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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.
Closed-loop response
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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)
Closed-loop response
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)
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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)
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)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
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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
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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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Figure P8-5 (p. 477)
i 8 6
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Figure P8.6
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Figure P8-7 (p. 479)
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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.
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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.
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a. Active
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vibration
absorber((c)1992
AIAA);
b. control
system
blockdiagram
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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
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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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