Modeling and Simulation of Quarter Car Semi Active
Suspension System Using LQR Controller
K. Dhananjay Rao1 and Shambhu Kumar2
1
School of Engineering and Technology, Centurion University, Odisha, India
kdhananjayrao@gmail.com
2
Jadavpur University, Kolkata, India
kumarshambhu544@gmail.com
Abstract. In this paper design of the Linear Quadratic Regulator (LQR) for
Quarter car semi active suspension system has been done. Current automobile
suspension systems use passive components only by utilizing spring and damp-
ing coefficient with fixed rates. The vehicle suspension systems are typically
rated by its ability to provide good road handling and improve passenger com-
fort. In order to improve comfort and ride quality of a vehicle, four parameters
are needed to be acknowledged. Those four parameters are sprung mass accele-
ration, sprung mass displacement, unsprung displacement and suspension def-
lection. This paper uses a new approach in designing the suspension system
which is semi-active suspension. Here, the hydraulic damper is replaced by a
magneto-rheological damper and a controller is developed for controlling the
damping force of the suspension system. The semi-active suspension with con-
troller reduces the sprung mass acceleration and displacement hence improving
the passengers comfort.
Keywords: Linear Quadratic Regulator (LQR), Bryson’s Rule of Tuning, Quar-
ter car semi active suspension system.
1 Introduction
A vehicle suspension system performs two major tasks. It should isolate the vehicle
body from external road disturbances for the sake of passenger comfort and control
the vehicle body attitude and maintain a firm contact between the road and the tyre to
provide guidance along the track. A Basic automobile suspension that is known as a
passive suspension system consists of an energy storing element normally a spring
and an energy dissipating element normally a shock absorber [10].
The main weakness of the passive suspension is that it is unable to improve both
ride comfort and safety factor simultaneously. There is always a trade-off between
vehicle ride comfort and safety factor [2, 5, 9]. To improve the ride comfort, the safe-
ty factor must be sacrificed, and vice versa. One way to overcome such a problem, the
car suspension system must be controlled.
© Springer International Publishing Switzerland 2015 441
S.C. Satapathy et al. (eds.), Proc. of the 3rd Int. Conf. on Front. of Intell. Comput. (FICTA) 2014
– Vol. 1, Advances in Intelligent Systems and Computing 327, DOI: 10.1007/978-3-319-11933-5_48
442 K.D. Rao and S. Kumar
Thus to design and analyze the car suspension system controller, high fidelity ma-
thematical model for capturing the realistic dynamic of a car suspension system is
necessary [7, 8].
In this paper, a semi-active suspension system is proposed [1, 7]. The semi-active
suspension system is developed based on the passive suspension system. A variable
MR Damper is installed parallel with the passive suspension. This MR Damper is
controlled by LQR controller.
2 Quarter Car Semi Active Suspension System Modelling
The mathematical modelling of a two degree of freedom quarter car body for a semi-
active suspension system is being carried out by using basic laws of mechanics.
Modelling of suspension system has been taking into account the following
observations.
• The suspension system modelled here is considered two degree of freedom
system and assumed to be a linear or approximately linear system for a quar-
ter cars.
• Some minor forces (including backlash in vehicle body and movement, flex
in the various linkages, joints and gear system,) are neglected for reducing
the complexity of the system because effect of these forces is minimal due to
low intensity. Hence these left out for the system model.
• Tyre material has damping property as well as stiffness.
Fig. 1. Quarter car semi active suspension model
Modeling and Simulation of Quarter Car Semi Active Suspension System 443
Fig. 2. Free body Diagram
From Figure 2, we have the following equations,
0
(1)
(2)
Where,
Ms = mass of the wheel /unsprung mass (kg)
Mu = mass of the car body/sprung mass (kg)
r = road disturbance/road profile
Zr = wheel displacement (m)
Zs = car body displacement (m)
Ks = stiffness of car body spring (N/m)
Kt = stiffness of tire (N/m)
Cs = damper (Ns/m)
After choosing State variables as,
Where,
=Suspension Deflection
=Tyre Deflection
=Car body Velocity
=Wheel Velocity
From equation (1), we have
444 K.D. Rao and S. Kumar
From equation (2), we have
Disturbance caused by road roughness,
Therefore,
State space equation can be written as form,
0 0 1 1 0 0
0 0 0 1 0 1
= / 0 / + U+ W (3)
/ 1/ 0
/ / / / 1/ 0
Where,
0 0 1 1
0 0 0 1
A= / 0 / /
/ / / /
0 0
0 1
B= Bw=
1/ 0
1/ 0
1 0 0 0
0 1 0 0
C= , 0 0 0 0
0 0 1 0
0 0 0 1
Table 1. Parameters used in system simulation
S.N. Parameter Symbol Quatities
1 Mass of vehicle body Ms 504.5kg
2 Mass of the tyre and suspention Mu 62kg
3 Coefficient of suspension spring Ks 13100N/m
4 Coefficient of tyre material Kt 252000
N/m
5 Damping coefficient of the Cs 400
dampers N-s/m
Modeling and Simulation of Quarter Car Semi Active Suspension System 445
The parameter values are taken from [7] and are listed in Table 1.
3 LQR Controller Design
Consider a state variable feedback regulator for the system given as
K is the state feedback gain matrix.
The optimization procedure consists of determining the control input U, which mi-
nimizes the performance index J. J represents the controller input limitation as well as
the performance characteristic requirement. The optimal controller of given system is
defined as controller design which minimizes the following performance index.
1
2
The matrix gain K is represented by:
The matrix P must satisfy the reduced-matrix equation given as
0
Then the feedback regulator U
Fig.3 shows the block diagram using LQR controller,
x1, x2, x3, x4
Road profile
Quarter car semi active
Control output
(Step input)
suspensionsystem
Linear Quadratic
Regulator
Fig. 3. A schematic Diagram for LQR controller Design
The LQR controller has a function to adjust the damping coefficient of the variable
shock absorber in order to keep the car body always stable. Adjustable process is
based on the characteristic of the road surface.
446 K.D. Rao and S. Kumar
3.1 Bryson’s Rule for Tuning
The selection of Q and R determines the optimality in the optimal control law [3]. The
choice of these matrices depends only on the designer. Generally, preferred method
for determining the values for these matrices is the method of trial and error in simu-
lation. As a rule of thumb, Q and R matrices are chosen to be diagonal. In general, for
a small input, a large R matrix is needed. For a state to be small in magnitude, the
corresponding diagonal element should be large. Another correlation between the
matrices and output is that, for a fixed Q matrix, a decrease in R matrix’s values will
decrease the transition time and the overshoot but this action will increase the rise
time and the steady state error. In the other condition, where R is kept fixed but Q
decreases, the transition time and overshoot will increase, in contrast to this effect the
rise time and steady state error will decrease.
Here LQR control strategy is used for controller. Then the weighing matrices Q
and R have to be determined. When not knowing Q and R values, a rule of thumb,
Bryson’s rule, may be give them values according to following equations [3,4].
1
The maximum value of state is found by simulating with no input. R can initially
set to 1 and then tuned by finding maximum input when a controller is included in the
simulation.
Using this method matrices Q and R are obtained as follows:
0.000865 0 0 0
0 1.8114 0 0
,
0 0 0.011 31 0
0 0 0 65.03
1
However, by simulating with the gain obtained from this, results shows little im-
provement in damping. These weigh matrices are not so optimal; to get better result
we tune Q and R manually, and found that a dramatically different Q and R gave far
better result.
After tuning finally we choose Q and R values are as following:
0.000865 0 0 0
0 1.8114 0 0
0 0 0.01131 0
0 0 0 65.03
0.000009
Modeling and Simulation of Quarter Car Semi Active Suspension System 447
4 Simulation Results
2
Passive
1.5 Semi active
0.5
Bod y Position
-0.5
-1
-1.5
-2
0 5 10 15
Time (seconds)
Fig. 4. Time response of vehicle body position
-0.01 Passive
Semi Active
-0.02
Suspension Deflection
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
0 5 10 15
Time (seconds)
Fig. 5. Time response of vehicle suspension Deflection
-3
x 10
0
-0.5 Passive
Semi active
-1
Wheel Deflection
-1.5
-2
-2.5
-3
-3.5
-4
0 2 4 6 8 10 12 14 16
Time (seconds)
Fig. 6. Time response of vehicle wheel deflection
0.01
Passive
Semi active
0.005
W heel Position
-0.005
-0.01
-0.015
0 2 4 6 8 10 12 14 16 18
Time (seconds)
Fig. 7. Time response of vehicle wheel position
448 K.D. Rao and S. Kumar
5 Conclusion
Implementation of Linear Quadratic Regulator control strategy in linear system of
semi active suspension for a half car model is studied successfully. The designed
matrix for feedback gain is also presented. The crucial step is to vary the value of
matrix Q and matrix R. It is because there is effect at the transients output if matrix Q
too large and there also effect at the usage of control action if matrix R is too large.
Finally comparison between semi-active and passive suspension system is pre-
sented and their dynamic characteristics are also compared. It has been observed that
performances is improved in reference with the performance criteria like settling time
and Peak overshoot for body acceleration, wheel deflection, wheel position, suspen-
sion deflection and body position. This performance improvement in turn will in-
crease the passenger comfort level and ensure the stability of vehicle.
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