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. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

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

Finding Breakin/Breakaway Points Find K so that the characteristic equation 1+K L(s)=0 has repeated roots 1. Rewrite the characteristic equation as (A(s), B(s) are polynomials): 2. Solve the characteristic equation for K as:   3. Solve the equation:   solutions are candidates of breakin and/or breakaway points

Determine Breakin/Breakaway Points Need to find K so that the characteristic equation 1+K L(s)=0 has repeated roots Write 1+K L(s) = 0 as: where A(s) and B(s) are polynomials independent of K If s is a root of f(s), then If s is a repeated root of f(s), then we have in addition So, solutions are candidates of breakin and breakaway points

Example   Characteristic equation:

Example     +        

Breakin and Breakaway Points  

Multiple Breakin/Breakaway Points     +  

Complex Breakin/Breakaway Points     +  

Departure Angles Departure angle: Example: 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: Example: Angle along which the root loci enter the open-loop zeros   Example: Angle condition:

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 fi 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:   +

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 Method To find K so that the characteristic equation has solutions on the imaginary axis, we let s=j

Method One: Routh’s Criterion Closed 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) Closed loop poles are solutions of characteristic equation   s is on the root locus if and only if +

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 +

Pole Zero Cancellation (cont.) Correct root locus a branch starting and ending at s=1 + +

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 +