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 2 INC 342

BP 3 INC 342

Example

) 2 )( 1 ( ) 5 )( 3 ( ) ( ) (      s s s s K s H s KG

Find breakaway, break‐in points

15 8 61 26 11 ) 15 8 ( ) 2 3 ( ) 2 3 ( ) 15 8 ( 1 ) ( ) (

2 2 2 2 2 2

                   s s s s ds dK s s s s K s s s s K s H s KG

Condition of poles then solve for s s = ‐1.45, 3.82 is breakaway and break‐in points

BP 4 INC 342

Example

sketch root locus and find angel of departure

• f complex poles

x x x

• ‐3

‐2 ‐1 1

Using MATLAB with Root Locus

• tf
• pole
• zero
• rlocus
• pzmap
• sisotool

BP INC 342 5

6

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

7

Gain adjustment

• Higher gain, smaller steady stead error,

larger percent overshoot

• Reducing gain, smaller percent overshoot,

higher steady state error

BP INC 342

8

Improving transient response

• Point A and B have the

same damping ratio.

• Starting from point A,

cannot reach a faster response at point B by adjusting K.

• Compensator is

preferred.

BP INC 342

Y=‐mx m = tan(acos(damping ratio)) θ

BP 9 INC 342

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

100 %

2

1 /

 

         

e OS

BP 10 INC 342

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

100 %

2

1 /

 

         

e OS

BP 11 INC 342

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 12 INC 342

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