

P.. 4' 4.)4.), 3L.U'.V C4)L~ x 'L 2 Lx44a,"4.a 3

644 .2 646j 1 64

Cl) .4 C4 44.131

p a= N) in. C a

-9 WN NAl N cof N N N

IL NAl to pa 4r NA CD MO 9, Go a 4,Drp 0t4" 91 ) X99 of N 'a W) 0u ryCA . " a

4 m- N lot 1." N M IN'al-4 I ,

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m~ K' AN LL L 1 wp I Wt-) &) rh 73 to u LlW* Li L- ri 24o k9 4'D I - tv ID

.4 1y C.N )..YCY4V~~'a t.J k4fr I., L.i

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F'4 - 'K I -IY

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'0 C. a1'a M aac m mkv

Wf W, V, )4.J V)a Wla k,, t W ,W ) V WV1 ,

La;

'4 w4 Q4 t), -WI, e a

Z) La L )D DC)D p= -4i C30.. Pr PwP i c t! c tW

a m. a3 a~34 a 33. m34 a34 I.m- a.4aj.w4 W Li.J4L L L(O 4.44. w ~ .. .44 w4 L.. w. .J~. -,) .4.-

w CD W- 4 CD4 ~ 4 W " CD4 U944 Lo& 4i Lo. 044 Cq C CD4

aL Li w -w- w-- I), w) (2 L LoV) W NI-'I4i l 1 1I-4- r C 2 4- '

a_ JLL w . 4.. 61- ai z e. w~ ZLt Uj Aj w w..f wll U Z

L~,.,g p. 1t4 U'' L. I~ ofl WI 'aa 4p. 0)' . x (.x x4

Nw Lo% -- s 0 r- r- C,~l r- V)9-6)W )- W

4 s l CN C C Y l IN CY Y C

.-.- w 4 .9 9P- *P U *u a (1 C l r ,r

cm CV ( N M(4 v ; V 4(r4V yI

LI. 8Lii(4

C~(~~(4OUCj(C l If-

C" C3-. wb (tlf~f cViV -. &~ CVJ mLu (fItfv IUU',Ut.t

co- -- p J ' L ' D I D wV 4 J U I UV w w

00 D 0C -,I Nr0 (47 (4Uf 0 M.nIL t, 4- - a V I I V 1 l 11 C . 1 1

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'VI V. 14.AV ! ! v .: . v n i ) w u- L Iw V , i i iU0 ) ( , ,U

IL a c ,a a II ' 4 L)) a a a aa (4 (4(4 t, MSa

m) D- aa

-3Uo4 C)1. a. C1 L.I. u

nt caatawt. awn a a ba a aacc

.- 4N *401 w U*t ~-Id .JLLi,( e a e-'( e ( V

_J zA. 7) (' -1- -Dt Z-f, wa .Li 29 r L 4

u~. Sa. (.C .1 N Le4. VX.afl. aa .

(4 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ L Xfk,> 394--- 4 NJJ J NN a'f- m ~ -

, IV

l IV pl imin

C., I, bI

V,'I

Ir w

IV IV .-. 11 1 Vm \

(V 3L

C, CUC C2 Inm 6C3%

.e~~~~~~ tv 44 .1 40C 1 4I V 1 .A

6).V L- t

IV LI)

41~ to W)t-,inI

V7 -Caq . 'aI~ 0 4 - "-SiP*CCCaCC a C C L L P U C?' NP P~t 31 4 . 1 1. ' CC C

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Ii I~

* -h- - i.p~IC',-.

* ~', i5~ a, Cs, w.0.

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PP C~, .. SE±I. P)(', U~- ~ *50. -CIa ~C) U.' ~ gi~v~ '4,0 0

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a J~-.'- C-) I.~

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a /20.0, -44F0*09 -loPOfle, -4C 0 0 015 - 17 6 e16,9 243o774Pg 43Fo'04F9 211 .411f

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CAT A T F S /32oCr A TA TS T rP /I Ole

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004040

SU8ROJTINC LINPLT(XTYTI YT?,YT3,NXt,'YTSYlI'l~YIMAXv

1Y2MINY2MAXY3MIAlqY3MAX)

DESCRIPTIONTI1S ROUTINE GENERATES ON-LINE PRINTER OL3TS FOR

11 2, OR 3 CURVTS. THF TA3LE OF INDEPENDENT VARIABLES

MUST IE EVENLY SPACEDo

INPUT

1 YT TABLE OF INDEPENDENT VALUES. MUST BEFVENLY SDACEO.

2 YT1 TABLE OF DEPENDENT VALUES FOR FIRST CURVE.

3 YT2 TABLE OF DEPENDENT VALUES FOR SECOND CURVE.

4 YT3 TA3LE OF DEDEN)ENT VALUES FOR THIRD CURVE.

5 NX NUMBER Or POINTS IN XT.

6 NYTS NUM9ER 0 CURVES TO 3E PLOTTED.

(NYTS=lt 29 OR 3)

7 YIMIN LOWER LIMIT OF YTI SCALE.

B YIMAX UPPER LIMIT OF YTI SCALE.

IF Y1MIN =Y1AX 9 THIS ROUTINE WILL

CALCULATE SCALE VALUES.

9 Y2MIN LOWER LIMIT Or YT2 S:ALE.

10 Y2MAX UPPER LIIT OF YT2 SCALE,

IF Y2MIN =Y2MAX 9 THIS ROUTINE WILLCALCULATE SCALE VALUES,

11 Y3MIN LOWER LIMIT OF YT3 SCALE.12 Y3MAX UPPER LIMIT OF YT3 SCALE.

IF YiMIN =Y3MAX 9 THIS ROUTINE WILL

CALCULATE SCALE VALUES.

OUTPUTOV-LINE PRINTER PLOTS

REMARKSIF A PLOT OF 1 CURVE OR A PLOT OF 2 CURVES IS

DESIRED, THE VARIABLES NOT NEEDED MUST BE DUMMYVARIABLES IN THE CALL STATEMENT.

EXAMPLE,..TO PLOT 1 CURVE

CALL LINPLT(XV1,YV1,DUMMYDUMMY,100,1-IOOI.OO.,OoO-,O.J

4,

41,

SUBIOUTINE L1IJPLT(XTYT1 ,YT2,YT3,VXNYTSY1MIjgy3MAX, 101 1C FlVY2A93INYMX 101 2

OIMFNSIO'V XT(1),YTi(1).YT2(i),YT3c1)WR(ARRc1O1) 101 3DIME'JSION TT(4),DLM(3) ,SCA(3) ,SCALE(&) .A8C(3) 101 4!DIM7NSION YMIN(3)9YMAX(3)vYLL(33 ,YUL(3) 101 5DATA BL'(,fOT/IH 91H*/ 101 6DATA ABC/IHAo1HbvlHC/ 101 7DATA TT /29092*095&09iO.0 /101 8

C IVTTIALI?E 101 9DO 200 11=193 101 10IF(Il *GT. 'JYTS) G0 TO 300 101 2iGO TO (10920930)9 11 101 12

10 YMN=Y1MIN 101 13YX=Y1 9AX 101 14GO TO 50 101 15

20 YMlVJY2MIN 101 16YMX=Y2MAX 101 17GO TO 50 101 18

30 YMN=Y3M1N 101 19YMX=Y3MAX 101 20

50 YMIN(11)=1.OE*20 101 21YMiAXCII )=-1.OE*20 101 22DO 60 IziNX 101 23IF(Il EQ. 1) Y=YT1(1) 101 24IFAII oEO. 2) Y=YT2(1) 101 25IFCII .EQ. 3) Y=YT3(f) 101 26Y F4INC II)=AMI NiCY MlN(II), Y) 101 27

60 YMAX( II)=AMAXi(YI4AX(11).Y) 101 281F(YMNI .EQ. YMX) GO TO 70 101 29YLL(1I) =YMN 101 30YUL(I1) = YMkX 101 31GO TO 140 101 32

c SET SCALES 101 3370 D=ABSCYMAX(I11)-YtiINCIl)) 101 34

IF(9 .eIE. 0.0) GO TO 72 101 35O 0.01*ABS(YMAXCIf)) 101 36IFCD .EQ. 0.0) D = 1.0 101 37

72 Li ALOGIO(D) 101 38IF(0 *LT* 1.0) LI Li-i 101 39TEST =.5*10.0**CFLOAT(Li-8)) 101 4000 75 1=1,4 1o1 41R= TT(I) * 10.0**FLOAT(LI) 101 42IF(R .GE. 0) GO TO 80 101 43

75 CONTINUE 101 4480 IF(YMIN(I1) eNE. 0.0) GO TO 90 101 45

YLLCII)=0.O 101 46YUL( I )=R 101 47GO TO 140 101 48

90 IF(YMAXCII) *NE* 0.0) GO TO 100 101 4995 YUL(II)0C.0 101 50

YLL(11X)-R 101 51GO TO 140 101 52

100 P=.5*(YMIN(I1 )+YMAX(1I)) 101 53P D'0.001*R*SIGN(1*09P) 101 54L2 0 101 55

42

IF(P .NEo 0.0) L2 =ALOGIO(APS(P)) 101 56IF(ABS(P .LT. 1.0) 12r12-I 101 57IP~ztD..5*1O0..*FLOAT(L?) )/ilo.**FLDAT(L2) 101 58IF(IP .LE. 0) IP=IP-1 101 59I110 YLLE 11):FLQAT(IP)*10.0*.FLOAT(L2)-.5. 101 60lr(vLL(1I) .GT. YMI1N(II)) GO TO 125 101 61IF(YMINUII) .GTe 0.0) YLL(I1)=AmAX1(0.OYLL(II)) 101 62YUL( II)=YLL( I1)+R 101 63IF(YUL(II) &LT. YMAX(1I)) GO TO 135 101 64IF(YMAX(II) *LT. 0.0 *AND. YUL(11) *GT. 0.0) 5O TO 95 101 65IF(YUL(Il)*YLL(II) .GE. 0.0) rv0 TO 130 101 66DO 120 1=1910 101 67TMP1=YLL(II)+.1R*FLOAT(l) 101 seIF(ABS(TMP1) .Lre TEST) GO rO 1310 101 69

120 CONTINUE 101 70125 IP=IP-1 101 71

GO TO 110 101 72130 IF(YUL(11) oGE9 YMAX(II) G3 TO 140 101 73

IF(YMAX(I1)-YUL(II) .LE9 9O05*R) GO TO 140 101 74135 R = 2.0*R 101 75

0O TO 110 101 76140 :)LM(II)=(*UL( II)-YLL(II))/5.0 101 77

SCA (II) :YLL (II) 101 78c PRINT CURVE MAX AND MIN VALUES 101 79

150 IF(II *EO.1) WRITE(69160) 101 80160 FOR4tAT(1H1) 101 81

WRITE(69170) IIA3C(JI),YMIN(1IJ) YMAXlI) 101 82170 FORMAT(1X97HCURVE Y911,1XIOHDENOTE-3 BYslXXA1,4Y94HMIN=1PE~o.3, 101 83

12X94H? AX:IPE1O.3) 101 84200 CONTINUE 101 85

C PRINT CURVE SCALES 101 86300 WRITE(69310) 101 87310 FORMAT(1H0) 101 88

DO 350 11=1,3 101 89IF(II 9GTo NYTS) GO TO 360 101 90SCALE(1)=SCA( H) 101 9130 320 I=296 101 92SCALEd )=SCALE(I-1)+DLM( II) 111 93IF(ASSCALE(JH) eLT. TEST) SCALEd!) : 0. 10~. 94

320 CONTINUE 101 95330 OR1TE(69340) ABC(31)s (SCALE(1),1:1,6) 101 96340 FORMAT( IX96HSCALE ,A1,10X,10E10.3,1OX,1'"E10.3,10X,1PE10.3,10X, 101 97

11PE1O.3,1OX,1PE1O.3,1OX,1PE1O.3) 101 98350 CONTINUE 301 99360 PXP=NX*1O 101 100

WRITE(69365) 101 101365 FORMAT(1HT) 301 102

OX=XT(2)-XT(1) 101 10330 800 1=1,NXP 101 104WRKARR (1) :DOT 101 10500 375 JJ=29101 101 106J:,JJ T01 107WRKARRCJ)=BLK 101 1081F(MOD( (J-1).10).EQ.0) VRKARI(J)=30T 101 1091rf1oEga3) WRKARR(J)00OT 301 110

43

IF(MOO( (I-1),5).EQ*0) WRKARR(J)=DOT 101 111

375 CONTIN~UE 101 1121E(I*STeNX) GO TO 760 101 113

400 00 420 11=1,3 101 114IF(II .GT. NYTS) GO TO 720 101 115IF(1I.EG.1 ) Y=YT1( I) 101 116

IF( I1.EO.2) Y=YT2( I) 101 117

I'CII.EO.3) Y=YT5(1) 101 118

NP=1009(Y-YLL(1I))/(YUL(II)-YLL(II))*.. 101 119IF(NP *GT. 101) NP:1O1 101 120

IF(NP .LT. 1) NP=I 101 121

Wq'(Aqq(NP)=ABC(!J) 101 122

420 CONTINUE 101 123

C PRINJT LINE OF DESIRED PLOTS 101 124

720 X:XT(1) 101 125

IC(1.EOo1) GO TO 740 101 126IFCMOOUI1-1)10)*E.l) GO TO 740 101 127

JRITE(69730) WRKARR 101 128730 FORMATC2OX,1O2I) 101 129

GO TO 800 101 130

740 JRITE(69750) X9WRKARR 101 131

750 FORMAT(10X,1IPE1O.391*01*1) 101 132

SO TO 800 101 133760 X=XT(NX)+FLOAT(I-NX)*DX 101 134

IF(MOODUI-l),10).EO.0) GO TO 8201013WRITE(69730) JRKARR 101 136

900 CONTINUE 101 137

GO TO 830 lo1 138

820 WRITE(6,750) XsWRIKARR 101 139

530 WRITE(69835) 101 140

835 FORMAT(IHS) .101 141

RETURN 101 142

END 101 143

44

J

SUBROUTINE SESOMI(XNNBMSMNlDRE GRKIHLD, ICoIDtIS)

DESCRIDTIONTHIS SU9ROUTINE WILL SOLVE AN N By N SYSTEM OF SIMULTANEOUSEQUATIONS wITH AN ARBITRARY NUMBER Or RIGHT HAND SIDES ORINVERT A MATRIX OF ORDER No It: THE PROCESS, THE RANK OFTHE MATRIX AND ITS DETERMINANT ARE EVALUATED. THE METHODUSE) IS THAT OF GAUSS-JORDAN WITH TOTAL PIVOTING IF DESIRED.

INPUTI x FIRST LOCATION OF INPUT COEFFICIENT MATRIXvX(11)

AUGMENTED BY N9 RIGHT HAND SIDES. FOR MATRIXINVERSE, X IS FIRST LOCATION OF THE MATRIX TO BEINVERTED. I.E. X(I1). X MJST BE DIMENSIONEDTC (MN1,MN1.N?) IN THE CALLING PROGRAM IN EITHERCASE.

2 N NUMBER OF SIMULTANEOUS ElUATIONS TO BE SOLVED*OR ORDER OF MATRIX TO BE INVERTED.

3 NB NB = NUMBER Or RIGHT HAN4D SIDES FOR SIMULTANEOUSEgUATION SOLUTION. N5B = N FOR MATRIX INVERSE.

4 MS MS = 0 FOR SIMULTANEOUS EDUATION SOLUTION.MS = 1 FOR MATRIX INV.ERS -.

5 MN1 ROW DIMENSION OF X AS DEFINED IN CALLING PROGRAM.6 wORK WORKING ARRAY DIMENSIONED AS FOLLOWS IN CALLING

PROGRAM... DRK(MNINB).7 IHLD WORKING ARRAY DIMENSIONED AS FOLLOUIS IN CALLING

PROGRAM... IHLD(MN1).9 IC IC:=1 PIVOTING BY RD.' ONLY. NORMALLY SUFFICIENT.

IC=09 PIVOTING BY ROW AJD CCLUMN. RUNS LONGER.9 ID IO=1, DETERMINANT CALCUALTED.

ID=0, DETERMINANT NOT DESIRED.10 is IS=lq MATRIX IS NOT SCALED PRIOR 10 VANIPULATION.

IS=O, EACH MATRIX ELEMENT IS S:ALED PRIOR TO MANIP.

OUTPUTI X X(l1) THROUGH X(Nl) CONTAIN FIRST SOLUTION

VECTOR. X(192) THROUGH X(N,2) CONTAIN SECONDSOLUTION VECTOR, ETC. FOR 4ATRIX INVERSE, THEARRAY X CONTAINS THE INVERSE MATRIX.

2 D DETERMINANT OF INPUT X.3 R RANK OF INPUT X,

4 E ERROR CHECKE=0 O.K.E=l MATRIX OF COEFFICIENTS IS SINGULAR.E=2 SOLUTION IS ATTEMPTED BUT 2'UATIONS MAY BE

SINGULAR DR ILL CONDITIONED.

REMARKSTHIS SUBROUTINE wILL RUN FASTER WITH IC=l AND IS=I. THE VALUEIC SHOULD BE SET TO 0 ONLY IN RARL CASES WHERE EXTREME ILL-CONDITIONING IS EVIDENT AND IS SHOULD BE SET TO 0 ONLY WHENELEMENTS OF ONE ROW OF THE MATRIX IS MUCH GREATER THAN THEELEMENTS OF OTHER ROWS, CAUSING A FALSE E:2.9 INDICATOR.

45

SUBROUTINE SESOMI(XNNBMSMN1,ORtEWORKIHLOICIO.IS) F01 1

DIMENSION X(MI9t)vJORK(1)qIHLD(1) F01 2

C DOUBLE PRECISION X9ORK0Y93sSUMX1 F01 3THE FOLLOING 9 CARDS ARE TEPOIARY MOVIFICATIONS TO ALLOW F01 4

EXISTING CALLS TO SESOMI (USIJG 10 ARGUMENTS) TO WORK PIOPERLY* F01 5

ANY CALLS NOW MADE SHOULD INCLUDE ALL 13 ARGU4ENTS. F01 6

J = LOCF(IC) F01 7

IF(j .GTo 64 )GO TO 50 F01 8

IIC = C F01 -9

IIO = 1 F01 10

IIS = 0 F01 11

GO TO 51 F01 12

50 IIC = IC F01 13

lID = ID F01 14

lIS = IS F01 15

51 Xl = 1. F01 16E=O. F01 17

R=O. F01 18

IF(IIC .NE. 0)GO TO 211 F01 19

DO 21 1=1,N F01 20

21 IHLD(I)=I F01 21

211 CONTINUE F01 22

IF(MS)69496 F01 23

6 NN=NN FO1 24

NB=N F01 25N=NI FO1 26

DO 14 1=1N FOI 27

DO 14 J=MNtNN F01 2814 X(ItJ)=O, F01 29 i

DO 15 1=19N F01 30J: I+N F1 31

15 X(lJ)=l. F01 32

GO TO 16 F01 33

4 NN=N+NB F01 3416 JJ=NN F01 35

NNN=N-1 F01 36D=0. F01 37

IF(lID *NE. 0)0=1. F01 38

IF(IIS .NE. O)GO TO 361 F01 39

DO 36 I=1N FOI 40

Y=X(I 1) FO 41

DO 35 J=29N F01 42

IF(ABS(Y).LT.ABSCX(TJ)))Y=X(IJ) FO1 43

35 CONTINUE F01 44

D=D*Y F01 45

00 36 J=19NN F01 46

36 X(IlJ)=X(IOJ)/Y F01 47

351 CONTINUF FO1 48

DO 5 I=I#N FO0 4Q

KK=N-I 0 bU

IF(KK)10OO26 F01 51

2 1I(IIC o.NE. O)GO TO 261 FOI 52

LL=KK*1 FOI 53

IJJ:l F01 54SL: F01 55

46

W04'(I)=X(1,1) FBI 56DO 17 I1,1LL FBI 57DO 17 Jl,9LL FOI 58IF(A3S(WORK(1))-A8S(X(IIJ)))1b,17,17 FBI 59

18 WOR.((1)=X( II J) FBI 60L=J*I -1 F01 61i J~j F01 62

17 CONYTINJE FBI 63IFi IJJ-1 ' 2,2*19 FBl 64

19 DO 20 11=19N F01 65Y=X (I1,1) FBl 66XCII, 1) :X (I JJJ) VBI 67

20 XC1IITJJ)=Y V01 68IY=THLO( I) F01 69IHLD(1)=IHLD(L) F01 70IHLD(L)=IY VBI 71D=-) VBI 72

261 IJJ~l V01 73Y=X (1,1) F01 74

2 DO 1 L=19KK F01 75lV(ABS(Y)-A83S(X(L-1.1)))7,plq1 FBI 76

7 IJJ=L.1 F01 77Y=X(L+191 ) VBl 78

I CONTIMlE VBI 79IF(IJJ.EQ.1) GO TO 10 V01 80D=-D V 0l 81

DO 9 J=1,JJ FBI 82

X(IVJ)=XCIJJ$J) F01 8'.9 X(IJJgJ)=Y Vl8

10 JJ=JJ-1 FBl 86

D0*X (1,1) V01 87IFCXC1,1)*EO.0.)GO TO 8 FBI 88

31 IF(ABS(ABS((X1-XU1,1))/X(1)-l.).LT.1.E-7)E=2. VBI 89X1=x(1,11) V01 90

11 R=R~l. F01 9100 12 J1,sjj VBI 92

12 JORK(J) X( 19J.1)fx(1,1) V01 93KK=JJ+1 Va. 94IF(NNN.E~o.)GO TO 33 F01 95DO 3 K=1,NNN V01 96

DO 3 J=29KK FBI 973 XC(,gJ-1)=X(Kllj)-X(K+1,1).hiORK(J-1) F19

33 DO 5 J=1,jj VBl 995 X(Ngj)=WORK(J) FBI 100

IV(11C *NE. 0)GO TO 13 V01 101NN=N-1 VBI 102IFCkNN.EG*O)GO TO 13 V 0l 10330 22 1=1,NN F01 104L 1.1 F01 105DO 22 J=LN F01 106IV(IHLD(I )-lHLO)(J))22q22.23 FBI 107

23 IY=IHL:)(I) VBl 108IHL3(1I)=IHL)(J) FBI 109IHLD( J) :IY VBI 110

47

DO 25 K-1,NB F01Ill1Y:X(IIK~) F01 112X(1,K):X(J*K) F01 113

25 X(J,'O:Y P01 12422 CONTINUE FOl 11513 RETURNJ F01 1168 ZI. F01 117

RETURN F01 1.18END F01 119

4R

APPENDIX B

SAMPLE RUNS

-- 49 -*

Several complete sample runs are presented in this Appendix in orderto furnish examples of the output which results when this program is runon a stand-alone basis. The plant used for the examples is a simplifiedautopilot loop described by aerodynamic transfer function and compensatordata taken at various times along a nominal trajectory (see Reference 7).For each case, the NAMELIST input section, the X matrix (before and afterinversion), output data as listed in Table 3, an line printer plotsshowing output Y and disturbance state estimates zl, z2 are given.

The disturbances used in each run are as follows:

(a) Run 1: w(t) = 1.

(b) Run 2: w(t) = 1. + 0.5t

(c) Run 3: w(t) = 1.5 + 0.5e"25t

For runs (a) and (b), where the disturbance is of the form w(t) = C Ctthe disturbance is modeled as

o

V = z =(1, 0) (:zz

i:

0 2D z D+ a [0 11]Qi.) + 0.

For run (c), the disturbance is of the form w(t) = Co + cleat and ismodeled as

-= (, ) (z)

z 2

0 a z2

51

RUN 1

52

tit

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N La

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tolit

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0 . 4MV. , Mcm N r C2 N 4400 4 Dc 0 U 4c, 10C'0

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DISTRIBUTION

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Defense Technical Information CenterCameron StationAlexandria, Virginia 22314 14

14i

CommanderUS Army Material Development and Readiness CommandATTN: DRCRD 1 A

DRCDL 15001 Eisenhower AvenueAlexandria, Virginia 22333

US Army Materiel Systems Analysis Activity .ATTN: DRXSY-MP 2Aberdeen Proving Ground, Maryland 21005

lIT Research InstituteATTN: GACIAC 110 West 35th StreetChicago, Illinois 60616

DRSMI-LP, Mr. Voigt 1DRSMI-R, Dr. Kobler 1

-RPR 3-RG 1-RGN 8-RPT (Reference Copy) I-RPT (Record Set) 1

-EX, Mr. Owen 1

77