1

The dynamic model and control algorithms for the

Active Suspension System
Giacomo Corradini, Student ID number 236873

Abstract—The suspension system can be classified as a passive, both conditions of smoothness and stability when studying
semi-active, and active suspension, according to its ability to add control algorithms for active suspension.
or extract energy. Road surface roughness is the leading cause of The main content of this paper is to implement two dif-
vehicle oscillation and the suspension system is used to dampen
these oscillations. The objectives of the active suspension system ferent control systems, based on the dynamic equations of
are to improve the ride quality and handling performance within the quarter-car model. This paper also compares the active
a given suspension stroke limitation. This paper will analyse suspension performances, given by different control systems,
the aspects of passive and active suspensions for a quarter- with the passive suspension when the car hits a speed hump
car model and construct an active suspension control for a road profile.
quarter-car model subject to excitation from a road profile.
Algorithms like PID (proportional–integral–derivative) and LQR II. S USPENSION PERFORMANCE
(linear quadratic regulator) will fit the linear model.
The suspension system in a vehicle aims at fulfilling the
requirements of both comfort, for passengers, and road han-
I. I NTRODUCTION dling, for the driver. These two aspects, however, characterise
in a general way what the suspension system should provide

V EHICLE suspension main task is to separate passenger

and vehicular body interactions from oscillations gener-
ated by road abnormalities whilst still maintaining continuous
to the driver and the vehicle. Hence, they must be translated
into physical quantities that can be observed, to evaluate the
suspension performance. The most important parameters are:
wheel-road contact. Generally, traditional suspension consist- • Ride comfort: is the ability of the vehicle suspension
ing of springs and dampers is referred to as passive suspension, system of insulating passengers, and payloads, from
while if the suspension is externally controlled it is known as vibrations caused by the road profile roughness. It is
a semi-active or active suspension. A good suspension system directly correlated to the acceleration of the sprung mass:
should provide good vibration isolation, i.e. small acceleration Z̈s . To provide better comfort, vertical acceleration must
of the body mass, and a small suspension travel, which is the be minimised and oscillations must be highly attenuated.
maximal allowable relative displacement between the vehicle • Body motion: which is known as bounce, pitch and roll
body and various suspension components. of the sprung mass created primarily by cornering and
An active suspension system has the ability to store, dis- braking manoeuvres. It is directly correlated to the motion
sipate and introduce energy into the system. It may vary its of the sprung mass: Zs . To provide better performance
parameters depending on operating conditions. The passive the sprung mass should remain in the nominal position
suspension system is an open-loop control system. It is de- in order to avoid oscillations of the vertical force causing
signed to achieve certain conditions only. The characteristic oscillations of the longitudinal and lateral forces which
of passive suspension are fixed and cannot be adjusted by any means bad performance. This quantity is mainly impor-
mechanical part. The problem of passive suspension is that it tant in racing cars contrarily in passenger cars.
cannot be designed as heavily damped or too hard, otherwise, • Vehicle handling: can be interpreted as the capability
it will transfer a lot of road input or it could throw around of the vehicle to correctly respond to the behaviour that
the car due to the unevenness of the road. Additionally, if it the driver is trying to impose, also under critical driving
is lightly damped or too soft, the suspension will reduce the conditions. Thanks to its deflection, the suspension sys-
vehicle’s stability in turns, change lane or even swing the car. tem is capable of guaranteeing continuous wheel-ground
Therefore, the performance of the passive suspension depends contact, required to control the vehicle. Minimising the
on the road profile. displacement between sprung and unsprung mass, thus
An active suspension system improves vehicle ride comfort, imposing a more rigid behaviour for the suspension
generating impact force that is transmitted to the sprung and system, will prevent tyre-road contact loss and will
unsprung masses. If the value of this force is small, the provide increased road handling, improving safety. This
system’s response will not be good. This means that the car’s requirement can be related to the physical quantities of
vibration has yet to be improved. If the impact force is bigger, suspension travel defined as the difference between the
the ride comfort can be further improved. However, a more sprung mass motion and the unsprung motion: Zs − Zus
significant impact force will cause a change in the dynamic
load at the wheel. Once the value of the dynamic force at the III. P HYSICAL AND MATHEMATICAL MODELLING
wheel is reduced to zero, the wheel may be lifted off the road, A quarter-car dynamic model is commonly used in studies
and instability will occur. That’s why, it is difficult to satisfy of oscillation control for suspension systems. This model,
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Fig. 3. Active suspension control system

Fig. 1. Passive Suspension Quarter Car Model

0 1 0 −1
    
x˙1 x1
x˙2   −k s −bs
0 bs  x2 
  =  ms ms ms  +
x˙3   0 0 0 1  x3 
ks bs −kus −(bs +bus )
x˙4 m mus mus mus
x4
us
  (2)
0 0
 0 1  ˙
 
 ms  Zr
 −1 0  Fa
bus 1
mus − mus

To perform the simulations the dynamic system described in

(2) has been modelled using SIMULINK as shown in Figure 3,
where the inputs are the derivative of the road profile and the
actuator force, while the outputs are the state space variable
Fig. 2. Active Suspension Quarter car model and the sprung mass acceleration.

IV. C ONTROLLER DESIGN

Figure 1, includes only two masses: the sprung mass, ms , and
A. LQR
the unsprung mass, mus . The suspension system is modelled
as a spring and a shock absorber. The tires are also modelled The linear time-invariant system (LTI), is described by
similarly to the suspension system. equation (2). Consider a state variable feedback regulator:
For vehicles with an active suspension system, an actuator u = −Kx (3)
is located between the sprung mass and the unsprung mass,
Figure 2. where K is the state feedback gain matrix. The optimisation
The equations describing the dynamics of a quarter-car procedure consists of determining the control input u, which
model are shown as follows: minimises the performance index. The performance index J
represents the performance characteristic requirements, as well
as the controller input limitation. The optimal controller of the
ms Z¨s = bs Z˙us − bs Z˙s − ks (Zs − Zus ) + Fa given system is defined as controller design, which minimises
mus Z¨s = −bs Z˙us − bus Z˙s + bs Z˙s + bus Z˙r (1) the following performance index:
Z ∞
−ks (Zus − Zs ) − kus (Zus − Zr ) − Fa J= (x′ Qx + u′ Ru) dt (4)
0
In order to analyse the model, the state-space model The matrix Q ∈ R and R ∈ Rn , where m is the number
m
describing the active suspension system will be created using of states x and n is the number of control input u, are
the two equations of motion found in (1). The state variable weighting matrices, in particular, the matrix Q is used to
representing the system are: penalise bad performances, any non-zero state x adds a non-
negative amount to J, while the matrix R is used to penalise
x1 = Zs − Zus suspension travel the control input, any non-zero u adds a non-negative amount
x2 = Z˙s sprung mass velocity to J.
x3 = Zus − Zr wheel’s deflection The gain matrix K is represented by:
x4 = Z˙us wheel’s vertical velocity
K = R−1 B ′ P (5)
The matrix P must satisfy Riccati’s reduced matrix equation:
The state-space model can easily be written in the matrix
form shown below: A′ P + P A − P BR−1 B ′ P + Q = 0 (6)
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Fig. 4. LQR controller

Fig. 6. Speed hump profile

Parameters Value Description

ms 234 [kg] Total sprung mass
mus 43 [kg] Total unsprung
masses
ks 26000 × 4 [N/m] Parallel of 4 suspen-
Fig. 5. PID controller sion stiffness
bs 1544 × 4 [N sec/ m] Parallel of 4 suspen-
sion damping
Then the feedback regulator is: kus 100000 × 4 [N/m] Parallel of 4 tyre
stiffness
u = −(R−1 B ′ P )x (7) bus 0 × 4 [N sec/ m] Parallel of 4 tyre
damping
The SIMULINK model of the LQR controller described in which represent the parameters of E-AGLE Trento Racing
(7) is shown in Figure 4. Team’s vehicle, Fenice.
The response of the system is tested considering a speed
hump profile, which corresponds to Figure 9 present in the
B. PID paper [1]. The speed hump profile has been modelled using
the following equation:
The PID controller can be written in the following form:

0.04 × (1 − cos(7πt)) tstart ≤ t ≤ tstop
Z
d zr =
u(t) = Kp e(t) + Ki e(t) dt + Kd e(t) (8) 0 otherwise
dt
Where the variables tstart and tstop are defined depending
where: e(t) = r(t) − y(t) is the tracking error. on the vehicle velocity. In this paper three different simulations
have been done considering three different velocities:
The desired closed loop dynamics can be obtained by
adjusting the three parameters Kp , Ki and Kd , often iter- • 30 km/h → tstart = 1 [s] tstop = 1.09 [s]

atively with ”tuning” and without specific knowledge of a • 40 km/h → tstart = 1 [s] tstop = 1.045 [s]

plant model. Stability can often be obtained using only the • 72 km/h → tstart = 1 [s] tstop = 1.025 [s]

proportional term. The integral term permits the rejection of The velocities of 30 and 40 km/h simulate a passenger car
a step disturbance. Meanwhile, the derivative term provides when hitting a speed hump in the road, while the velocity of
damping or shaping of the response. 72 km/h simulates a formula-SAE car when hitting a curb in
The SIMULINK model of the PID controller described in the circuit.
(8) is shown in Figure 5. In Figure 6 the speed hump profile considering the vehicle
velocity of 72 km/h is represented.

V. S IMULATION RESULTS AND DISCUSSION B. Parameters tuning

The control variables used in the two controllers are
A. Parameters setting
different, for the PID controller only the suspension travel
The parameters considered in the simulations are: and the sprung mass acceleration are controlled, while for the
LQR controller, all the state space variables are controlled by
 4

the feedback regulator. D. Time domain analysis

Figure 15 illustrates the time response of the suspension
The tuning of the parameters has been done differently for travel, which is related to vehicle handling. Regarding the
the two controllers. PID controller, both the peak and the oscillations have been
reduced by only some percentage points, 2.63% and 1.62%
For the PID controller, the MATLAB function pidtune has respectively, and also the settling time has been reduced.
been used, which given the transfer function and the type Regarding the LQR controller, we had a worse performance
of controller (PID in this case) returns the values of the regarding the peak, it increases its magnitude when the
parameters (Kp , Ki and Kd ), then following a ”trial and suspension travel is negative, of about 21.32%, while the
error” procedure a series of simulations have been made in oscillations and the settling time have been reduced.
order to set the parameters which maximise the performance
parameters. The parameters obtained are described in the Figure 16 illustrates the time response of the sprung mass
following table. acceleration, which is related to ride comfort. Regarding the
PID controller, there is a reduction in the peak of about
Control variable Kp Ki Kd 2.26%, while there is a slight worsening of the oscillations of
zs − zus 3.1953e+05 5.2392e+03 3.8195e+06 about 1.82%. Regarding the LQR controller, both the peak
z¨s 160 1.2673e+04 0 and the oscillations have been strongly reduced by about
20.68% and 13.71 % respectively. For both active suspension
controllers, the settling time has been reduced.
Also for the LQR controller, a series of simulations have
been made in order to choose the proper weighting matrices
Figure 17 illustrates the time response of the sprung mass
Q and R, where Q is a diagonal positive definite and R is
motion, which is related to body motion. Regarding the
a positive constant, starting by setting the elements on the
PID controller, both the peak and the oscillations have been
diagonal of the matrix Q and the positive constant R to 1.
reduced by about 10.99% and 6.26 % respectively. Regarding
Using the MATLAB function lqr it calculates the gain matrix
the LQR controller, both the peak and the oscillations
K given the weighting matrices Q, R and the state space
have been strongly reduced by about 35.77% and 29.96%
matrices A and B. Then as done with the PID controller
respectively. For both active suspension controllers, the
following a ”trial and error” procedure the parameters have
settling time has been reduced.
been adjusted in order to minimise the quadratic cost function
J. The chosen parameters are:
  Parameters Max Passive Max LQR KP Irms KP Imax
1 × 1e10 0 0 0 z¨s 55.10 47.55 20.68% 13.71%
 0 1 × 1e8 0 0 zs 0.06 0.04 35.77% 29.96%
Q=  (9) zs − zus 0.04 0.047 1.31% -21.32%
 0 0 1 0
0 0 0 1 TABLE I
K EY PERFORMANCE INDICES (KPI) FOR LQR CONTROLLER
R=1 (10)

C. Simulations Parameters Max Passive Max PID KP Irms KP Imax

z¨s 55.10 56.10 2.26% -1.82%
The SIMULINK scheme used to perform the overall simu- zs 0.06 0.05 10.99% 6.26%
lation is reported in Appendix B. zs − zus 0.04 0.038 2.63% 1.62%
To evaluate the performance of the active suspension sys-
TABLE II
tem, described in Section II, with respect to the passive K EY PERFORMANCE INDICES (KPI) FOR PID CONTROLLER
ones two different performance indices have been defined,
one to evaluate the percentage reduction of the oscillations,
Equation 11, and one to evaluate the percentage reduction of E. Frequency domain analysis
the overshoot, Equation 12.
  To better evaluate the performances of the active suspension
|rms(Xact )| system a frequency domain analysis has been performed. In
KP Irms = 1 − % (11)
|rms(Xpass )| Figure 7, 8 and 9 the bode plot of the transfer functions for
  the suspension travel, sprung mass acceleration and sprung
max(|Xact |) mass motion with respect to the derivative of the road input
KP Imax = 1 − % (12)
max(|Xpass |) respectively are represented.
Where Xact and Xpass are the physical quantities of the The bode plot of the suspension travel (Figure 7) shows a
active and passive system respectively to evaluate. reduction in the peak for both the controller used. The LQR
Appendix A reports all the results obtained in the simulation controller additionally shows a worsening in the low-frequency
considering the vehicle velocity of 72 km/h, and all the range.
performance indices are reported in Table I for the LQR The bode plot of the mass acceleration (Figure 8) shows a
controller and in Table II for the PID controller. reduction in the peak for both the controller used, in particular
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Fig. 7. Bode diagram of suspension travel Fig. 10. Zero-pole map of suspension travel

Fig. 8. Bode diagram of sprung mass acceleration Fig. 11. Zero-pole map of sprung mass acceleration

for the LQR controller. The LQR controller additionally shows

a worsening in the high-frequency range. The PID controller F. Stability analysis
shows an overall improvement along all the frequency-domain
contrary to the LQR controller. The stability for a continuous time linear system is given if
all the poles of the transfer function have negative real part.
For the bode plot of the mass motion (Figure 9), hold
the same considerations done for the bode plot of the mass In Figure 10, 11 and 12 are reported the zero-pole map
acceleration. of the transfer functions related to the physical quantities
The bode plots of the three transfer functions don’t show a associated with the suspension performances.
strong reduction but an overall improvement, especially in the As we can see from the three zero-pole maps all three
peak, due to the fact that parameter tuning has been done in transfer functions have poles with negative real part, and thus
order to improve all the suspension performance. can be concluded that are stable.

Fig. 9. Bode diagram of sprung mass motion Fig. 12. Zero-pole map of sprung mass motion
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A PPENDIX A
S IMULATION P LOTS

Fig. 13. Active Suspension Half car model

Fig. 15. Suspension travel with bump road

Fig. 14. Active Suspension Full car model

VI. E XTENSION TO H ALF CAR AND FULL CAR MODEL

Up to now, we only considered the quarter-car vehicle

model (Figure 2). This simple model, however, does not fully
represent the rigid-body motions of a real car. To better study
the dynamics of the suspension system the model can be
expanded to the Half-dynamic model, or the Full-dynamic
model which takes into account both the roll and pitch motion
of the car.
The Half-Dynamic Model of a vehicle, described in Figure
13. It can be roll model or pitch model. This model considers
the influence of the vehicle’s body roll or pitch angles depend- Fig. 16. Sprung mass acceleration with bump road
ing on which half of the model is being studied. This model
split the unsprung masses into two masses and the suspension-
tyre system into two equivalents, given by the parallel of two
suspension-tyre systems.
The Full-Dynamic Model of a vehicle, described in Figure
14. Overall, this model is quite complex. However, this model
provides all the necessary elements to evaluate the vehicle’s
oscillation.

VII. C ONCLUSION

The simulation results show that an active suspension sys-

tem reduces the oscillations, as shown in Table I and II, giving
better performance than the passive suspension. The peak
performance instead depends on the type of controller used.
In conclusion, from the simulation results, active suspension
with an LQR or PID controller can be considered one of the
valid solutions. Fig. 17. Sprung mass motion with bump road
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A PPENDIX B
SIMULINK SCHEME

Quarter-Dynamic Model PID

Fa_PID

PID

zs-zus
input_force
zs_dd

D1
D1
zr_d
FIlter
zr
Road disturbances

Fa_LQR
Quarter-Dynamic Model LQR

Feedback gain
input_force State
input_force State

Fa_passive

Quarter-Dynamic Model passive

Fig. 18. SIMULINK scheme simulation

R EFERENCES
[1] A. Kanjanavapastit and A. Thitinaruemit, “Estimation of a speed hump
profile using quarter car model,” Procedia - Social and Behavioral
Sciences, vol. 88, pp. 265–273, 10 2013.