Modeling and Control of A Car Suspension System Using P, PI, PID, GA-PID and Auto-Tuned PID Controller in Matlab/Simulink
Modeling and Control of A Car Suspension System Using P, PI, PID, GA-PID and Auto-Tuned PID Controller in Matlab/Simulink
Modeling and Control of A Car Suspension System Using P, PI, PID, GA-PID and Auto-Tuned PID Controller in Matlab/Simulink
ISSN: 2458-925X
Vol. 3 Issue 3, March - 2017
Abstract— This paper presents the application of Nowadays, different types of controllers are
P, PI, PID, GA-PID and Auto-tuned PID controllers being used to control the car suspension system such
to control the vibration of the 1/4 car suspension as adaptive control, Linear Quadratic Gaussian (LQG)
system. An open loop response of the car control, H-infinity, Proportional (P) controller,
suspension system is developed using equations Proportional Integral (PI) controller, and Proportional
of the 1/4 car suspension system, state space Integral Derivative (PID) controller [6-8]. In this paper,
model and transfer function model built in the ¼ car suspension system is modeled using
Matlab/Simulink. The results of the open loop Simulink blocks. Also, P, PI, PID, Genetic-Algorithm
response reveal that the system is under-damped (GA) PID and Automatic-tuned PID controllers are
for a disturbance unit step input W. Also, the car designed to control the vibration of the car suspension
takes unacceptably long time for it to reach the system using Matlab/Simulink.
steady state, that is, about 50 seconds way
beyond the design requirements of 5 seconds.
However, full implementation of PID controller to
the suspension system causes the design
II. MODELING OF QUARTER CAR SUSPENSION
requirements to be met. Also, GA-PID controller is
found to produce better results of suspension SYSTEM
control than PID, and Auto-tuned PID controllers.
The car suspension system is one of the
Keywords—PID; GA-PID; Auto-tuned PID; impressive challenging problems in terms of
State-Space; Transfer function; Matlab/Simulink; controlling the system. When designing the car
I. INTRODUCTION suspension system, a ¼ car model (one of the four
wheels) is used to simplify the problem to a one
Suspension systems are the most important dimensional spring-damper system [9]. The schematic
part of the vehicle affecting the ride comfort of representation of the quarter car suspension model is
passengers and road holding capacity of the car, as pictured in figure (1). Table1 depicts the model
which is crucial for the safety of the ride. Moreover, parameters. Moreover, in developing the
increasing progress in automobile industry demands mathematical model of the quarter car, only the mass
that highly developed vehicle models with better riding movements on the vertical axis is considered ignoring
capabilities to enhance passenger comfort be the rotational movement of the vehicle.
developed. The aim of the advanced vehicle
suspension system is to provide smooth ride and
maintain the control of the car over cracks and on
uneven pavement of roads. Suspension system
modeling has an important role for realistic control of
vehicle suspension [1-3].
Designing a good suspension system with
optimum vibration performance under different road
conditions is an important task. Over the years, both
passive and active suspension systems have been
proposed to optimize the vehicle quality. Passive
suspension uses conventional dampers to absorb
vibration energy and do not require extra power [4].
Whereas active suspension systems capable of
producing an improved ride quality use additional
power to provide a response-dependent damper [5].
Fig.1.Quarter car Suspension Model
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Journal of Multidisciplinary Engineering Science Studies (JMESS)
ISSN: 2458-925X
Vol. 3 Issue 3, March - 2017
TABLE1. PARAMETERS FOR QUARTER CAR SUSPENSION analyze the behavior of the suspension system. In this
MODEL work, the road disturbance (W) will be simulated by a
Parameter Parameter Parameter Parameter step input and this step could represent a car coming
Description Symbol Value Unit out of pothole [10]. A feedback controller has to be
designed so that the output (x1-x2) has an overshoot
1. Mass of less than 5% and settling time shorter than 5 seconds.
sprung mass m1 2500 kg For example, when the car runs onto a 0.1 m high
step, the car body will oscillate within a range of ±
2. Mass of 0.005 m and return to a smooth ride within 5 seconds.
Un-sprung m2 320 Kg
mass
III. MATHEMATICAL MODEL OF QUARTER CAR
3. Stiffness SUSPENSION
coefficient of k1 80,000 N/m To derive the dynamic governing equations of the
the ¼ car suspension system, Newton’s second law is
suspension used for each of the two masses in motion and
Newton’s third law for the interaction of the masses.
4. Vertical The dynamic equations are as shown:
stiffness of k2 500,000 N/m
the tire m1𝑥̈ 1 = −c1(𝑥̇ 1−𝑥̇ 2) −k1(x1−x2) + U (1)
5. Damping
coefficient of c1 350 Ns/m
m2𝑥̈ 2 = c1(𝑥̇ 1−𝑥̇ 2) +k1(x1−x2) + c2 (Ẇ − 𝑥̇ 2) +
the k2(W−x2) –U (2)
suspension
6. Damping
coefficient of c2 15,020 Ns/m where all the values of the constant parameters, m1,
the tire m2, k1, k2, c1and c2 in both equations are given in
table1.
7. Vertical A.
displacement x1 - Equations 1 and 2 are second order differential
of the sprung equations of the active suspension system of the car.
mass Solving this system of equations poses a lot of
difficulties, so therefore, the system is solved and
8. Vertical verified using Matlab Simulink software based on the
B.
displacement x2 - following approaches.
of the Building the car suspension system
unsprung equations in Matlab/simulink;
mass
Using the ‘state-space’ model and
9. Controller C.
output (force) U - Using the ‘transfer function’ approach.
which is to be
controlled
10. Road D.
W - A. Modeling/Building the system equations using
excitation
Matlab Simulink blocks
Equations 1 and 2 are built together using Matlab
A. Design requirements Simulink blocks to represent an implementation of the
A good car suspension system should have car suspension system in simulink. This is shown in
satisfactory road holding ability, while still providing figure (2).
comfort when riding over bumps and holes in the
road. When the car is experiencing any road
disturbance (that is, pot holes, cracks, and uneven
pavement), it is expected that the car body dissipates
its oscillatory motion quickly. Now, since the distances
x1-W is very difficult to measure, and the deformation
of the tire (x2-W) is negligible, the distance x1-x2
instead of x1-w is used as an estimated output to
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Journal of Multidisciplinary Engineering Science Studies (JMESS)
ISSN: 2458-925X
Vol. 3 Issue 3, March - 2017
A = m1 s c1 s k 1 c1 s k 1
2
c1 s k 1 m s2
2
c1 c2 s k 1 k 2
(6)
Fig.3. Open loop step response, body sprung mass
displacement ∆ = det
m1 s
2
c1 s k 1 c s k
c1 s k 1
m s c c s k k
1 1
2
2 1 2 1 2
(7)
or
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ISSN: 2458-925X
Vol. 3 Issue 3, March - 2017
Find the inverse of matrix A and then multiply with G2(s) = num1/den1=
inputs U(s) and W(s) on the right hand side as the −3.755𝑒007𝑠^3 − 1.25𝑒009𝑠^2
following: 800000𝑠^4 + 3.854𝑒007𝑠^3 + 1.481𝑒009𝑠^2 + 1.377𝑒009 + 4𝑒010
x1 ( s ) = 𝟏 (14)
∆
x2 ( s)
m2 s
2
c1 c2 s k 1 k 2 c s k Now, the process transfer function represented by
c1 s k 1 m s c s k equation (13) can be simulated as an open-loop
1 1
2
1 1 1 system (without any feedback control) to control input
U. The simulink model is shown in figure (2), and
U ( s)
s figure (5) below shows the open-loop response of the
c
2 k 2
W ( s ) U ( s ) process transfer function, which is obtained by
considering only the disturbance input W(s) = 0.1 m,
(9) and U(s) = 0.
x1 ( s ) 𝟏
=∆
x2 ( s)
2
s
m2 s c 2 2 k 2
c c s c k c k s k k
2
m c s m k c c s
c1 k 2 c2 k 1s k 1 k 2
1 2 1 2 2 1 1 2
m1 s
2 2
1 2 1 2 1 2
U ( s )
W ( s )
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Journal of Multidisciplinary Engineering Science Studies (JMESS)
ISSN: 2458-925X
Vol. 3 Issue 3, March - 2017
𝑥̇ 1 0 1 0 0
𝑥̈
[ 1 ]= cc1 2
0
c c c c c
1 1 1 2 1
c 1
𝑦̇1 mm 1 2 m m m m m
1 1 2 2 1 m 1
Fig.7: Closed loop of car suspension system
𝑦̈1 c 2
0
c c c
1 1 2
1
m 2 m m m
1 2 2
k 2
0 k k k
1 1 2
0
[ m 1 m m m
1 2 2 ] A. PID controller
𝑥1 A proportional-integral-derivative controller (PID
0 0
𝑥̇ 1 1 cc controller) is a generic control loop feedback
1 2 U
[𝑦 ] + [ ] mechanism (controller) widely used in industrial
1 m 1 mm
1 2
W
control systems [13]. A PID controller attempts to
𝑦̇1
0
c 2
correct the error between a measured process
m 2
variable and a desired set point by calculating and
1
1
k 2
then outputting a corrective action that can adjust the
m 1 m 2 m 2 process accordingly, to keep the error minimal [14]. A
(20) block diagram of the PID controller is as depicted in
figure (8).
𝑑𝑅(𝑠)
C(s) = kpR(s) + ki∫ 𝑅(𝑠)𝑑𝑡 + kd (21)
𝑑𝑡
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ISSN: 2458-925X
Vol. 3 Issue 3, March - 2017
error. Hence, from equation (21), the PID controller for for the GA-PID and Automatic Tuned PID simulation
the car suspension system can be given as: model as illustrated in figure (12a and b).
(22)
A. P Controller with Car Suspension System
P controller is mostly used in first order
Thus, the general response of the proportional,
processes with simple energy storage to stabilize the
integral and derivative controller is as shown in table2.
unstable process. The main usage of the P controller
is to decrease the steady state error of the system. As
TABLE2. RESPONSE OF PROPORTIONAL, INTEGRAL AND the proportional gain factor K increases, the steady
DERIVATIVE CONTROLLER state error of the system decreases. The simulink
model of the P controller with the car suspension
Closed Rise time Overshoot Settling Steady system is as pictured in figure (9a). The main purpose
loop time state error
response of this implementation is to obtain the desired
Kp Decrease Increase No Decrease response of the system. The value of the Kp used for
change the simulation is 1664200. The result of the
simulation is shown in figure (9b).
Ki Decrease Increase Increase Eliminate
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ISSN: 2458-925X
Vol. 3 Issue 3, March - 2017
Fig.10b: PI controller output response to 0.1 m input The analysis of the Car Suspension System is
further investigated by comparing the simulation of the
PID, GA-PID and the Auto-tuned PID controllers.
From figure (10b), the PI simulation results show Figure (12a) below shows the combined simulink
an overshoot of 0.026 m for a unit step input of 0.1 m model for PID, GA-PID and the Auto-Tuned PID. For
and a settling time of about 3.6 seconds, suggesting the combined simulation, put the value of Kp, Ki and
that the PI controller could not meet the design Kd, and also put the value of gains found by GA-PID
requirements adequately. Also, it must be noted that and the auto-tuned PID controller block as mentioned
without derivative action, a PI-controlled system is earlier in the text. The simulation result is as pictured
less responsive to real and relatively fast alterations in in figure (12b).
state and so the system will be slower to reach set-
point and slower to respond to perturbations than a
well-tuned PID system.
C. PID Controller
The PID controller calculation involves three
separate parameters, and is accordingly sometimes
called three-term control. The main purpose of the PID
controller performance for the car suspension system
is to get the desired response of the system within
expected times. The Simulink model of the Car
Suspension system using PID Controller is as depicted
in figure (11a) and the results of the simulation are as
shown in figure (11b). The values of Kp, Ki and Kd
used are 1664200, 1248150 and 416050 respectively.
Fig.12a: PID, TUNNED PID, GA-PID Controllers Simulink
Model
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Vol. 3 Issue 3, March - 2017
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