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2. PID Controller 1) Standard form of PID controller

2. PID Controller 1) Standard form of PID controller I action: Increase system type, reduce steady-state error. Additional pole at the origin  destabilizing D action: Additional zero  increase stability Increase damping (cf) general form.

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2. PID Controller 1) Standard form of PID controller

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  1. 2. PID Controller • 1) Standard form of PID controller • I action: Increase system type, reduce steady-state error. • Additional pole at the origin  destabilizing • D action: Additional zero  increase stability • Increase damping • (cf) general form

  2. 2) Time-response sensitivity to parameters • Assume depends on the parameter , i.e. • How to relate to ? • Def: Logarithmic sensitivity function • The change in the output with respect to percentage change in the parameter.

  3. 3) Tuning method of PID controllers • Eiegler-Nichols tuning rule • Stability margin tuner • (cf) There are many methods based on optimization techniques also • there are many AI techniques, pattern recognition techniques. • (i)Z-N tuner • Process reaction curve method • Ultimate gain method • Basically, auto-tuner consists of two stages i.e. identification stage and tuning stage

  4.  Process reaction curve method • Assumption on model : • Parameter achieve from unit step response. • Tuning to achieve decay ratio of 25%.

  5.  Ultimate gain method • Identification of oscillation point  gain tuning

  6. (ii) Stability margin tuner • Identification of one point in plane. •  Tuning based on margin. • 4) Integrator Antiwindup • Consider a PI control system.

  7. Typical response :

  8. How to tackle this problem? • Turn off the integral action as soon as the actuator saturates! • Antiwindup PI

  9. • See Ex. 4.12

  10. 3. Steady-state Tracking and System Type • vs. system type: • : position-error constant = • : velocity constant = • : acceleration constant = • (cf) Disturbance rejection vs system type •  same as vs. system type.

  11. (cf) Truxal’s Formula : • when is of type 1. • with • Let (ramp), then

  12. Using L’Hopital’s rule, • Assume T(s) = • Then

  13. 4. Stability •  BIBO stability • Stable iff every bounded input results in bounded output. • Condition for BIBO stability • pf): ) •  • 

  14. ) ( BIBO unstable if is unbounded.) • choose • then (note) • BI results in

  15.  Asymptotic stability • 입력이 없는 경우 어떤 초기 조건이건 관계없이 • 출력이  0 as t   • (Remark) LTI system 일 경우 • BIBO  asymptotic stability •  Re{pole}<0 • Then, the natural question is how to check the stability without • calculating the poles?

  16.  The Routh-Hurwitz Stability Criterion • Consider the characteristic equation of an n-order system : • Routh array :

  17. Case 1. Assume no element in the first column is zero. • The number of positive real roots of a(s) is equal to the number of sign change in first column. • Ex 1) . • Then Routh array is : •  There are two positive real roots.

  18. Ex 2) . • ⅰ) If stable • ⅱ) If there are two unstable roots.

  19. Ex 3). If the first column in R-H table is • Number of • Unstable roots : 2 1 4

  20. Case 2. If the first element in one of the rows is zero, including non-zero element in that row. • Ex 4) . • and •  there are two unstable roots.

  21. Ex 5) .  unstable K.

  22. Case 3. If an entire row of the Routh array is zero. • Ex 6) .

  23. Ex 7). Characteristic equation

  24. System이 stable한 조건은 • Ex 8)

  25. 실제로 • 1) • 2) •  “4개의 허수근”+안정근

  26. Ex 9)

  27. Now as •  “2개의 불안정근”+”2개의 허수근”+안정근 • 확인 :

  28.  Relative Stability • s-plane에서 특성방정식의 근들의 실수값이 적어도 보다 좌측에 존재하기를 원할 때, 즉 • Ex) • (HW) 4장 연습문제. 6, 11, 16, 18, 21, 26, 32, 39, 44, 49

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