Exact Analysis of a Nonlinear Feedback Control System subjected to Sinusoidal Input 利用統計を見る

(1)

Exact Analysis of a Nonlinear Feedback Control System

subjected to Sinusoidal Input

SeiichiroMAEZAWA

Th・’ ?Eeq・・n・y resp…e・f・f・cdb・・k・・n…1・y・t・m，。n・。i。i。g。。。。li。。a， el，m，。、。f piecewisc−linear type subjected to sinusoial input is exactly obtained by a new method ・tili・i・g・pP・・p・i・t・F…i・・se・i・・e・pan・i・n． Thi・m・th・d i・i・p・in・ipl・th・，am。 a， th。t i・t・・duced by th・a・th・・t…1・・the stc・dy・ib・ati・n・i・・n un・ymm・t・i・al piecewi，c−li。，a， system subjected to a harmonic exciting force． It has the followmg main features： 1）Li・・ali・i・g th・…li・・a・differe・ti・1・q・ati・n g・v・mi・g th・dynami・b。havi。。。f th， systcm by・xp・nding th…nli・・a・p・・t・f th・・utp・t・f th・piecewi・e−li・・a・d。m。nt i・t。 aFourier series with the same period as of the given sinusoidal input． 2）Obt・i・i・g th・f・・m・1・・1・ti・n・f th・・li・・alized・quati・nby，eg・，di。g th。・b・ve−m・・ti・n・d・・nli・・a・p・・t・・if・it were an・x・iti・g f・・ce f。。m with。ut． This s。1。ti。。 contains unknown coefficients of the Fourier expansion above assumcd． 3）Determining these unknown coefficients from the conditions that the above obtained solution satisfies the given piecewisc−linear characteristics of the system． Certain process ・f・・nverg・n・y imp・・v・m・nt by m・a…f・e・i・・t・a・・f・・m・・i・・m・y p・・v・h・ghly・ffec・i・・ in this step． As an example of the piecewise−linear element， we treat an element with a dead zone， and give the procedure of analySis for a feedback control system with such an element． Some nUmerical results and diagrams for the frequency response of the system are added． Also

afew examples of wave forms for certain amplitudes and frequencies of the sinusoidal

mput are shown．

These procedures of analysis and their results may contribute to the removal of

uncertainties involved in the describing function analysis usually apPlicd to obtain the frequency response of the nonlinear control systcm．

1．Introduction

An automatic control system contains oftcn a

number of nonlinear elements， caused by im・

perfect mechanisms of the elements， such that having dead zone， saturation chafacteristics and solid friction etc．， or caused by the intentional use of special mechanism as in the case of optimum nonlinear contro1． In such cases， the analysis of various dynamic behavior of the system becomes highly complicated compared with that in the linear control systems． If the system is originally， or approximately， gov・ erned by 2nd ordcr equation， we apply the phase plane method and draw trajectories of the rep・ resentative point on the phase plane to investigate the behavior of the transient response or the exi− stense of the limit cycles． Alternatively， we rcsort to thc describing function analysis to investigate the frequency and transient response of the system． This procedure is almost the same as the transfer function method to obtain the frequency response in the case of the linear systems． However， the rcsults of analysis may involve some quantitative or evcn qualitative errors・because the describing

28

(2)

Exact Analysis of a Nonlinear Feedback Control system subjected to Sinusoidal Input f皿ction analysis is based on sevcral assumptions and apProximations・ If a nonlinear element is subjected to a sinu− soidal input， x（り．．XI sinω彦，（1・Ol） the ensuing outputツ（t）will generally contain a constant term and higher harmonics in addition to the fundamcntal harmonic of the frequcncy equal to the input， so that Pt（t）＝Yo十yl sin（ωτ一∼ρ1）十y2 sin （2ωt−92 ）十Y3 sin（3ωZ−∼ρ3）十・・・… （1，02） The describing function or nonlinear gain 2V for this elcment is defined in this case 1）

N≡莞・一元ρ1．（1，・3）

This describing function is the counterpart of the transfer function， in which the diffcrential operator p is substituted for 元ω to represent the frequency characteristics of thc linear element， and it is utilized to apply the Nyquist criterion for stability． Thc describnig function is dependent not only on the frequencyωbut also on the amlitude Xof the input． This situation makes the analysis for the nonlinear system far more complicated and cumbersome than for the linear system． The difficulties of the describing function ana・ lys三s are not limited to these． As the omitting of higher harm皿ics and constant term implies some inaccuracies in the method， we are always con．

fronted with the uncertainty throughout the

analysis．2）For example， in the case of a feed back control system the input to the nonlinear element nlay， as the output does， contain higher hermonics，

and without highcr harmonic components， thc

problem as to the wave form of the output is out of power of the analysis． Aiso， the discarding

both of subharmonics and superharmonics may

sometimes lead to even quite erroneous results about resonance phcnomena． So it is urged to obtain the exact behabior of the nonlincar control systcm subjected to a sinu・ soidal input， in order to eliminate the above men− tioned uncertainties and inaccμracies involved in the transfer function mcthod． If the nonlinear element in the system con− cerned is of a piecewise’linear type， i．e． its characte一

29

r三stics is composed of several lincar parts， as for example， in the case of saturation， of dead zone， or of backlash， the exact behavior of the feed back control systcm， containing such nonlinear elements， subjected to a sinusoidal input can be obtained to any desired degree of accuracy， with the method of analysis by Fourier scries similar to the method introduced by thc author to solve thc steady forced vibration of an unsymmetrical piece− WiSe・linear syStem 3）． The general features of this me thod consist in， 1） Linealizing the given nonlinear differential equation governing the dynamic behavior of thc system by expanding the nonlinear part of the output of the piecewise−linear elcment concerned・ into a Fourier series with the same period as of the sinusoidal il〕put・ 2）Obtaining the formal solution of thus Iineali・ zed equat三〇n by rcgarding the above・mentioned nonlinear part as if it were an cxciting force

from without． This solution contains unknown

coefficients of the Fourier expansion above assumcd． 3）Determining these unknown coefficients from

the conditions that the formal solution above

obtained satisfies the g三vcn piecewise−linear char・ acteristics of the system． Some kind of conver− gency improvement by mcans of series transform・ ation may prove advantageous in this steP 2．Procedure of Analysis The block・diagram of the system to be treated is given in Fig．1， where N is a nonlinear element containing， e g．，adead zone，L is the cquivalent element of several linear ones；r（t），c（り，and e（の are the input， output， and aCtuating signal rcspectively；m （t）is the output from the non． linear clemcnt concerned． The】inea「element・quite arbitrary in form， can be treated equally by our method of analysis， but， for thc sake of definltcncss， its transfer・function is assumed to be k／s（1十Ts）， where k andアare given constants． N L r（t）十 e（t）

Aトm（t）S（暗。）

Fig．l Block Dlagram for Nonlincar

Feedback Control System

c（七） o

(3)

Oct． 1959 Reports of the Facutly of Engineering Yamanashi Univcrsity

No．10

The differential cquation governing the dynamic bchavior of the system can be written as （ア￡；＋昔）・（t）一叶（t）一・（t）｝，……（2。1） where the nonlinear function f（x）is defined as follows，（Flg 2a）

㌶惣：f惣司一・（2・2）

！ eo ’l 狽W⑩k an司 eo Fig．2a Diagram for f（x）

Flg．2b

Diagram for F（x） and represents the piecewise・linear characteristics with dcad zone as idealized． If we take the actuating signal e （の＝ r （t）＿ c （の … （203） as the dependent variable in above equation， and utilize the function F（X）dcfined as follows，（Fig． 2b）

霊；：9−；°；−1≡1／…（2。4）

F（x）＝x＋1，，’ xく＿1，∫

the equation（201）is transformed into ‘i…＋2ζω・2／＋t・le・F（・／・・）一（‘；＋2ζω・急）・（t）・…（2・5）

wbere

ω・−V芋 …（2・6）

is equal to the undamped circular natural frequency of the systcm for infinite large arnplitude， andζ is the damping ratio and given by l l

ζ一2／戻ア． ’”（207）

If we appiy to the systen a sinusoidal input signal r（り＝Acosωム …（208） the differential equation becomcs

30

C、ξ＋2ζω・劣＋ω1 e・F（・／・・）＝−Aω2〆コ＋μ；…（ω・一γ）・（’”2・09）

where

γ ＝ tan−1 Ft、。，（210） μω ＝ 2ζ／9，（211）

and

9 ＝ ω／ωo：being frequcncy ratio．（2．12） We measure the phase angleθfrom the middle point of the intervalθo during which e is greater than eo，so that

…∴：∴／∴∴｝（213）

This phase angleθis behind the input signal by the phase lag angle α， or

θ＝ωt一α．（2．14）

We callθo／πthe staying time ratio， where 2θo is the interval the system staシs in the nonlinear region and 2πthe whole period．（Fig．3） re 〈t） e 〈t） e。 m（t） Fig 3 1nput’ （t），Actuating Signal e （t）and Output of the Nonlinear Element m（t）ド Changing the indcpendent variable from t toθ，

our equation（209）becomes

d2e de 92 十2ζ9 十eo F（e／eo） de dθ2 ＝＝A92〆1十μω2cos（θ十α十π一γ）（215） The third term in the left side of（2．15）may be considered as the nonlinear p∬t of the restoring force and is designated as 9（θ） ≡ eoF（θ／θo）（216）

We assume the Fourier serieS expanslon of

9（θ），taking into account its symmetric nature， as

(4)

Exact Analysis of a Nonlinear Feedback Control Syste血subjected to Sinusoldal tnput

e・＝AC°STC°Sα一Ω・Σ・・ff−＋μ。・／疲C・・丁

、 nエ1，8，5…

・−A・i・2・inα＋6・Σ髭s裟π圭b齢・i・誓

n＝1，3，5…

Putting

τ剥⑳＋欝雲耀…当

and utllizing two equations of Xn＝ an／T，二Yn＝ bn／1「（n ＝ 1，3，5−・）， the expres＄ion of the actuating signal e （’） in （2・20） θ0

。＿．。＝τC°Sθ一C°S丁

一；Σ二：C警壼；劉・・…一…

n＝3，5… ロ

ー嚥穿鵠辮i・…；l

And similarly， two conditions（2．22）are transformed into

A…α・r｛1＋；弓等s斐云…1sinρ1｝

． sin nOO

A・’nα一イ☆Σ黎S砦π譜゜￥／Ln，i。￡

2 ％＝1，3，5… From these two equations， we can determine α the“amplitude，，（of the cosine component of the behin actuating signalθ（t）） and the“phase lag angle’， r

9（θ）認・巴c°s〃θ＋b” sinn°）（2．17）★9（θ）＝θ一讐≦θ≦豊／

㌔5）’㌢｛咽皿｛；1に∵∵蕊π＋∵1’9）

Ω・ f＋2ζΩ傷一Ω・醐＋i・t。・…（θ＋・＋・一・） S・1・i・gf・・m・11y・h・・li・・a・i・ed・qua・i・n（21・8）・ we obtain

議㍑c°s”θ＋b”s’n〃θ）ぽ18）・−A…（θ＋・）一±Σ噺警芸1ラs曇（”θ一gn）

The conditions of the piecewise−1inear π＝1’3’5… ’

。h。，a。、eri，、i。， t。 b。，a、i，fi。d by g（θ）d。，i。g。ne ’”（22°）

wh・1・peri・d l−・＋芸≦・≦・＋芸）・・e★ w蕊一一，，、。／n，。≧，。≧。．（221）

The conditions（2．13）are fulfilled by following two equations θo l Cln・C・Sgn −bn singn nθO （222）and the following non・dimensional coefficients， can be rcwrittcn as 望゜）

竺L

θ。2 Ci・θ）

百

（of the middle point of the

dthe maximum point of

r（t））as follows：一（222）．．． i223）（2・24）．、一 i225）．・・@（226） Staying interVal the input signal

v｛1＋誓一一）じ

Σ誓（…i・9・＋… C・・9n）

nロユ，3，5… ． nθ0 2

；；il｝

（227）

31

(5)

（）ct． 1959 Repofts of the Facutly of Engineering Yamanashi Uhivers三ty

_No．10

‘

and

tanα ＝ Mn

一Σ

92 nm1，3，5… （Xn singn 十夕n COSgn） sin nOo 2 s三n旦9＿ 2 （228） 1＋等、（・・c・…一ツ・si・S・，・）

where

仏一

рP子砺，／。，（n ＝ 1，3，5…）

Next， transforming（223）into

・・／r−…♀−6・Σ答C繋醤ψ㌃

π■3，5…

and combiming this with（227），we have

， COS』三勉 2 （229）（230）

v｛i

＋普

（Xl cosg1 一夕l sing）1 ）．A／eo＝ …与一Σ等（X・ C・・gn一ツ・・助）…η1° ヵ■3，5… （231） Lastly・the conditions of the piecewise・linear characteristics（2・19）bccome equivalent to the following ones， θo Σ（Xn COSnθ＋．y・Sinnθ）＝COSθ一COS 2 n−1，3，5… 一；，Σ響≡慧≒㌃（ nθo…nθ 一・°・−2一） π■3，5… 一±Σ瓢譜㌃（ si血．鋤．、in“θ＿＿旦，inθ 、in竺・）

n類3，5… 2．

f。r＿θ・≦θ≦θ・ 2 − − 2 ・ and Σ（Xn cosnθ十Yn sinnθ）＝0， nエ1，3，5…

f・・÷＜・＜・一一告（2．32）

一多h二二鳩ご1：還㌻㌧1−｛灘轡忽i：：1：

f・lfill・d by th・・ymm・t・i・p・・p・・ty・f th・F・u・i・・ val・…fth・n…dim・n・i・nal…ffi・i・nt・xn， Yn

・eri・・（2・17）… mpl・t・th…1・ti・n・f…wh・1・p・・bl・m．

・

｝ヨ鷺ピ輌＋万⑭）

i：雀｝2

Now， qulte similarly as the procedure for determination of the unknown coefficients in the Fourier series expressing a given function， we multiply cosnθand sin nθto the both sides of the first equation of（232）and integrate from一π／2 toπ／2 for the left hand side and from一θo／2 to θo／2 for the right hand side respectively． Then we can obtain a infinite set of simultaneous Namely， for given set of two parametcrs， i． e．

the staying time ratioθo／πand the frequency

ratio 9， we obtain a set of infinite numbers of coefficients xn and Yn （n＝1，3…一・） as a solution

of thesc equations・Then， for the actuating

signal， the amplitude T and the phase lag anglc

αare given by（227）and（228）respectively and

the wave form is obtained from（225）．

32

(6)

Exact Analysis of a NOnlinear Feedback Control Systom subjected td Sinusoidal Input At the same time， thc corresponding amplitude of the input signal is given by（2．31）．This mcans that the ratio of the input amplitude to the half dead zone interval A／eo is obtained step by step as a function ofθo／πand 9． By inverting this functional relation， we can obtain the staying time ratioθo／π for the given combination of input amplitude A and frequencyωfor the given system（givenωo and eo）． We plot this functional relation into a diagram beforehand， from which，コ_{if the input amplitude and frequency are glven} initially， the corresponding staying time ratioθo／π can bc read， and from this we start the above ★ mentioned procedure・

9（・）／r−S｛ξ・m・・c・・伽元1）πθ＋

m＝O The relations between the coefficientsξ2m＋1，η2加＋1 ツ，、（n・＝＝ 1．3・5……）・fth・f・・m・・…（232）a・e ea・ily・bt・i・・dbym・晦1yi・g・…θandsm ・igh・・id・・f（3・1）・・d i…g…i・g f・・皿一θ・／2・・θ0／2・・d・h…qua・i・g th・・e・e・ults t°、一一x・ a”d T一γ・・e・pecti・・ly・ These relations are m （2m−十一1） π

一一：…顎（−1）伽旱；）・π・一。，ξ…・

m＝o θ02 （302）

Yn−÷…〃1°蕊一1）m．（2切蒜；が一〆伽刊

（n ＝1，3，5・・・… ， m ＝ 0，1，2・・・… ） F。，、h。r，。。、“θ。nd、i。 nθ。an be exp・essed・・th・F…ier・e・i・・i・・h・i・…val（」嬰く・＜；）・・ follows nθO

−一梁＋華三（−1）㍑華鞠）・・2−〆ド（2”ま1已

（3、03） ”θo ・i…一℃／，．．2一芸（−1）・（21e＋i8。・一がsi「i（2”≒。1）π・ 2 k＝o θ02 If we substitute these expressions（3・02）and （3．03）for xn， Yn and cosnθ， sinnθ in the right side of（232）and equate the result to the new

adopted series（301），we obtain two different

cxprcssions for the same Fourier series in the

i・…val」2＜θ＜；f・・9（・）／r・…h・

★ 3．Convergency Improvement by

Series Transformation The direct determination of the non−dimensional coefficients xn， Yn （n＝1，35……）from the

cquations（232）as mentioned in thc foregoing

paragraph is so cumbersome in most cases that it is almost impossible to be accomplished， because of the poor convergcncy of the assumed serics− （217）・ To avoid this impasse， we resort to the method

of convergency improvement by series trans・

furmation． Namcly， we cxpand the nonlinear part g（θ） of the output into a ncw Fourier series in the interval（＿θo／2≦θ≦θo／2）as follows：

⑳・・si・伽

ｦ1）πθ｝

（3，01）（〃￠＝0，1，2……）of this ncw series and thosc xn， ’ 〃θ to the π

…ffi・i・・…f…（2m十1

@θo）π・and・i・（2物

ﾜ1）π・ Inust be the samc for the two scries in both sides． These conditions glve following infinite set of simultaneous linear equations ofξ2m＋1・η2m＋1 as

unknowns：

33

(7)

（）ct． 1959 _{RepOfts of the Facutly of Engineering Ya血anashi University}

No．10

ξ・k・1−・・k・・＋十自（・・te・・，・m・・ξ一＋δ・・。・，・m。、励．、） mm・O rl・k・・一撰（・・…，・m・・ξ・m・1＋ζ・・。1，…1η・肌．、） m＝・O

（ゐ＝0・1・2…… ）（3（n）

where （−1）kc・・》°

＋1＝で早・｛ge・2’th．M｛一／’，’L）−2’−2一二T ・（3・°5）

（ゐ＝ 0，1，2・・… ．） and

−・・＝5・（−1）舞昔三、百鼎瓢・）（r）一……（3・6）

δ一・…1弓上2諾竺μ・Σ（ n2（1＋COS”θ0）μ・2＋が）（μんL〃2）（㌦，−n，）．…．．（3。7） n＝3，5……

一…・㍉：（−1）（2m＋1）・・Σ。（，。、＋。，1（侮，一“，）・

n−3，5．・・…

・｛㎜慧÷一2≡讐）｝・…・・（3・8）

ζ……m・1−一。…。（−1）m・’e・X（，。、＋n，）n（㌦，一可ア・ nコ3，5・・・…

・｛竺tl；一一〃（1十COsnθolUic 2−n2）｝， ……（3・・9）

iettu，μk and伽in thcse formulas being notations for

・・−2ζ／9…一（2々吉1）π，μm−（2地蒜’）π（k・〃・一…1・2……） ……（3・1・）

respectively． These coefficents γ2ic＋1，2m＋1，δ2ic＋1，2m＋1，ε2ic＋1， zm＋1 and ζ2κ＋1，2m＋1 are all functions of the f・eq・…y・・ti・9・・d th・・t・yi・g tim・・ati・θ・／π・Th・・um・f th・i・fi・i…e・i・・exp・e・si・g・a・h of them can be condensed into a single term as follows，

一一一

磨ﾟ；簗ご11二fM

為峠仇 ……（3．11）

γ 一一μ、・』μ。・一。…。｛τ÷三：；sθiμ、＿1）「＋（／、、ic，？ptT。、），剖……（3」2）

δ・刷・m・1一警・・（2m＋1）γ・…，・一・ k≒ 〃z ・・・… （313）

δ・刷・k・・一；μ・（2々＋1）・・・…，・・k・・ ……（3・14）

34

(8)

Exaet θ0 2μωcot 2 M ＋L（。、・一嘱丁一M，・T・＋i，t。7 1atO ε2刷？k＋’＝μ、（letiC 2十μω2）

＋2一 c・t一穿…h弓一…h・pw（9一θ・Σ

θ・αic・＋μ…〃L’ ・・c・…丁一 …（3・16）

＋（㌍t焦，一士ガ

4，k．＿．1−一

凵i−1）m・〃μ≠、ご鵬．＋μ≒M｝

え ≒ m ・・・… （3．17）

C，ic＋＿一

gj，一÷μ，÷三⊇＋μ≒M｝……（318）

whcrc

…… i3．19）

N三『≒2ぽh弓 ……（32・）

Dcduction of these formulas is briefly explaied in the appendix A． Usually， the convergency of thus transformed series arc vcry good and it is far easicr to solve equations（304）than to solve those deduded from（232）輌n the last paragraph to give xn and Yn．The errors committcd by neglecting the higher terms ofξ2．＋1，η2m＋1 are usually so small that the first approximation i． e． discardlng all terms overξ3 andη3 and retaining only ξ1 and η1，is almost sufficiently accurate for most engineering eurposes． Onceξ2m＋1 andη2批＋1 determined， the non−dlmen− sional cueffielents xn and Yn can be calculated from（302），and the problem ls sald to be completely solved， as explained in the last paragraph． 4．Results of analysis in tlle case of no damping If we putζ＝Oorμω＝Ofor the case of no damping， the infinite set of equations to giveξ2m＋1 andη2ηし＋1 reduces to

ξ・…一・・…＋誌・一，・一・ξ・肌・・ ……（4．・1）

η2兎＋1＝ O le，．m ＝ O，1 2、3・・・… where 1 8 1 十 cos θo

γ2刷2’C＋1＝μ、・一πθ。（μ差L1）・ ……（4・02）

・・…・…・一一。；。（−1）・・弓綜（宗1Cls監一1）．…．．（4．。3）

〃峠〃z

As these coefficients are independent of the frequency ratio 9・「the procedure of analysis

substantlally simplifies itself・ As the second apProximation， we retain onlyξ1 andξ3・whlch are to be determined from 35 Amlysis of a Nonlinear Feedback Control System subjected to Sinusoidal Input

(9)

Oct． 1959 Repofts of the Facutly of Engineeing Yamanashi UniversitK

_No．10

Oく1 i．o 0．5 （92一γ11） ξ1一γ13 ξ3＝Ω2α1 一γ31 ξ1十（92一γ33） ξ3＝S2k）α3 ｝ ●0．10 ●◎．05 θく3 ‘イ・ノ㌔ o・10 イ｛3ノ’t”3’ 0・05 （4 04）

g o

O O．2 、．」ト O・6 0・8 1・0 0

−一『㌦・。．2・．、。．6。．8i．。

一一一く残 F三9・4Di・g・am f・・α・a・d・…Fun・・i・…f Fig・5Di・g・am f・・C・・ffi・i・・t・γ、、，γ、3，γ、1，

th・Stayi・g Tim・R・ti・θ・／・ andγ33 a・F・ncti・n・・f th・St・yi・g

Time Ratioθo！π The c・・ffi・i・nt・γ・1・r・3・γ…γ・・and・…tant・α1，α・a・e・h・wn in Fig．4 and Fig．5， and th。i， valu，， are tabulated in Appendix B． Onceξ1 andξ3 determined from（4．04），we calculate xn（n＝1，3，5……）

from

Xn−

e…

U㌫自一（芸トーn26・｝一（4・・5）

Thcn， in the case when the input is sinusnldal and expressed by

r（t）＝A…ωZ； ……（4．06）

wh・・e amplit・d・Ai・glv・・acc・・di・g・・th・・t・y・ng tim・・ati・θ。／・by th・f・U・wi・g・xp・essi…

．4＝。。 ΩLx・

92 c°s穿＋彦蕊ll・叉旦，．．．…（4．・7）

we obtain， for the output c（の

c（t）＝A

hρ・c°sθ＋泊三Ω・元嘉」°snθ｝， ……（4．。8）

for the actuating signal e （の θ（彦）＝A

o一・与゜sθ土三，籔．ヅ， ……（4。9）

and for the output of the nonlinear element with dead zone

m（t）＝一嵩㌣一θ＋撫⇒，＿（4．1。）

36

鬼

て、イ、3 イ，，

(10)

Exact Analysjs of a Non・linear Feedback Control System subjected to Sinusoidal Input The numerical results of practicing the above mentioned procedures are shown in the following several figures・ c／e。 i 10、O 8，0 6．0 4．0 2．0

0

−5、0

9 鵬

烏o’ ko’ ∠ふ ’ ∼05．0 0 5・o

−A／e。

Examples of the waveforms of the inputア（t）， the actuating signal θ （t）， the output of the nonlinear element m（t）and the total output of the system are shown for two cases in Fig．8and Fig・9， for∫2＝1．2， A／eo＝1．Ol16， θo／π＝0・1・and for 9＝0．4， A／eo＝L682，θo／π＝05， respectlvely． 1（t） e（t）．《 → オ

m（t）×10

Fig．60utput Amplitude Gap Ratio C／eo versus

Input various Imput Frequency Ratios

Amplitude Gap Ratio A／eo for 一c（t）×1000

Fig．6 shows the relation between input．and

output amplitude as multiples of the half dead zone distance eo，from which we can obtain Fig．7 showing the total gain l（：／A l versus frequency ratio g for various input amplitude・ The solid line in Fig．7means that the output signal is in the same phase with the input and コ． the broken・line means that they are ln oPPoslte phases・ c／A t Fig． 2。0 1，0 Fig． 0 0 2．0 4．0 6．0 ＿＿一・Ω

70verall G・i・C／A・…u・E・eq・・n・y R・ti・9f・・

Input Amplitude−Gap Ratios A／eo

37 l Ol 戟@−1 o．6 ’ 、工一一 1L ’ ， ’ 一『 ’ m ！．一@ ．5

@0十

。。塩川d帯一L・ pl“1 ！ 1 O Illll l ／吟dマo 相一． ₁

％一一

‘日 1 ！ i ／一一一一一゜二1．5− 一一一

”ぷ

1 ／ ’ ． ’ ， ’ f 一一一

芯

“ 1／1，／’ 「一一一レ／／ 8Waveforlns for the input r（t）＝Acosωt， the actuating signal e （彦）．the output of the elemcnt with dead zone m（のand the total output c（t）of the system in Fig．1 for the frequency ratio 9 ＝・ 1．2 and the amplitude・gap−ratio A／eo＝1．0116 and the residing time ratio θo／7τ ＝ 0・1 5．Conclusians We have given the exact analysis of a nonlinear feedback ．． COntrOl SyStem cOntammg a non・linear element with dead zone subjected to a sinusoldal input． The method of analysis is quite the same as that 8．O introduced by the author to investigate the steady forced の

_{varlous vibrations in an unsymmetrical}

bilinear system induced by a

(11)

Oct．1959

R・p・rt・・f th・F・・utly・f E・gi・ee・ing Y・皿ana・hi・Univer・ity

_No．10

妖七） t e（七）い｛ eo o． eo も m（七） t C（七） t ｲ zone・the frequency region 3 ＞ 9 ＞ 1／3 are almost safe from such resonance and our・results are confined to this region．

The superharmonic resonance can be treated

without any further alternations from the bases

of our analysis， while for the subharmonic

’resonance， the assumed Fourier series expressing the output of the nonlinear element must have the period e． g．3times longer thaロthat of the input signal． The problems of subharmonics and superhar・ monics， however， will be reserved as the subjects of further study．

6．Acknowledgements

The author is deeply grateful to Prof， S． Fujii of Tokyo University for his heartful encourage− ments． His thanks are also due to Mr． K．Ozawa and Mrs． A． Murata． who performed painstaking numerical computations・

References

Fig．9Waveforms for the input r（t）＝A cosω’， the actuating signal e （t）， the output of the element with dead zone m（のand total output c（t）of the system in Fig l for the frequency ratio 9＝04 and the amplitude・gap・ratio A／eo ＝1．682and the staying time ratioθo／π＝0．5 harmonic exciting force． For the sinusoidal input， the corresponding output・ the actuating signal， the output of the nonlinear element with dead zone and the total output of the system are given a， full Fourier series expressions． Determination of the coefficients of these Fourier series is the clux of our problem and is facilitated great ly by somc convergency

improvement by means of apPropriate series

transformation．

We have given numerical results of the

improved second approximation， i e． the first two term approximation for the transformed series， which is considered as sufficiently accurate for ． ■ most engmeermg purpose in the frequeney region

where any subharmonic and superharmonic

resonance may not occur． In our case of dead 1）Cosgriff， R． L、 Nonlinear Control System， McGraw−Hill，1958， p．171． 2）Truxa｜， J・G・， Control S ystem synthesis， McGraw・Hill，1955， Chap．10．

3）Maezawa， S．， Forced and Self・exclted

Vibrations in various Nonlinear Systems Having

Discontinuous Linear and Nonlinear Charac−

teristics， Report of the Faculty of Engineering Yamanashi University， No，9，1958， p．61∼81． 3b）Maezawa， S．， Steady Forced Vibrations

in Systems Having Unsymmetrical Bilinear

Charactcristics， lst，2nd and 3rd Rcport， Pamphlets for Adressses held before Meetings of J． S・M・E・， April 1958，0ct．1958 and April l959 respectively．

38

(12)

Exact Analysi50f a Nonlinear Feedback Control System subjected to Sinusoidal Input Appendix A． Summing up of the series for various coefficient5 1）At first， some examples of Fourier series needed below are given；

1＋，。，nθ＿。・i・hμ（；一θ・）＋・i・hμそ

三、，3，、ζ1、±τ一4μ …h・チ

0≦θo≦π ……（A，Ol）

1＋。。，。θLπ・i・・（；一θ・）＋・i・・チ

三、β㍑．：・2−一石・…；

μ凸妄integer，0 ≦ θo≦ π ・・一・・（A，02）

。，i。。θ。 π…hμ（S一θ・）

三㌫〃・＝4 …h・；

0＜θo＜π ……（A，03）

。，i。。θ＿。…μG一θ・）

三1，，㍍・一一丁…μ吾

μ峠硫9ぴ，0＜θ・＜π ……（A・04）

Σ七培竿一詮、・岬…9…一（2”θナ1）π・（〃一αL2……）……（A…）

n■1，3，…… 2）Deduction forγ2k＋1，2m＋1

・・…・…・−

Q，。（−1）・・m篇｝Σ、μ。，＋謬え・C讐θ1戊（㌦・一。つ

n＝＝3，5・・・… Σ（ n2ﾊω2十r・2）（

o去当））（，m，一。、）一一（μ。、＋μ鍔（，㍉＋x、m’i）Σ1，ま・c漂

n＝s，5…… nエtS，5……

＋（㌦、一。浩2（μ。，＋，めΣ5吉鵬θ゜

n＝3，5…… ＋（ μ7π2μバー㌦2）（μ・2＋㌦2）Σ嘉よ当゜・ n＝3，5……

一一

iμ・＋誌一十『；一㌶≡一’誌・讐｝

＋（ μ〃2伽Lμ〃2）（μ・2＋μ元2）（−1誌当）＋（μ、・一諾2（，。・＋，m・）（−1−int・cgsl°）＿＿ 1 十 cosθo ＿＿ μω π ハ4’ 一（1十μω2）（・kし1）（μm2−1）（ltt・2＋． ieti・ 2）（・・2＋μ肌’z’）4

∴一一一

ｻ（−1）⇒∵ll鷲1；2iM

k≒m 39

(13)

Oct． 1959 Reports of the Facutly of Engineering Yamanashi University

_No．10

This is the same as（3，11） 3）De寸uction forγ批＋1，21e＋1 γ一・一一π；。Σ（ 722 （1十COS nθ．）μ。2＋〆）（μ兎2一μ2）・ n＝＝3，5・・・… −8−（ Ptt。21tω 2十μ戸），Σ7t・c縣1° πθo n＝3，5・・・… ノー（ 2 μωμω2十μ〃2）・Σ’誌響゜ nエ3，5……

＋，。呉バΣ（漂認

n・・w，5…… 8 7tθ，

1十COSθ0

＿−

（1＋μ。川（Ptk 2−1）2 Ftt。2 T （μん2＋μ。2）24μ。十 πθo ・i・hμ・（ 8Ptk 2 π 2 Y．8LL2 μω2十FtiC 2 一θ・）＋ sinh iCttU 2 cosh？tttU 2

一研，。㌻π砺／（1十COSθOl＋μω2）（μ2−1）・ 1

（＋（ lttωμ充2十μω2）、÷・∫ This is the same as（3、12） Deductions for other coefficients are quite similar to above so we may dispense with writing down here・ Appendix B． Table for variou8 Constants and Coefficient8 θo／π α1 α3

O．○ _．OOOOO 一．．OOOOO

0．1 ．01270 ．00047

γ11 γ13 γ31 γ33

．00000 ．OOOOO ．OOOOO ．OOOOO

．00839 ．OOO53 ．OOOO6

0．2 ．05046

_一…18・

p

，02727 ．OO409 ．OOO45

．OOlO9 ．OO430

O．3 ．11220 一．00382 ．04804 ．Ol286 ．OO 143 ，OO956

O，4 ．19620 一．OO621 ．06376 ．02744 ．OO305 ．01691

0．5 ．30011 一．00857 ’．06987 ．04632 ．00515 ．02645

0．6 ．42097 一．01039 ．06464 ．06535 ．OO729 ．03838

O．7 ．55537 一．OllO9 ．04938 ．07922 ．OO880 ．05286

O．8 ．69947 一．0］OO4 ．02842 ．07901 ．00878 ．06998

O．9 ．84913 一．OO657 ．OO886 ．05576 ．OO620 ．08957

1．0 _1．00000 _{一．00000} _．00000 ．○OOOO ・・・…

P ・11111

Table I Numerical Values of Various Constants and ξ1，andξ3

40

Exact Analysis of a Nonlinear Feedback Control System subjected to Sinusoidal Input 利用統計を見る