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Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state
NONLINEAR FEEDBACK CONTROL FOR OPERATING A NONISOTHERMAL CSTR NEAR AN UNSTABLE STEADY STATE D D BRUNSt and J E BAILEY Department of Chemical Engmeenng, Umverslty of Houston, Houston, TX 77004,U S,A (Recewed
2 March 1976,accepted 24 May 1976)
Abstrae-Appbcat~on of relay feedback controllers to mamtam process states near unstable steady states 1s extended to a secondordersystem.the CSTR w&t an ureverslble,exotbernucreaction The controlscheme1s successfulprovided the bandanth (the reactor temperature devotion wtuch causes relay swltcbmg) ISnot chosen too large relative to the dnve levels (the deviation m coolant temperature caused by relay swttchmg) In tlus context Tsypkm’s analytical methods for approxunate solution of tlus and higher order probtems are elaborated and tested Tsypkm’s techmque proves supenor to JJescmbmg Function analysts 111accuracy and m bounds on the maxamum
pemnsslble bandwidth INTRODUCTION
Reactor aperation near an unstable steady state can provide an optunum comprormse between conversion and selectlvlty or between catalyst actlvlty and longevrty These and other examples of performance advantages at unstable states were reviewed m an earlier paper [l] Also considered m that work were feedback control stratemes for mamtammg the reactor state near the desved unstable steady state Although conventional hnear controllers are sometnnes adequate for thy task (see [2]), occas~onaliy practical dficulties prevent Implementation of these control methods [ 1,3] An alternative control strategy employmg no&near feedback control IS avadable [l] Usmg a relay wtth hysteresis as the control element, the closed loop system IS designed to yield a stable, small amphtudelmnt cycle m the neighborhood of the desued operatmg pomt In many Instances, such a response IS an acceptable substitute for operation at the desired pomt Two approxnnate analyt+ cai methods, Descnbmg Function (OF) analysis and Tsypkm’s method, are aviulable for design of the relay Based on apphcation of both of these methods to a first order system it appears that Tsypkm’s method yields far more accurate results than the DF approach[l] Consequently, emphasis m tis work wti be lmd on Tsypkm’s techmques Besides dlustratmg apphcation of the method to a more mterestmg and complex example than m[l], such emphasis WIII also permit more thorough presentation of the avadable mathematical tools based on Tsypkm’s work This 1s an vngortant conslderatlon smce this useful approach 1s apparently unfamti to most chenucal engmeers For the one-dnnenslonal CSTR (contmuous flow stu-red tank reactor) consldered in the orq3mal exposition of the nonhnear control scheme [l], many of the dynanuc features can be ngorously determmed from a now standard graphcal techmque By exammmg a van Heerden-type dragram, with removal rate and appearance rate simultaneously plotted agamst concentration, the
of Tennessee,
Knoxvdle,
REACTOR
MODELS AND RELAY CONTROL DBSIGN
dynamic CSTR model 1s usually wrrtten m the followmg dunensionless form The
$7 = 1 -
3
+
DafiW
Engmeermg, TN 37916, U S A
y 257
ey(l-“*)
+ v(W - 9)
(2)
where W, 9, LPIand t are the dImensIonless reactant concentration, reactor temperature, coolant temperature, and time, respectively, and the standard dunenslonless parameters are defined m the Notation Multiple steady stat+ aflSe m the system for certam sets of parameter values, mcludmg the followmg values which wdl be employed III ths work Da=OO637,
Wurrent address Chemical and Metallur~cal University
SW of the time dertvative of concentration at any concentration follows duectly T~LS feature makes design of the relay control element &a&forward It has been estabhshed for some tune, however, that ih~~graphcal remhtion of process dynanucs 1s not correct for systems described by two or more state varmbles Tlus rases the followmg questions Can the nonhnear feedback control strategy, which rehed so heavdy on the van Heerden warn for mterpretation and development m its 1rntlaIexposition. be adapted and apphed to second and h&er order processes7 What new pttfalls anse when there are two or more state varrables? As an i~tial exploration of these Issues, thus work considers the two state vanable reactor most deeply embedded m the annals of chemical reactor theory the CSTR ~th external coohng m wbch a single vreverslble exothermlc reaction occurs The next section recapitulates the standard model for this system, wluch IS then reduced to the lmeanzed form reqmred m the approxunate control design procedure
=377,
@=0167
v=O1356,
Q&=1048
(3)
D
258
D BRUNS and 3 E
We the general dynanucal features of this reactor have already been rather thoroughly mvesmted[4], tt ts mstructive m the current context to examme the behavior of the process 111the phase plane The control oblective IS mamtenance of the system state m the nerghborhood of the unstable steady state labelled S 111i%g 1 why% corresponds to Q = 9, Thus 1s to be achieved through mampulation of % m a fashion to be outhned next. Notice m FQ 1 that the separati, whch 1s mdscated by a dashed hne, 1s never crossed by a trajectory and, us a sense, repels all states not on the separati Focusmg now on a small neighborhood of the phase plane near S, consider the lntluence of small changes ~u~(u, > 0) m 0 Increasmg (ec to a value Q+ = c
+ &,
(4)
BAILEY
that the motion of the state so achieved remams near the om unstable state S, the generahzed control obJectWe mentioned above WINbe fuliilled Further mvesQ@on of tlus matter follows formahzation of the control strategy The control approach just descllbed ISpractical only If it can be reduced to a set of rules, preferably IIIfeedback form For tlus reactor, the previous phase plane scenarro suggests the followmg scheme mvolvmg feedback of reactor temperature
I
%
Q=
for
O<$-y*
and
~~-y*C~<~,+Y*
‘p1_ for
%>%+y*
c-y3 If
(any%
=>o dr
(6)
sMts the separatnx to the left, wlule the separatnx moves to the r&t when ‘11IS decreased to c_=%~-c
(3
(see FQ 2) Clearly, for the situation shown m Fig 2 to occur, u, must be chosen sufficiently small so that three steady states exist for both %+ and LpL_Smce separatrrces tend to repel the state, It follows that, vvlth Q = Q_, states between the %- and %+ separatices v&l eventually move towards the 91, separatrw If Q IS switched to %!+ before the 9+ separatnx 1s crossed, the state will ultunately return towards the ceC_separam SwWhmg back to Q_ at an appropIlate pomt followed by sllltable alternation between 91_ and ‘p1,would then trap the state forever between the correspondmg separatices Provided
These are sunply the definmg equations for a relay with hysteresis m which a switch from %_ to Q, occurs when 0 decreases to iV*- y *, and the reverse sw&ch happens when 3 increases to the value 9, + y * In order to mvestigate Uus control scheme analyt~ally, hnemtion of the reactor model IS necessary As &scussed m the earher paper on tlus toplc[l], the hnemtion approxunation Is not expected to introduce serious error so long as devlatrons of varmbles which appear m the model are sulliclently small Lmeanzmg the model of eqns (1) and (2) about the unstable state (W,, %, 4!&) yields dw dt = a11w + &zY (7 1)
.
UNSTABLE
STEADY
STATE
-----SEPARATRIX
096 DIMENSIDNLESS Fu
1 Phase
104
I IO
I 16
TEMPERATURE
plane dlustrat~oo of CSTR dynamtcs for the parameters of eqns (3) The control mamtenance of the system state near S, the unstable steady state
objectwe
IS
Nonlmear feedback control for operatmg a nomsothermal CSTR near an unstable steady state
_
108
107
106
SEPARATRIX
DIMENSIONLESS
109
TEMPERATURE
XGg 2 Separatnces correspondmg to %+ and %_ contine system tralectones to the band between them by appropruxte swtihmg between %+ and Qc_ Thus, small changes m 41 may be used to mamtam the process state near S The pomts labelled S+ and S_ are unstable steady states correspondmg to %r+and I- respectively The dotted traJectones correspond to system response near the %I- separatnx when 91 = I_. wlule the lmed traJectone8 demonstrate system dynarmcs near the 9, separati when 9r = 9s+
dy ;ii=a2,w+a,y+bu,
(7 2)
where w, y and I( are the devlatlons of reactant concentration, reactor temperature, and coolant temperature from thev respective values at the unstable steady state S of mterest The coefficients m these equations are obtamed 111standard fashion Asswung zero uuti values for w and y pernuts development from eqns (7) of the followmg transfer function relatmg the Laplace transform Y of reactor temperature deviations to the transform U of coolant temperature devu&ons
Y(s)
W -ad
Uo=sZ-s(a,1+a22)+a,,a,-a,,a,,
k G(s)
(8)
With ths represent&on
of the lmear reactor model, the overall structure of the closed loop control system is ready recapitulated in a block dmgram (Rg 3) AP?ROXUbtATE JJMIT CYCLE -ATION BY ‘ISWEWS MRITIOD
If the control scheme m Fw 3 a&eves the antxlpated effect descllbed above, osctllatrons m reactor temperature and concentration in the vicuuty of S(w = y = 0) WIU result Whde aualyt~al solutions for any such osc~&~t~ons are not possible for the ongmal nonhnear model, exact solutions for oscillatory states as well as necessary and sufliclent local stabMy conditions are available when the hnearued process descnption IS employed The basis for such analyses IS Tsypkn?s method[5,6] with a few addItional mgre&ents [7j
RELAY
LINEARIZED SYSTEM
Y(t)
U(t)
“r GW
Fii 3 Block wrn
I
of the nonlmcarfeedback control scheme The nonlmear control element IS a passive relay
w&b hysteretus, and the process has been approxunated by a Imear~A model to pernut control system aaalysur by Tsypkm and Descnbw Fun&on methods
D D BRUNS and 3 J BAILEY
260
Smce these mathematical tools had not appeared m the chenucal engmeenng hterature untd the previous paper on Uus control approach [ 11,It 1s appropmte to lscuss some aspects of tl~s method m detad Such attention IS necessary not only because these techruques are novel but also because, when applrcable, they appear to be much more powerful than the relatively famdlar Descnbmg Function (IX) analysis Tsypkm theory necessary con&tlons for an osculation m a system such as tis with a passive, symmetric relay (see [51) were cited m an earlier paper (eqns (24)-(26) III [l]) apphcation of these to the present problem yields
y&) waveform, which, accordmg to Tsypkm theory[6,7], may be wrrtten as
hn
[G&nil,)
y,(t) = q m .=P n
e-‘]
+ G-u(t)
(14)
Substltutmg the partial fraction expansion of G(= G1) into eqn (14) and usmg the summation formulae of Gelb and Vander Velde ([6], p 197) yields the desired result
(9 1) where (9 2)
g,ta,
q)
=acosh(m1/2)-cosh(ml/2q* cash (m/2)
Irslnh(vD-al))
g2(Q,‘) = 4
(10 1)
With Gs.= hm G(s)
(10 2)
I-
For this problem G, 1s zero so G, 1s Identical to G Expandmg G m eqn (8) m partial fractions followed by use of Gelb and Hen&son’s summation formulae (see [61, p 194) permits rewntmg of condlhons (9) XI the form
-panh 1
z!-&-z$&!&-~ c
2
(11 1)
where A, and A2 are the elgenvalues of the hneanzed system (1e the roots of the denonunator of G m eqn (8)) which have been assumed here and m the followmg to be d&mct Other formulae result for the case of repeated euenvalues The numbers ~~and K~are defined L-. OY K = HA, - a,,) K* = b;;;-;;;’ 1 64,-A*) ’
(12)
To determme the existence of osclllatlons 111the feedback system of Fe 3, eqn (11 1) ISsolved for the real frequencies a, (I = 0, 1,2, ) of the oscdlatlons, and the resultmg solutions must be tested 111mequahty (11 2) Proof of the existence of an osctiatory state y, with frequency Q from eqns (11) requves vefication of the “contuuuty condition” With a passive, symmetnc relay tlus con&tion is Y,wcY*
for
O=tCg,
1
(13)
where tune f = 0 is defined by a swatch of the relay to the posltlve dnve level Apphcabon of Ws test requires the
’
(16 1)
where & means that only odd n are to be mcluded m the summation, and G,(s) IS defined by Gl(s) = G(s) - G,
a~)
I) cash (lr7//2)
(16 2)
Fmally, there remams the question of stab&y of oscillatory solutions obtamed by the above procedures The theorems for mves0gatmg stabtity are summarized m the Append=, and proofs of these new results are aven in deli elsewhere171 In order to determme which case of the stability theorem apphes here, It is necessary to check the SI~RSof y before and after the swltchmg tune t = lr/fi, From eqn (14) and the theory of Fomer senes the followmg expressions for these denvatives may be obtamed [6,7]
For the present reactor example, apphcafion of the summation formula already mentioned allows the needed denvatives to be wntten convemently as y*(g)
= +,tanh$+K.tanh~-b] (18 1)
Y*(g)
= Y,(C)
+2&b
(18 2)
These expressions may be used to determine E=yn[n(g)]
%n[Y(Z)]
(19)
m the passive relay stabtity equation 1+
#?[G,(s)] + G-3 = 0 )I
(20)
Nonhnear feedback control for operatmg a nomsothermal
where 9 denotes the z-transform operator For talus reactor control example w&h G from eqn (8), the script-bracketed coefficient m eqn (20) may be wntten as PI
(21) where Ti = am, Local stab&y determmation now follows from the roots of eqn (20) accordmg to the theorem m the Appendur TsYPxlNOsclLLAmRY ExAcTAImDEsc~
$oLrJTIONs AM) coMPANsoNs mJNcrIoNREsuLTs
WITE
Analytical inves@ahons of eqns (11) reveal that for u, > 0 and any set of parameters for which S is a saddle pomt or node, If a real solutmn Q to eqn (11 1) exists, It always satisfies eqn (112) and It IS umque provided the followmg necessary and sufficient con&tion IS met
sgn hII=
sgn CKJ or
(23)
I4 > 14
where the algentralues A, and A2 arc labelled such that j11,ic ]&I Under the above con&tions of umqueness, further analysis shows that If ay/& = L, > 0, a passive relay with hysteresis must be employed for the controller If ctitrol IS to be aeheved For equations analogous to eqns (11) and the control system of Fig 3 contammg an active relay, It IS agam found that any real solution to the equality always satisfies the mequality v&h conQhon (23) provldmg a necessary and sutkent cntenon for umqueness When con&fion (23) is satisfied and aylau = b < 0, rt can be demons&a&d that the controller must be an acWe relay element If a peflo&c solution Is to exist Apphcatmn of these results to the steady state S of Interest (parameters of eqns 3) reveals that when a real solution to eqn (11 1) exists, It IS umque and satisfies Table
1 Sunulation
Cond&on (112) Table 1 mdicates the frequenc=s i& obtamed from eqn (11 1) for four Merent sets of control parameters In each case, stabMy of the Tsypkm SoEtlhon was vetied by numerical evaluation of the roots of eqn (20) Also shown m Table 1 IS a,, the frequency of the oscfiation obtamed by sunulabon usmg the fill nonlmezar model of eqns (1) and (2) m place of 1t.s hneanzed descnption, eqn (8) nrrP IS the frequency md~cated by Descnbmg Funtion analysis, an approprmte techmque, whnzhm add&on to hneanzation of the system, mvolves “harmomc hnmzation” of the relay control elezment (see[l, 71) Comparrsons among the various Q and L suggest that system lmearrzatlon does not mtroduce senous error, wMe &screpancles between 0, and fL reveal weaknesses in the harmomc hnatIon approxlmatmn These findmgs are consistent wxth those m[l] m wluch a sunpler system was consldered Analogous conclusions emerge from the oscdlafion amphtudes computed usmg smmlatmn (Ad, Describmg Function analysis (A&, and Tsypkm’s method (Ab) for each set of control parameters To determme Ae, the waveform y,(t) must be evaluated usmg eqns (15) and (16) F-e 4 dlustrates one osctiatory response so obtamed as well as the correspondmg DF and smmlatmn solution for tis example Agam the supenonty of Tsypkm analysis to the DF techmque is clear NoWe that,
_ -----
F@
I I
“C
II ___---__
TSYPKW SIMULATION
4 Oscfiation waveforms sven by DF. Tsypkm, smulation calculatmns for y* = 0 00521 and IJ== 0 0216
(sun), Tsypkln (O), and Descnbmg Function (DF) results for oscillation frequency (aZ)for a vmety of bandwIdth (y*) and dme level (L) values CASE
261
CSTR near an unstable steady state
III
lx __
0 0108
0 0216
y*x 103
2 61
I45
5 21
2 90
A Sld IO3
3 01
I 48
6 05
2 96
Aox103
3 00
I 46
6 01
2 95
Ao,x IO’
4 30
I89
8 61
3 78
a SIM
0 893
165
0 694
I 65
amphtude
(A) and
and
D D BRUNS and J E BAILEY
262
111contrast to both approxnnate responses, the not&near srmulatmn result is shghtly asymmetrical about the y = 0 baschne, mctcatmg a sbght d@acement 111the mean reactor temperature from ?& The overshoot evident m the smmlatlon waveform m Rg 4 does not occur for first order systems ms unportant feature is perhaps best put m proper perspective by exammmg the correspondmg phase plane presentation (Fe 5) Notice that as the relay bandwdth IS increased, there w more of an overshoot m both temperature y and concentration w before the traJect.ory reverses and heads away from the repellmg separatnx The cruetal nature of tis me&a is dramatically tllustrated by Curve 3 where the system state closely approaches the a_ separatnx before the control level IS switched to u = -Us The traJectory then follows near the separatnx and approaches the S_ unstable steady state very closely before bemg pushed away If the system were operated m such a fashion, even smalI hsturbances of suJ3cient dun&on m feed temperature or concentration could cause the state of the system to be moved across the separam and control would be lost The unportance of higher order system dynarmcs 1s also shown m the relation&p between the separatrices and traJectones If the bandw&h IS too large-, the corrcspondmg traJectory wdl escape the regon contamed by the 4~ and Q, separatrices,, and the control scheme fads Consequently, a maJor ObJectWein deslgntng nonhnear feedback controls for thy and other h@er order systems IS determmabon of the upper lumt on the bandvvldth whch pernnts reahzatlon of the desired oscdlatory state Such lmuts are most conveniently formulated m terms of the temperature deviations at the unstable steady states S+ and S_ correspondmg to ‘%!, and 4!L respectively According to the lmeanzed model, these deviations are my, where Ye =
bucall (alla22- a12a21)
‘1,. -1 w ‘\
IO
aUNSTABLE
(24)
STEADY
- ---
SEPARATRICES
-
LIMIT
STATE!
CYCLES
Evidently, because of second order system dynanucs, the ratio of swrtchmg band width y * to yc a=$
(25)
must be less than umty to preserve confrdl and obtam a peno&c state near S So long as con&tion (23) ISsatified, lmphclt Mere&ation of eqn (11 1) shows that as y * 1s mcreased, & decreases Tummg tis around, the upper lmut of bandwdth yL wbch provides an oscillatory response according to Tsypkm theory ISobtained by computmg the lout of eqn (11 1) as 0 approaches zero (a smular statement holds for an active relay) The result, followmg substitutions for A.I)At, K, and K~ m terms of the lmeanzed model coefficients, 1s
Notice that ths lumt on bandvvldth 1s a function of steady state operatmg con&tlons For those of thus example and thus all cases of Table 1, the value of S_, IS 0 644 The exact values for the upper hrmt of 8 determmed by simulation are 0 617 for cases I and II and 0 632 for cases III and IV Contrastmg VW&~ vvlth the accuracy of the Tsypkm approxuuation ISthe Descnbmg Functton result numerical stties of the DF equations appropriate to Ws problem revealed S_,, = 0 434 for all cases of Table 1
CONCLUDE’JG DISCUSSION
accurate anaIy&al upper lumt on relay bandwidth gtven m eqn (26) ISa strong attestauon to the power of the Tsypkm approach for design of no&near feedback controllers for chenucal process operation near an unstable steady state. In thus paper, all maJor aspects of Tsypkm theory mcludmg several recent extensions[7] have been described,, and the methods have been apphed to a classical system m chemical reactor theory Another unportant extension of [ 1] provided here ISreplacement of the van Heerden &agram mterpretatlon of the control scheme with a phase plane view Thus latter prespectwe is amenable to relatively dvect extension to h&er order systems In fact, investtgatmg apphcation of the control idea espoused here and m[ 1] to considerably more complex chemical processes 1s an mterestmg area for The
future study One key problem m tis connection 1s the relationship between aggregation or lumping approxunations and the smtablhty of the resultmg feedback relay controller Acknowledgements-The
Fw 5 Controlsystem osc~&tlonsm the phase plane show the effect of lncreasmg the bandwIdth. y*. for II,= 0 0216 and y* = 0 00290 (Case I of Table 1, 6 = 0 5 6,, Curve 11, y+=OOO521 (Case II of Table 1, 6=09 dmuDp, Curve 2), y* = 0 00843 (6 = 6_,
curve 3)
authors were supported in part by NSF, by the Drepfus Foundation, and by the Uruverslty of Houston D D Bruns gratefullyacknowledgesfinancialsupport from a Stella L Ebrbardt Fellowshp and an NSF Graduate Trameeshp NOTATION
amphtude SB heat transfer area
A
oscfflation
Noolmear feedback control for operatmg a nomsothermal CSTR near an unstable steady state
cctefficlents m the lmeanzed reactor
model reactant concentration spectic heat DamMhler number (= V.kO e-* /F) activation energy feed and efauent volumetnc flow rate reactor transfer function (eqn 8) functions defined by eqns (16) defined m eqns (10) z-transform of G(s) advanced z-transform of G(s) heat transfer coefficient magm=y part Ii-1 pre-exponential factor gas law constant real part Laplace transform parameter unstable steady state slgnum function (=F (real dunenslonless tune tm=)/V) T reactor temperature TCool coolant temperature u dunenslonless coolant temperature deviation u Laplace transform of u Q dunenslonless coolant temperature (= T,,/T,) V reactor volume W dunenslonless concentration devmtion W dunensionless concentration (= c Ic,) reactor temperature Y dunensionless devlatlon Y Laplace transform of y 3 dunenslonless reactor temperature (= TIT,) z-transform parameter z-transform operator advanced z-transform operator
Greek symbols B
dunenslonless
admbak
temperature
263
Subscripts DE f max S
Sllll
0 +. -
pertamngto relay dnve levels descnbmg functmn feed con&bon pertammg to the zth solution from the Tyspkm method mkcates maxunum value which allows control to be acheved pertammg to the reference unstable steady state s simulation the fist oscdlatory solution from Tyspkm’s analysis corresponrlmg to positive and negative coolant temperature deviations
Superscripts +,-
*
r&t and left hand lmuts, respectively pertammg to the svvltchmg pomts of the relay
REFERENc!Es
[ll Bruns D D and Badey J E , Chem Engng - - SCI 1975 30 7S5 [2] Ans R and Amundson N R , Cfiem Engng Scr 19587 132 131Westertexu K R . Chem Enana - - Scr 1962 17 423
i4i Uppal A ,kay W OH and Poore A B , Chem Engng
cient (=
density
j$$-)
1974
Hsu J C and Meyer A U , Modem Control PnncrDfes and _ Applxatrons Mck-aw-Hdl.New York 1%8 161 _ _ Gelb A and Vander Velde W E . Multde-Inout Lhwcnbnw Functions and Nonlrnear System krgn- Mc&aw-H111, New
[5l
York 1968 [7] Bruns D D , Ph D Thesis, Umversdy of Houston, Houston, Texas 1974
Stab&y analysts of Tsypkm approximate solutwns The theory of lmear sampled-data systems provides the
followmg theorem TheoremA 1 For a closed loop sampled-data system m the
form of Fw Al, let the lmear element of the system, G(s), be gwen by an ordmary tune-mvanant lmear Mere&al equation whose states are completely controllable and observable wltb K, the gam constant of the system and 7. the samphng penod If (1) for all m, (9,(z,m) has the same set of poles as S?*(z,0) = g(z), where L(z, m) and (4(z) are the advanced z-transform and z-transform of G(s) respectively[5]. and If (2) all the zeros, z,, of 1+ K.%(z) = 0
energy dunensionless activation (=E/RTI) dlmenslonless parameter defined m eqn (25) heat of reaction number evaluated usmg eqn (19) defined m eqns (12) linearized system elgenvalues dnnenslonless heat transfer coeffi-
SCJ
29 %7
(Al)
he m IzIC 1, the ongm of the state space IS stable If any of the zeros he m lzj > 1 the ongm wdl be unstable
1 Fa Al
Schematic block dmgram of lmear sample data feedback system
By appropnately choosmg 7. and K. [7]. further analysis [5,7] leads to the development of Theorem A2 Theorcrn A2 The local asymptotic stabtity of an oscdlation of
264
D D
BRUNS
and3
frequency t2, ISto be determmed for a relay controlled feedback system contauung a symmetnc reiay with hysteresis havmg drwe levels of CU, and a hnear transfer functmn, G(s), havmg at least as many poles as zeroes Usmg the nomenclature of FM Al, 7. and K, are taken as
E
BALIZY
with c defined accordrelay element [sgn[e,(g)]
to the passive or achve nature of the
sgn[e($)]
forapasswerelay
l =
(A21 and (A31
-sgn [e,(g)]
sgfa [ e,(g)]
for an active relay
‘hen, the penodw solutzon Hrlllbe asymptotically stable to small disturbances d all condhons of Theorem Al are satisfied except for a smgle zero at 2 = -1
2 March 1976,accepted 24 May 1976)
Abstrae-Appbcat~on of relay feedback controllers to mamtam process states near unstable steady states 1s extended to a secondordersystem.the CSTR w&t an ureverslble,exotbernucreaction The controlscheme1s successfulprovided the bandanth (the reactor temperature devotion wtuch causes relay swltcbmg) ISnot chosen too large relative to the dnve levels (the deviation m coolant temperature caused by relay swttchmg) In tlus context Tsypkm’s analytical methods for approxunate solution of tlus and higher order probtems are elaborated and tested Tsypkm’s techmque proves supenor to JJescmbmg Function analysts 111accuracy and m bounds on the maxamum
pemnsslble bandwidth INTRODUCTION
Reactor aperation near an unstable steady state can provide an optunum comprormse between conversion and selectlvlty or between catalyst actlvlty and longevrty These and other examples of performance advantages at unstable states were reviewed m an earlier paper [l] Also considered m that work were feedback control stratemes for mamtammg the reactor state near the desved unstable steady state Although conventional hnear controllers are sometnnes adequate for thy task (see [2]), occas~onaliy practical dficulties prevent Implementation of these control methods [ 1,3] An alternative control strategy employmg no&near feedback control IS avadable [l] Usmg a relay wtth hysteresis as the control element, the closed loop system IS designed to yield a stable, small amphtudelmnt cycle m the neighborhood of the desued operatmg pomt In many Instances, such a response IS an acceptable substitute for operation at the desired pomt Two approxnnate analyt+ cai methods, Descnbmg Function (OF) analysis and Tsypkm’s method, are aviulable for design of the relay Based on apphcation of both of these methods to a first order system it appears that Tsypkm’s method yields far more accurate results than the DF approach[l] Consequently, emphasis m tis work wti be lmd on Tsypkm’s techmques Besides dlustratmg apphcation of the method to a more mterestmg and complex example than m[l], such emphasis WIII also permit more thorough presentation of the avadable mathematical tools based on Tsypkm’s work This 1s an vngortant conslderatlon smce this useful approach 1s apparently unfamti to most chenucal engmeers For the one-dnnenslonal CSTR (contmuous flow stu-red tank reactor) consldered in the orq3mal exposition of the nonhnear control scheme [l], many of the dynanuc features can be ngorously determmed from a now standard graphcal techmque By exammmg a van Heerden-type dragram, with removal rate and appearance rate simultaneously plotted agamst concentration, the
of Tennessee,
Knoxvdle,
REACTOR
MODELS AND RELAY CONTROL DBSIGN
dynamic CSTR model 1s usually wrrtten m the followmg dunensionless form The
$7 = 1 -
3
+
DafiW
Engmeermg, TN 37916, U S A
y 257
ey(l-“*)
+ v(W - 9)
(2)
where W, 9, LPIand t are the dImensIonless reactant concentration, reactor temperature, coolant temperature, and time, respectively, and the standard dunenslonless parameters are defined m the Notation Multiple steady stat+ aflSe m the system for certam sets of parameter values, mcludmg the followmg values which wdl be employed III ths work Da=OO637,
Wurrent address Chemical and Metallur~cal University
SW of the time dertvative of concentration at any concentration follows duectly T~LS feature makes design of the relay control element &a&forward It has been estabhshed for some tune, however, that ih~~graphcal remhtion of process dynanucs 1s not correct for systems described by two or more state varmbles Tlus rases the followmg questions Can the nonhnear feedback control strategy, which rehed so heavdy on the van Heerden warn for mterpretation and development m its 1rntlaIexposition. be adapted and apphed to second and h&er order processes7 What new pttfalls anse when there are two or more state varrables? As an i~tial exploration of these Issues, thus work considers the two state vanable reactor most deeply embedded m the annals of chemical reactor theory the CSTR ~th external coohng m wbch a single vreverslble exothermlc reaction occurs The next section recapitulates the standard model for this system, wluch IS then reduced to the lmeanzed form reqmred m the approxunate control design procedure
=377,
@=0167
v=O1356,
Q&=1048
(3)
D
258
D BRUNS and 3 E
We the general dynanucal features of this reactor have already been rather thoroughly mvesmted[4], tt ts mstructive m the current context to examme the behavior of the process 111the phase plane The control oblective IS mamtenance of the system state m the nerghborhood of the unstable steady state labelled S 111i%g 1 why% corresponds to Q = 9, Thus 1s to be achieved through mampulation of % m a fashion to be outhned next. Notice m FQ 1 that the separati, whch 1s mdscated by a dashed hne, 1s never crossed by a trajectory and, us a sense, repels all states not on the separati Focusmg now on a small neighborhood of the phase plane near S, consider the lntluence of small changes ~u~(u, > 0) m 0 Increasmg (ec to a value Q+ = c
+ &,
(4)
BAILEY
that the motion of the state so achieved remams near the om unstable state S, the generahzed control obJectWe mentioned above WINbe fuliilled Further mvesQ@on of tlus matter follows formahzation of the control strategy The control approach just descllbed ISpractical only If it can be reduced to a set of rules, preferably IIIfeedback form For tlus reactor, the previous phase plane scenarro suggests the followmg scheme mvolvmg feedback of reactor temperature
I
%
Q=
for
O<$-y*
and
~~-y*C~<~,+Y*
‘p1_ for
%>%+y*
c-y3 If
(any%
=>o dr
(6)
sMts the separatnx to the left, wlule the separatnx moves to the r&t when ‘11IS decreased to c_=%~-c
(3
(see FQ 2) Clearly, for the situation shown m Fig 2 to occur, u, must be chosen sufficiently small so that three steady states exist for both %+ and LpL_Smce separatrrces tend to repel the state, It follows that, vvlth Q = Q_, states between the %- and %+ separatices v&l eventually move towards the 91, separatrw If Q IS switched to %!+ before the 9+ separatnx 1s crossed, the state will ultunately return towards the ceC_separam SwWhmg back to Q_ at an appropIlate pomt followed by sllltable alternation between 91_ and ‘p1,would then trap the state forever between the correspondmg separatices Provided
These are sunply the definmg equations for a relay with hysteresis m which a switch from %_ to Q, occurs when 0 decreases to iV*- y *, and the reverse sw&ch happens when 3 increases to the value 9, + y * In order to mvestigate Uus control scheme analyt~ally, hnemtion of the reactor model IS necessary As &scussed m the earher paper on tlus toplc[l], the hnemtion approxunation Is not expected to introduce serious error so long as devlatrons of varmbles which appear m the model are sulliclently small Lmeanzmg the model of eqns (1) and (2) about the unstable state (W,, %, 4!&) yields dw dt = a11w + &zY (7 1)
.
UNSTABLE
STEADY
STATE
-----SEPARATRIX
096 DIMENSIDNLESS Fu
1 Phase
104
I IO
I 16
TEMPERATURE
plane dlustrat~oo of CSTR dynamtcs for the parameters of eqns (3) The control mamtenance of the system state near S, the unstable steady state
objectwe
IS
Nonlmear feedback control for operatmg a nomsothermal CSTR near an unstable steady state
_
108
107
106
SEPARATRIX
DIMENSIONLESS
109
TEMPERATURE
XGg 2 Separatnces correspondmg to %+ and %_ contine system tralectones to the band between them by appropruxte swtihmg between %+ and Qc_ Thus, small changes m 41 may be used to mamtam the process state near S The pomts labelled S+ and S_ are unstable steady states correspondmg to %r+and I- respectively The dotted traJectones correspond to system response near the %I- separatnx when 91 = I_. wlule the lmed traJectone8 demonstrate system dynarmcs near the 9, separati when 9r = 9s+
dy ;ii=a2,w+a,y+bu,
(7 2)
where w, y and I( are the devlatlons of reactant concentration, reactor temperature, and coolant temperature from thev respective values at the unstable steady state S of mterest The coefficients m these equations are obtamed 111standard fashion Asswung zero uuti values for w and y pernuts development from eqns (7) of the followmg transfer function relatmg the Laplace transform Y of reactor temperature deviations to the transform U of coolant temperature devu&ons
Y(s)
W -ad
Uo=sZ-s(a,1+a22)+a,,a,-a,,a,,
k G(s)
(8)
With ths represent&on
of the lmear reactor model, the overall structure of the closed loop control system is ready recapitulated in a block dmgram (Rg 3) AP?ROXUbtATE JJMIT CYCLE -ATION BY ‘ISWEWS MRITIOD
If the control scheme m Fw 3 a&eves the antxlpated effect descllbed above, osctllatrons m reactor temperature and concentration in the vicuuty of S(w = y = 0) WIU result Whde aualyt~al solutions for any such osc~&~t~ons are not possible for the ongmal nonhnear model, exact solutions for oscillatory states as well as necessary and sufliclent local stabMy conditions are available when the hnearued process descnption IS employed The basis for such analyses IS Tsypkn?s method[5,6] with a few addItional mgre&ents [7j
RELAY
LINEARIZED SYSTEM
Y(t)
U(t)
“r GW
Fii 3 Block wrn
I
of the nonlmcarfeedback control scheme The nonlmear control element IS a passive relay
w&b hysteretus, and the process has been approxunated by a Imear~A model to pernut control system aaalysur by Tsypkm and Descnbw Fun&on methods
D D BRUNS and 3 J BAILEY
260
Smce these mathematical tools had not appeared m the chenucal engmeenng hterature untd the previous paper on Uus control approach [ 11,It 1s appropmte to lscuss some aspects of tl~s method m detad Such attention IS necessary not only because these techruques are novel but also because, when applrcable, they appear to be much more powerful than the relatively famdlar Descnbmg Function (IX) analysis Tsypkm theory necessary con&tlons for an osculation m a system such as tis with a passive, symmetric relay (see [51) were cited m an earlier paper (eqns (24)-(26) III [l]) apphcation of these to the present problem yields
y&) waveform, which, accordmg to Tsypkm theory[6,7], may be wrrtten as
hn
[G&nil,)
y,(t) = q m .=P n
e-‘]
+ G-u(t)
(14)
Substltutmg the partial fraction expansion of G(= G1) into eqn (14) and usmg the summation formulae of Gelb and Vander Velde ([6], p 197) yields the desired result
(9 1) where (9 2)
g,ta,
q)
=acosh(m1/2)-cosh(ml/2q* cash (m/2)
Irslnh(vD-al))
g2(Q,‘) = 4
(10 1)
With Gs.= hm G(s)
(10 2)
I-
For this problem G, 1s zero so G, 1s Identical to G Expandmg G m eqn (8) m partial fractions followed by use of Gelb and Hen&son’s summation formulae (see [61, p 194) permits rewntmg of condlhons (9) XI the form
-panh 1
z!-&-z$&!&-~ c
2
(11 1)
where A, and A2 are the elgenvalues of the hneanzed system (1e the roots of the denonunator of G m eqn (8)) which have been assumed here and m the followmg to be d&mct Other formulae result for the case of repeated euenvalues The numbers ~~and K~are defined L-. OY K = HA, - a,,) K* = b;;;-;;;’ 1 64,-A*) ’
(12)
To determme the existence of osclllatlons 111the feedback system of Fe 3, eqn (11 1) ISsolved for the real frequencies a, (I = 0, 1,2, ) of the oscdlatlons, and the resultmg solutions must be tested 111mequahty (11 2) Proof of the existence of an osctiatory state y, with frequency Q from eqns (11) requves vefication of the “contuuuty condition” With a passive, symmetnc relay tlus con&tion is Y,wcY*
for
O=tCg,
1
(13)
where tune f = 0 is defined by a swatch of the relay to the posltlve dnve level Apphcabon of Ws test requires the
’
(16 1)
where & means that only odd n are to be mcluded m the summation, and G,(s) IS defined by Gl(s) = G(s) - G,
a~)
I) cash (lr7//2)
(16 2)
Fmally, there remams the question of stab&y of oscillatory solutions obtamed by the above procedures The theorems for mves0gatmg stabtity are summarized m the Append=, and proofs of these new results are aven in deli elsewhere171 In order to determme which case of the stability theorem apphes here, It is necessary to check the SI~RSof y before and after the swltchmg tune t = lr/fi, From eqn (14) and the theory of Fomer senes the followmg expressions for these denvatives may be obtamed [6,7]
For the present reactor example, apphcafion of the summation formula already mentioned allows the needed denvatives to be wntten convemently as y*(g)
= +,tanh$+K.tanh~-b] (18 1)
Y*(g)
= Y,(C)
+2&b
(18 2)
These expressions may be used to determine E=yn[n(g)]
%n[Y(Z)]
(19)
m the passive relay stabtity equation 1+
#?[G,(s)] + G-3 = 0 )I
(20)
Nonhnear feedback control for operatmg a nomsothermal
where 9 denotes the z-transform operator For talus reactor control example w&h G from eqn (8), the script-bracketed coefficient m eqn (20) may be wntten as PI
(21) where Ti = am, Local stab&y determmation now follows from the roots of eqn (20) accordmg to the theorem m the Appendur TsYPxlNOsclLLAmRY ExAcTAImDEsc~
$oLrJTIONs AM) coMPANsoNs mJNcrIoNREsuLTs
WITE
Analytical inves@ahons of eqns (11) reveal that for u, > 0 and any set of parameters for which S is a saddle pomt or node, If a real solutmn Q to eqn (11 1) exists, It always satisfies eqn (112) and It IS umque provided the followmg necessary and sufficient con&tion IS met
sgn hII=
sgn CKJ or
(23)
I4 > 14
where the algentralues A, and A2 arc labelled such that j11,ic ]&I Under the above con&tions of umqueness, further analysis shows that If ay/& = L, > 0, a passive relay with hysteresis must be employed for the controller If ctitrol IS to be aeheved For equations analogous to eqns (11) and the control system of Fig 3 contammg an active relay, It IS agam found that any real solution to the equality always satisfies the mequality v&h conQhon (23) provldmg a necessary and sutkent cntenon for umqueness When con&fion (23) is satisfied and aylau = b < 0, rt can be demons&a&d that the controller must be an acWe relay element If a peflo&c solution Is to exist Apphcatmn of these results to the steady state S of Interest (parameters of eqns 3) reveals that when a real solution to eqn (11 1) exists, It IS umque and satisfies Table
1 Sunulation
Cond&on (112) Table 1 mdicates the frequenc=s i& obtamed from eqn (11 1) for four Merent sets of control parameters In each case, stabMy of the Tsypkm SoEtlhon was vetied by numerical evaluation of the roots of eqn (20) Also shown m Table 1 IS a,, the frequency of the oscfiation obtamed by sunulabon usmg the fill nonlmezar model of eqns (1) and (2) m place of 1t.s hneanzed descnption, eqn (8) nrrP IS the frequency md~cated by Descnbmg Funtion analysis, an approprmte techmque, whnzhm add&on to hneanzation of the system, mvolves “harmomc hnmzation” of the relay control elezment (see[l, 71) Comparrsons among the various Q and L suggest that system lmearrzatlon does not mtroduce senous error, wMe &screpancles between 0, and fL reveal weaknesses in the harmomc hnatIon approxlmatmn These findmgs are consistent wxth those m[l] m wluch a sunpler system was consldered Analogous conclusions emerge from the oscdlafion amphtudes computed usmg smmlatmn (Ad, Describmg Function analysis (A&, and Tsypkm’s method (Ab) for each set of control parameters To determme Ae, the waveform y,(t) must be evaluated usmg eqns (15) and (16) F-e 4 dlustrates one osctiatory response so obtamed as well as the correspondmg DF and smmlatmn solution for tis example Agam the supenonty of Tsypkm analysis to the DF techmque is clear NoWe that,
_ -----
F@
I I
“C
II ___---__
TSYPKW SIMULATION
4 Oscfiation waveforms sven by DF. Tsypkm, smulation calculatmns for y* = 0 00521 and IJ== 0 0216
(sun), Tsypkln (O), and Descnbmg Function (DF) results for oscillation frequency (aZ)for a vmety of bandwIdth (y*) and dme level (L) values CASE
261
CSTR near an unstable steady state
III
lx __
0 0108
0 0216
y*x 103
2 61
I45
5 21
2 90
A Sld IO3
3 01
I 48
6 05
2 96
Aox103
3 00
I 46
6 01
2 95
Ao,x IO’
4 30
I89
8 61
3 78
a SIM
0 893
165
0 694
I 65
amphtude
(A) and
and
D D BRUNS and J E BAILEY
262
111contrast to both approxnnate responses, the not&near srmulatmn result is shghtly asymmetrical about the y = 0 baschne, mctcatmg a sbght d@acement 111the mean reactor temperature from ?& The overshoot evident m the smmlatlon waveform m Rg 4 does not occur for first order systems ms unportant feature is perhaps best put m proper perspective by exammmg the correspondmg phase plane presentation (Fe 5) Notice that as the relay bandwdth IS increased, there w more of an overshoot m both temperature y and concentration w before the traJect.ory reverses and heads away from the repellmg separatnx The cruetal nature of tis me&a is dramatically tllustrated by Curve 3 where the system state closely approaches the a_ separatnx before the control level IS switched to u = -Us The traJectory then follows near the separatnx and approaches the S_ unstable steady state very closely before bemg pushed away If the system were operated m such a fashion, even smalI hsturbances of suJ3cient dun&on m feed temperature or concentration could cause the state of the system to be moved across the separam and control would be lost The unportance of higher order system dynarmcs 1s also shown m the relation&p between the separatrices and traJectones If the bandw&h IS too large-, the corrcspondmg traJectory wdl escape the regon contamed by the 4~ and Q, separatrices,, and the control scheme fads Consequently, a maJor ObJectWein deslgntng nonhnear feedback controls for thy and other h@er order systems IS determmabon of the upper lumt on the bandvvldth whch pernnts reahzatlon of the desired oscdlatory state Such lmuts are most conveniently formulated m terms of the temperature deviations at the unstable steady states S+ and S_ correspondmg to ‘%!, and 4!L respectively According to the lmeanzed model, these deviations are my, where Ye =
bucall (alla22- a12a21)
‘1,. -1 w ‘\
IO
aUNSTABLE
(24)
STEADY
- ---
SEPARATRICES
-
LIMIT
STATE!
CYCLES
Evidently, because of second order system dynanucs, the ratio of swrtchmg band width y * to yc a=$
(25)
must be less than umty to preserve confrdl and obtam a peno&c state near S So long as con&tion (23) ISsatified, lmphclt Mere&ation of eqn (11 1) shows that as y * 1s mcreased, & decreases Tummg tis around, the upper lmut of bandwdth yL wbch provides an oscillatory response according to Tsypkm theory ISobtained by computmg the lout of eqn (11 1) as 0 approaches zero (a smular statement holds for an active relay) The result, followmg substitutions for A.I)At, K, and K~ m terms of the lmeanzed model coefficients, 1s
Notice that ths lumt on bandvvldth 1s a function of steady state operatmg con&tlons For those of thus example and thus all cases of Table 1, the value of S_, IS 0 644 The exact values for the upper hrmt of 8 determmed by simulation are 0 617 for cases I and II and 0 632 for cases III and IV Contrastmg VW&~ vvlth the accuracy of the Tsypkm approxuuation ISthe Descnbmg Functton result numerical stties of the DF equations appropriate to Ws problem revealed S_,, = 0 434 for all cases of Table 1
CONCLUDE’JG DISCUSSION
accurate anaIy&al upper lumt on relay bandwidth gtven m eqn (26) ISa strong attestauon to the power of the Tsypkm approach for design of no&near feedback controllers for chenucal process operation near an unstable steady state. In thus paper, all maJor aspects of Tsypkm theory mcludmg several recent extensions[7] have been described,, and the methods have been apphed to a classical system m chemical reactor theory Another unportant extension of [ 1] provided here ISreplacement of the van Heerden &agram mterpretatlon of the control scheme with a phase plane view Thus latter prespectwe is amenable to relatively dvect extension to h&er order systems In fact, investtgatmg apphcation of the control idea espoused here and m[ 1] to considerably more complex chemical processes 1s an mterestmg area for The
future study One key problem m tis connection 1s the relationship between aggregation or lumping approxunations and the smtablhty of the resultmg feedback relay controller Acknowledgements-The
Fw 5 Controlsystem osc~&tlonsm the phase plane show the effect of lncreasmg the bandwIdth. y*. for II,= 0 0216 and y* = 0 00290 (Case I of Table 1, 6 = 0 5 6,, Curve 11, y+=OOO521 (Case II of Table 1, 6=09 dmuDp, Curve 2), y* = 0 00843 (6 = 6_,
curve 3)
authors were supported in part by NSF, by the Drepfus Foundation, and by the Uruverslty of Houston D D Bruns gratefullyacknowledgesfinancialsupport from a Stella L Ebrbardt Fellowshp and an NSF Graduate Trameeshp NOTATION
amphtude SB heat transfer area
A
oscfflation
Noolmear feedback control for operatmg a nomsothermal CSTR near an unstable steady state
cctefficlents m the lmeanzed reactor
model reactant concentration spectic heat DamMhler number (= V.kO e-* /F) activation energy feed and efauent volumetnc flow rate reactor transfer function (eqn 8) functions defined by eqns (16) defined m eqns (10) z-transform of G(s) advanced z-transform of G(s) heat transfer coefficient magm=y part Ii-1 pre-exponential factor gas law constant real part Laplace transform parameter unstable steady state slgnum function (=F (real dunenslonless tune tm=)/V) T reactor temperature TCool coolant temperature u dunenslonless coolant temperature deviation u Laplace transform of u Q dunenslonless coolant temperature (= T,,/T,) V reactor volume W dunenslonless concentration devmtion W dunensionless concentration (= c Ic,) reactor temperature Y dunensionless devlatlon Y Laplace transform of y 3 dunenslonless reactor temperature (= TIT,) z-transform parameter z-transform operator advanced z-transform operator
Greek symbols B
dunenslonless
admbak
temperature
263
Subscripts DE f max S
Sllll
0 +. -
pertamngto relay dnve levels descnbmg functmn feed con&bon pertammg to the zth solution from the Tyspkm method mkcates maxunum value which allows control to be acheved pertammg to the reference unstable steady state s simulation the fist oscdlatory solution from Tyspkm’s analysis corresponrlmg to positive and negative coolant temperature deviations
Superscripts +,-
*
r&t and left hand lmuts, respectively pertammg to the svvltchmg pomts of the relay
REFERENc!Es
[ll Bruns D D and Badey J E , Chem Engng - - SCI 1975 30 7S5 [2] Ans R and Amundson N R , Cfiem Engng Scr 19587 132 131Westertexu K R . Chem Enana - - Scr 1962 17 423
i4i Uppal A ,kay W OH and Poore A B , Chem Engng
cient (=
density
j$$-)
1974
Hsu J C and Meyer A U , Modem Control PnncrDfes and _ Applxatrons Mck-aw-Hdl.New York 1%8 161 _ _ Gelb A and Vander Velde W E . Multde-Inout Lhwcnbnw Functions and Nonlrnear System krgn- Mc&aw-H111, New
[5l
York 1968 [7] Bruns D D , Ph D Thesis, Umversdy of Houston, Houston, Texas 1974
Stab&y analysts of Tsypkm approximate solutwns The theory of lmear sampled-data systems provides the
followmg theorem TheoremA 1 For a closed loop sampled-data system m the
form of Fw Al, let the lmear element of the system, G(s), be gwen by an ordmary tune-mvanant lmear Mere&al equation whose states are completely controllable and observable wltb K, the gam constant of the system and 7. the samphng penod If (1) for all m, (9,(z,m) has the same set of poles as S?*(z,0) = g(z), where L(z, m) and (4(z) are the advanced z-transform and z-transform of G(s) respectively[5]. and If (2) all the zeros, z,, of 1+ K.%(z) = 0
energy dunensionless activation (=E/RTI) dlmenslonless parameter defined m eqn (25) heat of reaction number evaluated usmg eqn (19) defined m eqns (12) linearized system elgenvalues dnnenslonless heat transfer coeffi-
SCJ
29 %7
(Al)
he m IzIC 1, the ongm of the state space IS stable If any of the zeros he m lzj > 1 the ongm wdl be unstable
1 Fa Al
Schematic block dmgram of lmear sample data feedback system
By appropnately choosmg 7. and K. [7]. further analysis [5,7] leads to the development of Theorem A2 Theorcrn A2 The local asymptotic stabtity of an oscdlation of
264
D D
BRUNS
and3
frequency t2, ISto be determmed for a relay controlled feedback system contauung a symmetnc reiay with hysteresis havmg drwe levels of CU, and a hnear transfer functmn, G(s), havmg at least as many poles as zeroes Usmg the nomenclature of FM Al, 7. and K, are taken as
E
BALIZY
with c defined accordrelay element [sgn[e,(g)]
to the passive or achve nature of the
sgn[e($)]
forapasswerelay
l =
(A21 and (A31
-sgn [e,(g)]
sgfa [ e,(g)]
for an active relay
‘hen, the penodw solutzon Hrlllbe asymptotically stable to small disturbances d all condhons of Theorem Al are satisfied except for a smgle zero at 2 = -1