E. Approache s t o Proces s Contro l

3.1 Basi c Contro l Concept s an d Definitions

Central to the control of any entity or system is tha concept of 'feedback'. On a formal basis, it is defined as tha process whereby the difference between the output of a system and a reference input is used to maintain a prescribed relation between that output and the reference. Feedback is, of course, an everyday phenomenon. Relative to engineering, its us* for the control of mechanical systems is generally considered to date from James Watt's invention of the ball- governor to control his newly developed steam engin*.

Cont rod systems c an be broadly c at eg or Ized accord Ing to the use that is made of feedback. Figure 3. 1-1 shows a generally accepted scheme In which systems are classified as manual, closed-loop, or open-loop. Us ing manual control , a human be ing compares the output of the proce ss to that dos lred and then add us t s the actuator so as to ma inta In that output as spec If led. The operator 1s us Ing I cod back in that he or she I fret noni tors the inst ruuentat Ion and then comb inc s that inf ormat i on wl th knowledge of the pl ant dynamic s to de termini the appropr late s igna 1 I or the actuator .

Closed-loop control 1s s lm i 1 ar to manual operat ton except that the p lant ' s output 1s used d free t 1y In the de termlnat ton of the c on- trol signal. This 1s shown In the niddle portion of the I lguro. The ' reference output ' ref lects the do s lred or spec if led manner in which the plant la to be operated. This is compared to the measured output which is the s lgnal obtained fron the instrument at ion. The dlf I erence between these two quantities is the ' actuating error s lgnal '. The control s lgnal , which is sent to the actuator, 1s a funct ton of this error . The error and control s ignal s could be computed by o lther ana- log or dig 1ta1 computers . Or, as was the case with the ball-governor for the ste am eng inc , they could be determined by mechan Ical ne ans . Under c 1osed- I oop cont ro 1, the s igna 1 to the actuator i s generated as the process evolves . Hence all owanco 1s automat lcally nado for any perturbations that may at feet the p lant ' s dynamics . One I Inal po Int that is worth noting about c losod-loop contro 1 is that the control s lgna 1 1s a tune t ton of the measured ratho r than the actual output of the plant. Hence , if the Instruments I all , so will the control sys-

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OUTPUT

MANUAL CONTROL”

CLOSED - LOOP CONTROL

OPEN - LOOP CONT ROL

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tem. This may not matter during manual operation because human beings c an sonnet Innes recogn i ze and compensate I or I a i led sensors . Howeve r, under closed-loop condie ions, the sensor' s erroneous output will be used. The solution eo this problem is sensor validation which is d i s cus s ed in sec t ion s 10 . 2. I and 10 . k . 1 o I t h i s re po r t .

Open-loop control is shown in the lower portion of the figure. The d 1 s t i ng u i s hing fe at url o I t h i s ap p ro ac h i s t hat no u se i s made o I feedback. Rather , a p rede t e rmined ref e rence t raj ec tory is spec i I ied . For example, an accurate model of the process in question could be constructed and s lmul at ions performed unt il the sequence of control s i gnat s needed to generate the des trod output was i dent 1 I rod . Open- loop control way or may not result in the desired plant behavior. Clearly if the mode I usod to detarmlno the cram ectory is in error, then so clit tho conzro I act ion. However, even ulth a perfect mode 1, the results nighe be poor. There in , for exemp In, no way to allow for unan£lc lpated change a In the plant . Deep ice Ice drawbacks, open-loop cont rod 1s eatens lve ly s tud led and , aa d lscus sed be low, som• sophlst i- e axed cont rod st rateg tee use it .

3.2 Contro l Hethodolosios

Enttmeratod h•ro aro the maJ or cont ro I mezhodo I og les that are now In uso . As general references, the textc ' St ate Functions end Linear Control Systems' by Schultz and Holsa ( HcGraw-H111, 1967 ) , 'Control System Dea lgn: An Introduce ton to State-Space Methods ' by Fr Redland ( HcGrau-Hlll, l98G ) , ' Hodorn Control Eng Sneering ' by Ogat a ( Prent Ice- Hall , 1970 ) and 'Process Dynsmlcn and Control ' by Seborg, Edgar, and He 11 lchamp ( John Wiley R Sona, 1989 ) are recommended . Th• def In It tons g Ivan in sect ton 3 . 2. 3 . I of thls raport are from th• I mat of the so .

3.2. 1 Progort1onal - Integral-Der1va t Le a Contr a 1

Proportional-Integral-Derlvmtlvc or P-I-D control lm the moot commonly ueed method for thc autoemtcd operst[oo of procasm mystoms. Using thim appromcA, tAc oy*tez itzclf Is treated ao s 'block box'. That ia, the control s{#na1 {m determined oolely by cowp•rImon of the maasurmd aod reference outputs. Informxtloo on the dynamics of th• proceao, ovcn {f sv•ilxb1c, is not uxzd. Flgurc 3.2.1-1 is • block dl&gfaB O§’ th0 GpQrod€#. fhe efKOK Si#nHl âS t2k0n dS the dââfOK0RCe bmtvaen t#• rzf•r•nc• and mexoured outputs of tha procasm. If the controllm* output im ooh proportioosl to this crror, than th• control action Im tmrmsd 'proport*onzl'. If th• controller output almo con- tains • turn that im a function of the accumulated error, than it lm deoignatoQ •m 'proportional-integral'. Pin*lly, if the controller output conolstm of termm raprzmenting both the integral aod the rate ot change of the error mignal am well as the error signal ltoclf, theo it im refa*red to xm 'proportional-integral-derivative' or P-I-D.

It lz lnmtructlva to examine the m•themxt1cs of thc P-I-D control aQ KOAGh. ROI 9iBQllCit/ dSSUB€ thst th0 2CtUBl nd DO2&UKOd Qldnt oqtputs xre Identical. Th•t iz, the inxtrumaotc arc functioning per- fectly. Suppoxm thct the coxtrollzr is strictly proPortioo•l an that

CON T ROLL ER

SV ST E M

ACTUATOR		PROCESS

Figur e 3. 2. l- l Propor t ional-Integral-Deriva t iv e Control

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t he s y s t em and t he e one ro 11 e r c an b e re p re se n t e d re s pe c t i ve 1 y by t he I o 1 low ing re I at i on s :

n ( t ) - Hu ( t )

u ( t ) - Ke ( t )

where: n ( t ) is the plant output,

H denotes the process dynamics,

u ( t ) is th0 COnt[Ol Si#Rdl,

K is the gain of tho controller, and

e ( t ) is the orror signal.

Also note that the error zignsl iS defined as:

( 3 . 2 . 1 - 1 )

( 3 . 2 . I -2 )

whore n ( t ) 1s tho re fa renew out:put . Comb In ing trio above rel at lone

( 8 . 2 . \ -4 )

Zhlc sinple rotation Illustrates the enormous advantages to uslng feedback and closed-loop control. Spec if lcally, note that by naxlng the gsln job Lz zaz $L $ 1 arg• , tho quant lty ( HK/ ( 1 +HK ) ) w111 be dr lvon to unity and th• obsorvod plant output , n ( I ) , vl11 bocomo oqual to the reference or deulrod output, ii ( t ) . Ot epoclsl slgnlflcanco 1s I.hat thlu resutt la acñtevod ulthout any knowledge of eho process dyn mlcc

aod that it *s valid for sny refzrznce output. Morcovzr, zwzn II the process dyn*m*cc chsngc or arc pzrturbxd, it should bc poxxiblz to ma*ot•in tA• output of th• procexx at the d•m*r•d valu*. Hence th• app••l of cloz•d-loop control.

Unfortunmtcly, it Iz oot pommiblc to achi•v• all of th• theormt{- ca1 cdvaatage• asaoc 1 ated oLttz the proport 1 ooa1 cont ro 1 la r dascr 1bed abov•. For ooe th1ng, tIt• coat ro1 1er ga In ccn not ba zaad• arb Ie rar 1 1y'

large. So doing will make’ the syxt*m incresmlngly zzocltlvc to small pmrturbst{ono aod mml eventually csuxe Instability. Them crmztez an immediate problem bmcaus•, for fInIto valumm of K, the guaAt{ty ( HK/ ( l+HK ) ) will b• lesa than uo{ty and heoc• n ( t ) will never qultc agusl â ( t ) . Thia im x m•jor dlzadvantago of proport[ooxl control. Such syztmms exhibit an of€a•t betwoeo th• dex*r•d and oboervzd out- puts. ThIz problem cso bz rect[flod by add*oq {ntegrsl act{oo to the controller. That *m, inataad of a control signal that im d*rzctly

proportion al to the error signal as in EQuation 3. 2. 1-2, the control signal iS of the form.

u ( t ) - Ke ( t ) T. e ( t ) dt ( 3 . 2 . I -6 )

where the quantity T; is the integral time. The inverse of the integral timo, which is termed the 'reset rate' has a physical inter- pretation. It is the number of occasions per minute that the propor- tional part of the control action is duplicated. The advantage o2 adding an integral term to the control actioa is that the control sig- nal can have a non-zero valua even though tha error is zero. Thus, tha use of Integral actlo* resolves tha offset problem. Any differ- e*ca between tha desired and observed outputs will eventually cause the total crror to accumulate to tha point where it drives system response to the specified value. Tha drawback to integral action is that {t may induca oscillations as the system converges on tha speci- fied output.

Cont rod for po rf ortnanco may be I urthor Improved by add ing a do r lvat lvo to ria to tho control s lgnal . 'fhun ,

a ( t ) dt + KTg ( 3.2. L- y )

dt

’The quantity Th im the darIv*tiva Um*. Phyzlcclly, it Im tha Interval by vhich tha proportIon•l pxrt o# the control •ctIon is advanced. Darlvativa action is anticipatory and thus ltm usa may reduco osc 11 I at tone . However , IN aiay a loo arnpll I y s lgnal no Iss .

In summary , the advantages of the F-I-D approach are that the tmchoiqua can ba applied without reg#rd to th• procama dyn*micm, that its bGSlC OORdOQtS 2fO FOGdll/ UndGIStOOd, 2Hd thAt lt CSA bO lBQlO- mantod using aithar an•log or digital ogulpment. Its dIm•dvantageo arm that thc molactloo of the control p•rxmzterm ( the gain and th• Integral and dorlvstive tlmem ) im empIrIcaI and thxt themc p•ramaterz will b• valid £or ooly ooc p•rticul•r tranmiant. Thua the ucc of tha P-I-D approach for non-linasr or tLoa-dalxyad systaoms im impractical. Another drzwbac¥ im thmt the tachnlqua can only hmodl• ooa Input and ono output. AlSo, thc theory of P-I-D control doao not provide tha bam[m for d•t•rminin# symtcm stability.

3 . 2 . 2 TcaasCar . Func t lon g A Q R » oach _

The use of trsomfar functions in control cymtzm daoigo rmpraxentz a major advance ov•r the P-I-D approach bacxuzm th• myotou Im no longer traxtmd am m blxck box. Am m rasult, th• stability of th• con- t¥O§§Qd 9#AtBM CGV bO GDOl/A€ñ 2nñ thOOf€tiCS##/ V2# ñ BWtñOñ# C2n b0 usad to smlact the gxin coafficiant. Howcvar, wharcxs thc P-I-D approach wxx io thaory spplicablz to any proccxz, us• of tha transfer

function approach ia halted to linaxr, tims-*nv•riant mymtama.

A tranDfar function is daflncd sc the ratio of th• Lzplxc• trans-

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form of the system output to the Laplace transform of the system input. FOr a given system, it is obtained by first tdentifying the plant'x describing differential equation, than taking the Laplace transform, and finally rearranging terms so as to obtain the ratio of the output to the input. This process results in a transfer from the tame domain ( differential equation ) to the frequency domain ( transfer function ) . As an illustration of the utility of the approach, suppose that the system ( actuator plus process ) shown in Figure 3.2.1-1 was actually described by the following second-order, linear differential equation.

( 3.2.2-1 )

Taking the Laplace transform of this equation subject to the assump- tion that all initial conditions aro zero yiolds:

( 3.2 . 2-2 )

Hence, the tran*far function of thc prococs ia:

( 3 . 2 . 2-3 )

Further cuppo*z thst x control schzm* iz to bz designed which user the sama fezdbcck •ppro#ch sc did the P-I-D controller. That is, only tha output of the syzten is fed back. The proczsm i• illuztratod in Figure 3.2.2-1. The closad-loop behavior of the system Ix described by tha relation:

( R ( o ) - Y ( m ) ] ( X ) [

Hence, the clozzd-loop transfer function Is:

R ( z ) a’+s+K

This can b• written in factored form ac:

( 3 . 2 . 2 -4 )

( 3.2.2-5 )

( 3 2 . 2-6 )

ohero , us lag the qu•drat lc foriaul a, the quant It Iss a and b are de te r-

a. - -0.5+0.5 4 i b - -0. 5-0.5 4 ( 3.2.2- 7 )

Finally, by taking the [nva*me Laplace transfer of ( 3.2.2-6 ) , the closed-loop bmhav[o* of tha *ymtew can be daducad. So doing,

Huch can be do ternilned about syst«n sz ab11 1ty and re sponse us Ing t h i s

t F wo u r 3 2 2 Tran fe Fu c

A o h t Co ro e r e s

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analytic solution for the behavior of the closed-loop syxtem. First noto thaC tha v@luo of the gain can range from zero to infinity. If the gain lies between 0.00 and 0.25, then the quantities a and b, which aha the roots of the closed-loop transfer function, ar& negative real numbers. If the value of the gain exceeds 0.25, then thosa roots are complex numbers with negative real parts. Hence, for any value of the gain, the exponential terms ( EXP ( at ) and EXP ( bt ) ) wilL dio out as time increases. The system is therefore unconditionally stable. ( Note : For othor systems, it is possible that for a certain range of gains, tha roots would ba positive implying unbounded exponential growth and instability. ) Another important feature to nota is that if the valu• of tha gain exceeds 0.25, tha system response will bc under- damped mzanLng that them output will oscillate about its dzsirzd value bafor• zattl*ng out. If the gain l* equal to 0.25, tha spztem should attsin the dcxlrad r•spons• without ovarxhoot. If the gain im less thsn 0.25, than there will still bc no ovmrshoot but thc raxponcc tima will be longar th#o is perhaps necessary.

The abovo enampl o shone that by Inc Ind Ing a aodo I ot the system In the controller, nuch Ins ight can be achlavod. Us Ing the P-I-D approach, tho only guidance raco lvad vat to iuako the gain aa 1 argo an pose ible . But vJth the trannfor function approach, It 1s posn ible to ident i fy tho rnago of gains for wh I ch tha synteo wi 11 be stab I o and to select a gain within that range to achlavo dos trod response character- istics. ( Hoto : Tho syston shoau In Figure 3. 2. 2-1 wea analyzed by obtaining the ctosed-loop response, Equation 3. 2. 2-8. In roat ltd, this was not necessary hoc suse us Ing m•thoda such as ' root-locus ' or Routh's stability criteria, the saae information could have been obtained directly from who transfer function. This 1s a further st rongth of the toehn lquo . )

In s •ry, tho trsnafor function approach can be applied to linear, tlmo-Invariant systems. It provides s ra£lonal scene for anslyalng syaeou stabll ftp and for spoc If King conta In syctem rasponuo characeorJctlcs. Ie can be lmplomontod uc Ing either d lg £t at or analog oqulpaont . Its d1••dv•ntagos •ro that lz 1s llalzad to I lnoar sys- tem, they lk troaks only s Ing to- Input/s Ing lo-output syueamn, and that it does not address cort ala fundamonLal lsauos uuch ss oh•t const 1- cueos a propor cholco of control signal. filao, the propor funct ton Ing

of th• contzollor im dopmodaot on tha accuracy of the modal chores ao reprouont Int the zystoin during tho das lgn stage. This rslsos a numbor of lncunn Inc ludlng hodo 1 vat Jdae ton and no lntenanco .

3.2.3 9t I c S B th d

Stxt•-op•c• mathodm •r• arguably th• most muccaomful of the vari- ous control tczholquwm not sV•[lsblc. Them methodology vmm thc onc UGed tO dOS§#D th0 GOhtfOl S/Gt0BS #Of the LQOIIO SQRCOCIS#t 2Dd ât iS

tho oon nov iisod for mnny aerospace applications. Tho bus IG concept

originated In tho l9G0s and, although nearly throo docadoa hav• passed, the mthodo logy 1s st It I ref erred to by the ml snoaor 'aodorn control thoory' . As vlZh tho transf or funct ton approach, the st:rength of st•Zo-sp•co innehods 1s tho uso of a system modol . Onl7 1nato•d of

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using a single differential equation to relate system output to system

input, a set o£ I irst order different'Lu L equations is used. Thus,

£u1 I advent age i s t aken o f the know ledge g ive n by the i nte rnal s t ru e - t ure of the system. For example, in the proceed ing sect ion of th lg re po rt , a s ys tern c har ac te r i zed by the to 11 out ng e quat ion was ana I yzed :

• ( t ) • r ( t ) + y ( t ) ( 3.2.3-1 )

This is a second ordor differential equation where y is the output and u is the input. Using the state-space approach, this is rewritten as two first order differential equations. The stato variables, hore denoted by the symbols x, and x„ are defined as:

( 3.2.3- 2 ( a ) )

x, - y		( 3.2.3- 2 ( b ) )
Thus,
x, - x, X g - -X g	+ U	( 3.2.3- 3 ( z ) ) ( 3.2.3- 3 ( b ) )

7he second, of the s• equations is obt a ined by so lv Ing ( 3 . 2. 3 - I ) for the quantity y and than substituting x for y. Equations 3. 2.3-3 cap bq or it ten In nietr in I ono to I acll 1tat• so Int lori on a dlg it a 1 machine:

( 3.2.3-4 )

Note thGt by cubBtltUtiDg ( 3.2.3-3 ( b ) ) ing thet x, equal n y end thee x equal n y, equctlon 3 . 2. 3-1 1s regen- erated. 'th::n , both represent at lone ( 3. 2. 3- I and 3. 2 . 3 -3 ) prov ido

deoc*[ptionm o2 the mymt•m. Thc cdv•ntzge smmoclmted with the statm- sp•c• zpp*o•c#' im th•t by uminp • s•t of n flrmt order diff•rmntI•l equmtioom to d•mcrib• mn n orde* system, th• 1nto*naI dynom{co of thGt 9/fitWfi bOCOB8F G€00EBlbl0. F1#UK0 â.7.â-i iS A bloCb diA#EEmD 11- lustrxtlny the v•lu• of this spprozoh for coxtroll•r design. Inxtoad of f•mding back mcrmly th• output of the procaos, each state voriablc is smoIg#md z faedbcck zoefflclant. ( Note : Th• mymbol 'h' is used to denote the fmedbazk coofficientm Ln the f{gurm. ) Honcm, for an nth order system, x dogromm ot freedom mr• introduced. Them provides the COhtKOâ 0R#lID€0f 'WEU ‹bROEBOUS fl0BlblââtjF• lD Q2Kt1CUl2K, b/ th0 judtcioux cho{o• of the foedbxck or gcin coo2fic{entm xsooolstod with aach of the mtstm varLablem, the sh#pc of th• system's response zs well as Ito stability cmn bm spocif{md. Thor• are other sdv•ntagem as well to the use of mtste-xp•ca methods. Us{nq th*m *ppromch, control- lers cgn bo dom{gnmd for system thst hcvc multiple Inputs and out-

.

F i g u r e 3 . 2 . 3 - I S P a t e - S p a c e Ap p r o a c h t o Co n t r o 1 1 e r D e s i g n

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puts . A 1so, as is d i scuased be 1 or in sec t ion 3 . 2 . 3 . 1 o I this re po rt , the se ate-space rue t hodo logy address es I undament at issues such as the c r i ter i a by which It c an be de te rmi ned i I a sys tern 'L s in I ac I con t ro I - lab ie. 7he technique is however not wi thout its drawbacks . For one thing , res use is 1. imi ted to I ine ar sys tems . For ano ther, i t may not be pos s I ble to g ive a phys ie at ne an ing to e ac h o I t he s t ate var i a b 1 es and even if it is, there is no guarantee that the variable can be measured. For example, relative to a nuclear reactor, precursor consent rat ions are s t at e var i ab le s t hat c anno t be d i rece ly o bso rve d .

3 . 2 . 3 . I Con t r o I I a b i I L L g an d 0 be e rva b i 1 i t v

The concepts of 'controllabLlity' and 'observability' were first introduced by Xalman io thc 1950s. Thc former concarnz tha capability of the control signal ( i.e., tha input to the controller ) to affect each of the state varIabl*s. fhz latter Ix thc capability of each mt•tz variable to affect thc output. On a fomal bzxlc, thama ara defined ax›

( i ) Controllobility : A oyotae im said to bo 'coot*ollable' {f any initial state x ( 0 ) cao ba transferred to aoy final otata x ( tg ) io a finita tim0, tg } O, by some control, o.

( 2 ) Obs e rvabl l 1tx : A syntem 1s aa Id to be ' observable ' If ov o ry nt ate x ( 0 ) can be exact lb do terinlned f rom mnsnuromont u of the output y ovar a fIoIte iotarvxl of tI:ne, O t tg.

Host phys ical sy s tame are both cost ro 11 able and obse rvab le . Xeoce the oeed to ve 1£y thaae propert tec o£ten appears to be oC only’ acadezalc 1oterest . Xowever, such ahould oot be th• case, espec la 1 1y’ wheo coos1der1og cozap1ex systems. Tvo 1ssuec are o£ concern. F1rst, whl1e 1t 1c true that most phy'c 1ca1 spstezns are cozztro11 able, the 1r assoc 1ated mztheaat1ca1 nodelc aay’ 1ack that property'. Th1s most o£tan occurc wheo a 1 1naar 1zed aode1 1s used to represaot a oon-1 1near sy'steza. 'Zhe second lcsue 1a tbat sozae system zaay ba on1y part 1a1 Ip cootro11ab1e. 7b1s 1a a complex prob1eza that Izac oot y'et beeo £u1 1y'

addressed oo c tlbeor•t*cc* bcm*m. Spec*flcslly, the f rmc] def{o{ti o

of cootro11sb*1Ity provldmz no loformstion on the d•grze to which s

beGGNB0 BEAM G§€t0REN SDd thGlK AB¥OCl2tOd COntKOâ BOChMDiGBO LU1fi11 the dmf*nitIon of contro11•bIlIty in thc x•ncx th•t givcn enough tima and w{tA no r•str*ctIon oo th• allowed trsjmctorlex, it ic possible to Cducc the d€&âK€d chEA$€ Ucân# thO GQ0€lflGd CODtFOl d0YlC€. YOt, th0

resu1t1og coqt ro 1 actloo aay ba oC poor qua1 1ty’. for examp 1e, suppose

th•t m cootro11•r for • opmcocraft nuclzmr reactor {s to b• con- atructad in which thc romctor power I:m to bm adjusted in rccpoooz to z temperature moooor. Horcover, suppoao that them oeocor ’must bm lo€Gted GOmB dlStGDc0 f¥DB the ¥0SGtOK Go thGt radl&tIoD dAea#G effects zr• m*n*ulzzd. Is the xyzt•m cootroll•blzt Clearly, if the rzzctor power warm to change, then xo wuld the tamper•tur• of the rxxctor coolant. But the zeneor ic 1oc•tzd on • boox and heoce the tampcraturc of thc coolant in which it Iz Immcrs•d will oot change until h••t hss bean conducted along th• aotlre Izx#tR of th• boom.

- 26-

Thus, thera will be a significant time delay a ssociated with the use of such a sensor for control and the resulting control action may be oscLllatory. The system is controllable according to the definition. Nevertheless, the resulting power trajectory may exhibit u ndesirable features.

Concerns related to controllability and in particular to thg need to refine the concept so that the controllor is capable of producing not only the desired end-state but also an acceptable trajectory have been a major aspect of the HIT program in the closed-loop digital con- trol of reactor power. ?uch hag baeo #ccomplishzd. In particular, tha cooc•pt of 'fe*s1bIlity of control' has been defined and the rate of change of reactivity ( as opposed to the reactivity itself ) hac been rec ogn I rod ss the appropr lat.o me airs of cont roT . Both of tho so con- cepts sra discussed io detsll io Chapter Four of thic report. For the przseot, it iz sufflclaot to note that 'fazaIb{l*ty of cootrol• is, llkc cootrollxblllty, s property of the controller. Obsarvxncc Impllzc that the showed ststec of a syst*m arm reztractzd such that it will alwxyc b• poxzlble to hslt a traoclent oo dcmsnd. ReIat*wz to nuclear r•actorz, completed work st HIT hsz provLdzd a practical

method for both del Ining and lncorporat Ing sucL a property In conero t- he •n for reactor neutron Ie powor. Th• soloctlon of th• raz• of change of reactlvlzy as th• control algnal b•arn on this 1szu• as well. Spoelf ical ly, iz 1s a dlreckly control I abl • queue lty In Who sense that lk can b• altered on domand. In contract, ro•c£ lv lty can only be chanted over sons I In loo Interval of th.

( ) 3. 2. 3 . 2 Sunvnar r o f S t she-Spac e Ap p roach

In n cry, tho st •to-spaco approach 1s curr•nz ly th• most of do ly used k•chn1quo for tho d•s lgn of control lore for complex systems. Its advent •g•a Pro th•e It can be used both eo s••uro system szablll ty and zo ach1•v• a des lred system zr•J octory. Hor•ov•r iz lz capable of hand1 tag a›u1t1p1•-1oput/uu1t1p1•-output spctezas. A1so, 1tc voder1y'1og theory’ addracaea €vadaaeota1 lscuea svcb as srbct cooct1tutec a cozt- tro11ab1e sy'cteza. Et• d1cadvantcge• are thct 1ts uce Ie restr1ated to 11oear sycteaa, tbat 1t• liap1ezaentac too zaay requ1re 1zt£ozaat ton on vcr1ab1es tbct cannot he iaaasvred, aod that It does oot address the degree to wb1cb • system away be cootro11cb1e. A1so, the so1ut too oC zaoct state-cpaa• des1go prob1ezas requires the uce of a d1g 1ta1 coza-

3. 2. $ A l Control

Control system p•rIornanco In conn1d•rod opt1n•1 In chs sens• that the romu*t*ng trajectory either alnIe[meo or max{aizes a dos lgnor-spec If led perforn•neo Index. Por example, It may b• des lrod zo coup lets a tr•ns felon In mlnlnun tlmo or v1th as 11tt ie fuel con- suupz lori as possible. The real lzst lori of optimal control 1s schleved by ualng a state-upsco repros•ntat1on. Th•t 1s, the syston ( process plus sctuazor ) 1s described us Ing a s•t of I trek ord•r dlfforena 1st equal I one. In zh• st ste-spae• approach, each uzat• v•r1ablo ( 1. o . , the 1ndep•ndenz ver I ables In th• descr lblng squat lone ) 1s ssslgnod a

-27-

feedback coefficient or gain. The user then selects the desired shape or trajectory of tho system's response and solves for the gains so as to achieve that rosponso. Tho burden of selecting the app ropriate trajootory is therefore on the designer. With optimal control, tho designer meroly selects a performance criteria. The system's gain is chosen so as to minimize ( or, if appropriate, maximize ) this index. Thus, the designer is relieved of a major burden. Unfortunately, another one is imposed in that the mathematics of the ’approach are often intractable. For example, if the Pontryagin approach to optimal behavior is used, thon it becomes necessary to solvo a set of partia: differential equations with split boundary conditions. That is, some conditions are known that apply to the initiation of the transient and soma to itx tarmin*tioñ, but there is not a complatm sat at oither the start or and o€ tha trBnslont. Menca, thc problem must be colved by itzrat{on. Figure 3.2.4-1 illustrates the problem. The objmctivm is to identify z trajectory that optimicam noma porform*oca indox. Assumc that tha optimal trajectory [s the on• indicated by the brokon I inc. The curven denoted by the letters ' a' and 'b' represent uuccos- slvo attonptn to Ident fly tho optimal solution. The separate p race a of each curve are generated by marching forward from the initial eondlt form and backoarda I rom the I Inal ones with th• hop• of the tvo places J o In Ing to form a u Ing lo path. This process la ltera¥ lvo often with nany uoccess lve sttempta be Ing requ I rod bet ore Who dev lred path la found. 'thus, Yh• noma Ion to an optimal control problem 1s calcu- latlon- Intensive. 'fhJa meana that, unless the problem 1s very s tmp ie, the calcul at tons will kako longer than the t ime that In ayat labl o during esch sampl Ing interval. Hence, many opt lmal control problems are ao lved in the lr out irony prior to lnlt i at ing the ae tual cont ro I actlon. The solution to an optimal control problw 1s therefore usually not a c losed-form law but rather a sequence of s lgnal s that are to be applied •t excA sampling Interval. The drawback to thlm approach Ic thst thim xoquenco will be valid only for thc boundary

cond talons that worn init Sally spoetf led. to provlctons cxl st for

var 1 at 1ons 1n thoae cood1t 1on• and th•z• Ie no cont 1ououc feedback oC me*cured sIgn•l•. lx othcr wozdm, them ic • fom of open-loop con- trol. If the mymtcx model im pmrf•ct snd lf th• boundary conditions havc oot changed, th•a th• rzzultlnp control •ctioo will bz xz d•- mirzd. Oth•rwizm, mn lacorrcct tr•jmctory will b• generated. Altcr- nztlvam to the split bouodzry condition problem ac formulated by Pontryzgln do cxlmt. For example, Bcllmax'c dynxalc programming

*voldm tAat problmn. However, it too lm c•lculatioa-intzocive. ThIc im the wzWknzcm of the optimal •pprosch to zycton control.

A major zccomplizhmcnt of the HIT-SNL proqr*m that lm described in them rcport im that x xmthod was fouod to obtain cloxad-loop, tiaz- opt{mml control laws for the sdjuctmcnt of x rzxctor'c neutronic power. DetSllm Grp glvco in Ch•ptzrm Four Snd Twelve of this rcport.

la suazsary, the advaotag• o£ opt 1zac1 coat ro 1 1s that It a1 1o'U's th0 COOtKOl 0R#lD00f tO ObtdlO S dOSlKBd f0SQOhDO 'NlthoNt hAViR8 tO specify the dctclls of th0 trap&ctory. ( Note • In th& lAnguxgc of th€ state-space approach, "w1thout havlog to s•1act th• 1ocat 1on• oC the closed-loop polzm.’ ) Another advxntagm Iz th•t the tcchniqum is

T UE tc I

F tir e 3 . 2 . k - I Su e c e s s 1 v e fi p p ro x rn a t i o n s t o a n O p t i m a 1 P a t h

-2 9-

applicable to non-linear systems. Disadvantages are that the mathe- matics associated with the technique are often intractable and this in turn may cause resulting control action to be open-loop with no provi- s i one for feed back.

3.3 Trend s an d Unresolve d Issue s i n Contro l Syste m Desien

The brief roview of control methodologies given in section 3.2 of this report considorsd the P-I-D approach, the use of transfer func- tions, state-space methods, and optimal control. There are of course many variations on these basic techniques as well as other distinct methodologies. However, the ones examined here are the dominant approaches . Several t rends and unresolved is sues are apparent I rom the review. These bear d i roct ly on the des ign of cont rod systems for couples Techno I og les such aa nuc ie ar re ac to re .

First:, mere as In c •• fiance is being placed upon system code l s . The E-I-D approach avoids use of a model. Transfer functions are thense lves a mode I with only the system Input and output access ibl e . Both st ate-space and opt inal cont rol as sumo the ex i stonce of accurate

models repl etc with dot at 1 s of the system' s inner sCructure . As a general observation, It. appears that the nathema£lcs available for ue 1 l lz ing a modo 1 has out str ipped the capab i 1 ity to de1 ineaCo such models. That is, the limiting factor in taking advantage of the benef Its of advanced control concepts 1s , In many respects , the d if- f1cult.y associated with obtaining accurate representations of the plant . In part ten far, If chose control techniques are to be appl red co systems for ahlch safety 1s of paramount importance , then issues such as mode 1 valldat; lori , cal i br at ion, and malnt enance should be ad- dre ssed on a I ormal bas is .

A second issue im that of partial coEtrollability. This was dis- cussed in section 3.2.3.1 of thic report. The issue is of particular importance in the control of systems that are either non-linear or Uma-delxyzd. In such systems the influence of ona vxrizblz upon another may vzry zubstzntlally ax tha result of small changes in the control signal.

A third issue 1s that of non-linear control. Host control nethodo •e •• •• e l n L e Z t ded for use on I inear sys£ens . Moreover, even If a pare lcular cont ro 1 approach is capable of c re at ing the non- I inoar

c aso , lc In o£cen applied co a 11ne ar Ized nod• 1 in order to nln imi ze nathomae ical complexity. Thln pracz Ice 1s certainly acceptable for so at Ie n ltuaz lone in whlch the obj ect lve of the control act ton is to nain£aln a spec If led condlz Ion. But It 1s not accepcablu for tran- stents . The nov decad•-old revoluC Ion In dlg ltal techno 1 zz * • s mad• It pose ible to app ly enormous comput Ing powo r to ind iv idua I cont ro 1

loops. Advant z• should be taken of this new resource to design con£ro lie re In terms of the non- I Inc ar s yscem.