Discrete-Time PID (n-2) Stage PD Cascade Controllers with First Order Hold and Delayed First Order Hold Discretizations

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1 Proceeing of the International MultiConference of Engineer an Comuter Scientit 7 Vol I IMECS 7 March Hong Kong Dicrete-Time PID (n- Stage PD Cacae Controller with Firt Orer Hol an Delaye Firt Orer Hol Dicretiation Channarong Chiengtee Pittaya Pannil Praart Ukakimaarn an Thanit Triuwannawat Abtract---Thi aer rooe a technique to eign the (n- tage PD (Proortional-Derivative controller cacae with the PID (Proortional-Integral-Derivative controller in accorance with nth orer lant The Continuou-Time (CT eign i firtly reviewe to how the avantage of the Kitti metho The rooe technique i bae on the Kitti metho in combination with the ue of Firt Orer Hol (FOH to icretie the CT lant an Delaye Firt Orer Hol (DFOH to icretie the CT controller for obtaining the roer Dicrete-Time (DT controller tructure The imulation reult confirm that the rooe eign technique can be alie to the DT framework with better ecification than it wa execte Inex Term Continuou-Time / Dicrete-Time PID (n- PD controller Firt Orer Hol Delaye Firt Orer Hol I I INTRODUCTION t i known that mot inutrial lant are tye an conit of three to five firt orer lag or ea time lu one firt orer lag [] However the PID controller i roerly alie to a tyical econ orer lant only In orer to control a thir orer ytem to obtain the given ecification an analytic PIDA (Proortional-Integral- Derivative-Acceleration controller eign technique i then rooe [] For a thir or higher nth orer lant a eign metho bae on root locu technique for the PID (n- tage PD cacae controller in CT framework ha been reente [3] Thi eign technique i aime to atify the eire ecification without trial an error Then the forwar controller i emloye to ecreae the overhoot an the controlle ytem tructure become two egree of freeom (-DOF control ytem a hown in Fig R( G f ( X ( G ( c G ( P Fig Structure of the -DOF control ytem D( Y ( For DT framework three generation of thee PID (n- tage PD cacae controller have been rooe recently The firt eign for the DT PID (n- tage PD cacae controller i uing Zero Orer Hol (ZOH icretiation metho [4] while the econ one i uing Tutin or bilinear icretiation metho to eign the controller in - lane [5] The thir concet to eign DT controller i alo uing Tutin but it i require to tranform the CT eigne controller from -lane to -lane [6] In orer to be an alternative metho for DT controller eign thi aer reent an effective eign technique uing FOH an DFOH icretiation a well a uing Kitti metho The MATLAB imulation reult for verifying the controller erformance are alo inclue II METHODOLOGY Fig how the te for eign of igital control ytem [7] which are major te; lant moeling an controller eign Manucrit receive December 9 6; revie January 9 7 C Chiengtee i with Deartment of Intrumentation an Control Engineering Faculty of Engineering King Mongkut Intitute of Technology Lakrabang Bangkok Thailan 5 ( atom_@hotmailcom P Pannil i with Deartment of Intrumentation an Control Engineering Faculty of Engineering King Mongkut Intitute of Technology Lakrabang Bangkok Thailan 5 (hone : ; fax: ; ittaya@kmitlacth P Ukakimaarn i with Deartment of Intrumentation an Control Engineering Faculty of Engineering King Mongkut Intitute of Technology Lakrabang Bangkok Thailan 5 ( kuraar@gmailcom T Triuwannawat i with Deartment of Intrumentation an Control Engineering Faculty of Engineering King Mongkut Intitute of Technology Lakrabang Bangkok Thailan 5 ( ktthanit@gmailcom ISBN: ISSN: (Print; ISSN: (Online Fig Ste of the igital control ytem eign A Problem Statement From a block iagram of Fig 3 we nee to fin the PID (n- tage PD cacae controller K( or K( for the lant G( o that the given eire ecification coul be accetably achieve IMECS 7

2 Proceeing of the International MultiConference of Engineer an Comuter Scientit 7 Vol I IMECS 7 March Hong Kong R ( E ( U ( Y ( K ( G ( Fig 3 Block iagram of tyical control ytem From the given eire ecification in term of the Percent Overhoot (PO the aming ratio i P O P O ζ ln = π ln = 69 (7 B Continuou Time Framework Let the nth orer lant G( be controlle by the cacae controller K( their tranfer function i aume to be Kn G( = N ( T ( T ( T = ; n= 3 N = ( ( 3( 6 The tranfer function of the PID controller can be tate a Ki ( ( KPID ( = K K = K i ( where K i a roortional gain K i i an integral gain an K i a erivative gain Hence the PD controller tranfer function i K ( = K K = K ( (3 PD ( From the given Settling Time {t (±%} the uname natural frequency i ( ζ ln ωn = ζ = 369 ra/ec (8 t Hence one of the ominant cloe-loo ole i locate at = 8 j (9 The oen-loo tranfer function without at i ( 3( 6 KGwo( = ( ( 3 ( = 69 j3= ( The oen-loo tranfer function for the PID (n- tage PD cacae controller K( an the lant G( can be given by The angle from the ero to i ( KGwo arg[ ] = π arg ( = 737 ( PID Controller ( n PD K i ( ( K ( Kn KG( = ( ( ( nth orer Plant ( 3( 6( = K ( ( 3( 6 N (4 By uing Kitti metho = 3 an = 6 are firtly aigne then fin only an K from the root locu angle an magnitue conition a follow KG( =± (k π k = KG( = The eire ecification to be eigne are uually ecifie in term of tranient an teay tate reone characteritic of the control ytem to a unit-te inut exhibite by a air of comlex-conjugate ominant cloeloo ole ζω ω ζ ± = n ± j n a follow: (5 The location of the ero can fin from Im( = Re( = 789 ( tan arg[ ] Now it i imlie that ( ( 3( 6( 789 KG( = = 8 (3 ( ( 3( The oen-loo gain K can be foun from the magnitue conition of the root locu technique a follow: K = = (4 ζπ ζ Percent Overhoot ( P O = e % = 5% ln( ζ Settling Time ( t = = ec ζωn ( ± % (6 To ecreae the overhoot caue by aing the ero ( to the oen-loo tranfer function KG( the forwar controller can be tate a K f ( = (5 ISBN: ISSN: (Print; ISSN: (Online IMECS 7

3 Proceeing of the International MultiConference of Engineer an Comuter Scientit 7 Vol I IMECS 7 March Hong Kong Imag Axi Real Axi Fig 4 Plot of root loci in -Plane The DT tranfer function of the CT lant G( with the amling time follow: T (ec/amle i icretie by FOH i a ( G( = Z G( (7 T Hence from ( yiel F( ( G( = Z (8 T ( ( 3( 6 where Amlitue Time (ec Fig 5 Unit te reone The overall ytem i then aroximate a if it i a tanar econ orer ytem a follow: Y ( K( = R( ( K( 8854 = ζ ωn ωn (6 Fig 4 how the lot of root loci in -Plane The unit te reone with an without the forwar controller are hown in Fig 5 reectively C Dicrete-Time Framework To eign the DT controller the CT lant (or CT ytem an CT controller can be icretie by FOH [9] an by DFOH [] reectively Then we eign the DT controller in the ame way a the CT framework a b c e F( ( ( 3 ( F( = ( ( 3 ( 6 Then T 7 Z{ F( } = 8 ( 34 ( T 3T ( e 54 ( e 6T 54 ( e Finally we have 7 ( G( = T 8 34 T T ( e ( ( 3T 6T 54 T ( e 54 T ( e For T = / 5 ec/amle we obtain 3 β3 β β β G( = T 3T 6T ( e ( e ( e β = T e = 98 6 β = T e = 94 6 β = 354 6T e = β = 84 Then 5 ( 9539( 968( 97 G( = ( 98( 948( 8869 (9 ( ( ( (3 Fig 6 Dicretiation In thi work to obtain the tructure of DT PID (n- Stage PD cacae controller for a thir orer lant the ISBN: ISSN: (Print; ISSN: (Online IMECS 7

4 Proceeing of the International MultiConference of Engineer an Comuter Scientit 7 Vol I IMECS 7 March Hong Kong DFOH [] i alie Bae on the DFOH the eire DT tranfer function can be tate a K ( K ( = ( ZL (4 T Here Ki K( = K K ( K K Then L 3 ( 3 = b b b b 3 K( b3 b b b = T T K( b b = T T T 3 ( b3 b K ( = T T b T b T ( 3 T ( T ( Finally we have b T b T ( 3 T ( T ( 3 β3 β β β K( = ( β3 b3 β 6 4 T T b = β T 6 T T b β b ( a ( b ( c K( K ( (5 (6 (7 From ( an (6 the oen-loo tranfer function ue to eign the DT PID (n- tage PD cacae controller can be written a ( a ( b ( c K ( G( = K ( (8 5 ( 9539( 968( 97 ( 98( 948( 8869 By uing Kitti Metho to eign the cacae controller K( let = 958 an = 8969 Then a the oen-loo tranfer function without ( c i ( 958( 8969 KGwoc ( = K ( ( 9539( 968( 97 ( 98( 948( b (9 The eire ecification for eign of the controller K( are given in (6 Then the ominant cloe-loo ole in - Plane i T = e = 958 j43 T = 5 ec/ amle (3 Then the neceary angle of the oen-loo tranfer function without the ero ( c at the ominant cloe-loo ole i KGwoc ( = 36 j57 (3 arg [ KGwoc ( ] = 438 From the angle conition of the root locu metho the angle from c to can be written a [ ] π [ KGwo ] arg = arg ( = 5756 c c (3 Since the angle of the ero ( c i le than 9 then c i locate at the left han ie of a follow c Im( = Re( = 93 (33 tan arg ( [ c] Another require arameter i the controller gain K which can be foun from the magnitue conition of the root locu metho a follow K = K( G( = 9683 (34 Finally the controller tranfer function can be tate by ( a ( b ( c K ( = K ( (35 ( 958( 8969( 93 = 9683 ( The root loci for the DT controller with the amling time T = 5 ec/ amle are hown in Fig 7 While the unit te reone are hown in Fig 8 reectively Imag Axi Real Axi Fig 7 Root Loci in -Plane ISBN: ISSN: (Print; ISSN: (Online IMECS 7

5 Proceeing of the International MultiConference of Engineer an Comuter Scientit 7 Vol I IMECS 7 March Hong Kong Amlitue Imag Axi Amlitue No of Samle x Fig 8 Unit Ste Reone Real Axi Fig 9 Root Loci in -Plane 5 5 No of Samle Fig Unit Ste Reone For the amling time T = /5 ec/amle the correoning root loci an unit te reone are hown in Fig 9 an Fig reectively Where the lant tranfer function the controller tranfer function an the ominant cloe loo ole are a follow: III CONCLUSION The eign of PID (n- tage PD cacae controller in CT framework ha been ecribe to oint out the aim of Kitti metho which rovie that all eire ecification to be eigne can be achieve without trial an error te in the eign roce However the original eign bae on Kitti metho ue the forwar controller for ecreaing uneire overhoot Nowaay the forwar controller i rarely or never ue becaue there i alternate way to ecreae the maximum ercent overhoot by increaing the controller gain to be greater than the eigne value o that the lot of root loci are alway towar the real axi along circular hae If the amling time i enough all eire ecification are eaily obtaine REFERENCES [] Davi W Peen A New Look at PID Controller Tuning Tranaction of the ASME Journal of Dynamic Sytem Meaurement an Control Vol Set 994 [] Seul Jung an Richar C Dorf Analytic PIDA Controller Deign Technique for a Thir Orer Sytem Proceeing of the 35 th Conference on Deciion an Control Kobe Jaan December 996 [3] Thanit Triuwannawat Kitti Tiraeth Jongkol Ngamwiwit an Michihiko Iia PID(n- tage PD cacae controller for SISO ytem SICE 98 Proceeing of the 37 th SICE Annual Conference International eion Paer [4] Pittaya Pannil Sukiri Kanchanaomranvong Praart Ukakimaarn Thanit Triuwannawat an Kitti Tiraeth Dicrete PID (n- tage Cacae Controller for SISO Sytem SICE Annual Con- ference The Univerity Electro- Communication Jaan Augut - 8 [5] Krit Smeritak Praart Ukakimaarn Thanit Triuwannawat an Prera Lavanrakai Bilinear Dicrete PID (n- tage PD Cacae Controller for SISO Sytem th International Conference on Control Automation an Sytem in ICC (International Convention Center Jeju Ilan Korea Oct 7- [6] Suralak Charoonote Praart Ukakimaarn Pittaya Pannil an Thanit Triuwannawat Dicrete PID (n- Stage PD Cacae Controller Prooe by Kitti Proceeing of the 4 th IIAE International Conference on Inutrial Alication Engineering 58-6 B-Con Plaa Beu Jaan March 6-36 [7] Hirohi Fujimoto "General Framework of Multirate Samling Control an Alication to Motion Control Sytem" PhD Diertation The Univerity of Tokyo ubmitte in December an ublihe in March [8] Ryoo Nagamune MECH468/55P: Moern Control Engineering / Founation in Control Engineering Lecture Note # 6 Deartment of Mechanical Engineering Univerity of Britih of Columbia (UBC Canaa 8/9 [9] Jee B Hoagg Seth L Lacy R Scott Erwin an Denni S Berntein Firt-Orer-Hol Samling of Poitive Real Sytem An Subace Ientification of Poitive Real Moel Proceeing of the 4 American Control Conference Boton Maachuett June 3 - July 4 [] Maayohi Tomiuka Tu-Chin Tao an Kok-Kia Chew Analyi an Synthei of Dicrete-Time Reetitive Controller Journal of Dynamic Sytem Meaurement an Control Tranaction of the ASME Vol Setember ( 98595( 996( 6 G( = ( 998( 994( ( 995( 989( 993 K( = ( 3 = 996 j443 (36 ISBN: ISSN: (Print; ISSN: (Online IMECS 7