Ali Karimpour Associate Professor Ferdowsi University of Mashhad

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1 LINEAR CONTROL SYSTEMS Ali Karimpour Aociate Profeor Ferdowi Univerity of Mahhad

2 Root Locu Technique Topic to be covered include: Property and contruction of complete root loci. (Cont.) Effect of adding pole and zero on root locu. Effect of moving pole and zero. Root contour. Dr. Ali Karimpour May 3

3 مثال : مکان کامل ریشه ها را در سیستم زیر بیابید. Example : Draw the complete root loci of following ytem. ( )( 3) ( ) Rule : Specify the equation exactly in the tandard form. قانون اول: سیستم را دقیقا بصورت زیر استاندارد کنید. ( 3) ( ) ( )( 3) ( 9) 3 Dr. Ali Karimpour May 3

4 Example : ( 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 صفرهاي f() را مشخص کنید. مکان ریشه در = و در ±= روي صفرهاي f() قرار دارد. قانون f() روي قطبهاي Dr. Ali Karimpour May 3

5 Example : ( 3) ( 9) Rule 3: Define the real axi ection for poitive and negative value of. قانون 3: محور حقیقی را براي مقادیر مثبت و منفی مشخص کنید Dr. Ali Karimpour May 3

6 Example : ( 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 و محل مجانبها قانون 4: تلاقی مجانبها. (m ) 3, n p nz m, n p nz Dr. Ali Karimpour May 3

7 Example : ( 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 May 3

8 ( 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. 8 به قانون دیگري نیاز داریم. Dr. Ali Karimpour May 3

9 The Root Locu procedure f ( ) نحوه رسم مکان ریشه ها Rule 7: Find the arrival angle and departure angle. Departure angle. قانون 7: زوایاي ورود و خروج را تعیین کنید. Zero pole Arrival angle. Zero pole 9 Dr. Ali Karimpour May 3

10 Rule 7: Find the arrival angle and departure angle. قانون 7: زوایاي ورود و خروج را تعیین کنید (8 (8 tan tan ) 5 ) Dr. Ali Karimpour May 3

11 Rule 7: Find the arrival angle and departure angle. قانون 7: زوایاي ورود و خروج را تعیین کنید (8 (8 tan tan ) 5 ) Dr. Ali Karimpour May 3

12 Dr. Ali Karimpour May 3 هبساحم هشیر ناکم يور 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

13 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 ) n p 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. 3 Dr. Ali Karimpour May 3

14 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 4 Dr. Ali Karimpour May 3

15 Adding pole inide the loop ( a) ( a)( b) اضافه کردن قطب داخل حلقه -a b -b -a 5 Dr. Ali Karimpour May 3

16 Adding zero inide the loop اضافه کردن صفر داخل حلقه ( a) ( b) ( a) -a b -b -a 6 Dr. Ali Karimpour May 3

17 Effect of moving pole اثرات حرکت قطبها Conider following equation: ( b) ( a) ( b) ( a) Let b= but a=, 9, 8, 3, ( ) ( a) 7 Dr. Ali Karimpour May 3

18 Effect of moving pole. اثرات حرکت قطبها Let a= ( ) ( ) ( ) 4.5 Aymptote are 3, ( a) ( ) (3 a)( ) ( ) ( 3 a)( ) ( a) ( a) ( a 3) a ( a 3) a a 9 Let a= ? 9 8 Dr. Ali Karimpour May 3

19 Effect of moving pole اثرات حرکت قطبها Let a=9 ( ) ( 9) 9 ( ) 4 ( a 3) Aymptote are, a a 9 ( 9 3) ? 9 9 Dr. Ali Karimpour May 3

20 Effect of moving pole اثرات حرکت قطبها Let a=8 ( ) ( 8) 8 ( ) 3.5 ( a 3) Aymptote are, a a 9 ( 8 3) no real? 9 Dr. Ali Karimpour May 3

21 Effect of moving pole. اثرات حرکت قطبها Let a=3 ( ) ( 3) 3 ( ) ( a 3) Aymptote are, a a 9 ( 3 3) no real? 9 Dr. Ali Karimpour May 3

22 Effect of moving pole. اثرات حرکت قطبها Let a= ( ) ( ) ( ) ( a 3) Aymptote are, a a 9 ( 3 3) no real? 9 Dr. Ali Karimpour May 3

23 Effect of moving pole. اثرات حرکت قطبها a= 3 9 a= Dr. Ali Karimpour May 3

24 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 ( ) f4 ( ) Dr. Ali Karimpour May 3

25 Example : Draw the root contour for following ytem. 3 مثال : کانتور ریشه ها را براي سیستم زیر بیابید. Step : Let = then Aymptote are: 3 ( ) 3, 3 = ( ) 3 Angle of departure i: = - = θ.5 ( ) / 3 6 = 5 Dr. Ali Karimpour May 3

26 Example : Draw the root contour for following ytem. 3 مثال : کانتور ریشه ها را براي سیستم زیر بیابید. Step : Retore the value of 3 Let: = = Let: = Let: = = - = = 6 Dr. Ali Karimpour May 3

27 Exercie تمرینها A unity feedbac ( negative ign ) control ytem ha an open loop tranfer function G( ) (.)(.5) Setch the complete root loci, and find the correponding when the root loci croe jw axi. 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,.) 7 Dr. Ali Karimpour May 3

Lecture 7 CONTROL ENGINEERING. Ali Karimpour Associate Professor Ferdowsi University of Mashhad

Lecture 7 CONTROL ENGINEERING. Ali Karimpour Associate Professor Ferdowsi University of Mashhad CONTROL ENGINEERING Ali Karimpour Aociate Profeor Ferdowi Univerity of Mahhad Dr. Ali Karimpour Sep 5 Root Locu Criteria Topic to be covered include: Root locu criterion. Root loci (RL). Complement root