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Chapter9 Describing Function Method
DI INTRODUCTION
The describing function method is useful in determining stability of a nonlinear system. This method is classified as frequency-response method and is based on an analysis that neglects the effect of harmonics in the system. This approximation will be most successful in a system containing sufficient low-pass filters. Thus, in general, this method improves in accuracy with higher-order systems and a complimentary to phase plane technique.Let us consider single-loop servo-mechanism having a nonlinear element, N and a linearelement, Gas shown in Figure 9.1. The linear element has a transfer function G. It is a frequency sensuive function but does not depend on amplitude. The nonlinear element, N depends only on amplitude and is frequency insensitive.
r e mN
G(s) c
Relay or nonlinear Linear element element
Figure 9.1 Feedback control with a nonlinear element.
Let the input to the nonlinear element be sinusoidal, i.e. e = E sin wt. In general, the transfer function of the nonlinear element can be expressed as
m =/(e) (9.1) Equation (9.1) shows a functional relationship between e and m. It will be found moreconvenient in this development to represent the relationship with an expression consisting of twoports-a quasilinear gain and a distortion term.
284
SI? u;:i:uM ;:iq Ul?O mdino ;:itp JO iuouoduroo J1?1U;:iumput1J;:i4.L 'u U!S 3 = a mdur ll1? JOJ mdmo Sl! JO iueuoduroo oruourraq pnusnrepnnj ;:itp ounnrorap01 pannbar S! l! '1u;:iw;:i1;:i .rn;:iunuou 1? JO UO!PUOJ 8u!qosap ;:i41 8undwoo lOJ ';:iioJ;:iJ;:i4.Ltn ,\ou;:inb;:i.IJ JO mdur ;:i41 01 ro i\ou;:inb;:i.IJ JO indrno ;:i41 8upu;:is;:iid;:iJ srosaqd JO oi?~ =
1~)
O.\\lJO 1:uow ~
(1 6)(9'6)
1psna4 ainpaoord aAoqi? ;:i4.L
souroooq iuouoduroo J1?1u;:iumpun1 a41 pue S!SA(i?lll? JO osodmd;:itp lOJ p;:i1:>d18;:iu ;:iq Ul?O UJ JO iueiuoo O!UOUUlH.J. ;:i41 SUO!l!PUOO qons 1apU[l 'ft?P!OSOU!S SU!l?W;:lJ (1)x ;:i1 'tC JO iuouoduroo [1?1u;:iumpullJ ;:itp ssooord ;:itp U! pun mo p;:i1;:i1m ;:i.11? tC JO so1uoW.I1?4 ;:itp Ill? 11?41 Aoi?m:>ol? JO ;:iai8;:ip poof 1? 01 poumsse aq ueo 1! '(sw;:i1si[;:l 11?;'.lUH ;'.lt(l JI '(S')Q 'lU;:lW;:l{;} .Jl?;:lU!l ;:itp 48nOllp mdur Sl! 01)[Ol?q p;:iJ S! N 1u;:iw;:i1a .Il?aU!{UOU JO tu mdino ;:itp '(O = .1 ;:ii) mdm [l?W;:iJx;:i JO aou;:isqi? ;:itp UJJ(T) SO:> 1 El + J(T) U!S IV= UI
soruotnreqqns St? n;:iM St? soruourrsq 1;:i4814 a1ru;:iu;:i8 iou soop N 8U!wnssv
f -0 1tZ:9p (9u)soo (9w)w = "u. ui I
f -0 1tZ:9p(9u)u1s(9u)w = uVuz I
f-=0 ltZ:9p(9)w 0vui I,!JJdOO 1anod ;:i.re ru8 s , ig '1 El pue 'v s , ''v '1 v ov ;:iia4M... + J(T) SO:> "a + J(T) soo I El + ... + J(T) U!S 'v + J(T) U!S 1 v + ov = UI
uo ,(
-.faU;}~.roaun ~
'Pl\\ dwa1s.(,S'l:>d[&S!4.1 uSi\\O[[OJ SI? souss ianod JO SUJl;}l ut possardxa ;:iq OS(l? Ul?O UI 1uaw/ u!)sa1:>en4:> 1nchno
(:v)i(
W:;:~O(S
onsuaioareqo ../Sg1BU!PJOO:> UB!Sglltl::) u1 uo:>un1 ~U!q::isgp = ~"(~J _ T ~ ,'.q ll = f) SO:> ,'.q ll =~ Wv Wv
J19 :!Jll f)pf)U!SW -=19-u Z
J f'o+u 3 ll 19 3 xf)pf)U!SW- -+f)pf)U!SW -=(3)8'6-uz I 19-:u 1
~6Z PO\l\&W UO\\:IUR~ GU\IUJ:IS9Q
292 AdvancedControl Systems
The output is given by
O, 0 1XJ - u '(XJ U!S X)tulX) > lL > XJ > 1XJ W = ((} )t\IX)> X) > 0 '(XJ U!S X)UI:SMOCTOJ Sl? paquosop oq ueo mdmo ;;i4.L '')/ SJ UOJPUrlJ 2u1q::is;;ip ;;i41 ll?4l OS 'mdtn c-;;i41 soum tu S! mdmo ;;i41 '(v > x) uo!2;;iJ reouq a41 U! pouuuoo oq 01 paurnssa s~ mdui a4J.
(a)/(T)
(q)(I---,------------ w-
uo .------=u_,,z:L.......,/1--r+---...,.:.:.----,10
':o -JLZ,
O!JS!J:Jl::ll?.IU4J !Od!OQ
w+ w+(8).l :>!Jsp;11oe11140 ..-z
1t'(XJSOOX)UIS+. XJ)--zJ;
= b~.,.,, JO1t-z = g = X) punu
aa-z > x o
(6Z"6) [ ( gsoos + XJSOO ~ - )p + (X)z; U!S- gz; U!S)-(XJ- g)z;] ~ =
(3)8 =(3)q + (3)8 = ~
[( g SO:> s + XJ SO:>~ - ) p + (XJZ U!S - gz, U!S) - (XJ - g)z] ~ = ~: = (3)8 'os
u
[ ( -=z ) f} D z z DJ(1ro)p1rou!S --S "ff 3+(1ro)p1rou1sf3--(1ro)p(1roz;soo-1)- fa uu ff ITT '.FJ)f ff P[{(lro)p1rou!s(~ =S }H}1+ro)p1rou!s ~ -1ro ,msmH]: ~ 'v 10
{ gsoo( ~ -s )+(XJSOO- gsoo) ~}Ht+ [{(XJZU!s-gz; U!S)3-(x:>-g).nJ] ~ =
a uu a ff t[ (Jro)p J(IJ U!S(!:._- s) '.FJ')f J + (1(1)) p J(IJ U!S(!:._- J(IJ U!S '3) 'ff l1t =
0 uxop 1(1) U!SUI f - =Zill p
0 xunp J(IJ U!S UI f - =ui Iiusuoduroo ;::)St?4du1 = 1v 'MONo = (3)8 ';::)10p1;;il!.L 01;::iwwi\s ;::)Al?M-Jfell SJ indino ;::itU
296 Advanced Control Systems
Again, from Eq. (9.30), two cases arise:Case I: Saturation nonlinearity (i.e. D/2 = 0, a= 0) Substituting D/2 = 0 and a= 0 in Eq. (9.21),l, X tu = 1gJIC: I
(P"6)
("6)
uz 5 J(T) 5 (ff - lt(;) o I(ff - 1LZ,) 5 J(T) 5 (XJ - 1L) w-(XJ + 1L) 5 /(T} 5 (ff - lt) o =.{(ff - 1L) 5 J(I} 5 X) w+X) 5 J(T} 5 0 o
L6Z po1naw uon3un:1 fiU!Q!J3sao -
298 AdvancedControl Systems
Therefore, m1 = amplitude of the fundamental component of the output
=~A~ +B~
and ~1 = tan-1 (~:)
keq = describing function in Cartesian coordinates
= ~A~+ B~ Larg~1
D X>-= 2D0, X!lS!];JPRJeq::> ;JIU 41gg1 Jeg8 g41 U;};}Ml;}q dl:?8 dlfll!J 01 onp SJl:?g8 U! Sln:>::JO 'A[reJ;JU;}8 'qSl:?ppeg
1/.WJ'fJq :ht?fZVl/ KVJ"H
z-x a
(xz) uis-7 xu.}
H t- . HP W= b:>"_'O H
unnqo ;}M '(~z-6) b:3U! G = H 8u!)m!lsqns a = H 'gSl:?::> S!~ U] "(::>)Ql "6 ;:im8ttJ U! UMOqS S! ::>!lS!J;:!l::JRJl:?q:> Al:?f;JJ gq.LS!S;}Jg1s,x a
z a GP
'o
b~ W'la
66Z po1nawuon3un;jllU!Q!J3saa
300 Advanced Control Systems
where /3 = sin-1(1 - b/X)Let us rewrite Eq. (9.7)1 2ir
x -b/2, 0 $ cot $ ;r/2x -b/2, ;r/2 $cot$ (Tr - /3)y= x +b/2, (Tr - /3) $ cot $ (2Tr - /3)-X + b/2, 3;r/2 $ (27r - /3)x-b/2, (2Tr - /3) s cot $ 2Tr
(9.40)B1 = - J mcoscot d(cot)Tr 0
2 ir= - J mcoeox d(cot)no
2irf= ~
(x sin cot - !!..)cos cot d(cot) + ~ r(x _!?..)cos cot d(cot)Tr 0 2 Tr a 2
+-2 aJ
( X sin cot+-b)mcoscot d(cot)ti ir-f3 2
-c=- x osTr
2 /3
Now rewriting Eq. (9.8), we haveI 2irA1 = - J m sin cot d(cot)Tr 0
211"=- J msin cot d(cot)Tr 0= ~ nf (x sincot-!?..)sincot d(cot) +~ nJ'3(x _!?..)sin cot d(cot)Tr 0 2 n n/2 2+ ~ j (x sin cot_ !?_)sin cot d(cot)Tr n-{3 2
Therefore,
=~A2 +B2 k 1 I eq X
[from Eq. (9. l)]
fl?:l!l~l:) d4l dSOF)Ud lOU seop (ro()o (3)N JO 101d rejod d4l J! pddlUUlt.mg S! WdlSAS .IUdU![UOU1? JO Al!(!q1ns doo] pasop d4l 'WdlSAS ll?dU!]UOU lOJ B!Jdl!JO lS!nbAN cures dl\1 au!pUdlX3
WI
n6 dlng!d U! UM04S S! S!l\J:(o!'l-) iuiod (EO!l!JO ;}4l dSO(::>Ud lOU S;}Op [(ro()9 S! 101d ;}41 W;}lSAS ){Oeqp;Y.)J Al!Un lOJ] (rof)H (00()9 JO 101d rejod ;}ljl J! p;};}lut?Jeng S! ieqp;Y.)J Al!Un 10do = (ro()H (ro.f)!) + T
(s:N - = (ro!)9 JO
0 = (ro.f)9(3)N + l
S;}WOO;}q UO!JEOb;} 0!1S!];}l0U.Ie40 d4l ";}J0jdld4.1
(ro!)a(:J)N + T (ro!)"8 (ro!)a(:J)N = (ro!):)
,(q U;}A!g S! umurop AOU;}Ob;}lJ U! UO!l:>UOJ l;}JSUUJl doo] pcsop d4J. wdJSASdoo] pasop u JO ured p.rnMJOJ ;}41 U! ;}Je (ro()9 UO!l:>UnJ 1;}1sue11 n gu!AU4 1ue1d JUdU!f u pun N'lU;}W;}jd lUdU!(UOU u ;}l;}H T6 dlng!d U! UM04S se W;}lSAS Je;}U!(UOU O!SUq ;}41 l;}P!SUOO sn 1;}1
NOll:JNill !>NUffil:JS3G .!IO SWllll NI vnIB.LnI::) A..LITU:IV.LS ID
(W6) = b~)f 10z-q >X o
~0 po1nawuou:iun~liu1qp:1saa
302 AdvancedControl Systems
point (-1, JO). Therefore, the condition for closed loop stability of a nonlinear system can be stated as follows:l. If -(llkeq) locus is completely enclosed within G(jw) locus, as shown in Figure 9.13(a), then the system is unstable.2. If -(llkeq) locus is not enclosed by i.e. G(jw) locus, as shown in Figure 9.13(b), then the system is stable.3. If -(llkeq) locus is such that it has few intersection with G(jw) locus, as shown inFigure 9.13(c), then the system exhibits limit cycle.
Im Im Im
---+-----!--Re
(a) Unstable
Dt'"";"'-(1/keq) Locus
(b) Stable
-( l/keq) Locus
(c) Limit cycle
Figure 9.13 Nyquist plot of nonlinear system.
In Figure 9.13(a), the complete -(llkeq) locus is inside the G(jw) locus. Therefore, the closed loop system is unstable for all values of the input amplitude M, whereas in Figure 9.13(b), the complete -(llkcq) locus is outside the G(jw) locus. Therefore, the closed loop system is stable for all values of the input amplitude M. In Figure 9.13(c) it is observed that -(1/keq)locus intersects the G(jw) locus at points A and B, respectively. Therefore, part of -(llkeq) locus is outside the G(jw) locus and so that part is stable. But the part of -(1/keq) locus, which is inside the G(jw) locus, is unstable. So it is a case of limit cycle.
EXAMPLE 9.2 For the system shown in Figure 9.14, where an ideal relay is connected with a plant having G(s) = lls(s + l)(s + 2). Determine whether a limit cycle exists and if exists,determine the amplitude and frequency of the limit cycle.
r(t) = 0 x Ti
yG(s)
c(t)
Figure 9.14 Example 9.2.
Solution: The describing function of an ideal relay [as per Eq. (9.18)] is
k =4Meq n:EHere, M = 1 unit.
01n S! (ro!)!) JO ired ,(1eu!8-ew!'d4l 'UO!l::>'dSJ'dlU! JO lUJOd 'dlU l'v' ''dP!S 'dA!ll!8'dU Sl! uo srxn {l'!'dJ 'd4l l[l!M 'd UOJl::>'dSJ'dlU! JO nnod eS! 'dl'dl,P pua sixe Jl'l'dl 'dAJll!8'dU ll1 S! iojd (b~)(/l)- ll!lp poxrosqo 'dq uao l! 'sts 'dJn8!d WOld
.ozz- o86'ZSC:- o68'9C:- o86T- o95:'l9I- o06- ;>[lJUll JlJUOJ 8u1q::>S'dp 'd4.L
--=--!/lt l
0 poinaw UO!}:JUO::J 6U!Q!J:IS90
304 Advanced Control Systems
Now,
1 (1 - Jro)(2 - Jro) 2 - J3ro - ro2G(jro) = = = ---=-------Jro(l + Jro)(2 + Jro) JltJ(l + ro2)(4 + ltJ2) JltJ(l + ro2)(4 + ltJ2)
Separating G(jltJ) into real and imaginary parts, we obtain
2 - (1)2Im G(jro) = =0Jro(l + ro1)ltJ(4 + ltJ2)
or
Now,
Re G(jro) = -----=J3-ro----
(1)2 = 2 =} (1) =J2
-3 3-----=--=--Jro(l + ltJ2 )(4 + ro2)
Therefore, at the point of intersection
1 n:E l
(1+2)(4+2) 18 6
-3-=--=--keq 4 6
E = _i_ = 0.212 unit and frequency= 1.414 rad/s.6n:So, the coordinate of Pis (-0.212, JO) and is very much within the critical point (-1, JO).Therefore, a stable limit cycle of magnitude of 0.212 unit at a frequency of 1.414 rad/s exists.
EXAMPLE 9.3 For the system shown in Figure 9.16 where a relay with saturation type nonlinearity is connected .with a plant having G(s) = [l/s(s + l)(s + 2)]. Determine whether a limit cycle exists.
r(s)
x +lie:-=
~
yc(s) G(s)
Figure 9.16 Example 9.3.
Solution: The describing function for saturation, for the given set of parameters, is given by
JXJl
The describing function is real; therefore, -(1/kcq) plot will be on the negative side of the real axis. It will start from (-1,JO) and move towards negative infinity as X assumes large value'
1s!xa iou saop apk) l!W!( 'aJopJa4J. s!XI? reaJ a41 41!M uouoosronn ou aq IJ!M;iJ;i41 oOL'l sauroooq a18ur. ;i eqd ;)41 'oo f--- ro sn A(a1uwp1n aJow S! a18ul? oseqd 8u!puodsaJJO::>a41 put? sascaJ::>u! to pue o08T- ! a18ue oseqd a41 'news AJaA JO o = ro ua4M 'aJOpJal[J.z(w) 1_ uc1- =
- = a12lun oseud21( zizro + I )rof
I
pUl?
= apm!u8t?w au
lfo!Sl?:IJJU( SO:>Oj+ -UUOO S! uof2~u Jl?U![ snU! )/ U!l?8 JO J!]![dwe 8U!)l?JOll?S l? :m4M 'OZ'6 m8!d U! UM04S W1SAS 41 JO::{ 96 3'1dWVX3
SlS!X SfPl?J ('!:f a!) Pli>I JO ,(ouanb.IJ I? 11? nun ZIZ'OPll1!U8l?W JO ap,(o l!urrr 1qe1s 'E? 'JOJJ4..L co.r '1-) iurod (l?O!lO 41 U!lfl!M qonur AJl
The plot of -( llkeq) starts from (-1, JO) and travels along the negative real axis for increasing was shown in Figure 9.21. The phase angle of
G(jw) =- n -tan-1(ro)-tan-1(2ro)2
At the point of intersection [i.e. at (-1, JO)] with the negative real axisLG(jw) = -180 = x
Im
Increasing, X Re
Figure 9.21 Describing function plot.
Therefore, - n -tan-1(ro)-tan-1(2ro)=n2
tan -i(ro+2ro)
=-nor
or or
Now,
G(Jw) =
1-2ro2 2l -2cJ = 0
1to = J2 rad/s
K = k(-Jw)(l-Jw)(2- Jw) K(2- J3w-w2)
So,
Jw(1+Jw)(2+Jw) ro2(l+ro2)(4+ro2) ro2(1+ro2)(4+ro2)
=111K(2-ro2)ReG(jro)=(02 (1 + (02 )( 4 + (02)Substituting w = 11J2, we obtain K = 2.0 = largest value of K.Any value of K more than 2.0 will cause limit cycle. Therefore, for K = 5, the plot fordescribing function will intersect the negative side of real axis beyond (-1, JO) as shown inFigure 9.21. This intersection will take place at w = 11J2, rad/s. The point of intersection on the
[s1s!x;;i soop ;;ii::i.-b l!WH pus sirun ~'9 JO opruqdnra uu pua stpBJ oz: JO i\::iu;;inb;;iJJB lB S!XB fB;;!J JO opts ;;!A!lB~fau ;;i41 uo iojd (ba)f/I)- JO UO!}::l;;!SJ;;!lU! ;;iq IT!M ;;)l;;l4.1 -snv](,;;ip,(::i 1rwrr i\u-e ;;iq ;;iJ;;i41 [[!A\ w;;i1si\s ;;i41 JO Al!!!GBlS ;;itp ssnostrj ;;iw-es ;;i41 urauraisJ;;i1;;iw-eJ-ed J;;i410 (s~z:oo + I)(sro + nvsz = (s)o .roptsuoo 'urojqord ;;iAoq-e ;;i41 JOd n('lS!X;;> lOU soop ;;ip,(::i lffil!l OS pua S!XB fB;;!J JO opts ;;!A!ll38;;iu ;;i41 uo iojd (b~)f/l)- JO UO!P;;!Sl;;!lU! ou oq [HM ;;iJ;;i4.L 'nrarpenb p1 S! i\ruo S! iojd HD se ;;irq-e1s oq [[!M w;;iisi\s ;;i4.L -suv](,;;ip,(::i lfWH ,\uu ;;iq ;;i1;;i41 U!A\ w;;iisi\s ;;i41 JO Al
