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1. RESEARCH INVENTY: International Journal of Engineering and Science
ISBN: 2319-6483, ISSN: 2278-4721, Vol. 1, Issue 10
(December 2012), PP 54-58
www.researchinventy.com
Stability Improvement in Automobile Driving Through Feedback
Loop and Compensator
1
Dr. Sourish Sanyal, 2Prof. Raghupati Goswami, 3Prof. Amar Nath Sanyal
1
Assistant Professor, College of Engineering & Management, Kolaghat, Pu rba Midnapur
2
Professor, Ideal Institute of Technology, Kalyani Sh ilpanchal Kalyani, Nadia
3
Professor, Academy of Technology, Aedconagar, Adisaptagram, Hooghly
Abstract − The paper gives the mathematical model of an automobile along with its driver. The automobile has
been modelled from the standpoint of mechanics and the man driving the car has been modelled from the
standpoint of physiology. The later contains a time delay. The directional control through steering has been
considered as a closed-loop process. Analytical treatment of the control system has been made by replacing the
delay element by a zero based on binomial approximation and the stability of the system has been ensured for
the given gain. MATLAB tools have been used for the analysis. However the design specs could not be fulfilled.
To compensate the system rate feedback was tried but desired results could not be obtained. So a lag type
compensator was inserted in the forward path which could successfully meet the requirements. The design was
made using SISOTOOL o f MATLAB.
Keywords − Automobile, Driver, Feedback Loop, Time Delay, Compensator
I. Introduction
Driving an automobile is, in essence, a feedback process [1,2]. This is an automatic control system in
which the driver is to maintain a desired direction which dynamically varies. The feedback is through the eyes
of the driver [4] which is transmitted to the brain. The error signal produced by the brain is amp lifie d through
muscles of the driver which controls the steering wheel. The automob ile is modeled fro m the point of view of
mechanics. The hu man transfer function, modelled by physiologists [8], is to be included as an element of the
system.
II. Description Of The System
The block diagram of the system is given in fig. 1. The human transfer function includes a time delay
element. The proportional feedback (unity) has been augmented with a rate feedback. The task is to maintain the
desired direction in presence of the time delay. There must be adequate stability marg in and the time-do main
response must be acceptably good [3].
Desired d irection, R(s) Human Transfer Auto- Actual d irection, C(s)
Function mobile
∑
K (1  2s)e0.1s 2
s
(1  10s)(1  0.05s)
Rate Feedback
Kd s
Fig 1 Driving an automobile: block diagram
Design specificati ons:
The control system must meet with the fo llo wing specifications:
 The steady state error must be less than 1%
 The maximu m overshoot must be limited to 4 %
 The settling time must be less than 15 sec.
 The gain margin must be more than 30 db and the phase margin mo re than 60 o .
54

2. Stability Improvement in Automobile Driving Through Feedback Loop and Compensator
III. Mathematical Approach
The transfer function of the system inclusive of the hu man element is given below:
C (s) 2 K (1  2s)e0.1s
G(s)   ; H (s )  1 (1)
R( s) s(1  10s)(1  0.05s)
The delay element does not affect the gain. It only causes a phase lag of ( -0.1ω) rad.
This problem can be dealt with by linear control system analysis in two ways viz.
a. By excluding the time -delay and taking the transfer function as [1]:
2 K (1  2s)
G1 ( s) 
s(1  10s)(1  0.05s) (2)
0.1( j )
The (-1+j.0) point has to be shifted to: 1.e The solution can be made using Nyquist plot. The Nyquist
0.1( j )
plot of G1 (jω ) must not enclose the point [1.e ] and the phase margin is to be measured from the line
joining this point with the origin.
b. Replacing the time-delay by an equivalent zero in the right half of s -plane (b ionomial appro ximation) [2],
as given in eqn. 3.
C ( s) 2 K (1  2s)(1  0.1s)
G( s)  
R( s) s(1  10s)(1  0.05s) (3)
The second method is more convenient and it does not give rise to much error. For ease and flexibility in
handling the problem, we have used MATLAB tools for analysis.
IV. Analysis Using Matlab-Tools
Now, MATLA B tools are being used for analysis and design of the control system. At first, we shall
consider the uncompensated system [5,6].
a. The uncompensated system
The forward path gain has been chosen as 3.75 by cut and try process. The time delay has been
0.1s
approximated as: e  1  0.1s ; its effect is to add a zero in the right half of the s -plane. The t-do main
response to a step input is given in fig. 2 and the corresponding Bode plot in fig. 3. It is observed that the peak
overshoot is 14.8% and the peak time is 2.17 sec. The rise time and the settling time are 0.73 s ad 5.51 sec,
respectively. The gain margin is 16.3 db and the phase crossover occurs at 13.7 r/s. The phase margin is 62.6o
and the gain crossover occurs at 1.58 r/s.
System: sysmfb
Peak amplitude: 1.15
Overshoot (%): 14.8
Step Response
At time (sec): 2.17
1.2
System: sysmfb
Settling Time (sec): 5.51
1
System: sysmfb
0.8
Rise Time (sec): 0.73
0.6
Amplitude
0.4
0.2
0
-0.2
0 1 2 3 4 5 6 7 8
Time (sec)
Fig. 2. Step Response of the uncompensated system.
55

3. Stability Improvement in Automobile Driving Through Feedback Loop and Compensator
It is noted that the uncompensated system gives a highly oscillatory response. This is not acceptable.
Also the gain margin is below the specified value. So we add a rate feedback: K d = 0.07. The forward path gain
has been kept at the same value.
Bode Diagram
Gm = 16.3 dB (at 13.7 rad/sec) , Pm = 62.6 deg (at 1.58 rad/sec)
100
Magnitude (dB) 50
0
-50
-100
270
225
Phase (deg)
180
135
90
-3 -2 -1 0 1 2 3
10 10 10 10 10 10 10
Frequency (rad/sec)
Fig 3. Bode Plot of the uncompensated system
b. Addi tion of rate feedback
In order to fulfill the design specifications, we are now adding a rate feedback. The aim is to reduce the
peak overshoot. The adjusted value of rate feedback coefficient, K d = 0.07. The t-domain response of the system
with rate feedback is given in fig. 4 and the corresponding Bode plot in fig. 5 [5,7].
System: sysmfb
Peak amplitude: 1.13
Overshoot (%): 13.1 Step Response
1.2 At time (sec): 2.41
System: sysmfb
Settling Time (sec): 5.75
1
System: sysmfb
0.8
Rise Time (sec): 0.854
0.6
Amplitude
0.4
0.2
0
-0.2
0 1 2 3 4 5 6 7 8 9
Time (sec)
Fig. 4 Step Response with rate feedback
It is observed that the peak overshoot has only slightly reduced from 14.8% to 13.1% but the peak time
has increased from 2.17 sec to 2.41 sec. The rise time has also increased fro m 0.73 sec to 0.854 sec. The settling
time is now 5.75 sec. as against 5.51sec for the uncompensated system. It is noted that the addition of rate
feedback has made the system sluggish but there has been no appreciable reduction in the percent overshoot.
The gain marg in has reduced. It is now 13.6 db as compared to 16.3 db for the uncompensated case. The phase
margin has slightly increased- it has become 68.9o as against 62.6o fo r the earlier case.
56

4. Stability Improvement in Automobile Driving Through Feedback Loop and Compensator
Bode Diagram
Gm = 13.6 dB (at Inf rad/sec) , Pm = 68.9 deg (at 1.59 rad/sec)
100
80
Magnitude (dB)
60
40
20
0
-20
270
Phase (deg)
225
180
-3 -2 -1 0 1 2 3
10 10 10 10 10 10 10
Frequency (rad/sec)
Fig. 5: Bode plot with rate feedback
Increasing the coefficient of rate feedback will tend to make the overshoot lower but the system will be
unacceptably sluggish. So we have to think about cascading a compensator in the forward path. As the
specifications have not been fulfilled even by using rate feedback, we are adding a lag compensator in series in
the forward path and exclude the rate feedback.
c. Addi tion of a compensator in the forward path
The design is being made by using SISOTOOL of MATLAB. The design configuration is given in fig.
6. The gains, poles and zeroes of the fixed and tunable elements are given in table -I. [5]
Fig. 6: The system with co mpensator cascaded in the forward path
Table-I
Items Tunable elements Fi xed elements
C F G H
Gain 0.144076 1 -3 1
zeroes -0.081837 [] [10; -0.5] []
Poles -0.250987 [] [0 ; -20; -0.1] []
The root locus, closed and open loop Bode plot of the lag -compensated system are g iven in fig.7 and
the time domain response against step input and the open loop Bode plot are given in fig. 8.
Root Locus Editor f or Open Loop 1 (OL1) Open-Loop Bode Editor f or Open Loop 1 (OL1)
20 40
0.78 0.64 0.460.24
0.87
0.93 20
10
0.97
0.992 0
30 25 20 15 10 5
0 Requirement:
0.992 -20 GM > 20
0.97 PM > 30
-10
0.93 -40
0.87
0.78 0.64 0.460.24 G.M.: 33.2 dB
-20 -60
Freq: 13.9 rad/sec
-30 -20 -10 0 10 20 30
Stable loop
Bode Editor f or Closed Loop 1 (CL1) -80
0
315
-50 270
225
-100
360 180
180 135
P.M.: 71.1 deg
Freq: 0.313 rad/sec
0 90
-2 0 2 4 -2 0 2 4
10 10 10 10 10 10 10 10
Frequency (rad/sec) Frequency (rad/sec)
Fig 7 Root locus, closed and open loop Bode plot of the lag-co mpensated system
57

5. Stability Improvement in Automobile Driving Through Feedback Loop and Compensator
For the lag-compensated system, the peak overshoot is now only 3.71% and the peak time 10.5 sec.
The settling time is 13.6 sec. The gain margin is 33.2 db and the phase margin is 71.1 o . It is noted that the
system with lag co mpensator cascaded in the forward path has been able to meet all the design specifications.
Step Response
1.5 System: Closed Loop r to y
I/O: r to y
Peak amplitude: 1.04
Overshoot (%): 3.71
At time (sec): 10.5
1
System: Closed Loop r to y
I/O: r to y
System: Closed Loop r to y
Settling Time (sec): 13.6
I/O: r to y
Rise Time (sec): 5.15
Amplitude
0.5
0
-0.5
0 2 4 6 8 10 12 14 16 18 20
Time (sec)
Fig 8 Time response and open loop Bode of the lag-co mpensated system
V. Conclusion
The paper deals with an automobile driving. The automobile has been modeled along with its driver.
The combination has been treated as a closed loop system. The automob ile has been modeled as an integrator
and a gain. The driver has been modeled by a human transfer function, developed from the standpoint of
physiology [1,2,8]. It contains a time delay element.
The system has been analyzed by MATLAB tools [5,6,7]. It has been found that the uncompensated
system fails to fu lfill the design specifications. Then an adjustable rate feedback has been added. But it also fails
to fulfill the specifications, rather it makes the system sluggish. Hence rate feedback has been omitted and a lag
compensator has been added in the forward path. The design has been made by using SISOTOOL of
MATLAB7.9. The peak overshoot now comes down below 4%, settling time below 15 sec. and the
requirements of gain and phase marg in are fulfilled. So, design approach through SISOTOOL appears to be the
best.
References
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Journal of Emerging T echnology and Advanced Engineering, Vol. 2, Issue 10, October, 2012, pp 192-196.
[2] S. Sanyal, R.K.Barai, P.K. Chattopadhyay, R.N. Chakrabarti, Design of Compensator for a Hydrofoil Ship, International Journal
of Emerging T echnology and Advanced Engineering, Vol. 2, Issue 9, September- 2012, pp 266-271.
[3] S. Sanyal, R.K.Barai, P.K. Chattopadhyay, R.N. Chakrabarti, Design of Attitude Control System of a Space Satellite,
International Journal of Advanced Engineering Technology, Vol. III, Issue II, April-June, 2012, pp 13-16.
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2012.
[5] M. Gopal, Modern control system theory, 2nd Ed., New Age International.
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Control (MTIC10), CIEM, Kolkata, March 11-12, 2010.
[8] D. Patranabis, Sensors and Transducers, PHI, 2008.
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