lec 10. root locus analysis ii more about the root locus –breakin and breakaway points...
TRANSCRIPT
![Page 1: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/1.jpg)
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.
![Page 2: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/2.jpg)
-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)
£££
![Page 3: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/3.jpg)
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:
![Page 4: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/4.jpg)
ExampleCharacteristic equation:
![Page 5: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/5.jpg)
Example
+
![Page 6: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/6.jpg)
Breakin and Breakaway Points
![Page 7: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/7.jpg)
Multiple Breakin/Breakaway Points+
![Page 8: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/8.jpg)
Complex Breakin/Breakaway Points
+
![Page 9: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/9.jpg)
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?
![Page 10: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/10.jpg)
Arrival Angles• Arrival angle:
– Angle along which the root loci enter the open-loop zeros
Angle condition:
Example:
![Page 11: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/11.jpg)
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)
![Page 12: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/12.jpg)
Repeated Zeros/Poles Case
If we have repeated open loop zeros/poles, then there are multiple arrival/departure angles associated with them
Example:+
![Page 13: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/13.jpg)
-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?£ £ £
+
![Page 14: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/14.jpg)
Direct MethodTo find K so that the characteristic equation has solutions on the imaginary axis, we let s=j
![Page 15: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/15.jpg)
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:
![Page 16: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/16.jpg)
Angle Condition (review)
s is on the root locus if and only if
+
Closed loop poles are solutions of characteristic equation
![Page 17: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/17.jpg)
Magnitude Condition
For a point s known to be on the root locus, what is the corresponding K?
![Page 18: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/18.jpg)
Pole-Zero Cancellation
Root locus suggest the closed loop system is stable for all K>0
+
![Page 19: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/19.jpg)
Pole-Zero Cancellation (cont.)In practice, parameter inaccuracy may result in a slightly different system
+
![Page 20: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/20.jpg)
Root Locus with Positive Feedback
Characteristic equation
+
Angle condition: a point s is on the root locus if and only if
![Page 21: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/21.jpg)
Rules for Plotting a Root Locus with Positive Feedback
![Page 22: 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](https://reader034.vdocuments.us/reader034/viewer/2022051622/5697c0061a28abf838cc595d/html5/thumbnails/22.jpg)
Example
+