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Chap 6 The Compensation of the linear control systems

Chap 6 The Compensation of the linear control systems. P553. §6-1 Introduction 6.1.1 definition of compensation 6.1.2 types of compensation §6-2 The basic controller operation analysis 6.2.1 PI D controller ---active compensation 6.2.2 phase-lead controller

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Chap 6 The Compensation of the linear control systems

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  1. Chap 6 The Compensation of the linear control systems P553

  2. §6-1 Introduction 6.1.1 definition of compensation 6.1.2 types of compensation §6-2 The basic controller operation analysis 6.2.1 PI D controller ---active compensation 6.2.2 phase-lead controller 6.2.3 phase-lag controller 6.2.4 phase lag-lead controller §6-3 Cascade compensation method of Root loci §6-4 Cascade compensation method of frequency- Domain §6-5 Feedback compensation passive compensation controller Chap 6 The Compensation of the linear control systems

  3. 6.1 Introduction 6.1.1 What is compensation or correction of a control system ? solution

  4. This closed-loop system can be stable. 6.1 Introduction Definition of the compensation: Wemake the system stable by increasing a component. This procedureis called the compensation or correction. increasing a component ,which makes the system’s performance to be improved, other than only varying the system’s parameters, this procedure is called the compensationor correction of the system.

  5. 6.1 Introduction Compensator: The compensator is an additional component or circuit that is inserted into a control system to compensate for a deficient performance. 6.1.2 Types of the compensation

  6. 6.1 Introduction (1) Cascade(or series) compensation (2) Feedback compensation (3) Both series and feedback compensation (4) Feed-forward compensation (1) Cascade(or series) compensation Features : simple but the effects to be restricted.

  7. 6.1 Introduction R(s) C(s) - - - - - - R(s) C(s) C(s) R(s) (2) Feedback compensation Features: complicated but noise limiting, the effects are more than the cascade compensation. (3) Both cascade and feedback compensation Features: have advantages both of cascade and feedback compensation.

  8. R(s) + - For input For disturbance(voice) F(s) + R(s) C(s) - C(s) 6.1 Introduction (4) Feed-forward compensation Features: theoretically we can make the error of a system to be zero and no effects to the transient performance of the system. Demonstration:

  9. For input + R(s) C(s) - Question: actually the could not be easy implemented especially maybe the G20 is variable. 6.1 Introduction But no effect to the characteristic equation: 1+ G10G20 = 0

  10. R(s) + - For disturbance(voice) F(s) Question: actually the could not be easy implemented especially maybe the G20 is variable. And the F(s) could not be easy measured. C(s) 6.1 Introduction Also no effect to the characteristic equation: 1+ G10G20 = 0

  11. N(s) Where: - + E(s) R(s) C(s) - + GCR Determine GCR and GCN , make E(s) to be zero. Fig.6.1.7 6.1 Introduction example For the system shown in Fig.6.1.7: Solution : Thinking: if r(t) = n(t) = t , Determine GCRand GCN , make essto be zero — as a exercise.

  12. 6.2 Operation analysis of the basic compensators 6.2.1 Active Compensation PID controller － active “compensator”. Transfer function:

  13. ＋ ＋ R(s) C(s) G(s) ＋ － PD controller

  14. 6.2 Operation analysis of the basic compensators Effects of PD controller: 1) PD controller does not alter the system type; 2) PD controller improve the system’s stability (to increase damping and reduce maximum overshoot); 3) PD controller reduce the rise time and settling time; 4) PD controller increase BW(Band Width) and improveGM(Kg),PM(γc), and Mr . － bring in the noise !

  15. ＋ ＋ R(s) C(s) G(s) ＋ － 6.2 Operation analysis of the basic compensators PI controller

  16. 6.2 Operation analysis of the basic compensators Effects of PI controller: 1) Increase the system’s type－clear the steady-state error ; 2) reduce BW(Band Width) and GM(Kg), PM(γc) and Mr ; beneficial to the noise limiting, not beneficial to the system’s stability.

  17. R(s) C(s) G(s) Transfer function: + - 6.2 Operation analysis of the basic compensators PID controller PID controller have advantages both of PI and PD.

  18. R2 R2 C ur R1 ur _ R1 _ u0 u0 + C + PD controller PI controller R2 C2 ur R1 _ u0 + C1 PID controller Circuits of PID

  19. For example: Disk driver control system

  20. solution How to get？ Shown in 6.3 detail.

  21. 6.2.2 Passive compensation controllers Types of passive compensation controller

  22. 6.2.2 Passive compensation controllers Zero and pole

  23. z p Bode plot Compensation ideal: make ωm to be ωc ! Effects are similar to PD.

  24. Circuit of the phase-lead controller

  25. Zero and pole 6.2.2 Passive compensation controllers

  26. Bode plot Effects are similar to PI. Compensation ideal: Make 1/τto bein the lower frequency-band and far from ωc!

  27. Circuit of the Phase-lag controller

  28. Zero and pole 6.2.2 Passive compensation controllers

  29. Bode plot Effects are similar to PID. Compensation ideal: First make the phase-lag compensation－to satisfy ess and compensate a part of γc . secondmake the phase-lead compensation－to satisfy the transitional requirements.

  30. Circuit of the Phase lag-lead controller

  31. 6.2 Operation analysis of the basic compensators 6.2.3 Comparing active compensation controllers and passive compensation controllers

  32. Fig.6.3.1 6.3Cascade compensation by Root loci method 6.3.1 Phase-lead compensation (P569) Example 6.3.1: solution The root lociof the systemshown in Fig.6.3.1 Analysis: unstable. phase-lead compensation

  33. Fig.6.3.2 6.3Cascade compensation by Root loci method

  34. (1) Maximum α method (2) Method based on the open-loop gain Fig.6.3.3 6.3Cascade compensation by Root loci method There are two approaches to determine zc and pc .

  35. For this example we choose the Maximum α method: Fig.6.3.3 6.3Cascade compensation by Root loci method In terms of the sine’s law:

  36. 6.3Cascade compensation by Root loci method The root locus of the compensated system is shown in Fig.6.3.4 Steps of the cascade phase-lead Compensation: Fig.6.3.4 (1) Determine the dominant roots based on the performance specifications of the system: (2) plot the root locus of the system and analyze what compensation device should be applied. Root locus of the compensted system

  37. (3) Determine the angle φc to be compensated: (4) calculate θ andγ: (5) calculate zc and pc In terms of the sine’s law : 6.3Cascade compensation by Root loci method (6) plot the root locus of the compensated system and make validity check.

  38. 6.3.2 Phase-lag compensation using the root locus (P577) -10 Fig.6.3.5 Example 6.3.2: Solution: The root locus of the system is shown in Fig.6.3.5.

  39. Fig 6.3.6 6.3.2 Phase-lag compensation using the root locus (P577) The detail of the root-loci is shown in Fig 6.3.6.

  40. Fig 6.3.6 6.3.2 Phase-lag compensation using the root locus (P577)

  41. 6.3.2 Phase-lag compensation using the root locus (P577) Validate…… Steps of the cascade phase-lag Compensation: (1) Determine the dominant roots based on the performance specifications of the system: (2) plot the root locus of the system and analyze what compensation device should be applied. If the phase-lag Compensation be applied:

  42. 6.3.2 Phase-lag compensation using the root locus (P577) (5) plot the root locus of the compensated system and make validity check. 6.3.3 Phase lag-lead compensation by the root locus method Basic ideal:

  43. 6.3.3 Phase lag-lead compensation by the root locus method First: make the phase-lead compensation－to satisfy the transitional requirements. Second: make the phase-lag compensation－to satisfy ess requirements. Exercise: Make compensation using PD and PI for example 6.3.1 and example 6.3.2

  44. 6.4Cascade compensation by frequency response method 6.4.1 Phase-Lead Compensation using Bode diagram

  45. 6.4Cascade compensation by frequency response method

  46. 6.4Cascade compensation by frequency response method

  47. －40dB/dec Fig.6.4.1 6.4Cascade compensation by frequency response method Example: solution:

  48. －40dB/dec Fig.6.4.1 6.4Cascade compensation by frequency response method Exercise: • Make validity check for this example. • 2. Make compensation using PD for this example.

  49. －20dB/dec 6.32 －20lgβ 2 －40dB/dec －900 －1800 6.4.2 Phase-Lag Compensation using Bode diagram Example: solution: Fig.6.4.2

  50. －20dB/dec 6.32 －20lgβ 2 －40dB/dec －900 －1800 6.4.2 Phase-Lag Compensation using Bode diagram Fig.6.4.2 Validate……

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