MECH431_topic3_root_locus - K G(s Chapter 6 Root-locus...

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K G(s) + - MECH 431 Root Locus Analysis Slide 1 Chapter 6 Root-locus Analysis
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K G(s) + - MECH 431 Root Locus Analysis Slide 2 Introduction The Root Locus (RL) is the locus (i.e., location) of roots of the characteristic equation of the closed-loop system as a specific parameter (usually, gain K ) is varied from 0 to infinity. The RL shows the contributions of each open-loop pole or zero to the locations of the closed-loop poles.
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K G(s) + - MECH 431 Root Locus Analysis Slide 3 Introduction The RL method is valuable when designing a linear control system since it indicates the manner in which the open-loop poles and zeros should be modified so that the response meets system performance requirements. By using the RL, it is possible to determine the value of the loop gain K that makes the damping ratio of the dominant closed loop poles as prescribed.
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K G(s) + - MECH 431 Root Locus Analysis Slide 4 Angle and magnitude conditions • Given: + - G(s) C(s) H(s) R(s) To find the poles of the characteristic equation: Angle: Magnitude: • Therefore:
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K G(s) + - MECH 431 Root Locus Analysis Slide 5 Angle and magnitude conditions The values of s that satisfy both the angle and magnitude conditions are the roots of the characteristic equation (i.e., closed loop poles). A locus of the points in the complex domain satisfying the angle condition alone is the root locus. The roots of the characteristic equation corresponding to a given gain value are found from the magnitude condition.
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K G(s) + - MECH 431 Root Locus Analysis Slide 6 Angle and magnitude conditions The closed loop transfer function becomes: • Given: If K 0: poles of closed loop = poles open loop If K : poles of the closed loop = zeros of open loop
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K G(s) + - MECH 431 Root Locus Analysis Slide 7 Angle and magnitude conditions For example s A 1 -p 1 A 3 -p 3 A 4 -p 4 A 2 -p 2 B 1 -z 1 * Note the symmetry about the real axis.
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K G(s) + - MECH 431 Root Locus Analysis Slide 8 Example 6-1 • Given: Sketch the root-locus plot Determine the value of K such that the damping ratio of a pair of dominant complex-conjugate closed- loop poles is 0.5.
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K G(s) + - MECH 431 Root Locus Analysis Slide 9 Example 6-1 • Given: The angle condition gives:
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K G(s) + - MECH 431 Root Locus Analysis Slide 10 Example 6-1 • Given: The magnitude condition gives:
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K G(s) + - MECH 431 Root Locus Analysis Slide 11 Example 6-1 1) Determine the root loci on the real axis : The first step is to locate the open loop poles (i.e., K =0): 0 -1 -2
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K G(s) + - MECH 431 Root Locus Analysis Slide 12 Example 6-1 1) Determine the root loci on the real axis : Select a test point on the positive real axis, and look at the angles subtended by the vectors from the poles of the open loop to s: 0 -1 -2 Test point s , Can a closed loop pole s exist here???
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K G(s) + - MECH 431 Root Locus Analysis Slide 13 Example 6-1 1) Determine the root loci on the real axis : Select a test point on the negative real axis, between 0 and -1 0 -1 -2
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K G(s) + - MECH 431 Root Locus Analysis Slide 14 Example 6-1 1) Determine the root loci on the real axis : Select a test point on the negative real axis, between -1 and -2 0 -1 -2
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