Vibration control of active vehicle suspension system using fuzzy logic ...

Fuzzy Inf. Eng. (2010) 4: 361-387 DOI 10.1007/s12543-010-0056-3 ORIGINAL ARTICLE

Vibration Control of Active Vehicle Suspension System Using Fuzzy Logic Algorithm Essam Allam · Hesham Fath Elbab · Magdy Abdel Hady · Shawki Abouel-Seoud

Received: 18 March 2010/ Revised: 30 August 2010/ Accepted: 21 November 2010/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2010

Abstract Vehicle suspension along with tires and steering linkages is designed for safe vehicle control and to be free of irritating vibrations. Therefore the suspension system designs are a compromise between ride softness and handing ability. However, this work is concerned with a theoretical investigation into the ride behavior of actively suspended vehicles. It is based on using fuzzy logic control (FLC) to implement a new sort of active suspension system. Comparisons between the behavior of active suspension system with FLC with those obtained from active systems with linear-quadratic regulator (LQR), ideal skyhook system and the conventional passive suspension systems. Results are introduced in such a way to predict the benefits that could be achieved from a fuzzy logic system over other competing systems. Furthermore, a controller is designed and made by using results of FLC system, theoretical inputs are used to examine the validity of this controller. Moreover, comparison between actual outputs from this controller with those obtained theoretically is made to judge the validity of the controller. The results indicate that the controller has a good capability in simulation of the theoretical model. Keywords Body acceleration · Suspension working space · Dynamic tire deflection · Fuzzy control · Active suspension · Skyhook · Passive suspension

1. Introduction During the past thirty years a number of linear and non-linear suspension system models have been proposed to study the effect of suspension design on ride and handling improvements. The models range in complexity from simple quarter vehicle models to complex multi degree of freedom models. The assumptions made in deriving the equations of motion of the linear suspension models are: 1) the sprung mass Essam Allam () · Hesham Fath Elbab · Magdy Abdel Hady · Shawki Abouel-Seoud Automotive and Tractors Engineering Department, Helwan University, Cairo, Egypt email: [email protected]

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Essam Allam · Hesham Fath Elbab · Magdy Abdel Hady · Shawki Abouel-Seoud (2010)

is considered to be a rigid mass as are similarly all unsprung masses, 2) the available working space is large enough so that the suspension always operates without contacting the bump stops, 3) the tires assumed to be in constant contact with the road surface and are modeled as linear springs (sometimes damping elements are included), 4) the front and rear suspensions are modeled as linear spring and damper units with the possible addition of anti-roll bars. The next problem is to combine the input, described either in the frequency or time domain, with the vehicle model to obtain the output of interest [1, 2]. The performance of passive suspension systems have been studied by using either two mass quarter car model or various configurations for a three mass system representation consisting of an additional mass attached to the sprung mass and/or the unsprung mass. It is concluded that it is possible to reach a high degree of ride comfort without deterioration in the contact between wheels and road by using a dynamic absorber of mass equal to the wheel mass, but incorporating an auxiliary mass equal to the unsprung mass seems to be impractical. Furthermore, the validity of some of these linear models using actual measured road roughness profiles. The roughness inputs and the acceleration at the body connection points were measured for different quality roadways. Using these measurements, they showed that the seven degree of freedom model predicts more accurately the vehicle acceleration in the low frequency range (up to 10 Hz). They attributed the discrepancies which appeared over 10 Hz to excitation from tire unevenness [3-6]. The optimization problem essentially treated the suspension system design as an optimal control problem either with full state feedback capabilities or limited state feedback active systems using a gradient search technique or a slow active suspension. One of the practical difficulties in using the active system is the need to measure the body and wheel positions relative to the road. This difficulty is due to the fact that most of the possible methods of measurements are either too expensive or unreliable on all types of road surfaces. In the most important practical limitation is the need to measure the road profile height. Therefore, there has been a movement to use the concept of the limited state feedback active suspension systems. In these systems, the classical optimal control law based on full state feedback is replaced by one involving limited state feedback-omitting, in particular, the ground input information and one which may involve measurement errors were the first to formulate an optimization problem of the active suspension using a three dimensional vehicle model [7-10]. The performance of quarter car model controlled by a fuzzy inference system strategies with either two inputs (suspension working space and sprung mass velocity) and ignored the unsprung mass and tire contact with road or with three inputs (sprung mass acceleration, sprung mass velocity and suspension working space), the results were compared with those of the sky hooks model. The investigations made by setting up a controller based on fuzzy logic control with two inputs (sprung mass velocity and unsprung mass velocity), while fuzzy rules were made to minimize a defined performance index. Moreover, the theoretical investigation made by setting up a controller based on neuro-fuzzy logic control with two inputs (sprung mass acceleration error and the rate of error with reference to sky hooks body acceleration). The results were compared with those of active suspension system with fuzzy logic con-

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troller and also of active suspension system with linear quadratic regulation (LQR) controller. The neuro-fuzzy active suspension was shown to be the best performing system without any increase in the system complexity. The results were compared with those of active suspension system with LQR controller. The analysis show a great development in performance [11,12]. However, the aim of this work is concerned with a theoretical investigation into the ride behavior of actively suspended vehicles. It is based on using fuzzy logic control (FLC) to implement a new sort of active suspension system. Results achieved from the system with FLC are compared with those obtained from active systems with linear control theory, ideal skyhook system and the conventional passive suspension system. Results are introduced in such a way to predict the benefits could be achieved from fuzzy logic system over other competing systems. Furthermore, a controller is designed and made by using results of FLC system, theoretical input is used to examine the validity of this controller. Comparison between actual outputs from this controller with those obtained theoretical is made to judge the validity of the controller. 2. Nomenclature A BA C C2 C3 DT D K1 K2 M1 M2 Rc S WS ts V y1 y2 y3 λ ωc

Road input amplitude, m Body acceleration, m/s2 Coefficient of power spectral density Suspension damping coefficient, Ns/m Skyhook damping coefficient, Ns/m Dynamic tire deflection, m Tire stiffness, N/m Suspension stiffness, N/m Unsprung mass, kg Sprung mass, kg Road roughness coefficient Suspension work space, m Sampling time, s Vehicle velocity, m/s Body acceleration as an output, m/s2 Suspension working space as an output, m Dynamic tire loading as an output, m Road wave length, m Sprung mass natural frequency, rad/s

0.05 5 1300 20000 192000 20000 45 310 4 × 10−4 0.01 10.46 7

3. Theoretical Considerations Passive Suspension Systems This section consists of three parts; the first part introduces simulation of the road disturbance, while the second part concerns with the modeling of the quarter car pas-

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sive suspension system and its analysis. Finally the third part discusses the skyhook suspension model analysis. Road Disturbance Many researches have been made in order to represent the road surface disturbance is so complex to make a single model to reflect all variation of profile on road with different velocities. In this work the road input is represented as a sinusoidal signal. Referring to Ref [3], the road roughness as a sinusoidal signal is represented as follow: 2πV t), (1) λ where A is the amplitude = 0.05m, V is the vehicle velocity = 20m/s, λ is the wavelength of the road profile = 12.56m. x0 = A sin(

Passive Suspension System Analysis A quarter car suspension models are used to represent the passive suspension system. It is consisted of sprung and unsprung masses connected together by a spring and damper. The tire is modeled by a linear spring. Using Fig.1, matrices of masses, forces, stiffness and damping coefficients are as follow:

Fig. 1 Passive suspension model ⎡ ⎤ ⎡ .. ⎤ ⎡ ⎤⎡ ⎤ ⎢⎢⎢ M1 0 ⎥⎥⎥ ⎢⎢⎢ Z1 ⎥⎥⎥ ⎢⎢⎢ 1 −1 ⎥⎥⎥ ⎢⎢⎢ f1 ⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ ⎢⎣⎢ .. ⎥⎥⎦ = ⎢⎢⎣ ⎥⎥ ⎢⎢ ⎥⎥ 0 M2 Z2 0 1 ⎦ ⎣ f2 ⎦ or

(2) ..

MXDD Z = M f f , where MXDD is the matrix of the system masses, M f is the connection matrix, Z is the vector of the system coordinates, f is the passive suspension and the tire force vector. The most convenient form to describe the forces x is

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f = −Mk d − Mc d,

(3)

where Mk and Mc are (n × n) diagonal matrices containing the stiffness and damping coefficients respectively. In the case of the quarter car suspension model, these two matrices are as follows: ⎡ ⎡ ⎤ ⎤ ⎢⎢ K1 0 ⎥⎥⎥ ⎢ ⎥ ⎥⎥⎦ , Mc = ⎢⎢⎢⎢⎣ 0 0 ⎥⎥⎥⎥⎦ . Mk = ⎢⎢⎢⎣ (4) 0 K2 0 C2 The vector d includes the relative displacement across the connecting elements and is given by: ⎤ ⎡ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ Z1 − x0 ⎥⎥⎥ ⎢⎢⎢ 1 0 ⎥⎥⎥ ⎢⎢⎢ Z1 ⎥⎥⎥ ⎢⎢⎢ −1 ⎥⎥⎥ ⎥ ⎢ ⎢ ⎥⎥ ⎢⎢ ⎥⎥ + ⎢⎢ ⎥⎥ x0 d = ⎢⎣ ⎥=⎢ Z2 − Z1 ⎦ ⎣ −1 1 ⎦ ⎣ Z2 ⎦ ⎣ 0 ⎦ or

(5) d = M f T Z + MDu x0 . Substituting by Equation (5) and (3) in Equation (2), the results may be written as: ..

.

.

M xDD Z +M xD Z +M x Z = MuD x + Mu x0 ,

(6)

M xD = M f Mc M Tf , M x = M f Mk M Tf ,

(7)

0

where

MuD = −M f Mc MDu , Mu = −M f Mk MDu .

For the quarter car suspension model used in this work, these matrices are given as: ⎤ ⎡ ⎢⎢⎢ C2 −C2 ⎥⎥⎥ ⎥⎦ , ⎢ M xD = ⎣ −C2 C2 ⎤ ⎡ ⎢⎢⎢⎢ K1 + K2 −K2 ⎥⎥⎥⎥ Mx = ⎣ ⎦, −K2 K2 ⎡ ⎤ ⎡ ⎤ ⎢⎢ 0 ⎥⎥ ⎢⎢ K1 ⎥⎥ MuD = ⎢⎢⎣ ⎥⎥⎦ , Mu = ⎢⎢⎣ ⎥⎥⎦ . 0 0

(8)

The second order Equation (7) may be transferred into a first order equation by defining new state variable x. and the new result may be written as: .

.

x = Ax + B2 x0 + B3 x0 ,

(9)

where A, B2 and B3 are constant matrices which are related to M xDD , M xD , M x , MuD and Mu by: ⎤ ⎡ ⎥⎥⎥ ⎢⎢ 0 I ⎥⎦ , A = ⎢⎢⎣ −1 −1 −M xDD M x −M xDD M xD ⎤ ⎤ ⎡ ⎡ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢ ⎢⎢⎢ 0 0 ⎥ ⎥⎦ . ⎢ = , B B2 = ⎢⎢⎣ ⎦ 3 ⎣ −1 −1 M xDD Mu M xDD MuD

(10)

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For the quarter car suspension model shown in Fig.1, these matrices are given by: ⎡ ⎤ ⎢⎢⎢ 0 0 1 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥ ⎢⎢⎢ 0 0 0 1 ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥ A = ⎢⎢⎢⎢ −(K2 + K1 ) K2 −C2 C2 ⎥⎥⎥⎥⎥ , ⎢⎢⎢ M1 M1 M! M1 ⎥⎥⎥ ⎢⎢⎢ K2 −K2 C2 −C2 ⎥⎥⎥⎥⎦ ⎢⎣ M2 M M M ⎡ ⎤ 2 ⎡ 2⎤ 2 ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ ⎢⎢ 0 ⎥⎥ ⎢ ⎥ ⎢ ⎥ B2 = ⎢⎢⎢ K1 ⎥⎥⎥ , B3 = ⎢⎢⎢⎢ ⎥⎥⎥⎥ . ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎣ ⎥⎦ ⎢⎢⎣ M1 ⎥⎥⎥⎥⎦ 0 0

(11)

The vector of state variables x is related to the vector Z by x1 = Z1 , x2 = Z2 , x3 = Z3 , x4 = Z4 . The output coordinates in the state variable vector x and the road input x0 may be combined into various outputs variables. The result may be written as: .

.

y = τ xD x +τ x x + τuD x + τu x0 . 0

(12)

To indicate this transformation, the following outputs are chosen based on the quarter car suspension model shown in Fig.1. 1. Body acceleration,

..

.

Z 2 = x4 ;

(13)

S WS = x1 − x2 ;

(14)

DT D = x1 − x0 .

(15)

2. Suspension working space,

3. Dynamic tire deflection,

So ⎡ . ⎤ ⎡ ⎤ . ⎡ ⎤ ⎡ . ⎤ ⎡ ⎤ ⎢ x1 ⎥ ⎡ ⎤ ⎢ x1 ⎥ ⎡ ⎤ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ x4 ⎥⎥⎥ ⎢⎢⎢ 0 0 0 1 ⎥⎥⎥ ⎢⎢⎢⎢⎢ . ⎥⎥⎥⎥⎥ ⎢⎢⎢ 0 0 0 0 ⎥⎥⎥ ⎢⎢⎢⎢⎢ ⎥⎥⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢ ⎥ ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ x2 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ x2 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ . ⎥ ⎥ ⎥ ⎢⎢⎢ x1 − x2 ⎥⎥ = ⎢⎢⎢ 0 0 0 0 ⎥⎥⎥ ⎢⎢⎢ . ⎥⎥ + ⎢⎢⎢ 1 −1 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥ + ⎢⎢⎢ 0 ⎥⎥⎥ x0 + ⎢⎢⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ x0 . ⎢⎣ ⎥⎦ ⎥⎦ ⎢⎣ ⎢⎣ ⎥⎦ ⎢⎢⎢ x3 ⎥⎥⎥ ⎢⎣ ⎥⎦ ⎢⎢⎢ x3 ⎥⎥⎥ ⎢⎣ ⎥⎦ x1 − x 0 −1 0 0 0 0 ⎢⎣ . ⎥⎦ 1 0 0 0 ⎢⎣ ⎥⎦ 0 x4 x4

(16)

In order to carry out the ride analysis in response to road input for the vehicle models, several possible methods are available to solve the previous set of equations of motion. One of them is based on using the difference theories. The basic idea is as follows: . . The equation x = Ax + B2 x0 + B3 x0 is written in the form:

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xk+1 = eAT xk + (eAT − I)A−1 B2 x0k + (eAT − I)−1 B3 x0k ,

(17)

where eAT : Exponentional matrix, T : Sampling time, I: Unit matrix. Sky-hook Suspension Model A skyhook suspension model represents a system which is not real and could not be achieved in practice. Nevertheless, it is used to represent an ideal suspension system. This system is similar to a conventional passive suspension system (see Fig.2) where the spring and unsprung masses are connected together by a spring and damper and the tire is presented by a linear spring. The only difference is that the sprung mass is hooked to a rigidly via a passive damper element.

Fig. 2 Skyhook suspension model The equations of motion may be written as: ⎡ ⎤ ⎤ ⎡ .. ⎤ ⎡ ⎡ ⎤⎢ f ⎥ ⎢⎢⎢ M1 0 ⎥⎥⎥ ⎢⎢⎢ Z1 ⎥⎥⎥ ⎢⎢⎢ 1 −1 0 ⎥⎥⎥ ⎢⎢⎢⎢ 1 ⎥⎥⎥⎥ ⎥⎥ ⎢⎢ .. ⎥⎥ = ⎢⎢ ⎥⎥ ⎢⎢⎢ f2 ⎥⎥⎥ ⎢⎢⎣ 0 M2 ⎦ ⎣ Z2 ⎦ ⎣ 0 1 −1 ⎦ ⎢⎢⎣ ⎥⎥⎦ fs or

(18) ..

.

MXDD Z = M f f , f = −Mk d − Mc d, where Mk and Mc are (3×3) matrices containing the stiffness and damping coefficients respectively. In the case of the quarter car suspension model, these two matrices are as follows: ⎡ ⎡ ⎤ ⎤ ⎢⎢⎢ K1 0 0 ⎥⎥⎥ ⎢⎢⎢ 0 0 0 ⎥⎥⎥ ⎢⎢ ⎥⎥ ⎥⎥ ⎢⎢ Mk = ⎢⎢⎢⎢ 0 K2 0 ⎥⎥⎥⎥ , Mc = ⎢⎢⎢⎢ 0 C2 0 ⎥⎥⎥⎥ . ⎢⎣ ⎥⎦ ⎥⎦ ⎣⎢ 0 0 0 0 0 Cs

(19)

The vector d includes the relative displacement across the connecting elements and is given by:

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⎡ ⎤ ⎡ ⎡ ⎤ ⎤ ⎢⎢⎢ Z1 − x0 ⎥⎥⎥ ⎢⎢⎢ 1 0 ⎥⎥⎥ ⎡ ⎤ ⎢⎢⎢ −1 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ Z1 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ d = ⎢⎢⎢ Z2 − Z1 ⎥⎥⎥ = ⎢⎢⎢ −1 1 ⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ + ⎢⎢⎢ 0 ⎥⎥⎥ x0 ⎢⎣ ⎥⎦ ⎢⎣ ⎢⎣ ⎥⎦ ⎥⎦ Z2 0 − Z2 0 0 −1 or

(20) d = M f T Z + MDu x0 . Substituting by Equation (18) and (20) in Equation (17), the results may be written

as ..

.

.


(21)

⎤ ⎤ ⎡ ⎡ ⎢⎢ C2 −C2 ⎥⎥⎥ ⎢ K + K2 −K2 ⎥⎥⎥ ⎥⎦ , M x = ⎢⎢⎢⎣ 1 ⎥⎦ , M xD = ⎢⎢⎣ −C2 C2 −K2 K2 ⎡ ⎤ ⎡ ⎤ ⎢⎢ 0 ⎥⎥ ⎢⎢ K1 ⎥⎥ MuD = ⎢⎢⎣ ⎥⎥⎦ , Mu = ⎢⎢⎣ ⎥⎥⎦ . 0 0

(22)

0

The second order Equation (21) may be transferred into a first order equation by defining new state variable x, as explained earlier to be in the form: .

.

x = Ax + B2 x0 + B3 x,

(23)

⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ 0 0 1 0 ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ 0 0 0 1 ⎢⎢⎢ ⎥⎥⎥ + K ) K −C C −(K 2 1 2 2 2 ⎥⎥⎥ , A = ⎢⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ M1 M1 M! M1 ⎥ ⎢⎢⎢ K2 −K2 C2 −(C2 + C s ) ⎥⎥⎥⎥⎦ ⎢⎣ M2 M2 M2 M2 ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ ⎢⎢ 0 ⎥⎥ ⎢ ⎥ ⎢ ⎥ B2 = ⎢⎢⎢ K1 ⎥⎥⎥ , B3 = ⎢⎢⎢⎢ ⎥⎥⎥⎥ . ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎣ ⎥⎦ ⎢⎢⎣ M1 ⎥⎥⎥⎥⎦ 0 0

(24)

0

where

The output coordinates will remain unchanged as in the case of a passive suspension system, see Equation (16). Active Suspension System with LQR Controller Analysis It consists of sprung and unsprung masses connected together by a spring, damper and actuator. The unsprung mass interacts with the ground and between them there is a spring. Using Fig.3, matrices of masses and stiffness, damper and forces are used for analysis without any difficulties to formulate the equations, as follows:

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Fig. 3 Active (LQR) suspension model

⎤ ⎡ .. ⎤ ⎡ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ M1 0 ⎥⎥⎥ ⎢⎢⎢ Z1 ⎥⎥⎥ ⎢⎢⎢ 1 −1 ⎥⎥⎥ ⎢⎢⎢ f1 ⎥⎥⎥ ⎢⎢⎢ −1 ⎥⎥⎥ ⎥⎥ ⎢⎢ ⎥⎥ + ⎢⎢ ⎥⎥ u ⎥⎥⎦ ⎢⎢⎣ .. ⎥⎥⎦ = ⎢⎢⎣ ⎢⎢⎣ 0 M2 Z2 0 1 ⎦ ⎣ f2 ⎦ ⎣ 1 ⎦ or

(25) ..

MXDD Z = M f f + M f 2 u, where M f 2 is the connection matrix. .


(26)

where Mk and Mc are (2×2) matrices containing the stiffness and damping coefficients respectively. These two matrices are as follows: ⎡ ⎡ ⎤ ⎤ ⎢⎢⎢ K1 0 ⎥⎥⎥ ⎢⎢⎢ 0 0 ⎥⎥⎥ ⎢ ⎢ ⎥ ⎥⎥ . Mk = ⎢⎣ (27) ⎥ , Mc = ⎢⎣ 0 K2 ⎦ 0 C2 ⎦ The vector d includes the relative displacement across the connecting elements and is given by: ⎤ ⎡ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ Z1 − x0 ⎥⎥⎥ ⎢⎢⎢ 1 0 ⎥⎥⎥ ⎢⎢⎢ Z1 ⎥⎥⎥ ⎢⎢⎢ −1 ⎥⎥⎥ ⎥ ⎢ ⎢ ⎥⎥ ⎢⎢ ⎥⎥ + ⎢⎢ ⎥⎥ x0 d = ⎢⎣ ⎥=⎢ Z2 − Z1 ⎦ ⎣ −1 1 ⎦ ⎣ Z2 ⎦ ⎣ 0 ⎦ or

(28) d = M f Z + MDu x0 . T

Substituting by Equation (26) and (28) in Equation (25), the results may be written as ..

.

.

M xDD Z +M xD Z +M x Z = MuD x + Mu x0 , 0

where

(29)

370


M xD

⎤ ⎤ ⎡ ⎡ ⎢⎢⎢ C2 −C2 ⎥⎥⎥ ⎢⎢⎢ K1 + K2 −K2 ⎥⎥⎥ ⎥ ⎥⎦ , ⎢ ⎢ =⎣ ⎦ , Mx = ⎣ −C2 C2 −K2 K2 ⎡ ⎤ ⎡ ⎤ ⎢⎢ K1 ⎥⎥ ⎢⎢⎢⎢ 0 ⎥⎥⎥⎥ MuD = ⎣ ⎦ , Mu = ⎢⎢⎣ ⎥⎥⎦ . 0 0

(30)

The second order Equation (29) may be transferred into the first order equation by defining new state variable x, as explained earlier, to be in the form .

x = Ax + B2 x0 + Bu u,

(31)

where A, B2 and B3 are constant matrices which are related to M xDD , M xD , M x , MuD and Mu by: ⎤ ⎡ ⎥⎥⎥ ⎢⎢ 0 I ⎥⎦ , A = ⎢⎢⎣ −1 −1 −M xDD M x −M xDD M xD ⎤ ⎤ ⎡ ⎡ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢ ⎢⎢⎢ 0 0 ⎥ ⎥⎦ , ⎢ = , B B2 = ⎢⎢⎣ ⎦ 3 ⎣ −1 −1 M xDD Mu M xDD M f 2 ⎤ ⎡ ⎢⎢⎢ 0 0 1 0 ⎥⎥⎥ ⎥⎥ ⎢⎢⎢ ⎢⎢⎢ 0 0 0 1 ⎥⎥⎥⎥ ⎥⎥ ⎢⎢⎢ A = ⎢⎢⎢⎢ −(K2 + K1 ) K2 −C2 C2 ⎥⎥⎥⎥⎥ , ⎢⎢⎢ M1 M1 M! M1 ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥ K −K 2 2 C 2 −C 2 ⎥ ⎢⎣ ⎦ M2 M2 ⎡M2 ⎤M2 ⎡ ⎤ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢ ⎥ B2 = ⎢⎢⎢⎢⎢ K1 ⎥⎥⎥⎥⎥ , B3 = ⎢⎢⎢⎢⎢ −1 ⎥⎥⎥⎥⎥ . ⎢⎢⎢ ⎢⎢⎢ M1 ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎣ M1 ⎥⎥⎦ ⎢⎢⎢⎢ 1 ⎥⎥⎥⎥ ⎣ ⎦ 0 M2

(32)

(33)

The output coordinates in the state variable vector X and the road input x0 may be combined into various outputs variables. Therefore the result may be written as: .

.

y = τ xD x +τ x x + τuD x0 + τu x0 .

(34)

To indicate this transformation, the following outputs are chosen based on the active suspension quarter car model shown in Fig.3. Body acceleration, .. . (35) Z 2 = x4 . Suspension working space, S WS = x1 − x2 .

(36)

DT L = x1 − x0 .

(37)

Dynamic tire deflection, So

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371

⎡ . ⎤ ⎡ ⎤ . ⎡ ⎤ ⎡ . ⎤ ⎡ ⎤ ⎢ x1 ⎥ ⎡ ⎤ ⎢ x1 ⎥ ⎡ ⎤ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ x4 ⎥⎥⎥ ⎢⎢⎢ 0 0 0 1 ⎥⎥⎥ ⎢⎢⎢⎢⎢ . ⎥⎥⎥⎥⎥ ⎢⎢⎢ 0 0 0 0 ⎥⎥⎥ ⎢⎢⎢⎢⎢ ⎥⎥⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢ ⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ x2 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ x2 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ . ⎢⎢⎢ x1 − x2 ⎥⎥⎥ = ⎢⎢⎢ 0 0 0 0 ⎥⎥⎥ ⎢⎢⎢ . ⎥⎥⎥ + ⎢⎢⎢ 1 −1 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ + ⎢⎢⎢ 0 ⎥⎥⎥ x0 + ⎢⎢⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ x0 . ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎢⎢ x3 ⎥⎥⎥ ⎢⎣ ⎥⎦ ⎢⎢⎢ x3 ⎥⎥⎥ ⎢⎣ ⎥⎦ x1 − x0 −1 0 0 0 0 ⎢⎣ . ⎥⎦ 1 0 0 0 ⎢⎣ ⎥⎦ 0 x4 x4

(38)

Active Suspension System with Fuzzy Controller Fuzzy Logic Control (FLC) is a problem-solving control system methodology that lends itself to implementation in systems ranging from simple, small, embedded micro-controllers to large, networked, multi-channel PC or workstation-based data acquisition and control systems. It can be implemented in hardware, software, or a combination of both. FLC provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. FLC approach to control problems mimics how a person would make decisions, only much faster. Active Suspension with Fuzzy Controller Analysis It consists of sprung and unsprung masses connected together by a spring, damper and actuator. The unsprung mass interacts with the ground and between them there is a spring, as seen in the Fig.4, matrices of masses and stiffness, damper and forces are used for analysis without any difficulties to formulate the equations as follows:

Fig. 4 Active (FLC) suspension model ⎤ ⎡ .. ⎤ ⎡ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ M1 0 ⎥⎥⎥ ⎢⎢⎢ Z1 ⎥⎥⎥ ⎢⎢⎢ 1 −1 ⎥⎥⎥ ⎢⎢⎢ f1 ⎥⎥⎥ ⎢⎢⎢ −1 ⎥⎥⎥ ⎥⎥⎦ ⎢⎣⎢ .. ⎥⎥⎦ = ⎢⎢⎣ ⎢⎢⎣ ⎥⎥ ⎢⎢ ⎥⎥ + ⎢⎢ ⎥⎥ u 0 M2 Z2 0 1 ⎦ ⎣ f2 ⎦ ⎣ 1 ⎦ or

(39)

..

MXDD Z = M f f + M f 2 u, where M f 2 is the connection matrix number two, .


(40)

372


where Mk and Mc are (2×2) matrices containing the stiffness and damping coefficients respectively. In the case of the quarter car suspension model, these two matrices are as follows: ⎡ ⎡ ⎤ ⎤ ⎢⎢⎢ K1 0 ⎥⎥⎥ ⎢⎢⎢ 0 0 ⎥⎥⎥ ⎥⎥ , Mc = ⎢⎢⎣ ⎥⎥ . Mk = ⎢⎢⎣ 0 K2 ⎦ 0 C2 ⎦

(41)

The vector d includes the relative displacement across the connecting elements and is given by: ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎢⎢⎢ Z1 − x0 ⎥⎥⎥ ⎢⎢⎢ 1 0 ⎥⎥⎥ ⎢⎢⎢ Z1 ⎥⎥⎥ ⎢⎢⎢ −1 ⎥⎥⎥ ⎥⎥ = ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ + ⎢⎢ ⎥⎥ x0 d = ⎢⎢⎣ Z2 − Z1 ⎦ ⎣ −1 1 ⎦ ⎣ Z2 ⎦ ⎣ 0 ⎦ or

(42) d = M f T Z + MDu x0 . Substituting by Equation (42) and (40) in Equation (39), the results may be written

as ..

.

.


(43)

⎤ ⎤ ⎡ ⎡ ⎢⎢ C2 −C2 ⎥⎥⎥ ⎢ K + K2 −K2 ⎥⎥⎥ ⎥⎦ , M x = ⎢⎢⎢⎣ 1 ⎥⎦ , M xD = ⎢⎢⎣ −C2 C2 −K2 K2 ⎡ ⎤ ⎡ ⎤ ⎢⎢ K1 ⎥⎥ ⎢⎢ 0 ⎥⎥ MuD = ⎢⎢⎣ ⎥⎥⎦ , Mu = ⎢⎢⎣ ⎥⎥⎦ . 0 0

(44)

0

where

The second order Equation (43) may be transferred into the first order equation by defining new state variable x, as explained earlier to be in the form .

x = Ax + B2 x0 + Bu u,

(45)

where A, B2 and B3 are constant matrices which are related to M xDD , M xD , M x , MuD and Mu by: ⎤ ⎡ ⎥⎥⎥ ⎢⎢ 0 I ⎥⎦ , A = ⎢⎢⎣ −1 −1 −M xDD M x −M xDD M xD ⎤ ⎤ ⎡ ⎡ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢ ⎢⎢⎢ 0 0 ⎥ ⎥⎦ . ⎢ = , B B2 = ⎢⎢⎣ ⎦ 3 ⎣ −1 −1 M xDD Mu M xDD M f 2

(46)

For the quarter car suspension model show in Fig.4, these matrices are give by:

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⎡ ⎢⎢⎢ 0 ⎢⎢⎢ ⎢⎢⎢ 0 ⎢⎢⎢ + K1 ) −(K 2 A = ⎢⎢⎢⎢ ⎢⎢⎢ M 1 ⎢⎢⎢ K2 ⎢⎣ M2 ⎡ ⎤ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ B2 = ⎢⎢⎢⎢⎢ K1 ⎥⎥⎥⎥⎥ , ⎢⎢⎢ ⎥ ⎢⎢⎣ M1 ⎥⎥⎥⎥⎦ 0

373

⎤ 1 0 ⎥⎥⎥ ⎥⎥ 0 1 ⎥⎥⎥⎥ ⎥ −C2 C2 ⎥⎥⎥⎥ , ⎥⎥ M! M1 ⎥⎥⎥ C2 −C2 ⎥⎥⎥⎥⎦ ⎡M2 ⎤M2 ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ ⎢⎢ ⎥⎥ B3 = ⎢⎢⎢⎢⎢ −1 ⎥⎥⎥⎥⎥ . ⎢⎢⎢ M1 ⎥⎥⎥ ⎢⎢⎢ 1 ⎥⎥⎥ ⎢⎣ ⎥⎦ M2

0 0 K2 M1 −K2 M2

(47)

The output coordinates will remain unchanged as in the case of an active suspension system with LQR, seen in Equation (38). 4. Fuzzy Logic Controller The fuzzy rule sets usually have several antecedents that are combined by using fuzzy operators, such as AND, OR and NOT, though again the definitions tend to vary: AND, in one popular definition, simply uses the minimum weight of all the antecedents, while OR uses the maximum value. There is also a NOT operator that subtracts a membership function from 1 to give the “complementary” function. There are several different ways to define the result of a rule, but one of the most common and simplest is the “max-min” inference method, in which the output membership function is given the truth value generated by the premise. Rules can be solved in parallel in hardware, or sequentially in software. The results of all the rules that have fired are “defuzzified” to a crisp value by one of several methods. There are dozens in theory, each with various advantages and drawbacks. The “centroid” method is very popular, in which the “center of mass” of the result provides a crisp value. Another approach is a “height” method, which takes the value of the biggest contributor. The centroid method favors the rule with the output of greatest area, while the height method obviously favors the rule with the greatest output value. The example as in Fig.5 demonstrates max-min inferencing and centroid defuzzification for a system with input variables “x”, “y”, and “z” and an output variable “n”. Notice how each rule provides a result as a truth value of a particular membership function for the output variable. In centroid defuzzification the values are OR’d, that is, the maximum value is used and values are not added, and the results are then combined by using a centroid calculation. Fuzzy control system design is based on empirical methods, basically a methodical approach to trial-and-error. The general process is as follows: • Document the system’s operational specifications and inputs and outputs. • Document the fuzzy sets for the inputs. • Document the rule set.

374


Fig. 5 Centroid defuzzification using MAX-MIN inferencing • Determine the defuzzification method. • Run through test suite to validate system, adjust details as required. • Complete document and release to production. The design of a fuzzy controller for the active suspension system, where inputs are a body acceleration error and rate of its change when referred to that of the theoretical model “skyhook model”. The block diagram of this control system appears as Fig.6. There are two input variables of body acceleration error and rate of its change, and a single output variable, the actuating force setting. The suspension bodies oscillate from upwards to downwards, so the actuating force can be positive or negative. The fuzzy set mappings are shown in Fig.7. The linguistic terms are defined as follows: NB: Negative Big. NM: Negative Medium. NS: Negative Small. ZE: Zero. PS: Positive Small. PM: Positive Medium. PB: Positive Big.

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Fig. 6 Fuzzy controller schematic

Fig. 7 The fuzzy logic active suspension The rule set includes such rules as Table 1. Rule 13: If Error is NM AND Delta Error is PM, then Actuating Force is Z. Rule 12: If Error is NM AND Delta Error is PS, then Actuating Force is NS.

375

376


Table 1: Fuzzy logic controller rules set. ΔBAerror

BAerror

NB NM NS Z PS PM PB

NB NB NB NM NM NS NS Z

NM NB NM NM NS NS Z PS

NS NM NM NS NS Z PS PS

Z NM NS NS Z PS PS PM

PS NS NS Z PS PS PM PM

PM NS Z PS PS PM PM PB

PB Z PS PS PM PM PB PB

Rule 20: If Error is NS AND Delta Error is PM, then Actuating Force is PS. Rule 19: If Error is NS AND Delta Error is PS, then Actuating Force is Z. In practice, the controller accepts the inputs and maps them into their membership functions and truth values. These mappings are then fed into the rules. If the rule specifies an AND relationship between the mappings of the two input variables, as the examples above do, the minimum of the two is used as the combined truth value; if an OR is specified, the maximum is used. The appropriate output state is selected and assigned by a membership value at the truth level of the premise. The truth values are then defuzzified. 5. Controller with Error Input As mentioned early, the FLC depends on feedback of known measured input to generate a control output. The input is error in body acceleration. This means simply that the system requires a direct measurement of the body acceleration via a suitable accelerometer. The road input is feed to the central processing unit (CPU) of the controller to evaluate the ideal acceleration. This acceleration is compared with the measured acceleration signal to evaluate the error. The CPU also evaluates the rate of error (Δ error). These two inputs are fed to the controller and the output is the control force u. The graphical user interface (GUI) is used to generate the membership functions for input and output as shown in Fig.8. The input is the body acceleration error with 7 membership functions as follows: NB: Negative Big error. NM: Negative Medium error. NS: Negative Small error. Z: Zero error. PS: Positive Small error. PM: Positive Medium error. PB: Positive Big error.

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Fig. 8 Indicates a schematic presentation of the FLC And the universe of discourse is from −1 to 1 and the error signal is modified before entering the controller through gain and this gain is used to tune the fuzzy inference system. The second step is to start up the output and it is a single output (actuated force) and has 7 membership functions: NB: Negative Big force. NM: Negative Medium force. NS: Negative Small force. Z: Zero force. PS: Positive Small force. PM: Positive Medium force. PB: Positive Big force. The universe of discourse is from −1 to 1 and an actuating force signal is modified before entered the plant through a gain and this gain is used to tune the fuzzy inference system. The third step is to generate the lookup table however, this is the most difficult part of the problem due to the need of experience and the great nonlinearity of the system, and for example it is not easy to define how much force is needed for a big negative error and a small positive change in error. In order to overcome this problem the performance is evaluated for each crisp output and used to tune the controller. Table 2: Summarizes the optimum set of FLC rules. Body acc. error

NB

NM

NS

Z

PS

PM

PB

Actuated force

B

NM

NS

Z

PS

PM

PB

For instant, if the error of the body acceleration is NS (Negative Small), the controller output will be NS (Negative Small). On the other hand, if the error is PM (Positive Medium), the control output will be PM (Positive Medium) in Table 2. To evaluate the performance parameters of FLC active suspension system, a simulink model is introduced to combine the control with the system. Fig. 9 shows this combination. The road input is fed to the skyhook model to generate the ideal acceleration.

378


This acceleration is then used to determine the error. At time t=0, the error is set equal to zero. As mentioned above in the previous section, the error is used to evaluate control signal u; this control signal is used to evaluate the system performance of an FLC active suspension system.

Fig. 9 The fuzzy logic active suspension model with two inputs represented by simulink The first controller is based on using two inputs, error and change of error of body acceleration. Seven membership functions are used to describe each of these inputs. A centroid method is employed to generate crisp value of the output control signal. In the second controller type is simplified to use only error of body acceleration is the single input. GUI is used to build up computer programs of the generated systems. The results are generated for the two designed fuzzy logic controllers and include time histories of body acceleration, suspension working space and dynamic tire deflection. 6. Controller Design For theoretical study, the fuzzy logic control has the capability of improving further ride comfort and road holding parameter without any increase in the used working space. However, there is a need to examine the validity of this controller this experimentally. In principle at least, one needs to build up a test rig including a suspension system with a fuzzy logic controller, suitable excitation presentation, various needed sensors and a complete measuring system. This is however, far behind the scope of this work. Here the ides is simplified to the following steps: 1. Introducing a practical and simple design of controller. 2. Testing the controller through a defined and known input. 3. Evaluating the controller response to this input. 4. Comparing between achieved results with those calculated theoretically using the same excitation. 5. Evaluating the controller effectiveness.

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In this stage, the designed program is loaded to the interface and then to the microcontroller using a programmer interface shown in Fig.10 and Fig.11. The design of the controller is only introduced based on one input i.e., the error in the body acceleration. This controller uses only the error in body acceleration as input and generates it in response the required control the signal. A complete set of computer programs are used using Visual Basic language introduced to drive this controller. A simulink model is made to evaluate the relationship between the body acceleration and control force needed. The controller is tested for various acceleration inputs. Besides, experimental outputs are compared with those obtained theoretically using same inputs. Therefore, good agreements between experimental and theoretical outputs were found.

Fig. 10 Indicates a schematic presentation of the FLC

Fig. 11 The programmer connected to PC by serial

380


7. Results and Discussion Fig.12 to Fig.15 show the time histories for the performance parameters of the passive, skyhook, active with LQR and active with FLC (one and two inputs) suspension systems respectively. To highlight the benefits that could be achieved from using fuzzy logic control active suspension system over other competing systems, the passive suspension is used here as a baseline system, while the skyhook suspension system is used to represent the ideal system. Under these bases, a direct comparison between LQR active suspension system with FLC active suspension system is obtained.

Fig. 12 Body acceleration of all competing system

Fig. 13 Dynamic tire deflection of all competing system The root mean square (RMS) of the time histories for the performance parameters of body acceleration (BA), suspension working space (SWS) and dynamic tire deflection (DTD) are collected and presented in Fig.16 to Fig.18 respectively. In terms of RMS value, the following remarks can be done as follows:

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Fig. 14 Suspension working space of all competing system One input

382


Two input

Fig. 15 Performance of the (FCL) active suspension system 1. The FLC active suspension system with signal input approaches closely the system with two inputs. This is an important finding since it shows that the fuzzy logic control can still perform well even for the case of using limited feedback information. 2. The FLC active suspension systems approach the performance level of ideal skyhook model, this finding is also important, because a part of fuzzy controller takes input from this ideal system, and tune the active suspension to reduce the error and hence to perform very closely to this ideal system. 3. Fuzzy logic control shows that significant improvements can be achieved in comparison with LQR active suspension system. In terms of body acceleration, the improvement appears as a reduction in RMS value by 41.4%; at the same usage of suspension working space. Also, FLC reduces the dynamic tire

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Fig. 16 Body acceleration of all competing system

Fig. 17 Suspension working space of all competing system deflection RMS value by 55.5% when comparing with LQR active suspension system. 4. In comparison with passive suspension system, LQR active suspension system reduces body acceleration, suspension working space and dynamic tire deflection by 27.5%, 28.2% and 23.8% respectively. These improvements become 57.5%, 28.2% and 66.2% in case of using FLC system. In terms of system compiling, FLC appears to be much more easily to realize in practice. It depends on using an accelerometer to measure the body acceleration. A micro-controller with a suitable processor would be enough to evaluate the control

384


Fig. 18 Dynamic tire deflection all competing system of all competing system signal. On the other hand, the LQR active suspension system needs to have feedback of body acceleration and a relative displacement transducer to measure body to wheel displacement as well as a sensor or a state estimator to measure or estimate the road input. Therefore, it could be stated that the suspension performance level of FLC active suspension system is accomplished with less difficulty to be applied in practice when compared with LQR active suspension system. A simple and practical design of fuzzy logic controller is introduced. This controller uses only the error in body acceleration as input and generates in response the required control the signal. The results generated are designed to linearize the relationship between controller input/output responses as shown in Fig.19 and Fig.20.

Fig. 19 BA and actuating force for active suspension system with FCL controller This approximation is used to evaluate the system performance and compared with the obtained based on an actual nonlinear form. Results are collected and presented in Fig.21.

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Fig. 20 BA and actuating force for active suspension designed controller system with FCL controller

Fig. 21 Performance parameters with FCL and the designed controller

This figure compares directly between linearized system and actual FLC system.

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In terms of body acceleration, suspension working space and dynamic tire deflection, no much difference can be seen between linear and nonlinear systems. Under this finding, the linear model is used to design the controller. 8. Conclusion 1. Fuzzy logic control shows that significant improvements can be achieved in comparison with LQR active suspension system at the same usage of suspension working space: In terms of body acceleration, the reduction in RMS value by 41.4%; in terms the dynamic tire deflection rms value by 55.5%. 2. The FLC active suspension system with signal input approaches closely the system with two inputs. Moreover, the FLC active suspension systems approach the performance level of an ideal skyhook model. 3. An actual fuzzy logic controller is made and used in evaluating experimental results. The controller uses only error in body acceleration as an input and generates in response to the required control crisp values. 4. The controller is tested for various acceleration inputs. Experimental outputs are compared with those obtained theoretically using same inputs. Good agreements between experimental and theoretical outputs were found. Acknowledgements I would like to thank the staff and research students within the Advanced Materials Research Institute, Matria Engineering College, College-Cairo, Egypt. References 1. Horton D (1986) An introduction to ride analysis in vehicle dynamics. Research Report DAC12, Dept. of Mechanical Engineering, Leeds University 2. International Standard ISO 2631, (1974), (1978), (1978/Al), (1982) Guide for the evaluation of human exposure to whole-body vibrations 3. Ryba D (1974 a) Improvements in dynamics characteristics of automobile suspension systems (Part I, Two Mass System). Vehicle System Dynamics 3: 17-46 4. Ryba D (1974 b) Improvements in dynamics characteristics of automobile suspension systems (Part II, Three Mass System). Vehicle System Dynamics 3: 55-98 5. Healey A J, Nathman E, Smith C C (1977) An analytical and experimental study of automobile dynamics with random roadway input. Transaction of the ASME, Journal of Dynamic Systems, Measurement, and Control, December: 284-292 6. Sharp R S, Hassan S A (1984) The fundamentals of passive automotive suspension system design. Society of Environmental Engineers Conference on Dynamics in Automotive Engineering, Cranfield Inst. Tech. : 104-115 7. Abdel Hady M B (1989) The effect of active suspension control on vehicle ride behaviour. Ph.D. thesis 8. Abdel Hady M B, Crolla D A (1989) Theoretical analysis of active suspension performance using a four-wheel vehicle model. Proc. Instn. Mech. Engrs 203: 125-135 9. Eldemerdash S M (1995) A practical hydro-pneumatic slow active suspension. Engineering Research Journal 4: 160-172

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10. Abouel-seoud S A, Abdel-tawaab A (1998) A theoretical study of the hydro-pneumatic semi-active suspension system with preview. Heavy Vehicle Systems, Int. J. of Vehicle Design, Vol. 5, No. 2 11. Abdel Hady M B (2003) A fuzzy controller for automotive active suspension systems. Society of Automotive Engineers, Inc. 2003-01-1417. 12. Abdel Hady M B (2003) A neural fuzzy approach for controlling active vehicle suspension systems. International Mechanical Engineering Congress IMECE2003-42875