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Simulation of an Active Suspension
Using PID Control
Darlan Ferreira de Sousa
Suzana Moreira Avila
Faculdade UnB-Gama
Universidade de Brasilia - Brazil
Summary
Suspension
Objective
Mathematical Formulation
Numerical Results
Conclusion
Suspension
A vehicular suspension system has
the main purpose of adapting the
car's behavior based on two main
parameters, comfort and stability.
This performance is related to
features like stiffness and damping
coefficient of the suspension
components such as springs,
dampers and actuators.
Objective
This work studies the performance of an active suspension, with a quarter car
vehicle model using a PID controller.
Controllability and observability properties are analysed in a way to verify
good control performance.
The analysis is carried out using MATLAB and SIMULINK toolbox capabilities.
Mathematical Formulation Suspension systems can be modeled
approximately by spring-mass-dashpot
system of two degrees of freedom.
It is called a quarter car model, where
the sprung mass Ms is attached by the
suspension, modelled as a spring and a
damper, to the unsprung mass Mu.
The spring stiffness is given by ks and
the damping coefficient is bs.
When considering an active suspension,
the automatic actuator is connected
and is modelled by a control force u.
The tire is represented by a spring
with stiffness kt.
Quarter car model
Mathematical Formulation
The motion governing equation can be rewritten in a state space form:
Where A is the state space matrix, B is the input matrix, C is the output
matrix, D is the direct transmission matrix and U is the input of system, to
the quarter car system presented in matrices A and B
Mathematical Formulation
Automatic Control - PID control (proportional-integral-derivative) compares
the real value of the output quantity with the reference value (target value),
determines the deviation and produces a control signal which will reduce the
deviation to zero or a small value. The transfer function in this case is given
by:
𝑃(𝑠)
𝐸(𝑠)= 𝐾𝑝 1 +
1
𝑇𝑖𝑠+ 𝑇𝑑𝑠
Where Kp represents the proportional gain, Td represents the time derivative
and Ti the integral time. The first term of the transfer function corresponds
to proportional control, integral control to the second and so on
Mathematical Formulation
The open loop block diagram used to analyze the passive suspension:
Mathematical Formulation
In active suspension case the block diagram is a closed-loop diagram:
Numerical Results
Numerical Results
Controlability
The controllability matrix rank was 4 indicating that the system is controllable.
Numerical Results
Observability
In the case of observability property, three different cases were studied:
a) measuring only x1;
b) measuring only x2;
c) measuring both x1 and x2.
In all cases the rank of the observability matrix was 4 indicating that the system is
observable no matter measuring only one output or both of them.
Numerical Results – Passive Suspension
It can be observed an
overshoot of 53% and a
suspension time response of 3,1
s.
These values can be minimized
to improve passengers comfort
by the installation of an active
suspension.
Sprung mass displacement time history when
subjected to a step load profile
Numerical Results – Active Suspension
Sprung mass displacement time history – step road
profile.
Active suspension improve the
system behavior with an overshoot
of only 29%, reducing the maximum
displacement on 45,3%.
Suspension time response reduced
from 3,1 s to 2,3 s (25,8%).
-0,02
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0,18
0 1 2 3 4 5 6 7 8 9 10
Dis
pla
cam
ent
(mete
r)
Time (sec)
Step
Open Loop PID Tune
Numerical Results
Sprung mass displacement time history – harmonic
road profile
It can be observed that the
maximum displacement reduced
about 83,8% comparing active to
passive case.
And also on the steady state
response a good improvement on
performance is achieved.
-0,015
-0,01
-0,005
0
0,005
0,01
0,015
0 1 2 3 4 5 6 7 8 9 10
Dis
pla
cam
ent
(mete
r)
Time (sec)
Harmonic
Open Loop PID Tune
Numerical Results
Sprung mass displacement time history – White
noise road profile
it can be noticed that also in this
case active PID suspension achieves
a very good performance.
When comparing maximum sprung
mass displacement a 72,5%
reduction is reached.
-0,03
-0,02
-0,01
0
0,01
0,02
0,03
0 1 2 3 4 5 6 7 8 9 10
Dis
pla
cam
ent
(mete
r)
Time (sec)
White NoiseOpen Loop PID Tune
Conclusion
The system showed up to be controllable and observable.
Comparing the performance of an active suspension, designed with a PID
controller, with the passive one, it can be observed a considerable
improvement on efficiency for all the road profiles considered.
In the case of the harmonic road profile a reduction of 83,8% on the maximum
sprung mass displacement was found out.
Conclusion
Setting the PID parameters through Tune Simulink tool, is merely a basis for
designing the controller.
It is recommended verify what is the influence of the PID gains on the
behavior of the controller to make a more detailed analysis.
It has been found that the action of PID control showed satisfactory results
improving the performance of the suspension system, however active
suspensions still have a very high cost of manufacturing, installation and
maintenance compared to passive suspensions, that is the reason it is not
widespread in series productions of vehicles.
