Robust PID Controller Design for Performance Based on Ultimate Plant Parameters

Transcription

1 Proeedng of the 8th IC Sympoum on obut Control Degn, Brtlv, Slovk epubl, July 8-, 5 obut PID Controller Degn for Performne Bed on Ultmte Plnt Prmeter Š. Buz,. ozáková, V. Veelý ulty of Eletrl Engneerng nd Informton Tehnology, Slovk Unverty of Tehnology n Brtlv, Ilkovčov 3, S-8 9 Brtlv, Slovk epubl (e-ml: tefn.buz@tub.k, len.kozkov@tub.k, vojteh.veely@tub.k btrt: The pper del wth new robut PID ontroller degn method bed on ntegrtng requrement on trnent performne nto the populr frequeny-domn Zegler-Nhol degn pproh. The developed method provde upport to the degner by onvertng dentfed ultmte plnt prmeter nto PID ontroller prmeter ung vrble weght tht depend on epeted mmum overhoot m nd ettlng tme t of the loed-loop tep repone. n etenon of th method propoed whh enble to degn PID ontroller gurnteeng robut tblty nd nomnl performne for the unertn plnt modeled ung untrutured unertnty. ulfllment of performne pefton gurnteed for the nomnl model well for the wort-e one. The mn dvntge of the propoed method dret ntegrton of performne requrement n the degn proedure. Effetvene of the propoed robut PID degn method verfed by multon nd eperment on the rel plnt. eyword: performne evluton, overhoot, ettlng tme, ultmte frequeny, phe mrgn. INTODUCTION Quk omputton of ontroller prmeter nd mple lgorthmton re mn ttrbute due to whh the frequeny repone Zegler-Nhol method (Zegler nd Nhol, 94 wdely ued for tunng PID ontroller mplemented n ndutrl ontrol loop. However, t loed degn method not llowng the degner to modfy the performne wth repet to the pef tehnologl proe (Buz et l., 4. There re everl mlr method etendng the Zegler- Nhol frequeny method (Mvoy nd Johnon, 967, (tknon nd Dvey, 968, (Tnhm, 989. ordng to the orgnl Zegler-Nhol frequeny method the PID tunng rule re of the form PID =(P,T,T d =(,, 3 T = =(.6,.5,.5T, where (,, 3 re weght of ultmte plnt prmeter T nd,, (Pettt nd Crr, 987 propoe three ettng (,, 3 =(,.5,.5, (.5,,.67, (.67,,.67. The frt two ledng to underdmped nd perod repone of the output vrble, repetvely, nd the thrd one to repone on the perodty border. ther thn fed vlue (rbog nd lnl, 996 propoe ntervl.3,.6,.3,.46, 3.33,.469, however, wthout ny reommendton wth repet to epeted performne. However, ement of epeted performne heved by PID ontroller tuned ordng to thee method very ppromte nd only repreenttve (Åtröm nd Hägglund,. To remove th drwbk, the propoed modfed frequeny repone Zegler-Nhol method llow to heve pefed mmum overhoot m %,5% nd ettlng tme t 7/,/ of the loed-loop repone to the etpont tep hnge, where the plnt rtl frequeny. ulfllment of performne pefton gurnteed for the nomnl model well for the wort-e one. The pper orgnzed follow: Seton derbe the ll frequeny repone Zegler-Nhol method nd demontrte t modfton wth repet to trnent performne requrement. heved performne eed nd modfed Zegler-Nhol tunng rule for vrou vlue of mmum overhoot nd ettlng tme re provded. In Seton 3, n et etenon of the new method propoed yeldng robut PID degn rule for unertn ytem modelled ung untrutured unertnty. The propoed method w verfed v multon on benhmrk emple nd on rel plnt - DC motor wth vrble lod torque; the reult re ummrzed n Seton 4.. PID CONTOE DESIN. Zegler-Nhol frequeny repone method: prnple nd nly The frequeny domn Zegler-Nhol method (94 dret engneerng method wth ft rejeton of the dturbne z(t beng mot frequently ted n tehnl lterture. To degn ontroller, only two hrtert prmeter of the unknown plnt re to be dentfed. ely wth Hytere w(t e(t u(t PID ontroller 3 ( - SW Idel ely g.. eedbk ontrol loop. Copyrght 5 IC 387

2 Conder the feedbk loop n g. ; put the PID ontroller n proportonl mode (SW=3 nd nree the gn of the ontroller (= untl the output ehbt pertent ollton; from them, the rtl perod T nd the relted rtl gn re red. If onderng the tndrd ntertng form of the PID ontroller T ( d, ( T where proportonl gn, T, T d re ntegrl nd dervtve ontnt, repetvely, oeffent of P, PI nd PID ontroller re lulted ordng to the Tble. Tble. PID tunng rule ordng to the Zegler-Nhol frequeny repone method Controller T T d T p P,5 - - T PI,45,8T -,4T PID,6,5T,5T,85T elton n the lt olumn of Tble n be ued to etmte the domnnt loed-loop dynm T p (Åtröm nd Hägglund, 995. ordng to the Zegler-Nhol frequeny repone method, f the open-loop trnfer funton wth the proportonl ontroller (g. ( j ( j ( j ( j ( t the lmt of ntblty, t n be epreed n polr form ordng to the Nyqut ondton j8 ( j e, (3 where =/T the rtl frequeny of the plnt. rom ompron of ( nd (3 t = nd for = reult the omple equton j8 e ( j, (4 whh epree poton of the plnt rtl pont C=(j wth oordnte {,/,-} on the negtve hlf- of the omple plne. Th pont roed by the frequeny hrtert of the unknown plnt. If we ubttute the Zegler-Nhol PID ontroller tunng rule from Tble nto the frequeny repone trnfer funton of the PID ontroller ( j j Td (5 T nd onder rtl frequeny, we obtn the omple number ( j,6 T j 8 T,66e T,5. T. j5, wth mgntude dependng olely on the rtl gn of the plnt, nd ontnt rgument. Hene, the PID ontroller degned by the Zegler-Nhol method move the rtl pont C of the plnt wth oordnte (4 nto the fed poton n the omple plne (j =(j (j ( j e j8 j5 j55,66 e,46e,6 j, 8 whh wll be one pont of the open-loop Nyqut plot (j under the degned PID ontroller (ee g... Prnple of the modfed frequeny repone Zegler-Nhol method for pefed performne The preented modfed veron of the Zegler-Nhol method ntegrte performne requrement nto t ll veron (Buz et l., 4. The PID ontroller tuned ung the derved modfton of the Zegler-Nhol tble whh nlude eprte rule for djutng ontroller oeffent for: mmum overhoot m {%,%,%,3%,4%,5%}, ettlng tme t {7/,/,3/,6/,9/,/ }. Prnple of the propoed modfton ont n movng the dentfed rtl pont of the plnt C=(j =[-/,j] ung PID ontroller nto the omple plne pont (j =+jy whh wll be pont of the Nyqut plot (j of the degned openloop (ee g. b. Th ompenton rred out t rtl frequeny of the plnt. Coordnte nd y pefyng the future poton of the rtl pont C t wll depend on the epeted performne pefed by the degner n term of m nd t. Mthemtlly, th ompenton n be derbed by the open-loop trnfer funton t : ( j ( j ( j jy. (6 (j - (j g.. Illutrton of movng the rtl pont C nto ZN =[-,6-j,8] (by Z-N method; b =[+jy] (by the propoed method. fter ubttutng oordnte of the rtl pont C nto (6, the ontroller trnfer funton (j turn nto omple number ( j ( j jy. (7 ( j If equtng (7 nd the PID ontroller frequeny trnfer funton (5, ontroller oeffent n be obtned from the omple equton t = C ZN / (j Im (j e CZN PID rtg (j. (8 jy j T d T (j b C To lulte PID ontroller oeffent, followng relton reultng from (8 re ued - _ y C / (j (j Im y PID e 388

3 ; Td y. (9 T Two of the unknown prmeter our n (9b: T nd T d. To obtn unmbguou oluton, ntrodue new vrble rto of the ntegrl nd dervtve ontnt β=t /T d nd ubttute for T =βt d n (9b. fter mple mnpulton, the dervtve ontnt n be obtned by olvng qudrt equton n T d d T y T. ( d Conderng (9, oluton of ( provde followng relton to lulte PID ontroller oeffent T ; T T d d y y. ( 4 fter ubttutng =π/t nto (9 nd (b nd hoong =4 we obtn ; y y T T T ; Td. ( 4 fter mll modfton, fnl relton for lultng PID ontroller oeffent re obtned (Tble. Tble. elton for lultng PID ontroller oeffent by ll nd modfed Zegler-Nhol method Method T T d Cll,6,5T,5T Z-N Modfed Z-N y y T y y T 4 4 Note tht the rtl pont of the plnt n be dentfed ung the well-known (rely eperment (Åtröm nd Hägglund, 995. Whle ordng to Zegler-Nhol, the PID ontroller prmeter re omputed ung the formul PID =(P,T,T d =(, T, 3 T wth fed weght (,, 3 =(,6;,5;,5 on rtl prmeter nd T, the propoed new method provde tne n onvertng the dentfed rtl prmeter of the plnt nto PID ontroller oeffent ung vrble weght (,, 3 gven ; y y ; 3, 5, (3 whh depend on poton (proportonl gn nd on the rto y/ (ontnt T nd T d. Tunng formul of the modfed Zegler-Nhol method re n Tble. We k the queton: Wht the relton between the oordnte of the pont (j =+jy nd the epeted mmum overhoot m nd ettlng tme t (Buz et l., 4?.3 Performne evluton nd reton of the modfed Zegler-Nhol tunng rule It well-known from the ontrol theory tht tftory performne heved f n the mddle frequeny rnge the lope of the open-loop mgntude Bode plot - db/dede nd tht of the phe Bode plot -66 /dede. If we dept the poton of the pont (j =+jy n open-loop Bode plot oordnte, we obtn two pont: =[log(j,log ], nd =[rg(,log ]. ordng to g. 3 ther oordnte re : log ( j log y, (4 : rg ( rtg y. (5 It uppoed tht both the mgntude nd phe Bode plot of (j re known only n the pont, yet they onform to the bove performne requrement. Then trght lne n be drwn wth the lope =- db/dede png through wth oordnte (4 epreed by the equton log ( j log y (log log. (6 rom (6 the open-loop mgntude roover frequeny n be etmted (g. 3. or = the mgntude log(j =; fter ubttutng nto (6 we obtn log y (log log. (7 Ung (7, the mgntude roover n be epreed log y log. (8 The trght lne wth lope =-66 /dede png through wth oordnte (5 epreed by the equton y 66log log rg ( rtg (9 enble to etmte the open-loop phe mrgn M (g. 3 M rg. ( g. 3. Identfton of M nd from the ppromton of the mddle frequeny rnge of (j. fter ubttutng for = nd (9 nto ( we obtn M log log rtg y 66. ( The ettlng tme n be etmted ordng to the relton t log (j rg( -8, ( M rtg(y/ =- db/dek. =-66/dek. log log 389

4 Imgnry Imgnry where the urve ftor 3 for perod loed-loop repone (rbbre et l., nd π,4π for olltory output vrble repone. nly of PID ontroller degn for benhmrk emple (Åtröm nd Hägglund, h reveled tht the loedloop performne tftory f, (3 where.5,.95 (Buz et l., 4. eft-hnd-de of the equton t (4 obtned by ubttutng (3 nto ( defne new performne meure, the o-lled reltve ettlng tme t, (5 whh epree the rel ettlng tme weghted by the rtl frequeny of the ontrolled plnt. It dmenonle quntty whh enble to epre epeted loed-loop dynm for plnt wth vrou dynm. mnpulton of the eponent n (8 yeld the equton y, (6 t M t y t M t M (3 M wth entre n C T =-M t /[M t -] nd rd T =M t /-M t obtned for vrou M t nd hene for vrou mmum overhoot η m n the ene of (9. Intereton of depted Hll rle (orrepondng to mmum overhoot wth onentr rle Ω (orrepondng to ettlng tme n the omple plne determne oordnte (+jy of the erhed trget pont of the open loop frequeny repone (j. The et of M T rle for mmum overhoot m =% to 5%, nd Ω rle for ettlng tme for γ/τ = / =.5,.65,.8 nd.95 re depted n g Nyqut Dgrm % 4% 6% Nyqut Dgrm % whh fter qurng n be mnpulted to obtn the equton of rle el / -6 3% % % el y y (7 wth the rdu = / =γ/τ pefyng the reltve ettlng tme (5 for the ontnt urve ftor. Prmeter = / =.5,.65,.8 nd.95 ubttuted nto (7 defne the et of onentr rle Ω of ettlng tme entered n the orgn of the omple plne (Buz et l., 4. If the loed-loop mgntude plot T(j non-monoton funton of ngulr frequeny,.e. f t h mgntude pek (Åtröm nd Hägglund, 995 M t ( j upt ( j up, (8 ( j the mmum overhoot m of the loed-loop tep repone n be etmted ordng to,8m t T( m %. (9 T( If the ontroller nlude the ntegrtor hnnel of the ontrol error then T( =. The qured mgntude M t y (3 y fter ubttutng for (j =+jy nd = nto (8, fter ome mnpulton gve the equton of Hll rle M T g. 4. Croeton of M T nd Ω rle ued to determne oordnte,y of the pont (j =+jy for the weght (3. If n (9 the equlty tfed, the Nyqut plot of (j jut touhe the Hll rle M T, n e of nequlty t vod the re delneted by M T. Intereton of M T nd rle n be lulted by olvng the et of equton (7 nd (3 wth repet to,y: 8 m, y m. (3 The oordnte (3 pefy the pont n the omple plne, nto whh moved the dentfed rtl pont C of the plnt t the rtl frequeny. fter ubttutng (3 nto the weght (3 we obtn the modfed Zegler-Nhol tunng tble (Tble 3. Tble 3. Modfed Zegler-Nhol tble for requred mmum overhoot nd (reltve ettlng tme m % % % 3 % 4 % 5 % T.535T.875T.7948T.687T.683T.5777T T d.634t.76t.987t.77t.546t.444t T.83T.4364T.7T.9885T.48T.468T T d.5t.359t.475t.497t.537t.567t 39

5 3. OBUST PID CONTOE DESIN Conder the unertn ytem modeled by nomnl model nd untrutured unertnty. Hene, the ultmte prmeter (, beome vryng prmeter hrterzng the plnt. Meurement of ultmte prmeter under ndvdul unertnte n be performed ung repeted rely dentfton eperment (Åtröm nd Hägglund, 995. et of ultmte pont of the unertn plnt obtned (g. 5 rg ( ( j rg ( j... ( j n rg ( n C ( j e... n j ( n / /... / n e j. (33 Number of repeted eperment n= p, where p the number of vryng tehnologl quntte of the plnt. Phe dely of eh ultmte pont -8 ; nomnl ultmte prmeter re obtned ordng to m/ mn /, m mn. (34 two-dmenonl polytope of the unertn plnt n the oordnte (/, depted n g. 5b. 3. Symmetrton wth repet to the nomnl ultmte frequeny nd plnt dentfton By wthng between SW= (del rely nd SW= (rely wth hytere n the loop n g., ultmte pont C = (j nd ulry pont E = (j e, repetvely re meured. ulry pont ehbt phe hft ppromtely 5-8 wth repet to C = (j. Meured C nd E re ued to fnd out the lope S nd S of the mgntude nd phe plot repetvely, n (j. Then, lner ppromton of the Bode plot of the unertn plnt ppled, nd mgntude (j nd phe rg ( re lulted t the nomnl ultmte frequeny ordng to g. 7. g. 7. Dperon rle M for, b mutul poton of the pont,, e, e, o, o. If SW=, tble lmt yle re n the loop n (g. wth generl frequeny,.e. the open-loop ( for =...n meet the Nyqut ondton log ( j, rg (. (35 g. 5. mly of Nyqut plot (j, b unertnty polytope of ultmte prmeter (/,, =...n. The ontrol lw generted by the robut PID ontroller rob ( degned for the nomnl pont (j tully rre out the mppng :{ : = rob (j } of the et of dentfed pont (j enrled by M wth the rdu onto the et of pont (j delneted by M, nd lulte the rdu ( of the dperon rle M whh enloe the pont (j of the Nyqut plot o to gurntee fulfllment of the robut tblty ondton (g. 6. rom the equvlent trnfer funton of the rely wth hytere reult 4B log ( j log, C rg ( rn, (36 C where the wdth of hytere. Bode plot of the open loop ytem (= ( I ( re obtned follow log ( j log ( j log ( j, (37 rg ( rg ( rg (. (38 fter ubttutng (35 nd (36 we obtn the fnl epreon log ( j 4B C log, rg ( rn. (39 C Coordnte of dentfed ultmte pont nd pont n ther vnty re olleted n Tble 4. Tble 4. Coodnte of pont C, nd E, =...n g. 6. Dperon rle M nd M ; the mppng :{ }. ordng to (33, to eh dentfed pont C orrepond dfferent ultmte, =...n; thu poton of C ( determned ung ymmetrton wth repet to the nomnl ultmte frequeny. Ultmte pont C n ulry pont E n e C Ce log (j log log (j e log 4B 4B rg ( rg ( e rn C e 39

6 To dentfy the ultmte pont C zero hytere = djuted. rom the lmt yle, the mpltude C nd the perod T of rtl ollton re found out, ultmte frequeny =/T. fterwrd, the hytere =.5 djuted to dentfy the pont E. The mpltude C e nd perod T e re gn found out from the orrepondng lmt yle, meured frequeny of E e =/T e. rom meured pont C nd E wth o-ordnte C : ;log ; ;, (4 C e E : e;log ; rn e; e e (4 4B Ce we n lulte lope of Bode plot n g. 7b of the unertn plnt ordng to e log e log, e. (4 log e log fter ubttutng nto (4 oordnte from Tble 4 the reultng lope re obtned C e log log, (43 log e log 4B rn Ce. (44 log log e The objetve to fnd oordnte of pont of the dperon Nyqut plot of the plnt (j t nomnl ultmte frequeny OI j rg ( l joi ( j ( j e e (45 from the lope S nd S of the dperon Bode plot of the unertn plnt, nd the oordnte of dentfed ultmte pont,, n logrthm le. O, O, re oordnte of the pont O on dperon Bode plot t the nomnl ultmte frequeny. Coordnte O, O reult from g. 7b O O C S (log log, (46 C S(log log. (47 fter ubttutng men oordnte (4 nd (4 we obtn O log S (log log, (48 (log log. (49 O S ordng to (45, oordnte of pont on the dperon Nyqut plot t nomnl ultmte frequeny re O log log log log j... log log n log n ( j e... n n logn log. (5 3. obut PID Controller Tunng ule The nomnl pont C = (j loted on the rel of the omple plne (g. 6; t oordnte re ( j jb j. (5 Obvouly, (j =( +b,5 =/, ( =-π. du of the dperon rle n be found mmum dtne between the -th dentfed pont (j nd the nomnl pont (j b b m =,,...,n, (5 where nd b re oordnte of (j = +jb, =,,...,n. The robut ontroller hft the nomnl pont (j nto =(,jy. Coordnte (,jy re pefed from nomnl performne requrement nd the robut tblty ondton derved n Seton 3.3. obut ontroller tunng rule re hown n Tble ; oordnte =, y=y of the pont re to be ued. 3.3 obut tblty ondton bed on the modfed frequeny repone Zegler-Nhol method The robut PID ontroller trnform the dperon rle M of the plnt (n blue nto the rle M ; the whole fmly of open-loop Nyqut plot re png through M. The ontroller h to gurntee tht the green rle M vod the pont (-,j. Theorem (Suffent ondton for robut tblty under PID ontroller. Conder n unertn ontnuou-tme tble dynm ytem derbed by untrutured unertnty. The PID ontroller ( tuned ordng to the rule n Tble (lt row gurntee robut loed-loop tblty f the followng ondton tfed. (53 Proof: The proof n be performed ordng to g. 8. g. 8. rphl nterpretton of the robut tblty ondton. 39

7 The dtne + (j n g. 8 n be epreed ung the one rule for the trngle (-,,,,,, o,. (54 rom the trngle (,,, epreed follow o y. (55 fter ubttutng (55 n (54 we obtn y. (56 urther, ordng to g. 8 nd (56. (57 y Trnformton between the rd nd ( j (58 reult from the epreon of ompenton of the nomnl ultmte pont of the plnt C =[,-/,j] nto the pont =[,,jy ] n the followng form ( j ( j y. (59 ( j du of the rle M n be lulted from (58 nd (59 ung the relton. (6 y fter ubttutng (57 nd (6 n the generl robut tblty ondton <+ (j we obtn y y, (6 from whh we obtn fter ome mnpulton y. (6 Settlng tme mplemented n the robut ontrol lw mlrly n the orgnl method ung y. (63 Subttutng (63 nto (6, fnl form of the robut tblty ondton obtned, (64 whh mthe wth the ondton (53 to be proved..h.. of (64 the tblty degree q. (65 To fulfll the robut tblty ondton, the entre of the rle M h to be loted on ny of the ettlng tme rle Ω orrepondng to /, where the mpltude ( +y / ontnt, nd t the me tme outde the hthed re n the omple plne delneted by (64. Coordnte (,jy of the nomnl pont re hoen wth repet to the requred nomnl performne n term of mmum overhoot m nd reltve ettlng tme ordng to the relton m m 8, y (66 derved n Seton. In ddton, the lulted h to tfy the robut tblty ondton (64. Coeffent of the PID ontroller gurnteeng robut tblty re lulted ung the rule n the lt row of Tble. The Hll rle M T relted wth m depted n g. 8 (red plot. rom (58 nd (59, the rdu n be epreed ung the rto of frequene /. (67 rom (67 reult tht the rdu of the dperon rle proportonl to the rd of reltve ettlng tme rle (dotted plot. Hene, dereng ettlng tme nree unertnty of open-loop Nyqut plot. 4. VEIICTION O THE POPOSED METHOD Emple. et u degn PID ontroller ( for the benhmrk emple ( (68, 3 ung the modfed frequeny repone Zegler-Nhol method; the ontrol objetve to heve requred performne pefed n term of mmum overhoot m =%, %, % nd 3% nd requred reltve ettlng tme τ =7,, 3 nd 6 of the loed-loop tep repone. ll dt needed for the degn of PID ontroller for the plnt ( long wth heved performne meure vlue (mrked wth n the lot olumn re ummrzed n Tble 5. g. prove hevng of the dered performne meure. Tble 5. Summry of gven nd heved performne meure vlue nd orrepondng PID ontroller oeffent Sytem m / [rd/] T T d m / ( % 73,,433,38,96 % ( % 73,,75,36,79 9,7 % ( % 73, 4,347,88,7 8,6 % ( 3 % 73, 4,979,48,6 7,4 % ( 7 73,,8475,366,9 6,85 ( 73,,87,5,3 9,36 ( 3 73,,53,6,55,84 ( 6 73,,755,7,8 5, Cloed-loop tep repone for vrou = t =7 = t = = t =3 = t = Tme [] Tme [] g. 9. Cloed-loop tep repone for ( nd vrou vlue of nd η m Cloed-loop tep repone for vrou m m =% m =% m =% m =3% 393

8 u(t u(t Emple. et u pply the propoed method to ontrol phyl model of DC permnent mgnet motor; ontrolled vrble w the peed nd plnt nput w rmture voltge generted ung the Mtlb-eltme Workhop ontrol ytem. To ene the output thogenertor ued. The dturbne ffetng the motor operton the lod torque z(t depted n the feedbk loop n g.. The robut ontrol objetve w to gurntee m =5% nd τ = for the nomnl model, nd mn =5% nd τ N =5 for the worte model of the DC motor. w(t e(t MTB u(t TW - TCHO- ENETO DC MOTO g.. Control loop wth the DC motor. eultng loed-loop tep repone re depted n g. ; g. prove tht PID ontroller degned by the modfed frequeny repone Zegler-Nhol method gurntee the requred performne meure vlue C [] Obtned performne for the nomnl model: m=5%, = C Tme [] C [] g.. Tme repone of the DC motor..5 Obtned performne for the wort-e model: mn=5%, N= g. how grphl nterpretton of the PID ontroller tunng n frequeny domn. the dperon Nyqut plot of the -.5 DC motor were not vlble, only both dperon rle were - plotted (M wth the rdu nd the both rle delnetng prohbted re: M S wth rdu S t the lulted C [] nomnl ultmte frequeny =.99 rd. -. The dperon rle M of the DC motor h rdu =,. Dperon rle M wth the rdu =.6536 entered n (j =(,y =(-.8,-.4. or t, red-olor Hll rle of overhoot M TN n be plotted wth the rdu TN =7,7 entered n (-4.6,j whh orrepond to mn_obt. =3,7% for the wort-e model of N (j, hene performne requrement re met for the wort-e model. The overhoot Hll rle M T pe through the enter of M nd h rdu T =4,6 orrepondng to the nomnl m_obt. =4,6%, hene requred performne gurnteed for the nomnl model well. rom the rd of ettlng tme rle of the nomnl ( nd the wort-e ( N model reult the orrepondng reltve ettlng tme τ S =4,4 nd τ SN =8,6, whh gn prove fulfllment of the requred performne. - z(t w(t w(t u(t u(t Performne meure: mn=5%, regn=, m=5%, reg=5 M S + M S M T Im N M g.. obut PID tunng for the DC motor: grphl nterpretton n the omple plne. S CNOWEDMENT Th reerh work h been upported by the Sentf rnt geny of the Mntry of Eduton of the Slovk epubl, rnt No. /4/. EEENCES Åtröm,.J. nd Hägglund, T. (995. PID Controller: Theory, Degn nd Tunng, nd Edton, Intrument Soety of mer, eerh Trngle Prk, ISBN Åtröm,.J. nd Hägglund, T. (. Benhmrk Sytem for PID Control. IC Workhop on Dgtl Control PID', pp. 8-8, Terr, Spn, prl 5-7,. tknon, P. nd Dvey,.. (968. Theoretl pproh to the Tunng of Pneumt Three-term Controller. Control, Mrh, pp Buz, Š., ozáková,., Veelý, V. nd ozák, Š. (4. Novel PID Controller Degn Methodology for Spefed Performne Ung Ultmte Plnt Prmeter. In: Proeedng of the 9th World Congre of the IC 4: Cpe Town, pp ISBN rbog, D., lnl,. (996. Tunng PID Controller Prmeter Ung Tbu Serh lgorthm, Proeedng of the IEEE Interntonl Conferene on Sytem, Mn nd Cybernet, pp Mvoy, T.J. nd Johnon, E.. (967. Qulty of Control Problem for Ded-tme Plnt. Indutrl nd Engneerng Chemtry Proe Degn nd Development, Vol. 6, pp Pettt, J.W. nd Crr, D.M. (987. Self-tunng ontroller, US Ptent No. 4,669,4. Tnhm, B. (989. Tunng PID ontroller, Control nd Intrumentton, September, pp Zegler, J.. nd Nhol, N.B. (94. Optmum Settng for utomt Controller, SME Trnton, Vol.64, pp M N (j (j M TN M M + e (j 394