1. INC 341PT & BP INC341 Root Locus (Continue) Lecture 8

2. INC 341PT & BPINC 341PT & BP Sketching Root Locus (review) 1.Number of branches 2.Symmetry 3.Real-axis segment 4.Starting and ending points 5.Behavior at infinity

3. INC 341PT & BPINC 341PT & BP Refining the sketch 1.Real-axis breakaway and break-in points 2.Calculation of jω-axis crossing 3.Angels of departure and arrival 4.Locating specific points

4. INC 341PT & BPINC 341PT & BP Break-in point Breakaway point 1. Real-axis breakaway and break-in points point where the locus leaves the real axis point where the locus returns to the real axis

5. INC 341PT & BPINC 341PT & BP set s = σ (on the real axis) Breakaway point Break-in point

6. INC 341PT & BPINC 341PT & BP Example Find breakaway, break-in points Condition of poles then solve for s s = -1.45, 3.82 is breakaway and break-in points

7. INC 341PT & BPINC 341PT & BP Another approach without derivative

8. INC 341PT & BPINC 341PT & BP 2. Calculation of jω-axis crossing Imaginary axis is a boundary of stability use Routh-Hurwitz criterion!!! Imaginary axis crossing

9. INC 341PT & BPINC 341PT & BP Review of Routh-Hurwitz “the number of roots of the polynomial that are in the right half plane is equal to the number of sign changes in the first column”

10. INC 341PT & BPINC 341PT & BP Example From the closed-loop transfer function, find an imaginary axis crossing

11. INC 341PT & BPINC 341PT & BP Substitute K=9.65 in s 2 to find the value of s A complete row of zeros yields imag. axis roots

12. INC 341PT & BPINC 341PT & BP 3. Angles of departure and Arrival Fact: root locus starts at open loop poles and ends at open loop zeros Assume a point on the root locus close to a complex Pole, the sum of angles to this point is an odd multiple of 180.

13. INC 341PT & BPINC 341PT & BP Angel of arrival (zero) Angel of departure (pole)

14. INC 341PT & BPINC 341PT & BP Example sketch root locus and find angel of departure of complex poles x x x o -3-2 1

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16. INC 341PT & BPINC 341PT & BP 4. Calibrating root locus Search a given line for the point yielding a summation of angles equal to an odd multiple of 180. Gain at this point = pole length/zero length

17. INC 341PT & BPINC 341PT & BP At r=0.747 Intersection with damping ratio line Coordinate on Damping line = (rcosθ, rsinθ) Try r = 0.5, 1, 0.8, 0.7, 0.75, 0.725, ….. ζ=cosθ θ Y=-mx M = tan(acos(damping ratio))

18. INC 341PT & BPINC 341PT & BP Example sketching root locus What is the exact point and gain where the locus crosses the imag. Axis? Where is the breakaway point? What range of K that keep the system stable?

19. INC 341PT & BPINC 341PT & BP Transient Response Design via Gain Adjustment Find K that gives a desired peak time, settling time, %OS (find K at the intersection) Use 2 order approx. and consider only dominant pole

20. INC 341PT & BPINC 341PT & BP The third pole can be ignored (a gives a better approx. than b cause the third pole is further to the left) Zero closed to the dominant poles can be cancelled by the third pole (c gives a better approx. than d)

21. INC 341PT & BPINC 341PT & BP Example Find K that yields 1.52% overshoot. Also estimate settling time, peak time, steady-state error corresponding to the K Step I: 1.52% overshoot  ζ=0.8 Step II: draw a root locus

22. INC 341PT & BPINC 341PT & BP Step III: draw a straight line of 0.8 damping ratio Step IV: find intersection points where the net angle is added up to 180*n, n=1,2,3,…

23. INC 341PT & BPINC 341PT & BP Step V: find the corresponding K at each point Step VI: find peak time, settling time corresponding to the pole locations (assume 2 nd order approx.) Step VII: calculate Kv and ss error Note: case 1 and 2 cannot use 2 nd order approx. cause the third pole and closed loop zero are far away. In case 3, the approx. is valid.

24. INC 341PT & BPINC 341PT & BP Generalized Root Locus K is fixed, vary open loop pole instead!!! Creating an equivalent system where p1 appears as the forward path gain.

25. INC 341PT & BPINC 341PT & BP Try to get a general TF

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27. INC 341PT & BPINC 341PT & BP Using MATLAB with Root Locus tf pzmap rlocus sgrid sisotool