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1              Design via Root Locus•   How to use the root locus to design cascade compensators to    improve the steady ...
2                          IntroductionImproving transient response•   transient response can be improved with the additio...
3Improving Steady-State Error•   steady-state error can be improved with the addition of integration in the    forward pat...
4Improving Transient Response and Steady-State Error                                 Steady State•   By using dynamic comp...
5Compensator configurations to meet transient and      steady state      steady-state error specificationsCascadeconfigura...
6           Improving Steady-State Error            via Cascade Compensation             i C     d C          tiThere are ...
7        1. Ideal Integral Compensation (PI controller) to                    Improve Steady-State Error                  ...
8Implementation of ideal integral compensator                                                    … zero can be adjusted by...
9Problem Given the system operating with a damping ratio of 0.174, show thatthe addition of the ideal integral compensator...
10        Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wile...
11        Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wile...
12    2. Lag compensation to Improve Steady-State Error•     Does not use pure i      D                 integration       ...
13•      If the lag compensator pole and zero are close together, the angular       contribution of the compensator to poi...
14Problem Compensate the system, to improve the steady-state error by a factor of10 if the system is operating with a damp...
15 Solution:Uncompensated error (fU           d       (from previous example):                              i         l )A...
16        Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wile...
17         Improving Transient Response via              Cascade CompensationThere are two techniques:1. Ideal derivative ...
18    1. Ideal Derivative Compensation (PD controller)             to Improve Transient Response•    the original system c...
19Zero at -3                     Dr Branislav Hredzak             Control Systems Engineering, Fourth Edition by Norman S....
20Zero at -4                     Dr Branislav Hredzak             Control Systems Engineering, Fourth Edition by Norman S....
21•   The damping ratio is unchanged (0.4), hence the percent overshoot is the    same for all three cases•   More negativ...
22Implementation of ideal derivative compensator                                                         Dr Branislav Hred...
23Problem Given the system, design an ideal derivative compensatorto yield a 16% overshoot, with a threefold reduction in ...
24                  Sum of the angle of open-loop poles to the design point is 275.60Imaginary part:                      ...
25        Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wile...
26result needs to be verified bysimulation i l ti                   Dr Branislav Hredzak           Control Systems Enginee...
27        Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wile...
282. Lead compensation to Improve Transient Response•   consists of a pole and a zero•   if the pole is farther from the i...
29•    infinite number of lead compensators could be used to meet the                                p     transient respo...
30Problem Design lead compensator that will reduce the settling time by a factor of2 while maintaining 30% overshoot.     ...
31 Let zc= - 5The resulting angle is -172.690hence the pole must contribute -7.310                                        ...
32Second order approximation OK              pp                                        Dr Branislav Hredzak               ...
33         lmproving Steady-State Error and               Transient Response•   First    Fi t we design for transient resp...
341. PID Controller Design to lmprove Steady-State         Error and Transient Response                                   ...
35Design procedure1.   Evaluate the performance of the uncompensated system to determine     how much improvement in trans...
36      2. Lag-Lead Compensator Design to lmprove       Steady State       Steady-State Error and Transient ResponseDesign...
37             Physical realization of compensationActive circuit realization                                             ...
38        Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wile...
39        Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wile...
40Passive circuit realization                                      Dr Branislav Hredzak                              Contr...
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09 elec3114

  1. 1. 1 Design via Root Locus• How to use the root locus to design cascade compensators to improve the steady state error steady-state• How to use the root locus to design cascade compensators to improve the transient response• How to use the root locus to des g cascade compensators to ow o e oo ocus o design co pe sa o s o improve both the steady-state error and the transient response• How to realize the designed compensators physically Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  2. 2. 2 IntroductionImproving transient response• transient response can be improved with the addition of differentiation• the compensated system will have a root locus that goes through the desired pole location Improvement: - response B is faster than response A, while the overshoot is the same Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  3. 3. 3Improving Steady-State Error• steady-state error can be improved with the addition of integration in the forward path. p Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  4. 4. 4Improving Transient Response and Steady-State Error Steady State• By using dynamic compensators, compensating networks can be designed that allow to meet both transient and steady state error specifications steady-state simultaneously Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  5. 5. 5Compensator configurations to meet transient and steady state steady-state error specificationsCascadeconfiguration Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  6. 6. 6 Improving Steady-State Error via Cascade Compensation i C d C tiThere are two techniques:1. Ideal integral compensation – uses a pure i integrator. It reduces the steady-state error to zero2. Lag compensation – does not use pure integration. It p places the pole near the origin. It does not reduce the error p g to zero. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  7. 7. 7 1. Ideal Integral Compensation (PI controller) to Improve Steady-State Error Steady State• Steady-state error is improved by placing an open-loop pole at the origin Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  8. 8. 8Implementation of ideal integral compensator … zero can be adjusted by varying K2/K1• Since the ideal integral compensator has both proportional and integral control, it is given the alternate name PI controller Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  9. 9. 9Problem Given the system operating with a damping ratio of 0.174, show thatthe addition of the ideal integral compensator reduces the steady-state error to h ddi i f h id l i l d h dzero for a step input without appreciably affecting transient response. Thecompensating network is chosen with a pole at the origin to increase the systemtype and a zero at -0.1, close to the compensator pole, so that the angular d l h l h h lcontribution of the compensator evaluated at the original, dominant, second-order poles is approximately zero. Thus, the original, dominant, second-orderclosed-loop poles are still approximately on the new root locus. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  10. 10. 10 Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  11. 11. 11 Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  12. 12. 12 2. Lag compensation to Improve Steady-State Error• Does not use pure i D integration i• Uses passive networks• The pole and zero are placed to the left, close to the origin h l d l d h l f l h i i static error constant new static error constant Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  13. 13. 13• If the lag compensator pole and zero are close together, the angular contribution of the compensator to point P is approximately zero degrees. degrees• K is virtually the same for the uncompensated and compensated systems, since the lengths of the vectors drawn from the lag compensator are approximately equal and all other vectors have not changed appreciably.• Improvement is the steady-state error is given by a lag compensator with a pole that is not at the origin will improve the static error constant by a factor equal to zc/pc Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  14. 14. 14Problem Compensate the system, to improve the steady-state error by a factor of10 if the system is operating with a damping ratio of 0 174 0.174. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  15. 15. 15 Solution:Uncompensated error (fU d (from previous example): i l )A tenfold improvement means a steady-state error of p yLet us select Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  16. 16. 16 Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  17. 17. 17 Improving Transient Response via Cascade CompensationThere are two techniques:1. Ideal derivative compensation – uses a pure differentiator2. Lead2 L d compensation – d ti does not use pure differentiation t diff ti ti Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  18. 18. 18 1. Ideal Derivative Compensation (PD controller) to Improve Transient Response• the original system can be made faster by adding a single zero to the forward path• Disadvantage of ideal differentiation: differentiation of high frequency noise leads to large unwanted signals Zero at -2 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  19. 19. 19Zero at -3 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  20. 20. 20Zero at -4 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  21. 21. 21• The damping ratio is unchanged (0.4), hence the percent overshoot is the same for all three cases• More negative real part of dominant poles, hence shorter settling time• Imaginary parts are larger hence smaller peak times larger,• Improvement in steady state error (due to increase of Kp) Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  22. 22. 22Implementation of ideal derivative compensator Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  23. 23. 23Problem Given the system, design an ideal derivative compensatorto yield a 16% overshoot, with a threefold reduction in settling time. y , gSolution 16% overshoot → ζ = 0.504 3.320 Ts (new) = = 1.107 3 Real part: Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  24. 24. 24 Sum of the angle of open-loop poles to the design point is 275.60Imaginary part: Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  25. 25. 25 Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  26. 26. 26result needs to be verified bysimulation i l ti Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  27. 27. 27 Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  28. 28. 282. Lead compensation to Improve Transient Response• consists of a pole and a zero• if the pole is farther from the imaginary axis than the zero, the angular contribution of the compensator is still positive and thus approximates an p p pp equivalent single zero• can be implemented using passive components• less sensitive to noise• during design we arbitrarily select either a lead compensator pole or zero Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  29. 29. 29• infinite number of lead compensators could be used to meet the p transient response requirement However during the design we have to be aware of the static error constant, the gain, second order approximation. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  30. 30. 30Problem Design lead compensator that will reduce the settling time by a factor of2 while maintaining 30% overshoot. overshoot Solution 30% overshoot → ζ = 0.358 Ts (new) = 3.972 / 2 = 1.986 s ωd = −2.014 tan(110.980 ) = 5.252 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  31. 31. 31 Let zc= - 5The resulting angle is -172.690hence the pole must contribute -7.310 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  32. 32. 32Second order approximation OK pp Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  33. 33. 33 lmproving Steady-State Error and Transient Response• First Fi t we design for transient response and then design for steady-state d i f t i t d th d i f t d t t error• If we d i an active PD controller f ll design ti t ll followed by an active PI controller, db ti t ll the resulting compensator is called a proportional-plus-integral-plus- derivative (PID) controller• If we first design a passive lead compensator and then design a passive lag compensator, the resulting compensator is called a lag-lead compensator t Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  34. 34. 341. PID Controller Design to lmprove Steady-State Error and Transient Response Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  35. 35. 35Design procedure1. Evaluate the performance of the uncompensated system to determine how much improvement in transient response is required.2. Design the PD controller to meet the transient response specifications. The design includes the zero location and the loop gain.3. Simulate the system to be sure all requirements have been met.4.4 Redesign if the simulation shows that requirements have not been met.5. Design the PI controller to yield the required steady-state error.6. Determine the gains, Kl, K2, and K3.7. Simulate the system to be sure all requirements have been met.8. Redesign if simulation shows that requirements have not been met. d i i l i h h i h b Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  36. 36. 36 2. Lag-Lead Compensator Design to lmprove Steady State Steady-State Error and Transient ResponseDesign procedure1.1 Evaluate th E l t the performance of the uncompensated system to determine how f f th t d t t d t i h much improvement in transient response is required.2. Design the lead compensator to meet the transient response g p p specifications. The design includes the zero location, pole location, and the loop gain.3.3 Simulate h Si l the system to be sure all requirements have been met. b ll i h b4. Redesign if the simulation shows that requirements have not been met.5.. Evaluate the steady-state e o performance for the lead-compensated v u e e s e dy s e error pe o ce o e e d co pe s ed system to determine how much more improvement in steady-state error is required.6. Design the lag compensator to yield the required steady-state error.7. Simulate the system to be sure all requirements have been met.8. Redesign if the simulation shows that requirements have not been met. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  37. 37. 37 Physical realization of compensationActive circuit realization Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  38. 38. 38 Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  39. 39. 39 Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  40. 40. 40Passive circuit realization Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.

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