Ali Karimpour Associate Professor Ferdowsi University of Mashhad
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1 AUTOMATIC CONTROL SYSTEMS Ali Karimour Aociate Profeor Ferdowi Univerity of Mahhad
2 Root Locu Technique Toic to be covered include: Root locu criterion. Root loci (RL). Comlement root loci (CRL). Comlete root loci. Proerty and contruction of comlete root loci. Ali Karimour Nov
3 r + e - ( ) c M c( ) ( ) r( ) = = =75 = Amlitude Ste Reone (ec).5.5 =75 = = = (ec) (ec) (ec).5 3 Time (ec) Ali Karimour Nov
4 Root locu مکان ریشه ها Root locu, how the oition of root of the following equation for different value of مکان ریشه ها موقعیت ریشه هاي معادله زیر را بر حسب نشان می دهد. f ( ) مقادیر مختلف Root loci (RL) Comlement root loci (CRL) Comlete root loci R R R 4 Ali Karimour Nov
5 مثال : مکان کامل ریشه ها را در سیستم زیر بیابید. Examle : Draw the comlete root loci of following ytem. ( ) - 5 Ali Karimour Nov
6 The Root Locu rocedure نحوه رسم مکان ریشه ها Rule : Secify the equation exactly in the following form. قانون اول: سیستم را دقیقا بصورت زیر بیان کنید. f ( ) چند شاخه در مکان ریشه ها How many branche in root loci? It i : No. of branche max( m, n) 6 Ali Karimour Nov
7 The Root Locu rocedure f ( ) نحوه رسم مکان ریشه ها Which oint lie on the root loci? چه نقاطی بر روي مکان ریشه قرار دارند Condition of angle شرط زاویه f ( ) ( ) odd number of radian f ( ) even number of radian Condition of magnitude شرط اندازه f ( ) f ( ) 7 Ali Karimour Nov
8 The Root Locu rocedure f ( ) نحوه رسم مکان ریشه ها Examine the condition of magnitude and condition of angle on following examle. شرط زاویه و شرط اندازه را در مثال زیر بررسی کنید. ( ) - 8 Ali Karimour Nov
9 The Root Locu rocedure f ( ) نحوه رسم مکان ریشه ها Rule : Secify the ole and zero of f(). The root loci lie on the ole of f() for = and lie on the zero of f() for =± : قطب و صفرهاي f() را مشخص کنید. مکان ریشه در = و در ±= روي صفرهاي f() قرار دارد. قانون f() روي قطبهاي ( ) - 9 Ali Karimour Nov
10 The Root Locu rocedure f ( ) نحوه رسم مکان ریشه ها Rule 3: Define the real axi ection for oitive and negative value of. قانون 3: محور حقیقی را براي مقادیر مثبت و منفی مشخص کنید ( ) - Ali Karimour Nov
11 The Root Locu rocedure f ( ) نحوه رسم مکان ریشه ها Rule 4: Find the aymtote and centered of aymtote for oitive and negative value of. number of و محل مجانبها قانون 4: تلاقی مجانبها را براي مقادیر مثبت و منفی تعیین کنید. aymtote ( ) n n z (m ) n n n m n z z m,,,... m,,,... Aymtote center n i i i n n z n z z i - Ali Karimour Nov
12 Examle : Draw the comlete root loci of the following ytem. مثال : مکان کامل ریشه ها را در سیستم زیر بیابید. ( 5)( ) Rule : Secify the equation exactly in the tandard form. قانون اول: سیستم را دقیقا بصورت استاندارد بیان کنید. Clearly it i o Ali Karimour Nov
13 ( 5)( ) Rule : Secify the ole and zero of f(). The root loci lie on the ole of f() for = and lie on the zero of f() for =± : قطب و صفرهاي f() را مشخص کنید. مکان ریشه در = و در ±= روي صفرهاي f() قرار دارد. قانون f() روي قطبهاي 5 3 Ali Karimour Nov
14 ( 5)( ) Rule 3: Define the real axi ection for oitive and negative value of. قانون محور حقیقی را براي مقادیر مثبت و منفی مشخص کنید :3 5 4 Ali Karimour Nov
15 ( 5)( ) Rule 4: Find the aymtote and centered of aymtote. Aymtote center n z 5 3 i i i i n n n z z 5 و محل مجانبها قانون 4: تلاقی مجانبها. (m ) 3 5,, n nz m 4,, n 3 3 nz 5 We need another rule. به قانون دیگري نیاز داریم. 5 Ali Karimour Nov
16 The Root Locu rocedure f ( ) نحوه رسم مکان ریشه ها Rule 5: Find the brea oint. f ( ) قانون 5: نقطه شکست را بیابید. brea oint 6 Ali Karimour Nov
17 ( 5)( ) Rule 5: Find the brea oint. ( 5)( ) f ( ) قانون 5: نقطه شکست را بیابید. 7.89,.? 5? We need another rule. 7 به قانون دیگري نیاز داریم. Ali Karimour Nov
18 The Root Locu rocedure f ( ) نحوه رسم مکان ریشه ها Rule 6: Find the cro of root locu with imaginary axi by Routh Hurwitz criteria. قانون 6: نقطه تلاقی با محور موهومی را توسط روش روت هرویتز تعیین کنید. 8 Ali Karimour Nov
19 نقطه تلاقی با محور موهومی ( 5)( ) Rule 6: Find the cro of root locu with imaginary axi :6 قانون ( 5)( ) j 5 75 j j 9 Ali Karimour Nov
20 ( 5)( ) Ali Karimour Nov
21 ( 5)( ) rlocu(,[ 5 5 ]); hold on; rlocu(-,[ 5 5 ]) Root Locu 4 Root Locu Imaginary Axi - Imaginary Axi Real Axi Real Axi Ali Karimour Nov
22 مثال 3: مکان کامل ریشه ها را در سیستم زیر بیابید. Examle 3: Draw the comlete root loci of following ytem. ( )( 3) ( ) Rule : Secify the equation exactly in the tandard form. قانون اول: سیستم را دقیقا بصورت زیر استاندارد کنید. ( 3) ( ) ( )( 3) ( 9) Ali Karimour Nov
23 Examle 4: ( 3) ( 9) Rule : Secify the ole and zero of f(). The root loci lie on the ole of f() for = and lie on the zero of f() for =± : قطب و صفرهاي f() را مشخص کنید. مکان ریشه در = و در ±= روي صفرهاي f() قرار دارد. قانون f() روي قطبهاي Ali Karimour Nov
24 Examle 4: ( 3) ( 9) Rule 3: Define the real axi ection for oitive and negative value of. قانون 3: محور حقیقی را براي مقادیر مثبت و منفی مشخص کنید Ali Karimour Nov
25 Examle 4: ( 3) ( 9) Rule 4: Find the aymtote and centered of aymtote. Aymtote center n z ( 3) i i i i n n n z z 3.5 و محل مجانبها قانون 4: تلاقی مجانبها. (m ) 3, n nz m, n nz Ali Karimour Nov
26 Examle 4: ( 3) ( 9) Rule 5: Find the brea oint. (3 f ( ) 9)( 3) ( ( 3) ( 3 3 قانون 5: نقطه شکست را بیابید. 9) ( 3) 9) Ali Karimour Nov
27 ( 3) ( 6: نقطه تلاقی با محور موهومی (9 Rule 6: Find the cro of root locu with imaginary axi 3 (9 ) 3? قانون 5 3 We need another rule. 7 به قانون دیگري نیاز داریم. Ali Karimour Nov
28 Exercie تمرینها - The tranfer function of a ingle-loo control ytem are given a: G( ) H ( ) T ( )( 3) Contruct the root loci of the Zero of +G()K() for - <T d < - The oen loo tranfer function of a unity-feedbac (negative ign). ytem i: G( ) K (. e )( ) d Contract the comlete root loci of the characteritic equation. 8 Ali Karimour Nov
29 Exercie تمرینها 3- The oen loo tranfer function of a unity-feedbac (negative ign) ytem i: G ( ) Contruct the comlete root loci of the characteritic equation for Let n=, n=, n=3 and n=4 ( K 5) 4- The oen loo tranfer function of a unity-feedbac (negative ign) ytem i: K( )( 3) G( ) ( ) a) Contruct the root loci for - <K<, with α=5. b) Contruct the root loci for - < α <, with K =5. n 9 Ali Karimour Nov
30 Exercie تمرینها 5- The oen loo tranfer function of a unity-feedbac (negative ign) ytem i: 5 G( ) ( )( ) Contruct the root loci for << 6- Conider following ytem Contruct the root loci for < < 3 3 Ali Karimour Nov
31 Exercie تمرینها 7- Conider following ytem 3 For =, = nd = contruct the root loci for < <. 8- Contruct the root loci of the cloed loo ole of the following ytem for <a< (Midterm ring 8). R() C() a 3 Ali Karimour Nov
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