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Lecture 19

Lecture 19 covers the root locus method, which illustrates how closed-loop poles change with varying parameter K. It details the angle and magnitude conditions necessary for plotting the root locus, along with steps for drawing it, such as locating open-loop poles and zeros, determining asymptotes, and finding break-away points. The lecture also includes detailed examples to demonstrate the application of these concepts.

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Danny Chicas
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9 views16 pages

Lecture 19

Lecture 19 covers the root locus method, which illustrates how closed-loop poles change with varying parameter K. It details the angle and magnitude conditions necessary for plotting the root locus, along with steps for drawing it, such as locating open-loop poles and zeros, determining asymptotes, and finding break-away points. The lecture also includes detailed examples to demonstrate the application of these concepts.

Uploaded by

Danny Chicas
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Lecture 19: Root Locus Continued

1. Angle and magnitude condition


review

ENGR 4220/5220, Lecture 19


2. Detailed root locus examples

1
Root Locus Review
• The root locus is a plot of how the closed-loop
poles move for different values of a parameter K
• The root locus can be drawn based on the open-
loop transfer function

ENGR 4220/5220, Lecture 19


2
Root Locus Review
Therefore, any point s on the root locus must satisfy
the following two relations for some value of K

• Angle Condition • Magnitude Condition

• Which translates to:


• Which implies that:

branches of the root locus


from from start at open-loop poles
OL OL and end at open-loop zeros
zeros poles
Rules for Drawing the Root
Locus
1. Locate OL poles and zeros in the s-plane
2. Determine root locus on the real axis
3. Approximate the asymptotes of the root
locus

ENGR 4220/5220, Lecture 19


4. Approximate break-away and break-in
points
5. Determine angles of departure and arrival
6. Find Imaginary axis crossings
4
Detailed Example
• Plot the Root Locus for the following system

STEP 1: Locate open-loop poles and zeros


Detailed Example (continued)
(branches start at OL poles and end at OL zeros)

I
m

R
e
Step 2
Determine root locus on real axis (use angle condition
and recall angle from conjugate pairs cancel)

in general, branches lie to the left of an odd number


of poles and zeros
Step 3
Approximate asymptotes
# of asymptotes = # of zeros at infinity
one zero at infinity two zeros at infinity

three zeros at infinity four zeros at infinity


Step 4
Approximate break-away/break-in points
(this root locus does not have any)

ENGR 4220/5220, Lecture 19


In general, occur when branches come together
• Break-aways tend to occur between poles on
real axis
• Break-ins tend to occur between zeros on
real axis (including zeros at “infinity”)
9
Step 5
Determine angles of departure/arrival
use angle condition to test points around poles
(departure) and around zeros (arrival)
Step 6
Find imaginary axis crossings
points on imag axis have zero real part, s = jω
sub s = jω into and solve for ω and K
Step 6 (continued)

Can use this approach to find gains to achieve other


closed-loop pole locations (substitute s = -σ ± jω)
Detailed Example #2
• Plot the Root Locus for the system with open-loop transfer
function
Detailed Example #2 (continued)
I
m

R
e
Detailed Example #2 (continued)
Detailed Example #2 (continued)

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