- Email: [email protected]
Robust Stability Evaluation of Sampled-Data Control Systems with a Sector Nonlinearity
2d-1O 2
Copyright @ 1996 1FAC Illh Triennial Wortd Congress. San Fmncisco, USA
ROBUST STABILITY EVALUATION OF SAMPLED-DATA CONTROL SYSTEMS WITH A SECTOR NONLINEARITY YnshifllIui OKUYAMA .. nu Fumiaki TAKEMORI
Faculty of Engin ee ring,
TOtiU1'i
(/ni1.lersity,
.{-fOl , Koyama- ch Q Minami, Tott ori 680, Japan E-mail: oku @i ke. to Uori-u .ac .jp
Abst ract: This pa per describes a rouust stability e valuation of sampled-dat.a con Lro l syste ms containing a st' ctor nonlinearity in t.he fo rward path . The res ul t of this pa.per is derived fr oll! j be norm condit ion in the freque u cy domain by exte nding the Popov's crite rion t heory. Some lemmas a re presented, Cl t.beorem for f 2-stability is proved , and the st.abilit.y margin (a.llo\',°a ble sector) of 3. nonlinear clcHlt.'nt is explicit ly exhibited . This t.heorem is Vri.lirl o nly for da.sses of nonlinear sampled-data co nt l o l 5ystems that satisfy some of assumptions. A th eorem for t.he validity of extended A.ize rmall '5 ('.o nj(~c tlJre is abo presf~nt e d . Some numeri<:a.l examples are given to illu strate t he ~e res ult. ~.
K e ywords: Robust st.ability, nonline arity. samp led-data control. tell1.'>, Popo\' crit.erioH.
discre t.~tim c
sys-
1. INTRODUCTION
2. EQUIVALENT TRANSFORMATION
It wo uld 1I 0 t. hp. n.n overstat.emellt t.o say t hat. all control systems ill practice are composed of J\on linear elemeot,s. However , the design of a co ntro l 5,Ys t,I'm is IIsuall y perfo rmed for lin ea r t.ime- invariant. systems. As a result. , cont.rol syst.ems designed in t.his maJln er a re nol always gu ar:lnt.eed, even in t.erms of stability. This paper dea ls with t.he sta bilit.y of nonlinear sam plf'd-data cont rol systems as a nat.ural expansiun of t.11(' Pop ov crite ri o n in continuo us tim ~ ill tlw frequency domain (Desoe r and Vidya~:lagar, 1975: Hf1.I"I"is and Valenca, 198:Q, cla rifi es some p r o bl e Jll~ in di~nd(' t.imf', and givt's th e robust s t ability (oudit-iuJl as a. regiuH wiIcl"'.' all Hllccrta-in nonlinear cha.racterist.i c. is permit.t.ed ill the explicit. form . Th e: st.[I.bilit.y c. ritel'ion i ~ valid oIl ly whell th e tlOnlill Cr.1r samp Jeddata control s y s t(~ m~ :'> atisfy certain a,;.;sumptions. rn t his SC ClSC , it "flight. no t. h ~ a c.omp let.e so lut ion t o t his probIt> m. Howeve r , th e res ult contain:,; the previously prese nted ro bus t s ta.bili ty (·.ritf:'" rioIl of continuous-t.ime syste lltS (O ku y.a nHl , 1988) a nd t. ha t. o r lill c
The eontl'Ol system to b e cOHsidered is a sampled-dat a control system with t ime-illv;ui aut (louliw:,at characteris tics N(· ) as s how n in Fig. I. Here, 1t is t.ht~ zero-ord crhold wm
a. N(e)
= [( re + n( e )).
( 1)
relative 1.0 a uominalliuearile<.l gaiu I(. Huwever, t.he nonlinear t.erm n(·) is uncertain and confin ed within the s(~c.t or, i.e , t.he region represf· nted by t.h e in equality
in(') i ::; aie].
for
,,> o.
(2)
By rearranging the non linear sampled-data control syst.em , Fig. 2 i!:> obl.a.in ed , wher ~ G(z) i!) t he z-transform of G( ... ) togeth er with 'i.' ero-ordcr holrl . In t. hi s figure , ea c.h variable T , e. IV , . .. ind ica.tes t he :'>f'qnen ce '·(k). c(k). w(k) ,· · ill tlis n e le-time. Consider uew sequenCt'S "';,,(k) a nd w~,(A· ) whi c.h si:t.fisfy 1,11t"' following eq uat ion ("oJ)(·.erning t,he input and o utput sequenr.es
3438
When SIl"h ,..(k) is used. t.he expressed as
e(k) and w(k) of this nonlinear element nC):
e;,,(k) "W~,(k)
= fm(k) + q. :"elk)lh, = wm(k) - "q' :"e(k)lh,
= ,(k)+;lk-l).
w(k) = n(e(k)) = h(k) - a)e(k)
(4)
= "'(k)+~'(k-l)
w",lk)
= elk) -
,-:;. ---:l
1-'
(5)
+
c'
+
Fig. :3 Nonlinear subsystem.
d
" +
-!
_.-
+
". I -
N(r)
Fig. 2 Equivalent. nonlinear discrete-t.ime system. The z-transform of [;<]8. (.'3.) and (4) is a.." follows: ( l+z
-1'
)'lz)=(I+o
-1
)'lz)+"'I'
11 - z-l)e(z)
(1+o-')""lz)= 11 + ,")"'I.z) -2" •.
h
(" )
,
(l_Z~')'(Z)
(7)
The relationship between thesl' {'(luat.ioIls is t;}lOwn by the block dia.gram given in Fig. :3. Tll this figure, f, is defined as 2 1_-0- 1 hlz) := - .-;-----;(8) h 1+ z I ' This corresponds t.o the bilinea.r transformation approximation bet.ween :: and ~ when relating 6 to the Laplace t.ransform variable sfar thE' rOlltillllOus-tilllc t>Yt>tem. Define a new nonlinear function for 11.(.) of Eq. (2) such as flc) := Ill,) -+ Of. ( 9)
Though this function belongs to the first and third quadrant, considering the equivaknt. linear characteristic which varies wit.h tiw disc:rl'te-Limf:' k = 1 1 2, ... , il, can be written as
0:0 }(k)
0=
J(."llkk)) c')
<: 2".
"i")
d
+
Cl,)
+
--.----. -.J
c
y
"'I,)
1 + q6
e(k - 11.
CI·I
+
, - - , w'
C'
"
(11)
on the basis of Eqs. (9) and (10).
and .il.c( k) is the backward diffen'IH'e of error
:"e(k)
inequality (2) can be
(3)
where q is some non-negative number, em(k) and wm(k) are neutral points of the sequences c(k) and w(k), c",(k)
~ector
To avoid diffi(,~1l1t prohlems peculiar to a nonlincar sampled-data cont.rol syst.em le.g., Kalman, 1957), the following assumption is formulated with regard to the nonhnear c.haracteristics to be considered. [ASSUluption-l] The error sequences e(k) passes the origin. Specifically, the l'elatconship ,(k - I) = ,rk) is maintained whenever e(k-l)e(k) < O. This means that. the line between coordinates (elk - 1),f(e(k - I)) and (cl k).I(, (k))) by linea.r interpolation also passes the origin. This assumption is not so iIL;lccessible. If the sampling period is shorter t.han t.he tnlllsient. response of the system, the variation of the error ~e( k) is also expected to be small when t.he sequences passes t.he origin. Hence, Assumption-l t.hat is described a.bove might be satisfied. Even if tIle sa.mpling period j" relatively long, nonlinear characteristics, which are lin<::ar around the origin and form a broken (polygonal) line as shown in numerical examples. satisfy Assumptiot,-l. Ba"sed
Oil
t.he aboYr' premise. the following properties
of each variable of the subsystem are revealed. [Lenuua-l] For some posit.ive integer lv', the following equation holds:
(12) where 11 ,112,1\0' denotes the Euclidean norm written by
(The proof is olnit.ted on account. of t.he limited spacc) [Lenuna-2] If t.he following: inequality is satisfied in respect. of the inner product. of the neutral point of Eq. (9) a.nd the backward differPl,,,e series of error:
(\ 0)
( wmlk)
3439
+ 't(m(k)
:"elk) )N
2: 0,
(\3)
the following is
where
obta.in(~d:
1
(14)
erN)
Here, f(JV) indicates the difference between the area calculated on the a.ssumption that the error sample points follow a straight line t and t.he t.otal art'Cl of the trapezoid formed by sample point.s on Lhe a,ctual nonlinear curve. (The proof is omitted becau~e of t.he limiLed space.)
iY
L
.l:dk)JC,(k).
k=!
(Proof) Based
Eqs. (3) and (4).
Oil
u'llC;n(k)II~N
3, CLASSES OF NONLINEAR SAMPLED-DATA SYSTEMS
-llw;;'(k)lli.N
2 ' 2nq "'m ( k ) +"'m (k) ,Il.e (k) ) N =" , Iic m(k)112.N-llwm(k)112.N+T·(
Therefore, Eq. (14) is obt.ained. The left side of Eq. (13) can tw expressed as follows ill terms of t.he non linear function f(·) of Eq. (9). [Leuuua-3] For any step N, t he following equation is valid: (wm(k)+"em(k), ll.e(k))N =
1
)C . . . . . . . . . :- ,-. '"
: Pk
Iot k ) "
•
:
~
I""
Nonlinear sampled-data systems which satisfy 0 at. any step N, i.e., fork) 0 (k L 2,···, N) are classified int.o Class SIThe condition fork) = 0 is eSI."blished when ,(k) of Eq. (11) becomes a. positive ('.onsLmt '}. In other ,"vords, nOIllinear sampled-data syst.ems which belong to SI refer to linear samplcd-tbt.a syst.ems.
C
Fig. 4 Nonlinear fee).
,,'
,/
""l
/ . (2)) _ _ _ ".'_ _ __
,//'
-
11
"
./ /'
/0(3)
Fig. 5 Clockwise sample points. The tot.al area of the trapezoid 0'( iV) is rewritten into the following. [Lenlnla-4] For any step N. , rr(Iv)
1 = 2'(f(r(N))e(N)
f(r(O)I«O))
Inac-
sa1Ilpled-da~a
(1) A set of non linear sampled-data systems, of which point. (e(k),j(e(k»)) traces the same point on t.he nonlinear curve whether error e( k) tends t.o increase or decrease, belongs to Class Se. The response of ::;ampled-da1.a. systeITlB of which point (e(k),J(c(k))) exactly traces the sarne point on the non.. linea.r clII've seldom occurs. ht general, the fulfillment of (1) is expected from systems similar to continuous-Lime ones, which aff' cbara.c:tcrizc(: by a very short sampling pni()(l It or very slow respon~;e regarded as ~e(k) - O. (2)
.
() elk-I) e(k) elk+l)
consid~rably
cessible, some of the followillg non linear systems :"at.isfy it.
((N)
.
.
re~ponse.
AltholilSh this Assumption-2 seems
LU(e(k))+f(c(k-l)))Il.e(k).
fHk-I)) .... ·~:>l':~I:
[Assumption-2] The total area of the trapezoid allowing for the signs for coordinate (e(k), fleCk))), (k = 0,1,2,···, JV) which traces a nonlincar curve is always non-negative, i.e .. ( 18) ".(N) :': 0 regardless of the transient
,,:
(15 ) (Proof) This lemma. can be easily proved by directly substit.uting Eq. (r» and t.he del1nin,u; equation for inner products. If ".(N) is used for t.he right. side of Eq. (16), it. can be shown that 0'(1\/) is the toLal area of the trapezoid formed by the sample point (f(c(k-l)),f(e(k))) on the noulincar curve f{t) and t.he error :"l.ep \\'idLh ~[(k) as shown ill fig. 4. fie)
I(,(kJ)
(17)
k=l
for any q 2: O. Here. the inner product of xt{k) and 2',(k) is defined as
( '·il k). ,,,,(k) IN =
N
= 2' Lf'lk)' L'.e(k).
+ erN),
(16)
=
=
=
(3) Nonlinear sampled-data syst.ems which satisfy «N) ~ o.•t. auy step N i,e., fork) . L'.c(k) ~ 0 (k = 1,2.··· . ,IV), are classifif.'d int.o Class 5,.. As illust.rated iu Fig. 4. fo(t) . L'.e(k) ~ 0 always describes: when point. (e(k),J(e(k))) is in the first. quadrant., it. t.urns right for ~e(k) > 0 when deviating from the linear chc\ract.crist.ic; iL t.urns left for .6.c(k) < 0 whell dcviatin~ from thf' linear charact.eristic; as a wholf', a clockwis(' transitinn of response on the non linear C:lll·ve. The response also shows a clockwise transition when point. (e(k).f(c(k))) is located in the third quadrant.. The point always turns righl. for ..1.e(k) < 0, or turns left for L'.,.(k) > 11.
3440
4. ROBUST STABILITY CONDITION Apply the subsystem ill Fig. 3 inst.pad of non linear charaderistics n(·) in Fig. 2. The closed-loop characLerislies arc described as follows for :::-transfurm of series c~(k) and w;rJk), ami neutral point,s of t.ht' reference and disturbance inputs 1"m(h;), dm(k):
':n I') ~ LI~. q. z)rml z )+M (a, g, z )dm(z )-H(", g. z )w~, (,,),
sampled-data control systenL is defined to be stable (strictly speaking, t'2 stable for neutral points). Here, 11 ' 11, denotes norm 11 ' II"N [('I' N ~ ,x, Though it was described thal a neutral point is £2 stable~ is t.here a. possibility oJ" l2-stability for the output sequence y(k)'! It is n(Jt generally a.pproved. If Ily(k)11 < 'X,, then Ily",(k)11 < 'Xl, However, the opposite of it cannot be guaranteed.
(19)
where
H ( 0:, q, z) = .,--,'(",1,.:-+,-'q,:.:f('"cz '7))"7[(",0",(z",,)" l + (l + r>q6(z))f{0(z)
[Lemma-5] If t.he nominal linear sampled·data control system is stable, i.e., (20)
qli(z) = 1 + (1 +l +aqb(z))/{G(z) ,
.IVI(", '1, z)
= l + (I + ,y,/6(z))KG(z) '
(1
+ g/i( z) )G( z)
<
00,
<: CM < 00,
(25)
(26)
(22)
it can be proved that the comrol system is sta.ble in the meaning of the Definition-I (Eq. (26) corresponds to the small gain theorem for H "Xl-norm). (The proof is omitted becalls.> of the limited space.) By substituting z
= ejwh , Eq
then Eq. (20) is rewritten as
Il(a:
where F(::) is the inverse of t.he cOlllplementary sensitivit.y function, that is, t.h~ ;'stabilit.y robustness measure". The frequeHcy respollst's of tllesf' equations can be easily obtained by substit.uting :: = cjw /). Especially, Eq. (24) corresponds to the one in which inverse-Nyquist. locus is moved only by 1, so it is possible to usc the locus also as a guide for control system design. The frequency response of b(.:) 1S (l purely imaginary number shown as follows:
2 si IL.4.' h *iwh) =j_. =jp(w). h l+cof-,wh
IM(rt,q,z))1
(21 )
If t,he following notation is used:
1 F(z) = 1 + KG(z)'
CL
for [zl ?:: 1 and the non linear sampled-data. control system satisfies the Assumption."! and 2, for any q, when the follo'wing inequality is .... al id:
and
L(n,'1,z)
IL(o,q,z))1 <:
,q,
C]wh) =
(23) can be written a.">
Jwh l + q/i(e ) F( c]W") + "qa( eiwh) ,
(27)
By putting this Eq, (27) il,to Eq, (26) and solving the equatioll as for Q: explicit.ly, the following theorem concerning the robust stability can be given, because 8(ei "h) = jp(w) from El[, (24),
[TheoreUl.l] If q ?:: 0 exists in which the sector parameter 0' in uncertain nonlillear term n(.) satisfies the following inequalit.y, then th,~ nonlinear sampled-data control system in Fig. 1 is st;~ble: a
'1p(w)1j;(w)+~'p'(w)tp'(w) + IF(riWh)I', (28)
(24)
When considering Hte frequency re:::pOl1se, the following a.'3sllmptions arc provided. [Assllmptioll-3] Frequency I.J.,' 'which should. be considered in this control system is a. range which satisfies Shannon's samplmg theorem as with usual linear sa.mpled-data co11trol systems To examine the st.abilit.y of t.he nonlinear sampled-data control system of interest., tll(' foJlowilLg preparations are provided. [Definition-I] When Ilr",(klll, <: c, < 00 and Ily",(k)ll, <: c y < "", iLre oat.isfied for any Ilrm(k)ll, <: er < x' and Ildm(k)ll:! :S Cd
where 'i'(w) = 21(1"(e]Wh)), (Proof) This theorem c.an Le proved from Lemma-6. Based on I~q, (26) and (27),
,,2(1
+ '!'r,' (w)) < ,,'(w) + (~)(w) + aqp(w))'.
where X(w) = 'I?(F(e iwh )), TI,erefore
n' - 2qp(w)tj;(w)a -IF(riw"W < 0, Eq. (28) is obtained as for et" > O. Eq. (28) in Theorem-l is fC'f all w considered and a certain q. Therefore, if a. max·min of t}(q,w) can be obtained by some method, then E:q. (28) can a.lso be written as (y
3441
<
1J(qa,W(J) =
IIl.ilX
,
min1](q,(.•.;). w
(29)
6. NUME RICAL EXAM PLES
5. RELA TION TO LINEA R TIME -INVA RIAN T SYSTE M Conside ration with th e- linear gain band of th e stabilit y
of the nonlillc"ar syst ~m h as been known for long time as Aizerm an 's conject ure (A t hert.oll , 198 1; Grujic, 198 1). Howeve r , so me at tent,ion is neces:';(tI'Y for the discrete time s ystem a..;; desrrih ed in th is pa per , because it is a cOll clnsi on und er Assump tioJl-l , :2 and 3 m entione d ab ove, but a similar descrip tion is I>(,ssible. [Theor em-2] If t.he righ t side uf Eq. ( 29) , i.e .,
are represe nted in Fig . 6, fi g. 7 and t' ig. 8. The stabilit y ronditio n uf di screk-t ime cont rol systems i.lS. for samplin g period h = 0.1 i ~ as follows:
o < I(
"
is satisfied in th e sadd le p oint ; (30)
(35)
< \.821 ,
when lin ~ar gaiu feedbac k is taking place. The calculation rc,,, lt for q of ry(q , w) f,,, the system of Eq. (34) 1.0 is ;1.i1 s hown in Fig. 6. Ther ewit.h nomilLal gRin }{ 0.82 1, whi ch agrees to fore , when h ::: 0. 1, T/(t/n .wu) , Theore m-2 's validity case this In . the res ult. of 8q . (:)5) is obvious .
=
lJ( Qa , ;..Jo);::; wax mi u1/(rJ, "";) q
[Exarn pl(!-IJ Results off.he calcul atio n for a cont roJled f{ system (34) C(s) = 5(> + 1)'
=
When a nonliru."at· charact eristic is a polygon al line as Eq . (28) of Theorc m- I u ecomes eClual to th e robust sta.bilit y (.anditi on prov ided for t. he liucal" time-in variant sample d-d a t.a system . (Proof) If Eq. PO) is calcula ted f<>r rh e right side of F,q . (28) , ,.hen (3 1)
Ohviou siy, ?}( Q0 1 Wn) > O. In additi~)JJ , since from Assllmp t.ion-3 . p("-'o) > O. T!I (,H ~ VJ(wo ) all ,
Wo
= O.
<
W~h
After
Eq. (32) corres ponds to th~ stabilit y conditio n determ ined for .. he ti me- iu variant di sr. n~te- tim.(' system by the lin ear gain band . i.e. , t.he Nyqlli s t. cond ition for thp. disc.ret.e -t.ime syste m . Theore m-2 mean:> t.ha.t. it is ~ llffi cienl t.o fLpply the robust. s ta bility conditio n Eq . (~12) of tht- lillear t.ime-in variant syst.em for t he ll onlinea r sample d-d at. a coniro l system, whir.h come!:! into qu( ~s ti o n wlH~ n Eq . (30) is e fTt~ de d . of C lass b ~ lon ging to Classes S ~ aIld S7" to ~ ay nothin g vali<1 always not l,Ll'f': (29) . F.q of n J1ln.x·mi 5/. How{'ver, es exampl al numeric . Then (30). l~q. point saddle a~ th£' a.'S co unter ex ample~ of AizerwaJl 's conjer.tll re ex te nded int.o the Itonline ar sample d-dat.a syst.em which satisfies Theore m-2 an d t.he one whi ch do ps not satisfy Theore m:2 are s how l! .
=
shown in Fig. 7(a) : the sedor lies between P""1 0.6 and L.82. In su('h a. (ase, th1· s t.ability condif.ioll of Eq. K2 ( 35) is sa1.isfif'd . Fig. 7(h ) S\}(IWS the tran sient. res po nses of c( k) all ll 0'( N) . In this ~xalnple, ASSuIllption- 2 is satisfied . and it can be said t hat. j he system belon gs to Class Sr at kw; !. ('ol1('e rning t.his [("s ponse. ''''hen a ll ou line ar charact eris t ic is a polygo ual line as ~ h ow ll in Fig. 8(a) , the S0ctor lie~ \wtween /"1 :.= 0.7 and 1{ 2 = 2.48 . In this case, I.he st.ability rondit loll of Eq . (35) is not sal.i5flCd. The t.ransi ent respons es of error seq Ul'Il ('.e e(k) an d trap ezoidal area u( N) are ru; s how n in F ig. X( b).
=
[Exaln ple-2] A counter exampl e of Aizerm a n's c.onject ure whidl is ext.e nd ed to th e discrete -time system wil! he shown . Re8 ults of th e ra.lclll at ion for a ('.ou1.roll ed SYSl.CIO
u(.)
~
1\( - >' +8+2 ) $2(. + 4)
(36)
are repr{'~ c nt ed in Fig. 9 an d Fig. 10. Th e stabilit. y co ndition of disr.rete -t.ime ('.(.·u trol systems fo r samplin g pe rio d h = 0.0r, is as foll ows :
o<
I,
<
(:37 )
1.756,
when lin ear gain feedback i.;; r aking (JI C1.ce. The CC1.lculation result for 'I of f/(q,"-') fo r the syst em of Eq . (36) with nomina l gain [( == 1.0 becomes as sh own in Fig. 0.25 1. The meanin g of 9. ""hen h :::: 0.05. 11(Qol wo) thi8 result is delicatE' . \Vhcll a noulin ear charact eristi c is a polygon al line as shown in Fig. 10( a) t.he sed.or lie~ be l.wel' ll ,,' 1 = 0.6 iLnd 1\2 :::: 1.6. Though th~ gain hand satisfi es t.h e st.a bility conditlu n of linear syst.ems in Eq. (37), TluX>l'e m - 1 is not. valid . T he respom. l' of e(k ) aud O'( k ) at this t.in ", oscillat es as s hown in Fig . IO(b) . Th at is, EXCLlllp le- 2 might corresp ond to a counter examp le of Aizerm all's conjt"ctul'e ('xtend ed into discret.e -tim e syst.em.
=
1
Since tlw rohust- s l.ahi lity (ondit.io ll which corresp onds to q = 0 in t hese s},8t t'rns. i. e.,
(J3 ) is Cl stab ility conditio u for t.he gencral time-va rying system , it is ap provf:'d as a su ffi cie nt co ndi t ion in au y of n u meri cal ex amples helow .
3442
7. CONCLUSIONS This paper has analyzed, in the frequency domain: the problem of the robust stabilit.y in the nonlinear sampleddata control system as a natural expansion of the Popa\' criterion for continuous time, Thcn. t.he allowable region of an uncertain l10nlinear characteristic has been given explicitly. Howcvn, t.he st.ability criterion (Theorem1) is valid only for t.he system,:.; to satisfy the previous AssulTIpt.ion-l, 2, and:L Naturally, Assumption-l is valid for a non linear ('hal'acteri~ti(' such as the pn:vious exa.mple wit.h a. lineal' dw.racterist.ic around t.he origin. For Assumption-a, a sampling period h which allows it to remain approximately va.lid can be chosen, as in the case of the usual lineal' sampled-data control system. Therefore, t.his assumption is not. highly significant. Aft.er all, the validit.,y· of t.he result. of t.hi.':! paper depends on whet.her Assumpt-ion-2 holds or not. However, as is obvious from t.1lt' prcvi(\us t.wo examples, it. is unquest.ionable that. Assurnpt.ion-2 is considered to be valid for ;1. usual Ilonlinear (,Ul've whieh is bf!nt. slightly.
REFERENCES Athert.Oll. D. P. (19tH) . .':-r'ta,bilit!/ of lv'onlinear· System, Research Studies Press. Desoer, C'. A. and fvI. Vidyasagar (H)75). Feedback S:r;stem: [nput-Output Propertu_s, Academic Press. Grujic. L. T. (19tll). On Ab,olu\c St.ability and the Aizerma.1l Conjecture. A'lLtomatiw, pp. :l3.5-349. Rarri" C . .I. and.J. M. F:. Val,'un, (1983). The Stability of Input-Output D:IJ'fJ(J.'mimi System,,>, Academic
0.5
- 0,5
(h) Pig. 7 Error sequence and trapczoidal area for Exa.mple-I.
r
1
(j.5
i'f____ _/=/~~~=-+-----" (_b_)
Fig. 8 Error sequence and trapezoidal area for Example-I.
"Jn::!n
Pt'es~.
Kalman, R. E. (l9;}7). Nonlill(';:u Aspect.s of SampledData Cont.rol SysLems. Pruc. vftAr Symp. on lv'onlineal· Circuit ;1n(l./ysls, pp 27:~-:H:l. OkuywIIa, Y. (1988). Rohm;t, StabilJty of Feedback Control System:; Conta.ining all r Ilrcrtain N onlinearity. Trans. of the i-ru;titu/( of ,''';ystons, Control and InfOT·m.a_twn EnghlffT8, pp. D-16.
'I(q, __">
" h=O.~::=OI}l
h=O.Ose h _ 0.1
, r=-
,
,
,
•
q ,
I
APPENDIX
Fig. 9 l/(q,W) cul'v'c:s for Exarnple-2. I~ ~
/
I
'" (a) 1-C+--~--2 ----~-,---~---;cq-;,1
(b) Fig. 10 Error sequenrc .cllld trapezoidal area for EXillrlplc-2.
Fig. 6 '11('1,'..1.1) curvps for Example-I.
3443
Copyright @ 1996 1FAC Illh Triennial Wortd Congress. San Fmncisco, USA
ROBUST STABILITY EVALUATION OF SAMPLED-DATA CONTROL SYSTEMS WITH A SECTOR NONLINEARITY YnshifllIui OKUYAMA .. nu Fumiaki TAKEMORI
Faculty of Engin ee ring,
TOtiU1'i
(/ni1.lersity,
.{-fOl , Koyama- ch Q Minami, Tott ori 680, Japan E-mail: oku @i ke. to Uori-u .ac .jp
Abst ract: This pa per describes a rouust stability e valuation of sampled-dat.a con Lro l syste ms containing a st' ctor nonlinearity in t.he fo rward path . The res ul t of this pa.per is derived fr oll! j be norm condit ion in the freque u cy domain by exte nding the Popov's crite rion t heory. Some lemmas a re presented, Cl t.beorem for f 2-stability is proved , and the st.abilit.y margin (a.llo\',°a ble sector) of 3. nonlinear clcHlt.'nt is explicit ly exhibited . This t.heorem is Vri.lirl o nly for da.sses of nonlinear sampled-data co nt l o l 5ystems that satisfy some of assumptions. A th eorem for t.he validity of extended A.ize rmall '5 ('.o nj(~c tlJre is abo presf~nt e d . Some numeri<:a.l examples are given to illu strate t he ~e res ult. ~.
K e ywords: Robust st.ability, nonline arity. samp led-data control. tell1.'>, Popo\' crit.erioH.
discre t.~tim c
sys-
1. INTRODUCTION
2. EQUIVALENT TRANSFORMATION
It wo uld 1I 0 t. hp. n.n overstat.emellt t.o say t hat. all control systems ill practice are composed of J\on linear elemeot,s. However , the design of a co ntro l 5,Ys t,I'm is IIsuall y perfo rmed for lin ea r t.ime- invariant. systems. As a result. , cont.rol syst.ems designed in t.his maJln er a re nol always gu ar:lnt.eed, even in t.erms of stability. This paper dea ls with t.he sta bilit.y of nonlinear sam plf'd-data cont rol systems as a nat.ural expansiun of t.11(' Pop ov crite ri o n in continuo us tim ~ ill tlw frequency domain (Desoe r and Vidya~:lagar, 1975: Hf1.I"I"is and Valenca, 198:Q, cla rifi es some p r o bl e Jll~ in di~nd(' t.imf', and givt's th e robust s t ability (oudit-iuJl as a. regiuH wiIcl"'.' all Hllccrta-in nonlinear cha.racterist.i c. is permit.t.ed ill the explicit. form . Th e: st.[I.bilit.y c. ritel'ion i ~ valid oIl ly whell th e tlOnlill Cr.1r samp Jeddata control s y s t(~ m~ :'> atisfy certain a,;.;sumptions. rn t his SC ClSC , it "flight. no t. h ~ a c.omp let.e so lut ion t o t his probIt> m. Howeve r , th e res ult contain:,; the previously prese nted ro bus t s ta.bili ty (·.ritf:'" rioIl of continuous-t.ime syste lltS (O ku y.a nHl , 1988) a nd t. ha t. o r lill c
The eontl'Ol system to b e cOHsidered is a sampled-dat a control system with t ime-illv;ui aut (louliw:,at characteris tics N(· ) as s how n in Fig. I. Here, 1t is t.ht~ zero-ord crhold wm
a. N(e)
= [( re + n( e )).
( 1)
relative 1.0 a uominalliuearile<.l gaiu I(. Huwever, t.he nonlinear t.erm n(·) is uncertain and confin ed within the s(~c.t or, i.e , t.he region represf· nted by t.h e in equality
in(') i ::; aie].
for
,,> o.
(2)
By rearranging the non linear sampled-data control syst.em , Fig. 2 i!:> obl.a.in ed , wher ~ G(z) i!) t he z-transform of G( ... ) togeth er with 'i.' ero-ordcr holrl . In t. hi s figure , ea c.h variable T , e. IV , . .. ind ica.tes t he :'>f'qnen ce '·(k). c(k). w(k) ,· · ill tlis n e le-time. Consider uew sequenCt'S "';,,(k) a nd w~,(A· ) whi c.h si:t.fisfy 1,11t"' following eq uat ion ("oJ)(·.erning t,he input and o utput sequenr.es
3438
When SIl"h ,..(k) is used. t.he expressed as
e(k) and w(k) of this nonlinear element nC):
e;,,(k) "W~,(k)
= fm(k) + q. :"elk)lh, = wm(k) - "q' :"e(k)lh,
= ,(k)+;lk-l).
w(k) = n(e(k)) = h(k) - a)e(k)
(4)
= "'(k)+~'(k-l)
w",lk)
= elk) -
,-:;. ---:l
1-'
(5)
+
c'
+
Fig. :3 Nonlinear subsystem.
d
" +
-!
_.-
+
". I -
N(r)
Fig. 2 Equivalent. nonlinear discrete-t.ime system. The z-transform of [;<]8. (.'3.) and (4) is a.." follows: ( l+z
-1'
)'lz)=(I+o
-1
)'lz)+"'I'
11 - z-l)e(z)
(1+o-')""lz)= 11 + ,")"'I.z) -2" •.
h
(" )
,
(l_Z~')'(Z)
(7)
The relationship between thesl' {'(luat.ioIls is t;}lOwn by the block dia.gram given in Fig. :3. Tll this figure, f, is defined as 2 1_-0- 1 hlz) := - .-;-----;(8) h 1+ z I ' This corresponds t.o the bilinea.r transformation approximation bet.ween :: and ~ when relating 6 to the Laplace t.ransform variable sfar thE' rOlltillllOus-tilllc t>Yt>tem. Define a new nonlinear function for 11.(.) of Eq. (2) such as flc) := Ill,) -+ Of. ( 9)
Though this function belongs to the first and third quadrant, considering the equivaknt. linear characteristic which varies wit.h tiw disc:rl'te-Limf:' k = 1 1 2, ... , il, can be written as
0:0 }(k)
0=
J(."llkk)) c')
<: 2".
"i")
d
+
Cl,)
+
--.----. -.J
c
y
"'I,)
1 + q6
e(k - 11.
CI·I
+
, - - , w'
C'
"
(11)
on the basis of Eqs. (9) and (10).
and .il.c( k) is the backward diffen'IH'e of error
:"e(k)
inequality (2) can be
(3)
where q is some non-negative number, em(k) and wm(k) are neutral points of the sequences c(k) and w(k), c",(k)
~ector
To avoid diffi(,~1l1t prohlems peculiar to a nonlincar sampled-data cont.rol syst.em le.g., Kalman, 1957), the following assumption is formulated with regard to the nonhnear c.haracteristics to be considered. [ASSUluption-l] The error sequences e(k) passes the origin. Specifically, the l'elatconship ,(k - I) = ,rk) is maintained whenever e(k-l)e(k) < O. This means that. the line between coordinates (elk - 1),f(e(k - I)) and (cl k).I(, (k))) by linea.r interpolation also passes the origin. This assumption is not so iIL;lccessible. If the sampling period is shorter t.han t.he tnlllsient. response of the system, the variation of the error ~e( k) is also expected to be small when t.he sequences passes t.he origin. Hence, Assumption-l t.hat is described a.bove might be satisfied. Even if tIle sa.mpling period j" relatively long, nonlinear characteristics, which are lin<::ar around the origin and form a broken (polygonal) line as shown in numerical examples. satisfy Assumptiot,-l. Ba"sed
Oil
t.he aboYr' premise. the following properties
of each variable of the subsystem are revealed. [Lenuua-l] For some posit.ive integer lv', the following equation holds:
(12) where 11 ,112,1\0' denotes the Euclidean norm written by
(The proof is olnit.ted on account. of t.he limited spacc) [Lenuna-2] If t.he following: inequality is satisfied in respect. of the inner product. of the neutral point of Eq. (9) a.nd the backward differPl,,,e series of error:
(\ 0)
( wmlk)
3439
+ 't(m(k)
:"elk) )N
2: 0,
(\3)
the following is
where
obta.in(~d:
1
(14)
erN)
Here, f(JV) indicates the difference between the area calculated on the a.ssumption that the error sample points follow a straight line t and t.he t.otal art'Cl of the trapezoid formed by sample point.s on Lhe a,ctual nonlinear curve. (The proof is omitted becau~e of t.he limiLed space.)
iY
L
.l:dk)JC,(k).
k=!
(Proof) Based
Eqs. (3) and (4).
Oil
u'llC;n(k)II~N
3, CLASSES OF NONLINEAR SAMPLED-DATA SYSTEMS
-llw;;'(k)lli.N
2 ' 2nq "'m ( k ) +"'m (k) ,Il.e (k) ) N =" , Iic m(k)112.N-llwm(k)112.N+T·(
Therefore, Eq. (14) is obt.ained. The left side of Eq. (13) can tw expressed as follows ill terms of t.he non linear function f(·) of Eq. (9). [Leuuua-3] For any step N, t he following equation is valid: (wm(k)+"em(k), ll.e(k))N =
1
)C . . . . . . . . . :- ,-. '"
: Pk
Iot k ) "
•
:
~
I""
Nonlinear sampled-data systems which satisfy 0 at. any step N, i.e., fork) 0 (k L 2,···, N) are classified int.o Class SIThe condition fork) = 0 is eSI."blished when ,(k) of Eq. (11) becomes a. positive ('.onsLmt '}. In other ,"vords, nOIllinear sampled-data syst.ems which belong to SI refer to linear samplcd-tbt.a syst.ems.
C
Fig. 4 Nonlinear fee).
,,'
,/
""l
/ . (2)) _ _ _ ".'_ _ __
,//'
-
11
"
./ /'
/0(3)
Fig. 5 Clockwise sample points. The tot.al area of the trapezoid 0'( iV) is rewritten into the following. [Lenlnla-4] For any step N. , rr(Iv)
1 = 2'(f(r(N))e(N)
f(r(O)I«O))
Inac-
sa1Ilpled-da~a
(1) A set of non linear sampled-data systems, of which point. (e(k),j(e(k»)) traces the same point on t.he nonlinear curve whether error e( k) tends t.o increase or decrease, belongs to Class Se. The response of ::;ampled-da1.a. systeITlB of which point (e(k),J(c(k))) exactly traces the sarne point on the non.. linea.r clII've seldom occurs. ht general, the fulfillment of (1) is expected from systems similar to continuous-Lime ones, which aff' cbara.c:tcrizc(: by a very short sampling pni()(l It or very slow respon~;e regarded as ~e(k) - O. (2)
.
() elk-I) e(k) elk+l)
consid~rably
cessible, some of the followillg non linear systems :"at.isfy it.
((N)
.
.
re~ponse.
AltholilSh this Assumption-2 seems
LU(e(k))+f(c(k-l)))Il.e(k).
fHk-I)) .... ·~:>l':~I:
[Assumption-2] The total area of the trapezoid allowing for the signs for coordinate (e(k), fleCk))), (k = 0,1,2,···, JV) which traces a nonlincar curve is always non-negative, i.e .. ( 18) ".(N) :': 0 regardless of the transient
,,:
(15 ) (Proof) This lemma. can be easily proved by directly substit.uting Eq. (r» and t.he del1nin,u; equation for inner products. If ".(N) is used for t.he right. side of Eq. (16), it. can be shown that 0'(1\/) is the toLal area of the trapezoid formed by the sample point (f(c(k-l)),f(e(k))) on the noulincar curve f{t) and t.he error :"l.ep \\'idLh ~[(k) as shown ill fig. 4. fie)
I(,(kJ)
(17)
k=l
for any q 2: O. Here. the inner product of xt{k) and 2',(k) is defined as
( '·il k). ,,,,(k) IN =
N
= 2' Lf'lk)' L'.e(k).
+ erN),
(16)
=
=
=
(3) Nonlinear sampled-data syst.ems which satisfy «N) ~ o.•t. auy step N i,e., fork) . L'.c(k) ~ 0 (k = 1,2.··· . ,IV), are classifif.'d int.o Class 5,.. As illust.rated iu Fig. 4. fo(t) . L'.e(k) ~ 0 always describes: when point. (e(k),J(e(k))) is in the first. quadrant., it. t.urns right for ~e(k) > 0 when deviating from the linear chc\ract.crist.ic; iL t.urns left for .6.c(k) < 0 whell dcviatin~ from thf' linear charact.eristic; as a wholf', a clockwis(' transitinn of response on the non linear C:lll·ve. The response also shows a clockwise transition when point. (e(k).f(c(k))) is located in the third quadrant.. The point always turns righl. for ..1.e(k) < 0, or turns left for L'.,.(k) > 11.
3440
4. ROBUST STABILITY CONDITION Apply the subsystem ill Fig. 3 inst.pad of non linear charaderistics n(·) in Fig. 2. The closed-loop characLerislies arc described as follows for :::-transfurm of series c~(k) and w;rJk), ami neutral point,s of t.ht' reference and disturbance inputs 1"m(h;), dm(k):
':n I') ~ LI~. q. z)rml z )+M (a, g, z )dm(z )-H(", g. z )w~, (,,),
sampled-data control systenL is defined to be stable (strictly speaking, t'2 stable for neutral points). Here, 11 ' 11, denotes norm 11 ' II"N [('I' N ~ ,x, Though it was described thal a neutral point is £2 stable~ is t.here a. possibility oJ" l2-stability for the output sequence y(k)'! It is n(Jt generally a.pproved. If Ily(k)11 < 'X,, then Ily",(k)11 < 'Xl, However, the opposite of it cannot be guaranteed.
(19)
where
H ( 0:, q, z) = .,--,'(",1,.:-+,-'q,:.:f('"cz '7))"7[(",0",(z",,)" l + (l + r>q6(z))f{0(z)
[Lemma-5] If t.he nominal linear sampled·data control system is stable, i.e., (20)
qli(z) = 1 + (1 +l +aqb(z))/{G(z) ,
.IVI(", '1, z)
= l + (I + ,y,/6(z))KG(z) '
(1
+ g/i( z) )G( z)
<
00,
<: CM < 00,
(25)
(26)
(22)
it can be proved that the comrol system is sta.ble in the meaning of the Definition-I (Eq. (26) corresponds to the small gain theorem for H "Xl-norm). (The proof is omitted becalls.> of the limited space.) By substituting z
= ejwh , Eq
then Eq. (20) is rewritten as
Il(a:
where F(::) is the inverse of t.he cOlllplementary sensitivit.y function, that is, t.h~ ;'stabilit.y robustness measure". The frequeHcy respollst's of tllesf' equations can be easily obtained by substit.uting :: = cjw /). Especially, Eq. (24) corresponds to the one in which inverse-Nyquist. locus is moved only by 1, so it is possible to usc the locus also as a guide for control system design. The frequency response of b(.:) 1S (l purely imaginary number shown as follows:
2 si IL.4.' h *iwh) =j_. =jp(w). h l+cof-,wh
IM(rt,q,z))1
(21 )
If t,he following notation is used:
1 F(z) = 1 + KG(z)'
CL
for [zl ?:: 1 and the non linear sampled-data. control system satisfies the Assumption."! and 2, for any q, when the follo'wing inequality is .... al id:
and
L(n,'1,z)
IL(o,q,z))1 <:
,q,
C]wh) =
(23) can be written a.">
Jwh l + q/i(e ) F( c]W") + "qa( eiwh) ,
(27)
By putting this Eq, (27) il,to Eq, (26) and solving the equatioll as for Q: explicit.ly, the following theorem concerning the robust stability can be given, because 8(ei "h) = jp(w) from El[, (24),
[TheoreUl.l] If q ?:: 0 exists in which the sector parameter 0' in uncertain nonlillear term n(.) satisfies the following inequalit.y, then th,~ nonlinear sampled-data control system in Fig. 1 is st;~ble: a
'1p(w)1j;(w)+~'p'(w)tp'(w) + IF(riWh)I', (28)
(24)
When considering Hte frequency re:::pOl1se, the following a.'3sllmptions arc provided. [Assllmptioll-3] Frequency I.J.,' 'which should. be considered in this control system is a. range which satisfies Shannon's samplmg theorem as with usual linear sa.mpled-data co11trol systems To examine the st.abilit.y of t.he nonlinear sampled-data control system of interest., tll(' foJlowilLg preparations are provided. [Definition-I] When Ilr",(klll, <: c, < 00 and Ily",(k)ll, <: c y < "", iLre oat.isfied for any Ilrm(k)ll, <: er < x' and Ildm(k)ll:! :S Cd
where 'i'(w) = 21(1"(e]Wh)), (Proof) This theorem c.an Le proved from Lemma-6. Based on I~q, (26) and (27),
,,2(1
+ '!'r,' (w)) < ,,'(w) + (~)(w) + aqp(w))'.
where X(w) = 'I?(F(e iwh )), TI,erefore
n' - 2qp(w)tj;(w)a -IF(riw"W < 0, Eq. (28) is obtained as for et" > O. Eq. (28) in Theorem-l is fC'f all w considered and a certain q. Therefore, if a. max·min of t}(q,w) can be obtained by some method, then E:q. (28) can a.lso be written as (y
3441
<
1J(qa,W(J) =
IIl.ilX
,
min1](q,(.•.;). w
(29)
6. NUME RICAL EXAM PLES
5. RELA TION TO LINEA R TIME -INVA RIAN T SYSTE M Conside ration with th e- linear gain band of th e stabilit y
of the nonlillc"ar syst ~m h as been known for long time as Aizerm an 's conject ure (A t hert.oll , 198 1; Grujic, 198 1). Howeve r , so me at tent,ion is neces:';(tI'Y for the discrete time s ystem a..;; desrrih ed in th is pa per , because it is a cOll clnsi on und er Assump tioJl-l , :2 and 3 m entione d ab ove, but a similar descrip tion is I>(,ssible. [Theor em-2] If t.he righ t side uf Eq. ( 29) , i.e .,
are represe nted in Fig . 6, fi g. 7 and t' ig. 8. The stabilit y ronditio n uf di screk-t ime cont rol systems i.lS. for samplin g period h = 0.1 i ~ as follows:
o < I(
"
is satisfied in th e sadd le p oint ; (30)
(35)
< \.821 ,
when lin ~ar gaiu feedbac k is taking place. The calculation rc,,, lt for q of ry(q , w) f,,, the system of Eq. (34) 1.0 is ;1.i1 s hown in Fig. 6. Ther ewit.h nomilLal gRin }{ 0.82 1, whi ch agrees to fore , when h ::: 0. 1, T/(t/n .wu) , Theore m-2 's validity case this In . the res ult. of 8q . (:)5) is obvious .
=
lJ( Qa , ;..Jo);::; wax mi u1/(rJ, "";) q
[Exarn pl(!-IJ Results off.he calcul atio n for a cont roJled f{ system (34) C(s) = 5(> + 1)'
=
When a nonliru."at· charact eristic is a polygon al line as Eq . (28) of Theorc m- I u ecomes eClual to th e robust sta.bilit y (.anditi on prov ided for t. he liucal" time-in variant sample d-d a t.a system . (Proof) If Eq. PO) is calcula ted f<>r rh e right side of F,q . (28) , ,.hen (3 1)
Ohviou siy, ?}( Q0 1 Wn) > O. In additi~)JJ , since from Assllmp t.ion-3 . p("-'o) > O. T!I (,H ~ VJ(wo ) all ,
Wo
= O.
<
W~h
After
Eq. (32) corres ponds to th~ stabilit y conditio n determ ined for .. he ti me- iu variant di sr. n~te- tim.(' system by the lin ear gain band . i.e. , t.he Nyqlli s t. cond ition for thp. disc.ret.e -t.ime syste m . Theore m-2 mean:> t.ha.t. it is ~ llffi cienl t.o fLpply the robust. s ta bility conditio n Eq . (~12) of tht- lillear t.ime-in variant syst.em for t he ll onlinea r sample d-d at. a coniro l system, whir.h come!:! into qu( ~s ti o n wlH~ n Eq . (30) is e fTt~ de d . of C lass b ~ lon ging to Classes S ~ aIld S7" to ~ ay nothin g vali<1 always not l,Ll'f': (29) . F.q of n J1ln.x·mi 5/. How{'ver, es exampl al numeric . Then (30). l~q. point saddle a~ th£' a.'S co unter ex ample~ of AizerwaJl 's conjer.tll re ex te nded int.o the Itonline ar sample d-dat.a syst.em which satisfies Theore m-2 an d t.he one whi ch do ps not satisfy Theore m:2 are s how l! .
=
shown in Fig. 7(a) : the sedor lies between P""1 0.6 and L.82. In su('h a. (ase, th1· s t.ability condif.ioll of Eq. K2 ( 35) is sa1.isfif'd . Fig. 7(h ) S\}(IWS the tran sient. res po nses of c( k) all ll 0'( N) . In this ~xalnple, ASSuIllption- 2 is satisfied . and it can be said t hat. j he system belon gs to Class Sr at kw; !. ('ol1('e rning t.his [("s ponse. ''''hen a ll ou line ar charact eris t ic is a polygo ual line as ~ h ow ll in Fig. 8(a) , the S0ctor lie~ \wtween /"1 :.= 0.7 and 1{ 2 = 2.48 . In this case, I.he st.ability rondit loll of Eq . (35) is not sal.i5flCd. The t.ransi ent respons es of error seq Ul'Il ('.e e(k) an d trap ezoidal area u( N) are ru; s how n in F ig. X( b).
=
[Exaln ple-2] A counter exampl e of Aizerm a n's c.onject ure whidl is ext.e nd ed to th e discrete -time system wil! he shown . Re8 ults of th e ra.lclll at ion for a ('.ou1.roll ed SYSl.CIO
u(.)
~
1\( - >' +8+2 ) $2(. + 4)
(36)
are repr{'~ c nt ed in Fig. 9 an d Fig. 10. Th e stabilit. y co ndition of disr.rete -t.ime ('.(.·u trol systems fo r samplin g pe rio d h = 0.0r, is as foll ows :
o<
I,
<
(:37 )
1.756,
when lin ear gain feedback i.;; r aking (JI C1.ce. The CC1.lculation result for 'I of f/(q,"-') fo r the syst em of Eq . (36) with nomina l gain [( == 1.0 becomes as sh own in Fig. 0.25 1. The meanin g of 9. ""hen h :::: 0.05. 11(Qol wo) thi8 result is delicatE' . \Vhcll a noulin ear charact eristi c is a polygon al line as shown in Fig. 10( a) t.he sed.or lie~ be l.wel' ll ,,' 1 = 0.6 iLnd 1\2 :::: 1.6. Though th~ gain hand satisfi es t.h e st.a bility conditlu n of linear syst.ems in Eq. (37), TluX>l'e m - 1 is not. valid . T he respom. l' of e(k ) aud O'( k ) at this t.in ", oscillat es as s hown in Fig . IO(b) . Th at is, EXCLlllp le- 2 might corresp ond to a counter examp le of Aizerm all's conjt"ctul'e ('xtend ed into discret.e -tim e syst.em.
=
1
Since tlw rohust- s l.ahi lity (ondit.io ll which corresp onds to q = 0 in t hese s},8t t'rns. i. e.,
(J3 ) is Cl stab ility conditio u for t.he gencral time-va rying system , it is ap provf:'d as a su ffi cie nt co ndi t ion in au y of n u meri cal ex amples helow .
3442
7. CONCLUSIONS This paper has analyzed, in the frequency domain: the problem of the robust stabilit.y in the nonlinear sampleddata control system as a natural expansion of the Popa\' criterion for continuous time, Thcn. t.he allowable region of an uncertain l10nlinear characteristic has been given explicitly. Howcvn, t.he st.ability criterion (Theorem1) is valid only for t.he system,:.; to satisfy the previous AssulTIpt.ion-l, 2, and:L Naturally, Assumption-l is valid for a non linear ('hal'acteri~ti(' such as the pn:vious exa.mple wit.h a. lineal' dw.racterist.ic around t.he origin. For Assumption-a, a sampling period h which allows it to remain approximately va.lid can be chosen, as in the case of the usual lineal' sampled-data control system. Therefore, t.his assumption is not. highly significant. Aft.er all, the validit.,y· of t.he result. of t.hi.':! paper depends on whet.her Assumpt-ion-2 holds or not. However, as is obvious from t.1lt' prcvi(\us t.wo examples, it. is unquest.ionable that. Assurnpt.ion-2 is considered to be valid for ;1. usual Ilonlinear (,Ul've whieh is bf!nt. slightly.
REFERENCES Athert.Oll. D. P. (19tH) . .':-r'ta,bilit!/ of lv'onlinear· System, Research Studies Press. Desoer, C'. A. and fvI. Vidyasagar (H)75). Feedback S:r;stem: [nput-Output Propertu_s, Academic Press. Grujic. L. T. (19tll). On Ab,olu\c St.ability and the Aizerma.1l Conjecture. A'lLtomatiw, pp. :l3.5-349. Rarri" C . .I. and.J. M. F:. Val,'un, (1983). The Stability of Input-Output D:IJ'fJ(J.'mimi System,,>, Academic
0.5
- 0,5
(h) Pig. 7 Error sequence and trapczoidal area for Exa.mple-I.
r
1
(j.5
i'f____ _/=/~~~=-+-----" (_b_)
Fig. 8 Error sequence and trapezoidal area for Example-I.
"Jn::!n
Pt'es~.
Kalman, R. E. (l9;}7). Nonlill(';:u Aspect.s of SampledData Cont.rol SysLems. Pruc. vftAr Symp. on lv'onlineal· Circuit ;1n(l./ysls, pp 27:~-:H:l. OkuywIIa, Y. (1988). Rohm;t, StabilJty of Feedback Control System:; Conta.ining all r Ilrcrtain N onlinearity. Trans. of the i-ru;titu/( of ,''';ystons, Control and InfOT·m.a_twn EnghlffT8, pp. D-16.
'I(q, __">
" h=O.~::=OI}l
h=O.Ose h _ 0.1
, r=-
,
,
,
•
q ,
I
APPENDIX
Fig. 9 l/(q,W) cul'v'c:s for Exarnple-2. I~ ~
/
I
'" (a) 1-C+--~--2 ----~-,---~---;cq-;,1
(b) Fig. 10 Error sequenrc .cllld trapezoidal area for EXillrlplc-2.
Fig. 6 '11('1,'..1.1) curvps for Example-I.
3443