Understanding Feedback Control Systems: Key Concepts and Design Approaches

1. Class July 8th ~ 15th: • HW#4 & #5 due7/8 • ExamIII on 7/10 • ExamIV on 7/24 • FINAL on 7/29 • Lab#3 & #4 (group of 3~4) • #3: G244 on 7/10 & 7/15 2:30PM  report due 7/17 • #4:optional:on 7/17 & 7/12  report due 7/24

2. Steady-State Error: e(∞) E = R-C = R- T∙R =[ 1 - T]∙R

3. The Design of Feedback Control System <Performance of a feedback control system > • Output response • Damping ratio / natural frequency / Setting time / overshoot / … • Stability / Steady-state error Design requirements: desired system characteristics (ζ & ωn)  Pole positions U(s)

4. EX: PID Control U(s) T (s) R(s) C(s) Design of G(s): PID controllers • Gain selections for KP, KI, & KD so that output C(s) follows reference input R(s) • Also, should consider the stability & dynamic characteristics • Design requirements: desired system characteristics (ζ & ωn)  Pole positions

5. Root Locus • Analytical approach using the C.E. (denominator of T.F. = 0) • Graphical approach usingKGH(s) multiplication of the control, plant, and sensor dynamic equations T(s) • Applications of the root locus  analysis & design of a control system

6. Root Locus: pole positions as K= 0  ∞ • Analytical approach using the C.E. (the denominator of the closed-loop T.F. = 0) • Graphical approach using KGH(s), open-loop T.F. multiplication of the control, plant, and sensor dynamic equations

7. Original Pole/Zero Positions (K=0) & Case 0 < K < ∞ C. E. 1 + KGH(s) = 0 in vector representation, KGH(s) = -1 • Magnitude: |KGH(s)| = 1 • Phase Angle: (2k+1)•1800for k = 0, 1, 2,…

8. Original Pole/Zero Positions (K=0) & Case 0 < K < ∞

9. Root Locus Sketch using open-loop T.F. KGH(s) • Locate the open-loop poles & zeros • Locate the segments (root loci) • Begins at a pole & ends at a zero • Locus lies to the left of an odd number of poles/zeros • Symmetrical w.r.t. the real axis

10. Root Locus Sketch: using open-loop T.F. KGH(s)

11. Root Locus Sketch

12. Root Locus Sketch

13. Root Locus Sketch

14. Root Locus Sketch

15. Root Locus Sketch: stability as K increases

16. Root Locus Concept • Graphical approach using KGH(s)  multiplication of the control, plant, and sensor dynamic equations C. E. 1 + KGH(s) = 0 in vector representation, KGH(s) = -1 • Magnitude: |KGH(s)| = 1 • Phase Angle: (2k+1)•1800for k = 0, 1, 2,…

17. Root Locus Concept • C. E. Q(s) = 1 + KGH(s) = s2 + 10s + K = 0 • Magnitude: |KGH(s)| = 1 • Phase Angle: (2k+1)•1800for k = 0, 1, 2,…

18. Root Locus Complete Analysis – Step by Step • Graphical approach using KGH(s) • Analytical approach using the C.E. (denominator of T.F. = 0)

19. Root Locus Sketch

20. Root Locus Sketch

29. Original Pole/Zero Positions (K=0) & Case 0 < K < ∞

30. Root Locus for a Control Design C. E. 1 + KGH(s) = 0 in vector representation, KGH(s) = -1 • Magnitude: |KGH(s)| = 1 • Phase Angle: (2k+1)•1800for k = 0, 1, 2,…  Should be held for all poles that the locations will be depended on k.

31. Original Pole/Zero Positions (K=0) & Case 0 < K < ∞

32. Feedback Control System Gc(s) Gp(s)

33. PD Control

34. Gc(s) Gp(s)

35. Gc(s) Gp(s)

38. = 2 (s+4)