Lecture 7 CONTROL ENGINEERING. Ali Karimpour Associate Professor Ferdowsi University of Mashhad
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1 CONTROL ENGINEERING Ali Karimpour Aociate Profeor Ferdowi Univerity of Mahhad Dr. Ali Karimpour Sep 5
2 Root Locu Criteria Topic to be covered include: Root locu criterion. Root loci (RL). Complement root loci (CRL). Complete root loci. Effect of adding pole and zero on root locu. Effect of moving pole and zero. Root contour. Dr. Ali Karimpour Sep 5
3 Root Locu Technique Topic to be covered include: Root locu criterion. Root loci (RL). Complement root loci (CRL). Complete root loci. Effect of adding pole and zero on root locu. Effect of moving pole and zero. Root contour. 3 Dr. Ali Karimpour Sep 5
4 Root locu مکان ریشه ها Root locu, how the poition of root of the following equation for different value of مکان ریشه ها موقعیت ریشه های معادله زیر را بر حسب نشان می دهد. f ( ) مقادیر مختلف Root loci (RL) Complement root loci (CRL) Complete root loci R R R 4 Dr. Ali Karimpour Sep 5
5 Root locu مکان ریشه ها Why root of +f()= i important for u?... 5 Dr. Ali Karimpour Sep 5
6 The Root Locu procedure f ( ) نحوه رسم مکان ریشه ها Which point lie on the root loci? چه نقاطی بر روی مکان ریشه قرار دارند Condition of magnitude f ( ) Condition of angle R f ( ) R شرط اندازه شرط زاویه f ( ) or f ( ) R f ( ) or f ( ) R 6 Dr. Ali Karimpour Sep 5
7 The Root Locu procedure f ( ) نحوه رسم مکان ریشه ها Rule : Specify the equation exactly in the following form. f ( ) قانون اول: سیستم را دقیقا بصورت زیر بیان کنید. How many branche in root loci? چند شاخه در مکان ریشه ها Rule : Specify the pole and zero of f(). The root loci lie on the pole of f() for = and lie on the zero of f() for =± قانون : قطب و صفرهای f() را مشخص کنید. مکان ریشه در = روی قطبهای f() و در ±= روی صفرهای f() قرار دارد. Rule 3: Define the real axi ection for poitive and negative value of. :3 7 قانون محور حقیقی را برای مقادیر مثبت و منفی مشخص کنید Dr. Ali Karimpour Sep 5
8 The Root Locu procedure f ( ) نحوه رسم مکان ریشه ها Rule 4: Find the aymptote and centered of aymptote for poitive and negative value of. قانون 4: مجانبها و محل تالقی مجانبها را برای مقادیر مثبت و منفی تعیین کنید. number of aymptote np n z Rule 5: Find the brea point. f ( ) (m ) n n n m p m,,,... قانون 5: نقطه شکست را بیابید. brea point Rule 6: Find the cro of root locu with imaginary axi by Routh Hurwitz criteria. قانون 6: نقطه تالقی با محور موهومی را توسط روش روت هرویتز تعیین کنید. p n z z m,,,... Aymptote center n p p i i i n p n z n z 8 z Dr. Ali Karimpour Sep 5 i
9 The Root Locu procedure f ( ) نحوه رسم مکان ریشه ها f ( ) قانون : سیستم را دقیقا بصورت مقابل بیان کنید. قانون : قطب و صفرهای f() را مشخص کنید. مکان ریشه در = روی قطبهای f() و در ±= روی صفرهای f() قرار دارد. قانون 3: محور حقیقی را برای مقادیر مثبت و منفی مشخص کنید number of f ( ) :4 9 قانون و محل مجانبها تالقی مجانبها را برای مقادیر مثبت و منفی تعیین کنید. aymptote np n z (m ) n n brea point n m p p n z z m,,,... m,,,... Aymptote center n p p i i i n p n z n z :5 z Dr. Ali Karimpour Sep 5 i قانون قانون نقطه شکست را بیابید. 6: نقطه تالقی با محور موهومی را توسط روش روت هرویتز تعیین کنید.
10 Example : Draw the complete root loci of the following ytem. مثال : مکان کامل ریشه ها را در سیستم زیر بیابید. ( ) - Dr. Ali Karimpour Sep 5
11 Example : Draw the complete root loci of the following ytem. مکان کامل ریشه ها را در سیستم زیر بیابید. ( 5)( ) 7.7 j مثال : j Dr. Ali Karimpour Sep 5
12 Example : Draw the complete root loci of the following ytem. ( 5)( ) Dr. Ali Karimpour Sep 5
13 ( 5)( ) rlocu(,[ 5 5 ]); hold on; rlocu(-,[ 5 5 ]) Root Root Locu Locu 4 4 Root Root Locu Locu Imaginary Axi Imaginary Axi - - Imaginary Axi Imaginary Axi Real Real Axi Axi Real Real Axi Axi 3 Dr. Ali Karimpour Sep 5
14 Example 3: Draw the complete root loci of following ytem. مثال 3: مکان کامل ریشه ها را در سیستم زیر بیابید. ( )( 3) ( ) Rule : Specify the equation exactly in the tandard form. قانون اول: سیستم را دقیقا بصورت زیر استاندارد کنید. ( 3) ( ) ( )( 3) ( 9) 4 Dr. Ali Karimpour Sep 5
15 Example 3: ( 3) ( 9) Rule : Specify the pole and zero of f(). The root loci lie on the pole of f() for = and lie on the zero of f() for =± : قطب و صفرهای f() را مشخص کنید. مکان ریشه در = و در ±= روی صفرهای f() قرار دارد. قانون f() روی قطبهای Dr. Ali Karimpour Sep 5
16 Example 3: ( 3) ( 9) Rule 3: Define the real axi ection for poitive and negative value of. قانون 3: محور حقیقی را برای مقادیر مثبت و منفی مشخص کنید Dr. Ali Karimpour Sep 5
17 Example 3: ( 3) ( 9) Rule 4: Find the aymptote and centered of aymptote. Aymptote center n p p z ( 3) i i i i n p n n z z 3.5 قانون 4: و محل مجانبها تالقی مجانبها. (m ) 3, n n n m p p n z z, Dr. Ali Karimpour Sep 5
18 Example 3: ( 3) ( 9) Rule 5: Find the brea point. (3 f ( ) 9)( 3) ( ( 3) ( 3 3 قانون 5: نقطه شکست را بیابید. 9) ( 3) 9) Dr. Ali Karimpour Sep 5
19 ( 3) ( 6: نقطه تالقی 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. 9 به قانون دیگری نیاز داریم. Dr. Ali Karimpour Sep 5
20 The Root Locu procedure f ( ) نحوه رسم مکان ریشه ها Rule 7: Find the arrival angle and departure angle. Departure angle. قانون 7: زوایای ورود و خروج را تعیین کنید. Arrival angle. Zero pole Zero pole Dr. Ali Karimpour Sep 5
21 Rule 7: Find the arrival angle and departure angle. قانون 7: زوایای ورود و خروج را تعیین کنید (8 (8 tan tan ) 5 ) Dr. Ali Karimpour Sep 5
22 Rule 7: Find the arrival angle and departure angle (8 (8 tan tan ) 5 قانون 7: زوایای ورود و خروج را تعیین کنید. ) Dr. Ali Karimpour Sep 5
23 Dr. Ali Karimpour Sep 5 3 هبساحم هشیر ناکم یور Calculation of on the Root Loci ) ( f ) ( f Condition of magnitude m i i n j j z C p n j j m i i p z C f ) ( Let
24 Summary Rule : Specify the equation exactly in the form. f ( ) Rule : Specify the pole and zero of f(). The root loci lie on the pole of f() for = and lie on the zero of f() for =± Rule 3: Define the real axi ection for poitive and negative value of. Rule 4: Find the aymptote and centered of aymptote. Aymptote center Rule 5: Find the brea point. n p p i i i n p n z n z z i (m ) np nz m np nz Rule 6: Find the cro of root locu with imaginary axi by Routh Hurwitz criteria. Rule 7: Find the arrival angle and departure angle. 4 Dr. Ali Karimpour Sep 5
25 Root Locu Technique Topic to be covered include: Root locu criterion. Root loci (RL). Complement root loci (CRL). Complete root loci. Effect of adding pole and zero on root locu. Effect of moving pole and zero. Root contour. 5 Dr. Ali Karimpour Sep 5
26 Adding pole and zero inide the loop اضافه کردن صفر و قطب داخل تابع انتقال حلقه r + e - G() c r Adding pole to the loop e - G() b c r Adding zero to the loop e - G() b c 6 Dr. Ali Karimpour Sep 5
27 Adding pole inide the loop اضافه کردن قطب داخل حلقه ( a) ( a)( b) -a b -b -a 7 Dr. Ali Karimpour Sep 5
28 Adding zero inide the loop اضافه کردن صفر داخل حلقه ( a) ( b) ( a) -a b -b -a 8 Dr. Ali Karimpour Sep 5
29 Root Locu Technique Topic to be covered include: Root locu criterion. Root loci (RL). Complement root loci (CRL). Complete root loci. Property and contruction of complete root loci. Effect of adding pole and zero on root locu. Effect of moving pole and zero. Root contour. 9 Dr. Ali Karimpour Sep 5
30 Effect of moving pole اثرات حرکت قطبها Conider following equation: ( b) ( a) ( b) ( a) Let b= but a=, 9, 8, 3, ( ) ( a) 3 Dr. Ali Karimpour Sep 5
31 Effect of moving pole. Let a= and aymptote... اثرات حرکت قطبها ( ) ( ) Breapoint Departureangle Dr. Ali Karimpour Sep 5
32 Effect of moving pole Let a=9 اثرات حرکت قطبها ( ) ( 9) 9 ( ) 4 ( a 3) Aymptote are, a a 9 ( 9 3) ? 9 3 Dr. Ali Karimpour Sep 5
33 اثرات حرکت قطبها Effect of moving pole Let a=8 ( ) ( 8) 8 ( ) 3.5 ( a 3) Aymptote are, a a 9 ( 8 3) no real? 9 33 Dr. Ali Karimpour Sep 5
34 Effect of moving pole. Let a=3 اثرات حرکت قطبها ( ) ( 3) 3 ( ) ( a 3) Aymptote are, a a 9 ( 3 3) no real? 9 34 Dr. Ali Karimpour Sep 5
35 Effect of moving pole. اثرات حرکت قطبها Let a= ( ) ( ) ( ) ( a 3) Aymptote are, a a 9 ( 3 3) no real? 9 35 Dr. Ali Karimpour Sep 5
36 Effect of moving pole. اثرات حرکت قطبها a= 3 9 a= Dr. Ali Karimpour Sep 5
37 More tudy on root locu criterion Remar #: Let f() be bi-proper: f ( ) Jump in root locu Example 5: Draw the complete root loci of following ytem Dr. Ali Karimpour Sep 5
38 More tudy on root locu criterion Example 4: Draw the complete root loci of following ytem Dr. Ali Karimpour Sep 5
39 More tudy on root locu criterion Example 5: Univerity entrance exam 388 Find root locu for < ( ( 5)( 6)( 3) 4) Root Locu 6 4 Imaginary Axi Real Axi 39 Dr. Ali Karimpour Sep 5
40 More tudy on root locu criterion Example 6: Univerity entrance exam 39 A unity poitive feedbac ytem ha a loop tranfer function a: ( 5)(.5) G( ) ( )( 5) Draw root locu for > Imaginary Axi Root Locu Real Axi 4 Dr. Ali Karimpour Sep 5
41 Root Locu Technique Univerity entrance exam 393 Example 7: Univerity entrance exam 393 Root locu of a unity negative feedbac ytem for > i hown. Suppoe undamped ocillation period i π 3, find the teady tate error to unit tep. ) 7/9 ) 7/64 3)64/9 4) 64/7 4 Dr. Ali Karimpour Sep 5
42 Root Locu Technique Topic to be covered include: Root locu criterion. Root loci (RL). Complement root loci (CRL). Complete root loci. Property and contruction of complete root loci. Effect of adding pole and zero on root locu. Effect of moving pole and zero. Root contour. 4 Dr. Ali Karimpour Sep 5
43 Root contour (Multiple parameter variation ( ) کانتور ریشه ها ( تغییرات چند پارامتر Remember root loci f ( ) Suppoe: Q ( ) P ( ) P ( ) and are parameter Step : Put one variable equal to zero. Let u = Q ) P ( ) ( Step : Retore the value of Q( ) ( ) ( ) P P P ) ( f ( ) Q( ) f( ) P ( ) Q ( ) P ( ) ( f ( ) f43 ( ) Dr. Ali Karimpour Sep 5
44 Example 8: Draw the root contour for following ytem. 3 Step : Let = then Aymptote are: ( ) 3, 3 مثال 8: کانتور ریشه ها را برای سیستم زیر بیابید. 3 = ( ) 3 Angle of departure i: = - = θ.5 ( ) / 3 6 = 44 Dr. Ali Karimpour Sep 5
45 Example 8: Draw the root contour for following ytem. 3 Step : Retore the value of مثال 8: کانتور ریشه ها را برای سیستم زیر بیابید. 3 Let: = = Let: = Let: = = - = = 45 Dr. Ali Karimpour Sep 5
46 Exercie - A unity feedbac ( negative ign ) control ytem ha an open loop tranfer function Setch the complete root loci, and find the correponding when the root loci croe jω axi. - The tranfer function of a ingle-loop control ytem are given a: G ( ). H ( ) G( ) (.)(.5) ( )( 3) Contruct the root loci of the Zero of +G()H()= for - <T d < 3- The open loop tranfer function of a unity-feedbac (negative ign) ytem i: K G p ( ) ( 5) Contruct the complete root loci of the characteritic equation for Let n=, n= and n=3. n T d 46 Dr. Ali Karimpour Sep 5
47 Exercie 4- The open loop 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. 5- The open loop tranfer function of a unity-feedbac (negative ign) ytem i: 5p G( ) ( )( p) Contruct the root loci for <p< 6- Conider following ytem Contruct the root loci for < < 3 47 Dr. Ali Karimpour Sep 5
48 Exercie 7- Contruct the root loci of the cloed loop pole of the following ytem for <a< (Midterm pring 8). R() C() a 8- Conider following ytem 3 For =, = nd = contruct the root loci for < <. 48 Dr. Ali Karimpour Sep 5
49 Exercie 9- Find the root-locu graph for the following ytem. Anwer : - The open loop tranfer function of a unity-feedbac (negative ign) ytem i: G( ) ( p) Contruct the root loci for <p< (Final 39) 49 Dr. Ali Karimpour Sep 5
50 Exercie - The open loop tranfer function of a unity-feedbac (negative ign) ytem i: G( ) K (. e )( ) Contruct the complete root loci of the characteritic equation. - The open loop tranfer function of a unity-feedbac (negative ign) ytem with PD controller i: G( ) ( K p K d ) Setch the root loci for different value of K p and K d. (Let Kp=,,5,.) 5 Dr. Ali Karimpour Sep 5
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