Lecture 14 ME 176 7 Root Locus Technique
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This document provides background information on root locus analysis and techniques for sketching root loci. It discusses rules for determining the number and symmetry of root locus branches, where the locus begins and ends, and how to find breakaway/break-in points and jw-axis crossings. The document concludes by posing a problem to sketch the root locus for a given transfer function. The key information is that root locus analysis graphically shows how closed-loop poles vary with gain and provides insights into stability and transient response.
Lecture 14 ME 176 7 Root Locus Technique
- 1. ME 176 Control Systems Engineering Root Locus Technique Department of Mechanical Engineering
- 2. Introduction : Root Locus "... a graphical representation of closed loop poles as a system parameter is varied, is a powerful method of analysis and design for stability and transient response." "...real powere lies in its ability to provide for solutions for systems of order higher than 2." Department of Mechanical Engineering
- 3. Background: Control Systems Open Loop vs. Closed Loop Department of Mechanical Engineering
- 4. Background: Complex Number Vector Representation Representation of complex Number: Cartesian Form : Vector Form : Vector complex arithmetic: Department of Mechanical Engineering
- 5. Background: Complex Number Vector Representation Example Department of Mechanical Engineering
- 6. Background: Root Locus "... is the representation of the paths of the closed-loop poles as the gain is varied." Department of Mechanical Engineering
- 7. Background: Root Locus Given Poles and Zeros of a closed-loop transfer function KG(s)H(s) a point in the s-plane is on the root locus for a gain K if : angles of zeros minus angles of poles add up to (2k+1)180 degrees. K is found by dividing the product of pole lengths to that of zeros. Department of Mechanical Engineering
- 8. Sketching: Root Locus Rule 1 of 5 : The number of branches of the root locus equals the number of closed-loop poles. 2 poles therefore 2 branches Department of Mechanical Engineering
- 9. Sketching: Root Locus Rule 2 of 5 : The root locus is symmetrical about the real axis Symmetry of segments between: Branch 1 : -5 onward Branch 2 : -5 onward Department of Mechanical Engineering
- 10. Sketching: Root Locus Rule 3 of 5 : On the real axis, for K > 0 the root locus exists to the left of an odd number of real-axis, finite open-loop poles and/or finite open-loop zeros. Root Locus Exists: Left of -3 : zero number 1 Left of -1 : pole number 1 Department of Mechanical Engineering
- 11. Sketching: Root Locus Rule 4 of 5 : The root locus begins at the finite and infinite poles of G(s)H (s) and ends at the finite and infinite zeros of G(s)H(s). Root Locus Exists: Starts on poles at real axis : -1, -2 Ends on zeros at real axis : -3, -4 Department of Mechanical Engineering
- 12. Sketching: Root Locus Rule 5 of 5 : The root locus approaches straight lines as asymptotes as the locus approaches infinity. Further, the equation of the asymptotes is given by : Angle is radians with respect to positive extension of the real axis. Department of Mechanical Engineering
- 13. Refining Sketch: Root Locus Real Axis Breakaway and Break-in : Breakaway and break-in points satisfy the relationship: where zi and pi are vnegative of the zero and pole values, respectively. Department of Mechanical Engineering
- 14. Refining Sketch: Root Locus jw - Axis Crossing : Use Routh-Hurwitz criterion,1) forcing a row of zeros in the Routh table to establish the gain; 2) then going back one row to the even polynomial equation and solving for the roots yields the frequency at the imaginary axis crossing. 1) 2) Department of Mechanical Engineering
- 15. Problem: Department of Mechanical Engineering
- 16. Problem: Sketch the root locus and its asymptotes for a unity feedback system that has the forward transfer function: Department of Mechanical Engineering