351_27435_EE411_2015_1__1_1_0 0 EE411 Lec 1,2,3 root locus.pdf

Introduction. What is root locus? Root locus is the locus (graphical presentation) of the closed-loop poles as a specific parameter (usually gain, K ) is varied from 0 to infinity . Why do we need to use root locus? We use root locus to analyze the transient response qualitatively. (E.g. the effect of varying gain upon percent overshoot, settling time and peak time). We can also use root locus to check the stability of the system..

Introduction. we use root locus to analyze the feedback control system. K in the feedback system is called a gain. Gain is used to vary the system in order to get a different output response..

E x a m p le. How the dynamics of the system (camera) changes as K is varied ?.

Drawing the root locus. Characteristic equation:.

Drawing the root locus. Next step is to plot the poles values on the s-plane by varying the gain, K , value..

Drawing the root locus. Join the poles with solid lines and you will get the shape of the locus (path).

Drawing the root locus. The process of drawing a root locus is time consuming. If the system is complex. An alternative approach is to s ketch the root locus instead of drawing the root locus..

Sketching the root locus. In order to sketch the root locus we must follow these 5 rules..

Example. 1) Sketch the root locus of the following system:.

Construction of root loci. S t e p - 1 : T h e f i r s t s t e p i n c o n s t r u c t i n g a r o o t - lo c u s p l ot is t o locate the open-loop poles and zeros of G(s)H(s) in s-plane..

Construction of root loci.

Construction of root loci Step-3 : Determine the of the root loci..

180 when k 1 300 w h e n k 2. 60 w h e n k 0. 3 0.

Construction of root loci Step-3 : Determine the asymptotes of the root loci..

- 5. - 4. - 3. - 2. - 1. 0. 1. 2. - 1. - 0. 5. 0.

Construction of root loci. Step-4 : Determine the breakaway point or break-in point . The breakaway or break-in points are the closed-loop poles that.

The breakaway point can now be determined as. 1. K s ( s 1 ) ( s 2 ).

- 5. - 4. - 3. - 2. - 1. 0. 1. 2. - 1. - 0. 5. 0.

Construction of root loci Step-5 : Intersection points with the imaginary axis..

Construction of root loci. S t e p -5 : D et e rm i n e t h e p o i n t s w h e r e r oo t l o c i c r o s s th e imaginary axis..

31.

- 7. - 6. - 5. - 4. - 1. 0. 1. 2. - 5.

E x a m p l e # 1. Consider following unity feedback system..

E x a m p l e # 1. The damping ratio of 0.5 corresponds to cos cos 1 cos 1 (0.5) 60.

?. 36.

E x a m p l e # 1. T h e v al u e o f K t h a t y i e l d s s u c h p o l e s is f o u n d f r o m t h e magnitude condition.

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E x a m p l e # 1. The third closed loop pole at K=1.0383 can be obtained as.

40.