ANN-based Acceleration Harmonic Identification for an Electro-hydraulic Servo System

Jianjun Yao, Dacheng Cong, Hongzhou Jiang, Zhenshun Wu and Junwei Han School of Mechatronics Engineering

Harbin Institute of Technology Harbin, Heilongjiang Province, China

travisyao@126.com

Abstract - Since� the dead zone phenomenon occurs in electro-hydraulic servo system, the acceleration output of the system corresponding to a sinusoidal input contains higher harmonic besides the fundamental response, causing harmonic distortion of the output acceleration signal. The output wave includes odd harmonics up to 11th harmonic. The method for harmonic identification based on artificial neural network (ANN) is proposed here. This method uses an Adaline neural network to identify the amplitude and phase of harmonics as well as the fundamental acceleration output on-line. The weights of the Adaline are adjusted according to the error between the actual and the estimated acceleration to yield the Fourier coefficients of the output wave. The simulation results show the validity of the analytical results and the ability of the algorithm to on-line identify all harmonics including the fundamental effectively with high accuracy.

Index Terms - Neural network, dead zone, odd harmonic, harmonic identification, harmonic distortion. �

I. INTRODUCTION

Nonlinearlites are inherent in many physical systems. Some examples of such nonlinearities are dead zone, backlash and friction. So many components of control systems have discontinuous nonlinear characteristics that can’t be locally approximated by linear functions. They often severely limit system performance, giving rise to undesired inaccuracy, oscillations or leading to limit cycles even instability. In linear system analysis, we assume that any non-zero input will cause the output to respond. For some physical devices in reality, the output is zero until the magnitude of the input exceeds a certain value. This phenomenon is called the dead zone nonlinearity, as in [1]-[6]. The dead zone phenomenon may occur in various components of control systems including sensors, amplifiers and actuators, especially in valve-controlled actuators, and in hydraulic components.

For electrohydraulic servo system, various mechanical, hydraulic and electric nonlinear characteristics and other typical nonlinearities are commonly present in the system. The parameters of hydraulic system, depending on the relationship among flow velocity, pressure and oil viscosity, vary heavily, so it is a typical nonlinear system. The electro-hydraulic servo system to be investigated here is a symmetric cylinder controlled by symmetric valve system. Assuming that the actuator can be linearly modeled as a third-order system, the dynamic model of the hydraulic system can be described as a

serial connection of a static dead zone nonlinearity and a linear system.

Because of the dead zone phenomenon, the acceleration output of the system, corresponding to a sinusoidal input, contains harmonics with frequencies that are integer multiples of the fundamental frequency besides the fundamental, which causes the acceleration harmonic distortion, as in [15]. Harmonics are steady-state periodic phenomena that produce continuous distortion of waveforms. These periodic waveforms are described in terms of their harmonic, whose magnitudes and phase angles are computed using Fourier analysis. The analysis permits a periodic distorted waveform to be decomposed into an infinite series containing fundamental frequency, second frequency, third harmonic, and so on. In order to cancel the harmonics, it needs harmonic measures against harmonic.

The number of harmonics, and their corresponding amplitudes, that are generated depends on the degree of nonlinearity of the systems. In general, the more nonlinearity, the higher the harmonics, and vice versa. Distortion measurements may be used to quantify the degree of nonlinearity of a system. Some common distortion measurements include Total Harmonic Distortion (THD), Total Harmonic Distortion + Noise (THD+N), Signal Noise and Distortion (SINAD), and Inter-modulation Distortion.

Most frequency domain harmonic analysis technique use discrete Fourier transform (DFT) or fast Fourier transform (FFT). The Fourier transform is used to transform samples of time domain into the frequency domain. The phenomena of aliasing, leakage and picket fence effects may lead to inaccurate estimation of harmonic magnitudes. The DFT suffers from inaccuracies due to the presence of random noise usual in the measurement process and tracking of signals with time varying amplitude and phase involves large errors. The application of Kalman filters and recursive LMS and RLS filters, as in [17-19], have been reported in the literature for tracking time varying signals embedded in random noise and decaying dc components. Both these filters suffer from large computational overhead and suitable values of covariance matrices and real-time implementation of these filters pose difficult problems.

Adaline network as an artificial neural network (ANN) is applied here used for on-line identifying harmonics from the system output acceleration. An Adaline, which is a single adaptive neuron, has a set of inputs, and a desired response signal. It has also a set of adjustable parameters called the

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Proceedings of the 2007 IEEEInternational Conference on Integration Technology

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weight vector. The weight vector of the Adaline generates the Fourier coefficients from a distorted signal using a nonlinear weight adjustment algorithm based on a stable difference error equation [20]. The identification task is completed on-line in real-time at each sampling interval.

II. PLANT DESCRIPTION

The hydraulic system with a symmetric cylinder and a symmetric valve is shown in Fig.1. Ref. [7] has computed the system open-loop transfer function. The external load and the dynamic characteristics of the valve are neglected. The simplified transfer function then is

p 2h

2hh

( )2( 1)

KG sss sζ

ωω

′ =+ +

(1) where K is the system open-loop gain, hω is the hydraulic natural frequency and hζ is the hydraulic damping ratio.

To take the dead zone of the actuating valve into account, a linear dead zone, as in [2]-[5], that is portrayed in Fig.2 and described by

r p r p r

l p r

l p l p l

( )0( )

m X b X bU b X b

m X b X b

� − ≥�= < <�� − ≤�

(2)

is added to the input of the linear model. So the hydraulic system can be described as a serial connection of a nonlinear dead zone and a third-order linear system, as shown in Fig.3.

State feedback compensation, as in [14-15], is introduced, as shown in Fig.4. fC and bC are designed for the system in the absence of the dead zone such that the closed loop is stable.

sp bp

iVq oVq

ip opA

iV oV

LFB

y

vx

m

Fig. 1 Schematic diagram of the hydraulic system.

U

pXrblb rm

lm

Fig. 2 Nonlinear dead zone.

p ( )G s′DZpYpX

p ( )G s

U

Fig. 3 Nonlinear hydraulic system model.

p ( )G s

bC

fC

y y��y�

r +

−

Fig. 4 System after compensated.

III. ADLINE FOR HARMONIC ESTIMATION

The system output waveform is assumed to be comprised of fundamental and harmonic components as

N

1( ) sin( )i i

iy t A iwt φ

== +��� (3)

where iA and iφ are the amplitude and phase of the harmonics, respectively; N is the total number of harmonic, and ω is the angular frequency of the fundamental component of the signal. To obtain the solution for on-line estimation of the harmonic, an adaptive neuron called Adaline, shown in Fig.5, as in [12,20], is applied. The Adaline is a single-layered neural network having n inputs and a single output which is the dot product of input x, and the weight vector w. The input/output relationship of the Adaline is linear at any given time. However, when its weights are adjusted on-line, the relationship between the input/output signals, as a function of time, is no longer linear. The error signal ( )e k is used to drive the adaptive algorithm. Clearly, when the network output closely matches the system output, the matching error will be quite small. So training is the process of tuning the weights of the Adaline so that its output matches the desired outcome.

To obtain the input variables for the Adaline, the signal given in (3) is written in the discrete form as

1 1 1 1( ) cos sin sin cos cos sin sin cosN N N N

y k A AA N A N

φ θ φ θφ θ φ θ

= + ++ +

�� � (4)

where s2 /k Tθ π= (5)

N is the order of the highest harmonic present in the signal, k the sample number or iteration count and sT is the sample time. Thus, the input vector to the Adaline is given by T( ) [sin cos sin 2 cos 2 sin cos ]k N Nθ θ θ θ θ θ= �x (6)

1x

2x

nx

Σˆ( )y k��

1w

2w

nw

�+−

( )y k��

Weightupdationalgorithm

( )e k

Fig. 5 Block diagram of an Adaline network.

399

The adaptation algorithm most often used to set the weights of the Adaline is the least-mean-squares (LMS) algorithm, as in [8-10] and [16], given by equations

Tˆk ky = w�� xk

ˆ( ) ( ) ( )e k y k y k= −�� �� (7)

1 ( )k k e kα+ = +W W xk

1 2 3 4 2 1 2 2( ) [ ( ) ( ) ( ) ( ) ( ) ( )]TN Nk W k W k W k W k W k W k+ += �W

where the following hold good at the kth sampling instant: kx is the input vector; ( )e k is the error; ( )y k�� is the actual

signal; ˆ( )y k�� is estimated signal; α is the learning parameter or step size that regulates the speed and stability of the adaptation. In the third equation of (7), a plus sign is used to insure the step is in the opposite direction of the gradient of the error with respect to the coefficients. Those equations describe the dynamic behaviour of weights.

LMS is an iterative gradient-descent algorithm that uses an estimate of the gradient on the mean-square error surface to seek the optimum weight vector at the minimum mean-square error point. The term kex represents the estimate of the negative gradient, and the adaptation constant α determines the step size taken at each iteration along that estimated negative gradient direction. The true negative gradient is given by the expected value of kex . If α is chosen properly such that small steps are taken, adaptation noise due to error in the gradient estimate is averaged out. When adapting with LMS on stationary stochastic processes, the expected value of the weight vector converges to a Wiener optimal solution. The LMS algorithm is important because of its simplicity and ease of computation, and because it dose not require off-line gradient estimations or repetitions of data. If the adaptive system is an adaptive linear combiner, and if the input vector kx and the desired response kd are available at each iteration, the LMS algorithm may be generally the best choice for many different applications of adaptive signal processing [11-13].

The error ( )e k between the actual signal and the estimated signal is brought down to zero, when perfect learning is attained and the weight vector will yield the Fourier coefficients of the signal. What’s more, the output of the Adaline is not zero, but rather stops adapting and should in theory be closely matched to the system output. If 0W is the weight vector after the convergence is reached, the Fourier coefficients are then obtained as

T1 1 1 1[ cos sin cos sin ]o N N N NA A A Aφ φ φ φ= �W (8)

The amplitude and phase of the ith harmonic are given by

2 20 02 1 2 2A W i W i= + + +� � � � � (9)

and

1 0

0

(2 2)tan(2 1)i

W iW i

φ − � �+= +� � (10)

For index i, values range from 0 to N, where i=0 denotes the fundamental response.

IV. HARMONIC ANALYSIS

In order to analyse the system acceleration harmonics, signal 0.001sin 40�t m is used as the command. Identification of the real plant gives the transfer function, as in [15].

3

p 5 2 4

18221 10( )(3.65 10 2.277 10 1)

G ss s s

−

− −

×′ =× + × +

Its sinusoidal response is shown in Fig.6 as its time domain response. Fig.7 is the equivalent response of Fig.6 in the frequency domain. Note that there are higher harmonics: third harmonic (60Hz), fifth harmonic (100Hz), seventh harmonic (140Hz), ninth harmonic (180Hz) and eleventh harmonic (220Hz), which causes the acceleration output distortion. Clearly, the third harmonic is in dominance. Thus the input vector of (6) is got for harmonic identification.

To determine the amount of nonlinear distortion that a system introduces, it is needed to measure the amplitudes of the harmonics that were introduced by the system relative to the amplitude of the fundamental. Harmonic distortion is a relative measure of the amplitudes of the harmonics as compared to the amplitude of the fundamental. If the amplitude of fundamental is 1A and the amplitudes of the harmonic are 2A (second harmonic), 3A (third harmonic), and so on, then the percentage total harmonic distortion (%THD) is given by

2 2 22 3 4

1

%THD 100A A A

A+ + +

=�

� (11)

0.0 0.1 0.2 0.3 0.4 0.5 0.6-20

-15

-10

-5

0

5

10

15

20

.

Acc

eler

atio

n (m

/s2 )

Time (s)

Actual Expected

Fig. 6 System sinusoidal response.

0 50 100 150 200 250-200

-150

-100

-50

0

50

Am

plitu

de (d

B)

Frequrncy (Hz) Fig. 7 Amplitude frequency characteristic.

400

V. HARMONIC IDENTIFICATION

From Fig.7 it is clearly seen that the acceleration output contains several odd harmonics; in other words, the frequency of the harmonics is odd multiples of the fundamental frequency. The input vector of the Adaline neural network then can be derived as

T( ) [sin cos sin3 cos 3 sin11 cos11 ]k wt wt wt wt wt wt= �x � Based on the basic theorem of the Adaline network, the harmonic identification scheme is obtained as shown in Fig.8. The system acceleration output and the input vector x are all used for (7). The summing junction compares its estimated signal, ˆ( )y k�� , against the actual system acceleration output to determine whether or not to regulate the Adaline network weights. When y�� reaches y�� , the weights will not be tuned and the result weights or the Fourier coefficients

T0 0 0 0[ (1) (2) (11) (12)]o W W W W= �W (12)

are obtained. The amplitude and phase of the harmonics, as well as the fundamental response, can then be easily calculated by using (9) and (10). Take third harmonic for an example

2 23 0 0(3) (4)A W W= + (13)

1 03

0

(4)tan(3)

WW

φ − � �=

� � (14)

Others can be obtained in a similar way. The amplitudes and the phases of the harmonics, including the fundamental frequency response, are shown in Fig.9 and Fig.10, respectively. The values of amplitude and phase are listed in the following two tables. They are computed in steady state. In some cases, the values, especially for the phases, are averaged if it has variations.

Plant(Fig.4)

90�

shift

cos wt

�cos11wt

cos3wt

�

�

Identification algorithm

Tˆk k ky = w�� x

ˆ( ) ( ) ( )e k y k y k= −�� ��

1 ( )k k ke kα+ = +w w x

Sine command

sin wt

sin3wt

sin11wt

0 (1)W

0(2)W

0(3)W

0(4)W

0(11)W

0(12)W

�

y��

Fig. 8 Harmonic identification configuration based on Adaline network.

0 1 2 3 4 5 60

2

4

6

8

10

12

14

16

.

Am

plitu

de (

m/s2 )

Time (s)

Fundamental response Third harmonic Fifth harmonic Seventh harmonic Ninth harmonic Eleventh harmonic

Fig. 9 Identified amplitude.

0 1 2 3 4 5 6-100

-80

-60

-40

-20

0

20

40

60

80

100

Phas

e (de

g)

Time (s)

Fundamental response Third harmonic Fifth harmonic Seventh harmonic Ninth harmonic Eleventh harmonic

Fig. 10 Identified phase.

TABLE I

AMPLITUDE Harmonic order Value (m/s2)

Fundamental response 15.793 Third harmonic 2.324 Fifth Harmonic 0.253 Seventh harmonic 0.127 Ninth harmonic 0.049 Eleventh harmonic 0.041

TABLE �

PHASE Harmonic order Value (deg)

Fundamental response -0.002 Third harmonic 55.158 Fifth Harmonic -88.975 Seventh harmonic -45.175 Ninth harmonic -31.753 Eleventh harmonic 28.793

As seen in Table I, the amplitude of the fundamental

response is 15.793, which is nearly equivalent to the theoretic value, because the reference input is 0.001sin 40�t m. What’s more, the amplitude of fifth harmonic is slightly bigger than the amplitude of the seventh; the difference of values between ninth and eleventh harmonic is very small. The estimated amplitude is very close to the value shown in Fig.7. However, the latter is off-line computed by FFT algorithm. The values of ninth harmonic and eleventh harmonic are very small, so these effects on the output acceleration are in less dominance compared to other harmonics. Using (11) the %THD can be obtained

2 2 2 2 22.324 0.253 0.127 0.049 0.041%THD =15.793

14.830%

+ + + +

=

The %THD for the acceleration signal computed from Fig.7 is 14.42%. It means that the match between the two %THDs is particularly good.

From Table �, the phase of the fundamental response is -0.002, a value very close to zero, which can be seen from Fig.6. There are no phase difference between the two curves plotted together in Fig.6, and the phase of the expected signal is zero. The phases of ninth harmonic and eleventh harmonic are over a relative wide range of variations compared with other curves plotted in Fig.10. This is due to the fact that they are in the lest dominance among those signals, which results in worse estimation.

401

0 1 2 3 4 5 6-6

-4

-2

0

2

4

6

Erro

r (m

/s2 )

Time (s) Fig. 11 Identification error.

Fig.11 is the identification error plot. It is the error

between the estimated and actual acceleration and is seen to decrease rapidly once the estimation algorithm is engaged. Note that the error flatness is well maintained to within 0.03± , approximately 0.19% related to the theoretic acceleration output. The residual error becomes very small 1.2 seconds after the estimation begins, and it means that the harmonic amplitude and phase are identified correctly. What’s more, the identification is computed on-line in real-time.

VI. CONCLUSIONS

The dead zone phenomenon occurs in electrohydraulic servo system. Its acceleration output corresponding to a sinusoidal input contains higher harmonics, which causes the acceleration signal distorted severely.

In this paper, an acceleration harmonic identification algorithm based on Adaline neural network has been developed. Several computer simulation tests are conducted to on-line identify the harmonic amplitudes, and phase. The developed identification structure was validated by comparing with the values computed by FFT. However, the FFT algorithm is operated on off-line. The proposed algorithm can on-line estimate harmonics, thus has good real-time performance.

The proposed identification technique is adaptive and capable of tracking the variations of amplitude and phase angle of the harmonics. The performance of this algorithm is superior for its good learning ability, high real-time performance and high speed recognition but simple structure. It can decompose each order of harmonic and detect amplitude and phase of any order of harmonic as well as fundamental wave in the same time.

In particular, the proposed identification algorithm does not need to know the priori knowledge of the system. The results thus presented here can be used to provide a basis for harmonic cancellation, which executes in real-time. The key point of harmonic identification here is to emphasize the real-time performance, less demanding computation and on-line computational efficiency.

It was found that Adaline is sensitive for the value of the learning rate. Small learning rate may lead to slow converge time, while large learning rate may cause Adaline to lose the ability of tracking the actual acceleration output, even instability.

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