ME 176 Control Systems Engineering Department of Mechanical Engineering Root Locus Technique
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ME 176Control Systems Engineering
Department of
Mechanical Engineering
Root Locus Technique
Introduction : Root Locus
Department of
Mechanical Engineering
"... 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."
Background: Control Systems Open Loop vs. Closed Loop
Department of
Mechanical Engineering
Background: Complex Number Vector Representation
Representation of complex Number:Cartesian Form :Vector Form :
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Mechanical Engineering
Vector complex arithmetic:
Background: Complex Number Vector Representation Example
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Mechanical Engineering
Background: Root Locus
"... is the representation of the paths of the closed-loop poles as the gain is varied." Department of
Mechanical Engineering
Background: Root Locus
Given Poles and Zeros of a closed-looptransfer function KG(s)H(s) a point in thes-plane is on the root locus for a gain K if : angles of zeros minusangles of poles add up to (2k+1)180 degrees. K is found by dividingthe product of pole lengths to that of zeros.
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Mechanical Engineering
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
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Mechanical Engineering
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Mechanical Engineering
Sketching: Root Locus Rule 2 of 5 : The root locus is symmetrical about the real axis
Symmetry of segments between:Branch 1 : -5 onwardBranch 2 : -5 onward
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 1Left of -1 : pole number 1
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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, -2Ends on zeros at real axis : -3, -4
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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 :
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Mechanical Engineering
Angle is radians with respect to positive extension of the real axis.
Refining Sketch: Root Locus Real Axis Breakaway and Break-in :
Breakaway and break-in pointssatisfy the relationship: where zi and pi are vnegativeof the zero and pole values,respectively.
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Mechanical Engineering
Refining Sketch: Root Locus jw - Axis Crossing :
Use Routh-Hurwitz criterion,1) forcinga row of zeros in the Routh table toestablish the gain; 2) then going backone row to the even polynomial equation and solving for the rootsyields the frequency at the imaginaryaxis crossing.
1)
2)
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Problem:
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Mechanical Engineering
Problem: Sketch the root locus and its asymptotes for a unity feedback system that has the forward transfer function:
Department of
Mechanical Engineering
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