Z-transformation in simulation of continuous system

Transcription

1 Z-tranformation in imulation of continuou tem Mirolav Kašpar, Alexandr Štefek Univerit of defence Abtract Motl ued method for continuou tem imulation i uing algorithm for numeric olving of differential equation tem Thee algorithm are uual more compute-intenive Paper i conidering b poibilit of uing dicrete method for olving continuou tem Thi kind of tem i fater and accurac i better then other alternative method (like Euler method of order) One part of experiment i compare of output from three imulation tpe: olving b RungeKutt 4order method olving b Euler order method olving b new method In paper there i compare of whole computing proce b numeric method and new method At the end there i anali of preparing and uing new method to olve continuou tem Main accent i done on computing time of all method and on output accurac too Stem imulation In tem imulation we are uing operator decribe of tem' block We need time dicreet for mot count We mut ue dicrete time, to imulate continue tem too We mut count input and output in ever tep of imulation, it mean count output value i, which depend on input value ui and inner tem tate xi

2 u i Block of tem g i i Inner tate xi Picture : One block of continue tem Stem gain i decribed b differential equation or b operator equation In one tep olving, the gain of block i decribed b tem of differential equation Thee equation are olved b numeric method 3 Numeric method Euler method i baic method for olving differential equation Thi method i fat, but their reaon have big mitake Thi mitake depend on method tep We olve differential equation b Euler method: ' f (, t) C tangent at point A A B t h t k t k+ t Euler method equation: h f t, ( t h) ( t), where h tep ize and f,t Picture : Euler method i continuou function

3 From picture we can ee, that mitake of thi method i o big (ize of abcia BC) It reaon, wh we ue more difficult method, motl higher order Mot ued method i Runge-Kutt 4order method Thi method ha ver good reaon at low computing tangent at point A, B, C F A B C t h t k t k+ t Picture 3: Runge-Kutt method 4order Runge-Kutt 4order olve the ame differential equation: ' f (, t) The count of thi method i different b olving partial reaon Each tep i divided into 4 part k až k4 Step reaon i count from thee four partial reaon k hf,t, k hf k, t h, k 3 hf k, t h, k hf, t h 4 k 3 New tep i counted b: k k k 3 k Thi method ha better reaon then Euler method, but it contain 4 time more count

4 4 Solving b uing Z-tranformation For olve continue tem b uing Z-tranformation, we mut ue time dicrete and block dicrete too To tem we mut inert ampler and haper Into block come dicrete input o we mut Z-tranformation There i the ame tem at picture 4 like tem at picture, but tranformed to dicrete ampler tem block harper u i T = 0,05 () G i e T Picture 4: Block of tem with ued Z-tranformation Gain of thi tem can be count a multiple of gain Gi(z) and gain of harper G -T - e ) Gi( ), ( G ( z) Z{ G i( ) - e -T }, where Z{} mean Z-tranformation and -T - e i haper gain For reaon of thi count i gain we ue definition of gain: z) G(z), u(z) and than we get difference equation: k n) a k n ) a k n ) a b u( k n ) b u( k n ) b u( k) n n k) We move difference equation b n tep back and get output k): k ) a k) a b u( k) b u( k ) b 0 0 k ) a m n u( k m) k n) Now we have difference equation that can be ued in imulation Input into thi difference equation are onl lat value of input and lat value of output

5 5 Experiment A a tem for experiment we take followed tem, that can be olved analtic (example i taken from literature page 04) Stem cheme: w - 0 u 0 (0 ) Picture 5: Solved tem Differential equation that decribe tem i: ' ' 0' 00 00w, unit pule repone of thi tem i : 5 e t (co( 75 t) 5 in( t)) Uing Z-method mean inert ampler and harper into tem The tem look like on picture 3 after that w T = 0,05-0 u 0 (0 ) e T Picture 6: Stem olved b Z-method We inert ampler and harper into tem and count dicrete tranfer function -T ( 00 - e 0044z G z) Z{ } 0 00 z 44z () After ue invere Z-tranformation the difference equation i k 44k 06065k 0044wk 00889wk

6 Repone for (t): Repone for (t),600,400,00,000 0,800 0,600 0,400 0,00 0,000 Z-tran Euler RungeKutt Analtic 0,00 0,5 0,30 0,45 0,60 0,75 0,90,05,0,35,50,65,80,95 Picture 7: Repone for (t) for each method Output error for repone for (t) 0,350 0,300 0,50 0,00 0,50 Z-tran Euler RungeKutt 0,00 0,050 0,000 0,00 0,0 0,0 0,30 0,40 0,50 0,60 0,70 0,80 0,90,00,0,0,30,40,50,60,70,80,90,00 Picture 8: Output error of repone for (t) for each method Second experiment ue rectangle input ignal with period T = 06

7 Repone for rectangle input,600,400,00,000 0,800 Z-tran 0,600 0,400 Euler RungeKutt 0,00 0,000-0,00-0,400 0,00 0,5 0,30 0,45 0,60 0,75 0,90,05,0,35,50,65,80,95 Picture 9: Repone for rectangle input for each method Output error for repone for rectangle input 0,300 0,50 0,00 0,50 0,00 Z-tran Euler 0,050 0,000 0,00 0,0 0,0 0,30 0,40 0,50 0,60 0,70 0,80 0,90,00,0,0,30,40,50,60,70,80,90,00 Picture 0: Output error of repone for rectangle input for each method Here in t poible to count tem analtic, o we ue Runge-Kutt 4order method a accurate input 6 Advantage and handicap of new method There i one big advantage and it i impl count of new method When we compare the number of count with Runge-Kutt 4order method we find that the ame equation need 4 time le count Next advantage i implif uing thi block in combination tem (dicrete and continuo)

8 When we compare new method and Euler method, we find that at ame number of count the new method i more precie At the other ide there i problem with automatic tranferring ever differential equation to difference equation (it reduce uing of new method onl to linear tem) Next handicap i firm tep With new method we mut ue firm tep, which i ame a ample period When we ue numeric olving of differential equation, we can count tem, where the tep period in t contant Although there are more handicap than advantage, we hould thing (motl in pecial application) about uing new method and ue Z-tranformation to difference equation and olving them 7 Bibliograph [] Krupka Zdeněk, Řeřucha Vladimír, Štefek Alexandr, Automatické řízení I a II, Vokoškolká učebnice, Vojenká akademie Brno 00, 00 Author: Ing Mirolav Kašpar, Doc Dr Ing Alexandr Štefek Katedra témů PVO, Fakulta vojenkých technologií Univerzita obran Kounicova 65, 6 00 Brno, ČR Czech Republic , mirolavkapar@unobcz, alexandrtefek@unbocz