root locus control system design.ppt

7/22/2019 ROOT LOCUS Control system Design.ppt

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Design of Control Systems

Cascade Root Locus Design

This is the first lecture devoted to the control system design. In the

previous lectures we laid the groundwork for design techniques

based on root locus analysis. At the end, root locus design will not

prove to be the best technique. That honor is reserved for the Bode

design method.

However

Root locus design is very intuitive. The root locus

provides a portrait for 0 K for all possible closed-

loop pole locations.

Proportional plus integral plus derivative (PID) control

is widely used in industrial applications. PID control is

best understood from the root locus perspective


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Compensation

The design of a control system is concerned with the arrangement of

the system structure and the selection of a suitable components andparameters.

A compensator is an additional component or circuit that is inserted

into a control system to compensate for a deficient performance.

Types of Compensation

Cascade compensation

Feedback compensation

Output compensation Input compensation


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PID Controllers

PID control consists of a proportional plus derivative (PD)

compensator cascaded with a proportional plus integral (PI)compensator.

The purpose of the PD compensator is to improve the transient

response while maintaining the stability.

The purpose of the PI compensator is to improve the steady state

accuracy of the system without degrading the stability.

Since speed of response, accuracy, and stability are what is neededfor satisfactory response, cascading PD and PI will suffice.


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The Characteristics of P, I, and D ControllersNote that these correlations may not be exactly accurate, because Kp, K i, and Kd are dependent

of each other. In fact, changing one of these variables can change the effect of the other two.

For this reason, the table should only be used as a reference when you are determining thevalues forKi, Kp and Kd.

Response Rise Time Overshoot Settling

Time

SS Error

KP Decrease IncreaseSmallChange Decrease

KI Decrease Increase Increase Eliminate

KD

Small

Change Decrease Decrease

Small

Change


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The Simplest form of compensation is gain compensation

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Root Locus for Simple Gain Compensator

0.5-1

0.5

Re (s)

Im(s)


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Lead/Lag Compensation

Lead/Lag compensation is very similar to PD/PI, or PID control.

The lead compensator plays the same role as the PD controller,

reshaping the root locus to improve the transient response.

Lag and PI compensation are similar and have the same response: to

improve the steady state accuracy of the closed-loop system.

Both PID and lead/lag compensation can be used successfully, and

can be combined.


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Lead Compensation Techniques Based on the Root-Locus Approach

From the performance specifications, determine the desired location for thedominant closed-loop poles.

By drawing the root-locus plot of the uncompensated system ascertainwhether or not the gain adjustment alone can yield the desired closed-looppoles. If not calculate the angle deficiency. This angle must be contributedby the lead compensator.

If the compensator is required, place the zero of the phase lead networkdirectly below the desired root location.

Determine the pole location so that the total angle at the desired rootlocation is 180o and therefore is in the compensated root locus.

Assume the transfer function of the lead compensator.

Determine the open-loop gain of the compensated system from themagnitude conditions.


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Lead Compensator using the Root Locus

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s

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ps

zssG

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s

KsGH

s

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c

co

ppo

nn

s

c

-1

p

s = -p = -3.6


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Adding Lead CompensationThe lead compensator has the same purpose as the PD compensator:

to improve the transient response of the closed-loop system by

reshaping the root locus. The lead compensator consists of a zero and

a pole with the zero closer to the origin of the s plane than the pole. Thezero reshapes a portion of the root locus to achieve the desired

transient response. The pole is placed far enough to the left that it does

not have much influence of the portion influenced by the zero.

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Root Locus for Simple Gain Compensator

Re (s)

Im(s)

Closed-loop poles

3

-3


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Adding a Lag Controller

A first-order lag compensator can be designed using the root locus. A lag

compensator in root locus form is given by

where the magnitude ofzo is greater than the magnitude ofpo. A phase-lag

compensator tends to shift the root locus to the right, which is undesirable.

For this reason, the pole and zero of a lag compensator must be placed

close together (usually near the origin) so they do not appreciably changethe transient response or stability characteristics of the system.

o

o

ps

zssG

)(


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How does the Lag Controller Shift the Root Locus to the

Right?

Recall finding the asymptotes of the root locus that lead to the zeros at

infinity, the equation to determine the intersection of the asymptotes along

the real axis is:

When a lag compensator is added to a system, the value of this intersection

will be a smaller negative number than it was before. The net number of

zeros and poles will be the same (one zero and one pole are added), but the

added pole is a smaller negative number than the added zero. Thus, theresult of a lag compensator is that the asymptotes' intersection is moved

closer to the right half plane, and the entire root locus will be shifted to the

right.

zerospoles

zerospoles


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Control ModesThere are many ways by which a control unit can react to an error

and supply an output for correcting elements.

The two-step mode: The controller is just a switch which isactivated by the error signal and supplies just an on-off correctingsignal. Example of such mode is the bimetallic thermostat.

The proportional mode (P): This produces a control action that isproportional to the error. The correcting signal thus becomes

bigger the bigger the error. Therefore, the error is reduced theamount of correction is reduced and the correcting process slowsdown. A summing operational amplifier with an inverter can beused as a proportional controller.

The derivative mode: This produces a control action that isproportional to the rate at which the error is changing. When thereis a sudden change in the error signal the controller gives a largecorrecting signal. When there is a gradual change only a smallcorrecting signal is produced. An operational amplifier connectedas a differentiator circuit followed by another operational amplifierconnected as an inverter make an electronic derivative controllercircuit.


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The integral mode (I): This produces a control action that is

proportional to the integral of the error with time. Therefore, a

constant error signal will produce an increasing correcting signal.The correction continues to increase as long as the error persists.

Combination of modes: Proportional plus derivative modes (PD),

proportional plus integral modes (PI), proportional plus integral plus

derivative modes (PID). The term three-term controller is used forPID control.

The controller may achieve these modes by means of pneumatic

circuits, analog electronics involving operational amplifiers or by the

programming of a microprocessor or computer.


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DC Motor Speed ModelingThe DC motor has been the workhorse in industry for many reasons including

good torque speed characteristics. It is a common actuator in control systems.

It directly provides rotary motion and, coupled with wheels or drums andcables, can provide transitional motion. The electric circuit of the armature and

the free body diagram of the rotor are shown in the following Figure.

We develop here the transfer function of a separately excited armature

controlled DC motor.

Motor

Generator

Mechanical

energy

(T, )

VA

RA IA

VF

RF

LF

IF L

k

Jmotor & load

T

Field circuit Armature circuit


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Physical Parameters

Electrical Resistance R= 1

Electrical Inductance L = 0.5 H

Input Voltage V

Electromotive Force Constant K = 0.01 nm/A

Moment of Inertia of the RotorJ= 0.01 kg.m2

/s2

Damping Ratio of the Mechanical System b = 0.1 Nms

Position of the Shaft

The rotor and shaft are assumed to be rigid

The motor torque T is related to the armature current by a constant Kt

The back emf, e, is related to the rotational velocity

KVRidtdiL

KibJ

Ke

iKT

e

t


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Speed Control Speed Control by Varying Circuit Resistance: The operating speed can only

be adjusted downwards by varying the external resistance, Rext

Speed Control by Varying Excitation Flux:

Speed Control by Varying Applied Voltage: Wide range of control 25:1; fast

acceleration of high inertia loads.

Electronic Control.

rad/s

22

a

exta

aa

aaam

k

RR

k

Va

k

IRV

1

2

2

1

m

m


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Transfer Function

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)()()()()()(

KRLsbJs

K

V

sKsVsIRLssKIsbJss

Controller PlantR u


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Data Measurement

Once we have identified the transfer function of the system we may

proceed to the final two phases of the design cycle, the design of asuitable controller and the implementation of the controller on the

actual system. In the case of speed control of the DC motor, the

control will prove to be quite easy.

An important point to be highlighted here is that if we have a good

model of the plant to be controlled, and we already have identifiedthe parameters of the model, then the design of the controller is

easy.

Computer D/A Converter Power Amplifier DC Motor

Oscilloscope

Armature voltage

Tachometer


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Design Needs

The uncompensated motor may only rotate at 0.1 rad/sec with an input

voltage of 1 V. Since the most basic requirement of a motor is that it shouldrotate at the desired speed, the steady-state error of the motor speed should

be less than 1%.

The other performance requirement is that the motor must accelerate to itssteady-state speed as soon as it turns on. In this case, we want it to have asettling time of 2 seconds for example. Since a speed faster than the

reference may damage the equipment, we want to have an overshoot of lessthan 5%. If we simulate the reference input (r) by a unit step input, then themotor speed output should have:

Settling time less than 2 seconds

Overshoot less than 5%

Steady-state error less than 1%

Use the MATLAB to represent the open loop response


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PID Design technique for DC Motor Speed Control

Design a PID controller and add it into the system.

Recall that the transfer function for a PID controller is:

See how the PID controller works in a closed-loop system using theprevious Figure. The variable (e) represents the tracking error, thedifference between the desired input value (R) and the actual output (y).This error signal (e) will be sent to the PID controller, and the controllercomputes both the derivative and the integral of this error signal.

The signal (u) just past the controller is equal to the proportional gain (Kp)times the magnitude of the error plus the integral gain (Ki) times theintegral of the error plus the derivative gain (Kd) times the derivative of theerror.

s

KsKsKsK

s

KK IPDD

IP

2


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The PID Adjustment Steps

Use a proportional controller with a certain gain. A code should be

added to the end of m-file. Determine the closed-loop transfer function.

See how the step response looks like.

You should get certain plot.

From the plot you may see that both the steady-state error and the

overshoot are too large. Recall from the PID characteristics that adding an integral term will

eliminate the steady-state error and a derivative term will reduce theovershoot. Let us try a PID controller with small Kiand Kd.

The settling time is too long. Let us increase Ki to reduce the settling

time. Large Ki will worsen the transient response (big overshoot). Let us

increase Kd to reduce the overshoot.

See the plot now and see if design requirements will be satisfied.


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Root Locus Design Method for DC Motor Speed Control

Drawing the Open-Loop Root Locus

The main idea of root locus design is to find the closed-loop responsefrom the open-loop root locus plot. Then by adding zeros and/or

poles to the original plant, the closed-loop response can be modified.

We need the settling time and the overshoot to be as small as

possible. Large damping corresponds to points on the root locus near

the real axis. A fast response corresponds to points on the root locusfar to the left of the imaginary axis.

The system may be overdamped and the settling time will be about

one second, so the overshoot and settling time requirements could

be satisfied.

The only problem we may see from the generated plot is the steadystate error. If we increase the gain to reduce the steady-state error,

the overshoot becomes too large. We need to add a lag controller to

reduce the steady-state error.

root locus control system design.ppt

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