INC 342 Lecture 2: Root Locus (cont.) Dr. Benjamas Panomruttanarug - PowerPoint PPT Presentation

INC 342 Lecture 2: Root Locus (cont.) Dr. Benjamas Panomruttanarug Benjamas.pan@kmutt.ac.th

Sketching Root Locus (review) 1. Number of branches 2. Symmetry 3. Real ‐ axis segment 4. Starting and ending points 5. Behavior at infinity BP INC 342 2

Example   ( 3 )( 5 ) K s s  ( ) ( ) Find breakaway, break ‐ in points KG s H s   ( 1 )( 2 ) s s   2 ( 8 15 ) K s s    Condition of poles ( ) ( ) 1 KG s H s   2 ( 3 2 ) s s    2 ( 3 2 ) s s  K   2 ( 8 15 ) s s   2 11 26 61 dK s s then solve for s   0   2 8 15 ds s s s = ‐ 1.45, 3.82 is breakaway and break ‐ in points BP INC 342 3

Example sketch root locus and find angel of departure of complex poles 1 x x o ‐ 3 ‐ 2 ‐ 1 x BP INC 342 4

Using MATLAB with Root Locus • tf • pole • zero • rlocus • pzmap • sisotool BP INC 342 5

2 objectives for desired response 1. Improving transient response  Percent overshoot, damping ratio, settling time, peak time 2. Improving steady ‐ state error  Steady state error BP INC 342 6

Gain adjustment • Higher gain, smaller steady stead error, larger percent overshoot • Reducing gain, smaller percent overshoot, higher steady state error BP INC 342 7

Improving transient response • Point A and B have the Y= ‐ mx same damping ratio. m = tan(acos(damping ratio)) θ • Starting from point A, cannot reach a faster response at point B by adjusting K. • Compensator is preferred. BP INC 342 8

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   2       / 1     % 100 OS e BP INC 342 9

Example Find K that yields 1.52% overshoot. Also estimate settling time, peak time, steady ‐ state error corresponding to the K   2       / 1     Step I: 1.52% overshoot  ζ =0.8 % 100 OS e Step II: draw a root locus BP INC 342 10

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,… BP INC 342 11

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. BP INC 342 12