2008CCS13 Root Locus

8/6/2019 2008CCS13 Root Locus

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Root LocusCCS-532

Lecture - 13

Closed-loop poles

Plotting the root locus of a transfer function

Choosing a value of K from root locus

Closed-loop response

Key Matlab commands used in this tutorial:

cloop, rlocfind, rlocus, sgrid, step

DrPavan Chakraborty

IIIT-Allahabad

Indian Institute of Information Technology - Allahabad

http://www.library.cmu.edu/ctms/ctms/matlab42/rlocus/rlocus.htm


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Root-locus Analysis

Based on characteristic eqn of closed-looptransfer function

Plot location ofroots of this eqn Same as poles of closed-loop transfer function

Parameter (gain) varied from 0 to g

Multiple parameters are ok

Vary one-by-one Plot a root contour (usually for 2-3 params)

Quickly get approximate results Range of parameters that gives desired response


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The root locus of an (open-loop) transfer functionH(s) is a plot

of the locations (locus) of all possible closed loop poles withproportional gain kand unity feedback:

The closed-loop transfer function is:

and thus the poles of the closed loop system are values ofssuch that 1 + K H(s) = 0.

Closed-Loop Poles


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If we writeH(s) = b(s)/a(s), then this equation has the form:

Let n = order ofa(s) and m = order ofb(s)

[the order of a polynomial is the highest power ofs thatappears in it].

We will consider all positive values ofk.

In the limit as kp 0, the poles of the closed-loop systemare a(s) = 0 or the poles ofH(s).

In the limit as kp g (infinity), the poles of the closed-loopsystem are b(s) = 0 or the zeros ofH(s).


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No matter what we pick kto be,the closed-loop system

must always have n poles, where nis the number of poles

ofH(s).

The root locus must have n branches, each branch starts

at a pole ofH(s) and goes to a zero ofH(s).

IfH(s) has more poles than zeros (as is often the case),

m < n and we say thatH(s) has zeros at infinity. In this

case, the limit ofH(s) as s p g (infinity) is zero.

The number of zeros at infinity is n-m, the number of polesminus the number of zeros, and is the number of branches of

the root locus that go to infinity (asymptotes).


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Since the root locus is actually the locations of all possible

closed loop poles, from the root locus we can select a gain

such that our closed-loop system will perform the way wewant.

If any of the selected poles are on the right half plane, the

closed-loop system will be unstable.

The poles that are closest to the imaginary axis have the

greatest influence on the closed-loop response,

so even though the system has three or four poles, it maystill act like a second or even first order system depending

on the location(s) of the dominant pole(s).


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The closed-loop transfer function is:

and thus the poles of the closed loop system are values ofssuch that 1 + K H(s) = 0.

Closed-Loop Poles

0

)20)(15)(5(

7)20)(15)(5(

)20)(15)(5(

71 !

!

ssss

sKssss

ssss

sK

.20,15,5,0

07)20)(15)(5(

{

!

sfor

sKssss


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The plot shows all possible closed-loop

pole locations for a pure proportionalcontroller. Obviously not all of those

closed-loop poles will satisfy our

design criteria.


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The plot shows all possible closed-loop pole locations for a

pure proportional controller.

Obviously not all of those closed-loop poles will satisfy our

design criteria.

Choosing a value of K from the root locus

To determine what part

of the locus is

acceptable, we can use

the command

sgrid(Zeta,Wn) to plotlines of constant

damping ratio and

natural frequency.


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Its two arguments are the

damping ratio (Zeta) and

natural frequency (Wn)

[these may be vectors if you want to look at a range of

acceptable values].

In our problem, we need an overshoot less than 5%

(which means a damping ratio Zeta " 0.7)

and a rise time of 1 second (which means a natural frequency

Wn > 1.8).

Enter in the Matlab command window:

zeta=0.7; Wn=1.8; sgrid(zeta, Wn)

Choosing a value of K from the root locus


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Rise Time : defined as the 10% to 90% time from a

former setpoint to new setpoint.

The quadratic approximation for normalized rise time fora 2nd-order system, step response, no zeros is:

where is the damping ratio and 0 is the naturalfrequency of the network.

However, the proper calculation for rise time of a system

of this type is:

where is the damping ratio and n is the natural

frequency of the network.


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On the plot above, the two white dotted lines at about a 45

degree angle indicate pole locations with Zeta = 0.7;

in between these lines, the poles will have Zeta > 0.7 and

outside of the lines Zeta < 0.7.

The semicircle indicates pole locations with a natural

frequency Wn = 1.8; inside the circle, Wn < 1.8 andoutside the circle Wn > 1.8.


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Zeta " 0.7

Wn > 1.8


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Going back to our problem, to make the overshoot less than

5%, the poles have to be in between the two dotted lines,

and to make the rise time shorter than 1 second, the poleshave to be outside of the dotted semicircle.

So now we know only the part of the locus outside of the

semicircle and in between the two lines are acceptable.

All the poles in this location are in the left-half plane, so the

closed-loop system will be stable.

From the plot above we see that there is part of the root locus

inside the desired region. So in this case we need only a

proportional controllerto move the poles to the desired

region.


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Interactive

Root Locus in

MathScriptWindow


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Click on the plot

the point where

you want the

closed-loop pole

to be.

You may want to

select the pointsindicated in the

plot to satisfy the

design criteria.

You can use

rlocfind command in Matlab to choose the desired poles on

the locus:

[kd,poles] =

rlocfind(num,den)


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Note that since the root locus may have more than one

branch, when you select a pole, you may want to find out

where the other pole (poles) are.

Remember they will affect the response too.

From the plot we see that allthe poles selected

(all the RED "+")

are at reasonable positions.

We can go ahead and use

the chosen kd as ourproportional controller.


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In order to find the step response, you need to know the closed-

loop transfer function. You could compute this using the rules ofblock diagrams, or let Matlab do it for you:

[numCL, denCL] = cloop((kd)*num, den)

The two arguments to the function cloop are the numeratorand

denominatorof the open-loop system.

You need to include the proportional gain that you have chosen.

Unity feedback is assumed.



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If you have a non-unity feedback situation, look at the help

file for the Matlab function feedback, which can find the

closed-loop transfer function with a gain in the feedbackloop.

Check out the step response of your closed-loop system:

step(numCL,denCL)


As we expected, this

response has an

overshoot less than5% and a rise time less

than 1 second.


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Example: Solution to the Cruise Control

Problem Using Root Locus Method

Proportional ControllerLag controller

The open-loop transfer function for this problem is:

where

m=1000

b=50

U(s)=10

Y(s)=velocity output

The design criteria are:Rise time < 5 sec

Overshoot < 10%

Steady state error < 2%


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Recall from the Root-Locus Tutorial page, the root-locus plot

shows the locations of all possible closed-loop poles when asingle gain is varied from zero to infinity. Thus, only a

proportional controller (Kp) will be considered to solve this

problem. Then, the closed-loop transfer function becomes:

Also, from the Root-Locus Tutorial, we know that the Matlab

command called sgrid should be used to find an acceptable

region of the root-locus plot. To use the sgrid, both the damping

ratio (zeta) and the natural frequency (Wn) need to be

determined first.

Proportional controller


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The following two equations will be used to find the damping

ratio and the natural frequency:

where

Wn=Natural frequency

zeta=Damping ratio

Tr=Rise time

Mp=Maximum overshoot

One of our design criteria is to have a rise time of less than 5

seconds. From the first equation, we see that the naturalfrequency must be greater than 0.36. Also using the second

equation, we see that the damping ratio must be greater than

0.6, since the maximum overshoot must be less than 10%.


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Now, we are ready to generate a root-locus plot and use the

sgrid to find an acceptable region on the root-locus.

Create an new m-file and enter the following commands.

hold off; m = 1000;

b = 50; u = 10;

numo=[1];

deno=[m b];

figure hold;

axis([-0.6 0 -0.6 0.6]);

rlocus (numo,deno) sgrid(0.6, 0.36) [Kp, poles] = rlocfind(numo,deno)

figure hold;

numc=[Kp];denc=[m (b+Kp)];

t=0:0.1:20;

step (u*numc,denc,t) axis ([0 20 0 10])


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Running this m-file should give you the following root-locus plot.


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The two dotted lines in an angle indicate the locations of

constant damping ratio (zeta=0.6); the damping ratio is

greater than 0.6 in between these lines and less than 0.6outside the lines. The semi-ellipse indicates the locations of

constant natural frequency (Wn=0.36); the natural frequency

is greater than 0.36 outside the semi-ellipse, and smaller than

0.36 inside.

If you look at the Matlab command window, you should see a

prompt asking you to pick a point on the root-locus plot.

Since you want to pick a point in between dotted lines

(zeta>0.6) and outside the semi-ellipse (Wn>0.36), clickon the real axis just outside the semi-ellipse (around -

0.4).


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You should see the gain value (Kp) and pole locations in the

Matlab command window. Also you should see the closed-

loop step response similar to the one shown below.

With the specified gain

Kp (the one you justpicked), the rise time

and the overshoot

criteria have been met;

however, the steady-state error of more than

10% remained.


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To reduce the steady-state error, a lag controller will be added

to the system. The transfer function of the lag controller is:

The open-loop transfer function (not including Kp) now becomes:

Finally, the closed-loop transfer function becomes:

Lag controller

readthe"LagorPhase-LagCompensatorusingRoot-Locus"sectionin

http://www.library.cmu.edu/ctms/ctms/matlab42/extras/leadlag.htm#lagrl


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hold off;

m = 1000;

b = 50;u = 10;

Zo=0.3;

Po=0.03;

numo=[1 Zo];

deno=[m b+m*Po b*Po];figure hold;

axis ([-0.6 0 -0.4 0.4])

rlocus(numo,deno)


The pole and the zero of a lag controller need to be placed

close together. Also, it states that the steady-state error will

be reduce by a factor of Zo/Po. For these reasons, let Zo

equals -0.3 and Po equals -0.03.Create an new m-file, and enter the following commands.

sgrid(0.6,0.36)

[Kp, poles]=rlocfind(numo,deno)

figuret=0:0.1:20;

numc=[Kp Kp*Zo];

denc=[m b+m*Po+Kp b*Po+Kp*Zo];

axis ([0 20 0 12])

step (u*numc,denc,t)


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Running this m-file should give you the root-locus plot similar to

the following:

In the Matlab command

window, you should see

the prompt asking you

to select a point on theroot-locus plot.

Once again, click on the

real axis around -0.4.


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You should have the following response.

As you can see, the

steady-state errorhas

been reduced to near

zero.

Slight overshoot is aresult of the zero

added in the lag

controller.

Now all of the design criteria have been met and no further

iterations will be needed.


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Figure 8.1

a. Closed-

loop system;

b. equivalenttransferfunction



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Figure 8.2

Vector

representation

of complex

numbers:

a. s = W+j[;b. (s +a);

c. alternate

representationof(s +a);

d. (s +7)|sp5 +j2



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Figure 8.3

Vector

representation

of Eq. (8.7)


)2(

)1()(

!

ss

ssF


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Figure 8.4a. CameraManPresenterCamera System

automatically follows a

subject who wears infrared

sensors on their front and

back (the front sensor is alsoa microphone); tracking

commands and audio are

relayed to CameraMan via a

radio frequency link from a

unit worn by the subject.

b. block diagram.c. closed-loop transfer

function.

Courtesy ofParkerVision.



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Table 8.1

Pole location as a function of gain for the

system of Figure 8.4



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Figure 8.5

a. Pole plotfrom

Table 8.1;

b. root locus



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Figure 8.6a. Example system;

b. pole-zero plot ofG (s)



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Figure 8.7

Vector representation ofG(s) from Figure 8.6(a)

at -2+ j3



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Figure 8.8

Poles and zeros of a general open-loop

system with test points,Pi, on the real axis



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Figure 8.9

Real-axis segments of the root locus for thesystem of Figure 8.6



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Figure 8.10Complete root locus for the system of Figure 8.6



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Figure 8.11

System for

Example 8.2



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Figure

8.12Root locus

andasymptotes

for the

system of

Figure 8.11



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Figure8.13Root locus

example

showingreal- axis

breakaway

(-W1) and

break-inpoints (W2)



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Figure 8.14Variation of gain along the real axis for the

root locus of Figure 8.13



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Table 8.2Data for breakaway and break-in points for the root

locus of Figure 8.13



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Table 8.3

Routh table for Eq.(8.40)



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Figure 8.16

Unity feedback system

with complex poles



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Figure 8.17Root locus for system

of Figure 8.16

showing angle of

departure



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Figure 8.18

Finding and calibrating exact points on theroot locus of Figure 8.12



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Figure 8.19a. System forExample 8.7;

b. root locus sketch



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Figure 8.20Making second-order approximations



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Figure 8.21

System for

Example 8.8


Fi 8 22


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Figure 8.22Root locus for Example 8.8



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Table 8.4Characteristics of the system of

Example 8.8



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Figure 8.23Second- andthird-order

responses for

Example 8.8:a. Case 2;

b. Case 3



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Figure 8.24

System requiring aroot locus calibrated

with p1 as a

parameter



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Figure8.25

Root locus for

the system ofFigure 8.24,

with p1 as a

parameter



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Figure 8.26

Positive-feedbacksystem



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Figure 8.27a. Equivalent

positive-feedback

system for

Example 8.9;b. root locus



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Figure

8.28Portion of the

root locus for

the antenna

control system



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Figure 8.29Step response of thegain-adjusted antenna

control system



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Figure 8.30

Root locus ofpitch control

loop without

ratefeedback,

UFSS vehicle


Figure 8 31


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Figure 8.31Computer simulation of step response of pitch control loop

without rate feedback,UFSS vehicle



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Figure 8.32

Root locus ofpitch control loop

with rate

feedback, UFSS

vehicle



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Figure 8.33Computer simulation of step

response of pitch control loop

with rate feedback, UFSS

vehicle



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Figure P8-1

(p. 474)



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Figure

P8.2



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Figure P8.3


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Figure P8-4 (p. 476)


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i 8 6


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Figure P8.6


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Fi P8 9


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Figure P8.9



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Figure P8.10



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Figure P8.11



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Figure P8.12



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Figure P8.13

a.R

obotequipped to

perform arc

welding;(figure

continues)

Courtesy of FANUC Robotics.


Figure P8 13


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Figure P8.13

(continued)

b.block

diagram for

swing motionsystem

1967 H.L. Hardy.



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Figure P8.14

Block diagram ofsmoother

1985 Rockwell International.


a. Active


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vibration

absorber((c)1992

AIAA);

b. control

system

blockdiagram


P8 16


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P8.16

Floppy diskdrive:

a.physical

representation;

b.blockdiagram



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Figure P8.17

Simplified blockdiagram of pupil

servomechanism


Figure P8 18


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Figure P8.18Active suspension system

1985 ASME.



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Fi P8 20


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Figure P8.20

Pitch axis attitudecontrol system utilizing

momentum wheel



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Figure P8.21a. Combustor

with microphone

and loud speaker

((c)1995 IEEE);

b. block diagram

((c)1995 IEEE)



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Figure P8.22a. Wind turbines

generating

electricitynearPalm

Springs,

California;

(figure continues)

Jim Corwin/ Photo Researchers



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Figure P8.22(continued)b. Control loop

for a

constant-speedpitch-controlled

wind turbine

((c)1998 IEEE);

c. Drivetrain((c)1998 IEEE)



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2008CCS13 Root Locus

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