[5] Root-Locus Design, Mechatronics, Control Theory
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Root-Locus Design
The root-locus can be used to determine the value of the loop gain , which results in a
satisfactory closed-loop behavior. This is called the
proportional compensator
or
proportional controller
and provides gradual response to deviations from the set point.
There are practical limits as to how large the gain can be made. In fact, very high gains
lead to instabilities. If the root-locus plot is such that the desired performance cannot be
achieved by the adjustment of the gain, then it is necessary to reshape the root-loci by
adding the additional controller
G
to the open-loop transfer function.
G
must be
chosen so that the root-locus will pass through the proper region of the -plane. In many
cases, the speed of response and/or the damping of the uncompensated system must be
increased in order to satisfy the specifications. This requires moving the dominant
branches of the root locus to the left.
K
c
s
( )
c
s
( )
s
The proportional controller has no sense of time, and its action is determined by the
present value of the error. An appropriate controller must make corrections based on the
past and future values. This can be accomplished by combining proportional with integral
action or proportional with derivative action . One of the most common controllers
available commercially is the controller. Different processes are suited to different
combinations of proportional, integral, and derivative control. The control engineer's task
is to adjust the three gain factors to arrive at an acceptable degree of error reduction
simultaneously with acceptable dynamic response. The compensator transfer function is
PI
PD
PID
K
I
Gs
()
=++
D
K
Ks
(1)
c
P
s
For
PD
or
PI
controllers, the appropriate gain is set to zero.
Other compensators, are lead, lag, and lead-lag compensators. A first-order compensator
having a single zero and pole in its transfer function is
sZ
+
0
Gs
()
=
(2)
c
sP
+
0
The pole and zero are located in the left half s-plane as shown in Figure 1.
s
s
θ
θ
θ
θ
×
:
:
×
−
p
−
z
−
p
−
z
0
0
(a) Phase-lead (b) Phase-lag
Figure 1
Compensator phase angle contribution
1
For a given
1
= − is positive
if as shown in Figure 1 (a), and the compensator is known as the
phase-lead
controller
. On the other hand if
s
=+ , the transfer function angle given by
σ
j
ω
θ θθ
(
p
0
)
1
1
c
z
0
0
zp
<
0
z
>
p
as shown in Figure 1 (b), the compensator angle
0
θ = θ is negative, and the compensator is known as the
phase-lag controller
(
)
c
z
p
0
0
In general, the open-loop transfer function is given by
Ks z s z
(
+
)(
+
)
"
"
(
s z
+
)
)
1
2
m
KG s H s
() ()
=
(
spsp
+
)(
+
)
(
sp
+
1
2
n
where
m
is the number of finite zeros and
n
m
is the number of finite poles of the loop
transfer function. If
nm
>
, there are (
n
−
)
zeros at infinity. The characteristic equation
of the closed-loop transfer function is
1
+
KG s H s
( )
( )
= 0
Therefore
(
spsp
+
)(
+
)
"
"
(
sp
+
)
1
2
n
= −
K
(
szsz
+
)(
+
)
(
sz
+
)
1
2
m
From the above expression, it follows that for a point in the
s
-plane to be on the root-
locus, when 0
<<∞
, it must satisfy the following two conditions.
|
spsp
+
||
+
|
"
"
|
sp
+
|
or
1
2
n
K
=
| || | | |
product of vector lengths from finite poles
product of vector lengths from finite zeros
sz sz
+
+
sz
+
1
2
m
(3)
K
=
and
∑
∑
of zeros of
GsH s
( )
( )
−
angle of poles of
GsH s
( )
( )
=
r
(180),
r
= ± ±
1,
3,
"
or
m
n
∑∑
θ
−
θ
=
180 ,
r
r
= ± ±
1,
3,
"
(4)
zi
pi
i
=
1
i
=
1
The magnitude and angle criteria given by (3) and (4) are used in the graphical root-locus
design.
In addition to the MATLAB control system toolbox
rlocus(num, den)
for root locus plot,
MATALB control system toolbox contain the following functions which are useful for
interactively finding the gain at certain pole locations and intersect with constant
ω
circles. These are:
sgrid
generates a grid over an existing continuous s-plane root locus or pole-zero map.
Lines of constant damping ratioζand natural frequency
ω are drawn. sgrid('new') clears
the current axes first and sets hold on.
2
sgrid(Z, Wn)
plots constant damping and frequency lines for the damping ratios in the
vector Z and the natural frequencies in the vector Wn.
[K, poles] = rlocfind(num, den)
puts up a crosshair cursor in the graphics window
which is used to select a pole location on an existing root locus. The root locus gain
associated with this point is returned in
K
and all the system poles for this gain are
returned in
poles
.
rltool
or
sistool
opens the SISO Design Tool. This Graphical User Interface allows you
to design single-input/single-output (SISO) compensators by interacting with the root
locus, Bode, and Nichols plots of the open-loop system.
1. Gain Factor Compensation or P-Controller Design
The proportional controller is a pure gain controller. The design is accomplished by
choosing a value , which results in a satisfactory transient response. The specification
may be either the step response damping ratio or the step response time constant or the
steady-state error. The procedure for finding
K
0
K
is as follows:
0
• Construct an accurate root-locus plot
• For a given ζ draw a line from origin at angle
−
1
θ
=
cos
ζ
measured from
negative real axis.
• The desired closed-loop pole
s
is at the intersection of this line and the root-
locus.
• Estimate the vector lengths from
s
to poles and zeros and apply the magnitude
criterion as given by (3) to find
K
.
0
Example 1
The open-loop transfer function of a control system is given by
K
=
++)
(a) Obtain the gain of a proportional controller such that the damping ratio of the
closed-loop poles will be equal 0.6. Obtain root-locus, step response and the time-domain
specifications for the compensated system.
KGH s
()
ss
( ( 4
s
K
0
The root-locus plot is shown in Figure 2. For
ζ=
0.6
,
θ
−
= =
D
The line drawn at this angle intersects the root-locus at approximately,
1
cos
0.6
53.13
s
−+
0.41
j
0.56
.
1
The vector lengths from
s
to the poles are marked on the diagram
3
Figure 2
P-Controller Design
. From (1), we have
K
=
(0.7)(0.8)(3.65)
=
2.04
2.04
This gain will result in the velocity error constant of
K
=
=
0.51
. Thus, the steady-
4
1
1
state error due to a ramp input is
e
== = .
1.96
ss
K
0.51
v
The compensated closed-loop transfer function is
Cs
Rs
( )
2.05
=
+++
3
2
()
s
5
s
4
s
2.05
(b) Use the MATLAB Control System Toolbox functions
rlocus
and
sgrid(zeta, wn)
to
obtain the root-locus and the gain
ζ= . Also use the
ltiview
function to obtain
the system step response and the time-domain specifications.
K
for
0.6
0
The following commands
num=1;
den=[1 5 4 0];
rlocus(num, den);
hold on
sgrid(0.6, 1) % plots constant line zeta=0.6 & constant line wn=1
4
result in
Figure 3
Zoom in at the area of intersection, click at the intersection, hold and move the mouse at
intersection and adjust for Damping: 0.6. The gain is found to be 2.04. In addition, the
percentage overshoot and natural frequency are obtained, i.e.,
PO
=
9.48%
and
ω = .
To obtain the step response and time-domain specifications, we use the following
commands.
0.697
numc=2.04;
denc=[1 5 4 2.04];
T=tf(numc, denc)
ltiview('step', T)
The result is shown in Figure 4. Right-click on the LTI Viewer, use Chracteristics to
mark peak response, peak time, settling time, and rise time. From File Menu use Print to
Figure to obtain a Figure plot.
5
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