ME 176Control Systems Engineering

Department of

Mechanical Engineering

Root Locus Technique

Definition: Generalized Root Locus

Department of

Mechanical Engineering

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:

Department of

Mechanical Engineering

Definition: Root Locus for Positive Feedback

Department of

Mechanical Engineering

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:

Definition: Pole Sensitivity

Root Sensitivity:Change in pole location for changes in gain:

Department of

Mechanical Engineering

Find root sensitivity at s = -9.47 and -5 + j5. Also, calculate the change in pole location for a 10% change in K

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

Department of

Mechanical Engineering

Design: Root LocusCompensation may beapplied to systems in two types of configurations:

1. Cascade Compensation2. Feedback CompensationIdeal compensators use activeelemements.

Nonideal compensators use passiveelements.

Department of

Mechanical Engineering

Design: Root Locus - Improving Steady-State Error

Ideal Integrator Compensator - a compensator with pole at origin and zero close to the pole.

Department of

Mechanical Engineering

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.

Department of

Mechanical Engineering

Design: Root Locus - Improving Steady-State Error Ideal Integrator Compensator

Department of

Mechanical Engineering

"Closed-loop poles and gains are approximately the same, which indicates the transient response is about the same."

Design: Root Locus - Improving Steady-State Error Ideal Integrator Compensator

Department of

Mechanical Engineering

Design: Root Locus - Improving Steady-State Error Ideal Integrator Compensator

Department of

Mechanical Engineering

Design: Root Locus - Improving Steady-State Error Lag Compensation

Department of

Mechanical Engineering

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.

Department of

Mechanical Engineering

Design: Root Locus - Improving Transient Response Ideal Derivative Compensation

Department of

Mechanical Engineering

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.

Department of

Mechanical Engineering

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.

Department of

Mechanical Engineering

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:

Department of

Mechanical Engineering

Design: Root Locus - Improving Transient Response Ideal Derivative Compensation

Department of

Mechanical Engineering

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

Department of

Mechanical Engineering