control of an electro-hydraulic system using neuro-fuzzy modelling

CONTROL OF AN ELECTRO-HYDRAULIC SYSTEM USING NEURO-FUZZY MODELLING AND REAL-TIME LEARNING APPROACHES (Invited Paper)

P.J. Costa Branco J.A. Dente

Instituto Superior Técnico CAUTL - Mechatronics Laboratory Av. Rovisco Pais, 1096 Lisboa, Portugal {E-mail: [email protected]} Keywords: Neuro-Fuzzy Control, Fuzzy Modelling, Intelligent Control, Drive systems

Abstract Drive systems are fundamental parts of industrial processes. Although, their conventional models and projected controllers based on those models are becoming not enough appropriate and having a limited scope. So, it is important incorporate learning capabilities into drive systems in such a way that they improve their accuracy in real-time, becoming more autonomous agents with some “intelligence degree”. To investigate this challenge, this paper shows the development of a position control system using neuro-fuzzy techniques to an experimental electro-hydraulic system. The paper describes the acquisition of the neuro-fuzzy inverse-model of the system with some practical questions like the acquisition of a “good” training data set, and the choice of actuator significant variables. The extracted inverse model and its learning capabilities are used to design the actuator position controller based on the feedback-error-learning technique. Through a set of experimental results, we show the generalisation capacity of the controller, its real-time learning capability in extract relations not incorporated into the initial inverse-model, and therefore improve its tracking performance. At last, it is our aim in present this paper not only to describe the good benefits of incorporate neurofuzzy methodologies into drive control systems, but also discuss and pointed out certain control situations where this methodology reveals some drawbacks.

1

Introduction

Recent integration of new technologies involving new materials, power electronics, microelectronics, and information sciences, made relevant new demands in performance and optimisation procedures for drive systems. To handle command and control problems, the dynamic behaviour of a drive must be modeled taking into account all electromagnetic and mechanical phenomena at play. However, if one requires a precise and fast performance with the command laws being highly complex and non-linear,

the classical models become a limited representation of the real system. The existing models are not sufficiently accurate as their parameters are poorly known, and also because physical effects like for example thermal behaviour, magnetic saturation, friction, viscosity, being in general time-variants, they are difficult to account with necessary simplicity and accuracy. So, it is important to investigate the design of drive systems that can incorporate learning capabilities in such a way that their control system learns to improve its

accuracy in autonomous.

real-time

and

becomes

more

To study the addition of learning mechanisms into drive systems, we describe the implementation of a control system that uses neuro-fuzzy modelling and learning procedures to design a position controller to an electro-hydraulic actuator. The learning capabilities of the neuro-fuzzy models are employed to permit the controller get the actuator inverse model and so compensate possible unstructured uncertainties to improve actuator’s performance in trajectory following.

methods in the literature. Section IV describes the cluster-based method used to initialise the neurofuzzy modelling algorithm. Section V shows the experimental system and the procedures used to obtain a “good” training data set from the electrohydraulic system. In Section VI, we acquire the inverse-model of the actuator using the modelling algorithm presented in Section IV and the training set of Section V. Section VII describes the implemented control system based on the feedback-error-learning ideas and presents some experimental results. At last, Section VIII discusses some neuro-fuzzy control drawbacks appeared during certain control situations.

The paper starts presenting the actuator’s inverse modelling using the neuro-fuzzy methodology. In this way, the information about its dynamic behaviour is expressed in a multimodel structure by a rule set composing the neuro-fuzzy inverse-model. Each region of actuator’s operating domain is characterised by a rule subset describing its local behaviour. The neuro-fuzzy model permits that actuator’s information codified into it can be generalised, and use its neural-based-learning capabilities in the manner to permit modify and/or add knowledge to the model when necessary.

2

In the second part of the paper, it is presented the implementation of the learning control system to the electro-hydraulic actuator combining its neuro-fuzzy inverse-model with a proportional controller. This scheme results in a indirect compensation control based on the feedback-error-learning technique proposed by Kawato in [5], [15], and initially explored by the authors in an unsupervised way in [18] using fuzzy logic.

' ) into fuzzy sets represented by x ' = ( x1' , x2' , , xm membership functions in U. These functions are

The implemented system is constituted by real-time learning and control cycles. During these cycles, the inverse-model of the actuator uses its neural-basedlearning capabilities to extract relations not incorporated into the initial model, and even change itself to characterise possible new actuator’s dynamic. At each control cycle, the learning mechanism actualises if necessary the inverse-model, and generates a compensation signal to the actuator. The results show that the controller generalises its acquired knowledge for new trajectories, it can acquire and introduce in real-time new information about the system using the sensor signals, and compensates some nonlinearities in the electrohydraulic system to progressively reduce possible trajectory errors. The structure of the paper is as follows. Section II characterises the fuzzy system used in the paper. In Section III, we brief summarise the fuzzy modelling

The Fuzzy Logic System

A fuzzy logic system consists in three main blocks: fuzzification, an inference mechanism, and defuzzification process. This section briefly explains and characterises each block concerning the type of fuzzy system used in the paper. 2.1 Fuzzification Fuzzification is a mapping from the observed numerical input space to the fuzzy sets defined in the corresponding universes of discourse. The element fuzzifier maps a numerical value denoted by



denoted by µ i ( x 'j ) and are gaussian functions, as A j

expressed in equation (1).   ' i  1 x j − bj ' i µ i ( x j ) = a j exp −  i Aj 2   c j

2       

(1)

In this equation, 1≤j≤m refers to the variable (j) from m considered input variables; 1≤i≤nj considers the i membership function among all n membership functions considered for variable (j); a ij defines the . ; b ij maximum of each gaussian function of a ij = 10 is the centre of the gaussian function; and cij defines its shape width. 2.2 Inference Mechanism The inference mechanism consists in the fuzzy logic reasoning process that determines the numerical output corresponding to fuzzified inputs, concerning the fuzzy rules and used inference operators. The rule-base is composed by IF-THEN rules like R

(l )

(l)

: IF ( x1 is A1

(l)

and x2 is A2 and

THEN ( y is ω

(l)

)

 xm is Am(l ) ) ,

(l )

m  ∑cl =1 ∏ µ ( l ) x 'j .ω( l )    j =1 A j  ' Y(x ) = m  '  c  ∑ l =1 ∏ µ ( l ) x j    j =1 A j 

( )

where: R is the lth rule with 1 ≤ l ≤ c determining the total number of rules in the model; x1 , x 2 , x m and y are, respectively, the input and output system



( )

(l)

variables; A j are the antecedent linguistic terms (or fuzzy sets) in rule (l) with 1 ≤ j ≤ m being the

(5)

(l)

number of antecedent variables; and ω is the rule conclusion being, for that Takagi-Sugeno type rules, a real number usually called fuzzy singleton. Each IF-THEN rule defines a fuzzy implication between condition and conclusion parts denoted by (l )



(l)

(l )

expression A1 × × Am → ω . The implication operator used in the paper is the product-operator, as shown in expression (2). The right term ' ( l ) ( x ) in this expression represents the A1 × × Am

µ (l)



condition degree that is defined by equation (3). ' (l) ( l ) ( x1 , A1 × × Am →ω

µ (l )



= µ (l )

, xm' , y ' ) =



'

'

(2)

( l ) ( x ). µω( l ) ( y )

A1 × × Am

µ (l)

'

'

( l ) ( x ) = µ ( l ) ( x1 )∗



A1 × × Am

A1

∗ µ A(l ) ( xm' ) (3) m

The symbol " ∗ " in (3) is the t-norm corresponding to conjunction and in the rules. In this work, we used the algebraic product as the chosen t-norm operator.

3

Fuzzy Modelling

Basic principles of fuzzy models, also called in literature fuzzy modelling, were first introduced by Zadeh in [2]. First applications in modelling systems using fuzzy logic consisted initially in duplication of expert experience and/or control engineering knowledge to process control [22]. Although, this qualitative information can present limitations as the acquired knowledge usually present errors and even some gaps. Another source of information is the quantitative information. This is acquired by acquisition of numerical data from most significant system variables, and can be used together with the anterior qualitative information to complete it or even produce new information [3]. The acquisition of models using fuzzy logic is usually divided in two approaches as shown in Figure 1: a linguistic approach composed by relational and natural models; and a hybrid approach concerning the neuro-fuzzy models. Fuzzy Modelling

Since it is considered each rule conclusion ω

(l)

as a fuzzy singleton, the value of its

membership degree µ ( l ) ( y ' ) in expression (2) stays ω 1,0. So, final expression for the fuzzy implication degree (2) results in multiplication of each condition membership degree (3) and equals expression (4). µ (l )





Relational

Natural

Hybrid (linguistic variables) + numerical variables

Neuro-Fuzzy

'

(l ) (l ) ( x ) = A1 × × Am →ω

= µ (l)

Linguistic (linguistic variables)

Fig. 1. Fuzzy modelling approaches. m

(4)

' ' ( l ) ( x ).(1,0) = ∏ µ ( l ) ( x j )

A1 × × Am

j =1 A j

The product inference in equation (3) expresses the activation degree of each rule by measured condition variables, and equals the expression (4) for implication degree. The reasoning process combines all rule conclusions (l)

ω using the centroid defuzzification formula in a weighted form, as indicated by inference function in (5). This equation maps input process states ( x 'j ) to the value resulted from inference function Y( x ' ) .

The main difference between these approaches is related on how the knowledge representation is represented into the model. While linguistic approach describes the system behaviour through rules of type IF-THEN using only fuzzy sets (linguistic variables), the hybrid approach uses linguistic variables in the condition rule part (IF) and uses a numerical value in the conclusion part (THEN) being considered as a function of input variables [3], [4]. Linguistic modelling can be separated into two types: relational modelling and the modelization called natural. Relational modelling [25], [26], [27], [28], establishes a set of all possible rules based on an attributed linguistic partition for each input-output

variable. It computes for each rule the respective true value of how much that rule contributes to describe system’s behaviour. The set of all rules composes, in a computational way, a multidimensional matrix called relational matrix. Using the theory of relational equations [29], [30], each matrix element can be computed as the rule membership degree in extracted system’s model . Second type of linguistic modeling is denoted by natural modelling. It does not use relational equations to obtain the model. The rules are codified from information supplied by the process operator and/or from knowledge obtained from the literature. The first application examples of this type of modelling were the fuzzy controllers in [22], [23]. Fuzzy systems modelling based on the hybrid approach permits to employ learning techniques used by neural networks in the extraction of each rule [16], [17], [20]. The set of parameters composing rule’s condition part is the membership functions width and their position in the respective universe of discourse. In the conclusion part, the parameters are function terms that compute the rule conclusion. Fuzzy modelling techniques are providing new directions in drive systems modelling [31], [32]. These techniques permit obtain more complete models, and so design better controllers which can

provide to the existing drives some learning capabilities.

4

The Learning Mechanism

During the learning process there are used two data sets: one for the training stage and other to test the extracted fuzzy model. Initially, using the training set, we extract the model rules and their conclusion value through a cluster-based algorithm [19]. After, the model has its conclusion values tuned by a gradient-descent method [24] to produce the process neuro-fuzzy inverse-model. Since the test data set has examples not presented in the training stage, we use it to verify the generalisation model performance. Follow, this section summarises the algorithms used to acquire the actuator’s inverse-model. 4.1

Model Initialisation Using the Clusterbased Algorithm The first modelling stage of the electro-hydraulic actuator concerns the initialisation of each rule conclusion using the cluster-based algorithm. The cluster concept when used with fuzzy logic [33] associates to each data value a degree among zero and one that represents its membership degree in the rule. This allows each sample data belong to multiple rules with different degrees.

x2 NB

NM

ZE

( ) Rule region R l (cluster)

Induced Membership function PB

1

PM

x1

ZE

(l )

ω

y'

NM

NB

( x1' , x 2' )

(a)

y'

µ

PM

( x 1' ) . µ

NM

( x 2' )

y'

(b)

Fig. 2: (a) Set of examples selected from the training data to extract the rule with antecedents defined by fuzzy sets PM and NM. (b) Membership function induced by weighted output values y' into the specified rule region, and the (l ) computed conclusion value ω .

Figure 2 illustrates the cluster concept applied to a fuzzy system. Suppose, by simplicity, a system with two inputs denoted by x1 and x 2 , and one output variable y. As shown in the figure, each domain variable x1 and x2 is equally partitioned by symmetric triangular fuzzy sets.

A data set composed by system examples is acquired to be used in the training stage. Figure 2(a) displays the examples covering the domain. Each one is formed by a data sample like ( x1' , x 2' ) → y ' . The examples are grouped in clusters for each respective

rule R ( l ) . In the figure, we exemplify the acquisition of rule expressed in (6). IF [(x1 is PM) and (x2 is NM)] THEN [y is ω

(l )

First stage

R

] (6)

The rule condition part is characterised by fuzzy sets PM and NM. The conclusion part, characterised by numerical value ω , is extracted based on the examples contained into the domain region covered by the two fuzzy sets PM and NM. This set of examples is represented in Figure (2) by filled circles into the rule region R

(l )

of that rule R

(l )

.

Suppose an example ( x1' , x 2' ) → y ' inside the rule region. Its contribution degree is computed by the product of each condition membership degree in fuzzy sets PM and NM of specified rule region, as expressed in (7) and displayed in Figure 2(b). The computed contribution degree weights the '

corresponding output value y .

µ PM ( x1' ). µ NM ( x 2' )

(7)

The anterior operations are executed for each example inside the rule region. This composes a membership function defined for all output values '

y into the rule region, as Figure 2(b) illustrates to (l )

rule (6). The final conclusion value ω for rule (l ) is computed from the induced membership function using the centroid method. 4.2 The Neuro-Fuzzy Algorithm The neuro-fuzzy algorithm developed by Wang [24] uses the hybrid model developed by Takagi-Sugeno in [3]. In this type of model, the condition part uses linguistic variables, and the conclusion part is represented by a numerical value considered a function of system’s condition expressed in the numerical values of variables x1 , x2 , , xn (22). ω

(l)

(1)

ω(1)

Π

x

1

.. .

. ..

Π

Σ

. ..

x

2

R

Y

-

+

ω(c)

Π

.. .

y'

(c)

.

Using the fuzzy cluster concept, it is attributed to each example a certain degree of how much it belongs to that cluster or, in other words, how much each example contributes to the extraction of conclusion value ω

Inference mechanism

Π

(l )

(l )

Second stage

' ' = g ( x1 , x2 ,





' , xn )

(22)

The neuro-fuzzy algorithm uses membership functions of gaussian type as set in equation (1). With gaussian fuzzy sets, the algorithm is capable of use all information contained in the training set to calculate each rule conclusion, different when there are used triangular partitions.

Adjust of the conclusion values by the gradient-descent method

Fig. 4: Neuro-fuzzy scheme.

Figure 4 illustrates the neuro-fuzzy scheme for an example with two input variables ( x1 , x 2 ) and one output variable (y). In the first stage of the neurofuzzy scheme, the two inputs are codified into linguistic values by the set of gaussian membership functions attributed to each variable. The second stage calculates to each rule R ( l ) its respective activation degree. At last, the inference mechanism weights each rule conclusion ω (l ) , initialised by the cluster-based algorithm, using the activation degree computed in the second stage. The error between the model inferred value Y and the respective measured value (or teaching value) y ' , is used by the gradient-descent method to adjust each rule conclusion. The algorithm changes the ω (l ) values to minimise an objective function E usually expressed by the mean quadratic error (23). In this equation, the value

y ' ( k ) is the desired output value related with the condition

vector

x ' ( k ) = ( x1' , x 2' ,

 , xn' ) k .

The

element Y ( x ' ( k )) is the inferred response to the same condition vector x ' ( k ) and it is computed by equation (24). E=

[

]

2 1 Y ( x ' ( k )) − y ' ( k ) 2

m  ∑cl =1 ∏ µ ( l ) ( x 'j ( k )) . ω( l ) ( k )  j =1 A j  Y ( x ' ( k )) = m  ∑cl =1 ∏ µ ( l ) ( x 'j ( k ))   j =1 A j 

(23)

(24)

Equation (25) establishes the adjust of each conclusion ω (l ) by the gradient-descent method. The symbol α is the learning rate parameter, and k indicates the number of learning iterations executed by the algorithm. ω (l ) ( k + 1) = ω( l ) ( k ) − α

∂E

(25)

∂ω ( l )

(l)

using equation (25) The adjust to be made in ω can be interpreted as being proportional to the error between the neuro-fuzzy model response and the supervising value, but weighted by the contribution of rule (l), denoted by d inference be good or not. ω

(l)

( k + 1) = ω =ω

5

(l )

(l )

(l)

pressure difference ( Pl ) in the piston induces a force that moves it. The implemented experimental system permits to connect a variable load to the piston represented in the figure by the symbol Fx .

ω ref

'

(Y ( x ( k )) − y ( k )) d

(l)

Electronic inverter + PI Controller

'

(Y ( x ( k )) − y ( k )) d

(l )

Fig. 6: First subsystem composed by the electrical drive and the P.M. Motor. Pcircuit

The control results concern a laboratorial system composed by a permanent-magnet (P.M.) synchronous motor driving a hydraulic pump that sends fluid to move a linear piston. Figure 5 shows a diagram of the system that is composed by two control loops. The interior loop, in grey, is responsible for the PM motor speed control. The loop is composed by an electrical drive with a PI controller to command the motor speed. The exterior loop, in black, controls the piston position using a proportional controller that gives the motor speed reference to the electrical drive. Load

-

Proportional Controller

Piston position

Power Electronics

P.M. Motor

+

TH

Electrical drive

The Experimental System

+

ω

P.M. Motor

iq

(35)

∑ cl =1 ( d ( l ) )

Piston Position Reference Speed Reference

id

ω

b '

(k ) − α

operating in a pressure of 40 bar ( Pcircuit = 40bar ) . As the pump sends fluid ( q p ) to the piston, the

, to the final neuro-fuzzy '

(k ) − α

In Figure 7, we show the second subsystem that composes the electro-hydraulic actuator. The hydraulic pump is assumed to rotate at same speed as the motor (ω = ω p ) , with the hydraulic circuit

Hydraulic Actuator System

PI Controller

PM Speed Measured Piston Position

Fig. 5: Diagram of the experimental electro-hydraulic drive system.

Two subsystems compose the actuator. Figures 6 and 7 show their diagrams. The first subsystem in Figure 6 shows the electrical drive that controls the motor speed (ω) . The electronic inverter employs IGBT’s to generate currents i d and i q that command the P.M. motor (220V/ 1.2Nm/ ±3000 rpm). The speed controller is composed by a PI regulator. The motor load is denoted by TH , and it comes from the hydraulic pump connected to the motor.

v ω

Hydraulic pump

qp

Fx

Piston

Pl

y

ωp

Fig. 7: Second subsystem composed by the hydraulic system.

The electro-hydraulic system has its behaviour dominated by a nonlinearity localised into the hydraulic pump. This introduces a non-linear interface between the electrical system and the hydraulic actuator. Figure 8 displays the nonlinearity by an experimental curve that shows the relationship between motor speed signal (ω) , considered equal to the pump speed, and the piston linear speed ( v ) . The graphic shows an asymmetric dead-zone localised between the motor speed values of -700 r.p.m. and +900 r.p.m., and it displays hysteresys friction effects out of the dead-zone. When operating into the dead-zone, the piston stops as the fluid stream debited by the pump is near zero and the two actuator subsystems stay disconnected. Out of the dead-zone, the inclination of the two lines shows that the pump debits slightly more hydraulic fluid when rotating in one direction than rotating to the other.

about system physics. This knowledge is present when we model the actuator using electromechanical power conversion theory and hydrodynamic laws. As the system contains a great number of variables that can be chosen to characterise its dynamic, it is important make some hypothesis and simplifications to concentrate our attention to a small but representative variable set.

v [m/s]

0.02 m/s ~ -700

ω [r.p.m.] ~ +900

As shown before, the electro-hydraulic actuator is separated into two subsystems: the electrical drive and the hydraulic circuit with the pump and piston elements. If we consider these subsystems as blackboxes and make some considerations, as for example not consider relevant the contribution of the pressure signal in the circuit ( Pcircuit ) because it remains approximately constant during actuator’s operation, we can interpret the piston position signal (y) as a function of the reference signal ( y ref ), the motor

500 rpm

Fig. 8: Experimental curve showing the non-linear characteristic present in the electro-hydraulic actuator.

Figure 9 illustrates the piston asymmetric behaviour when the system operates in an open-loop (without the proportional position controller) for a sinusoidal reference to the motor speed (Figure 9(a)). We observe in Figure 9(b) that the piston moves more to one direction than to the other. Therefore, after some sinusoidal periods, it halts at the end of its course of 0.20m. This behaviour is caused by the non-linear characteristic with its asymmetry into the dead-zone, and even by the debit of more fluid for one pump speed direction that to the other.

speed (ω), and the linear speed of the piston (v). So, its direct model can be represented by relation (36). y = f ( y ref , ω, v )

(36)

To extract function f (. ) , it is necessary use some numerical data available from the system. For this, two different sets of experimental values were acquired to the modelling process, one set for training and other for testing.

5.1 Training Data Generation To obtain some relevant information to the training process, we started using theoretical knowledge 1500

0.20

1200 900 600

0.15

300

ω [r.p.m.] 0

y [m]

-300

0.10

-600 -900 -1200 -1500

0.05

0

5

10

15

20

25

30

0

5

10

15

20

25

30

Time [s] Time [s] (a) (b) Fig. 9: Actuator’s response for a sinusoidal reference signal with amplitude and frequency constants, and operating in an open-loop mode. (a) Motor speed signal (ω). (b) Piston position signal (y).

As described, the actuator has an asymmetric behaviour dominated by the presence of a non-linear characteristic. If we need to acquire some training data that characterises a significant part of the electro-hydraulic system’s operating domain, we can not use the system in an open-loop mode (see Figure 9) since we can not control where system will operate. So, to assure that the training data contain representative data and attenuate the non-linear characteristic effects, we used the actuator in a

closed-loop with a proportional regulator for a coarse control of piston position. The use of a coarse controller as the proportional one helps us to put in evidence the high non-linear character of the actuator, and better observe in the next sections the learning controller potentialities.

y

ref 0.2

y ref y

,y [m]

desacoupled from the electrical part. Although rotating, the pump debits almost no fluid into the hydraulic system, and so there is no sufficient pressure difference on the piston to move it.

0.1 0

0

2

4

6

8

10 Time [s]

12

14

16

18

20

2000 0 -2000

(a) Error [%] +20 +10 0 -10 -20

Dead-zone [-700, 900] 0

2

4

6 where the electrical { Regions and hydraulic system stay desacoupled

v [m/s] 0.2 0

2

4

6

8

10 Time [s]

12

14

16

18

20

2

4

6

8

10 Time [s]

12

14

16

18

2

4

6

20

(c) v [m/s] 0.2 0 0

0

Fig. 11: Picture detail of the pump speed and piston speed signals. It shows the effect of the dead-zone desacoupling the hydraulic part from the electrical one.

2000 0 -2000 0

0 -0.2

(b)

ω [r.p.m.]

-0.2

ω [r.p.m.]

2

4

6

8

10 Time [s]

12

14

16

18

20

(d) Fig. 10: Electro-hydraulic system behaviour when operating with the closed-loop proportional controller. (a) Reference signal evolution ( y ref ) and the piston position signal ( y ) . (b) Error signal evolution displayed in a percentage scale. (c) Evolution of the pump speed signal (ω) . (d) Evolution of the piston speed signal ( v ) .

Figure 10 presents the actuator’s evolution when it operates with the proportional closed-loop controller under a sinusoidal reference signal. As Figure 10(a) displays, the piston follows its reference signal with an asymmetric time-delay causing high tracking errors. As the pump dead-zone is more large for positive speeds, there is a larger delay in system’s response for up directions which causes the appearing of high positive errors (see Figure 10(b)). On the contrary, as negative dead-zone is shorter, the system responds more fast and so the error signal decreases for down directions causing small negative errors. If we correlate the pump speed signal displayed in Figure 10(c), with the respective piston speed signal in Figure 10(d), we recognise that there is a set of operating regions where, although pump rotates, the piston stays halted. Figure 11 shows a zoom of this behaviour. For the pump speed signal, we mark the speed interval corresponding to the dead-zone. Below, we mark the corresponding regions where the piston speed is zero. When the pump operates into the dead-zone, the hydraulic circuit stays

To complement with more objective information the theoretical knowledge about the experimental system, it is acquired some experimental data from the laboratorial system. This data set is used in the training stage, and it is composed by some system’s behaviour examples. Usually, to construct a training set, a Pseudo-Random Binary Signal (PRBS) is injected to the system in the manner that collected data spans almost all system’s operating domain. Although, this signal is not good to excite drive systems as pointed too in [7]. So, a good procedure is to use an excitation signal of sinusoidal type composed by different magnitudes and frequencies, but these into drive’s response limits. For the electro-hydraulic actuator, we used a sinusoidal signal as a reference for piston position with its amplitude ranging from 0 to 0.02m, the piston course limits, and frequencies among 0 and 1Hz, because for higher frequency values the actuator begins to filter the reference signal. The modelling scheme is described by a diagram in Figure 12. Initially, a data set with four system signals ( y ref , ω, v , y ) is acquired using the anterior training procedure. In Figure 13, we display the acquired training data set composed by the sinusoidal reference signal y ref , the piston position y , the hydraulic pump speed signal signal, v.

ω, and the piston speed

The fuzzy model is composed by 7 membership functions attributed to the reference signal y ref , 11 Electrohydraulic

y ref

membership functions to the piston position signal y, and 7 membership functions attributed to the piston speed v. The functions are of gaussian type, as

y

System

i

ω y ref

explained before, with their shape b j settled in 60%

v Training

of each partition interval in each variable (j).

y

Set

The first step in the modeling process uses the cluster-based algorithm to extract the initial fuzzy model. To verify the generalisation capability of this initial model, we use the test data set. Figure 14(a) shows the generalisation results displaying the

Learning Process

*

measure (ω) and inferred speed signals (ω ) , with Figure 14(b) showing the corresponding error evolution.

Neuro-Fuzzy Model

Fig. 12: Diagram scheme representing the modelling stages. y

ref

,y [m]

y

- those domain regions where a small number or even no examples were acquired, being that information not enough to extract a representative rule set for those regions;

ref

0.2 0.1

[m]

y 0

0

5

10

15

20

25

30

Time [s]

(a)

ω [r.p.m.] 2000 0 -2000 0

5

10

15

20

25

Through the error signal, we can observe that there are high errors to some operating regions. These are caused by:

30

Time [s]

- when the actuator operates into the dead-zone, it can not be defined an inverse functional relation and the model generates high prediction errors; - other errors appear by consequence of noise presence in acquired signals y and v, which can deviate the inferred pump speed values from their correct predictions within a certain degree. ω ω*

ω, ω * [r.p.m.] 2000

(b) v [m/s] 0.1 0 -0.1

1000 0 0

5

10

15

20

25

30

Time [s]

(c)

-1000 -2000

0

40

speed (ω) . (c) Piston speed ( v ) .

6

Neuro-Fuzzy Modelling of The Electro-Hydraulic Actuator

In this section, the actuator is modelled using the neuro-fuzzy algorithm based on training data set of Figure 13. The experiment consists in obtain the inverse model of the actuator that is represented by relation ω = f

*−1

( yref , y , v ) .

80

120

160

200

Time [s]

Fig. 13: The acquired training data set. (a) Reference and position signals ( yref and y ) . (b) Hydraulic pump

(a) Error [% ] 30 20 10 0 -10 -20 -30

0

40

80

120

160

200

Time [s]

(b) Fig. 14: Initial modelling results using the cluster-based algorithm to extract the actuator’s fuzzy inverse-model.

(a) Evolution of the measured (ω) and the inferred * pump speed signal (ω ) . (b) Error signal evolution.

7

Continuing with this experiment, we consider now the use of the gradient-descent method explained in section IV.D to fine adjust the anterior initial model. For the learning process, the parameters used by the algorithm were: a number of 50 iterations (K = 50) , a learning rate of 0.8 (α = 0.8) , and the same fuzzy model structure used in the cluster-based algorithm.

This section describes the implemented control system and shows experimental results of its application in an electro-hydraulic position controller.

The results obtained after tuning the initial rule conclusions are displayed in Figure 15. They show, compared with the results of Figure 14, the good tuning made by the neuro-fuzzy algorithm that reduces the error signal to about 10%. 2000 1000 0 -1000 -2000

0

40

80

120

160

200

Time [s]

(a) Error [% ] 30 20 10 0 -10 -20 -30 0

The neuro-fuzzy control system is based on the feedback-error-learning scheme. The neuro-fuzzy model adjusts each rule conclusion to minimise the mean quadratic error generated by the proportional controller (P), as indicated in equation set (37).

(

)

2  1 E = P ( y − yref )  2  ω( l ) ( k + 1) = ω( l ) ( k ) − α ∂E (l )  ∂ω

ω ω*

ω, ω * [r.p.m.]

The Neuro-Fuzzy Control System Implemented

(37)

Figure 16 shows in a diagram the controller. It was implemented in a Personal Computer (PC) with a 80386 CPU and an interface with A/D (analog to digital) and D/A (digital to analog) converters. All programming was done in C language, including the neuro-fuzzy algorithm and actuator signal’s acquisition and conditioning. The control system performs in two levels. The high level contains the learning mechanism responsible by actualisation of the information contained in the inverse relation. The low level constitutes the control system formed by the feedback-loop and a feedforward-loop composed by the fuzzy inverse

40

80

120

160

200

Time [s]

(b) Fig. 15: Modelling results obtained after tuning the initial model using the neuro-fuzzy algorithm. (a) Evolution of the measured (ω) and the inferred pump * speed signal (ω ) . (b) Error signal evolution.

It is important note that the number of iterations and the learning rate were chosen in the manner that the model does not start incorporate noise dynamics by overfiting. Another aspect is that for domain regions where the number of acquired examples is small, the neuro-fuzzy algorithm continues to present high inference errors because it had little or none information to do a good tuning and so extract significant rules. These points reveal the necessity of acquire in real-time information from the system. In this way, the learning mechanism can collect more information to correct and/or incorporate other rules into the model and so reduce its prediction errors.

relation ωcomp = f

*−1

( yref , v , y ) with its inference

mechanism producing the compensation signal ω comp . At each control iteration, the learning system collects the present values of the reference signal ( y ref ) , the piston speed ( v ) , and the current piston position ( y ) , through the disposable sensor set. These signals express actuator’s operating condition and make each model rule be actived in some degree (see expression (3)). After, the inference mechanism uses model rules with corresponding activation degrees and computes the compensation signal (ω comp ) to be added to the proportional controller command (ω p ) , as seen in Figure 16. The final signal, denoted by ω ref and equal

to

the

sum

of

ω comp

and

ωp

(ω ref = ω p + ω comp ) , is sent to the electrohydraulic actuator as its final command signal.

Neuro-fuzzy learning mechanism

Rules R (l)

v

. ..

y

ω(1)

Nº.1

. ..

Nº.2

ω(2)

Nº.3

ω(3)

High level

. .. . ..

yref

ω(c)

Nº c

.. .

Learning system

Rules y ref

Inverse fuzzy

y

relation

Feedforward-loop

v

Low level

ωcomp y ref

-

P +

+ ωp +

v

Electro-hydraulic

ω ref

y

actuator

Feedback-loop

Fig. 16: Diagram of the implemented neuro-fuzzy control system.

To close the controller cycle, the error signal generated by the proportional controller after the application of before computed compensation signal is used to adjust each rule. The inverse relation is then adjusted based on the performance attained by the compensation made with the anterior rule set and verified through the magnitude of the proportional controller signal. Each rule is adjusted proportionally to its anterior activation degree, interpreted as a measure of how much the rule contributed to the actual actuator’s performance. 7.1 Experimental Results The experimental results use a square wave as the reference position signal to the piston. Figure 17 shows the results for the first test. In this test, the actuator is controlled only through the feedback-loop through the proportional controller without any compensation signal. The results show the presence of an offset error signal between the reference position and that attained by the piston (Figure

17(a)). The asymmetric error evolution in Figure 17(b) is conditioned by the asymmetric dead-zone at the hydraulic pump. y

y ref y

,y [m] ref 0.2 0.1 0

0

10

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80

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Time [s]

(a) Assymetric error signals

Error [%] 25 0 -25 0

10

20

30

40 50 Time [s]

60

70

(b) Fig. 17: Experimental results obtained when the actuator operates only with the feedback-loop. (a)

Evolution of the reference signal ( y ref ) and the piston position signal ( y ) . (b) Asymmetric error evolution.

Experimental actuator system

y ref +

The results in Figure 17 as shown that the compensator absence in the control-loop gives high tracking errors. To a second test, we added the compensation signal generated by the neuro-fuzzy inverse-model to the command signal of the proportional controller. The new results are displayed in Figures 18(a) and 18(b). They use the compensation feedforward-loop with the initial extracted neuro-fuzzy model but with the learning mechanism off. The results show that addition of the compensator signal eliminates the error at the superior part of the reference signal, but makes a higher error value to the inferior part. The compensation signal distorted the proportional controller signal (ω p ) increasing the error for the

-

y

v

P y ref

y

Fuzzy inverse relation

ω comp Fig. 19: Diagram showing the use of the proportional controller signal to correct the inverse relation.

Figure 20(a) illustrates the piston approximation to its reference signal as rules are adjusted and the compensation signal is corrected. For this test, we used a low learning rate (α = 0.0005) to better visualise the adjust of the compensation signal.

inferior part. y ref y

y ref ,y [m] 0.2 0.1 0

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30

40 50 Time [s]

60

70

80

90

100

(a)

As the learning mechanism begins to actuate, the system slowly increases the pump speed, as verified in Figure 20(b), to send fluid to the hydraulic circuit and hence move the piston. The pump increases the speed until its magnitude leave out the dead-zone and then conduct the piston to the reference position reducing the error signal as shown in Figure 20(c). y

Error [%] 25 0 -25

,y

ref 0.2

y ref y

[m]

0.1 0

0

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50 60 Time [s]

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90

0

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100

The anterior results point out the necessity of a more precisely adjustment of the model rules that were made active when operating at the inferior region. The neuro-fuzzy learning mechanism is then introduced in the control system to acquire new signals in real-time and so correct the rules to better tune the inverse model. The system uses the proportional controller signal to adjust the rules as described in Figure 19, and therefore correct the compensation signal ω comp .

25

30

35

(a)

ω [r.p.m.] 2000 0 -2000 0

( y ) . (b) Error signal evolution.

20

With adjust of the compensation signal

Without adjust

(b) Fig. 18: Experimental results when the feedforwardloop is added to the actuator system but without the neuro-fuzzy learning mechanism. (a) Evolution of the reference signal ( y ref ) and the piston position signal

15 Time [s]

Dead -zone

5

10

Without adjust

15 Time [s]

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With adjust of the compensation signal

(b) [%]

Error 25 0 -25

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5

Without adjust

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15 Time [s]

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25

With adjust of the compensation signal

(c) Fig. 20: Actuator’s evolution showing the learning mechanism action to adjust the compensation signal. (a) Evolution of ( y ref ) and the piston position signal ( y ) .

(b) Hydraulic pump speed (ω) . (c) Error signal.

In Figure 21, we present the piston evolution when the feedforward-loop and the learning mechanism are inserted into the feedback control system. It is used a

higher learning rate with a value of α = 0.02 in the manner to have a fast transient but without overshoots. The results show the real-time tuning of the initial model until about 16 seconds with the compensation signal gradually correcting the error offset, and approximating the piston to the reference signal. y

ref 0.2

y ref y

,y [m]

9

Discussion: The Influence of the Learning Parameter’s α value in Controller’s Performance

The learning rate parameter α used for the square wave reference was established in the manner piston position does not have an overshoot, case for an α value to much high, and the piston does not have a transient response to much slow, in this case the learning parameter has a value considered low.

0.1 0

0

4

8

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16

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24

28

32

28

32

During the first experiments presented in Figure 20, it was used a learning parameter value of 0.0005 to evitate possible oscillations, and then verify the gradually correction effect of the rules and compensation signal too. The follow experiments employed a learning parameter of 0.02 which reduced significatively the tracking error in the electro-hydraulic position error (Figures 21 and 22).

Time [s] Adjust of the compensation signal

Evolution in steady-state

(a) Error [%] 25 0 -25 0

4

8

12

16 20 Time [s]

Adjust of the compensation signal

24

Evolution in steady-state

(b) Fig. 21: Actuator’s evolution with the adjustment in real-time of the model rules to correct the compensation signal. (a) Evolution of the reference signal ( y ref ) and the piston position signal ( y ) . (b) Error signal.

At last, the results in figure 22 show a zoom of two cycles of system’s evolution in steady-state after the model be adjusted and eliminated the error. y

ref 0.2

,y [m]

y ref y

0.1 0

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80

90

(a) Error [%] 25 0 -25 10

20

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100

Time [s]

(b) Fig. 22: Actuator’s evolution in steady-state using the neuro-fuzzy learning control system. (a) Evolution of the reference signal ( yref ) and the piston position signal

( y) . (b) Error signal.

Although, when the reference signal presents an accentuated variation, the dead-zone effect makes more critic the choice of the learning rate parameter α to controller’s performance. To describe these controller’s sensitivities, we discuss in this section a series of control situations that use a sinusoidal reference signal with different frequency values and using distinct α values.

100

Time [s]

0

During the results in response to the square wave reference, the dead-zone presence helps the controller action when it has to stop the piston. In this situation, there is no need to have pump negative speeds since the actuator begins to operate into the dead-zone and the piston stops. One other advantage of this control situation, is that the reference signal remains constant during large intervals, which helps additionally the controller’s performance in track the reference signal.

Two effects contribute to the presence of an offset in the error signal in steady-state. The first effect refers to the situation in which the model is incomplete and needs more information to complete the rules not extracted, or improve those rules already extracted but using a small quantity of information. The second effect, and this related with the non-linear characteristic of the electro-hydraulic drive system, corresponds to the dead-zone that actuates has a filter for all compensation actions that occurs into its interval. The results in Figure 23 illustrate the anterior effects. They are obtained with the feedback and feedforward loops acting together but without the learning mechanism. During the time interval 0 to about 21 seconds, the actuator is in the transient stage between operating only with the feedback-loop through the

(a)

proportional controller, and after, when the inverse model is added by the feedforward-loop.

Error [%] 10 0

As presented by error evolution in Figure 23(b), the introduction of the compensation signal produces a decreasing transient in the tracking error. Although, the error signal in steady-state attains a mean value causing an error offset. The offset appears when the actuator operates into the dead-zone where the nodefined inverse model generates a deteriorated compensation signal (Figure 23(c)) and, it appears when operating out of the dead-zone, from the extracted inverse model that demand for more information to complete and improve its rules. In this way, if the controller has now the information of system’s error coming from the proportional controller, it can complete its model and/or leave out of the dead-zone to eliminate the offset and so reduce the tracking error. y

0

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25 30 Time [s]

Transient interval + Without learning

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Dead -Zone 0

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50

(c) v [m/s] 0.1 0 -0.1 0

5

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25 Time [s]

(d)

45

Fig. 24: Actuator’s performance using the neuro-fuzzy controller with a sinusoidal reference signal, and a learning parameter value of (α = 0.02) . (a) Reference signal and the piston position. (b) Error signal evolution. (c) Pump speed signal. (d) Piston linear speed signal.

50

Steady-state interval + Without learning

(a) Error [%] 20 10 0 -10 -20 0 5

10

15

20

25 30 Time [s]

Decreasing transient

35

40

45

A. Case 1. In this first case, the results must be referenced with those obtained without the learning mechanism in Figure 23.

50

Steady-state interval

(b)

ω [r.p.m.] 2000 0 -2000 0

Dead -zone

5

10

15

20

25 30 Time [s]

Decreasing transient

35

40

45

50

Steady-state interval

(c) Fig. 23: Results that point out the presence of an offset in the error signal in steady-state. (a) Actuator evolution after the introduction of the compensation signal but without the presence of a learning mechanism. (b) Error signal evolution. (c) Pump speed signal.

Next, we present a series of operating situations where the reference signal has an accentuated variation being a sinusoidal signal. During the tests, we show the effects in controller’s performance when using different learning rates and reference signals with different frequencies. y

ref 0.2

,y [m]

y ref y

0.1 0

0

5

10

15

20

25 30 Time [s]

35

40

45

50

(b)

ω [r.p.m.]

0.1 0

0

Time [s]

y ref y

,y [m]

ref 0.2

-10

50

Figure 24 shows this case obtained for a sinusoidal reference signal with same amplitude and frequency as those used in Figure 23, and a value for the learning parameter of α = 0.02 , considered “ideal” in the anterior results for a square-wave reference signal. Considering the signal of the proportional controller, the rules are adjusted to increase the pump speed, leave the dead-zone, and complete the model approximating the piston to the reference signal. However, as the pump speed takes some time to leave the dead-zone, the piston remains stopped during these time intervals (see Figure 24(d)). When its speed reaches the boundary region of the deadzone, the friction effect makes the piston surpass the reference signal (see Figures 24(a) to 24(c)). Hence, the compensator uses now a negative error signal and adjusts the rules in the manner to invert the pump speed to stop the piston and drive it to the reference. The pump has to surpass all dead-zone to achieve negative speed values that act over the piston and take it to the reference position. However, the learning parameter value is not enough to get the compensation signal to the negative speed values out

We verify by Figure 25 and comparing it with results of Figure 24, that the actuator shows a better performance, although it continues to present a movement in steps since the system needs to increase the compensation signal and leave out of the deadzone. As the used α value results in slow learning, and the actuator can sometimes present oscillations, we increase in ten times the α value in the manner the system leaves more fast from the dead-zone and the piston does not stop. Next case presents actuator’s evolution with this new learning parameter.

of the dead-zone, and the pump speed stays into the dead-zone where the piston stays halted. At the same time, as the reference signal continues to change, the error increases, and the pump speed is increased again by the compensation signal until reach negative values out of the dead-zone. Therefore, the piston approaches again to the reference. The effect of anterior operating cycles when the reference signal goes up and goes down, makes the piston follow the reference in steps, as seen in Figures 24(a) and 24(b), and by piston speed evolution in Figure 24(d).

y

,y [m] ref 0.2

y ref y

0.1 y

,y [m] ref 0.2

y ref y

0

0

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60

0.1 0

80 100 Time [s]

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(a) 0

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80 100 Time [s]

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Error [%] 10

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(a)

-10

Error [%] 10

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0 -10

0

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100

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2000 0 -2000

(b)

ω [r.p.m.]

(b)

ω [r.p.m.]

200

Time [s]

Dead -Zone 0

2000 0 -2000

20

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20

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80 100 Time [s]

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(c) v [m/s] 0.1 0

(c)

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v [m/s] 0.1

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Time [s]

0 -0.1

120

(d) 0

20

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80 100 Time [s]

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(d) Fig. 25: Actuator’s performance using the neuro-fuzzy controller with a sinusoidal reference signal, lower frequency, and a learning parameter value of (α = 0.02) . (a) Reference signal and the piston position. (b) Error signal evolution. (c) Pump speed signal. (d) Piston linear speed signal.

B. Case 2. This case uses the same learning parameter value (α = 0.02) but a sinusoidal reference with lower frequency. We investigate now the controller’s performance when the reference signal changes more slowly, and then this allows the pump have time to leave out the dead-zone without the reference signal deviates to much from actual piston position.

Fig. 26: Actuator’s performance using the neuro-fuzzy controller with a sinusoidal reference signal, lower frequency, and a higher learning parameter value of (α = 0.2 ) . (a) Reference signal and the piston position. (b) Error signal evolution. (c) Pump speed signal. (d) Piston linear speed signal.

C. Case 3. The results in Figure 26 show actuator’s evolution for a reference signal with the same values to its magnitude and frequency as before, but with a much superior learning parameter value (α = 0.2 ) . The error signal between reference and piston position decreases (see Figures 26(a) and 26(b)) but presenting high oscillations around the reference. The pump and motor signal show often speed variations too (Figure 26(c)). Also, we observe that the position signal presents higher errors when the reference signal indicates an

has to learn more fast, or where it is not need to learn so quickly.

expansion and operates with positive angular speeds than on the contrary. This happens because the deadzone to positive speed values is larger than to negative ones. Therefore, the compensator action is more attenuated to piston expansion then to the contrary, increasing the position error. y

,y [m] ref 0.2

9

The neuro-fuzzy methodology is used in this paper to demonstrate the incorporation of learning mechanisms into control of drive systems. We believe that emerging technologies as neuro-fuzzy systems have to be used together with usual conventional controllers in the manner to produce more “intelligent” and autonomous drive systems. It is wrong put way all knowledge accumulated about the classical controllers and substituted them by emerging techniques as fuzzy systems, neural networks, or genetic algorithms. So, it is becoming important investigate control designs that permit a symbiotic effect between the old approaches and the new ones.

y ref y

0.1 0

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20 25 Time [s]

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(a) Error 10

[%]

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(b)

ω [r.p.m.] 2000 0 -2000

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20 25 Time [s]

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(c) v [m/s] 0.1 0 -0.1

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Conclusion

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25 20 Time [s]

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45

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(d) Fig. 27: Actuator’s performance using the neuro-fuzzy controller with a sinusoidal reference signal, higher frequency, and a higher learning parameter value of (α = 0.2) . (a) Reference signal and the piston position. (b) Error signal evolution. (c) Pump speed signal. (d) Piston linear speed signal.

D. Case 4. This last case illustrates actuator’s performance when the reference signal has a higher frequency, hence a larger variation, and we look to improve the compensator actuation increasing the learning rate from 0.02 to 0.2. The results displayed in Figure 27 show that the pump speed begins to present more oscillations, although system improves for expansion direction because it takes advantage of the dead-zone that diminishes the high speed oscillations. For down direction, as the dead-zone is smaller, the oscillations begin to appear more salient in the position signal. The four situations presented show that, for some operating conditions, it is important to have a certain adjustment of the learning parameter α related with the characteristics of the reference signal. Therefore, the learning system adapts to new situations where it

To the concretization and investigation of the anterior objectives, we presented in this paper a neuro-fuzzy modelling and learning approach to design a position controller to an electro-hydraulic actuator. The results presented indicate the ability of the implemented neuro-fuzzy controller in performing learning and generalisation properties to quite different movements than those presented during the training stage. It also demonstrated the compensation of nonlinearities in the electro-hydraulic system that deviated the feedback controller action to drive the piston position to its reference signal. The presence of a significant dead-zone into the electro-hydraulic system provokes the appearing of perturbations in actuator’s evolution. These perturbations are dependent of used learning parameter magnitude and dependent of the reference signal type for piston position.

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