Design of Compensator for Roll Control of Towing Air-Crafts

ISSN 2349-7815
International Journal of Recent Research in Electrical and Electronics Engineering (IJRREEE)
Vol. 3, Issue 4, pp: (1-6), Month: October - December 2016, Available at: www.paperpublications.org
Page | 1
Paper Publications
Design of Compensator for Roll Control of
Towing Air-Crafts
1
Avik Ghosh, 2
Sourish Sanyal, 3
Amar Nath Sanyal, 4
Raju Basak
1
Assistant Professor, Ideal Institute OF Engineering, Kalyani, Nadia, West Bengal
2
Professor, Techno India College, Salt Lake, Kolkata, India
3
Professor, Calcutta Institute OF Engineering & Management, Kolkata, India
4
Research Scholar, Jadavpur University
Abstract: It is a difficult task to make proper adjustment of towing vehicles, keeping the motion secured and
predetermined. In older days the control was manual. Now-a-days automatic feedback control systems are used.
The specifications are very stringent due to imposition of govt. and industrial rules. There are constraints on
steady state accuracy, transient performance and stability margins. The requirements are contradictory. If the
steady state accuracy is realized, the transient requirements and the stability margins cannot be maintained. It is
difficult to fulfil the requirements by modifying the feedback or adding feed-forward. It is expedient to add a
compensator in the forward or feedback path. In this paper, the design of a towing aircraft has been taken up. Its
block diagram and transfer function are given. The gain has been fixed up to keep the steady state error within
prescribed limits. The transient performance has been shaped and stability ensured by adding a lag compensator
of chosen parameters.
Keywords: Towing Air-Craft, Forward Path Gain, Feedback Path, Transient Performance indices, frequency-
domain analysis, and stability margins.
Symbols:
( ), ( )r s c s Actual roll angle/ command roll angle
K Forward path gain
,T  Parameters of the lag compensator
vK Velocity error constant
1. INTRODUCTION
Coupling two or more objects together so that they may be pulled by a power source is called towing. The towing power
source may be a vehicle, vessel, animal, or human, the load may be anything that can be pulled. The load and the power
source may be joined by a chain, rope or bar or other means to keep the objects together while moving. Towing may be as
simple as a tractor pulling a tree stump. In the extreme it may be a heavy duty task requiring enormous tractive force.
Standards imposed by authorities are now prevailing to ensure safety and interoperability of towing equipment. In the
same way, aircraft may tow one-another. Gliders carrying troop and cargo are frequently towed behind powered aircraft –
it remains a popular means of shifting a load [1,2,11]. At this stage, it has to be appreciated that towing needs close
control. An automatic control system against given constraints must be incorporated with all towing equipment
2. GUIDANCE AND CONTROL
The towing source and the load must match with each other while in motion. The roll of the power source is to be sensed
by internal gyroscopes and accelerometers, and corrections are to be made by automatic control systems employing
feedback. The steady state accuracy or precision level is specified which fixes up the system gain. The transient
performance is also specified.

ISSN 2349-7815
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It may not be possible to realize the desired performance and to meet the design specifications with an uncompensated
system. In such cases a suitable compensator has to be inserted in the forward path. The compensators may be of lag, lead
or lag-lead type [3]. Authors like S. Sanyal, A.N. Sanyal, R. Basak have used various types of compensators for designing
industrial devices to match the specifications [6,7,8,9,10]. Design and its optimization can be made after modelling the
system [4]. The matching has to be made in respect of steady state accuracy, time domain and frequency-domain
constraints. Not only matching the specifications but also system optimization remains the ultimate aim [5]. In this work,
quasi-optimal values have been found out by cut-and-try method.
3. BLOCK DIAGRAM
Fig. 1. Block diagram representation of the towing air-craft
It has been modelled as a linear continuous control system [4]. The dynamics of the aircraft has been given as a product of
an integrator and two first order lags. The actuator gain has been given as 100, and the amplifier gain has to be adjusted to
limit the steady state error. Unity feedback has been used. The design has been initially made without a compensator. As
the specifications cannot be fulfilled without a compensator, a properly designed compensator has been inserted in the
next stage [3].
4. SPECIFICATIONS
The following specifications are to be fulfilled.
The steady state error must be less than 1%
The transient overshoot is to be about 1%
The gain margin must be more than 20 db
The phase margin must be more than 60o
5. MATHEMATICAL DESCRIPTION
With reference to the block diagram, the forward path and the feedback transfer functions (without the compensator) are
given as [1,2,3]:
( ) 100
( )
( ) ( 36)( 100)
r s K
G s
c s s s s
 
 
; ( ) 1H s  (1)
The closed loop transfer function is given as:
3 2
( ) 100
( )
( ) 136 3600 100
c s K
M s
r s s s s K
 
  
(2)
pK has been taken as 1 as it is a unity feedback system. From the closed loop transfer function, it is noted that the system
is of 3rd
. order. The velocity error constant is given as:

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0
100 100 100
100/sec 3600
1 ( 36)( 100) 3600
v
s
K K
K Lims K
error s s s


      
 
(3)
Against this value of gain, the t-domain and f-domain performance has been evaluated using MATLAB [12,13,14] and
are given in fig. 2 and fig. 3.
We note that the uncomopensated system fulfills the specification on steady state accuracy but fails to fulfill the criteria
set up for transient performance and stability limits. So we must cascade a compensator of appropriate parameters.
In cascade compensation, the compensator is inserted in the forward path. The transfer function of the compensating
network is designed to provide additional lag, lead or a combination of both. Accordingly they are designated as lag, lead
or lag-lead compensator. All these networks are made up of two types of circuit elements viz. resistors and capacitors. A
lag compensator has been used in this case for the following reasons [3]:
Fig. 2. Step response of the uncompensated system
Fig. 3. Bode plot of the uncompensated system
Step Response
Time (sec)
Amplitude
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
System: systemfb
Rise Time (sec): 0.0225
System: systemfb
Peak amplitude: 1.76
Overshoot (%): 76.5
At time (sec): 0.0672
System: systemfb
Settling Time (sec): 1.21
Bode Diagram
Frequency (rad/sec)
-150
-100
-50
0
50
System: systemfrd
Gain Margin (dB): 2.67
At frequency (rad/sec): 60
Closed Loop Stable? Yes
Magnitude(dB)
10
0
10
1
10
2
10
3
10
4
-270
-225
-180
-135
-90
System: systemfrd
Phase Margin (deg): 8.01
Delay Margin (sec): 0.00273
At frequency (rad/sec): 51.2
Phase(deg)

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a) It is less expensive
b) Gives a smaller BW
c) The resulting system has less noise
d) The output response has less jitter
6. THE LAG NETWORK
The circuit configuration for the lag compensator is given [1,2] in fig. 4. In the phase lag network, the phase of the output
lags the phase of the input. It has a simple pole and a zero in the left half of the s-plane, with the pole to the right of the
zero.
Fig.4 A lag compensator
The lag compensator attenuates high frequency noise in the control loop. It also increases the steady state error
coefficients, thus reducing the error. The transfer function of the compensator is found to be:
(1 )
( )
1
c
c
K s
G s
s

 



, (4)
Where, 2 2R C  & 1 21 /R R   (with ref. to fig. 4). The values of  &  are to be adjusted to meet the design
requirements.  is generally chosen between 3 & 10. We choose the following transfer function for the compensator:
1(1 23 )
( )
1 230
c
s
G s
s


 , (5)
The parameters have been chosen by cut and try method: 1, 23, 10cK    
The forward path transfer function inclusive of the lag compensator is given as:
( ) 100 (1 23 )
( )
( ) ( 36)( 100)(1 230 )
r s K s
G s
c s s s s s

 
   (6)
With this lag compensator in cascade, the t-domain and f-domain performances have been evaluated using MATLAB [12,
13, 14] and are given in fig. 5 and fig. 6.
Now, we find that all the design specifications have been fulfilled.
R2
R1
1
C2.s
Ei(s) Eo(s)
Amp
lifier
Gain

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Fig. 5. Step response of the compensated system
Fig. 6. Bode plot of the compensated system
7. CONCLUSION
Towing is an operation in various industries which require close control. The automatic control system chosen for them
must satisfy stringent control requirements. The requirements are on steady state accuracy, t-domain and f-domain
performance. To maintain the specified steady state accuracy, the forward path gain has to be increased. This increase in
gain may make the system unstable or give rise to highly oscillatory response violating constraints on peak overshoot and
other t-domain variables as well as reducing the margins of stability. Therefore, a compensator has to be added either in
the forward path or in the feedback path. In this example of towing, a lag type compensator has been used in the forward
path for its various advantages. The parameters of the compensator have been chosen judiciously so that the t-domain and
f-domain specifications are matched. A cut and try process generally gives the desired performance (which may be noted
from the results obtained using MATLAB). However, computer programs may be used to find out the parameters of the
optimal compensator.
Step Response
Time (sec)
Amplitude
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
0.2
0.4
0.6
0.8
1
1.2
1.4
System: systemfb
Final Value: 1
System: systemfb
Peak amplitude: 1.01
Overshoot (%): 1.18
At time (sec): 0.322
System: systemfb
Rise Time (sec): 0.142
System: systemfb
Settling Time (sec): 0.227
Bode Diagram
Frequency (rad/sec)
-150
-100
-50
0
50
100
150
System: system
Gain Margin (dB): 22.7
At frequency (rad/sec): 60
Magnitude(dB)
10
-4
10
-2
10
0
10
2
10
4
-270
-225
-180
-135
-90
System: system
Phase Margin (deg): 69.3
Delay Margin (sec): 0.126
At frequency (rad/sec): 9.62
Phase(deg)

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REFERENCES
[1] K. Ogata, “Modern control engineering”, Pearson Education
[2] M. Gopal, “Modern Control System Theory” , New Age Int. Ltd.
[3] S.M. Shinners, “Modern control system theory and design”, John Wiley and Sons.
[4] A.M. Law and W.D. Kelton, “Simulation, modeling and analysis”, McGraw-Hill, New York, 2nd
.Edition, 1991.
[5] K. Deb, ”Optimization for engineering design”, PHI, 2010
[6] S. Sanyal et al, “The motor cycle and rider-modeling and analysis of the control system”, International Journal of
Emerging Technology and Advanced Engineering, Vol. 2, Issue 10, October- 2012, pp-192-196
[7] S. Sanyal et al, “Stability improvement in automobile driving through feedback loop and compensator”, Research
Inventy- International Journal of Engineering and Science, ISSN: 2278-4721, Vol. 1, Issue 10, December, 2012, pp-
54-58 (available online).
[8] A.N. Sanyal et al, “Pitch attitude control of a booster rocket”, Research Inventy: International Journal of
Engineering and Science, ISSN: 2278-4721, Vol.2, Issue 7 (March 2013), pp 8-12 www.researchinventy.com
(indexed by Copernicus)
[9] S. Sanyal et al, “Roll attitude control of a space vehicle”, Research Inventy: International Journal Of Engineering
and Science, ISSN: 2278-4721, Vol. 4, Issue 3 (March 2014), www.researchinventy.com (indexed by Copernicus)
[10] R. Basak, “Design of intelligent soft arm control”, International Journal of Science, Engineering and Technology
Research (IJSETR), Vol. 4, Issue 7, July 2015
[11] Free encyclopedia of Wikipedia
[12] Herniter M.E., “Programming in MATLAB“, Thomson- Vikas publishing House.
[13] J.J. D’Azzo, C.H. Houpis and S.N. Sheldon, “Linear control system analysis and design with MATLAB”, 5e, Marcel
Dekker Inc. New York, BASEL
[14] A.J. Grace, N. Laub, J.N. Little and C. Thomson, “Control system tool box for use with MATLAB”, User Guide,
Mathworks, 1990.

Design of Compensator for Roll Control of Towing Air-Crafts

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