In the Name of Allah, the Most Beneficent, the Most Merciful Root Locus Design Techniques II. Due: Monday 02 October 2006.

Transcription

1 Page 1 of 7 A11 I the Name of Allah, the Most Beefiet, the Most Meriful Root Lous Desig Tehiques II Due: Moday Otober 6. Before 5: m Name Dr WAWY Solutio Setio XX ID No XXXX 1. Give the uity feedbak system i Figure 1, where K( s + 6) G ( s) ( s + )( s + 3)( s + 5) 1.1. The system is oeratig with a domiat ole damig ratio of.77. Desig a PD otroller so that the settlig time is redued by a fator of. Comare the trasiet ad steady-state erformae of the uomesated ad omesated systems. Combie your alulatios betwee maual ad MATLAB. Submit at least: Root lous of uomesated system showig the arameters Root lous of omesated system showig the arameters SIMULINK blok diagram of both omesated ad uomesated system Ste resose lot that has both the uomesated ad omesated resose We a start with the root lous of the uomesated system. %Root Lous um oly([-6]); 6 de oly([- -3-5]); systf(um,de) rlous(sys) lear; Root Lous Here, settig ξ. 77 we a obtai the eessary arameters. K.57 % OS.7 ω 3.7 The, usig these arameters, we a obtai the settlig time. T s 1.73 Seods ξω.77(3.7) The desired erformae requires settlig time to be half of the uomesated system. T.5(1.73).865 s ew The ew erformae a be summarized as follows: Imagiary Axis Gai:.57 Pole: i Damig:.78 Overshoot (%):.7 Frequey (rad/se): 3.7 ξ.77 % OS.7 T s New ξω.865 ω.77(.865) 6.5 From these ew seifiatios, we a determie the ew oles: 1, 1, ξω ± jω 1 ξ.77(6.5) ± j form is: G s) K( s + z ) ( The PD omesator.6 ± j.6 θ θ θ Therefore, we have to determie a zero that auses the root lous to go to oit.6 ±. 6. 1, j θ 1.6 j BMM3613_AC_SEM67_Assigmet_11_Key Dr WAWY SEM I 6/7 Fakulti Kejuruteraa Mekaikal, KUKTEM

2 .6 θ 1 18 ta ( ) θ θ 18 ta ( ) θ θ 3 ta ( ) θ θ ta ( ) θ θ + θ ) ( θ + θ + θ ) (k + 1)18 ( 1 3 ( θ ) ( ) 18 θ.8 θ j Page of 7.6 ta(6.8) x x.58 Therefore, the zero loatio is: -(.6+.58)-7. Therefore, the PD omesator is: ( s) K( s + 7.) G The overall trasfer futio is: K( s + 7.)( s + 6) G ( s) ( s + )( s + 3)( s + 5) Next, we fid the ew root lous so that we a fid gai that math with our ew seifiatios. x Uomesated: K.57 ξ.77 % OS.7 T s 1.73 ω 3.7 Desig objetives: ξ.77 % OS.7 T s New ω Comesated K.68 ξ.77 % OS.33 T s.869 ω 6.51 Imagiary Axis Root Lous Gai:.68 Pole: i Damig:.77 Overshoot (%):.33 Frequey (rad/se): BMM3613_AC_SEM67_Assigmet_11_Key Dr WAWY SEM I 6/7 Fakulti Kejuruteraa Mekaikal, KUKTEM

3 Page 3 of 7 BMM3613_AC_SEM67_Assigmet_11_Key Dr WAWY SEM I 6/7 Fakulti Kejuruteraa Mekaikal, KUKTEM

4 Page of 7 Ste.57(s+6) (s+)(s+3)(s+5) G(s)_Origial Result.68(s+6)(s+7.) (s+)(s+3)(s+5) G(s)_New. For the uity feedbak system of Figure, where K G ( s) ( s + 3)( s + 5) θ 6 5 Figure : Uity Feedbak System.1. By had alulatio, show that the system aot oerate with a settlig time of /3 seod ad a eret overshoot of 1.5% with a simle gai adjustmet. Sie % OS 1.5% % OS l( ) ξ 1 % OS π + l ( ) 1 Next, we determie ω, T s ξω ω ξt With the kowledge of s l(.15).8 π + l (.15).8(.667) 1, ± ξω ± jω 1 ξ 6. 5 j 7.5 If the losed-loo oles are o the root lous, The, ( θ 1 + θ ) (k + 1)18.5 θ1 18 ta ( ) θ 18 ta ( ) ( ) (k + 1)18 ω ad ξ, we a determie the loatio of the losed-loo oles. Therefore, the losed-loo oles that orresod to 1.5% overshoot ad /3 seod settlig time is ot o the origial root lous. θ j BMM3613_AC_SEM67_Assigmet_11_Key Dr WAWY SEM I 6/7 Fakulti Kejuruteraa Mekaikal, KUKTEM

5 Page 5 of 7 5 Root Lous Imagiary Axis 3 1 Gai: 9.91 Pole: i Damig:.81 Overshoot (%): 1.9 Frequey (rad/se): We lot the uomesated root-lous usig MATLAB ad ofirm that at 1.5% overshoot ad, we have atural frequey is aroximately 5. This will ot give the settlig time of /3 seod... Desig a lead omesator so that the system meets the trasiet resose harateristis of a settlig time of /3 seod ad a eret overshoot of 1.5%. Seify the omesator s ole, zero ad the required gai. Submit at least: Root lous of uomesated system showig the arameters Root lous of omesated system showig the arameters SIMULINK blok diagram of both omesated ad uomesated system Ste resose lot that has both the uomesated ad omesated resose The form of lead omesator is, s + z G ( s) s + From revious results, we otie that we eed the agle otributio from the omesator zero ad ole to be: ( θ z θ ) Lookig at the diagram, ( θ z > θ ) so that the agle 6. a be obtaied The good hoie is to let z 5 so that there will be ole-zero aellatio. s + 5 G ( s) s + Now, we eed to obtai Sie z 5, θ z θ 1. 5 Now we obtai θ (1..5 θ ) 6. θ 56.3 By geometry, we a fid the loatio of θ 6 5 z θ z 3.5 j BMM3613_AC_SEM67_Assigmet_11_Key Dr WAWY SEM I 6/7 Fakulti Kejuruteraa Mekaikal, KUKTEM

6 .5 ta(56.3) 1.5 x x 3 9 The lead omesator is the, s + 5 G ( s) s + 9 Below is the omesated root-lous Page 6 of 7 7 Root Lous 6 5 Imagiary Axis 3 Gai: 9. Pole: i Damig:.8 Overshoot (%): 1.51 Frequey (rad/se): Notie that we a obtai the required atural frequey ad damig ratio at the gai of 9.. Below is the SIMULINK blok diagram Ste (s+3)(s+5) G(s)_Origial Result 9.(s+5) (s+3)(s+5)(s+9) G(s)_New BMM3613_AC_SEM67_Assigmet_11_Key Dr WAWY SEM I 6/7 Fakulti Kejuruteraa Mekaikal, KUKTEM

7 Page 7 of 7 With Lead omesator Uomesated Dari Hafshah r.a., kataya Rasulullah saw bersabda Ada lima maam biatag yag tidak berdosa jika sesaorag membuuhya. Yaitu: gagak, helag, tikus, kalajegkig da ajig yag berbahaya. Hadis riwayat Imam Bukhari. BMM3613_AC_SEM67_Assigmet_11_Key Dr WAWY SEM I 6/7 Fakulti Kejuruteraa Mekaikal, KUKTEM