Optimization of automobile active suspension system using minimal order

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 12, No. 3, June 2022, pp. 2378~2392
ISSN: 2088-8708, DOI: 10.11591/ijece.v12i3.pp2378-2392  2378
Journal homepage: http://ijece.iaescore.com
Optimization of automobile active suspension system using
minimal order
Sairoel Amertet Finecomes1
, Fisseha L. Gebre2
, Abush M. Mesene3
, Solomon Abebaw4
1
Department of Mechanical Engineering, Mizan Tepi University, Tepi, Ethiopia
2
Department of Mechanical Engineering, Defence Engineering Collage, Bishoftu, Ethiopia
3
Department of Mechanical Engineering, University of Gonder, Ethiopia
4
Department of Statistics, Mizan Tepi University, Tepi, Ethiopia
Article Info ABSTRACT
Article history:
Received Sep 23, 2019
Revised Nov 8, 2021
Accepted Nov 30, 2021
This paper presents an analysis and design of linear quadratic regulator for
reduced order full car suspension model incorporating the dynamics of the
actuator to improve system performance, aims at benefiting: Ride comfort,
long life of vehicle, and stability of vehicle. Vehicle’s road holding or
handling and braking for good active safety and driving pleasure and
keeping vehicle occupants comfortable and reasonably well isolated from
road noise, bumps, and vibrations are become a key research area conducted
by many researchers around the globe. Different researchers were tested
effectiveness of different controllers for different vehicle model without
considering the actuator dynamics. In this paper full vehicle model was
reduced to a minimal order using minimal realization technique. The entire
system responses were simulated in MATLAB/Simulink environment. The
effectiveness of linear quadratic regulator controller was compared for the
system model with and without actuator dynamics for different road profiles.
The simulation results were indicated that percentage reduction in the peak
value of vertical and horizontal velocity for the linear quadratic regulator
with actuator dynamics relative to linear quadratic regulator without actuator
dynamics was 28.57%. Overall simulation results were demonstrated that
proposed control scheme has able to improve the effectiveness of the car
model for both ride comfort and stability.
Keywords:
Linear actuator
Linear quadratic regulator
control
Minimal realization technique
active suspension system linear
DC motor
This is an open access article under the CC BY-SA license.
Corresponding Author:
Sairoel Amertet Finecomes
Department of Mechanical Engineering, Mizan Tepi University
Tepi, Region of Southern Nations Nationalities and peoples, Ethiopia
Email: sairoel@mtu.edu.et
1. INTRODUCTION
The suspension system is a mechanism that physically separates the car body from the car wheels,
and a complex vibration system having multiple degrees of freedom. It is the most important part of the
vehicle which heavily affects the ride quality and used to isolate the vehicle structure from shocks and
vibration due to irregularities of the road surface [1]. The main purpose of vehicle suspension system is to
minimize the vertical displacement, velocity or acceleration transmitted to the passenger which directly
provides ride comfort [2], [3]. Usually vehicle model could be quarter car, half car and full car models based
on types of suspension system and could be passive, semi active, and active suspension system based on
energy consumptions. In passive suspension system, the good ride quality is mainly achieved by an
appropriate choice of the springs and dampers. The parameters are generally fixed, with values compromised
to achieve a certain level of performance of the suspension system. Since the appropriate choice is depend on

Int J Elec & Comp Eng ISSN: 2088-8708 
Optimization of automobile active suspension system using minimal order (Sairoel Amertet Finecomes)
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the road surface, it has a limitation to satisfy to different types of road irregularities [4], [5]. Therefore, a
means of controlling to different types of road surface automatically should be researched. In contrast to
passive systems, active suspension systems can adjust their dynamic characteristics in response to varying
road conditions in real time [6], [7]. In order to design linear quadratic regulator controller, all states should
be measurable and are sensed by its own individual sensor. Fourteen sensors are used to get information from
each state which is bulky and requires high cost [8], [9]. Thus, a means of technique to reduce the number of
states and sensors to a minimal number without affecting the behavior of the original model system is
investigated. To have a good control performance, a good modeling of the system is necessary. As improving
the model from quarter car to half car model, from half car to full car model and including the sensors and
actuator dynamics in the system model, it will be more accurate [10]–[12]. Most of the researchers were
proven the effectiveness of different controllers for different car suspension system model without
considering the actuator dynamics, without considering the state reachable, and without considering the state
measureable in which the design of state feedback is impossible [13]–[18]. However, in this research paper, it
is proposed to design linear quadratic regulator (LQR) for active suspension system of automobile vehicles
based on active suspension system of automobile vehicles using reduced order car model by combing a linear
direct control motor actuator to reduce the vehicle body vibration and to settle within short period of time so
that the passengers feel comfort for more specific and concern [19]. At the end of this paper, the controller
performance will be evaluated under full car suspension system reduced order for different road profiles
conditions. The controller’s dynamic study is performed by evaluating the transient response during the
magnitude variations of road profile reference.
The remaining part of this paper is organized: section 2 describes mathematical model of full car
model with governing equation. The design of linear quadratic regulator controller and system analysis
carried out in section 3. Section 4 presents the results of simulation carried out for different road proﬁle.
Conclusion of the work will present in section 5.
2. MATHEMATICAL MODELING OF SUSPENSION SYSTEM
The vehicle active suspension system model utilized for numerical simulation was developed in the
MATLAB/Simulink environment and the presenters of the vehicle active suspension system are extracted.
The model, as shown in Figure 1 indicates that block diagrams of active suspension system, and its control
mechanism, Figure 2 depicted the schematic diagram of the active suspension system of automobile vehicle,
and Figure 3 utilizes three degree of freedoms for the motion of the car body in the space (pitch, roll, and
bounce), and four degree of freedom for wheel displacement (relative to the vehicle chassis) and rotations.
The mathematical models were developed based on some assumptions (the model considers only linearity’s
region, external factor (aerodynamics resistance) were not consider, and non-linearity properties of tires,
actuators were not considered).
Figure 1. Block diagram representation of the active suspension system [10]
2.1. Basic structure of active suspension system
Active suspension systems are mechatronics systems that control the vertical movement of the
vehicle body or chassis relative to the wheels. The active suspension systems considered in this paper is

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Int J Elec & Comp Eng, Vol. 12, No. 3, June 2022: 2378-2392
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applied for enhancing the ride comfort of automobile vehicles. Therefore, they are placed in between the
chassis and the wheels through attachment of their two ends to the car body and the wheels. In active
suspension system the vehicle’s up and down motion is controlled by linear quadratic regulator feedback
controller according to the road conditions; the controller output controls the actuator to compensate the road
oscillations and increases the vehicle stability as well as ride comfort. As a result of the road disturbance, the
vehicle body has been oscillated for some time. The sensors measure the amplitude of the vibration from the
equilibrium position. The linear quadratic regulator controller processed the electrical signal information
obtained from the sensors and it provides a control signal which controls the action of the actuator for fine
response in real time. The passive elements and linear direct control motor actuator generate forces which
counteract vertical, pitch and roll motions. Gradually, the purpose of the active suspension system is to
replace the classical passive elements by a controlled system, which can supply a regulated force to the
system. The active suspension system dynamically responds to the changing road surface due to its ability to
supply energy, which is used to achieve the relative motion between the body and wheel [9]. As shown in
Figure 2 the main purpose of suspension is supporting both roads holding and ride quality. Moreover,
suspension system affects on the vehicle handling too. Furthermore, it is very important to keep the road
wheel in contact with the road surface, for the suspension. There are different ways of attaching the wheels of
the car so that they can move up and down on their springs and dampers. The design of front suspension and
design rear suspension have some differences to the ability of opposite wheels to move independently of each
other. For front-wheel drive cars, rear suspension has few constraints and a variety of beam axles and
independent suspensions are used. For rear-wheel drive cars, rear suspension has many constraints and the
development of the superior but more expensive independent suspension layout has been difficult. Four-
wheel drive cars often have suspensions that are similar for both the front and rear wheels.
Figure 2. Schematic diagram of the active suspension system of automobile vehicle [10]
2.2. Mathematical models
Various types of car models such as quarter car model, half car model and full car model have been
used to simulate the performance of suspension systems. In the research studies the quarter car model is
frequently used because of its simplicity, however, the half car model shows more appropriate vertical
motion, including either the pitch or the roll effects. The full car model is the best accurate one, but it
requires more computation than the others as a result very few studies have been carried out based on it [20].
Lot of common vehicles today uses passive suspension system to control the dynamics of a vehicle’s vertical
motion as well as spinning (pitch) and tilting (roll) [21]. The design of a vehicle suspension is an issue that
needs a series of mathematical calculations. To study the vibrational characteristics of the vehicle and to
design the controller appropriately, a mathematical model of a dynamic system is defined as a set of various
equations that represents dynamics of the system accurately or at least, well. By applying Newton’s second
law motion and using the static equilibrium position as the origin, for the linear vertical displacement, pitch
angular displacement, and roll angular displacement of the vehicle body Y0, θ, and ϕ respectively from the
center of gravity the equations of motion for the system can be formulated. Once a mathematical model of
the system is obtained, various analytical and computer tool, MATLAB, can be used for design of linear
quadratic regulator controller and analysis.
2.3. Equations of motion
The free body diagram of the passive and active suspension systems is shown in Figure 3. While
modeling the system components the spring and the damper are assumed linear, i.e. the model is based on
elements of linear dynamic systems theory. Therefore, the overall equations are linear.

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Figure 3. Full vehicle model [10]
For rolling motion of the sprung mass system
𝐼𝑠𝑥𝜑𝑠𝑥 = −𝑏𝑓𝑇𝑓(𝑧𝑠1
̇ − 𝑧𝑢1
̇ ) + 𝑏𝑓𝑇𝑓(𝑧𝑠2
̇ − 𝑧𝑢2
̇ )
−𝑏𝑟𝑇𝑟(𝑧𝑠3
̇ − 𝑧𝑢3
̇ ) + 𝑏𝑟𝑇𝑟(𝑧𝑠4
̇ − 𝑧𝑢4
̇ )
−𝑘𝑓𝑇𝑓(𝑧𝑠1
̇ − 𝑧𝑢1
̇ ) + 𝑘𝑓𝑇𝑓(𝑧𝑠2
̇ − 𝑧𝑢2
̇ )
−𝑘𝑟𝑇𝑟(𝑧𝑠3
̇ − 𝑧𝑢3
̇ ) + 𝑘𝑟𝑇𝑟(𝑧𝑠4
̇ − 𝑧𝑢4
̇ ) (1)
For pitching motion of the sprung mass system
𝐼𝑠𝑧𝜑𝑠𝑧 = −𝑏𝑓𝑎(𝑧𝑠1
̇ − 𝑧𝑢1
̇ ) + 𝑏𝑓𝑎(𝑧𝑠2
̇ − 𝑧𝑢2
̇ )
−𝑏𝑟𝑏(𝑧𝑠3
̇ − 𝑧𝑢3
̇ ) + 𝑏𝑟𝑏(𝑧𝑠4
̇ − 𝑧𝑢4
̇ )
−𝑘𝑓𝑎(𝑧𝑠1
̇ − 𝑧𝑢1
̇ ) + 𝑘𝑓𝑎(𝑧𝑠2
̇ − 𝑧𝑢2
̇ )
−𝑘𝑟𝑏(𝑧𝑠3
̇ − 𝑧𝑢3
̇ ) + 𝑘𝑟𝑏(𝑧𝑠4
̇ − 𝑧𝑢4
̇ ) (2)
For bouncing motion of the sprung mass system
𝑀𝑠𝑍𝑠 = −𝑏𝑓(𝑧𝑠1
̇ − 𝑧𝑢1
̇ ) + 𝑏𝑓(𝑧𝑠2
̇ − 𝑧𝑢2
̇ )
−𝑏𝑟(𝑧𝑠3
̇ − 𝑧𝑢3
̇ ) + 𝑏𝑟(𝑧𝑠4
̇ − 𝑧𝑢4
̇ )
−𝑘𝑓(𝑧𝑠1
̇ − 𝑧𝑢1
̇ ) + 𝑘𝑓(𝑧𝑠2
̇ − 𝑧𝑢2
̇ )
−𝑘𝑟(𝑧𝑠3
̇ − 𝑧𝑢3
̇ ) + 𝑘𝑟(𝑧𝑠4
̇ − 𝑧𝑢4
̇ ) (3)
For each side of the wheel motion (vertical direction)
𝑀𝑢𝑓𝑧
••
𝑢1 = 𝑏𝑟(𝑧
•
𝑠1 − 𝑧
•
𝑢1) + 𝑘𝑓(𝑧𝑠1 − 𝑧𝑢1) − 𝑘𝑡𝑓𝑧𝑢1 + 𝑘𝑡𝑓𝑧𝑟1 (4)
𝑀𝑢𝑓𝑧
••
•
𝑠2 − 𝑧
•
𝑀𝑢𝑓𝑧
••
•
𝑠3 − 𝑧
•
𝑀𝑢𝑓𝑧
••
•
𝑠4 − 𝑧
•
where
𝑧𝑠1 = 𝑇𝑓𝜑𝑠 + 𝑎𝜃𝑠 + 𝑧𝑠, 𝑥1
̇ = 𝜑𝑠
̇ ≈ 𝑥8
𝑧𝑠1
̇ = 𝑇𝑓𝜑𝑠
̇ + 𝑎𝜃𝑠
̇ + 𝑧𝑠
̇ , 𝑥2
̇ = 𝜃𝑠
̇ ≈ 𝑥9
𝑧𝑠2 = 𝑇𝑓𝜑𝑠 + 𝑎𝜃𝑠 + 𝑧𝑠, 𝑥3
̇ = 𝑧𝑠
̇ ≈ 𝑥10
𝑧𝑠2
̇ = 𝑇𝑓𝜑𝑠
̇ + 𝑎𝜃𝑠
̇ + 𝑧𝑠
̇ , 𝑥4
̇ = 𝑧𝑢1
̇ ≈ 𝑥 11
𝑧𝑠3 = −𝑇𝑓𝜑𝑠 + 𝑎𝜃𝑠 + 𝑧𝑠, 𝑥5
̇ = 𝑧𝑢2
̇ ≈ 𝑥12
𝑧𝑠3
̇ = −𝑇𝑓𝜑𝑠
̇ + 𝑎𝜃𝑠
̇ + 𝑧𝑠
̇ , 𝑥6
̇ = 𝑧𝑢3
̇ ≈ 𝑥 13
𝑧𝑠4 = −𝑇𝑓𝜑𝑠 + 𝑎𝜃𝑠 + 𝑧𝑠, 𝑥7
̇ = 𝑧𝑢4
̇ ≈ 𝑥14
𝑧𝑠4
̇ = −𝑇𝑓𝜑𝑠
̇ + 𝑎𝜃𝑠
̇ + 𝑧𝑠
̇ (8)

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The values of are obtained by back substitution into the equation from the (1) to (8). After obtained the state
equation, we put into state matrix:
𝑥̇ = |
𝐴11 𝐴12
𝐴21 𝐴21
|𝑥(𝑡) + 𝐵𝑢(𝑡) + 𝐹(𝑡)
where
𝐴 = |
𝐴11 𝐴12
𝐴21 𝐴21
|𝑥(𝑡) = [14 × 1],𝑢(𝑡) = [4 × 1],
𝐵 = [14 × 1], 𝐹 = [14 × 4],𝐶 = [𝑒𝑦𝑒2 × 14], 𝐷 = [𝑧𝑒𝑟𝑜𝑠2 × 4]
3. SYSTEMANALYSIS AND CONTROLLER DESIGN
The active suspension system model has been developed in section two. As derived and stated in
section two, the model has fourteen (14) state variables. The state matrix (A) of the state space representation
of the system is fourteen by fourteen (14×14). Therefore, it needs 14 sensors to measure each state which is
not feasible practically. This is costly, so the system should be described with a minimal number of states and
this is achieved using a minimal realization principle. In addition, it is difficult to manipulate the system.
Therefore, to handle the system, the order of the system state space representation is reduced to eighth (8th)
by using minimal realization technique without affecting the characteristics of the original model system.
This is proved to be true by checking the response of the 14th order system and the 8th order system for the
same input disturbance of single bump as shown in the Figures 4(a) and 4(b). Due to the more state variables
are used to describe the system (redundancy of state variables), too much symmetry the system and the
system has physically uncontrollable components. It is difficult (not feasible) to implement the system
physically in real time application. Therefore, these problems are reduced using minimal realization
technique. A minimal realization principle is a means of describing the system with a small number of states.
Therefore, the system is implemented with a minimal number of components. Minimal realization technique
eliminates uncontrollable or unobservable state in state-space models, or cancels pole-zero pairs in transfer
functions. It describes the system with the minimum number of states. Thus, the obtained minimal realization
model has minimal order and the same response characteristics as the original model system. A minimal
realized system is both controllable and observable [22], [23]. When we split out the Figure 4(a) it looks as
displayed in the Figure 4(b). So, our conclusion was that there is no any deviation between the original
system and Minimal realization system. This implies that we can use the realization system instead of
original system.
As it can be observed in Figures 4(a) and 4(b) the simulation results of both the original model
system (14×14 matrixes) and minimal realized system (8×8 matrix) are the same. Therefore, this shows that
minimal realization technique doesn’t alter the behavior of the original model system. The six states
(velocities of the four wheels, and positions of coupling wheels) were removed, since they were not affect the
entire suspension system model due to velocities of the wheels are the same to the vehicle suspension system,
and the position of the coupling wheels (both front wheels, and rear wheels) are the same. The difference of
relative motion between them are zero. Therefore, the new state space model becomes as follow. A=[8×8],
B=[8×4], C=[2×8], D=[zeros(2)×8]. In order to design a linear quadratic regulator controller, the system must
be fully controllable. This is verified by determining the rank of the controllability matrix (A, B). Therefore,
the controllability matrix (A, B) of the active suspension system as shown in the appendix B has full rank (8),
which makes it fully controllable. It is also known that if the system is minimal realized, then it is fully
accessible and observable. Stability is an important property that a system is required to have. It is usually not
desirable that a small change in the input, initial condition, or parameters of the system produces a very large
change in the response of the system. If the response increases indefinitely with time, the system is said to be
unstable. The open-loop response of the system to a road profile 1 can be used to verify stability of the
system.
From the Figure 5(a) it can be observed that the response of the suspension system without feedback
controller is bounded. This shows that the system is stable because, for the bounded input the system is
producing bounded output. Furthermore, Figure 5(a) displayed that open-loop body displacement of the
suspension system model for road single bump road profile Figure 5(b), and open-loop body, pitch, roll, and
yaw displacement of the suspension system model for road single bump road profile. As it is observed in
Figures 5(a) and 5(b), the suspension system without active controller is stable, however, it needs an
improvement to be more comfort for passengers. Therefore, in order to enhance the performance a suitable

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controller must be designed. The controller generates an appropriate control signal to maintain the car body
at the desired equilibrium position in response to road disturbances. The controller designed will help the
system to be insensitive to the road disturbances. Therefore, the controller will try to make the system stable
and perform well regardless of the disturbance. The linear quadratic regulator is the extension of pole
placement technique that tends to find the control input so as to place the poles of the system at a desired
optimal position. The main idea in linear quadratic regulator control design is to minimize the quadratic cost
function of J given in (9) [24], [25].
𝐽 =
1
2
∫ [𝑥𝑇
(𝑡)𝑄(𝑡)𝑥(𝑡) + 𝑢𝑇
(𝑡)𝑅(𝑡)𝑢(𝑡)]
𝑡𝑓
𝑡0
𝑑𝑡 (9)
The LQR should minimize this cost function (performance index) while obtaining the state feedback
gains K that drives the system to the desired operating point. It turns out that regardless of the values of Q
and R, the cost function has a unique minimum that can be obtained by solving the following Algebraic
Riccati equation [25].
𝐴𝑇
𝑃 + 𝑃𝐴 − 𝑃𝐵𝑅−1
𝐵𝑇
𝑃 + 𝑄 = 0 (10)
By solving the above Riccati equation the positive-definite matrix P is obtained, thus the optimal gain (K)
and controller (u) are determined as (11):
𝑘 = 𝑅−1
𝐵𝑇
𝑃
𝑢(𝑡) = −𝑘𝑥(𝑡) (11)
(a)
(b)
Figure 4. Comparing the original and reduced automobile suspension system (a) comparing the original and
minimal realization system and (b) comparison the response of original and minimal realized systems

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(a)
(b)
Figure 5. Observing the effects of (a) open-loop body displacement of the suspension system model for road
profile 1 and (b) open-loop pitch angle response of the suspension system model for road profile 1
From Figure 5, we can observe that the effects of Figure 5(a) open-loop body displacement of the
suspension system model for road profile, and Figure 5(b) open-loop pitch angle response of the suspension
system model for road profile. The control signal given by (11) is the optimal control law. Therefore, if the
matrix K are determined so as to minimize the performance index, J, then (11) is optimal for any initial state
(0). The block diagram of the optimal LQR controller is displayed in the Figure 6.
Figure 6. Block diagram of LQR control scheme
As seen in the cost function, J, given in (8) in addition to the states x(t) and control signals u(t)there
are the weighting matrices Q and R. The parameters Q and R can be used as design parameters to penalize
the state variables and the control signals. The larger these values are, the more you penalize these signals.
The parameter Q penalizes the deviation of system states from the equilibrium. Basically, selecting a large
value for Q means you try to stabilize the system with the least possible changes in the states and large Q
implies less concern about the changes in the states. The parameter R penalizes the use of input control

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signal. If you choose a large value for R means you try to stabilize the system with less (weighted) energy.
This is usually called expensive control strategy. On the other hand, choosing a small value for R means you
don’t want to penalize the control signal (cheap control strategy). Q and R are weighting matrices and should
be positive semi-definite and positive definite, respectively. They are also symmetric matrices Q=QT; R=RT.
Normally, the Q and R matrices are chosen as diagonal matrices such that the quadratic performance index is
a weighted integral of squared error. The sizes of Q and R matrices depend on the number of state variables
and input variables respectively. If the system A, B is controllable, then we can place the Eigen values of the
closed loop system anywhere we want. That is extremely powerful, but in practice it is sometimes not
usefulness. Therefore, there is problem with placing system Eigen values. The main problem is we do not
have a great sense of the input that is required to accomplish the Eigen values of the closed loop system to
place anywhere. For example, if you try to make your system super-fast (i.e. placing of poles far way to the
left of the s-plane) it can require huge input, which the real physical system could not achieve. So just placing
system Eigen values from this fundamental reason to the appropriate position is required. Since the controller
is a regulator, it tries to drive each state to a constant set point. In this thesis, the controller drives each state
to zero. So, any value bigger than zero weather it is positive or negative is bad in this case. Therefore, the
controller drives each state (t) to zero as time tends to infinity. By choosing the value of Q and R we can
change the relative weightings of one state versus another. Since the number of state variables are eight, the
value of Q can be represented by the following eight by eight (8×8) matrix Q=10000 ∗[eye (8×8)]; R=0:001
∗[eye (8×8)]. All the diagonal elements penalize their correspondence state individually. The off diagonal
elements penalize combination of the states. Therefore, since the outputs of the system are combination of
these individual states, by penalize each individual state independently using its respective Q value and
observing the combination effect on the outputs the required performance can be achieved. In fact, penalize
one state has an effect on another but it is small. So, the value of Q is selected carefully and systematically.
The Figure 7(a) demonstrated the responses of four tires for each state variable for the Q and R values,
whereas, the Figure 7(b) showed the responses of roll, pitch, and yaw for each state variable for the Q and R
values respectively.
(a)
(b)
Figure 7. Response of each state variable for the Q and R values, (a) responses of four tires for each
state variable for the Q and R values and (b) responses of roll, pitch, and yaw for each state variable for the Q
and R values

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4. SIMULATION RESULTS AND DISCUSSION
This section discusses about simulation results of passive and active suspension systems for
different scenarios. Performance of the suspension system in terms of ride quality will be observed, where car
body displacement, velocity, pitch angle and angular velocity are considered as output parameters. Using the
vehicle data’s, the exact physical characteristics of that car can be determined by simulating the mathematical
model with the help of MATLAB software. In order to show the performance of the LQR with actuator
dynamics with tuned values Q and R matrices, different bump disturbances are applied to the system to
observe for different scenarios. In order to study the dynamic behavior of the vehicle and to analysis the
performance of the suspension system an external excitation input for the model is required. In this study, a
different type of sinusoidal function road profiles is used as excitation for simulation purpose. The road bump
profile in Figure 8 is appearing for1:5≤t≤1:75 sec for front right and left wheels and 2:06≤t≤2:31 sec for rear
right and left wheels. The width of each bump ‘(t) 0 in this case 0:25 sec indicates the duration of the road
bump at each wheel. From Figure 9, it can be seen that the peak value response of chassis displacement for
passive suspension system is0:06 m: It can be also observed that the peak value response of the chassis
displacement for active suspension excluding actuator dynamics in the system model is 0:055 m while that
for the active suspension with actuator dynamics included in the system model is 0:045 m for the same road
input and the same controller gains. Furthermore, the blue curve indicates the linear quadratic regulator with
actuator dynamics, the rose color showed the passive suspension system, and the black doted curve tells the
linear quadratic regulator without actuator dynamics at single bump road profile. The reduction
(improvement) in percentage for the displacement of the chassis can be calculated as (12), (13):
𝑟𝑝 = (
𝑝𝑣−𝑎𝑣
𝑝𝑣
) ∗ 100% (12)
𝑟𝑒 = (
𝑒𝑣−𝑎𝑣
𝑒𝑣
) ∗ 100% (13)
where rp=reduction in peak value from passive pv=passive value av=active (LQR) with actuator dynamics
value re=reduction in peak value from active excluding actuator dynamics ev=active (LQR) excluding
actuator dynamics value.
𝑟𝑝 = (
0.06−0.045
0.06
) ∗ 100% = 25%; 𝑟𝑒 = (
0.055−0.045
0.055
) ∗ 100% = 18.18%
Figure 8. Road input disturbance of a single bump
Thus, the chassis displacement (peak value) is reduced by 25% and 18:18% in case of an active
suspensionsystem with actuator dynamics which is included in the system model. This is a direct indication
of the superiority of active suspension system using LQR with actuator dynamics over passive suspension
system andactive suspension system without actuator dynamics. The settling time, as we can observe from
Figure 9 is 4:95 sec, 3:45 sec and 1:9 secfor passive suspension, active suspension excluding actuator
dynamics and active suspension including actuator dynamics respectively. Thus, reductions (improvements)
in settling time in activesuspension including actuator dynamics in the system model are 61:61% and 44:93%
as compared to passivesuspension and LQR excluding actuator dynamics respectively. Moreover, the Table 1
comparison of passive suspension system, linear quadratic regulator without actuator dynamics, and linear
quadratic regulator with actuator dynamics fordisplacement.

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Figure 9. Simulation result of comparison for body displacement using a single bump
Table 1. Comparison of PSS, LQR without actuator dynamics, and LQR with actuator dynamics
fordisplacement
PS PSS (rpm) LQRA LQRAO % LQRA over PSS % LQRA over LQRAO
PPR(m) 0.06 0.045 0.055 25% 18.18%
Ts(sec) 4.95 1.9 3.45 61.61% 44.93%
VPR(m/s) 0.65 0.4 0.56 38.46% 28.57%
Ts(sec) 4.95 1.7 3.45 65.66% 50.72%
OPR(rad) 0.0388 0.0263 0.0338 32.22% 22.19%
Ts(sec) 3.95 1.57 2.45 62.25% 35.92%
AVPR(rad/s) 0.3375 0.2625 0.2875 22.22% 8.7%
Ts(sec) 3.95 1.57 2.45 60.25% 35.95%
Where PS=performance specifications, PSS=passive suspension system, LQRA=LQR with actuator
dynamics, LQRAO=LQR without actuator dynamics, PPR=position peak response, (amplitude)(m)=Ts(sec)
settling time(sec), VPR(m=sec)=velocity peak response (amplitude)(m=s); OPR(rad) orientation peak
response (amplitude)(rad), AVPR(rad/sec)=angular velocity peak response (amplitude)(rad/s). As observing
from the simulation result of Figure 10, the peak overshoot of sprung mass velocity for passive suspension
system is 0.65 m/s. For the LQR without actuator dynamics and LQR with actuator dynamics are 0.56 m/s
and 0.4 m/s respectively. From these values, it is found that for active suspension system (LQR) with actuator
dynamics the peak value of the velocity of the sprung mass is reduced by 38:46% as compared to passive
suspension system. As compared to active suspension system (LQR) without actuator dynamics the reduction
is 28:57%. The passive suspension system and LQR without actuator dynamics have the same settling time
as of the displacement whereas the settling time for the LQR with actuator dynamics is 1:7 sec. Therefore,
the reduction settling time in active controller (LQR) with actuator dynamics which is included in the system
model is 65:66% as compared with passive system while compared with LQR without actuator dynamics is
50:72%. Furthermore, the blue curve indicates the linear quadratic regulator with actuator dynamics, the rose
color showed the passive suspension system, and the black doted curve tells the linear quadratic regulator
without actuator dynamics at single bump road profile.
Figure 10. Simulation result of comparison for body velocity using a single bump

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Similarly, in Figure 11 we can observe that, the peak values of the pitch angle for sprung mass of
passive suspension, LQR without actuator dynamics and LQR with actuator dynamics are 0.0388 rad,
0.0338 rad and 0.0263 rad respectively. 32.22% and 22.19% are the peak value reductions in LQR with
actuator dynamics as compared to passive suspension system and LQR without actuator dynamics
respectively. From the Figure 11, the settling time for the passive suspension system is 3.95 sec. For the LQR
without actuator dynamics the settling time is 2.45 sec. and for the LQR with actuator dynamics, it is
1.57 sec. Thus, the settling time reductions (improvements) are 60:25% and 35:92% for LQR with actuator
dynamics as compared to the passive and LQR without actuator dynamics respectively. From the Figure 11,
the settling time for the passive suspension system is 3:95 sec. For the LQR without actuator dynamics the
settling time is 2:45 sec and for the LQR with actuator dynamics, it is 1:57 sec: Thus, the settling time
reductions (improvements) are 60:25% and 35:92% for LQR with actuator dynamics as compared to the
passive and LQR without actuator dynamics respectively. The Figure 12(a) showed that the force generated
two wheels from front tires (actuators) in active suspension system at single road profile (bump). Further,
Figure 12(b) demonstrated that the force generated two wheels from front tires (actuators) in active
suspension system at single road profile (bump). Moreover, all the response of the tires is settles about
3.5 second; this showed that the proposed control algorithm is best fitting on the active suspension system.
The time interval 1.7 second to 3.5 second tell us the four tires, and roll, pitch, yaw is in the condition of
unstable. With sometimes later at 3.5 second all the active suspension systems are well settle.
Figure 11. Simulation result of comparison for pitch angular velocity using upward single bump
Figure 13 displayed that the different road profile applied on the front and rear wheels respectively.
The top most indicates that the road profile for the front wheels whereas the bottom most demonstrated that
the rear wheels. Furthermore, it means that Speed humps are in widespread use around the world. Despite
their effective performance in increasing safety, they cause considerable damage to vehicles and discomfort
to drivers and passengers. So, this road profiles are used as input for the system.
As it can be seen in Figures 14 for a two bumps input road profile with different amplitudes, the
amplitude and settling time of the results are different. For the high amplitude input road profile both the
amplitude and settling time of the simulation results are higher than as the input is the low amplitude road
profile which is expected. In addition to, the simulation results indicate that for existence of road disturbances
the vehicle body vibrates up and down from its equilibrium position which is zero whereas for the smooth
(absence) of road input disturbance the vehicle body will be remained in its equilibrium position.
Figure 15(a) displayed that the comparison of suspension system with actuator dynamics, without
actuator dynamics, and passive suspension system for angular position at double road bump profile, whereas
Figure 15(b) indicates that comparison of suspension system with actuator dynamics, without actuator
dynamics, and passive suspension system for angular velocity at double road bump profile. From the
Figures 15 we generalized as the proposed control is best fit for the regulation and optimization of the vehicle
suspension with actuator dynamics, and without actuator dynamics, compared to passive suspension system.
Furthermore, the proposed controller that is linear quadratic regulator is more effective on the suspension
system with actuator dynamics, as we compared to without actuator dynamics. From Figure 16 we infer those
effects of forces on actuators are demonstrated as Force generated from front actuators in active suspension
system using double bumps displayed in Figure 16(a) whereas Figure 16(b) indicated that Force generated
from rear actuators in ASS using double bumps.

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(a)
(b)
Figure 12. Effects of the forces on actuator (a) force generated from front actuators in active suspension
system using a single bump and (b) force generated from rear actuators in active suspension system using a
single bump
Figure 13. Road input disturbance of two road bumps
Figure 14. Comparison of the vehicle suspension system by considering actuator and without considering
actuator dynamics at double bump road profile

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2390
(a)
(b)
Figure 15. Effects of automobile vehicle suspension system on double bump road profile,
(a) simulation result of comparison for pitch angle using two bumps and (b) simulation result of comparison
for pitch angular velocity using two bumps
(a)
(b)
Figure 16. Effects of forces on actuator, (a) force generated from front actuators in ASS using double bumps
and (b) force generated from rear actuatorsin active suspension system using double bumps

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5. CONCLUSION
Active suspension system is one part of the essential mechatronic system of a vehicle system. In this
research paper, with some assumptions, the model of the active suspension system is developed and the state
feedback controller LQR is designed. By comparing the performance of the passive, LQR without actuator
dynamics and LQR with the actuator dynamics, the simulation results clearly indicate that LQR with actuator
dynamics can give lower amplitude and faster settling time. The reduced value of peak response will result in
less sprung-mass travel and hence, the reduced vibrations felt by the passenger. The less settling time will
quickly suspend the oscillations induced in the car body which will ensure better comfort to the passenger.
Therefore, the proposed LQR controller with actuator dynamics which is included in the system model is
more effective in the vibration isolation of the car body than the passive suspension system and LQR without
actuator dynamics. So, the proposed LQR controller with the selected weighting matrices Q and R is
acceptable. The proposed LQR control with actuator dynamics that is included in the system model gives
25% and 18.18% reduction in the peak value of vertical displacement as compared to passive and LQR
controller without actuator dynamics respectively, for the same road input and the same controller gains, thus
improving passenger comfort. It is found that for LQR controller with actuator dynamics, the peak value of
the velocity of the sprung mass is reduced by 38.46% compared to passive suspension system while
compared to LQR without actuator dynamics the reduction is 28.57% which guarantee better ride comfort.
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BIOGRAPHIES OF AUTHORS
Sairoel Amertet Finecomes received a BSc. degree in Electromechanical
Engineering from Hawassa University, Institute of Technology, Hawassa, Ethiopia in 2016,
MSc. degree in Mechatronics Engineering from Addis Ababa Science and Technology
University, Addis Ababa, Ethiopia, in 2019. He is currently Lecturer at Mizan Tepi
University,Tepi, Ethiopia. His professional activities have been focused in Software
developing, emerging technology, Robotic, Autonomous Technology, Mechatronic
systems design, Instrumentation and control. He can be contacted at email:
sairoelamertet23@gmail.com.
Fisseha Legesse Gebre (PhD) Academic Rank: Assistant professor. He has
done his Ph.D degree from Indian Institute of Technology Bombay (IITB) 2018. He
completed his M.Tech from IIT Madras in 2005 and B.Tech from Defence Engineering
College, Ethiopia in 2001. His research interests include additive manufacturing,
functionally gradient objects, welding, robotics, CNC and automation. He can be contacted
at email: fissehal@gmail.com.
Abush M. Mesenewas born in Wolaita, Ethiopia in 1992 G.C. He received his
B.Sc. degrees from Gondar University Institute of Technology in 2016. In 2016, he joined
the School of Mechanical Engineering at Institute of Technology at Gondar, in Ethiopia, as
Assistant Lecturer. He spent the 2016-2017 academic year as a teaching undergraduate
students. He currently has active collaborations with research center at institute of
technology in Gondar. Currently, he has M.sc on mechatronics engineering and working as
service engineer at buhlergroup specifically in Mechatronics activities and Lecturer at
university of Gondar. His activities currently focus on model, control and design of
electromechanical equipments. He can be contacted at email: beyu1216@gmail.com. For
further information on his linkedin homepage: https://www.linkedin.com/in/abush-
mohammed-mesene-2034411a5.
Solomon Abebaw received BSc degree in Statistics from University of Gondar,
Ethiopia. He has received his MSc degree in Biostatistics in 2016 from Jimma University.
Now he is lecturer and Head of StatisticsDepartment at MizanTepi University, College of
Natural and Computational Science. His Professional and Research activities have been
focused on any Statistical Modelling and Data Analysis. He can be contacted at email:
solabew@gmail.com. Further info on his homepage: https://www.mtu.edu.et/about-us.

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