On the use of the harmonic linearization method in the automatic control theory

MISSING IMAGE

Material Information

Title:
On the use of the harmonic linearization method in the automatic control theory
Series Title:
NACA TM ;
Physical Description:
6 p. : ; 26 cm.
Language:
English
Creator:
Popov, E. P ( Egor Paul ), 1913-
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington
Publication Date:

Subjects

Subjects / Keywords:
Automatic control   ( lcsh )
Nonlinear systems   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
This paper considers the use of harmonic linearization as applied to the analysis of nonlinear automatic control systems. This approach, which has been proven in practical engineering applications, involves the replacement of a nonlinear equation by a linear equation. In establishing the method a small parameter is considered which makes it possible to speak with some degree of approximation of the solution of the new equation to that of the given linear equation. Thus, the mathematics of nonlinear mechanics is brought to bear on the problem by explaining and extending the usefulness of automatic control theory and nonlinear systems.
Bibliography:
Includes bibliographic references (p. 6).
Statement of Responsibility:
by E.P. Popov.
General Note:
Cover title.
General Note:
"Report date January 1957."
General Note:
"Translation of "K voprosu o primenenii metoda harmonicheskoi linearizatsii v teorii regulirovaniya." From Doklady Akademii Nauk (SSSR), vol. 106, no. 2, 1956."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003874879
oclc - 45441727
System ID:
AA00009190:00001


This item is only available as the following downloads:


Full Text
W~~ 4M- f 40







L/ ~ I


NACA TIM 1406


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1406


ON THE USE OF THE HAR~MOM~IC LIH~EARIZAiTIOIJ METHOD IN THIE

AUTOMATIC CONTROL THEORY

By E. P. Popov

The method of harmonic linearization (harmonic balance), first
proposed by N6. M. Krylov and N. N~. Bogolyubov (ref. 1) for the approxi-
mate investigation of nonlinear vibrations, has been developed and re-
eived wide practical application to problems in the theory of automatic
control refss. 3 to 6). Recently, some doubt has been expressed on the
legitimacy of application of the method to these problems, and assertions
were made on the absence in them of a small parame~ter of any kind. Never-
theless, the method gives practical, acceptable results and is a simple
and powerful means in engineering computatations. Benee, the importance
of questions arises as to its justification. The underlying principle of
the method. is the replacement of the given nolinear equation by a linear
equation. In establishing the method, a small parameter is considered
whose presence makes it possible to speak, with some degree of approxi-
mation, of the solution of this new equation to the~ solution of the~ given
nonlinear equation. In an article by the author (ref. 7), certain con-
siderations were given on the presence of the small, parameter, but this
question has not as yet received a final answe~r. In the present report,
a somewhat different approach to the problem is applied that permits:
(a) establishing, in the clearest manner, the form of the presence of
the small parameter in nonlinear problems of control theory, solvable by
the method of harmonic linearization; (b) connecting it with previous
intuitive physical concepts (with the "filter property") an~d extending
the class of problems possessing this property; and (c) discussing various
generalizations of the method.

The free motion (transition process and autovibration) for a very
wide class of nonlinear systems of automatic control (ref. 7) are
described by differential equations of the form

Q(p)x + B(p)F(x,px) = O pa(1



~t"K voprosu o primenenii metoda harmonicheskoi linearizatsii v teorii
regulirovaniya." Doklady Akademii Nauk (SSSE), vol. 106, no. 2, 1956, pp.
211-214.





NACA TIM 1406


where Q(p) and B(p) are polynomials of any degree, of which some re-
q~uired properties will be established in the following paragraphs, and
F(x,px) is a given nonlinear function possible only with respect to as-
sumptions of the most general character. However, in problems of the
theory of control, no assumptions must be made as to the smallness of
the nonlinear function F(x,px) or to its small difference from a linear
function. In order to render explicit the form in which it would be
possible to write a small parameter in equation (1), we shall proceed as
follows.

Let equation (1) have a periodic solution or a solution approximately
periodic differing slightly from the sinusoidal. We write th~is solution
in the form
~x = + ey(t) X Sna (2>

where E denotes a, small parameter, and y(t) denotes an unknown bounded
function of time. In the case of the existence of a periodic solution,
we~ write


cy(t) = sC ak sin(iwt + 9k) (3)
k=2

We represent the gi-:en nonlinear function F(x,px) in the form

F(x,p) = F(x*C,pxQ) + [F(xlC + esy, px" + cpy) -F(x ,px )l (4)

Expanding the two components separately in a Fourier series, we obtain


F(xlpx) = qO + A + p i a k- k+ (5)

where qO, A., and B are the coefficients of the initial term? the sine,
and cosine terms, rexpectively, the cosine being replaced by p sin at
of the expansion of the function F(x",px") in a Fourier series. EFk
denotes all the higher harmonics of the expansion of F (x',px*) in a
Fourier series (they must not be considered small, since the nonlinearity
is not small) where we write

Fk = bk sin (kaut + Jlk) (k = 2, 3, .., b) (6)

ez'P denotes all the terms of the expansion in a Fourier series of all
the expressions shown in brackets in formula (4). This entire expression






NACA 00 1_406


is written with a small parameter which, according to equation (4), is
small if the derivatives aF/ax and aF/apx are finite. This expression
is also computed as smarll in the case of certain discontinuous nonlinear
characteristics (e.g., the Raleigh type where the preceding derivatives
at the points of discontinuity are delta functions). We may write
Sk = ECk sin(kwt + zlk) ( ,1 ,...= 7

We substitute equations (2) and equattion (4) in the given equation (1),
so that

Q(p)x* + Q(p)sy +


R(p)qq + R(p) A +,, p)snat (p)eI Fk + R(p)ef ,@k O~I= O (8)
k=2 k=0

Since the equation must be satisfied identically, we separately
equate to zero all coefficients with the same order harmonic. We note
that formula (8), in the case of the existence of a periodic solution,
is exact.

From the equating of the zeroth harmonics of equation (8), there is
obtained with an accuracy up to a: the relation


e3 =y- 26F(a sin u, amu cos u) du = O (u = cut) (9)

which is a certain general requirement for F(x,px).

From the equating of the first harmoics of equation (8), taking
account of equations (2) and equation (7), we have


a si at= A2 B2 i) Isin (rt + -r + B) -


Scl s ~c)(in)(i (rat + d1 + 8) (10)

where y and. 8 are arguments of th expressions A + iB and
R(io)/Q(ioj). On the basis of the3 exact eq~uatiojn (10), we obtain the!
following approximation (with an accuracy up to E):

a I I/.A2;-~B-i)~w I 42 d + P a re (11)






NACA TM 1406


It is here assumed that the polynomial. Q(]p) in equation (1) does not
hae purely imaginar roots.

From the equating of the higher harmonics of equation (8), consider-
ing equations (3), (6), and (7), we have

sak sin(kwt + 9k)


=-bk sinfSI"( kat t +k k)ec sinai~~fkat~u +d 4k ) (12)

where Bk denotes thre argument of the expression Rim/~k)
It isthus seen that if bk is not small, the magnitude
IR(ikmo)/Qiku)l should be of the order of e. The last component in
equation (12) will then be of the order e2. From the exact equation
(12), we obtain the following approximate equation (with an accuracy up
to e):


se =bkl~~ Qlk) k ek = (13)

Comparing formulas (13) and (113 we see that, for example, the wish
to have in the solution (see eqs. (2) and (3))


(sk2<< a2 (14)

leads to the need of satisfying, in the given equation (1), the following
requirement:

b~kw R 344 (A2 + B2 2IRiw (15)


where its satisfying in the concrete system can be checked after a is
obtained. The degree of Q(p) should, in, any case, be higher than that
of R(p). A particular case of the general expression (15) is intuitively
the earlier introduced "filter property" of the linear part.

Thus, condition (15) has been obtained anrd must be satisfied by the
coefficients of the given differential equation (1) in order that a pe-
riodic solution, if it exists, may be approximately determined in the
form of sinusoids in thre presence of a "strong" nonlinearity of F(x,px).






NACA TMl 14106


Th equation for its approximate~ determination according to equation (8)
with the substitution of sin wt =x*t/a, assumes the :form

Q(p) + R(p) q + p~ r xjt = 0 (16)

where

A na1 2xr~ i ,acsusnud




Q1 0 na JF(a sin u, am cos u)cos u du (17)


The replacement of equation (1) by equation (1_6) with its subsequent
investigation by the linear methods is called harmonic line~arizattion.
This is formally equivalent to following the mode of writin-g the initial
eqution (1) with a small pa-rameter, so tha~t

Q(p)x +- R(p) + p~ + f(x~p) = O (18)

where, according to equation (8), we may write


ef(x,p) =~, Rp Fk +s # -~ q+9-p)ey


where the first term of the three terms shown in brackets, takn sepa-
rately, is not small. The smallness of the function ef(x,p) is acquired
only with the factor R(p) owing to the property of equation (15).

There has thus been found the form of the presence of the small
parameter a in the equations of nonlinear systems of automatic control
required for the application of the method of harmonic linearization
(this also refers to the first approximation of the method of small pa-
rameter which, according to reference 7, coincides with the~ given method).
The writing of equation (1) with small parameter in the form of equation
(18) is valid in the region of the existence of eqcuation (2).

For definiteness we shall assume that the plnmasRp n
Q(p) are such that the characteristic equation (eq. (16)) has allpoitv
coefficients, and that the quotient of the division of the entire left
side of equation (16) by p2 + w2 satisfies the criterion of Burwits.
It may then be said that the given nonlinear system (eq. (1)) for the
existence of the periodic solution (eq. (2)) is close to the equivalent
linear system which is on the boundary of stability, except for the






NACA TM~ 1406


circumstance that the linear system is different for various periodic
solutions (because of a change in the coefficients q and ql from one
periodic solution to another, since q and ql depend on the amplitude
a and the frequency a of the required solution). Thus, evidently, the
equation of the~ first approximation (eq. (16)) is nonlinear if one speaks
of the combination of all possible periodic solutions for different values
of the coefficients of the polynomials R and Q (i.e., for different
parameters of the system), and is linear only for each given periodic
solution (along the given solution). Such is the property of the method
of harmonic linearization in its application to systems with "strong"
nonlinearity.

By writing the given equation (1) in .t;he form of equation (8) all
the variants of the method of harmonic balance in control theory and all
generalizations of the method also become easily understandable.

A similar reasoning may also be developed for the application of the
method of N. N. Bogolyubov (ref. 2) to the investigation of transient
vibrational processes in certain nonlinear systems of control.


REF EREH C E S

1. K~rylov, N. M., and Bogolyubov, N. N.: NJew Methods of Nonlinear Mechan-
ics. K~iev, 1934; Introduction to Nonlinear Mechanics. Kiev, 1937.

2. Bogolyubov, N. N.: Sborn. tr. Inst. stroiteln. mekh, AN~ Uk~rSSR, 10,
K~iev, 1949.

3. Goldfarb, L;. S.: Avtomat i telemekh, no. 5, 1947; M~ethod of Investi-
gation of Nonlinear Systems of Control Based on the Principle of
BHarmnonic Balance. Ch. XXXIII of Fundamentals of Automatic Control,
V. V. Solodovnikova, ed., 1954.

4. Popov, E. P.: Avtomat. i telemekh., no. 6, 1953; Izv. ANJ SSSR, OTNd,
no. 5, 1954; The Dynamics of Systems of Automatic Control, 1954,
pp. 593-663; 704-710; 716-721.

5. Popov, E. P.: DAN, 98, no. 3, 1954.

6. Popov, E. P.: DAN, 98, no. 4, 1954.

7. Po~povt E. P.: Izvi. AN SSSR, OTN, no. 2, 1955.


Translated by S. Reiss
National Advisory Committee
for Aeronautics


NACA Langley Futld, Va





~ .C~3 F
Nld,., ~N
~oo E m
ac~o, mo
c
mL~ 3
E'* C "
~e:
P~o oru
3 Um
~ O ~D~P~z?
N
lnWI~~~
n :3
- N ~E ~E6a
g 3 ~~~
m p 1
~co
a 9 ya,
U vr rrE4Z CI-
cl N D ,ejB


C,
~D-CO r
Nmoo eZCU
c~ E ~ vr d
90 ~~ vre
rdc~~
1~1' P~
E~b C m
c ~."
e o~ocu
~ up~~Z r
( o ~-L~& m~rrrl
n
'7 cnW$1
J I :E C~~ba
m a
E ~~fS U
rYln
O ~ICi O~
u m e~IaZ a~
cu o ,,E1


Y ZL~1 OEo
rdeon~aq.;m
m g~~ P
~3 C ~~pm~~m~
P- 36 u
ca
e~ldio lg,,e e
~Ld" d~d
,~~e~8q~m ~
oppu~~e
~c3
c.~ Id c ~ m
Z3 c ~r:.S ~ m I ~ E ~
eb,;;i 00
oa CLO
~g ~s e~~
CI~~0-~5
,,,,
o id blr O ~ b~j E
L ~E~) eU
~.~ e ~~ c~RC8~
"Le "" g
'J 3 P ~ 3 bL *Og
P) rd d e e E~, a
Sm~MO~up~m
~SE~i~2~
E~~ IdZ~
c~i c~ 03~~
~rra E
omO~P Eo
m~ ~~dS~gg~
C$EU M3~i P)
$ ~E~s~~
m~El~
,~Lau .~E~
y CI
B~PC, ;eEP~~
,,esaogr r
~3"
m ~sLrd
e ~e e
E~O~~-ly
COO 5~-
~~ u a~


rh3 ~4rd 5!m
N~e id ~C
~ E ~ g ~ ~ 99 P E
~o P md
,~jP~=
j~dOP 19~
~mmd $oe~D
~S1~CERE' LI
E~'C~PDIP~'
Oe~dEdm9C~~
e3c~P~m~lE~
cbllqgo
~",~I Se~~~
., $ 0~"
""'3 .p
o mM Em~PY1E
mC;c~ "SOI~
or
LLO~d'j~
pa a~~ M ol
13 e~E
L~oo~
5~ 9accs
""e~~~9~9
rd2E 01
%f TaS~09~
ppg U ~O01 Eg
e g EE k 1~
~15~E5,~~Ep
~pea~ c ~ Q
'~;eE"
a"~~010sg3r E
0 ~C41d eil
e aL~r;
C~01~ e~
~~~pS~&~f~p


ZP re
go: ooN
~O~o
qWoam
m ~d~ m
UNC&6~ILI
r: O
L~ ~
rd05.EFII
LEI~E rCI
Q)rl~4~0P:
( O ~d ~It~
UUC.2
~Zua
~O~oer~3
'I B e u z
7
~:3~~1~
~Z:J
oW3~
~~:88_43
qP
O~V~as_~N
PI~W&o:~~
m rd
r(3 ImQ
f~"~'14j"
~7W 0~
o r. N ~~ d
B~~a*~
oC~~o~
3 ro
le~D
za8~t;,~~


Z~ 5e
g~: B .E!N
~O- o d
(woa
m r~5 m
L~ ~~i E1D
am o
C 4
rde W ,
es
Wel gp:
4 O"x~vr
U ~v)
dzU~3s"
00d~,,
(u~=4m
~EE~uz
E~:O:~:~P~"
f I~eZ~
c;W~~'
U~~6~i~
",wtie~?,-
gPo~Pc N
P'~W90~A$
OV1
2(3 ~'""
e~m~:~rN
c~ awn g 3 ~P d
(c3ga~~u-
US~r8~l~eg
(aZW~y~e ~bl
ZZOS


"F
'03~
E~ m
UmO ~oi moE:
a~g
1~13
8,( a" "~
S O O"ON
~ C ~m~P~
"" o
C.WL~ N
OE ~' ~' 'El
O 1 ac>*~ re
~ ~coU
E~a 1D
m a rur
o ~~dg* O~
U m P~ZPIZ: CI
pj ~ c;~jE1


Cm I







29;:s aew P
U59 pm99: e
:*s == gO
0 .?5s s
.8.Q.. y:

r50. :

c E r


U .14





.a go me
rsqo~ag 9at
age*=> 8
50.12: 'is



Si? a, 4
"in th?


m~. m


SC O d~~U
a u 1 -r


8 .U ~F
Nldm.N LI~N
Eo vl
ULDO ~Q1 V1S
fic;S-- a- '3a to IDrr
E~* ~i Ld
s 3 9, ,Nr;

;i s~::
N LE C1Gfid
~ 1 sac*~
~~a
o* o,
ucouql~
vr ~T~Z CI
Cj C; c;~jE~

















i

e
3

O




al~
" 0


se


1~


8






So


Sri


S m


a


8





ag
go


18

4 =
mi


e a

li
z



















5,~
d~Od r
~~oaycila S
5: E ~ u a.p ~ a
d D O E~ p PP ~ 0 ~
plYP- Br M
'aPr~~S~
1 de 8U9C
u~C Rl~,~ca
~,O~lqg'U
C~~~ las co~
e U 1
O "
ebl oo
r. S a a O
rd~L: B
~ ,aa 050)
'~3a5 e
'd O O a~" E ~ a E r
cdO U
.ses* 4aa i
,, ~o~laR98p
~~aoj"ue-'Eo~
e 0 ap o-
ydaCeEC
a u
PP
EBeta ou
~'~ ajbr =;5 Z
$J "O~FOSao E
9 U EE g
a 03,~8~~E"
rr E c MSS ~ S
8'J"E53~1%~ '
1 L I B
* aao, e b q 5
"~ $E~ ie"
4 E5
,~,0~g~$ r c
ObOb
a ~s~~s,,2g
a88~~&Ss~E,
CI~UPJOC


eas F
Nrde .w a
ld
E1 e an

U mD 1D
o a, ,


U
rs3 3;* Od
N~de rd cc41
~eu~~eg~ g
~3' eaP~1 IZd
~',
~PP $ue~
R85 Ld
rr 9c
~20E
.eu ~:~
ei~us "-1E~
ebl
Ide ~5~~'"
:" B"'" '
,9 """4co%
"S
o a M I ~ bb~ E
~~es~ 'a Ua
c~
,, ~Rse,
~BellaBo- ~
,,
5 2 e -2 3 s
mMS~20 9~
rd
Z!e ,a o
e'E"id=ICI
gf rap og Id
m, o09eu B
E116u~
"'e~~~ld~
SQ)Q&86,d"6rr
1 a
r, mac Q
y~e~E, ;11.4
a1~8&0m0" r:
n 31M
m cc~~LCO,
ee~aJF 9c-jj
U a~


Z~ +e
~s"
~V"Dls~
(WB~m
C ~m e
3c*
9
eOr. r w
oP:
i2cl,
O d ~d~ VJ

M o s
o ~ org
~6,oz
EE '2"2~
o o
~4~i%
Lr* r~q ~I
~C~Orr*4 c~
o9V t a~ co
~gw r-pl
pl o, ~n rcl
vr~j,00~-.
E43~'~!"4~LY
c~aw 08
a~n o 8d
*O~C~~,eoYI
q Be
aZWyc~~~
ZZOSi 1L~ ~


o





ME
eO


nm -dn m


a s 2Y 3< 44or
It !




























B



U
z
e


a


a
s
U


Si

cm


8
u

O







Em


eSi
so

US


D
o
c
z
I
a
a
I
8








UNIVERSITY OF FLORIDA

3 1262 08106 651 5




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EPN7LLDTK_NVR50M INGEST_TIME 2012-03-02T21:05:24Z PACKAGE AA00009190_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES