A Quarter Active Suspension System Based Ground-Hook Controller

An alternative design technique for active suspension system of vehicle using a developved ground-hook damping system as a reference is proposed. The controller parameters are determined using Lyapunov method and can be tuned to precisely achieve the type of desired response which given by reference model. The simulation result show that the designed active suspension system based ground-hook reference model is able to significantly improve the ride comfort and the road holding compared with semi-active suspension.


Introduction
In recent years there has been a significant growth in the design of suspension systems by which should reduce vibration due to the external disturbances to the maximum possible degree [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Two criteria of good vehicle suspension performance are typically their ability to provide good road handling and increased passenger comfort. It is difficult to meet both requirements, since this involves opposing tendencies in the selection of the main parameters of suspension, i.e. stiffness and damping. Suspension should be rigid enough to effectively carry the static load, and soft enough to ensure the good isolation of vibrations. These two factors are related to the body vertical acceleration and the suspension deflection. If the suspension system is unable to absorb the vibration caused by the unevenness of the road surface, a significant deflection of the suspension will occur resulting unsafe condition of the vehicle. Moreover, the sprung mass of the vehicle will also exhibit a large vertical acceleration which can be the reason of a ride discomfort. In order to achieve a good ride quality, the performance of the suspension system plays an important role.
Since most of the vehicle suspensions do not operate with a single well defined roadway input, real suspensions are rarely optimal for any particular road and speed. On smooth roads the suspension will be stiffer than necessary and on rough roads softer that desirable. This leads to the concept of an adaptable suspension which can change its characteristics bases on measured performance. By using an adaptation algorithm, it is expected that the optimal performance can be maintained over the wide range of input conditions typically encountered by a vehicle. In this paper, an alternative design technique for active suspension system of vehicle using adaptive controller based skyhook damper model is proposed. The reference model and the controller systems are designed and determined using skyhook model and Lyapunov method, respectively. The control design was applied to two different suspension models, namely the body vertical acceleration and the suspension deflection models, each representing comfort and safety factors.

A Quarter model of suspension system
In most cases, dynamic systems are stimulated by the movement of anchorages. The suspension system of car is stimulated as a result of moving on the ruggedness of the road. A car is moving on a bumpy road in which some forces are exerted on the passengers as the result of the change of the length of springs and dampers. The schematic diagram of a quarter-vehicle suspension system under investigation can be presented as in Fig. 1, where ms and mu each represents the sprung mass and unsprung mass respectively, c a represents absorber coefficient, k a represents spring constants, k t represents the tire equivalent constants, z r represents surface roughness of road, z s represents vehicle body position and z u represent the tire vehicle position. The variable Q represents an applied actuator force for the active suspension. The values of the parameters in this model, which are taken from a typical passenger car, are collected in Table 1. However, the value of a k is unknown therefore it can be assumed that there is no limitation in generating the actuator force Q. Using Newton's second law, the equations of motion for the model in Fig. 1 By defining , and u s z z x 4 , then the equations (1) and (2) can be expressed in the state space form as follows: where, the matrices A, B, and C are given by.

A ground-hook damper as reference model
In order to specify the desired performance of the suspension system, the reference model is derived from quarter-vihicle ground-hook model (see Fig. 2). Groundhook proposed the control method Ground-hook that in which an assumed amortize has been incorporated between the sprung mass and the ground in order to reduce dynamic force of tire and to improve the conductibility of the car. In Fig. 2, The linear equation of the state for reference model Ground-hook from the order of four has been presented as follows: where, b c and g c are indicate a variable damper and a coefficient that connected between the unsprung mass and the fixed fictitious frame on the ground, respectively.
are the velocities of sprung mass, unsprung mass, and relative velocity between sprung mass and unsprung mass. Theoretically, ground-hook damper model will improve the responses of the unsprung system. The ideal ground-hook damper model is given by The generated forces by the groundhook model are added to the passive vehicle model which used as a reference on active suspension controller design.

Figure 2.
Ground-hook reference model of a quarter-vehicle suspension system.
In the state space form, equations (4) and (5) is written as where the matrices d A and d C are given by

Controller design
One of the main approach often used in adaptive control design is the model-reference adaptive system. Here, the desired performance is expressed in terms of a reference model, which gives the desired response to a command signal. The desired output of the model is compared to the output of plant, where its error is used to generate an actuating signal. One of most critical point in the design is how to make the error as small as possible. This will depend on the model and the command signal. If it would be possible to make the error equal to zero for all command signals, then perfect model-following would be achieved. Choosing the control law as follows: In order to make e goes to zero, then there must be exist a set of unknown constant gain * By substituting (11) into (10), we have where *

T T )
. In order to drive the adjustment law of the parameter, the Lyapunov function is introduced as Then the parameter adjustment law is obtained as (17)

Discussions
To investigate and observe the performance of the designed adaptive controller to changing dynamics of the vehicle suspension systems, two different types of input signals (i.e., sinusoid and random signals) representing different types of road to stimulate the dynamics of suspension of the vehicle have been identified. It is desired that when the vehicle runs on the three different road types, the controller will choose the coefficients of controller T such that the closed-loop output response of each model will be close to the model-reference response for any changes of the dynamic suspension system. Revisiting the ground-hook scheme in Fig. 2   c1=4000, c2=4000 c1=3000, c2=4000 c1=2500, c2=4000 c1=2000, c2=4000 c1=1000, c2=4000 Passive:c1=2000, c2=0 . The magnitudes of the frequency responses of the sprung-mass acceleration with respect to the road disturbance are shown in Fig. 4. The solid line and the dotted line represent the magnitude plots of the frequency responses of the semi-active (with ground-hook controller) and the active. In Fig. 4, the dashed line depicts a large improvement according to the other values. However, the dashed line shows a deterioration of ride comfort in the frequency range 4-8 Hz and the deterioration of noise & harshness in the frequency range of 12 Hz and higher, when compared with the passive suspension. If the results between semi-active and active suspension systems are compared in Figs. 5-9. It can be seen that, for the same input road disturbance in the form of random, the output response of semi-active suspension will follow the disturbance, whereas for the active suspension using the designed model reference adaptive technique then the output response only slightly react accordingly to the disturbance. The velocities and the accelerations of the sprung and unsprung masses (Figs. 6 -9), respectively, of the active suspension were highly reduce compate to the only semi-active suspension control design. The performance of the sprung mass displacement and sprung mass veocity were improved about 20% and 35%, respectively. The response of the sprung mass acceleration with the active one was poor at a certain point. This condition suspected as a accumulation of the sprung and unsprung masses amplitude. However, another performance improvement are achieved in the unspung mass responses (velocity and acceleration). At the same condition with the sprung mass response, the velocitiy and the acceleration with active one were highly reduces which indicate that the road holding performance is improved. This means that the designed active suspension performs better than the semi-active one when the vehicle is subject to different road disturbance.

Conclusions
The controller based Lyapunov method and ground-hook reference model for suspension system is designed. The effects of the road excitation frequency and road roughness in association with the ride comfort and the road holding of the vehicle were studied. The simulation result show that an active suspension system with the proposed control strategy is able to improve the ride comfort and the road holding, significantly compared with the conventional passive suspension systems as well with semi-active suspension.