Ali Karimpour Associate Professor Ferdowsi University of Mashhad
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1 LINEAR CONTROL SYSTEMS Ali Karimour Aoiate Profeor Ferdowi Univerity of Mahhad
2 Leture 0 Leture 0 Frequeny domain hart Toi to be overed inlude: Relative tability meaure for minimum hae ytem. ain margin. Phae margin. Nihol hart or gain hae lot. Stability analyi with gain hae lot. Bode lot. Stability analyi with Bode lot. Ste by te Bode lot ontrution. Dr. Ali Karimour Feb 0
3 Leture 0 Stability margin Stability I Not A YeNo Prooition It' not enough to know that a ytem i table or untable. If a ytem i ut barely table, then a mall gain in a ytem arameter ould uh the ytem over the edge, and you will often want to deign ytem with ome margin of error. If you're going to do that, you'll need ome meaure of how table a ytem i. To get uh meaure - and there are at leat two that are widely ued. Dr. Ali Karimour Feb 0
4 Leture 0 Stability margin R + - C Phae Margin Phae margin i the mot widely ued meaure of relative tability when working in the frequeny domain. ω=ω ω ω ω=ω 80 We define hae margin a the hae angle that the frequeny reone would have to hange to move to the - oint. : I the gain roover frequeny 4 Dr. Ali Karimour Feb 0
5 Leture 0 Phae margin omutation R + - C The hae margin, m i defined a follow: ω=ω 80 m 80 ω=ω ω ω How to derive Let 5 Dr. Ali Karimour Feb 0
6 Leture 0 Stability margin Phae Margin The two different Nyquit lot above would lead to two different hae margin. The ytem with the frequeny reone with the dahed line i le table. 6 Dr. Ali Karimour Feb 0
7 Leture 0 Stability margin Phae Margin The two different gain hown for the Nyquit lot below would lead to the ame hae margin. But do they have the ame tability margin? Clearly no. - Thu we need another meaure of relative tability. 7 Dr. Ali Karimour Feb 0
8 Leture 0 Stability margin R + - C ain Margin ain margin i another widely ued meaure of relative tability when working in the frequeny domain. ω=ω ω=ω 80 We define gain margin a the gain that the frequeny reone would have to inreae to move to the - oint. ω ω 80 : I the hae roover frequeny 8 Dr. Ali Karimour Feb 0
9 Leture 0 ain margin omutation R + - C The gain margin, M i defined a follow: M But gain margin i uually eified in db. ω=ω 80 M 0log ω=ω How to derive 80 ω ω Let or Im Dr. Ali Karimour Feb 0
10 Leture 0 Stability margin Phae and ain Margin Same Phae Margin - But different ain Margin 0 Dr. Ali Karimour Feb 0
11 Leture 0 Dr. Ali Karimour Feb 0 C + - R L S L L T L Frequeny rad T S L M M P Senitivity and omlementary enitivity eak
12 Leture 0 Senitivity eak R + - C M S 0 0 S 0 0 ω ω The enitivity eak, M i defined a follow: M max S Dr. Ali Karimour Feb 0
13 Let Nyquit hart olar lot ontrution Imaginary Axi R + - Nyquit Diagram 80 7 M? Leture Real Axi?. 6 PM 45?? C??? db Dr. Ali Karimour Feb 0
14 Nihol hart gain hae lot ontrution R + - Leture 0 C Let Nihol Chart 0log db.7 db.7 db 5.9 db 8.4 db. db.80 db 6.8 db 8.9 db 5.77 db Oen-Loo ain db Oen-Loo Phae deg?. 6 PM 45?? 80 7 M 4?.8 db Dr. Ali Karimour Feb 0
15 R + Bode lot ontrution - Leture 0 C Let log Magnitude db 9. db db db db db -80. db db db 8.9 db 5.77 db?. Phae deg 5 PM 48?? Bode Diagram Frequeny rade M 5?.8 db Dr. Ali Karimour Feb 0 00
16 Leture 0 Dr. Ali Karimour Feb 0 6 Ste by te bode lot ontrution Let k Ste : k Ste : 0log 0log 0log 0log 0log 0log 0log 0log k Ste : k k
17 Leture 0 Dr. Ali Karimour Feb 0 7 Ste by te bode lot ontrution Let k 0log 0log 0log 0log 0log 0log 0log 0log k k
18 Leture 0 Ste by te bode lot ontrution 0log 0log k k Bode Diagram Phae deg Magnitude db 0log k 0or80 Why 0 or 80? It deend to ign of k Frequeny rade 8 Dr. Ali Karimour Feb 0
19 Ste by te bode lot ontrution 0log 0log Leture 0 Let 0 40 Phae deg Magnitude db db de Frequenyrade Dr. Ali Karimour Feb 0
20 Leture 0 Ste by te bode lot ontrution 0log 0log 0 Phae deg Magnitude db db de Frequenyrade Dr. Ali Karimour Feb 0 0 0
21 0log Ste by te bode lot ontrution 0 0log Let 0 Leture 0 Phae deg Magnitude db db de Frequenyrade Dr. Ali Karimour Feb 0 00
22 Examle : Derive the M and PM of following ytem by ue of Bode lot. Leture 0 R C 0 Oen loo tranfer funtion i: Phae deg Magnitude db Frequenyrade Dr. Ali Karimour Feb 0 0
23 Examle : Derive the M and PM of following ytem by ue of Bode lot. Leture 0 R ? 6.5 rad 80? e PM 0 7. rad e C Phae deg Magnitude db M db Frequenyrade Dr. Ali Karimour Feb 0 0
24 Leture 0 Exerie : Derive the gain roover frequeny, hae roover frequeny, M and PM of following ytem by ue of Bode lot. R C Anwer :.5, 80, M and m 8 4 Dr. Ali Karimour Feb 0
25 Leture 0 Exerie The olar lot of an oen loo ytem with negative unit feedbak i hown. a Find the oen loo tranfer funtion. b Find the loed loo tranfer funtion. Nyquit Diagram.5 Imaginary Axi rad e Real Axi 50 anwer a : b : Dr. Ali Karimour Feb 0
26 Leture 0 Exerie The Bode lot of an oen loo ytem with negative unit feedbak i hown. a Find the oen loo tranfer funtion. b Find the loed loo tranfer funtion. 40 Bode Diagram 0 Magnitude db Phae deg Frequeny rade 00 anwer a : b : Dr. Ali Karimour Feb 0
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