Analysis of Lag Compensator Design for
Robotic Elbow Joint Control System
Using MATLAB
Htwe Htwe Thein Myint
Department of Electronic Engineering
Mandalay Technological University
Email: htwehtwetheinmyint@gmail.com
Abstract: The paper describes the result comparisons
that were developed for the phase lag compensator design
using MATLAB. The implementation of classical experiment
as MATLAB m-files is described. Flexible robot arm
controllers can be designed to gain insight into a variety of
concepts, including stabilization of unstable systems,
compensation properties, Bode analysis and Root locus
analysis. The analysis has resulted in a number of
important conclusions for the design of a new
generation of control support systems.
1. Introduction
The purpose of phase lag compensator
design in the frequency domain generally is to
satisfy specifications on steady-state accuracy
and phase margin. There may also be a
specification on gain crossover frequency or
closed-loop bandwidth. A phase margin
specification can represent a requirement on
relative stability due to pure time delay in the
system, or it can represent desired transient
response characteristics that have been translated
from the time domain into the frequency
domain.
The overall philosophy in the design
procedure presented here is for the compensator
to adjust the system’s Bode magnitude curve to
establish a gain-crossover frequency, without
disturbing the system’s phase curve at that
frequency and without reducing the zerofrequency magnitude value. In order for phase
lag compensation to work in this context, the
following two characteristics are needed.
If the compensation is to be performed
by a single-stage compensator, then it must also
be possible to drop the magnitude curve down to
0 db at that frequency without using excessively
large component values. Multiple stages of
compensation can be used, following the same
procedure as shown below.
More is said about this later. The gain
crossover frequency and closed-loop bandwidth
for the lag-compensated system will be lower
than for the uncompensated plant (after the
steady-state error specification has been
satisfied), so the compensated system will
respond more slowly in the time domain. The
slower response may be regarded as a
disadvantage, but one benefit of a smaller
bandwidth is that less noise and other high
frequency signals (often unwanted) will be
passed by the system.
The smaller bandwidth will also
provide more stability robustness when the
system has unmodeled high frequency dynamics,
such as the bending modes in aircraft and
spacecraft. Thus, there is a trade-off between
having the ability to track rapidly varying
reference signals and being able to reject highfrequency disturbances. The design procedure
presented here is basically graphical in nature.
All of the measurements needed can be obtained
from accurate Bode plots of the uncompensated
system. If data arrays representing the
magnitudes and phases of the system at various
frequencies are available, then the procedure can
be done numerically, and in many cases
automated.
The examples and plots presented here
are all done in MATLAB, and the various
measurements that are presented in the examples
are obtained from the various data arrays. The
primary references for the procedures described
in this paper are [1]–[6]. Other references that
contain similar material are [4] – [13].
2. Compensator Structure
The basic phase lag compensator
consists of a gain, one pole, and one zero. Based
on the usual electronic implementation of the
compensator [6], the specific structure of the
compensator is:
(1   s)
(1  α s)
1(s  z c )
G c (s) 
α(s  p c )
G c (s) 
(1)
where   R 2 C and α  (R 1  R 2 )
R2
In this case, because of α > 1, the pole
lies closest to the origin of the s-plane. The
characteristics of the phase-lag compensation
network are summarized in the Figure 1.
Generally this operational amplifier circuits are
provided
for
phase-lead
or
phase-lag
compensation networks. If R1C1 is greater than
R2C2, this circuit can be used as a phase-lead
compensator and if R1C1 is less than R2C2, this
may be a phase-lag compensator.
C2
C1
R1
_
+
R2
R4
R3
_
+
Figure 1. Lead or Lag Compensator Design
The major characteristics of the lag
compensator are the constant attenuation in
magnitude at high frequencies and the zero
phase shifts at high frequencies. The large
2
negative phase shift that is seen at intermediate
frequencies is undesired but unavoidable. Proper
design of the compensator requires placing the
compensator pole and zero appropriately so that
the benefits of the magnitude attenuation are
obtained without the negative phase shift
causing problems. The following article show
how this can be accomplished.
The first step in the design procedure is
to determine the value of the gain K c . In the
procedure, the gain is used to satisfy the steadystate error specification. Therefore, the gain can
be computed from
ess - plant
K x - required
(2)
Kc 

ess -specified
K x - plant
where ( ess ) is the steady-state error for a
particular type of input, such as step or ramp,
and K x is the corresponding error constant of the
system. Defining the number of open-loop poles
of the system G (s ) that are located at s  0 to
be the System Type N , and restricting the
reference input signal to having Laplace
transforms of the form R(s)  A s q the steadystate error and error constant are ;
(3)
where
K x  lim [s N G(s) ]
Kx
(s/z c  1)

(N req - N sys )
(s/p c 1)
s
(5)
where N req is the total required number of poles
at the origin to satisfy the steady-state error
specification, and N req is the number of poles at
the origin in G p (s) . At the stage of making the
bode plots of the design, the system whose
frequency response is being plotted.
3. Design Procedure for Compensated
Plant
 As N 1 q 
e  lim  N

s0 s  K 
x 

G c -lag (s) 
(4)
s0
For N  0 , the steady-state error for a
step input (q  1) is ess  A (1  K x ) . For N  0
and q  1 , the steady-state error is infinitely
Kx
(N req - N sys )
(6)
s
The compensated gain crossover frequency
ωx -compensate d is that frequency where
G(s) =
∠
s
Kc
(N req - N sys ) × G p (s)
= 180 + PM specified + 10
(7)
If G(jωx -compensate d ) is expressed in dB, then the
value of α is computed from
(
G ( jω x - compensate d ) dB
20
α = 10
and G(jω x -compensate d )
)
(8)
is expressed as an
dB
absolute value, the value of α is computed from
(9)
α  G(jω x -compensate d )
absolute - value
3.1. Design Calculation
The control system for the robot arm
actuator is shown in Figure 2(b). The controller
gain will be K(s  1) . The elbow joint actuator is
a two-phase motor and its gain will be 2 / s(s + 4) .
Joint torque control can be substantially
improved by sensory feedback. So in this model
the second order poles compensator is used as a
joint torque feedback sensor and its measuring
gain will be 3 /(s 2 + 2s + 5) .
large. For N  0 , the steady-state error is
for the input type that
ess  A K x
has q  N  1 . If q  N  1 , the steady-state error
is infinite. The calculation of the gain assumes
that the given system G p (s) is of the correct Type
N to satisfy the steady-state error specification.
Once
this
is
recognized,
the
compensator poles at s  0 can be included with
the plant G p (s) during the rest of the design of
(a)
the lag compensator. The values of K x and of
K c in would then be computed based on G p (s)
(b)
Figure 2(a). A robot arm actuated at the joint
elbow
(b). The control system of the robot arm
being augmented with these additional poles at
the origin.
Once the compensator design is
completed, the total compensator will have the
transfer function.
The elbow joint of the robot arm to be
controlled is described by the transfer function.
G p (s ) 
G1 (s)
1  G1 (s) H(s)
3
And then by solving this equation to get the
simplest term,
2K(s3  3s 2  7s  5)
Kc 
s  6s3  13s2  (20  6K)s  6K
4
2(s  1)(s  1  j2)(s  1 - j2)
(s  0.2609)(s  4.3083)(s  0.7154  j0.1969)(s  0.7154 - j0.1969)
G(s) 
We must assume the one root of
denominator for this control system will be s
because of very small value 0.2609. So this is a
Type 1 system, so the closed-loop system will
have zero steady-state error for a step input, and
a non-zero, finite steady-state error for a ramp
input (assuming that the closed-loop system is
stable). As shown in the next section, the error
constant for a ramp input is Kx-plant
 0.434812404. At low frequencies, the plant
has a magnitude slope of-20 db/decade, and at
high frequencies the slope is -60db/decade. The
phase curve starts at -0⁰ and ends at -90⁰.
The specifications that must be satisfied
are:
1. Steady-state error for a ramp input
e ss -specified ≤ 0.02 and
2. Phase margin PMspecified ≥ 90 .
These specifications do not impose
any explicit requirements on the gain crossover
frequency or on the type of compensator that
should be used. It may be possible to use either
lag or lead compensation for this problem, or a
combination of the two, but we will use the
phase lag compensator design procedure
described above. The following paragraphs will
illustrate how the procedure is applied to design
the compensator for this system that will allow
the specifications to be satisfied.
3.2. Compensator Gain Calculation
The given plant is Type 1, and the
steady-state error specification is for a
ramp
input, so the compensator does not need to have
any poles at s = 0. Only the gain needs to be
computed for steady-state error. The steady-state
error for a ramp input for the given plant is
2(s3 + 3s2 + 7s + 5)
K x = lim s
s(s + 4.3083)(s + 0.7154 + j2.1969)(s + 0.7154 - j2.1969)
s→0
= lim
s→0
10
4.3083(0.7154 + j2.1969)(0.7154
j2.1969)
 0.434812404
And then the steady-state error is
ess =
1
1
=
= 2.29984239
K x 0.434812404
The steady-state error of the plant is
very large. It seems to become the system will
be unstable at infinity. Since the specified value
of the steady-state error is 0.02, the required
error constant is Kx- required = 50.
Therefore, the compensator gain is
ess - plant
ess -specified
K x - required
K x - plant

2.299842394
 114.9921197
0.02

50
 114.9921197
0.434812404
This value for Kc will satisfy the steady-state
error specification, and the rest of the
compensator design will focus on the phase
margin specification.
3.3. The Bode Structure
The magnitude and phase plots for
KcGp(s) are shown in Figure3. The dashed
magnitude curve is for Gp(s) and illustrates the
effect that Kc has on the magnitude. Specifically,
K cG p (jω ) is 20 log 114.992119 7 = 41.21336159
dB above the curve for
G p (jω )
at all
frequencies. The phase curve is unchanged when
the steady-state error specification is satisfied
since the compensator does not have any poles at
the origin. The horizontal dashed line at -80⁰ is
included in the Figure to indicate the phase shift
that will satisfy the phase margin specification
(+10⁰) when the frequency at which this phase
shift occurs is made the gain crossover
frequency. The vertical dashed line indicates that
frequency.
Bode for Compensator Gain Kc and Gp
40
23.9062dB
magnitude(dB)
G p (s) 
Kc 
0
-40
14.141rad/sec
-80
-100 -4
10
10
-3
10
-2
-1
0
10
10
10
frequency(rad/sec)
1
10
2
10
3
10
Figure 3. Bode plots for the plant after the
steady-state error
specification has been satisfied
Note that the gain crossover frequency
of KcGp(s) is larger than that forGp(s); the
crossover frequency has moved to the right in
the graph. The closed-loop bandwidth will have
increased in a similar manner. The phase margin
has decreased due to Kc, so satisfying the steadystate error has made the system less stable in fact
increasing Kc in order to decrease the steadystate error can even make the closed-loop system
unstable. Maintaining stability and achieving the
desired phase margin is the task of the pole–zero
combination in the compensator. Our ability to
4
4
graphically determine values for ωx-compensated and
α obviously depends on the accuracy and
resolution of the Bode plots of K cG p (jω ) . High
resolution plots like those obtained from
MATLAB allow us to obtain reasonably
accurate measurements. Rough, hand-drawn
sketches would yield much less accurate results
and might be used only for first approximations
to the design. Being able to access the actual
numerical data allows for even more accurate
results than the MATLAB-generated plots. The
procedure that I use when working in MATLAB
generates the data arrays for frequency,
magnitude, and phase from the following
instructions:
w = logspace(N1,N2,1+100*(N2-N1));
[mag,ph] = bode(num1,den1,w);
[mag2,ph2] = bode(num2,den2,w);
semilogx(w,20*log10(mag1),w,ph1,w,20*log10
(mag2),w,ph2),grid
where N1=log10 (ωmax), N2= log10 ( ωmin),
and
num, den are the numerator and
denominator polynomials, respectively, of
KcGp(s) . For this example,
N1= -4;
N2= 4;
num1= 2* [ 1 3 7 5];
den1=[1 6 13 26 6];
num2=2*114.850658057*[1 3 7 5];
den2=[1 6 13 26 6];
The data arrays mag, ph, and w can be searched
to obtain the various values needed during the
design of the compensator.
3.4. Gain Crossover Frequency
As mentioned above, the compensated
gain crossover frequency is selected to be the
frequency at which the phase shift of
K cG p (jω ) = -80⁰. This value of phase shift was
computed from (7), which allows the phase
margin specification to be satisfied, taking into
account the non-ideal characteristics of the
compensator. From the phase curve plotted in
Figure 3, this value of phase shift occurs
approximately midway between 10r/s and 20r/s.
Since the midpoint frequency on a log scale is
the geometric mean of the two end frequencies,
an estimate of the desired frequency is ω=
√10*20=14.14r/s. Searching the MATLAB data
arrays storing the frequency and phase
information gives the value ω=14.14r/s (using
100 equally-spaced values per decade of
frequency), so the estimate obtained from the
graph is very accurate in this example.
3.5. Calculation α
The lag compensator must attenuate
K c G p (jω ) so that it has the value 0 dB at
frequency ω  14.141rad/s . From the magnitude
curve in Figure 3, it is easy to see that
is between 20 dB and 30 dB.
K c G p (j14.141 )
dB
K c G p (j14.141 )
dB
 20 log 10 114.9921197  20 log 0.434825989  20 log 14.14 2  1
 20 log 1  (0.4  14.14) 2  (0.2  14.142 ) 2 - 20log 14.14
- 20log (0.2321 14.14) 2  1 - 20log 1  (0.268  14.14)2  (0.1873  14.142 ) 2
 33 .97967146  23 .03065  32 .12726 - 23 .00698819 - 10.7084 - 31.5160 dB
 23.9062  24dB
( 24 / 20 )
  10
 15.84893192
An
accurate measurement of the graph would
provide a value of 41.21336159 dB for the
magnitude. The corresponding value of α is
calculated by using (9) and the result
is α = 15.84893192 ; which can be easily
implemented with a single stage of
compensation.
dB
3.6. Compensator Zero and Pole
Now that the values for ω x -compensate d
and α is obtained, it can be determined the
values for the compensator’s zero and pole [6].
14.14
z=
= 1.414
10
p=
z
1.414
=
= 0.0892
α 15.84893192
The values that is
and p c  0.0892 .
z c  1.414
compensator for this design is
obtained are
The
final
 s

114.9921197
1 
 1.414

G c-lag1 (s) 
 s

 1

 0.0892 
 114.9921197
(0.7072s  1)
(11.2086s  1)
 114.9921197 0.0631
 7.2555 
(s  1.414)
(s  0.0892)
( s  1.414)
( s  0.0892)
3.7. Evaluation of the Design
The frequency response magnitude and
phase of the compensated system Gc-lag (s) Gp(s)
are shown in Figure 4. The magnitude is
attenuated at high frequencies, passing through 0
db at approximately 14.14r/s, so the
compensated gain crossover frequency has been
established at the desired value. The
compensated phase curve is seen to be nearly
back to the uncompensated curve at ωx-compensated.
The
lag
compensator
is
contributing
approximately -4.9 at that frequency, so the
phase margin specification is satisfied. To
illustrate the effects of the compensator on
5
closed-loop bandwidth, the magnitudes of the
closed-loop systems are plotted in Figure 4. The
smallest bandwidth occurs with the plant Gp(s).
Including the compensator gain Kc > 1 increases
the bandwidth and the size of the resonant peak.
Significant overshoot in the time-domain step
response should be expected from the closedloop system with Kc =114.9921197. Including
the entire lag compensator reduces
the
bandwidth and the resonant
Magnitude (dB)
60
Phase (deg)
Bode Diagram
Gm = Inf , Pm = 96.3 deg (at 14.1 rad/sec)
Magnitude (dB)
20
0
-45
-90
-4
10
40
-2
10
10
Frequency (rad/sec)
0
10
2
Figure 5. Gain Margin and Phase Margin for
Lag Compensator 2
The drawbacks to using the third
compensator are the increased values of the
electronic components (a problem particularly
with the capacitors) and the increase in the
settling time of the ramp response. The time it
takes to reach the unit ramp response is a value
0.02 less than the input (e ss - specified ) that increases
20
0
-20
0
Phase (deg)
40
-20
0
60
-45
-90
-135
-3
10
Bode Diagram
Gm = Inf , Pm = 101 deg (at 14 rad/sec)
10
-2
-1
0
10
10
Frequency (rad/sec)
10
1
10
Figure 4. Gain Margin and Phase Margin with
lag Compensator 1
2
from approximately 15 seconds with G c-lag2 (s)
while the time it takes for G c - lag1 (s) is about 4
seconds to reach a value 0.02 ≥ e ss -specified . So it
is chosen G c - lag1 (s) that can be obtained a better
4. Compare with Other Compensator Designs
To see some of the effects of the value
of the compensator’s zero, a second lag
compensator was designed. The same
specifications were used; the only difference
between these designs is that the zero of the new
compensator was placed two decades in
frequency below the compensated gain crossover
frequency, rather than one decade. The value of
the compensator pole also changed as a result.
The new compensator is
 s

114.9921197
1)
0.1414


G c-lag2 (s) 
s


 1

 0.00892 
 114.9921197

(7.072s  1)
(112.086s  1)
7.2555(s  0.1414)
(s  0.00892)
This new compensator reduces the percent
overshoot to a step input and provides additional
phase margin. The added phase margin is due to
the fact that the compensator’s phase shift is
virtually zero at a frequency two decades
above z c . Both of these characteristics are good.
performance for this elbow joint control system.
Trade-offs such as this typically has to be made
during the design of any control system.
R.C. Dorf [6] presents a table showing
analog circuit implementations for various types
of compensators. The circuit for phase lag is the
series combination of two inverting operational
amplifiers. The first amplifier has input
impedance that is the parallel combination of
resistor R 1 and capacitor C1 and feedback
impedance that is the parallel combination of
resistor R 2 and capacitor C 2 . The second
amplifier has input and feedback resistors R 3
and R 4 , respectively. Assuming that the opamps are ideal, the transfer function for this
circuit as shown in Figure 1 is
Vout (s) R 2 R 4 (sR 1C1  1)


Vin (s)
R 1R 3 (sR 2C 2  1)
R 2 R 4 R 1C1 (s  1/R 1C1 )


R 1R 3 R 2 C 2 (s  1/R 2 C 2 )
Comparing with G c - lag (s) in (5) shows that the
following relationships hold:
K  (R 2 R 4 )/(R1R 3 )
τ  R1C1 , ατ  R 2C2

z c  1 / R1C1
6
p c  1/ R 2 C 2
[2]
α  z c /p c  R 2C2 /R1C1
To implement the compensator using
the circuit in [6], note that there are six unknown
circuit elements (R1 , C1 , R 2 , C2 , R 3 , R 4 ) and
three compensator parameters (K c , z c , pc ) .So,
three of the circuit elements can be chosen to
have convenient values. To implement the
original lag compensator G c - lag1 (s) in this thesis,
it
will
be
used
the
following
values C1  C2  0.47μ F and R 3  10 kΩ that
are assumed. Thus, these values are obtained for
the phase-lag compensator design.
[3]
[4]
[5]
[6]
R 1  1 /( z c C1 )  1 /(1.414  0.47 μ )  1.5047097415 
R 2  1 /( p cC 2 )  1 /( 0.0892  0.47 μ )  2.38526858 
Kc 
[7]
R 2R 4
C R
R
C
 2 4 , where 1  α 1
R2
C2
R 1R 3
αC 1R 3
R 4  (R 3 K c /α)  (10 K  114.992119 7)/15.8489 3192
 72.5551225 
[8]
5. Summary
In this example, the phase lag
compensator is able to satisfy both of the
specifications of the system. In addition to
satisfying the phase margin and steady-state error
specifications, the lag compensator also produced a
step response with shorter settling time. In
summary, phase lag compensation can provide
steady-state accuracy and necessary phase margin
when the Bode magnitude plot can be dropped
down at the frequency chosen be the compensated
gain crossover frequency. The philosophy of the
lag compensator is to attenuate the frequency
response magnitude at high frequencies without
adding additional negative phase shift at those
frequencies.
ACKNOWLEDGEMENT
I would wish to acknowledge the many
colleagues
at
Mandalay
Technological
University who have contributed to the
development of this paper. In particular, I would
like to thank Yai Win Aung, my brother and Wai
Mar, my closed friend, for their complete
support.
REFERENCES
[1]
Zhi Hong Mao, February 2008, Linear
Control Systems, Lecture 12: Phase-lead
and Phase-lag Compensators and April
2008, Lecture 21: Frequency Response
Design (3), University of Pittsburgh, PA.
[9]
[10]
[11]
[12]
[13]
Venkata Sonti, March 2008, Lecture 6:
Gain, Phase Margins, designing with
Bode Plots, Compensators, Indian
Institute of Science, India.
Katsuhiko Ogata, 1997, Modern Control
Engineering, 3rd Edition, Prentice Hall,
Upper Saddle River, NJ.[2]
Dr. K. P. Mohandas, 2006, Modern
Control Engineering, NIT Campus PO,
Calicut, printed by Sanguine Technical
Publishers, Bangalore, India.
MathWorks, 2001, Introduction to
MATLAB, the MathWorks, Inc.
Richard C. Dorf and Robert H. Bishop,
2001, Modern Control Systems, Ninth
Edition, Prentice Hall Inc, A Simon and
Schuster Company, New Jersey[1]
Lawrence E. Pfeffer, O. Khatib, and J.
Hake, August 1989, “Joint Torque
Sensory Feedback in the Control of a
PUMA Manipulator”, IEEE Transaction
on Robotics and Automation, VOL. 5,
NO.4.
J.J. D’Azzo and C.H. Houpis,
Linear Control System Analysis and
Design, McGraw-Hill, New York, 4th
edition, 1995.
MathWorks, 2001, SIMULINK, the
MathWorks, Inc.
MathWorks, 2001, What is SIMULINK,
the MathWorks, Inc.
David F. Griffiths, August 2001, An
Introduction to MATLAB (version 2.2),
Stockholm, Sweden.
Sathish K. Shanmugasundram, 2000,
Control
System
Design
for
an
Autonomous mobile Robot, PSG College
of Technology, India.
Natick, January 1990, MA 01760,
MATLAB User’s Guide, The Math
Works, Inc.