Download - Lecture 16 ME 176 7 Root Locus Technique
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ME 176Control Systems Engineering
Department of
Mechanical Engineering
Root Locus Technique
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Definition: Generalized Root Locus
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Definition: Root Locus for Positive Feedback 1. Number of branches.2. Symmetry3. Real-axis segments : even instead of odd 4. Starting and ending points.5. Behavior at infinity:
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Definition: Root Locus for Positive Feedback
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1. 4 branches.2. Symetric about real-axis.3. Left of even poles\zeros real-axis.4. Starting at poles ends at zeros.5. Infinity behavior:
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Definition: Pole Sensitivity
Root Sensitivity:Change in pole location for changes in gain:
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Find root sensitivity at s = -9.47 and -5 + j5. Also, calculate the change in pole location for a 10% change in K
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Design: Root Locus
Modifying behavior of the system iseffectively done using compensators,which are used to improve bothtransient response and steady state errors.
Compensators on feed forward path:1. Differentiator - addition of zeros toa system, improves transient response
2. Integrator - addition of poles to asystem, improves steady-state error
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Design: Root LocusCompensation may beapplied to systems in two types of configurations:
1. Cascade Compensation2. Feedback CompensationIdeal compensators use activeelemements.
Nonideal compensators use passiveelements.
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Design: Root Locus - Improving Steady-State Error
Ideal Integrator Compensator - a compensator with pole at origin and zero close to the pole.
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Design: Root Locus - Improving Steady-State Error Ideal Integrator Compensator - Compensator with pole at origin and zero close to the pole. - Implemented using a proportional-plus-integral controller (PI).- Needs the use of active circuits.
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Design: Root Locus - Improving Steady-State Error Ideal Integrator Compensator
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"Closed-loop poles and gains are approximately the same, which indicates the transient response is about the same."
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Design: Root Locus - Improving Steady-State Error Ideal Integrator Compensator
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Design: Root Locus - Improving Steady-State Error Ideal Integrator Compensator
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Design: Root Locus - Improving Steady-State Error Lag Compensation
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Design: Root Locus - Improving Steady-State Error Lag Compensation Transient Response: Steady State Error:Almost the same since the angleand magnitude adjustments fromcompensator pole and zero cancels.
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Design: Root Locus - Improving Transient Response Ideal Derivative Compensation
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Design: Root Locus - Improving Transient Response Ideal Derivative Compensation Dominant Pole Characteristics: - Higher negative closed-loop real parts, shorter settling time.- Higher imaginary closed-loop parts, smaller peak time.
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Design: Root Locus - Improving Transient Response Ideal Derivative Compensation Design ideal derivative compensator to yield 16% overshoot with a threefold reduction in settling time. Steps:- Get damping ration line from %OS.- Get dominant pole on damping ration line.- Find Ts from equation - 2nd order app - third pole must be more than 6 times as far from jw-axis as 2nd order pair.- Get wanted settling time 1/3 computed Ts.- Get real part of dominant pole from Ts equation.- Get imaginary part using geometry of damping ration line.
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Design: Root Locus - Improving Transient Response Ideal Derivative Compensation Design ideal derivative compensator to yield 16% overshoot with a threefold reduction in settling time. Steps:- Sum all angles of poles to new dominant pole location.- Missing zero is located where sum of angle is 180.- Use geometry to find:
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Design: Root Locus - Improving Transient Response Ideal Derivative Compensation
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Design: Root Locus - Improving Transient Response Lead CompensationFor passive networks a single zero cannot be produced. Compensator with a zero and pole results. When the poles are father from the imaginary axis than the zero, the equivalent angle of both pole and zero are positive and thus approximates a single zero. Differentiating lead compensators:- Static Error Constants- Gain- 2nd order approximation- transient response
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