2023 European Control Conference (ECC)

June 13-16, 2023. Bucharest, Romania

Application of constrained H∞ control for a half-car model of an active

suspension system equipped with inerter
Keyvan Karim Afshar1 , Roman Korzeniowski2 , and Jarosław Konieczny3

Abstract— In this study, a multi-objective robust H2 /H∞ To avoid the consequences of vibration, many techniques
controller for a half-car model of an active inerter-based have been introduced, such as isolating systems from vibra-
suspension system in the presence of external disturbance has tion, controlling systems, redesigning systems to change their
been investigated. Its main goal is to improve the inherent
trade-offs between ride quality, handling performance, and natural frequencies, using tuned mass dampers/absorbers,
suspension travel and to guarantee the allowable level for and more [3], [5]. Tuned mass dampers (TMDs) are widely
suspension stroke and energy consumption. Inerters have been used to suppress unwanted vibrations of various mechani-
widely used to suppress undesirable vibrations of various types cal structures, e.g., buildings, bridges, motorcycle steering
of mechanical structures. The advantage of inerters is that the systems, vehicle and train suspensions, landing gear suspen-
realized equivalent mass ratio (inertance over primary structure
mass) is greater than its actual mass ratio, leading to higher sions, etc [3], [5]. The classical TMD consists of mass on a
performance for the same effective mass. First, the dynamics linear spring, and as it is known that the classical TMD is
and state space of the active inerter-based suspension system particularly efficient at reducing the response of the main
for a half-car model have been achieved. In order to attain structure in principal resonance, but at other frequencies
the defined objectives, and ensure that the closed-loop system (even near the resonance frequency) it enhances the ampli-
achieves the prescribed disturbance attenuation level, the Lya-
punov stability function, and linear matrix inequality (LMI) tude of the system’s motion [3], [5]. Therefore, we must
techniques have been utilized to satisfy the multi-objective always encounter with a tradeoff between damping vibra-
robust H2 /H∞ criterion. Furthermore, to limit the gain of the tions most effectively at a particular frequency or achieving
controller, some LMIs have been added. In the case of feasibility, tolerable damping characteristics across a wide range of vi-
sufficient LMI conditions by solving a convex optimization bration frequencies. This problem is capable to be minimized
problem afford the stabilizing gain of the constrained robust
state-feedback controller. Numerical simulations show that the by novel TMDs containing inerters or magnetorheological
active inerter-based suspension system performs much better dampers, which are intensively developed nowadays. Inerter
than a passive suspension with inerter and active suspension is a device with two free-moving terminals whose gener-
without inerter. ated force is proportional to the relative acceleration of its
terminals (the idealized model). The proportional constant
I. INTRODUCTION
is called the inertance with the unit kilogram [6]. The
The primary goal of vehicle suspension systems devel- inerter possesses the effect of mass amplification and would
opment is to reduce the acceleration of the car body and provide much greater inertia compared to its own mass, thus
passengers while ensuring good contact between tires and increasing the inertia of the entire dynamic system rather than
the road. The suspension travel must also be limited in the increasing the mass [5], [6]. Due to its mechanical properties,
permissible working space. These objectives (ride comfort, it is therefore an efficient structure for damping vibrations.
road holding, and suspension travel) can be in opposition In [5], an inerter equipped with a continuously variable
with each other, and the design problem is to find a com- transmission (CVT) is proposed, which is able to achieve
promise between them [1]. In terms of the control structure the step-less and accurate change of inertance through the
for suspension systems, there are three main categories which varying transmission ratio of the CVT. In [3], both semi-
have been developed to attain the required performance of the active damper and semi-active inerter were applied in the
vehicle: passive, semi-active, and active suspension systems vehicle suspension system, and a feedback control approach
[2]. In many investigations, the active suspension system was proposed, that is capable to improve the ride comfort and
has been indicated as an effective method for improving road holding of the vehicle. In [1], the active inertia-based
suspension performance [3], [4]. Nowadays, research to suspension uses a controllable actuator to produce the desired
improve suspension performance focuses on two aspects: force. Compared to passive and semi-active suspensions, it
first, on the rational and precise design of advanced vehicle provides better dynamic performance, although it requires
suspension, and second, on the search for optimal control the most energy.
methods. On the other hand, to find a compromise between the
*This work was not supported by any organization
conflicting performances of the vehicle suspension system,
K. Karim Afshar, R. Korzeniowski, and J. Konieczny are with the many approaches have been proposed based on various
Department of Process Control, AGH University of Science and Technology, control techniques, such as fuzzy logic and neural network
Krakow, Poland.
1 afshar@agh.edu.pl control [7], [8], adaptive control [7], [8], model predictive
2 korzerom@agh.edu.pl control [9], [10], H∞ control [2], [9], [11], [12], etc. In
3 koniejar@agh.edu.pl particular, the application of H∞ control of the active vehicle

978-3-907144-09-1 ©2023 EUCA 1454

suspension system has been intensively investigated in the (semidefinite) matrix. Rn stands for the n-dimensional Eu-
context of robustness and damping of road disturbances. clidean space and the superscript T denotes matrix transpo-
Furthermore, it has been recognized that it is not only an ef- sition. I and 0 are utilized to indicate the identity and zero
fective way to compromise between conflicting performance matrices with appropriate dimensions, and diag {· · · } stands
requirements, but also to optimize either a weighted single for a block-diagonal matrix. Let k•k symbolize the induced
objective function with hard constraints or a multi-objective norm for matrices and the Euclidean norm for vectors. k•kL2
function [2], [12]. represents the L2 norm of a signal defined as kv(t)k2L2 =
In this paper, the active inerter-based half-car suspension 2
0 kv(s)k ds. Furthermore, k•kL∞ denotes the L∞ norm of
R∞
system is investigated based on the parallel-connected con-
a signal defined as kv(t)k2L2 = sup kv(t)k2 .
figuration, since this configuration is simple and space-saving t≥0
[6]. The tire and suspension deflections are limited by their
II. P ROBLEM F ORMULATION
peak values in the time domain [11]. Hence, the hard limits of
the suspension system are assessed employing a generalized Since the spring responds to displacements and the damper
H2 (GH2 ) norm (energy-to-peak) in the form of an LMI responds to velocities, the idea of the inerter is to act
[12]. While the H∞ control (energy-to-energy) is used to against accelerations. Accordingly, the inerter is connected in
optimize the ride comfort of the suspension system, which parallel to the spring and damper between the wheel and the
measures the accelerations of the body including both the chassis. The main function of the inerter is to counteract
heaving and pitching motions. Sufficient stability conditions the vibrations coming from the tire, thus improving the
and performance criteria are derived in the form of LMIs contact between the wheel and the ground [6]. The half-
employing the direct Lyapunov method. Moreover, additional car model of the active suspension system equipped with
LMIs are added to the original condition to reduce the gain of inerter, as shown in Fig.1, can be reduced by four degrees
the controller, which results in avoiding measurement noise of freedom considering the vertical and angular dynamics
amplification and saturation of the actuator. [13]. The model is assembled by one sprung mass (car body)
The main contributions of this work can be summarized that is connected to two unsprung masses (representing the
as follows: front and rear wheels) and includes heave and pitch modes
• In this paper, we purpose to design a multi-objective
of vibrations. The two front and rear unsprung masses are
robust H2 /H∞ controller for the active inerter-based free to move vertically and are confronted with the road
half-car suspension system that provides a compromise disturbance input.
between the basic performance requirements for vehicle
including road holding, ride comfort, suspension deflec-
tion, energy consumption, and structural constraints.
• For the first time, we were able to obtain the state space
of the active vehicle suspension system for the half-
car model with the presence of passive-inerter in its
dynamics and evaluate the performance of this system
using a constrained robust H∞ controller.
• High gain of the controller in the real implementation
of the active suspension system, can lead to major
problems such as noise amplification and saturation of
the actuator. To avoid such problems, some additional Fig. 1. Inerter based half-car model of active suspension system.
LMIs are presented to reduce the gain of the controller.
• The stability conditions are derived as linear matrix In Fig.1, ms indicates the mass of the car body; mu f and
inequalities (LMIs) and therefore the stabilization gain mur are the unsprung masses of the front and rear tires,
of the system is achieved by solving the convex opti- respectively. bs f and bsr denote the inertance of the inerters
mization problem. for the front and rear axles of suspensions, respectively;
The subsequent parts of this paper are structured into four cs f and csr represent the damping coefficients of suspension
sections. The description of the active inerter-based half- elements for the front and rear suspensions, respectively.
car suspension system is presented in Section II. Section ks f and ksr represent the stiffnesses of the suspension for
III contains the problem formulation for constrained robust the front and rear suspensions, respectively. Likewise, kt f
H∞ control based on the solvability of LMIs for the active and ktr are the front and rear tire stiffnesses, respectively;
inerter-based suspension system. In Section IV, the proposed u f (t) and ur (t) are the front and rear actuator force inputs,
controller is applied to the half-car suspension systems respectively. ϕ(t) denotes the pitch angle, and zc (t) is the
for performance evaluation. Finally, the conclusion of our displacement of the center of gravity, lϕ is the pitch moment
findings is given in Section V. of inertia about the center of mass, l1 is the distance between
Notation: The following nomenclature will be utilized the front axle and the center of gravity, and l2 is the
throughout this paper. The notation P > 0 (≥ 0) is used distance between the rear axle and the center of gravity.
to denote that P is a real symmetric and positive definite zs f (t) and zsr (t) represent the vertical displacements of the

1455
 " #T
front and rear body, respectively; zu f (t) and zur (t) denote the 0 α7 0 β7 0 κ7 0 ρ7
B=
vertical displacements of the front and rear unsprung masses, 0 α8 0 β8 0 κ8 0 ρ8
#T
respectively. zr f (t) and zrr (t) denote the road disturbance
"
0 0 0 0 −1 0 0 0
D=
inputs to the front and rear wheels, respectively. It is assumed 0 0 0 0 0 0 −1 0
that the tires are always in contact with the ground. " # " #
u f (t) żr f (t)
It is supposed that the characteristics of the suspension u(t) = , v(t) =
ur (t) żrr (t)
elements are linear, and the pitch angle ϕ(t) is small enough.
αi (ms , mu f , mur , bs f , bsr , cs f , csr , ks f , ksr , kt f , ktr , Iϕ , l1 , l2 )
Accordingly, the displacements of the sprung mass can be
βi (ms , mu f , mur , bs f , bsr , cs f , csr , ks f , ksr , kt f , ktr , Iϕ , l1 , l2 )
established by
κi (ms , mu f , mur , bs f , bsr , cs f , csr , ks f , ksr , kt f , ktr , Iϕ , l1 , l2 )
zs f (t) = zc (t) − l1 sin ϕ(t) ≈ zc (t) − l1 ϕ(t)
ρi (ms , mu f , mur , bs f , bsr , cs f , csr , ks f , ksr , kt f , ktr , Iϕ , l1 , l2 )
zsr (t) = zc (t) + l2 sin ϕ(t) ≈ zc (t) + l2 ϕ(t) (1) i = 1, 2, 3, 4, 5, 6, 7, 8
Assuming linear dampers, springs, and inerters, differen- As mentioned earlier, ride comfort, suspension deflection,
tial equations of motion can be calculated by using Newton’s and road-holding ability are the three most important perfor-
second law as follows mance criteria to consider when developing controllers for
ms z̈c (t) + bsr [z̈sr (t) − z̈ur (t)] + csr [żsr (t) − żur (t)] vehicle suspension systems. Therefore, the suspension out-
 
+ksr [zsr (t) − zur (t)] + bs f z̈s f (t) − z̈u f (t) puts are divided into two categories including the optimiza-
    (2)
+cs f żs f (t) − żu f (t) + ks f zs f (t) − zu f (t) tion output and the constraint output which are explained in
= ur (t) + u f (t) the following subsections.
Iϕ ϕ̈ (t) + l2 bsr [z̈sr (t) − z̈ur (t)] + l2 csr [żsr (t) − żur (t)]
  A. Ride comfort
+l2 ksr [zsr (t) − zur (t)] − l1 bs f z̈s f (t) − z̈u f (t)
    (3) Indeed, minimization of the vertical acceleration sensed
− l1 cs f żs f (t) − żu f (t) − l1 ks f zs f (t) − zu f (t)
= l2 ur (t) − l1 u f (t) by the rider is the paramount assignment of the suspension
    system, leading to ride comfort and less depreciation. In
mu f z̈u f (t) − bs f z̈s f (t) − z̈u f (t) − cs f żs f (t) − żu f (t)
    (4) other words, ride comfort is the general sensation of noise,
−ks f zs f (t) − zu f (t) + kt f zu f (t) − zr f (t) = −u f (t)
vibration and motion inside a driven vehicle and it impacts
mur z̈ur (t) − bsr [z̈sr (t) − z̈ur (t)] − csr [żsr (t) − żur (t)] the comfort, safety and health of the passengers. Hence, both
(5)
−ksr [zsr (t) − zur (t)] + ktr [zur (t) − zrr (t)] = −ur (t) the heave acceleration z̈c (t) and the pitch acceleration ϕ̈(t) of
It is noteworthy that the equations of motion for the the half-car suspension system are chosen as the first control
passive system can be reached by letting u f (t) = ur (t) = 0. output vector, that is " #
Defining eight state variables as follows q1 z̈c (t)
z1 (t) = (8)
x1 (t) = zs f (t) − zu f (t) , x2 (t) = żs f (t) q2 ϕ̈(t)
x3 (t) = zsr (t) − zur (t) , x4 (t) = żsr (t) where q1 and q2 are weighting√constants and, normally,
q1 is selected as 1 and q2 = q1 l1 l2 [2], [12]. The ride
x5 (t) = zu f (t) − zr f (t) , x6 (t) = żu f (t)
comfort performance of the suspension system is optimized
x7 (t) = zur (t) − zrr (t) , x8 (t) = żur (t) (6) by the concept of the H∞ control (energy-to-energy) to
where x1 (t) is the suspension deflection of the front car measure the body accelerations. As a performance measure,
body, x2 (t) denotes the vertical velocity of the front car body, the H∞ norm (or L2 -gain) leads to organize active suspension
x3 (t) represents the suspension deflection of the rear car enacting adequately in a wide range of shock and vibration
body, x4 (t) indicates the vertical velocity of the rear car body, environments and we intend to decrease the H∞ norm of
x5 (t) is the tire deflection of the front car body, x6 (t) denotes the suspension system from the disturbance v(t) to the
the vertical velocity of the front wheel, x7 (t) represents the controlled output z1 (t) to reduce heave and pitch vibrations
tire deflection of the rear car body, and x8 (t) is the vertical of the vehicle. z̈c (t) and ϕ̈(t) are given by Eqs. (2) and (3),
velocity of the rear wheel. Accordingly, by defining x(t) = respectively, and can be described as follows
[ x1 (t) x2 (t) x3 (t) x4 (t) x5 (t) x6 (t) x7 (t) x8 (t) ]T , z1 (t) = C1 x(t) + D12 u(t) (9)
the active inerter-based suspension system can be represented where   
by the following state-space equation q
 1
0 υ1 υ2 υ3 υ4 υ5 −υ2 υ6 −υ4
C1 =  
ẋ (t) = Ax (t) + Bu (t) + Dv (t) (7) 0 q2 µ1 µ2 µ3 µ4 µ5 −µ2 µ6 −µ4
  
where q1 0 υ7 υ8
D12 = 
 
0 1 0 0 0 −1 0 0
 
  0 q2 µ7 µ8
 α1 α2 α3 α4 α5 −α2 α6 −α4 
υi (ms , mu f , mur , bs f , bsr , cs f , csr , ks f , ksr , kt f , ktr , Iϕ , l1 , l2 )
 
0 0 0 1 0 0 0 −1 
 

  µi (ms , mu f , mur , bs f , bsr , cs f , csr , ks f , ksr , kt f , ktr , Iϕ , l1 , l2 )
 β1 β2 β3 β4 β5 −β2 β6 −β4 
A=
 
 i = 1, 2, 3, 4, 5, 6, 7, 8

 0 0 0 0 0 1 0 0 
−κ2 −κ4  B. Suspension deflection limitation
 
 κ1 κ2 κ3 κ4 κ5 κ6
 

 0 0 0 0 0 0 0 0  Vehicle suspension must be capable of support the static
ρ1 ρ2 ρ3 ρ4 ρ5 −ρ2 ρ6 −ρ4 weight of the vehicle. Accordingly, in order to avoid ride

1456
comfort deterioration and mechanical structural damage, the As aforementioned, we assume the case that all the state
active suspension controllers should be qualified to preclude variables x(t) can be measured, leading to the design of a
the suspension from hitting its travel limit [12]. Conse- constrained state-feedback H∞ controller.
quently, the suspension deflection requires to be limited y(t) = C3 x(t) , C3 = I (17)
within a suitable range induced by the constraint of the For the design of the state-feedback robust H∞ controller,
mechanical structure, which is defined as Consider the following control input
zs f (t) − zu f (t) 6 z f max , |zsr (t) − zur (t)| 6 zr max (10) u(t) = Ky(t) = Kx(t) (18)
where z f max and zr max are the maximum suspension travel where K is the state-feedback gain matrix to be designed,
hard limits, under any road disturbance inputs and vehicle such that, first, the closed-loop system without external
running conditions for front and rear tires, respectively. It is disturbance is asymptotically stable, and, second, under
not necessary that the suspension deflection space is minimal, zero initial condition the L2 gain (i.e., H∞ norm) of the
but only its peak value should be limited. As regards L∞ kz 2 2 2
norm of a signal in the time domain, it actually represents
closed-loop system guarantees
1 (t)kL2 < γ1 kv(t)kL2 for
all nonzero v(t) ∈ L2 0 ∞ , and some scalar γ1 > 0,

its peak value, that is q subject to the L2 − L∞ gain (i.e., GH2 norm) of the closed-
∆
kz(t)kL∞ = sup
t∈[0,∞)
zT (t)z(t) (11) loop system guarantees kz2 (t)k2L∞ < γ22 kv(t)k2L2 , and the
The L∞ norm of the suspension deflection output under prescribed constant γ2 > 0.
the energy-bounded road disturbance input is able to be
III. C ONSTRAINED H∞ C ONTROLLER DESIGN
optimized according to r
Z ∞
kv(t)kL2 =
∆
vT (t)v(t)dt < ∞ (12) The active inerter-based half-car suspension system can be
0
defined by the following state-space equations
That is, v(t) ∈ L2 0 ∞ , to perceive the hard require-


ẋ (t) = Ax (t) + Bu (t) + Dv (t)
ment for the suspension deflection. Actually, this is a gen-
z1 (t) = C1 x (t) + D12 u (t)
eralized H2 (GH2 ) or energy-to-peak optimization problem
[12]. z2 (t) = C2 x (t)
y (t) = C3 x (t)
C. Road holding ability x(t) = φ (t) (19)
In practical vehicle systems, during maneuvers such as where x (t) ∈ Rn is the state, u (t) ∈ Rm
is the input vector,
braking, accelerating, or cornering, there are many forces y(t) ∈ R p is the measured output, z1 (t) ∈ Rd1 and z2 (t) ∈ Rd2
acting on the wheel that can lift it off the ground and leads are the controlled outputs, φ (t) is a real-valued initial func-
to losing control of the vehicle, in either drive or steering tion, v (t) ∈ Rq is the external disturbance vector, matrices
senses. Hence, the dynamic tire load should not exceed the A, B, D, C1 , D12 , C2 , and C3 , are all constant real matrices
static tire load to ensure firm uninterrupted contact of both with appropriate dimensions.
front and rear wheels to the road [2], according to In this section, we will solve the problem of constrained
kt f (zu f (t) − zr f (t)) < Ff , |ktr (zur (t) − zrr (t))| < Fr (13) state-feedback robust H∞ controller for the active inerter-
where Ff and Fr are static tire loads of front and rear wheels, based suspension system. Theorem 1 presents the conditions
respectively. They can be calculated by under which the closed-loop system without external distur-
Ff + Fr = (ms + mu f + mur )g bance becomes asymptotically stable, and in the presence
of external disturbance a prescribed disturbance attenuation
Fr (l1 + l2 ) = ms gl1 + mur g(l1 + l2 ) (14)
level is achieved. This can be accomplished by minimizing
Such as suspension deflection, here we have another peak the H∞ norm and the generalized H2 (GH2 ) norm of the
value optimization problem that can be dealt with in the same system under the external disturbance v(t) to the controlled
manner. Consequently, the tire load and the hard constraints outputs z1 (t) and z2 (t), respectively.
on the suspension deflection are defined as the second control Assumption 1: In this paper, the external disturbance sig-
output, that is,   nal v(t) is considered toZbe square-integrable, that is
zs f (t)−zu f (t) ∞
 z f max  kv(t)k2L2 = kv(s)k2 ds < vmax < ∞
 zsr (t)−zur (t)  0
z2 (t) = 
 zr max
kt f (zu f (t)−zr f (t))

 (15) Assumption 2: In this paper, it is assumed that all the
 
 Ff  states of the system (Eq. (19)) can be measured, that is,
ktr (zur (t)−zrr (t))
Fr C3 = I .
Therefore, z2 (t) can be described as follows Theorem 1: Assuming positive constants γ1 , γ2 , LR , and
z2 (t) = C2 x (t) (16) LS , the linear suspension system in Eq. (19) with state-
where   feedback controller in Eq. (18), in the absence of external
1/z f max 0 0 0 0 0 0 0 disturbance is asymptotically stable and in the presence of
external disturbance satisfies kz1 (t)k2L2 < γ12 kv(t)k2L2 and
 
 0 0 1/zr max 0 0 0 0 0 
C2 =  
0 0 0 0 kt f /Ff 0 0 0  kz2 (t)k2L∞ < γ22 kv(t)k2L2 for v(t) ∈ L2 0 ∞ , if there
 

 
0 0 0 0 0 0 ktr /Fr 0 exist symmetric positive definite matrix X > 0 and matrix Y

1457
with appropriate dimensions, such that the following LMIs By setting γ1 = 7 , γ2 = 18 , LR = 106 , Ls = 104 and solving
hold
  the convex optimization problem formulated in Theorem 1
AX + XAT + YT BT + BY D XCT1 + YT DT12 using the YALMIP toolbox [16], the gain matrix of the
 

 DT −γ1 2 I 0 <0
 (20) controller is obtained

C1 X + D12 Y 0 −I −9513.24 −4736.85 1401.17 405.67
  KI = 
X XCT2 −4924.25 −875.04 573.78 −3215.12
 γ22
≥0 (21)

C2 X I 34116.68 91.34 −17090.31 159.01
γ12 
" # 11868.81 −65.54 30971.43 123.32
LR I YT
>0 (22) And for brevity, we will indicate the proposed controller
Y I
" # as Controller I hereafter. To assess the performance of the
LS I I proposed controller, acquired results will be compared to the
>0 (23)
I X results obtained by constrained robust H∞ control for active
In this case, if inequalities (20)–(23) have a feasible vehicle suspension without inerter [14], and will be denoted
solution, the stabilizing gain of the state-feedback controller as Controller II for brevity. The obtained controller gain with
(Eq. (18)) is given by K = YX−1 . design parameters γ1 = 7 , γ2 = 12 is as follows

Proof: The proof of the Theorem 1 is omitted, for the −67894.2 −11741.4 8624.57 1690.71
KII = 
sake of brevity. Inequality (20) is very similar to [14], which −24669.15 −3326.28 −40418.42 −9583.87

satisfies the asymptotically stable of the closed-loop system 21531.09 −26.72 −7585.85 303.78

without external disturbance, and second, under zero initial 10457.97 −415.68 72402.62 1028.39
condition, the L2 gain (i.e., H∞ norm) of the closed-loop According to ISO 2361, improving ride comfort is equiv-
2 2 2
system guarantees
kz1 (t)kL2 < γ1 kv(t)kL2 for all nonzero alent to minimizing the vertical and horizontal accelerations
v(t) ∈ L2 0 ∞ , and some scalar γ1 > 0. Inequality (21)

of a vehicle system. The human body is more sensitive to
is similar to [2] which assures that the closed-loop system vibrations in the frequency range of 4Hz to 8Hz in the
guarantees kz2 (t)k2L∞ < γ22 kv(t)k2L2 and the prescribed con- vertical direction and 1Hz to 2Hz in the horizontal direction
stant γ2 > 0. Inequalities (22) and (23) are similar to [2], [15] [14]. Therefore, we first focus on the frequency responses
and lead to the limitation of the gain matrix K. from the ground velocity to the heave and pitch accelerations
for the passive and closed-loop systems using the constrained
IV. A PPLICATION TO INERTER - BASED HALF - CAR robust H∞ state-feedback controllers. From Fig. 2, we can see
SUSPENSION CONTROL that the desired controller I and the controller II can provide
In this section, to illustrate the effectiveness of the pro- the lower value of the H∞ norm over the frequency range of
posed controller in Section III, we will apply it to the active 1Hz – 8Hz.
inerter-based half-car suspension system described in Section
II. The parameters of the inerter-based half-car model are
listed in Table I. The design constraints and the design
parameters for the controller are shown in Table II.

TABLE I
S YSTEM PARAMETER VALUES OF THE INERTER - BASED HALF - CAR
SUSPENSION MODEL .
Parameter Value Parameter Value
ms 690 kg Iϕ 1222 kg m2
mu f 40 kg mur 45 kg
ks f 18 000 N/m ksr 22 000 N/m
kt f 200 000 N/m ktr 200 000 N/m
cs f 1000 Ns/m csr 1000 Ns/m
bs f 6 kg bsr 6 kg
l1 1.3 m l2 1.5 m Fig. 2. Frequency responses for the open- and closed-loop systems

Performance of the half-car suspension system is capable

to be assessed by examining six response quantities, that is,
TABLE II
the sprung mass heave acceleration z̈c (t), the pitch acceler-
T HE VALUES OF DESIGN CONSTANTS CHOSEN IN THE CONTROLLER
ation ϕ̈(t), the suspension deflection x1 (t) and x3 (t) for the
DESIGN PROCEDURE .
front and rear wheels, respectively, and the tire deflection
Parameter Value x5 (t) and x7 (t) for the front and rear wheels, respectively.
z f max 0.035 m
zr max 0.035 m In the following sub-sections, we will utilize Shock (Bump)
Ff 4014.5 N and Vibration (Rough Road) road profiles to evaluate the
Fr 3580.5 N performance of the active inerter-based half-car suspension
q1 1
q2 1.3964 system with respect to vehicle handling, ride comfort, energy
consumption, and working space of the suspension.

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A. Bump response energy consumption of two s
active control methods:
Z T̄
Here, the bumps or potholes of relatively short duration ku(t)kL2 = u(t)T u(t)dt (25)
and high intensity that occur on a smooth surface have 0

been considered to reveal the characteristics of the transient Where T̄ = 2 s is the simulation time. Energy consumption
response, which isgiven by of two controllers is shown in Table III. It can be seen
from Table III, active suspension system with inerter has
 
 a
2 1 − cos( 2πv0
l t) ,0 ≤ t ≤ l
v0
zr f (t) = (24) a superior performance compared to the active suspension
l
 0 ,t > v0
without inerter, and the low gain of the Controller I results
Where a and l are the height and the length of the bump.
in lower energy consumption.
We choose a = 0.1 m, l = 2 m and the vehicle forward
velocity as v0 = 20 km/h. Here, the road profile zrr (t) for TABLE III
the rear wheel is supposed to be same as for the front wheel E NERGY CONSUMPTION OF ACTIVE CONTROLLERS
profile zr f (t) but with a time-delay of (l1 + l2 )/v0 . Controller Controller
The response of the half-car suspension system with I II
energy consumption (front) 423.34 592.45
inerter by using Controller I and without inerter by using energy consumption (rear) 519.45 636.1
Controller II, and passive suspension with inerter are com-
pared in Fig. 3. Fig. 3 displays the heave acceleration, pitch
acceleration, suspension deflection, and tire deflection for the B. Random response
front and rear wheels, respectively. The control efforts for the
Generally, it is capable to assume random vibrations as
front and rear wheels of the active controllers are also plotted
road disturbances, which are consistent and typically speci-
in Fig. 4. It can be seen from Fig. 3 that the Controllers I
fied as the random process. The ground displacement power
and II compared to the passive suspension system acquire
spectral density (PSD)
( is defined as follows
better responses. It is confirmed by the simulation results that
Sg (Ω0 )( ΩΩ )−n1 i f Ω ≤ Ω0
bump response quantities for heave and pitch accelerations of Sg (Ω) = 0 (26)
Sg (Ω0 )( ΩΩ )−n2 i f Ω > Ω0
active inerter-based suspension system are better than active 0

suspension without inerter. On the other hand, the required Where Ω0 = 1/2π stands for reference spatial frequency
control effort for Controller I is less than for Controller II, and Ω is a spatial frequency. The value of Sg (Ω0 ) denotes a
which is shown in Fig. 4. measure for the roughness coefficient of the road. n1 and n2
represent the road roughness constants.
In particular, if the vehicle is presumed to travel with a
constant horizontal speed v0 over a given road, it is capable
to simulate the force resulting from the road irregularities by
the following series
N
zr f (t) = ∑ sn sin(nω0t + ϕn ) (27)
n=1
p
Where sn = 2sg (n∆Ω)∆Ω, ∆Ω = 2π/L, and L is the
length of the road segment considered. The amplitudes sn
of the excitation harmonics are assessed from the road
spectra selected. Additionally, the value of the fundamental
temporal frequency ω0 is determined from ω0 = 2π L v0 . While
the phases ϕn are treated as random variables, following a
Fig. 3. Sprung mass heave and pitch accelerations, suspesion deflection
for the front and rear wheels, tire deflection for the front and rear wheels uniform distribution in the interval [0, 2π).
According to ISO2631 standards, road class D (poor
quality) Sg (Ω0 ) = 256 × 10−6 m3 , is selected as a typical

road profile. In this paper, n1 = 2, n2 = 1.5, L = 100, N f = 200

and the horizontal speed v0 = 36 m/s, are utilized to generate
the random road profile.
The Monte Carlo simulation is utilized to assess the prob-
abilistic characteristics of the random response. Therefore,
taking into account the random variable ϕn of the excitation
applied, the performance index of the Root Mean Square
(RMS) is determined by the expected values:
 Z T̄
1
J1 = E [z̈c (t)]2 dt (28)
T̄ 0
1 T̄
 Z 
Fig. 4. Control effort for the front and rear wheels J2 = E [ϕ̈(t)]2 dt (29)
T̄ 0
" Z   #
In order to qualitatively assess the control effort, the 1 T̄ x1 (t) 2
J3 = E dt (30)
following L2 norm value is employed to determine the T̄ 0 z f max

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 " #
x3 (t) 2
Z T̄  
1
J4 = E dt (31) to the active inerter-based half-car suspension system to
T̄ 0 zr max
" # minimize the influence of road disturbance on the system
kt f x5 (t) 2
Z T̄  
1 performance. It was observed that the active inerter-based
J5 = E dt (32)
T̄ 0 Ff
" suspension system for all performance requirements achieve
 #
ktr x7 (t) 2
Z T̄ 
J6 = E
1
dt (33)
better response compared to both active suspension without
T̄ 0 Fr inerter and passive suspension with inerter.
The RMS values for sprung mass heave acceleration J1 ,
pitch acceleration J2 , suspension deflection for the front J3 R EFERENCES
and rear J4 , tire deflection for the front J5 and rear J6 , [1] W. Sun, H. Pan, Y. Zhang, and H. Gao, “Multi-objective control for
respectively, are presented; where T̄ = L/v0 is the temporal uncertain nonlinear active suspension systems,” Mechatronics, vol. 24,
no. 4, pp. 318–327, 2014. Vibration control systems.
measurement period. For calculating RMS values, we have [2] K. K. Afshar and A. Javadi, “Constrained H∞ control for a half-car
considered T̄ = 5 in Eqs. (28)-(33) and the simulation has model of an active suspension system with actuator time delay by
been run randomly 100 times. predictor feedback,” Journal of Vibration and Control, vol. 25, no. 10,
pp. 1673–1692, 2019.
To validate the effectiveness of controller I in dealing with [3] M. Z. Q. Chen, Y. Hu, C. Li, and G. Chen, “Application of Semi-Active
the active suspension system based on inerter, the RMS ratios Inerter in Semi-Active Suspensions Via Force Tracking,” Journal of
JIIi (t)/JIi (t), JPi (t)/JIi (t), i = 1, 2, 3, 4, 5, 6 are calculated, Vibration and Acoustics, vol. 138, no. 4, 2016. 041014.
[4] J. Konieczny, W. Raczka, M. Sibielak, and J. Kowal, “Energy con-
where JIi (t) denote the RMS values of the active inerter- sumption of an active vehicle suspension with an optimal controller
based suspension system using Controller I, JIIi (t) denote the in the presence of sinusoidal excitations,” Shock and Vibration, 2020.
RMS values of the active suspension system without inerter [5] P. Brzeski, T. Kapitaniak, and P. Perlikowski, “Novel type of tuned
mass damper with inerter which enables changes of inertance,” Journal
by using Controller II, and JPi (t) are the RMS values of the of Sound and Vibration, vol. 349, pp. 56–66, 2015.
passive suspension system with inerter. [6] Y. Wang, X.-Y. Jin, Y.-S. Zhang, H. Ding, and L.-Q. Chen, “Dy-
Tables IV represent the results of RMS ratios for Con- namic performance and stability analysis of an active inerter-based
suspension with time-delayed acceleration feedback control,” Bulletin
troller I, Controller II, and passive suspension system for of the Polish Academy of Sciences: Technical Sciences, vol. 70, no. 2,
the poor (class D) quality road profile. The control efforts p. e140687, 2022.
for the front and rear wheels of the active controllers are [7] P. S. Kumar, K. Sivakumar, R. Kanagarajan, and S. Kuberan, “Adaptive
neuro fuzzy inference system control of active suspension system
also shown in Table IV. It can be seen from Table IV that with actuator dynamics,” Journal of Vibroengineering, vol. 20, no. 1,
the RMS ratios of Controllers I is always less than the pp. 541–549, 2018.
passive suspension system (the response ratio is more than [8] H. Taghavifar, A. Mardani, C. Hu, and Y. Qin, “Adaptive robust
nonlinear active suspension control using an observer-based modified
1). On the other hand, it can be seen that heave acceleration, sliding mode interval type-2 fuzzy neural network,” IEEE Transactions
pitch acceleration, and tire deflection for the front and rear on Intelligent Vehicles, vol. 5, no. 1, pp. 53–62, 2020.
wheels with Controller I acquire better response compared [9] S. Bououden, M. Chadli, and H. R. Karimi, “A robust predictive
control design for nonlinear active suspension systems,” Asian Journal
to Controller II. In addition, the required control effort for of Control, vol. 18, no. 1, pp. 122–132, 2016.
Controller I is less than for Controller II. [10] S. M. Moradi, A. Akbari, and M. Mirzaei, “An offline lmi-based robust
model predictive control of vehicle active suspension system with
TABLE IV parameter uncertainty,” Transactions of the Institute of Measurement
E NERGY CONSUMPTION AND RMS VALUES OF R ANDOM ROAD P ROFILE and Control, vol. 41, no. 6, pp. 1699–1711, 2019.
[11] H. Chen and K.-H. Guo, “Constrained H∞ control of active sus-
performance criteria JIIi (t)/JIi (t) JPi (t)/JIi (t)
heave acceleration 2.6241 2.2365 pensions: an lmi approach,” IEEE Transactions on Control Systems
pitch acceleration 2.1644 3.8632 Technology, vol. 13, no. 3, pp. 412–421, 2005.
front suspension deflection 0.81252 1.5082 [12] H. Du and N. Zhang, “Constrained H∞ control of active suspension
rear suspension deflection 0.84161 2.3661 for a half-car model with a time delay in control,” Proceedings of the
front tire deflection 2.3314 1.6733
rear tire deflection 2.2647 2.9368 Institution of Mechanical Engineers, Part D: Journal of Automobile
Pasive Controller I Controller II Engineering, vol. 222, no. 5, pp. 665–684, 2008.
energy consumption (front) — 118.59 169.85 [13] J. Konieczny, M. Sibielak, and W. Raczka, “Active vehicle suspension
energy consumption (rear) — 129.84 169.11 with anti-roll system based on advanced sliding mode controller,”
Energies, vol. 13, no. 21, 2020.
[14] H. Chen, Z. Y. Liu, and P. Y. Sun, “Application of Constrained H∞
Control to Active Suspension Systems on Half-Car Models,” Journal
V. C ONCLUSIONS of Dynamic Systems, Measurement, and Control, vol. 127, no. 3,
pp. 345–354, 2004.
In this paper, the performance of the active inerter-based [15] H. Gritli and S. Belghith, “Robust feedback control of the underac-
half-car suspension system has been investigated via con- tuated inertia wheel inverted pendulum under parametric uncertainties
strained robust H∞ controller. In order to improve the ride and subject to external disturbances: Lmi formulation,” Journal of the
Franklin Institute, vol. 355, no. 18, pp. 9150–9191, 2018.
comfort performance, the L2 gain of the heave and pitch [16] J. Lofberg, “Yalmip : a toolbox for modeling and optimization in
accelerations of the suspension system has been optimized. matlab,” in 2004 IEEE International Conference on Robotics and
Whereas, the generalized H2 (GH2 ) norm has been utilized to Automation (IEEE Cat. No.04CH37508), pp. 284–289, 2004.
insert the hard limits of the suspension and tire deflections
to guarantee that these performance criteria do not exceed
their pre-specified maximum values. Furthermore, two more
LMIs are added to the established sufficient conditions in
order to limit the gain of the controller. Finally, to validate
the effectiveness of the proposed approach, it is applied

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