Lec 10. Root Locus Analysis II

• More about the root locus– Breakin and Breakaway Points– Departure and Arrival Angles– Cross Points of j! Axis

• Magnitude Condition• Pole Zero Cancelation• Root Locus with Positive Feedback

• Reading: 6.1-6.5.

-6 -5 -4 -3 -2 -1 0 1 2-4

-3

-2

-1

0

1

2

3

4Root Locus

Real Axis

Imag

inar

y A

xis

Breakin/Breakaway Points

Breakaway points Breakin points

Breakin and breakaway points: where two or more branches of the root locus cross (repeated closed-loop poles)

£££

Finding Breakin/Breakaway Points

Find K so that the characteristic equation 1+K L(s)=0 has repeated roots

solutions are candidates of breakin and/or breakaway points

1. Rewrite the characteristic equation as (A(s), B(s) are polynomials):

2. Solve the characteristic equation for K as:

3. Solve the equation:

ExampleCharacteristic equation:

Example

+

Breakin and Breakaway Points

Multiple Breakin/Breakaway Points+

Complex Breakin/Breakaway Points

+

Departure Angles• Departure angle:

– Angle along which the root loci leave the open-loop poles

Example:

Test point s is on the root locus if

For a test point s very close to p1, this implies

1 is the departure angle from p1

Departure angle from p2?

Arrival Angles• Arrival angle:

– Angle along which the root loci enter the open-loop zeros

Angle condition:

Example:

General Departure/Arrival Angle

+

A general point s is on the root locus if and only if

Departure angle i from an open loop pole pi: (choose a test point close to pi)

Arrival angle i from an open loop zero zi: (choose a test point close to zi)

Repeated Zeros/Poles Case

If we have repeated open loop zeros/poles, then there are multiple arrival/departure angles associated with them

Example:+

-6 -5 -4 -3 -2 -1 0 1 2-4

-3

-2

-1

0

1

2

3

4Root Locus

Real Axis

Imag

inar

y A

xis

Points Where the Root Locus Crosses the j Axis

Determining the points where the root locus crosses the j axis is important because it gives on bound on K for the stability of the system

cross points?corresponding K?£ £ £

+

Direct MethodTo find K so that the characteristic equation has solutions on the imaginary axis, we let s=j

Method One: Routh’s CriterionClosed loop poles are solutions of the characteristic equation

Cross points occur correspond to boundary value of K for stability

Routh’s array:

Angle Condition (review)

s is on the root locus if and only if

+

Closed loop poles are solutions of characteristic equation

Magnitude Condition

For a point s known to be on the root locus, what is the corresponding K?

Pole-Zero Cancellation

Root locus suggest the closed loop system is stable for all K>0

+

Pole-Zero Cancellation (cont.)In practice, parameter inaccuracy may result in a slightly different system

+

Root Locus with Positive Feedback

Characteristic equation

+

Angle condition: a point s is on the root locus if and only if

Rules for Plotting a Root Locus with Positive Feedback

Example

+