Neural Network Model of a Vehicle Active Suspension System

Copyright © IFAC Real Time Programming, Shantou, Guangdong Province, P.R. China, 1998

NEURAL NETWORK SUSPENSION SYSTEM

c. Morgan, and

S. Yao

MODEL

OF

A

VEHICLE

ACTIVE

*

*School of Engineering and the Built Environment, University of Wolverhampton Wolverhampton, WVllSB, United Kingdom. Email: {C.Morgan;S.Yao}@wlv.ac.uk

Abstract. This paper describes about the simulation and performance identification of a zero-rate active suspension system which exhibits significant nonlinear behavour. Identification is achieved by a back-propogation neural network with 10 hidden layers. Comparison of simulation, experimental and identification results obtained by neural networks shows that the simulation and neural network can be used to predict the dynamic performance of the system effectively. Copyright © 1998 IFAC

Keywords. Simulation, Identification, Vehicle Active Suspension, Neural Network.

procedure and a neural network appropriate to system identification have been proposed and verified by comparing test and simulation results. The results indicate the potential to analyse and modify the control strategy of the system efficiently.

1. INTRODUCTION It is generally known that vehicle active or semiactive suspensions are superior to conventional passive suspensions with regard to ride comfort, road holding and handling (in general, under all conditions, including straight roads, bends, rough roads, aerodynamic effects, effects of wind on body, etc.). But such superior performance is often at the expense of increased system complexity, because the vehicle must be fitted with a series of devices such as sensors, force actuators, and associated hardware for realising the control law. On the one hand, more expensive and heavier vehicles result; on the other hand, vehicles become less reliable, owing to the greater number of parts that are liable to fail (Nagai, 1993).

2. ZRS SYSTEM MODELLING The ZRS is based on a buckled spring element that is coupled to a variable leverage system (Leighton, et aI, 1994), as shown in Fig.l.

Research has been carried on to limit or overcome these shortcomings to some extent (Darling, et aI, 1992; Fukami, et ai, 1994; Leighton, et ai, 1994; Mahajan, et ai, 1994; Rezeka, et ai, 1994; Truscott, et aI, 1994). Among which, a novel active suspension configuration, proposed by Leighton & Pullen (1994) , based on a buckled spring element coupled to a variable leverage system (" zero rate suspension (ZRS)") is capable of performing well as a fully active suspension with excellent ride quality and very low power consumption, compared with previously reported active suspensions offering similar levels of control (Leighton, et aI, 1994a; Leighton, 1994b).

Lsverage

Fig.1 Experimental Trailer Fitted with ZRS

A half car model with four degrees-of-freedom can be used to represent the trailer fitted with the ZRS. The trailer body motion can be described as the vertical and roll movement shown in Fig.2. In the figure, e represents the trailer body roll angle, the top rigid body mass is mo, and the wheels are represented by the two masses ml and m2'

This paper describes the simulation and identification, via Neural Networks (NN), of a vehicle active suspension system which is fitted to a two-wheeled trailer at the Engineering Division, University of Wolverhampton, UK. The simulation

129

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O X

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a

b

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1

~Il

- -m o

control matrix:

B=

-a b

°

-1

C,O.M.

o -1

The observation equation can be based on the practical installation of the sensors on the trailer and written as: c n

y=cx

C

12

(2)

where the observation matrix is: Fig.2 Analytical Model of the Trailer With ZRS

c= Based on dynamic analysis, the equations of motion for this model can be written as the state equation

= {x o e

excitation vector:

Xc

control vector: U

=

x tl

{xcI

= {Fal

0

1

b

0

-1

a

-1

-b

0

0 1 0

0

1

in which: mo is the mass of the body; m l , m2 are the masses of both wheels; I is the moment of inertia about the centre of the trailer body (C.O.M.); a, b are the distances between the centre of the sprung body and the wheels; Fal ,Fa2 are the forces of both

where: state vector X

1 -a

x r2 } T

actuator systems; kt l' kt2 are the stiffnesses of both

xcI

hub assemblies; COl' CO2 are damping factors between the body and tyres; ctl ' ca are damping factors of

x c2

x c2 } T

both tyres; Xo is the height of the trailer body; (J is the roll angle of the trailer; x tl ' X a are the displacements of both wheels; and x cl , x c2 are the road excitations

Fa2 }T

in the vertical direction at both wheels. mass matrix: M

=diag{nto,I,ml'~}

A buckled spring is the basic element in the ZRS. A general description and simulation results of the buckled spring element have been described previously (Yao, et aI, 1997).

damping matrix: Cc)} +C02

-cOla+Cozb -Col-Cm G! = -Co.a+emb Co/ +em b2 Cola -emb -Col cOla <;1 0 b 0 -Cm -em ca

The control system for the rig described by equations (1) and (2) is completely controllable and observable, according to modern control theory. That is, the arrangement of actuators and sensors are theoretically capable of controlling the motion of the trailer body.

stiffness matrix: K

=diag{O,

0, k tl , k t2 } 3. ZRS SYSTEM SIMULATION

excitation matrix:

° Bc: =

0

0

Based on the analysis above, a program based on the analysis above has been developed and written in MATLAB/SIMULINK, and runs on a PC. Fig.3 shows the data flowchart for simulating the suspension system and the actuator system.

0

0 0 0 c tl 0 k tl 0 0

°

c t2 0

k t2

Fig.4 compares the measured and simulated results

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for a free body vibration of the two wheeled trailer fitted with the ZRS. The responses are obtained as follows: (1) the trailer is placed in an equilibrium position and the data collection system is initiated; (2) one side of the trailer is displaced to a suitable position, and then released; (3) data collection begins at the instant that the trailer is released, accelerometer data is collected

Calculate System Basic Parameters

I

L..!:

_

figure that the simulation results predict this vibration accurately, which verifies the proposed model and its corresponding simulation program.

4.

ZRS SYSTEM IDENTIFICATION BY APPLYING A BACK PROPAGATION NEURAL NETWORK (NN)

There is great potential to apply NN's to system signal identification for a non-linear system such as the vehicle active suspension considered here. This will efficiently contribute to the verification of the system model and can subsequently used to realise optimum control.

Generate 80slc Parameters (by MATlAB Language)

r-=--- -------

4.1 Basic Construction ofa NN

Neural net models are specified by net topology, node characteristics and training or learning rules. The function and performance of a NN are determined primarily by its pattern of connectivity. In this sense there are feed forward NN's; and NN's incorporating feedback loops (Demuth, et aI, 1994; Monostori, et aI, 1992).

System Simulating

•(b.Y SIMULJ~K .for MAT~? _

_ _I

Fig.3 Flowing Chart of the Computation for Suspension Simulation

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j

The most commonly used feed forward network with analogy nodes is illustrated in Fig.5 (Demuth, et ai, 1994; Monostori, 1992). The network has n inputs and j neurons in hidden layers. In the network, each element of the input vector, X, is connected to each neuron input through the weight matrix W. The ith neuron has a sum that gathers its weighted inputs and bias to form its own scalar output Y through a transfer function, a(x ), that is:

+ .

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(l~

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·t················

:: :·:::·:::::::·:·:··::.:.:·:::·:::r:·::···:.::.:1::·::.::::::::::L:·:::::::::]::::::·::::·:::.:':':: ::::::::. (lIS

....•....•.....

0175 0

T

;

r T

r .

3

6

4

Y(j)

=

(J j

(L W(i, j). X (i) + b(i)) ;

(j=1,2, ... ,m).

(3)

where, er is a transfer function, and b is the sum of the biases. In this type of connection, the output for one layer of neurons comprises the input vector for the following layer. This process continues to the output layer.

Ture (lnXnii)

Fig.4 Simulated and Experimental Results of the Trailer Body Free Vibration (0 Experimental data; -- Simulation data)

The potential benefits of neural networks extend beyond the high computation efficiency provided by their massive parallelism.

automatically during operation. For this experiment, the left side of the trailer body is firstly kept stable at its equilibrium position, about 0.2 m higher than the centre of its wheels; then pushed down to 0.175 m higher than the wheel centre, the left side is then released. The trailer body will vibrate and gradually reach its stable position. It can be concluded from the

4.2 NN Model Identification by Forward Modelling

Off-line identification will be performed by presentation of a pre-defined number of previously obtained patterns. The patterns are made up of a series of input and corresponding output signals. The approach is much the same as in standard

131

operating range of the plant (Nabney, et aI, 1996).

identification, the only difference being that presentations of the data are repeated. This procedure is usually called network training.

X(1)

For the ZRS system, a set of input-output pairs is used to train the neural network by setting the number of neurons in the hidden layer to 10. The training result is shown by Fig.7.

w(1.1)

4.3 Validation for NN Model Input vector X

After a particular two-layer feed-forward neural network, at least implicitly, model has been arrived

y(m)

Input layer

Output Layer

Fig.5 Construction of NNs

20,---.,.------.-----.---....--......--...-----..,

The procedure of traInIng a neural network to represent the forward dynamics of a system is referred to as forward modelling (lrwin, et al, 1995). A structure for achieving this is shown in Fig.6. The neural network model is placed in parallel with the system and the error between the plant output yp and network output Ym (the prediction error) is used as the network training signal. In Fig.6, u is the input, and d 1 and d 2 are the disturbance signals to the system.

-10

·:.•.•...;.. .....•..•.•.;......••.•.•.•;..••.•.••••.•;...•.. . . . .. ··· .. ·

-15

-.......L--..4-----I-------' 150 200 250 300 350 400 450

-20'-----"--........ o 50 100

Fig.7. System Output (dotted) vs NN output(solid):Training u

15 10

·10 ·15 .20 '-----"-_---a.._--I..._'"""""-_I-oo-----"-_---a.._--"----' 600

650

700

750

800

850

900

950

1000

1050

Fig.8 System Output (dotted) as NN Output (solid) : Testing Fig.6 Identification by Neural Network Forward Modelling

that best describes the data by the back propagation algorithm, it is still needed to test whether this model is valid for the intended purpose. Such tests are known as model validation. Another set of input data is used to test if the outputs of the network model match the original system's outputs. This is shown in Fig.8 from which it can be seen that the network model fits the system accurately.

The neural networks is trained using the backpropagation algorithm, which has become the most commonly used algorithm in neural network applications(Drago, et aI, 1995). To train a network it is imperative that the training signals are obtained over a sufficiently wide

132

Fig.8 have indicated that the built-up NN is applicable to the development of an optimum control strategy for the ZRS. This work is currently underway.

4.4 Comparison with ARX Model The simplest input-output relationship for single input single output system is a linear difference equation (Ljung, 1987):

(3) It can be seen from Fig.9 that the neural network describes the system more accurately than the ARX model in that the network model can represent the non-linear characteristic of the system output which has been contaminated by noise.

where i represents the pure delay of the system. This is the familiar ARX model. Equation (4) can be re-written as:

A(q)y(k) = B(q)u(k - i) + e(t)

6. REFERENCES

(5)

Darling, J., Dorey, R. E. and Ross-Martin, T. J. (1992). 'A Low Cost Active Anti-roll Suspension For Passenger Cars'. Trans. ASME, Journal of Dyn. Systems, Meas., Control. Vo1.114, No.4, pp.599-605. Demuth, H. and Beale, M. (1994). Neural Network Toolboxfor Matlab. The Math Works, Inc., MA. Drago, G. P., et al. (1995). 'An Adaptive Momentum Back Propagation'. ibid, Vol. 3, No. 4, pp.213.. 221. Fukami, A., Yano, M., Tokuda, H., Ohki, M. and Kizu, R. (1994). 'Development of Piezo-electric Actuators and Sensors for Electronically Controlled Suspension'. Int. Journal of Vehicle Design. 15(3-5), pp.348-357. Irwin, G. W., et al. (1995). 'Neural Network Applications in Control'. lEE Control Engineering Series 53, pp. 113. Leighton, N.J. and Pullen, J. (1994a). 'Novel Active Suspension System for Automotive Application'. IMechE, Proc. Instn. Mech. Engrs., Part D: Journal of Automobile Engineering, Vo1.208, 1994, pp.243-250. Leighton, N.J. (1994b). 'Application of Advanced Modelling Techniques to Reduce Prototyping Time For a Novel Active Suspension System'. lEE Colloquium (Digest), 1994, No.218, pp.9/1-4. Ljung, L. (1987). System Identification: Theory for the User. Prentice Hall. Mahajan, S. and Redfield, R. C. (1994), 'Dynamic Performance Issues in Active, Energy-efficient Vibration Control Systems'. ASME, Transportation Systems, DSC-Vo1.541DE-Vo1.76, pp.53-67. Monostori, L. and Barschdorff, D. (1992). 'Artificial Neural Networks in Intelligent Manufacturing'. Robotics & Computer-integrated Manufacturing, Vo1.9, No.6, pp.421-437. Nabney,I.T. and Cressy,D.C. (1996). 'Neural Network Control of a Gas Turbine'. Neural Computing & Applications, Vol. 4, No. 4, pp.

To assess the relative performance of the NN model, the same set of input.. output training data, used to train the neural network, is also used to obtain an ARX model by least square parameter estimation. Fig.9 compares the outputs obtained by the system, network, and ARX models.

50

100

150

200

250

300

350

400

450

Fig.9 Outputs: System(point); NN (Solid,training);ARX (dotted)

5. DISCUSSION AND CONCLUSION (1) It can be seen from Fig.4 that the simulated system vibration is in close agreement with the experimental result. This verifies the application of the system modelling procedure and the developed program which can be used for further dynamic analysis of the open-loop and closedloop performance of the zero rate suspension, in respect of both static and dynamic properties of the buckled spring element. (2) The identification results shown in Fig.7 and

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