IDENTIFICATION OF ELECTRO-HYDRAULIC ACTUATOR USING ...s2is.org/Issues/v9/n1/papers/paper3.pdf · 1Faculty of Electrical Engineering, ... (UiTM), 40450 Shah Alam, Selangor, Malaysia

IDENTIFICATION OF ELECTRO-HYDRAULIC ACTUATOR

USING FRACTIONAL MODEL

1Nuzaihan Mhd Yusof,

2Norlela Ishak,

2Ramli Adnan,

3Yahaya Md. Sam,

2Mazidah Tajjudin and

2Mohd Hezri Fazalul Rahiman

1Faculty of Electrical Engineering,

2Frontier Materials and Industry Application, UiTM-RMI-CoRe FMIA,

University Teknologi MARA (UiTM), 40450 Shah Alam, Selangor, Malaysia

3Department of Control and Instrumentation, Faculty of Electrical Engineering.

Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia

Email: [email protected]

Submitted: Oct. 15, 2016 Accepted: Jan. 6, 2016 Published: Mar. 1, 2016

Abstract –Electro-hydraulic actuator (EHA system) identification is to describe the characteristic of

the system that useful for prediction or control system design. There are numerous methods of EHA

modeling but there has not been much model using fractional-order (FO) model. In this work,

integer-order (IO) model and FO model are developed to model EHA system. Output-error method

is used as the estimator for both model. The coefficient of IO model was first estimated and using

the estimated coefficient, the derivative order of FO model is estimated. These models has been

validated by comparison of error, coefficient of determination (R2), mean square error (MSE) and

correlation function. The results for the proposed model show improvement compared to the IO

model.

Keyword – system identification; EHA; modeling; fractional-order model; continuous-time transfer

function.

INTERNATIONAL JOURNAL ON SMART SENSING AND INTELLIGENT SYSTEMS VOL. 9, NO. 1, MARCH 2016

32

mailto:[email protected]

I. INTRODUCTION

Hydraulic servo-system can be found widely used in many engineering field. The application

covers from heavy industries to soft industries. From constructions and metal manufacturing

to agricultural and textile industries. The main advantage of hydraulic servo system as stated

in Pascal’s Law is its good ratio between force on one side and the weight of actuator at the

other side. This application is called direct drive construction.

A hydraulic control system can be considered as ‘simple’ hydraulic control system when the

data transmission is through the mechanical linkage and gear. However, for more complex

hydraulic control system that can take many form of data transmission, such as electronic,

electrical, pneumatic and optical, this hydraulic system can be inaccurate and difficult to be

controlled. As a part of solution, this system can be replaced with electro-hydraulic actuator

(EHA) system where the data transmission used is electronic. The main reasons of model

identification of EHA system is important are for control system design, prediction,

simulation, and fault detection.

EHA system also known for its nonlinearity behavior. The contribution of these

nonlinearities of EHA are from the compressibility of hydraulic fluid, complex flow

properties of valve and the friction occur in actuators. Thus the modeling of EHA can be

difficult. There have been many studies in hydraulic modeling particularly on EHA [1]–[5].

PRBS input is famously used as the perturbation signal in system identification. However, this

type of input is not suitable for hydraulic system because it only has two-level amplitude. For

hydraulic system identification, it is necessary to consider all the important amplitude. Thus,

the suitable perturbation signal is independent sequences with Gaussion or uniform

distribution, pseudo-random multilevel signal (PRMS) and chirp or multi-sine signal [6, p.

159]. Step and sine signal were used widely as the excitation signal for hydraulic system [4],

[5], [7].

Modeling of a physical process is necessary as it is the first step of system analysis. The main

reason of system identification is that the performance affect the quality and safety of a

component and the whole system. A good model is required in order to design and

implementing a control system.

Nuzaihan Mhd Yusof, Norlela Ishak, Ramli Adnan, Yahaya Md. Sam, Mazidah Tajjudin and Mohd Hezri Fazalul Rahiman, IDENTIFICATION OF ELECTRO-HYDRAULIC ACTUATOR USING FRACTIONAL MODEL

33

The utilization of fractional calculus get to be more extensive into different area, from light

application field, for example, biomedical, medicine, and chemistry [8]–[10] into heavy

application, for example, robotic and mechatronic [11]. Typically, FO model is applied in

viscoelasticity [12], [13], RC networks [14]–[18]and electrochemistry [15], [19], [20].

In many system and process, modeling of real system using non-integer model is reliably

more adequate than integer model and gives better estimate [21]. In addition, there are some

dynamic systems cannot be identified using linearization and exponential function, thus using

FO model is an additional degree of freedom that gives a better estimation to the responses.

The objective of this paper is to model EHA using continuous-time transfer function. Two

type of model structure will be used as continuous-time transfer function which are integer-

order (IO) model and fractional-order (FO) model. Moreover, fractional calculus offers a

great tool to define memory and hereditary effects in various substances [22]. In addition, the

long memory transients and infinite dimensional structure of real physical system modeling

has attract researches to apply fractional order dynamic models significantly [23], [24].

This paper is organized as follow: in Section II, the general concept of Fractional-Order (FO)

model is briefly described. The experimental setup is presented in Section III. Then, the

identification and estimation method are introduced in Section IV. In Section V the simulation

results followed by the discussions are presented. The conclusions are drawn in Section VI.

II. FRACTIONAL-ORDER MODEL

The concept of classical differentiation equation operator, as in Eq. (1),

d

Ddt

(1)

For dynamic system, the function f for the n-th derivative is defined as in Eq. (2),

( )

( )n

n

n

d f tD f t

dt (2)

In conventional calculus, n is an integer, different with fractional calculus, n can be as non-

integer number, as in Eq. (3). Fractional calculus is the generalization of integration and

differentiation of non-integer number. Thus, it should be called as non-integer calculus

because it includes non-integer number such as complex number [25].

121

21

2

( )( )

d f tD f t

dt (3)


34

This expression is as old as the traditional differential equation, which began late seventeenth

century in a letter discussion of two well-known mathematicians, l'Hopital and Leibniz. From

that point forward, the idea of fractional calculus has drawn consideration of numerous

popular mathematicians, including Euler, Laplace, Fourier, Riemann and Liouville [26].

Applying Laplace transform as in Eq. (4) in order to obtain the transfer function as presented

in Eq. (5).

( ) ( ) ( )a tD f t s f t s F s L L (4)

1 0

1 0

1 0

1 0

...( )( )

( ) ...

m m

n n

m m

n n

b s b s b sY sG s

U s a s a s a s

(5)

Where na and mb indicate pole and zero polynomial differential operator coefficients and n

and m signify the corresponding exponent (derivative order). There is a significant

difference between IO and FO models where n and m are in a form of integer. Meanwhile,

in FO model, they can be represented by non-integer numbers, known as fractional-order. In

several previous works, FO model have shown better performance in terms of accuracy and

flexibility [25], [27].

On account of a system of commensurate-order , the continuous-time transfer function is

given by

0

0

( )

( )

( )

mk

k

k

nk

k

k

b s

G s

a s

(6)

0 2

Where is the commensurate order, and it holds for every s -pole, the stability criterion of

Matignon’s stability theorem where “The fractional transfer function ( ) ( ) ( )G s Z s P s is

stable if and only if the following condition is satisfied in plane :

arg( ) , , ( ) 02

q C P

(7)


35

Where : qs . When 0 is a single root of P(s), the system cannot be stable. For q=1, this

is the classical theorem of pole location in the complex plane: no pole is on the closed right

plane of the first Riemann sheet.”

Thus,

arg( )2

ks

(8)

The highest order of the derivative order will be the multiplication between integer order, n

and commensurate order .

III. EXPERIMENTAL SETUP

The experimental hardware that was utilized as a part of these studies is an EHA system that

is indicated in figure 1. The hydraulic cylinder was held in vertical position. This is a difficult

issue as effect of gravity is considered. The EHA system includes single-ended cylinder kind

of actuator. The bidirectional cylinder has 150 mm stroke length, however, the unit

measurement is in inches. Thus the stroke length is 5.9 inches. The wire displacement sensor

is mounted at the highest point of piston rod. The flow of the fluid is controlled by electronic

control valve.

Figure 1: Electro-hydraulic actuator system

Figure 2 illustrates the schematic diagram of complete experimental set-up. The interfacing

between the computer and plant was done using Matlab Real-Time Workshop via Advantech

PCI-1716 interface card. Figure 3 represents the data collection block diagram using the

interface card. The main purpose of this interfacing is to collect the analogue measurement


36

and convert into digital data. The signal from controller to the plant also pass through the

interface card.

Data Acquisition

Board

PCI1716

Position

Transduser

Bidirectional

Cylinder

Proportional

Valve

Figure 2: Experimental setup for electro-hydraulic actuator system

Figure 3: Block diagram

IV. METHODOLOGY

The steps taken in conducting this experiment will be briefly described in this section. The

general procedure of system identification are as in figure 4 [28].


37

Figure 4: Flowchart of system identification

a. Data Collection

The input signal used in system identification plays an important role to trigger the system so

that the output signal contains good information as much as possible and ensure

identifiability. Thus, input signal can have a significant influence on the estimation result.

There are several type of recommended input signals used for exciting EHA system which are

[6, pp. 131–134];

Independent sequences with a Gaussian.

Multi-sine signal,

Chirp signal,

Pseudo-random multilevel signal (PRMS)

In this paper, multi-sine signal was used as the excitation signal. This signal is a combination

of several sinusoid with different frequencies and could give very good estimations at the

corresponding frequencies [29, p. 615]. Basically, multi-sine signal can be generated using

Eq. (9) as expressed,

1

( ) cosp

i i s

i

u k a t k

(9)


38

Where ia is amplitude, i is frequencies (rad/sec), st is the sampling time (sec) and k is

integer. For this particular study, three frequencies within the range of 0.01 Hz to 1 Hz will be

used in order to generate a multi-sine signal.

0 5 10 15 20 25 30 35 40 45 50-6

-4

-2

0

2

4

6

Voltage (

V)

Time (s)

0 5 10 15 20 25 30 35 40 45 502.5

3

3.5

4

4.5

5

5.5

6

Dis

pla

cem

ent

(Inches)

Time (s)

Figure 5: Input and output data

The amplitude of this system is -5 V and +5V . The amplitude must not be too high because

the actuator will be damaged. The saturated output also should be avoided since the data in

that saturated output is much less informative and redundant.

The sampling time was selected where it is long enough to cause an effect on output but not

too long which the output not give any more dynamic information. The sampling time was set

to 1ms so that all important responses were considered. And every sample was repeated by

three times. This is due to longer time for response observation.

The duration of the system to run was 50s and the total data was 50,000 sample. However, the

first 10s was removed because it is assumed during that time the plant was unstable. The data

also was converted into slower sampling time by selecting the data in 40ms sequence. Thus,

the total data was 1250 samples. After deducting the first 10s, the data size was 1000 samples.

This data then divided into two part equally. First part was used for model estimation and the

second part was used for model validation.

b. Model Structure Selection

The model structure used in this identification was basically the same which is differential

equation (DE) used in time-domain model. The first model was an ordinary IO model and the

second model was the extended integer-order model. Where we called FO model. The


39

difference between these models is the derivative order of the model. Based on first principle

model, third order structure was used for both model identification. The mathematical model

of both model was expressed in Eq. (10) and Eq. (11), respectively.

2 1

2 1 0

( )( )

( )IO

Y s KG s

U s a s a s a

(10)

02 1

2 1 0

( )( )

( )FO

Y s KG s

U s a s a s a s

(11)

Where K is the gain, na is the model coefficient, and n is the non-integer derivative order

for FO model.

c. Model Estimation

Before building IO model and FO model, the system is assumed to be a non-stochastic

system. The output-error estimator were used to compute the undisturbed input and the error

was not modeled. The objective is to estimate the deterministic signal. The initialization is

followed by nonlinear least-squares search updates to minimize the error. The algorithm used

was Lavenberg-Marquardt.

The FO model used Grunwald-Letnikov definition based solver to compute the output of the

model during time-domain simulations. The selected commensurate order was 0.9 because it

is close to integer 1.

d. Model Validation

Infinite-step-prediction was used to test both model where prediction of any stochastic model

is zero. Therefore, error model prediction was not computed in this test.

The model’s results were validated using several common validation approach which were

residual (Error), mean square error (MSE), coefficient of determination (R2). The

mathematical equation of all validation approaches are;

Residual

ˆError y y (12)

Where y is the actual output, and y is the simulated output.

Means-square error


40

2

1

1ˆ( )

n

i

MSE y yn

(13)

Where n is the number of data, y is the actual output, and y is the simulated output.

Coefficient of determination (R2) [29, p. 363]

2

2 2

2

2

ˆ1

y yR

y y

(14)

Where y is actual output and y is simulated output. The R2 is presented in percentage.

Another important validation technique is correlation function. The purpose is to measure the

predictability in a series where the concept is to compute the correlation between an

observation and each of the past observation. If there is at least one correlated observation in

the past, then the correlation can be utilized to predict future observation. In this study, auto-

correlation function (ACF) and cross-correlation function (CCF) were used for this purpose.

In order to perform this, the mean was first removed. ACF can be obtained by correlating a

signal with its own shifted while CCF can be obtained by correlating two different signals

with one of the signal was shifted. The confidence interval was set to 95% and 99%.

The purpose of conducting validation approach is to avoid from overparametrizing which

could lead to unnecessarily complex model or underparametrizing which results inaccurate

model. This validation method has been applied in modeling of EHA system [30]–[32].

Lastly, the path of flowchart in figure 4 was leading back to Loop 1 if the model validation is

not accepted. This step could repeat until Loop 3 until a satisfied result could be obtained.

V. RESULT AND DISCUSSION

In this section, the result of IO model and FO model are presented. First, the IO model

identification was conducted and the coefficient was estimated. Then, FO model used the

same coefficient from IO model and estimated its derivative order.

a. Integer-Order Model

The IO model identification with second order structure was obtained as in Eq. (10). The

transfer function of the model is expressed as,

2

0.025648

1.3461 2.7008 0.0052484IOG

s s

(15)


41

As shown in figure 6, the result from the simulation of IO model in Eq (15) is presented. The

simulated output is managed to approach the pattern of actual output quiet well and changed

when the amplitude. However, it is apparent that the output of IO model gives very little

accurate simulation with R2 at 82.35%.

0 50 100 150 200 250 300 350 400 450 5002.5

3

3.5

4

4.5

5

5.5

6

6.5

Time (40ms)

Dis

pla

cem

ent

(Inches)

Actual

Simulated

Figure 6. Model validation of IO model

Figure 7 shows the error obtain between actual output and simulated output. Through visual

inspection, it can be observed that the range of error is between -0.5436 and 0.2638 with the

mean square error of 0.2040.

0 50 100 150 200 250 300 350 400 450 500-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (40ms)

Dis

pla

cem

ent

(Inches)

Figure 7: Error from IO model


42

Figure 8 shows ACF and CCF of IO model. Note that the distance between two lags is equal

to the sampling time. Therefore one lag corresponds to 40ms. As can be seen from the figure,

the result of the ACF shows that all the observations of ACF and CCF exceed the confidence

intervals. This presents that each observation has significant correlation.

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

AC

F

Lag

-20 -15 -10 -5 0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

CC

F

Lag

Figure 8: ACF and CCF of IOM

b. Fractional-Order Model

The initial condition of FO model was set as in Eq. (16). Note that the commensurate order

was set to 0.9.

0.9 0

0.9 2 0.9 1 0.9 0

0.025648( )

1.3461( ) 2.7008( ) 0.0052484( )initialFO

sG

s s s

0

1.8 0.9 0

0.025648

1.3461 2.7008 0.0052484initialFO

sG

s s s

(16)

The FO model identification with second order structure was obtained as in Eq. (17). The

transfer function of the model is expressed as,

10

0.053672

1.1013 0.98247 3.3186

0.025648

1.3461 2.7008 0.0052484FO e

sG

s s s

(17)

The estimated derivative order, n and n shown in Eq. (17) approaching to the integer

order in Eq. (16).

The identification is more accurate than IO model with R2 at 88.68% and MSE at 0.1760. The

simulation of FO model are presented in figure 9. As shown in the figure, the simulation

pattern is similar with IO model. However, there has been a lack of accurate simulation during

first 50 samples. There are close simulations approaching to the actual output during the


43

sample 50 to 500. The simulated output also managed to follow the changes of the amplitude

direction.

0 50 100 150 200 250 300 350 400 450 5002.5

3

3.5

4

4.5

5

5.5

6

6.5

Time (40ms)

Dis

pla

cem

ent

(Inches)

Actual

Simulated

Figure 9. Model validation of FO model

The error of FO model is illustrated in figure 10. The range of the error is between -0.3669

and 0.2571.

0 50 100 150 200 250 300 350 400 450 500-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (40ms)

Dis

pla

cem

ent

(Inches)

Figure 10: Error from FO model

Figure 11 shows ACF and CCF of IOM. As can be seen from the figure below, the result of

the ACF and CCF shows a significant improvement of residual correlation even though all the

correlations exceeds the confidence interval.


44

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

AC

F

Lag

-20 -15 -10 -5 0 5 10 15 20-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

CC

F

Lag

Figure 11: ACF and CCF for FOM

The comparison of the parameter value can be presented as in Table 1. As seen in the table,

all the derivative orders, n of both models are similar except 2 where 2 of FO model is

closer to integer 1 instead of 2. This is because the commensurate order is 0.9. Thus, the

maximum order for FO model is 1.8.

Table 1: Comparison of parameter value

Model 2a 1a 0a 2 1 0 0b 0

IO -1.3461 2.7008 -0.0052 2 1 0 0.0256 0

FO -1.3461 2.7008 -0.0052 1.1013 0.9825 3.3190e-10 0.0256 0.0537

Difference - - - 0.8987 0.0175 3.3190e-10 - 0.0537

The comparison of the R2 and MSE is presented in Table 2. The FO model gives better

estimation than IO model. The value of R2 for both model are acceptable after considering the

position of the cylinder is in vertical, which is difficult to have a good model.

Table 2: Comparison of model validation

Model R2 MSE

IO 82.35 0.204

FO 88.68 0.176


45

VI. CONCLUSION

In this study, models identification of EHA system with continuous-time transfer function are

presented and multi-sine signal was used as the excitation signal. Same pre-processing

method was applied to both model so that the result was not biased. In addition, output-error

method was used as the estimator for both model. The FO model has shown a better

estimation than IO model where FO model could describe the nonlinearities that IO model

could not represent. For future recommendation, a modelling of EHA system using FO model

with variation of commensurate order in order to observe the effect of commensurate order to

the model stability.

ACKNOWLEDGEMENT

This work was conducted at the Faculty of Electrical Engineering, UiTM facilities with

financial support from 600-RMI-ST-FRGS 5/3 (85/2013). The authors would like to thank the

Ministry of Higher Education Malaysia for MyBrain15 sponsorship, Research Management

Institute (RMI), and UiTM for their support.

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IDENTIFICATION OF ELECTRO-HYDRAULIC ACTUATOR USING ...s2is.org/Issues/v9/n1/papers/paper3.pdf · 1Faculty of Electrical Engineering, ... (UiTM), 40450 Shah Alam, Selangor, Malaysia