LINEAR CONTROL SYSTEMS

Transcription

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

2 Time domain deign of control ytem Topic to be covered include: Introduction. Variou controller configuration. Different ind of controller. Controller realization. Controller deign in time domain. PID controller deign. PD controller deign. PI controller deign. Lag controller deign. Lead controller deign. 2

3 Time domain deign of control ytem Topic to be covered include: Introduction. Variou controller configuration. Different ind of controller. Controller realization. Controller deign in time domain. PID controller deign. PD controller deign. PI controller deign. Lag controller deign. Lead controller deign. 3

4 Variou controller configuration. r(t) + e(t) u(t) c(t) G c () ساختارهای کنترلی متفاوت 1 ساختار کنترلی سری compenation. 1 Serie or cacade G p () 2 کنترل فیدبک حالت control. 2 Statefeedbac r(t) + u(t) x(t) G () p D c(t) K 3 Forward compenation with erie compenation. (Two degree of freedom) 3 جبران سازی پیش رو با جبران سازی سری )دو درجه آزادی( r(t) + e(t) u(t) c(t) G ( c2 ) G ( c1 ) G p () 4

5 Variou controller configuration. 4 Feed forward compenation. (Two degree of freedom) Controller G ( c2 ) ساختارهای کنترلی متفاوت 4 کنترلر پیش خور )دو درجه آزادی( Controlled proce r(t) + e(t) + u(t) c(t) G ( c1 ) + G p () Controller 5

6 Serie Compenation Structure جبران سازی سری In thi Coure we conider the erie or cacade compenation. Different ind of controller PID controller. Lead lag controller H 2 Controller H Controller.. Intelligent Controller Controller In Thi coure In Graduate coure Adaptive Controller NN Controller.. 6

7 PID Controller کنترلر PID PID ha become almot univerally ued in indutrial control. Thee controller have proven to be robut and extremely beneficial in the control of many important application. PID tand for: P (Proportional) I (Integral) D (Derivative) The tandard form PID are: Proportional only: Proportional plu Integral: Proportional plu derivative: Proportional, integral and derivative: An Alternative form for PID G G G G G PIDerie P K p PI PD PID ( ) Ki ( ) K p K p Kd K ( ) K p ( ) K p K i i K d K d 1 Kd 1 K K d 1 d 7

8 Leadlag Compenator کنترلر پیش فاز پس فاز Cloely related to PID control i the idea of leadlag compenation. The tranfer function of thee compenator i of the form: G( ) a 1 1 or G( ) z p If a<1, then thi i a lag networ. Or (z>p) in other form. If a>1 thi i a lead networ. Or (z<p) in other form. 8

9 PID and Operational Amplifier کنترلر PID و تقویت کننده عملیاتی V G( ) V o i ( ) ( ) Z Z f i ( ) ( ) 9

10 PID and Operational Amplifier کنترلر PID و تقویت کننده عملیاتی V G( ) V o i ( ) ( ) Z Z f i ( ) ( ) G( ) V V out in ( ) ( ) R R f in K p G( ) V V out in ( ) ( ) R 1/ C RC K D Vout ( ) 1/ C G( ) 1/ RC Ki / V ( ) R in 10

11 PID and Operational Amplifier کنترلر PID و تقویت کننده عملیاتی G( ) Vo ( ) V ( ) i Z Z f i ( ) ( ) G( ) Vo ( ) V ( ) i R 2 1 R 1 1 C2 1 C 1 R2C2 R1C 1 1 ( ) ( ) R2C R C R C G

12 Time domain deign of control ytem Topic to be covered include: Introduction. Variou controller configuration. Different ind of controller. Controller realization. Controller deign in time domain. PID controller deign. PD controller deign. PI controller deign. Lag controller deign. Lead controller deign. 12

13 Effect of the PD control on the time repone. تاثیر کنترلر PD بر پاسخ زمانی R () + E() K P K D + + PD controller 2 n ( 2 ) n C() Derivative part can improve the ocillation. جمله مشتق می تواند رفتار گذرا را بهبود بخشد. c(t) Amplitude Amplitude e(t) Amplitude e(t) 2 1 Step Repone Time (ec) Step Repone Time (ec) Impule Repone Time (ec) 13

14 Effect of the PI control on the time repone. تاثیر کنترلر PI بر پاسخ زمانی R () E() + K P K I n ( 2 ) n C() PI controller Loop tranfer function without controller Loop tranfer function with controller G( ) 2 n ( 2 ) n G c ( ) G( ) K P K I 2 n ( 2 ) PI controller can improve error by increae the type of ytem by one 14 n

15 Tuning of PID Controller Becaue of their widepread ue in practice, we preent below everal method for tuning PID controller. Actually thee method are quite old and date bac to the Nonethele, they remain in widepread ue today. In particular, we will tudy. Time domain deign Frequency domain deign ZieglerNichol Ocillation Method ZieglerNichol Reaction Curve Method CohenCoon Reaction Curve Method تنظیم کنترلرهای PID See in appendix 15

16 Time domain deign R () + E() C() ( 25) طراحی حوزه زمانی بله Example 1: I it poible to et the value of uch that the damping ratio of complex pole be 0.707? Ye مثال 1: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم گردد ?

17 Time domain deign R () + E() C() ( 25) v طراحی حوزه زمانی بله Example 2: I it poible to et the value of uch that ramp error contant be 100? Ye مثال 2: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که ثابت خطای شیب معادل 100 گردد

18 Time domain deign طراحی حوزه زمانی Example 3: I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 3: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد R () + E() ( 25) C() 62.5 Clearly the deign i not poible???!!!??? Other controller 18

19 Tuning PD controller طراحی کنترلر PD Example 4: Determine the controller coefficient uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 4: در سیستم ضرایب کنترلر را بگونه ای تنظیم کنید که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد R () + E() P D ( 25) C() P v 1 D

20 Tuning PD controller طراحی کنترلر PD R () + E() P D ( 25) C() 1 D Tuning D by graphical method. D D D D D 0 20

21 Tuning PD controller طراحی کنترلر PD R () + E() P D ( 25) C() 1 D Tuning D by mathematical method... 1 D D

22 Tuning PD controller طراحی کنترلر PD R () + E() P D ( 25) C() P D Clearly v Step Repone 1.2 Why P.O. > 4.3% 1 Amplitude C( ) R( ) ( )

23 Tuning PI controller طراحی کنترلر PI Example 5: Determine the controller coefficient uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 5: در سیستم ضرایب کنترلر را بگونه ای تنظیم کنید که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد R () + E() P + + ( 25) C() R () + E() P I ( 25) C() I Clearly type of ytem i 2 o: v 23

24 طراحی کنترلر PI )ادامه( (Continue) Tuning PI controller We now need damping ratio of complex pole be حال نیاز داریم که نسبت میرائی قطبهای مختلط سیستم گردد. R () + E() P I ( 25) C() Root loci with proportional controller 1 P 0 ( 25) ( I / P) 1 P 0 ( 25) Let I / P No important change in root loci P? / 2 I P... P

25 Tuning PI controller (Continue) طراحی کنترلر PI )ادامه( Root loci with PI controller ( 2) 1 P 0 ( 25)

26 Tuning PI controller طراحی کنترلر PI R () + E() P I ( 25) C() P I P Clearly v Step Repone Why P.O. > 4.3% Amplitude C( ) R( ) 3 338( 2) Time (ec) Poleare: 2.37, j 26

27 Compare PI and PD controller مقایسه کنترلرهای PI و PD R () + E() P D ( 25) P D C() R () + E() P I ( 25) P I C() Step Repone With PI controller Amplitude With PD controller Time (ec) 27

28 Time domain deign of control ytem Topic to be covered include: Introduction. Variou controller configuration. Different ind of controller. Controller realization. Controller deign in time domain. PID controller deign. PD controller deign. PI controller deign. Lag controller deign. Lead controller deign. 28

29 Lag and Lead Controller Deign طراحی حوزه زمانی I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد R () + E() ( 25) C() 62.5 Clearly the deign i not poible???!!!??? Other controller 29

30 Lag controller deign procedure 1 Obtain the rootlocu (without controller) and determine the gain 0 to atify the deired damping ratio or... 2 Find the gain to atify the deired teady tate error (without controller). If i in conflict with 0 continue. 3 Evaluate the needed controller gain R () + E() رویه طراحی کنترلر پس فاز z z p p G() C() needed gain 0 Gain to Gain to atify the deired teady tate error atify the deired damping ratio or... 4 Chooe pole and zero of controller near origin uch that: 5 Chooe the controller a: G c ( ) 0 z p z p Why? needed gain What i near? 0 6 Chec the controller. 30

31 Lag controller deign Determine the controller coefficient uch that the damping ratio of complex pole be and ramp error contant be 100? R () + E() در سیستم ضرایب کنترلر را بگونه ای تنظیم کنید که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد z p ( 25) C() 1 Obtain the rootlocu (without controller) and determine the gain 0 to atify the deired damping ratio or

32 Lag controller deign Example 6: Determine the controller coefficient uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 6: در سیستم ضرایب کنترلر را بگونه ای تنظیم کنید که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد R () + E() z p ( 25) C() v Determine the gain to atify the deired teadytate error (without controller). If there i a problem then continue

33 Lag controller deign Example 6: Determine the controller coefficient uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 6: در سیستم ضرایب کنترلر را بگونه ای تنظیم کنید که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد 3 Evaluate the needed controller gain needed gain 0 Gain to Gain to atify the deired teady tate error atify the deired damping ratio or Chooe pole and zero of controller near origin uch that: z p needed gain 0 8 p 0.1 z Chooe the controller a: G c ( ) 0 z p

34 Lag controller deign R () + E() ( 25) C() 6 Chec the controller. v lim ( 25) Why? C( ) 312.5( 0.8) R( ) ( 25)( 0.1) 312.5( 0.8) C( ) R( ) ( 0.8) Poleare: j,

35 Deigning lag controller and it tep repone R () + E() ( 25) C() 1.4 Step Repone Amplitude Time (ec) 35

36 Lag controller deign Example 7: Deign a lag controller uch that the ramp error contant be 50 and P.O. be 16 %. مثال 7: کنترلی طراحی کنید که ثابت خطای شیب معادل 50 گردد و درصد فراجهش %16 گردد. R () + E() z p 1 ( 5)( 10) C() P.O. 16% 0.5 Why? Obtain the rootlocu and determine the gain 0 to atify the deired damping ratio or... 0?

37 Lag controller deign Example 7: Deign a lag controller uch that the ramp error contant be 50 and P.O. be 16 %. مثال 7: کنترلی طراحی کنید که ثابت خطای شیب معادل 50 گردد و درصد فراجهش %16 گردد. R () + E() z p 1 ( 5)( 10) C() v 50 2 Determine the gain to atify the deired teadytate error. If there i a problem then Continue. (without controller)

38 Lag controller deign Example 7: Deign a lag controller uch that the ramp error contant be 50 and P.O. be 16 %. مثال 7: کنترلی طراحی کنید که ثابت خطای شیب معادل 50 گردد و درصد فراجهش %16 گردد. 3 Evaluate the needed controller gain needed gain 0 Gain to Gain to atify the deired teady tate error atify the deired damping ratio or Chooe pole and zero of controller near origin uch that: z p needed gain p 0.01 z Chooe the controller a: G c ( ) 0 z p

39 Lag controller deign R () + E() ( 5)( 10) C() v 6 Chec the controller. lim ( 5)( 10) Root Locu C( ) 84( ) R( ) ( 5)( 10)( 0.01) 84( ) C( ) R( ) 4 84( ) Imaginary Axi Poleare: j, 0.37, Real Axi 39

40 Lag controller deign R () + E() ( 5)( 10) C() 1.4 Step Repone P.O. i not o Amplitude Tune the controller to derive the performance Time (ec)

41 Lag controller deign طراحی کنترلر پس فاز When the deign of lag controller i not poible? 41

42 Lead controller deign طراحی حوزه زمانی I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد R () + E() ( 25) C() 62.5 Clearly the deign i not poible???!!!??? Other controller 42

43 Lead controller deign procedure 1 From the timedomain pecification obtain the deired location of the cloedloop dominant pole. 2 Select the zero of controller. Place the zero on the real value of deired location of the cloedloop dominant pole or on the pole for polezero cancellation. R () + E() رویه طراحی کنترلر پیش فاز z z p p G() C() 3 Locate the compenator pole o that the angle criterion i atified. 4 Determine the compenator gain, uch that the magnitude criterion i atified. 5 Chooe the controller a: 6 Chec the controller. G c ( ) 7 If the overall repone rie time, overhoot and ettling time i not atifactory, chooe another location of the cloedloop dominant pole. 43 z p

44 Lead controller deign Example 8:I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 8: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد + R () E() z p z p ( 25) C() From the timedomain pecification obtain the deired location of the cloedloop dominant pole v v 44

45 Lead controller deign Example 8:I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 8: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد + R () E() z p z p ( 25) C() 2 Select the zero of controller. Place the zero on the real value of deired location of the cloedloop dominant pole or on the pole for polezero cancellation. Where?

46 Lead controller deign Example 8:I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 8: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد + R () E() z C() p ( 25) z p 3 Locate the compenator pole o that the angle criterion i atified. p... p 128 x

47 Lead controller deign Example 8:I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 8: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد + R () E() z p z p ( 25) C() 4 Determine the compenator gain, uch that the magnitude criterion i atified

48 Lead controller deign Example 8:I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 8: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد + R () E() z p z p ( 25) C() 5 Chooe the controller a: G c ( )

49 Lead controller deign Example 8:I it poible to et the value of uch that the damping ratio of complex pole be and ramp error contant be 100? مثال 8: در سیستم زیر آیا می توان را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم و ثابت خطای شیب معادل 100 گردد + R () E() z p z p 1 ( 25) C() 6 Chec the controller. Clearly v lim ( 25 ) v 73 It i not o we mut try other cloedloop dominant pole Tune the controller to derive the requeted performance 49

50 Exercie 1 In the following ytem deign a PD controller uch that that the damping ratio of complex pole be 0.6 and ramp error contant be 80. R () E() + G PID () ( 48.5) C() 2 In the following ytem deign a PD controller uch that that the damping ratio of complex pole be 0.6 and ramp error contant be 80. R () + E() G PID () 4 ( 5)( 10) C() 50

51 Exercie 3 In the following ytem deign a PI controller uch that that the damping ratio of complex pole be 0.6 and ramp error contant be 80. R () + E() G PID () ( 48.5) C() 4 In the following ytem deign a PI controller uch that that the damping ratio of complex pole be 0.6 and ramp error contant be 80. R () E() + G PID () 4 ( 5)( 10) C() 51

52 Exercie 5 Conider following tructure: Let the input impedance be generated by a reitor R 2 be in erie with a reitor R 1 and a capacitor C 1 ) that are in parallel, and let the feedbac impedance be generated by a reitor R f in erie with a capacitor C f. a) Show that thi choice lead to form a PID controller with high frequency gain limit a; b) Derive the parameter in the controller with repect to reitor and capacitor. 52

53 Exercie 6 In the following ytem deign a lag controller uch that the damping ratio of complex pole be 0.6 and ramp error contant be 80. R () + E() G lag () ( 48.5) C() 7 In the following ytem deign a lag controller uch that the damping ratio of complex pole be 0.6 and ramp error contant be 80. R () + E() G lag () 4 ( 5)( 10) C() 8 Deign a lead controller for exercie 6. 9 Deign a lead controller for exercie In the following ytem deign a controller uch that lead to zero teady tate error to tep and 4.3% P.O. (Final) 53

54 Exercie 11 In the following ytem deign a lag controller uch that that the ettling time be le than 3 ec. R () + E() G lag () 1 2 C() 12 In the following ytem deign a lead controller uch that that the ettling time be le than 3 ec. R () + E() () G lead 1 2 C() 13 In the following ytem deign a Lead controller uch that the damping ratio of complex pole be 0.4.(Final 1390) R () + E() G c () 1 2 C() 54

55 Supplementary Exercie 14 In the following ytem deign a PID controller with ZieglerNichol Ocillation Method R () E() + G PID () ( 48.5) C() 15 In the following ytem deign a PID controller with ZieglerNichol Ocillation Method R () E() C() + G PID () 4 ( 5)( 10) 55

56 Appendix: ZieglerNichol Deign طراحی زیگلر نیکولز Thi procedure i only valid for open loop table plant. OpenLoop Tuning CloedLoop Tuning According to Ziegler and Nichol, the openloop tranfer function of a ytem can be approximated with time delay and ingleorder ytem, i.e. where TD i the ytem time delay and T1 i the time contant. 56

57 Appendix: ZieglerNichol Reaction Curve Method(OpenLoop Cae) طراحی زیگلر نیکولز حالت حلقه باز For openloop tuning, we firt find the plant parameter by applying a tep input to the openloop ytem. The plant parameter K, TD and T1 are then found from the reult of the tep tet a hown in Figure. 57

58 Appendix: ZieglerNichol Reaction Curve Method(OpenLoop Cae) طراحی زیگلر نیکولز حالت حلقه باز K p K i K d P T 1 KT D PI PID 0.9T 1 KTD 1.2T 1 KTD 0.27T KT D T KT D T 1 K 58

59 Example 9: Following figure how the tep repone of an openloop tranfer function ytem. Deign a PID controller. So : K C, TD 5 ec, T1 20ec G( ) T e D

60 Example 9: Following figure how the tep repone of an openloop tranfer function ytem. Deign a PID controller. So : K C, TD 5 ec, T1 20ec G( ) T e D 1 20 K p K i K d T P KT D K P ( ) T PI KT D 1.2T 1 0. KT D PID T KT D 0.6T KT D T K K PI ( ) K PID ( )

61 ZieglerNichol Ocillation Method(Cloedloop) طراحی زیگلر نیکولز بروش نوسانی)حلقه بسته( Thi procedure i only valid for open loop table plant and it i carried out through the following tep Set the true plant under proportional control, with a very mall gain. Increae the gain until the loop tart ocillating. Note that linear ocillation i required and that it hould be detected at the controller output. Record the controller critical gain K c and the ocillation period of the controller output, T. Adjut the controller parameter according to Table 61

62 Appendix: ZieglerNichol Ocillation Method(Cloedloop) طراحی زیگلر نیکولز بروش نوسانی)حلقه بسته( K p K i K d P PI PID 0.5Kc 0.45Kc 0.6K c K 0.54 c T K 1.2 c 0.075K c T T 62

63 Example 10: Deign a PID controller for following ytem. Conider a plant with a model given by Find the parameter of a PID controller uing the ZN ocillation method. Obtain a graph of the repone to a unit tep input reference. 63

64 Solution حل Applying the procedure we find: K c = 8 and ω c = 3. T=3.62 Hence, from Table, we have K p Kc 0.6Kc 4.8 Ki Kd 0.075K ct 2.17 T The cloed loop repone to a unit tep in the reference at t = 0 i hown in the next figure. 64

65 Repone to tep reference پاسخ سیستم به پله 1.5 Step repone for PID control C PID ( ) Amplitude 0.5 C PID ( ) Time (ec) 65