Computer Control Systems

Transcription

1 Computer Control ytem In th chapter we preent the element and the bac concept of computercontrolled ytem. The dcretaton and choce of amplng frequency wll be frt examned, followed by a tudy of dcrete-tme model n the tme and frequency doman, dcrete-tme ytem n cloed loop and bac prncple for degnng dgtal controller.. Introducton to Computer Control The frt approach for ntroducng a dgtal computer or a mcroproceor nto a control loop ndcated n Fgure.. The meaured error between the reference and the output of the plant converted nto dgtal form by an analog-to-dgtal converter DC, at amplng ntant k defned by the ynchronaton clock. The computer nterpret the converted gnal yk a a equence of number, whch t procee ung a control algorthm and generate a new equence of number {uk} repreentng the control. y mean of a dgtal-to-analog converter DC, th equence converted nto an analog gnal, whch mantaned contant between the amplng ntant by a ero-order hold ZOH. The cacade: DCcomputer-DC hould behave n the ame way a an analog controller PID type, whch mple the ue of a hgh amplng frequency but the algorthm mplemented on the computer very mple we jut do not make ue of the potentalte of the dgtal computer!. econd and much more nteretng approach for the ntroducton of a dgtal computer or mcroproceor n a control loop llutrated n Fgure. whch can be obtaned from Fgure. by movng the reference-output comparator after the analog-to-dgtal converter. The reference now pecfed n a dgtal way a a equence provded by a computer. In Fgure. the et DC - plant - DC nterpreted a a dcreted ytem, whoe control nput the equence {uk} generated by the computer, the output beng the equence {yk} reultng from the /D converon of the ytem output yt. Th dcreted ytem charactered by a dcrete-tme model, whch 5

2 6 Dgtal Control ytem decrbe the relaton between the equence of number {uk} and the equence of number {yk}. Th model related to the contnuou-tme model of the plant. CLOCK et ek uk ut yt rt + - DC COMPUTER DC + ZOH PLNT CONTROLLER Fgure.. Dgtal realaton of an «analog» type controller CLOCK rk + ek uk ut yt - COMPUTER DC + ZOH PLNT DC yk DICRETIZED PLNT Fgure.. Dgtal control ytem Th approach offer everal advantage. mong thee advantage here we recall the followng:. The amplng frequency choen n accordance wth the bandwdth of the contnuou-tme ytem t wll be much lower than for the frt approach.

3 Computer Control ytem 7. Poblty of a drect degn of the control algorthm talored to the dcreted plant model. 3. Effcent ue of the computer nce the ncreae of the amplng perod permt the computaton power to be ued n order to mplement algorthm whch are more performant but more complex than a PID controller, and whch requre a longer computaton tme. In fact, f one really want to take advantage of the ue of a dgtal computer n a control loop, the language mut alo be changed. Th may be acheved by replacng the contnuou-tme ytem model by dcrete-tme ytem model, the contnuou-tme controller by dgtal control algorthm, and by ung dedcated control degn technque. The changng over to th new language dcrete-tme dynamc model make t poble to ue varou hgh performng control tratege whch cannot be mplemented by analog controller. The operatng detal of the DC analog-to-dgtal converter, the DC dgtal-to-analog converter and the ZOH ero-order hold are llutrated n Fgure.3..D.C. D..C. + Z.O.H. Z.O.H. T amplng perod Fgure.3. Operaton of the analog-to-dgtal converter DC, the dgtal-to-analog converter DC and the ero-order hold ZOH

4 8 Dgtal Control ytem The analog-to-dgtal converter mplement two functon:. nalog gnal amplng: th operaton cont n the replacement of the contnuou gnal wth a equence of value equally paced n the tme doman the temporal dtance between two value the amplng perod, a thee value correpond to the contnuou gnal ampltude at amplng ntant.. Quantaton: th the operaton by mean of whch the ampltude of a gnal repreented wth a dcrete et of dfferent value quanted value of the gnal, generally coded wth a bnary equence. The general ue of hgh-reoluton /D converter where the ample are coded wth bt or more allow one to conder the quantfcaton effect a neglgble, and th aumpton wll hold n the followng. Quantaton effect wll be taken nto account n Chapter 8. The dgtal-analog converter DC convert at the amplng ntant a dcrete gnal, dgtally coded, n a contnuou gnal. The ero-order hold ZOH keep contant th contnuou gnal between two amplng ntant amplng perod, n order to provde a contnuou-tme gnal.. Dcretaton and Overvew of ampled-data ytem.. Dcretaton and Choce of amplng Frequency Fgure.4 llutrate the dcretaton of a nuod of frequency f for everal amplng frequence f. It can be noted that, for a amplng frequency f = 8 f, the contnuou nature of the analog gnal unaltered n the ampled gnal. For the amplng frequency f = f, f the amplng carred out at ntant f t other than multple of a perodc ampled gnal tll obtaned. However f the amplng carred out at the ntant where f t = n, the correpondng ampled equence dentcally ero. If the amplng frequency decreaed under the lmt of f = f, a perodc ampled gnal tll appear, but t frequency dffer from that of the contnuou gnal f = f - f. In order to recontruct a contnuou gnal from the ampled equence, the amplng frequency mut verfy the condton Nyqut' theorem: f > f max..

5 Computer Control ytem 9 f = 8 f f = f f = f!! Fgure.4. nuodal gnal dcretaton n whch f max the maxmum frequency to be tranmtted. The frequency f = f max a theoretcal lmt; n practce, a hgher amplng frequency mut be choen. The extence of a maxmum lmt for the frequency that may be converted wthout dtorton, for a gven amplng frequency, alo undertandable when t oberved that the amplng of a contnuou-tme gnal a magntude modulaton of a carrer frequency f analogy wth the magntude modulaton n rado tranmtter. The modulaton effect may be oberved n the replcaton of the pectrum of the modulatng gnal n our cae the contnuou gnal around the amplng frequency and t multple. The pectrum of the ampled gnal, f the maxmum frequency of the contnuou gnal f max le than / f, repreented n the upper part of Fgure.5. The pectrum of the ampled gnal, f f max > /f, repreented n the lower part of Fgure.5. The phenomenon of overlappng alang can be oberved. Th correpond to the appearance of dtorton. The frequency /f, whch defne the maxmum frequency f max admtted for a amplng wth no dtorton, known a Nyqut frequency or hannon frequency.

6 3 Dgtal Control ytem Cae : f < f max o ƒ max Contnuou-tme pectrum Cae : f > f max ƒ pectrum of the ampled gnal ƒ 3 ƒ ƒ o ƒ max Contnuou-tme pectrum ƒ ƒ Overlappng alang of the pectrum dtoron! 3 ƒ Fgure.5. pectrum of a ampled gnal For a gven amplng frequency, n order to avod the foldng alang of the pectrum and thu of the dtorton, the analog gnal mut be fltered pror to amplng to enure that: f max f.. The flter ued are known a ant-alang flter. good ant-alang flter mut have a mnmum of two cacaded econd-order cell f max << / f. n example of an ant-alang flter of th type gven n Fgure.6. Thee flter mut ntroduce a large attenuaton at frequence hgher than / f, but ther bandwdth mut be hgher than the requred bandwdth of the cloed loop ytem generally hgher than open loop ytem bandwdth. Crcut of th type or more complex are currently avalable. Fgure.6. nt-alang flter

7 Computer Control ytem 3 In the cae of very low frequency amplng, frt a amplng at a hgher frequency carred out nteger multple of the dered frequency, ung an approprate analog ant-alang flter. The ampled gnal thu obtaned paed through a dgtal ant-alang flter followed by a frequency dvder decmaton thereby gvng a ampled gnal havng the requred frequency. Th procedure hown n Fgure.7. It alo employed every tme the frequency of data acquton hgher than the amplng frequency choen for the loop that mut be controlled the amplng frequency hould be an nteger dvder of the acquton frequency. yt nt-alang analog flter nt-alang dgtal flter y k T a T /D converter acquton frequency Under-amplng T = n.t a Fgure.7. nt-alang flterng wth under-amplng.. Choce of the amplng Frequency for Control ytem The amplng frequency for dgtal control ytem choen accordng to the dered bandwdth of the cloed loop ytem. Note that, no matter how the dered performance are pecfed, thee can alway be related to the cloed loop ytem bandwdth. Example: Let u conder the performance mpoed n ecton..6 on the tep repone maxmum overhoot 5%, re tme.75. The tranfer functon to be determned correpond to the dered cloed loop ytem tranfer functon. From the dagram gven n Fgure. we have deduced that the cloed loop tranfer functon mut be a normaled econd-order tranfer functon wth =.7 and = rad/. y mmedately ung the dagram gven n Fgure., t can be oberved that the bandwdth of the cloed loop ytem approxmately equal to f CL The rule ued to chooe the amplng frequency n control ytem the followng: H f 6to 5 f CL..3 where f : amplng frequency, : cloed loop ytem bandwdth f CL

8 3 Dgtal Control ytem Rule of Equaton..3 equally ued n open loop, when t dered to chooe the amplng frequency n order to dentfy the dcrete-tme model of a CL plant. In th cae f replaced by an etmaton of the bandwdth of the plant. For nformaton purpoe, Table. gve the amplng perod T = /f ued for the dgtal control of dfferent type of plant. The rule for choong the amplng frequency gven n Equaton..3 can be connected to the tranfer functon parameter. Frt- order ytem H T In th cae the ytem bandwdth f f T an attenuaton greater than 3 db ntroduced for frequence hgher than = /T = f. Table.. Choce of the amplng perod for dgtal control ytem ndcatve value Type of varable amplng perod or plant Flow rate 3 Level 5 Preure 5 Temperature - 8 Dtllaton - 8 ervo-mechanm. -.5 Catalytc reactor - 45 Cement plant - 45 Dryer 45

9 Computer Control ytem 33 y applyng the rule of Equaton..3 the condton for choong the amplng perod obtaned T = /f : T 4 T T..4 Th correpond to the extence of two to nne ample on the re tme of a tep repone. econd- order ytem H The bandwdth of the econd-order ytem depend on and on ee Fgure.. For example:.7 f.6 f y applyng the rule of Equaton..3, the followng relaton are obtaned between the natural frequency and the amplng perod T : and.5 T ; = T.75 ; =..6 The lower value correpond to the choce of a hgh amplng frequency and the upper value to the choce of a low amplng frequency. For mplcty' ake, gven that n cloed loop the behavor frequently choen a the dered behavor that of a econd order havng a dampng factor between.7 and, the followng rule can be ued approxmaton of Equaton..5 and..6:.5 T.5 ;.7..7

10 34 Dgtal Control ytem.3 Dcrete-tme Model.3. Tme Doman Fgure.8 llutrate the repone of a contnuou-tme ytem to a tep nput, a repone that can be mulated by a frt order ytem an ntegrator wth a feedback gan ndcated n the fgure. u t u G T + dy _ dt - I T y y t Fgure.8. Contnuou-tme model The correpondng model decrbed by the dfferental equaton dy dt G y t u t.3. T T or by the tranfer functon G H.3. T where T the tme contant of the ytem and G the gan. If the nput ut and the output yt are ampled wth a pecfed amplng perod, the repreentaton of ut and yt are obtaned a number equence n whch t or k now the normaled dcrete-tme real tme dvded by the amplng perod, t = t/t. The relaton between the nput equence {ut} and the output equence {yt} can be mulated by the cheme gven n Fgure.9 by ung a delay backward hft operator ymboled by q - : yt- = q - yt, ntead of an ntegrator. Th relaton decrbed n the tme doman by the algorthm known a recurve equaton or dfference equaton yt = -a yt- + b ut-.3.3

11 Computer Control ytem 35 u ut- b + yt y t - a delay q- t Fgure.9. Dcrete-tme model Let u now examne n greater detal the dcrete-tme model gven by Equaton.3.3 for a ero ntal condton y = and a dcrete-tme unt tep nput: t u t t The repone drectly computed by recurvely ung Equaton.3.3 from t = n the cae of dcrete-tme model there are no problem wth the ntegraton of the dfferental equaton lke n contnuou tme. We hall examne two cae. Cae. a = -.5 ; b =.5 The output value for dfferent ntant are gven n Table. and the correpondng equence repreented n Fgure.. Table.. tep repone of a frt-order dcrete-tme model a = -.5, b =.5 T yt y -, t Fgure.. tep repone of a frt-order dcrete- tme model a = -.5, b =.5 It oberved that the repone obtaned reemble the tep repone of a contnuou-tme frt order ytem whch ha been ampled. n equvalent tme

12 36 Dgtal Control ytem contant for the contnuou-tme ytem can even be determned re tme from to 9 %: t R =. T. From Table., one then obtan 3T 4T T.. Cae. a =.5 ; b =.5 Output value for dfferent ntant are gven n Table.3 and the correpondng equence repreented n Fgure.. Table.3. tep repone of a frt-order dcrete-tme model a =.5; b =.5 T yt n ocllatory damped repone oberved wth a perod equal to two amplng perod. Th type of phenomenon cannot reult from the dcretaton of a contnuou-tme frt order ytem, nce th latter alway a-perodc. It may thu be concluded that the frt order dcrete-tme model correpond to the dcretaton of a frt order contnuou-tme ytem only f a negatve.,5- y -,5 Damped ocllatng repone Fgure.. tep repone of a frt-order dcrete-tme model a =.5; b =.5 We now go back to the method ued to decrbe dcrete-tme model. The delay operator q - ued to obtan a more compact wrtng of the recurve dfference equaton whch decrbe dcrete-tme model n the tme doman t ha the ame functon a the operator p = d/dt for contnuou-tme ytem. The followng relaton hold: t For a potve a, th correpond to the dcretaton of a nd order ytem, wth a damped reonant frequency equal to.5 f ee ecton.3..

13 Computer Control ytem 37 q q d y t y t y t y t d.3.4 y ung the operator q -, Equaton.3.3 rewrtten a + a q - yt = b q - ut.3.5 Dcrete-tme model may alo be obtaned by the dcretaton of the dfferental equaton decrbng contnuou-tme model. Th operaton ued for the mulaton of contnuou-tme model on a dgtal computer. Let u conder Equaton.3. and approxmate the dervatve by dy dt y t T y t.3.6 T Equaton.3. wll be rewrtten a y t T y t T T G y t u t T.3.7 y multplyng both de of Equaton.3.7 by T, and wth the ntroducton of the normaled tme t = t/t, t follow that T G y t y t T u t.3.8 T T whch can be further rewrtten a: where + a q - yt+ = b ut.3.9 T a T ; G b T T hftng Equaton.3.9 by one tep, Equaton.3.3 obtaned. We pont out that, n order to repreent a frt-order contnuou model wth Equaton.3.9, the condton a < mut be verfed. a conequence, the amplng perod T mut be maller than tme contant T T < T. Th reult correpond to the upper bound n Equaton..4, ntroduced for amplng perod electon of a frt-order ytem a a functon of the dered cloed loop bandwdth.

14 38 Dgtal Control ytem If Equaton.3.6 the approxmaton of the dervatve, the dgtal ntegrator equaton can be drectly deduced. Thu, f normaled tme ued, Equaton.3.6 wrtten a d dt y py y t y t q y t.3. where - q - now equvalent to p. the ntegraton the oppote of the dfferentaton, one obtan: t y dt y y t.3. p q Multplyng both de of Equaton.3. by -q -, t follow that t - q - = yt.3. whch we can rewrte a t = t- +. yt.3.3 correpondng to the approxmaton of the ntegraton operaton by mean of the rectangular rule, a llutrated n Fgure. f contnuou-tme ued, Equaton.3.3 wrtten a t = t-t + T.yt. y t- yt t- t Fgure.. Numercal ntegraton.3. Frequency Doman The tudy of contnuou-tme model n the frequency doman ha been carred out conderng a perodc nput of the complex exponental type e jt = co t + j n t or e t wth = + j.

15 Computer Control ytem 39 For the tudy of dcrete-tme model n the frequency doman we hall conder complex ampled exponental,.e. equence reultng from complex contnuou-tme exponental evaluated at the amplng ntant t = k T. Thee equence wll thu be wrtten a j Tk e e T k ; ; k=,,3 nce the dcrete-tme model beng condered are lnear, f a gnal of a certan frequency appled to the nput, a gnal of the ame frequency, but amplfed or attenuated accordng to the frequency, wll be found at the output. Th ummared n Fgure.3. n whch H the tranfer functon of the ytem that expree the dependence of the gan and the phae-devaton on the complex frequency j. ut=e jt k ut=e T k DICRETE TIME YTEM yt=hje jt k yt=he T k Fgure.3. Frequency repone of a dcrete-tme ytem If the nput of the ytem n the form e T k, the output wll be y t H e T k.3.4 and repectvely y t H e T k e T H e T k e T y t.3.5 It thu oberved that hftng backward by one tep equvalent to multplyng e T by. Let now determne the tranfer functon related to the recurve Equaton.3.3. T In th cae k u t e and the output wll be n the form of Equaton.3.4. y alo ung Equaton.3.5 one obtan: T Tk a e H e b e T e Tk.3.6 from whch reult

16 4 Dgtal Control ytem T b e H.3.7 T a e We conder now the followng change of varable: e T.3.8 whch correpond to the tranformaton of the left half-plane of the -plane nto the nteror of the unt crcle centered at the orgn n the - plane, a llutrated by Fgure.4. Re < j X X Re > T e - X X - T Fgure.4. Effect of the tranformaton = e Wth the tranformaton gven by Equaton.3.8 the tranfer functon gven n Equaton.3.7 become H b.3.9 a Note that the tranfer functon n - can be drectly obtaned from the recurve Equaton.3.3 by ung the delay operator q - ee Equaton.3.5, and afterward by formally computng the rato yt/ut and replacng q - wth -. Th procedure can obvouly be appled to all model decrbed by lnear dfference equaton wth contant coeffcent, regardle of ther complexty. The ame reult can be alo derved by mean of the - tranform ee ppendx, ecton. We alo remark that the tranfer functon of dcrete-tme model are often wrtten n term of q -. It of coure undertood that the meanng of q - vare accordng to the context delay operator or complex varable. When q - condered a a delay operator, the expreon Hq - named tranfer operator. It mut be oberved that the repreentaton by tranfer operator can alo be ued for model decrbed by lnear dfference equaton wth tme varyng coeffcent

17 Computer Control ytem 4 a well. In contrat, the nterpretaton of q - a a complex varable - only poble for lnear dfference equaton wth contant coeffcent. T Properte of the Tranformaton e The tranformaton of Equaton.3.8 not bjectve becaue everal pont n the - plane are tranformed at the ame pont n the -plane. Neverthele, we are ntereted n the -plane beng delmted between the two horontal lne crong the pont [, j / ] and [, j / ] where = f =/T. Th regon called prmary trp. Re < j Re > 3 X - X 4 5 e T X X - T Fgure.5. Effect of the tranformaton = e on the pont located n the prmary trp n -plane The complementary band are outde the frequency doman of nteret f the condton of the hannon theorem ecton.. have been atfed. Fgure.5 gve a detaled mage of the effect of the tranformaton e T for the pont that are nde the prmary trp. ttenton mut be focued on an mportant apect for contnuou econd-order ytem n the form: < for whch the reonant damped frequency equal to half the amplng frequency: / The mage of ther conjugate pole, j

18 4 Dgtal Control ytem through the tranformaton correpond to a ngle pont placed on the real ax n the - plane and wth negatve abca. e T e One get:,, e e e e e e e T j T T j T T nce: Th the reaon why dcrete-tme model n the form of Equaton.3.3 uch a + a q - yt = b q - ut gve ocllatng tep repone for a > damped f a < wth perod T ee ecton.3.. Thee frt-order dcrete-tme model have the ame pole a the dcrete-tme model derved from econd-order contnuou-tme ytem havng a damped reonant frequency equal to /..3.3 General Form of Lnear Dcrete-tme Model lnear dcrete-tme model generally decrbed a.3. d t u b t y a t y n n n whch d correpond to a pure tme delay whch an nteger multple of the amplng perod. Let u ntroduce the followng notaton:.3. * q q q q a n.3. *... n a n q q a a q.3.3 * q q q q b n

19 Computer Control ytem 43 * q b b q... b n q n.3.4 y ung the delay operator q - n Equaton.3. and takng nto account the notaton of Equaton.3. to.3.4, the Equaton.3. decrbng the dcretetme ytem wrtten a q - yt = q -d q - ut.3.5 or n the predctve form by multplyng both de by q d q - yt+d = q - ut.3.6 Equaton.3.5 can alo be wrtten n a compact form ung the pule tranfer operator yt = Hq - ut.3.7 where the pule tranfer operator gven by H q d q q.3.8 q The pule tranfer functon characterng the ytem decrbed by Equaton.3. obtaned from the pule tranfer operator gven n Equaton.3.8 by replacng q - wth - d H.3.9 Pule Tranfer Functon Order To evaluate the order of a dcrete tme model repreented by the pule tranfer functon n the form of Equaton.3.9, the repreentaton n term of potve power of needed. If d the ytem pure tme delay expreed a number of ample, n the degree of the polynomal - and n the degree of the polynomal -, one mut multply both numerator and denomnator of H - by n n order to obtan a proper 3 pule tranfer functon H on the potve power of, where The pule tranfer operator Hq - can be ued for a compact repreentaton of the nput-output relatonhp even n the cae of q - and q - have tme dependng coeffcent. The pule tranfer functon H - only defned for the cae of q - and q - are wth contant coeffcent. 3 Th mean that the denomnator degree greater than or equal to the numerator degree.

20 44 Dgtal Control ytem n= max n, n + d n repreent the dcrete-tme ytem order the hgher power of a term n n the pule tranfer functon denomnator. Example : a b b H 3 n = max, 5 = 5 a b b H 4 5 Example : a a b b H n = max, = a a b b H One note that the order n of an rreducble pule tranfer functon alo correpond to the number of tate for a mnmal tate pace ytem repreentaton aocated to the tranfer functon ee ppendx C..3.4 tablty of Dcrete-tme ytem The tablty of dcrete-tme ytem can be tuded ether from the recurve dfference equaton decrbng the dcrete-tme ytem n the tme doman, or from the nterpretaton of dfference equaton oluton a um of dcreted exponental. We hall ue example to llutrate both thee approache. Let u aume that the recurve equaton yt = -a yt- ; y = y.3.3 whch obtaned from Equaton.3.3 when the nput ut dentcally ero. The free repone of the ytem wrtten a y = -a y ; y = -a y ; yt = -a t y.3.3

21 Computer Control ytem 45 The aymptotc tablty of the ytem mple lm y t t.3.3 The condton of aymptotc tablty thu reult from Equaton.3.3. It neceary and uffcent that a <.3.33 On the other hand, t known that the oluton of the recurve dfference equaton of the form for a frt-order ytem: T t y t K e K t.3.34 y ntroducng th oluton nto Equaton.3.3, and takng nto account Equaton.3.5, one obtan T T t ae Ke a K t.3.35 from whch t follow that e T e j T e T e jt a.3.36 For th oluton to be aymptotcally table, t neceary that = Re < whch mple that e T < and repectvely < or a <. However, the term + a - nothng more than the denomnator of the pule tranfer functon related to the ytem decrbed by Equaton.3.3 ee Equaton.3.9. The reult obtaned can be generaled. For a dcrete-tme ytem to be aymptotcally table, all the root of the tranfer functon denomnator mut be nde the unt crcle ee Fgure.4: + a a n -n = <.3.37 In contrat, f one or everal root of the tranfer functon denomnator are n the regon defned by > outde the unt crcle, th mple that Re > and thu the dcrete-tme ytem wll be untable. for the contnuou-tme cae, ome tablty crtera are avalable Jury crteron, Routh-Hurwt crteron appled after the change of varable w = + /- for etablhng the extence of untable root for a polynomal n the varable wth no explct calculaton of the root Åtröm and Wttenmark 997.

22 46 Dgtal Control ytem helpful tool to tet -polynomal tablty derved from a neceary condton for the tablty of a - -polynomal. Th condton tate: the evaluaton of the polynomal - gven by Equaton.3.37 n =, and n = - - mut be potve the coeffcent of q - correpondng to uppoed to be potve. Example: - =.5 - table ytem =.5 =.5 > ; - = +.5 =.5 > - =.5 - ; untable ytem = -. 5 < ; - =.5 >.3.5 teady-tate Gan In the cae of contnuou-tme ytem, the teady-tate gan obtaned by makng = ero frequency n the tranfer functon. In the dcrete cae, = correpond to T e.3.38 and thu the teady-tate gan G obtaned by makng = n the pule tranfer functon. Therefore for the frt-order ytem one obtan: b G a b a Generally peakng, the teady-tate gan gven by the formula b d G H H.3.39 n a In other word, the teady-tate gan obtaned a the rato between the um of the numerator coeffcent and the um of the denomnator coeffcent. Th formula qute dfferent from the contnuou-tme ytem, where the teady-tate gan appear a a common factor of the numerator f the denomnator begn wth. n

23 Computer Control ytem 47 The teady-tate gan may alo be obtaned from the recurve equaton decrbng the dcrete-tme model, the teady-tate beng charactered by ut = cont. and yt = yt- = yt-... From Equaton.3.3, t follow that and repectvely + a yt = b ut b y t a u t G u t.3.6 Model for ampled-data ytem wth Hold Up to th pont we have been concerned wth ampled-data ytem model correpondng to the dcretaton of nput and output of a contnuou-tme ytem. However, n a computer controlled ytem, the control appled to the plant not contnuou. It contant between the amplng ntant effect of the eroorder hold and vare dcontnuouly at the amplng ntant, a llutrated n Fgure.6. It mportant to be able to relate the model of the dcreted ytem, whch gve the relaton between the control equence produced by the dgtal controller and the output equence obtaned after the analog-to-dgtal converter, to the tranfer functon H of the contnuou-tme ytem. The ero-order hold, whoe operaton revewed n Fgure.7 ntroduce a tranfer functon n cacade wth H. DC + ZOH - H PLNT H DC Fgure.6. Control ytem ung an analog-to-dgtal converter followed by a ero-order hold ZERO ORDER HOLD T t + T t-t Fgure.7. Operaton of the ero-order hold

24 48 Dgtal Control ytem The hold convert a Drac pule gven by the dgtal-to-analog converter at the amplng ntant nto a rectangular pule of duraton T, whch can be nterpreted a the dfference between a tep and the ame tep hfted by T. the tep the ntegral of the Drac pule, t follow that the ero-order hold tranfer functon H ZOH e T.3.4 Equaton.3.4 allow one to conder the ero-order hold a a flter havng a frequency repone gven by H ZOH e j j n T / jt T j T e T / From the tudy of th repone n the frequency regon f f / /, one can conclude:. The ZOH gan at the ero frequency equal to: G ZOH = T.. The ZOH ntroduce an attenuaton at hgh frequence. For f = f / one get Gf / = T =.637 T d. 3. The ZOH ntroduce a phae lag whch grow wth the frequency. Th phae lag between for f = and - / for f = f / and hould be added to the phae lag due to H. The global contnuou-tme tranfer functon wll be T e H ' H.3.4 to whch a pule tranfer functon aocated. Table whch gve the dcrete-tme equvalent of ytem wth a ero-order hold are avalable. ome typcal tuaton are ummared n Table.4. The computaton of ZOH ampled model for tranfer functon of dfferent order can be done by mean of the functon: contdc.c clab or contdc.m MTL. The correpondng ampled model wth Z.O.H for a econd-order ytem charactered by and can be obtaned wth the functon ftpol.c clab or ftpol.m MTL 4. 4 To be downloaded from the book webte.

25 Computer Control ytem naly of Frt-order ytem wth Tme Delay The contnuou-tme model charactered by the tranfer functon G e H T.3.4 where G the gan, T the tme contant and the pure tme delay. If T the amplng perod, then expreed a = d T + L ; < L < T.3.43 where L the fractonal tme delay and d the nteger number of amplng perod ncluded n the delay and correpondng to a ampled delay of d-perod. From Table.4, one derve the tranfer functon of the correpondng ampled model when a ero-order hold ued H d b b a d b b a.3.44 wth T T LT T a e ; b G e T ; b Ge T e The effect of the fractonal tme delay can be een n the appearance of the coeffcent b n the tranfer functon. For L =, one get b =. On the other hand, f L = T, t follow that b =, whch correpond to an addtonal delay of one amplng perod. For L<.5T one ha b < b, and for L>.5T one ha b > b. For L=.5T b b. Therefore, a fractonal delay ntroduce a ero n the pule tranfer functon. For L >.5 T the relaton b > b hold and the ero outde the unt crcle untable ero 5. The pole-ero confguraton n the plane for the frt-order ytem wth ZOH repreented n Fgure.8. The term -d- ntroduce d+ pole at the orgn [H = b + b / d+ + a ]. L T 5 The preence of untable ero ha no nfluence on the ytem tablty, but t mpoe contrant on the ue of controller degn technque baed on the cancellaton of model ero by controller pole.

26 5 Dgtal Control ytem Table.4. Pule tranfer functon for contnuou-tme ytem wth ero-order hold H H G T b a T T / T ; b G e ; a e T / T Ge L ; T L T b a b T / T L / T b Ge e ; LT / T ; b G e ; T / T a e b b a a b ; b a ; a e T ; co T ; n T Fgure.9 repreent the tep repone for a ytem charactered by a pule tranfer functon H b.3.45 a b wth teady-tate gan = for dfferent value of the parameter a : a a = -. ; -.3 ; -.4 ; -.5 ; -.6 ; -.7 ; -.8 ; -.9

27 Computer Control ytem 5 b b < o o x +j x -a o - ero x - pole b b > -j Fgure.8. Pole-ero confguraton of the ampled-data ytem decrbed by Equaton.3.44 frt order ytem wth ZOH On the ba of thee repone, t eay to derve the tme contant of the correpondng contnuou-tme ytem, expreed n term of the amplng perod the tme contant equal to the tme requred to reach 63% of the fnal value. The preence of a tme delay equal to an nteger multple of the amplng perod only caue a tme hft n the repone gven n Fgure.9. tep Repone a = -..8 a = a = Tme t/t Fgure.9. tep repone of the dcrete-tme ytem b - /+a - for dfferent value of a and [b /+ a ]= The preence of a fractonal tme delay ha a a man conequence a modfcaton at the begnnng of the tep repone, f compared to the cae wth no fractonal tme delay. Exerce. umng that the ampled-data ytem model

28 5 Dgtal Control ytem yt = -.6 yt- +. ut- +. ut- What the correpondng contnuou-tme model? It nteretng to analye the relaton between the locaton of the pole a and the rng tme of the ytem. Fgure.9 ndcate that the repone of the ytem become lower a the pole of the ytem move toward the pont [, j], and t become fater a the pole of the ytem approache the orgn =. Thee conderaton can be appled to ytem wth everal pole. In the cae of ytem wth more than one pole, the term domnant pole ntroduced to charactere the pole or the pole that are the cloet to the pont [,j],.e. whch the lowet pole. Fgure. how the frequency repone magntude and phae of the frtorder dcrete ytem gven by.3.45 for a = -.8; -.5; -.3. It can be oberved that the bandwdth ncreae when the ytem pole approachng the orgn fater pole. We can alo remark that the phae lag at the frequency.5f -8 o due to the preence of the ZOH ee ecton.3.6. ode Dagram a =-.3 Magntude d a =-.8 a = Phae deg a =-.5 a =-.8 a = Frequency f/f Fgure.. Frequency repone magntude and phae of the dcrete-tme model b - / + a - for dfferent value of a and b.3.8 naly of econd-order ytem The pule tranfer functon correpondng to the dcretaton wth a ero-order hold of a normaled econd-order contnuou-tme ytem, charactered by a natural frequency and a dampng, gven by

29 Computer Control ytem 53 H d b b.3.46 a a where d repreent the nteger number of amplng perod contaned n the delay. The value of a, a, b, b a a functon of and for a pure tme delay = d T are gven n Table.4. It nteretng to expre the pole of the dcreted ytem a a functon of, and the amplng perod T or the amplng frequency f. From Table.4 the followng relaton are ealy found for < : a T co T j T j e T e e e a e T The pole of the pule tranfer functon root of the denomnator are found by olvng the equaton + a + a = - - = From the expreon of a and a the oluton are drectly derved: T, e T j T Note that the pole of the dcreted ytem correpond to the pole of the contnuou-tme ytem, j by applyng the tranformaton e T. For <, the pole of the dcreted ytem are complex conjugate and, conequently, ymmetrc wth repect to the real ax. They are charactered by a module and a phae gven by, e T e f f e f, T f

30 54 Dgtal Control ytem Imag x f f.4 Pole - Zero Map Real x Fgure.. The curve contant and T / f / f contant n the -plane for a econd-order dcrete-tme ytem Note that the pole locaton depend upon and T or / = f /f. That : = f, T / = f, f /f and n the -plane the followng curve can be drawn: = f T / = ff /f for = contant and = f for T / = ff /f = contant We mut remember ee Fgure.9 that n the -plane contnuou ytem the curve = contant are traght lne formng an angle = co - wth the real ax and the curve = contant are crcle wth radu thee two et of curve are orthogonal. In the -plane the curve = f T for = contant are logarthmc pral that are orthogonal n each pont to the curve : f for T = contant.

31 Computer Control ytem 55 Fgure. how the et of curve = f for T / = contant and = f T / for = contant correpondng to dfferent value of and T / repectvely f /f. We hould alo remember ee ecton.3. that for f / f / /, the correpondng pole n the - plane are confounded, = ±, and they are located on the egment of the real ax -, havng an abca coordnate equal to e. The tablty doman of the econd-order dcrete-tme ytem n the plane of the parameter a - a a trangle ee Fgure.. For value of a, a placed nde of the trangle, the root of the denomnator of the pule tranfer functon are nde the unt crcle. a a - Fgure.. tablty doman for the econd-order dcrete-tme ytem.4 Cloed Loop Dcrete-tme ytem.4. Cloed Loop ytem Tranfer Functon Fgure.3 gve the dagram of a cloed loop dcrete-tme ytem. The tranfer functon on the feedforward channel can reult from the cacade of a dgtal controller and of the group DC+ ZOH + contnuou-tme ytem + DC dcreted ytem.

32 56 Dgtal Control ytem H OL - r t - + y t dturbance pt Fgure.3. Cloed loop dcrete-tme ytem Let H OL.4. be the feed-forward channel tranfer functon wth b b b n n.4. where the coeffcent b, b... b d may be ero f there a tme delay of d amplng perod. In the ame way a for contnuou-tme ytem, the cloed loop tranfer functon connectng the reference gnal rt to the output yt wrtten a H CL H OL.4.3 H OL The denomnator of the cloed loop tranfer functon, whoe root correpond to the cloed loop ytem pole, alo called charactertc polynomal of the cloed loop..4. teady-tate Error The teady-tate obtaned for rt = contant by makng =, correpondng to the ero frequency = e T = for =. It follow from Equaton.4.3 that y H CL n b r n b r.4.4

33 Computer Control ytem 57 where H CL the teady-tate gan tatc gan of the cloed loop ytem. In order to obtan a ero teady-tate error between the reference gnal r and the output y, t neceary that H CL =.4.5 From Equaton.4.4 the followng condton are derved: n b and.4.6 In order to obtan =, - mut have the followng tructure: where - = - - ' ' n ' a... a n '.4.8 and thu the global tranfer functon of the feedforward channel mut be of the type H OL.4.9 ' It thu oberved that the feedforward channel mut contan a dgtal ntegrator n order to obtan a ero teady-tate error n cloed loop. Th tuaton mlar to the contnuou cae ee ecton..3 and nternal model prncple alo applcable to dcrete-tme ytem..4.3 Rejecton of Dturbance In the preence of a dturbance pt actng on the controlled output ee Fgure.3, the objectve to reduce t effect a much a poble, at leat n ome frequency regon. In partcular, the contant dturbance effect a tep, often called load dturbance, expected to be ero n teady-tate t. The pule tranfer functon, whch lnk the dturbance to the output, yp H OL for the contnuou-tme cae, yp - called output entvty functon. The teady-tate obtaned for =. It follow that

34 58 Dgtal Control ytem y yp p p where p the tatonary value of the dturbance. In order to acheve a perfect teady-tate dturbance rejecton, t neceary that yp = and thu =. Th mple that - mut have the form gven n Equaton.4.7, correpondng to the ntegrator nerton n the feedforward channel. mlarly, to the contnuou cae, a perfect teady-tate dturbance rejecton mple that the feedforward channel mut contan the nternal model of the dturbance the tranfer functon that produce pt from a Drac pule. n the contnuou-tme cae, t hould be avoded that an amplfcaton of the dturbance effect occur n certan frequency regon. Th the reaon why yp e -j mut be lower than a pecfed value for all frequence f = / f /. typcal value ued a upper bound yp e - j f Furthermore, f t known that a dturbance ha t energy concentrated n a partcular frequency regon, yp e j may be contraned to ntroduce a dered attenuaton n th frequency regon..5 ac Prncple of Modern Method for Degn of Dgtal Controller.5. tructure of Dgtal Controller Fgure.4 gve the dagram of a PI type analog controller. The controller contan two channel a proportonal channel and an ntegral channel that proce the error between the reference gnal and the output. In the cae of ampled-data ytem the controller dgtal, and the only operaton t can carry out are addton, multplcaton, torage and hft. ll the dgtal control algorthm have the ame tructure. Only the memory of the controller dfferent, that the number of coeffcent. Fgure.5 llutrate the computaton tructure of the control ut appled to the plant at the ntant t by the dgtal controller. Th control a weghted average of the meaured output at ntant t, t-,..., t-n.., of the prevou control value at ntant t-, t-, t-n and of the reference gnal at ntant t, t-,, the weght beng the coeffcent of the controller.

35 Computer Control ytem 59 nalog PI rt + - K K/T + + ut PLNT yt Fgure.4. PI analog controller Dgtal controller q q- rt t q q- ut PLNT yt - q t - r r q - Fgure.5. Dgtal controller Th type of control law can even be obtaned by the dcretaton of a PI or PID analog controller. We hall conder, a an example, the dcretaton of a PI controller. The control law for an analog PI controller gven by u t K [ r t y t].5. pt For the dcretaton of the PI controller, p the dfferentaton operator approxmated by q /T ee ecton.3., Equaton.3.6. Th yeld dx x t x t q px x t.5. dt T T

36 6 Dgtal Control ytem xdt T x p q x t.5.3 and the equaton of the PI controller become u t K q q KT T [ r t y t].5.4 Multplyng both de of Equaton.5.4 by - q -, the equaton of the dgtal PI controller wrtten a where q - ut = Tq - rt - Rq - yt.5.5 q - = - q - = + q - = Rq - = Tq - = K +T /T -K q - = r + r q whch lead to the dagram repreented n Fgure.6. Dgtal PI rt q - r + + r r r q q- q - ut PLNT yt Fgure.6. Dgtal PI controller Takng nto account the expreon of q -, the control gnal ut computed on the ba of Equaton.5.5, by mean of the formula ut = ut- - Rq - yt + Tq - rt = ut- - r yt - r yt- + r rt + r rt-.5.8 whch correpond to the dagram gven n Fgure.6.

37 Computer Control ytem 6.5. Dgtal Controller Canoncal tructure Dvdng by q - both de of Equaton.5.5, one obtan t r q q T t y q q R t u.5.9 from whch we derve the dgtal controller canoncal tructure preented n Fgure.7 three branched RT tructure. In general, Tq - n Fgure.7 dfferent from Rq -. T R / / PLNT rt ut yt + - Fgure.7. Dgtal controller canoncal tructure Conder H.5. a the pule tranfer functon of the cacade DC + ZOH + contnuou-tme ytem + DC, then the tranfer functon of the open loop ytem wrtten a R H OL.5. and the cloed loop tranfer functon between the reference gnal rt and the output yt, ung a dgtal controller canoncal tructure, ha the expreon P T R T H CL.5. where p p R P the denomnator of the cloed loop tranfer functon that defne the cloed loop ytem pole. Note that Tq - ntroduce one more degree of freedom, whch

38 6 Dgtal Control ytem allow one to etablh a dtncton between trackng and regulaton performance pecfcaton. We alo remark that rt often replaced by a dered trajectory y * t, obtaned ether by flterng the reference gnal rt wth the o-called hapng flter or trackng reference model, or avng n the memory of the dgtal computer the equence of the dered trajectory value. The dgtal controller repreented n Fgure.7 alo defned a RT dgtal controller. It a two degree of freedom controller, whch allow one to mpoe dfferent pecfcaton n term of dered dynamc for the trackng and regulaton problem. The goal of the dgtal controller degn to fnd the polynomal R,, and T n order to obtan the cloed loop tranfer functon, wth repect to the reference and dturbance gnal, atfyng the dered performance. Th explan why the dered cloed loop performance wll be expreed, f not, they wll be converted n term of dered cloed loop pole, and eventually n term of dered ero n th way the cloed loop tranfer functon wll be completely mpoed. In the preence of dturbance ee Fgure.8 there are other four mportant tranfer functon to conder, relatng the dturbance to the output and the nput of the plant. The tranfer functon between the dturbance pt and the output yt output entvty functon gven by yp.5.4 R rt T + - dturbance pt vt + ut + yt / / + + PLNT R + + bt noe Fgure.8. Dgtal control ytem n preence of dturbance and noe Th functon allow the characteraton of the ytem performance from the pont of vew of dturbance rejecton. In addton, certan component of - can be pre-pecfed n order to obtan atfactory dturbance rejecton properte. Thu, f a perfect dturbance rejecton requred at a pecfed frequency, - mut nclude a ero correpondng to th frequency. In partcular, f a perfect load dturbance rejecton n teady-tate.e. ero frequency dered, yp - mut nclude a term - - n the numerator, whch lead to a value of the gan

39 Computer Control ytem 63 equal to ero for =. Th coherent wth the reult gven n ecton.4.3., becaue a ero of yp - correpond to a pole of the open loop ytem. The tranfer functon between the dturbance pt and the nput of the plant ut nput entvty functon gven by R R up.5.5 The analy of th functon allow one to evaluate the nfluence of a dturbance upon the plant nput, and to pecfy a factor of the polynomal R - f the controller mut not react to dturbance concentrated n a partcular frequency regon. When noe added to the meaured output ee Fgure.8, mportant nformaton can be retreved by the tranfer functon that relate the noe bt to the plant output yt noe-output entvty functon. R R yb.5.6 the noe energy often concentrated at hgh frequency, attenton hould be pad n order to obtan a low gan of the tranfer functon j yb e n th frequency regon. For T=R, the entvty functon between r and y alo called complementary entvty functon defned a R R yb yr.5.7 Note that yr yp yb yp whch mple an nterdependence between thee entvty functon. Notce that ub -, the tranfer functon between the noe and the plant nput, equal to up -. nother mportant tranfer functon decrbe the nfluence on the output of a dturbance vt on the plant nput. Th entvty functon nput dturbanceoutput entvty functon gven by R yv.5.8

40 64 Dgtal Control ytem The mportance of th entvty functon that t enhance the poble mplfcaton of untable plant pole by the ero of R -. In order to clarfy th pont, let u conder the aumpton R - = - plant pole compenaton by controller ero and uppoe that the plant to be controlled untable - ha root outde the unt crcle. In th cae ] [ yv yb up yp Note that yp, up, yb are table tranfer functon f - choen n order to have table, that whle the entvty functon yv - untable. Th remark yeld to the followng general tatement: The feedback ytem preented n Fgure.8 aymptotcally table f and only f all the four entvty functon yp, up, yb or yr and yv decrbng the relaton between dturbance on one hand and plant nput or output on the other hand are aymptotcally table. The et of fve tranfer functon H OL -, yp -, up -, yb - or yr - and yv - alo play an mportant role n the cloed loop ytem robutne analy..5.3 Control ytem wth PI Dgtal Controller In th ecton the degn of dgtal PI controller wll be llutrated. The tranfer functon operator of the dcreted plant wth ero-order hold gven by q a q b q q q H.5.9 For the ake of notaton unformty, we hall often ue, n the cae of contant coeffcent, q - notaton both for the delay operator and the complex varable -.

41 Computer Control ytem 65 The - notaton wll be pecally employed when an nterpretaton n the frequency jt doman needed n th cae e. The dgtal PI controller charactered by the polynomal ee Equaton.5.6 and.5.7: Rq - = Tq - = r + r q -.5. q - = - q -.5. The cloed loop ytem tranfer functon wth repect to the reference rt n the general form gven by Equaton.5.. The charactertc polynomal Pq -, whoe root are the dered cloed loop ytem pole, eentally defne the performance. a general rule, t choen a a econd-order polynomal correpondng to the dcretaton of a econd-order contnuou-tme ytem wth a pecfed natural frequency and dampng and, for example, and can be obtaned on the ba of the dagram gven n Fgure. or. tartng from pecfcaton n the tme doman. The coeffcent correpondng to the polynomal Pq - are obtaned ether by converon table mentoned n Table.4, or by clab and MTL functon gven n ecton.3. In th cae, amplng perod T, natural frequency and dampng mut be pecfed. We recall that the relaton between and T mut be repected ee ecton.., Equaton..7:.5 T.5 ;.7.5. For a plant havng an equvalent dcrete-tme tranfer operator functon gven by Equaton.5.9, and the ue of a dgtal PI controller, the cloed loop ytem pole are gven by Equaton.5.3, and they are + a q - - q - + b q - r + r q - = + p q - + p q y rearrangng the term n Equaton.5.3 n acendng q - power, we get + a - + r b q - + b r - a q - = + p q - + p q For the polynomal Equaton.5.4 to be verfed, t neceary that the coeffcent of the ame q - power mut be equal on both de. Thu the followng ytem obtaned: a rb b r a p p.5.5

42 66 Dgtal Control ytem whch gve for r and r the reult p a r ; b r p a.5.6 b One can ee that the parameter of the controller depend upon the performance pecfcaton the dered cloed loop pole and the plant model parameter. y ung Equaton.5.7, one can obtan the parameter of the contnuou-tme PI controller: K r ; T Tr r r.6 naly of the Cloed Loop ampled-data ytem n the Frequency Doman.6. Cloed Loop ytem tablty In the cae of contnuou-tme ytem, t wa hown n Chapter, ecton..5, how to ue the open loop tranfer functon repreentaton n the complex plane the Nyqut plot n order to analye the cloed loop ytem tablty and the robutne wth repect to the parameter varaton or uncertante on the parameter value. The ame approach can be appled to the cae of ampled-data ytem. The Nyqut plot for ampled-data ytem can be drawn ung the functon Nyqut-ol.c clab and Nyqut-ol.m MTL 6. Fgure.9 how the Nyqut plot of an open loop ampled-data ytem ncludng a plant repreented by the correpondng tranfer functon H - = - / - and a RT controller. In th cae, the open loop tranfer functon gven by H j j e R e.6. j j e e j OL e The vector lnkng the plane orgn to a pont belongng to the Nyqut plot of the tranfer functon repreent H OL e -j for a pecfed normaled radan frequency = = f/f. The condered range of varaton of the radan natural 6 To be downloaded from the book webte.

43 Computer Control ytem 67 frequency between and correpondng to an unnormaled frequency varaton between and.5 f. Crtcal pont Im H - Re H - yp = + H e -j OL H e -j OL Fgure.9. Nyqut plot for a ampled-data ytem tranfer functon and the crtcal pont In th dagram the pont [-, j] the crtcal pont. Fgure.9 clearly how, the vector lnkng the pont [-, j] to the Nyqut plot of H OL e -j ha the expreon R yp H OL.6. Th vector repreent the nvere of the output entvty functon yp - ee Equaton.5.4 and the ero of - yp - correpond to the cloed loop ytem pole ee Equaton.5.3. In order to have an aymptotcally table cloed loop ytem, t neceary that all the ero of - yp - that are the pole of yp - be nde the unt crcle <. The neceary and uffcent condton for the aymptotc tablty of the cloed loop ytem are gven by the Nyqut crteron. For ytem havng table pole n open loop n th cae - = and - = the Nyqut tablty crteron tate table open loop ytem: The Nyqut plot of H OL - travered n the ene of growng frequence from to, leave the crtcal pont [-, j] on the left. a general rule, for the gven nomnal plant model - / -, polynomal Rq - and q - are computed n order to have R - = P where P - a polynomal wth aymptotcally table root. a conequence, for the nomnal value of - and -, nce the cloed loop ytem table, the open loop tranfer functon:

44 68 Dgtal Control ytem H OL R doe not encrcle the crtcal pont f - and - have ther root nde the unt crcle. In the cae of an untable open loop ytem, ether f - ha ome pole outde the unt crcle untable plant, or f the computed controller untable n open loop - ha ome pole outde the unt crcle, the tablty crteron : The Nyqut plot of H OL - travered n the ene of growng frequence from to, leave the crtcal pont [-, j] on the left and the number of counter clockwe encrclement of the crtcal pont hould be equal to the number of untable pole of the open loop ytem 7. Note that the Nyqut locu between.5 f and f the ymmetrc of the Nyqut locu between and.5 f wth repect to the real ax. The general Nyqut crteron formula that gve the number of encrclement around the crtcal pont N P CL P OL P CL P OL where the number of cloed loop untable pole and the number of open loop untable pole. Potve value of N correpond to clockwe encrclement around the crtcal pont. In order that the cloed loop ytem be aymptotcally table t neceary that N P OL. Fgure.3 how two nteretng Nyqut loc. If the plant table n open loop and the controller computed on the ba of Equaton.6.3 to obtan a dered table cloed loop polynomal P - th mean that the nomnal cloed loop ytem table too, then, f a Nyqut plot of the form of Fgure.3a obtaned, one conclude that the controller untable n open loop. Th tuaton mut be generally avoded 8, and th can be acheved by reducng the dered cloed loop dynamc performance by modfyng P -. 7 The crteron hold even f an untable pole-ero cancellaton occur. The number of encrclement hould be equal to the number of untable pole wthout takng nto account the poble cancellaton. 8 Note that there ext ome «pathologcal» tranfer functon - / - wth untable pole and/or ero that can be only tabled by controller that are untable n open loop.

45 Computer Control ytem 69 Im H Open Loop untable pole table Cloed Loop a - Re H table Open Loop Untable Cloed Loop b Fgure.3. Nyqut plot: a untable ytem n open loop but table n cloed loop; b table ytem n open loop but untable n cloed loop.6. Cloed Loop ytem Robutne When degnng a control ytem, one ha to take nto account the plant model uncertante uncertante of the parameter value or of the frequency charactertc, varaton of the parameter, etc.. It therefore extremely mportant to ae f the tablty of the cloed loop guaranteed n the preence of the plant model uncertante. The cloed loop wll be termed robut f the tablty guaranteed for a gven et of model uncertante. The robutne of the cloed loop related to the mnmal dtance between the Nyqut plot for the nomnal plant model and the crtcal pont a well a to the frequency charactertc of the modulu of the entvty functon. The followng element help to evaluate how far the crtcal pont [-, j] ee Fgure.3: Gan margn; Phae margn; Delay margn; Modulu margn. Gan Margn j The gan margn G equal the nvere of H OL e gan for the frequency correpondng to a phae hft = -8 ee Fgure.3. The gan margn often expreed n d. In other word, the gan margn gve the maxmum admble ncreae of the open loop gan for the frequency correpondng to = - 8. G for H j 8 OL e 8 8