Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 4, pp. 1151-1168, Warsaw 2011

CONTROL DESIGN OF SEMI-ACTIVE SEAT SUSPENSION

SYSTEMS

Igor Maciejewski

Tomasz Krzyżyński

Koszalin University of Technology, Division of Mechatronics and Applied Mechanics, Koszalin,

Poland; e-mail: igor.maciejewski@tu.koszalin.pl; tomasz.krzyzynski@tu.koszalin.pl

The paper deals with the control design of semi-active seat suspension
systems. A semi-active vibration control strategy basing on the inverse
dynamics of a spring or damper element and a primary controller is
studied. The optimisation procedure proposed in the paper makes it
possible tocalculate controller settingsandthese, in turn, todefinevibro-
isolation properties of semi-active suspension systems.

Key words: vibration damping, semi-active suspension, control system

1. Introduction

Passive seat suspensions amplify vibration at frequencies close to their natu-
ral resonance frequencies. The first natural frequency of typical passive seats
can be measured between 1 and 2Hz. Active suspensions require large power
supply and this is the main disadvantage of using such systems in practice
extensively. Semi-active suspensions consumemuch less power than active su-
spensions, therefore they have receivedmuch attention in the literature (Ballo,
2007). A desirable performance of suspension systems can be archived using
semi-active control, especially when some controllable dampers, like electro-
rheological (ER) ormagneto-rheological (MR) ones, are used (Du et al., 2005;
Maślanka et al., 2007; Spencer et al., 1997; Tsang et al., 2006).

The design of vibro-isolating systems, constructed and manufactured at
present, is a big challenge for engineers. This is due to opposite criteria that
are involved in the design process (Alkhatiba et al., 2004). For example, in
the automotive industry, it is desired to reduce vibration of the cabin floor
transmitted to the operator’s seat. On one hand, dynamic forces transmitted



1152 I. Maciejewski, T. Krzyżyński

from the cabin floor to the seat should approach zero to protect machine ope-
rator’s health. The Seat Effective Amplitude Transmissibility factor (SEAT)
provides a simple numerical assessment of the seat isolation efficiency (Griffin,
1996)

SEAT =
(ẍw)RMS
(ẍsw)RMS

(1.1)

where (ẍsw)RMS is the frequencyweighted rootmean square value of the simu-
lated input acceleration, (ẍw)RMS is the frequencyweighted rootmean square
value of the measured seat acceleration. On the other hand, the suspension
deflection should approach zero in order to ensure the controllability of wor-
king machines. The suspension travel can be a simple numerical assessment
of the seat performance as well. In this paper, the suspension travel is defined
by the maximum relative displacement of the suspension system. Its value is
calculated on the basis of the displacement signal in the time domain t as
follows

(x−xs)max =max
t
(x−xs)−min

t
(x−xs) (1.2)

where x is the seat displacement and xs is the displacement caused by input
vibration. Comfort criteria standardized for a selection of a trade-off between
the SEAT factor and the suspension travel (x−xs)max cannot be found in
the literature. However, the trade-off between conflicted requirements can be
selected with the help of a multi-criteria optimisation.

2. Control system design

An evaluation of the control algorithms and strategies is required for the con-
trol of feedback loops in the semi-active suspensions. In order to control loops
to work properly, the feedback loop must be properly tuned.Methods for tu-
ning feedback loops and criteria for judging the loop tuning should be defined
and used in modern control systems. The loop tuning can be achieved by
appropriate selection of the controller settings and those correspond to the
vibro-isolation properties of semi-active suspension systems.

2.1. Evaluation of primary controller

The simplified seat suspensionmodel, that is composed of a single degree
of freedom body mass, a linear spring and damper is used (Fig.1). Such a



Control design of semi-active seat suspension systems 1153

model has been extensively discussed in the literature and captures many es-
sential characteristics of a real seat suspension system. The passive subsystem
is applied to describevisco-elastic characteristics of the seat suspension system
(for example with an air-spring and shock-absorber). The active subsystem is
used to determine the desired force Fa that should be introduced into the
visco-elastic suspension system in an active way.

Fig. 1. Simplified model of the hybrid seat suspension system

The state spacemodel of the hybrid seat suspension system (Fig.1) can be
obtained by using the LFT (Linear Fractional Transformation) technique (Gu
et al., 2005) andby grouping signals into sets of external inputs and outputs as
well as into sets of controller inputs and outputs. Choosing the state variables
as: x1 :=x−xs; x2 := ẋ, the disturbance caused by road roughness: w1 :=xs;
w2 := ẋs and the external input force of the suspension system Fa, the state
space equation of the hybrid seat suspension can be written in the following
form

ẋ(t)=Ax(t)+B1w(t)+B2Fa(t) (2.1)

where

A=

[

0 1

− c
m
− d
m

]

B1 =

[

0 −1
0 d

m

]

B2 =

[

0
1
m

]

(2.2)

In order to satisfy the performance requirement, the acceleration of the
suspendedmass z1 := ẍ and the suspension deflection z2 :=x−xs are defined
as controlled outputs. The output equation reads

z(t)=C1x(t)+D11w(t)+D12Fa(t) (2.3)



1154 I. Maciejewski, T. Krzyżyński

where

C1 =

[

− c
m
− d
m

1 0

]

D11 =

[

0 d
m

0 0

]

D12 =

[

1
m

0

]

(2.4)

If the suspension deflection y1 :=x−xs and the velocity of the suspended
mass y2 := ẋ are measurable, then the measurement equation can be written
as follows

y(t)=C2x(t)+D21w(t)+D22Fa(t) (2.5)

where

C2 =

[

1 0
0 1

]

D21 =

[

0 0
0 0

]

D22 =

[

0
0

]

(2.6)

The controller is determined by formulating the state feedback control
problem in the following form:

Fa(t)=Ky(t)=KC2x(t) (2.7)

where
K=

[

K1 K2

]

(2.8)

is the output feedback gain vector to be designed.
However, the desired active force (Eq. (2.7)) has to be reproduced using

controlled elements, i.e. by the springwith variable stiffness or by the damper
with variable damping. In the semi-active suspension systems, it is well known
that the spring and damper forces depend not only on their control signals
but also on their actual working conditions, i.e. the actual deflection of the
spring x − xs or the actual velocity of the damper ẋ − ẋs. If the actual
spring deflection or the actual damper velocity are equal to nearly zero, then
their forces reach zero and any control signal can produce the desired force.
Therefore, the desired active force Fa, that can be reproduced in the semi-
active system in a better way, is calculated as follows

Fa(t)= gsKC2x(t) (2.9)

where gs is the gain-scheduling function that shapes the desired active force
to the actual working conditions of the spring or damper. This function is
defined as follows

gs =















∣

∣

∣

x−xs
(x−xs)n

∣

∣

∣ ← controllable spring
∣

∣

∣

ẋ− ẋs
(ẋ− ẋs)n

∣

∣

∣ ← controllable damper
(2.10)



Control design of semi-active seat suspension systems 1155

where (x−xs)n and (ẋ−ẋs)n are the nominal displacement andvelocity of the
controlled spring anddamper, respectively. InFig.2, graphical representations
of the functions described by Eqs. (2.10) are shown.

Fig. 2. Functions g
s
for the spring control (a) and the damper control (b)

As follows from Fig.2, the gain-scheduling function value equal to 1 is
achieved for the actual displacement/velocity of controlled spring/damper ele-
ment equal to the nominal value. In this case, the desired active force is not
modified by the gain-scheduling function and such the force is reproduced in
the semi-active system. If theactual displacement/velocity is greater two times
than the nominal value, the desired active force is increased also 2 times by
the gain-scheduling function (assuming a linear dependence). In this instance,
the higher force is reproduced in the semi-active system. If the actual displa-
cement/velocity is less than the nominal value, the lower force is reproduced
in the semi-active system similarly.

2.2. Evaluation of secondary controller

If the desired active force Fa is determined then it has to be partly repro-
ducedby the passive spring or damper element. This can be achieved using the
force tracking control system that adjusts the controllable spring or damper.
The force tracking control systemcanbehandledby applyingan internal force
feedback or else by applying a reverse model of the spring or damper element
(Maślanka et al., 2007). The second approach is employed in this study, and
in Fig.3 the graphical illustration of such principle is presented.

Theactual control signal u is calculated usinga reversemodel of the spring
or damper element in the following form

u=

{

f(x−xs,Fa) ← spring force control
f(ẋ− ẋs,Fa) ← damper force control

(2.11)



1156 I. Maciejewski, T. Krzyżyński

Fig. 3. Simplified models of the semi-active suspension system: with the controlled
spring force (a) and with controlled damper force (b)

where x− xs and ẋ− ẋs are the actual displacement and velocity of the
controlled spring and damper, respectively.

The spring displacement, the damper velocity and the desired active force
are the reverse model inputs. The model outputs are control signals to the
spring and damper which should reproduce the desired active force in the
semi-active system.Unfortunately, very often the force tracking control system
efficiency is lowered by a phase shift in the feedback loop (Maślanka et al.,
2007).This effectmightbecausedbyactuating time tc of the springordamper
element. Therefore, the proportional-derivative (PD) controller is applied in
order to speed up the overall control system. Finally, the output signal uc of
the PD controller, that controls the spring or damper element, is described as
follows

uc = tcu̇+u (2.12)

where u is the control signal calculated on the basis of the reverse model
(input to the PD controller).

The control signal uc sent to the spring or damper should be restricted
in the range of the minimum umin and maximum umax values. Therefore,
although the desired force Fa can be of any value, the calculated input signal
is constrained within the operating range

uc =















umin for uc <umin

uc for umin ¬uc <umax
umax for uc ­umax

(2.13)



Control design of semi-active seat suspension systems 1157

2.3. Formulation of overall control system

In Fig.4, the block diagram of the overall control system is presented.
If the desirable active force is obtained according to the primary controller
(Eq. (2.9)), then the desired force has to be approximately achieved by the
spring or damper element with the calculated input signal using the reverse
models (Eqs. (2.11)). In the control system proposed, any additional feedback
loop from the actual spring/damper force is not required, because the force
tracking is handled by applying the reverse model of the spring or damper
element. The actuating time of the spring or damper element is eliminated
by speeding up the control signal introduced to the the spring or damper
(Eq. (2.12)). Due to the actual constraint of the input signal to the spring
or damper element, the control signal is restricted within the range umin
and umax (Eq. (2.13)).

Fig. 4. Block diagram of the semi-active control for a seat suspension system

2.4. Multi-criteria optimisation of controller settings

1st step Suspended mass range

The vector of the suspendedmass is defined as follows

m= [m1, . . . ,mn−1,mn] mn−mn−1 = const n=1, . . . , i
(2.14)



1158 I. Maciejewski, T. Krzyżyński

Such a vector (in some software packages called the linearly spaced vector,
e.g. MATLABr) consists of equidistantly spaced elements whose the first
value is m1 and the last value is mn, with the total number of n. The vector
elements are defined as follows

m1 =mmin

. . .

mn−1 =mmin+
n−1
i−1

(mmax −mmin)

mn =mmax

(2.15)

2nd step Random starting points

The vector of controller settings is given as

K= [K1,K2, . . . ,Kp] p=1, . . . ,j (2.16)

where p is the number of chosen controller settings. The ranges of controller
settings have to be defined by the following vectors

Kl = [Kl1,Kl2, . . . ,Klp] ← lower bounds
Ku = [Ku1,Ku2, . . . ,Kup] ← upper bounds

(2.17)

where Kl and Ku are the vectors that contain the lowest and highest values
of chosen controller settings, respectively. Using an uniform random number
generator, the initial starting points may be expressed by

Ks1 =Kl1+(Ku1 −Kl1)rand(k)
Ks2 =Kl2+(Ku2 −Kl2)rand(k)
. . .

Ksp =Klp+(Kup−Klp)rand(k)

(2.18)

where k is the number of pseudo-random values (rand) drawn from the stan-
dard uniform distribution on the open interval (0, . . . ,1).

3rd step Minimization of particular optimisation criteria

The separateminimizing processes of the SEAT factor and the suspension
travel (x−xs)max are defined as follows

minKSEAT(K) ∀mn ∈ 〈mmin;mmax〉, n=1, . . . , i
minK(x−xs)max(K) ∀mn ∈ 〈mmin;mmax〉 n=1, . . . , i

(2.19)



Control design of semi-active seat suspension systems 1159

with the bounds of controller settings

Kl ¬K ¬Ku (2.20)

Such the optimisation procedure allows one to find extreme solutions which
minimize the particular optimisation criteria separately:

• criterion coordinates ((x−xs)max)max, (SEAT)min – minimum of the
SEAT factor,

• criterion coordinates ((x−xs)max)min, (SEAT)max – minimum of the
suspension travel.

The optimized systems are characterized by the best reduction of forces trans-
mitted to the isolated body (minimum SEAT) or by the best limitation of the
suspension travel (minimum (x−xs)max). InFig.5, a graphical representation
of minimizing particular optimisation criteria is shown.

Fig. 5. Minimization of the particular optimisation criteria at different values of
suspendedmasses

4th step Minimization of conflicted vibro-isolation criteria

In order to optimise both of the conflicted vibro-isolation criteria, minimi-
zation of the SEAT factor (primary criterion) is defined as follows

min
K
SEAT(K) ∀mn ∈ 〈mmin;mmax〉 n=1, . . . , i (2.21)

subject to the suspension travel (x−xs)max that is transferred into anonlinear
inequality constraint

(x−xs)max(K)¬ ((x−xs)max)c (2.22)



1160 I. Maciejewski, T. Krzyżyński

with the bounds of controller settings

Kl ¬K ¬Ku (2.23)

The value ((x− xs)max)c determines a constraint of the maximum relative
displacement of the suspension system. This value has to be included in the
range of the suspension travel defined by the extreme solutions

((x−xs)max)min ¬ ((x−xs)max)c ¬ ((x−xs)max)max (2.24)

where ((x−xs)max)min and ((x−xs)max)max are theminimumandmaximum
values of the suspension travel, respectively. An appropriate selection of the
constraint value ((x−xs)max)c allows one to choose the vibro-isolation pro-
perties of the seat suspension system. In Fig.6, a graphical representation of
minimizatioi of the conflicted vibro-isolation criteria is shown. Each optimisa-
tion procedure should be repeated for the randomly generated starting points
in order to find the global optimumof the seat suspension vibro-isolating pro-
perties.

Fig. 6. Minimization of the conflicted vibro-isolation criteria at different values of
suspendedmasses



Control design of semi-active seat suspension systems 1161

3. Example: Multi-criteria optimisation of the seat suspension in

semi-active vibration control

3.1. Optimisation object

In Fig.7, a physical model of the semi-active seat suspension system con-
taining a passive air-spring and a controllable magneto-rheological damper
is shown. The equation of motion of the semi-active seat suspension takes a
form similar as in the case of the passive seat suspensionmodel (Maciejewski,
2009a). In that system, the pneumatic spring is connected to the additional air
reservoir, therefore stiffness of the suspension system is rather low. However,
damping of the suspension system is controlled using theMR damper.

Fig. 7. Physical model of the semi-active seat suspension

The Bingham model presented by Spencer et al. (1997) is adopted in this
study for themagneto-rheological damper usingdata obtained experimentally.
In such a simplifiedmodel, the hysteresis loop of theMR damper is neglected
and the description of the MR damper force contains components from a
viscous damper and system friction only (cf. Fig.8). The force is given by the
following equation

Fmr = dmr
(ẋ− ẋs
δd

)

+αmr sgn
(ẋ− ẋs
δd

)

(3.1)

where dmr is the viscous damping coefficient, αmr is the scale factor of the
friction force, δd is the reduction ratio of the damper force. Basing on the
experimental data, the least-square approximationmethod is employed to de-



1162 I. Maciejewski, T. Krzyżyński

termine the appropriate parameters dmr and αmr for the analytical model

dmr = amru+ bmr αmr = emru
2+fmru+gmr (3.2)

where amr, bmr, emr, fmr and gmr are the polynominal coefficients expressed
in respect to the input signal u. These coefficients are evaluated by means
of additional MR damper measurements that were presented by Maciejewski
(2009b).

Fig. 8. (a) Binghammodel of the magneto-rheological damper, (b) force of the

magneto-rheological damper for control current 0.3A: simulation ( ),

measurement (.....)

If the Bingham model is determined then the desirable force Fa can be
realized by injecting an appropriate control signal into theMR damper in ac-
cordancewith the actual piston velocity of the damper u= f(Fa, ẋ−ẋs). The
control signal u is calculated from Eqs. (3.1) and (3.2) with the measurable
velocity ẋ− ẋs, and is given by

u=
−fmr sgn

(

ẋ−ẋs
δd

)

−amr
(

ẋ−ẋs
δd

)

+ sgn
(

ẋ−ẋs
δd

)√
∆

2emr sgn
(

ẋ−ẋs
δd

) (3.3)

with the function ∆ that is calculated as

∆=
(

fmr sgn
(ẋ− ẋs
δd

)

+amr
(ẋ− ẋs
δd

))2

− . . .

+4emr sgn
(ẋ− ẋs
δd

)(

gmr sgn
(ẋ− ẋs
δd

)

+ bmr
(ẋ− ẋs
δd

)

− δdFa
)

(3.4)

In Fig.9, the graphical representation of theMR damper reverse model is
shown. It should be noted that the MR damper is a semi-active device and
the desired force Fa can be realized only if this force and the damper velocity



Control design of semi-active seat suspension systems 1163

have the same sign. Then the calculated input signal of theMRdamper varies
in the range of 0A (minimum value) and 1A (maximum value) and depends
on the actual value of the desiredMR damper force and its actual velocity. If
the desired force and damper velocity have sign opposite to each other then
the input signal is settled to zero.

Fig. 9. Reverse model of the magneto-rheological damper u= f(F
a
, ẋ− ẋ

s
)

3.2. Decision variables and optimisation criteria

In order to enable controlling the vibro-isolation properties of a semi-active
suspension, the controller settings and their ranges are taken as:

• the proportionality factor of the relative displacement feedback loop
Ka1 =5-50 ·103N/m

• the proportionality factor of the absolute velocity feedback loop
Ka2 =0.5-5 ·103Ns/m

The optimisation of the controller settings is realized using a simulationmodel
of the semi-active system. The dynamic behaviour of the semi-active seat su-
spension ismodelled in theMATLAB-Simulink RO software package. Thenonli-
near ordinarydifferential equations (ODE) in themodel are solvednumerically
using the fixed-step (step time of 1ms)Bogacki-Shampine solver (Bogacki and
Shampine, 1989). The optimisation procedure is performed for selected input
vibrations that are specified in the International Standard (ISO7096, 2000)
for earth-movingmachinery in particular spectral classes. In order to find the
constrained minimum of the SEAT factor, a gradient-based optimisation al-
gorithm with the Sequential Quadratic Programming (SQP) method is used
(Gill et al., 2005).



1164 I. Maciejewski, T. Krzyżyński

3.3. Optimisation results

In Fig.10, the Pareto-optimal point distribution in conflicted criterion do-
mains are shown. These results are obtained for the selected spectral classes
of the excitation signal:

• EM3 – a signal at low frequencies and high amplitudes of vibration,
representative for wheel loaders,

• EM5 – a signal at middle frequencies and middle amplitudes of vibra-
tion, representative forwheel dozers, soil compactors onwheels, backhoe
loaders,

• EM6 – a signal at high frequencies and low amplitudes of vibration,
representative for crawler loaders, crawler dozers.

Fig. 10. Pareto-optimal point distribution for selected input vibrations: EM3 (a),
EM5 (b), EM6 (c), mass load on the suspension system m=50kg (◦),
m=75kg (2), m=100kg (⋄), m=125kg (∇), m=150kg (△)

In the examplepresented in this paper, tenPareto-optimal solutions are fo-
und for the following values of the suspendedmass: 50kg, 75kg, 100kg, 125kg
and 150kg. Each individual Pareto-optimal solution corresponds to a set of
the decision variables which define different vibro-isolation characteristics of
the semi-active suspension system.



Control design of semi-active seat suspension systems 1165

3.4. Simulation and measurement results

In Fig.11, point distributions (◦) corresponding to the simulated Pareto-
optimal solutions for the excitation signals: EM3, EM5 and EM6 are shown.
The compromising solutionsmarked by the black circles (•), having a constra-
int imposed on the suspension travel, are experimentally investigated in this
paper. The measurement results of the optimized semi-active suspension (∇)
and of conventional passive suspension (△) are presented in the same figure.
As follows fromFig.11, the semi-active control significantly improves the seat
suspension vibro-isolating properties for the excitation signals EM3 and EM5
(Fig.11a,b) and only slightly for the excitation signal EM6 (Fig.11c). Thema-
in improvement of the seat suspension dynamic behaviour is observed at the
low frequency range. At higher frequencies, the friction force in the suspen-
sion system is dominating. In this frequency range both systems, the passive
suspension and the semi-active suspension, yield almost the same response. In
Fig.12, the measured and simulated power spectral densities and transmissi-
bility functions of the semi-active seat suspension for chosen excitation signals
are presented.

Fig. 11. Simulated Pareto-optimal point distribution (◦) for selected input spectral
classes: EM3 (a), EM5 (b), EM6 (c), measured conventional passive (△) and
semi-active (∇) seat suspension systems, mass load on the suspension

system m=100kg



1166 I. Maciejewski, T. Krzyżyński

Fig. 12.Measured ( . . ) and simulated ( ) power spectral densities of the

semi-active seat suspension for selected excitation signals (.....): EM3 (a), EM5 (c),

EM6 (e), measured ( . . ) and simulated ( ) transmissibility functions of the

semi-active seat suspension for selected excitation signals: EM3 (b), EM5 (d),

EM6 (f), mass load on the suspension system m=100kg

However, themeasurement results show that the dynamic behaviour of the
semi-active seat suspensionat optimized controller settings is close to the inve-
stigated Pareto-optimal system for each input spectral class (cf. Fig.11). The
conflicted optimisation criteria, i.e. the SEAT factor and the suspension travel
(x−xs)max are close to the calculated Pareto-optimal point distributions. It
speaks well for the correctness of the proposed methodology for selecting the
control system of the semi-active seat suspension.



Control design of semi-active seat suspension systems 1167

4. Conclusions

The simulation and experimental results show that thepresentedmethodology
of the control system design allows one to define the overall structure of the
semi-active seat suspension. Moreover, the proposed optimisation procedure
assists to adjust the vibro-isolation properties of the semi-active suspension
system. A stiff suspension system can be transformed to a soft suspension
system by an appropriate selection of the controller settings. The proposed
methodology for selecting the control system of semi-active suspension allows
one to find desired dynamic behaviour of the seat for different requirements
defined by machine operators. Each Pareto-optimal controller setting ensures
the optimality of the control system for the conflicted vibro-isolation criteria.

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Projekt systemu sterowania semi-aktywnym układem zawieszenia

siedziska

Streszczenie

wpracyprzedstawiono sposóbprojektowania systemu sterowania semi-aktywnym
układem zawieszenia siedziska. Analizowano algorytm sterowania semi-aktywnego,
który bazuje na modelu odwrotnym elementu sprężystego lub tłumiącego oraz na
kontrolerze głównym. Procedura optymalizacji zaproponowana w pracy dodatkowo
wspomaga dobór nastaw regulatora, ktore to nastawy definiują właściwości wibroizo-
lacyjne semi-aktywnego układu zawieszenia.

Manuscript received November 8, 2010; accepted for print February 7, 2011