anfis & pi d controller design and comparison for overhead ... · nonlinear elements of the...

Indian Journal of Engineering & Materials Sciences Vol. 18, June 2011, pp. 191-203

ANFIS & PIλDµ controller design and comparison for overhead cranes

Hüseyin Arpacıa & Ö Faruk Özgüvenb aInformatic Technology Department, İnönü University , Malatya, Turkey

bElectrical and Electronics Engineering Department, İnönü University , Malatya, Turkey

Received 18 March 2010; accepted 21 March 2011

The study suggests a method of designing an “intelligent” digital control for maintaining the load masses and target displacement at predefined position on overhead crane system. The control works on the basis of modeling at position control and sway angle control. In this study, the ANFIS and PIλDµ control system are applied to crane. Heuristic rules derived with the membership functions then the parameters of membership functions are tuned by adaptive neuro-fuzzy inference system (ANFIS). MATLAB, SIMULINK and Fuzzy Logic TOOLBOX are the programming environments used for realization of the model. The principle aim in designing the control is to assure the fastest and best transition possible from a controller for overhead crane.

A non-linear model for an overhead crane system, which takes into account a combination of a trolley and a pendulum, is derived. The overall mathematical model obtained is simulated using MATLAB-SIMULINK. An adaptive neuro-fuzzy controller, which includes three rule bases, and PID, PIλDµ, used for position control, is successfully designed and implemented on the below simulated model. At the same time, in this performance, more rapid and less swing results are obtained for longer transportation distance with ANFIS control system, compared with the results of other studies.

Keywords: ANFIS, Fractional, PID, Fuzzy logic, Crane control, Modeling, Simulation

Overhead cranes used for transporting of load from one place to another. These type of cranes can handle huge loads and especially used in factories, ships, platforms, depots, dockyards. Different type of cranes used in different areas. Gantry cranes used in factories, load is attached to the trolley by a cable, and moving this load along production lines. Rotary cranes used especially in construction, transport of building materials to high point. In these areas transporting of heavy loads with cranes means saving time and hence costs. The purpose of controlling such a crane is to transport loads in the minimum time, safely, and without swing. If an operator operate manually the cranes more time will be consumed and recently because of the cranes becoming larger, the manual controlling is becoming difficult.

Recently, several researchers proposed many approaches for controlling an overhead crane. Because of the non-linear, multivariable, and including uncertanties structure, there are some difficulties in controlling of cranes. The most important point in controlling cranes is to deliver the load in the shortest time and minimizing the swing

angle. Manson1 majored on the transfer time. Karihaloo and Parbery2 tried to minimize a control force objective function instead of time. Sakawa and Shinde3 minimize the swing angle. The controller developed by Ridout4 feeds back the trolley position and speed and the load swing angle. An anti-sway control system combining a position control with a PI controller was proposed by Lee et al.5 Another feedback control of crane is proposed by Ohnishi et

al.6 For simultaneous travel and transverse motion for minimum time. Moustafa and Ebied7 derived a nonlinear dynamics model of an overhead crane. A linear feedback controller using full-state feedback is developed by Hurteau and Desantis8. Omar and Nayfeh9 developed a gain-scheduling controller, by which the transfer time is closer to optimal time and the load swing angle is small. An adaptive type of pole placement controller is used by Marttinen et al.10 Moustafa and Emara-Shabaik11 developed a nonlinear model including the effects of electrical transients in the driving motors. Yasunobu and Hasegawa12 applied predictive fuzzy control to a shipyard crane control for evaluation of safety, stop-gap accuracy, minimum swing, and minimum time. Nally and Trabi13 applied a distrubuted fuzzy logic controller to

___________________________ *Corresponding author (E-mail: [email protected])

INDIAN J. ENG. MATER. SCI., JUNE 2011

192

a bidirectional gantry crane. Another fuzzy controller is introduced by Suzuki et al.14 For transporting the load without swing. The fuzzy logic control is compared with linear quadratic gaussian control by Benhindjeb et al.15

In crane control the aim to transport the load from one place to another in short time with minimum swing. Some of the controllers mentioned above could control the transportation time and swing angle simultaneously and some of them try to control these two tasks separetely.

This paper proposes a new method for the design of an overhead crane control system. Considering nonlinear elements of the crane, we are to design a controller for crane with adaptive neuro-fuzzy ınference systems (ANFIS) for controlling to transport the load safely, rapidly, correctly, and without swing. Then the performance of this controller is compared with the performances of PID & PIλDµ controllers. The results obtained in this study have time saving and less swing than the results reported in the literature.

Principle of ANFIS Modeling

Fuzzy logic controller have played an important role in the design and enhancement of a great number of real time application. A fuzzy model is a system description with fuzzy quantities which are expressed in terms of fuzzy numbers or fuzzy sets. The proper selection of the number, the type, and the parameters of the fuzzy membership functions and rules are important for achieving the desired performance. ANFIS (adaptive neuro-fuzzy ınference system) combines the fuzzy qualitative approach with the neural network adaptive capabilities to achive a desired performance. The algorithm of a fuzzy system is as follows:

A. Fuzzification (i) Normalize of the universes of discourses for

the fuzzy input and output vectors. (ii) Choose heuristically the number and shape of

the membership functions for the fuzzy input and output vectors.

(iii) Calculate of the membership grades for every crisp value of the fuzzy inputs.

B. Fuzzy inference

(i) Complete the rule base by heuristics from the viewpoint of practical system operation.

(ii) Identify the active rules stored in the rule base.

(iii) Calculate the membership grades contributed by each rule and the final membership grade of the inference, according to the chosen fuzzyfication method.

C. Defuzzification

(i) Calculate the fuzzy output vectors, using an adequate defuzzyfication method.

(ii) Simulation test until desired parameters are obtained.

(iii) Hardware implementation.

Two models are popular (i) Mandami-style inference, expects all output membership functions to be fuzzy sets. It is intuitive, has widespread acceptance, is better suited to human input. It’s main limitation is that the computation for the defuzzification lasts longer

(ii) Sugeno-style inference, based on Takagi-Sugeno-Kang method of fuzzy inference. Formalize a systematic approach in generating fuzzy rules from an input-output data set, that expects all membership functions to be a singleton. It has computational efficiency, works well with linear techniques, works well with optimization and adaptive techniques, guaranties continuity of the output surface, is better suited to mathematical analysis.

A fuzzy model with Sugeno-style inference is presented as a collection of fuzzy rules in the following form:

If x is A, y is B and z is C then u= f(x,y,z),

A, B and C are fuzzy sets and u is a function of input variables. Consider a first-order Sugeno fuzzy inference system with two-fuzzy if-then rules (Figure 1).

1 1 1

1 1 1 1 1

2 2 2

2 2 2 2 2

Rule 1: If x is A , y is B and z is C then

. .

Rule 2: If x is A , y is B and z is C then

. .

f p x q y k z r

f p x q y k z r

= + + +

= + + +

pi, qi, ki and ri are design parameters to be determined during the training stage. In Fig. 1 a circle indicates a fixed node, a square indicates an adaptive node, the parameters of an adaptive node are changed during adaptation or training. As seen in the figure there is one input layer, three hidden layers, and one output layer. If Oli denotes the output of node i in layer l, the meaning of them are:

ARPACI & OZGUVEN: ANFIS & PID CONTROLLER DESIGN

193

Layer 1

In this layer all the nodes are adaptive and the output of each node i is the degree of membership of the input to the fuzzy membership function represented by the node. The node function of the ith node, if the bell membership function is used:

−+

==ii b

i

i

Ai

a

cx

xO2

1

1

1)(µ ; )(1

xOiAi µ=

.2,1=i and 3,4.i yOiBi ==−

)(2

1 µ ... (1) j

iO denotes the output of the ith node in the j

th

layer, Ai and iB are the fuzzy sets in parameter

form, x is the input to the node i, iii cba and ,, , are

the parameters for the membership function.

Layer 2

The nodes in this layer are fixed. They play the role of a multiplier. Calculates the firing strenght of a rule via multiplication. The output of these nodes:

.2,1 )()(2 === iyxwOii BAii µµ ... (2)

Layer 3

The nodes of this layer are also fixed and they perform a normalization of the firing strength from the previous layer. The output of each node:

.2,121

3 =+

== i ww

wwO i

ii ... (3)

Layer 4

The nodes in this layer are adaptive. With the following node function, the output of each node is the product of the normilized firing strength

( iw output of layer 3) with the parameter set

( iii rqp ,, ). These parameters are design parameters

and refered to as the consequent parameters.

.2,1)(4 =++== i yqxprwfwO iiiiiii ... (4)

Layer 5

There is a single fixed node in this layer. Compute the overall output by performing the function of a simple summer. Therefore, the output of this single node is:

.2,15 ===∑

∑∑ i

f

fw

fwO

ii

iii

ii

ii ... (5)

The learning rule is the backpropagation gradient descendent, which calculates the error signals recursively from the output layer backward to the input nodes. The task of the learning algorithm for this architecture is to tune all the modifiable parameters to make the ANFIS output match the training data. The overall output is a linear combinations of the modifiable parameters.

222222111111

222211112211

)()()()()()(

)()(

rwqywpxwrwqywpxw

yqxprwyqxprwfwfwf

+++++=

+++++=+=

... (6)

Fig. 1 Three input neuro-fuzzy system


194

Fractional Calculus and Fractional order PID

(PIλDµ )Controller

Fractional calculus is a generalisation of ordinary calculus. There are several definitions of fractional derivatives. The best known is the Grunwald-Letnikov definition given by

∑

−

=

−

→=−

−=

h

mt

jm

mjm

htadt

tfdjhtf

jm

htfD0

0

)()()1(lim)(α

… (7) h should approach 0 as m approaches

infinity.1)j-(m!

)1(

+Γ

+Γ=

j

m

jm is the Gamma function.

10 =

m ,

!

)1)...(1(

j

jmmm

jm +−−

=

Differentiation:

∑

−

=→

−+−Γ

+Γ−=

)(

00

)()1(!

)1()1(

1lim)(

h

at

j

j

hta jhtfjjh

tfDα

αα

α

… (8)

Integration: ( )

00

( )( ) lim ( )

! ( )

t a

h

a th

j

jD f t h f t jh

j

α α α

α

−

−

→=

Γ += −

Γ∑ … (9)

The general calculus operator is defined as

… (10)

Another common definition of fractional differointegral is the Riemann-Liouville definition

Integration: 11( ) ( ) ( )

( )

t

a t

a

D f t t f dβ ατ τ τα

−= −Γ ∫ … (11)

Differentiation:

11( ) ( ) ( ) , n-1 n

( )

tn

a t n

a

dD f t t f d

dt

β ατ τ τ βα

−

= − < ≤ Γ

∫

… (12)

The differential equation of fractional PID is:

( ) ( ) ( ) ( )p i t d tu t K e t K D e t K D e tλ µ−= + + … (13)

The transfer function is:

µ

λsK

s

KKsG d

i

pc ++=)( … (14)

Five parameters have to be designed in fractional order PID (PIλDµ) controller (Kp, Ki, Kd, λ, µ). Different design approaches proposed in a lot of papers. The most popular approaches are the S-shaped tuning rules and the critical gain based tuning rules. To design the parameters of fractional order PID by one of these methods, the following specifications, proposed by Monje et al.16, must be satisfied.

i. The gain-crossover frequency cgω is to have

some specified value:

( ) ( ) 0cg c cg p cg

G G dBω ω ω⇒ = … (15)

ii. The phase margin mϕ is to have some

specified value:

arg ( ) ( )m c cg p cg

G Gφ π ω ω = + … (16)

iii. To reject high-frequency noise, the closed-loop transfer function must have a small magnitude at high frequencies.

( ) ( )

1 ( ) ( )c h p h

c h p h

G j G jH

G j G j

ω ω

ω ω<

+ … (17)

iv. To reject output disturbances the sensitivity function must have small magnitude at low frequencies:

1

1 ( ) ( )c l p l

NG j G jω ω

<+

… (18)

v. To be robust in gain variations of the plant, the open-loop phase must be constant around the gain-crossover frequency:

arg ( ) ( )cg

c p

dG j G j

d ω ω

ω ωω =

… (19)

To design the controller with S-shaped tuning

rules, the plant is assumed as LseTs

KsG −

+=

1)( .

When a S-shaped response is obtained, graphically L and T can be determined. Depending on the varying

<ℜ

=ℜ

>ℜ

=

∫−

t

a

ta

d

dtd

tfD

0)()(

0)(1

0)(/

)(

ατ

α

α

α

αα

α


195

of L and T regularly, Kp, Ki, Kd, λ, and µ calculated from S-shaped tuning tables given elsewhere17.

To design the controller parameters with critical gain based tuning rules, the plant is inserted into a feedback control-loop with proportional gain. For a particular gain the system response will be oscillating. This gain is the critical gain (Kcr), and the period of the oscillation is the critical period (Pcr). According to the critical gain and critical period, the parameters Kp,

Ki, Kd, λ, and µ can be calculated from critical gain tuning tables given earlier17.

Mathematical Modeling of Crane Overhead crane system is modelled as a

combination of a trolley and a pendulum. Fig. 2 shows an overhead crane model. The physical representation of an overhead crane is shown in Fig. 3. To derive the equation of motion, the system parameters must be clearly defined. We modelled the load as a point mass. The parameters of crane are:

applied force (N)

mass of the trolley (kg)

mass of the load (kg)

length of the string (kg)

horizontal position of trolley (m)

velocity of trolley (m/s)

load swing angle (rad)

U

M

m

l

x

x

θ

θ

=

=

=

=

=

=

=

=

�

� angular velocity of load swing angle (rad/s)

velocity of load m (m/s)

acceleration of gravity (m/s)

v

g

=

=

The trolley and the load can be considered as point masses, the rope is considered as an inflexible rod with a length of l and the tension force is neglected, the trolley and the load are assumed to move in two dimensional, x-y plane. The crane is assumed to be rigid and the overhead trolley runs on frictionless rails. The control objective is to drive the trolley to transport the load safely from initial position to the destination in short time without resudual swing. For this aim four

parameters ( , ,x x θθ ��, ) have to converge to zero.

Using Lagrangian’s equations of motion, the model of overhead crane can be derived as follows. The forces acting on trolley are the applied force U, the inertial force )(txM �� , and the gravity force Mg . The

forces acting on the load are the gravity force mg

and the inertial force in the horizontal direction

))()(( tltxm θ�� + .

Balancing the forces in the horizontal direction,

0 : ( ) ( ) ( )xF Mx t mx t ml t Uθ= + + =∑ �� ... (20)

( ) ( ) ( )M m x t ml t Uθ+ + =�� ... (21)

Balancing the torque, or the moment, about the pivot-point of the load,

0T =∑ or ∑ = 0M

0)(sin)(cos))()(( =++ tmgltltltxm θθθ�� ... (22)

( )cos ( ) ( )cos ( ) sin ( ) 0mx t t ml t t mg tθ θ θ θ+ + =�� ... (23)

Equations (21) and (23) are related equations. Both equations involve )(tx and )(tθ , hence these two

equations can be expressed as one matrix equation. ( )

cos ( ) cos ( ) ( )

0 0 ( )

0 sin ( ) 1 0

M m ml x t

m t ml t t

x t U

mg t

θ θ θ

θ

+ +

=

��

��

... (24)

Fig. 2 Schematic diagram of an overhead crane

Fig. 3Physical representation of an overhead crane

IND

IAN

J. EN

G. M

AT

ER

. SC

I., JUN

E 2011

196

Fig. 4 The crane system modelled with SIMULINK


197

Inverting the matrix on the left hand side of the matrix, these two equations can be solved for )(tx��

and )(tθ�� . Figure 4 shows the MATLAB-SIMULINK

crane model according to the following equations.

2

( ) 1

( ) ( )cos ( ) cos ( )

cos ( )

cos ( ) sin ( )

x t

t ml M m t m l t

ml t ml U

m t M m mg t

θ θ θ

θ

θ θ

=

+ −

− − + −

��

��

... (25)

ANFIS Desing for Crane

This paper proposes a new method for the design of a controller with ANFIS for overhead crane (Fig. 5). The aim of controller designed for such cranes is to transport the load from one place to another in a short time and with small swing angle.

First we apply a PI controller to the overhead crane model. Then from this example the training data

( velocity angular velocityx errore === θ�� ,, ) were

obtained and saved in a file. From MATLAB-SIMULINK with the command anfisedit these datas were taken for train and the system were trained (Fig. 6)

At the end of the training the formed rules, the membership functions and the output coefficients were obtained with respect to Sugeno-style and saved as a fis file. Eventually each of rules have three antecedent

( velocity angular velocityx errore === θ�� ,, ), and

one consequent (u) parts. Memberships of Antecedents

( velocity angular velocityx errore === θ�� ,, ) are

seen in Fig. 7. And these rules and memberships are exported to the control system in SIMULUNK (Fig. 5.)

Fractional Order PID Desing for Crane In Fig. 8 the PIλDµ control system corresponding to

crane is shown. To implement the PIλDµ controller, the block which is in MATLAB-SIMULINK is used. A distance error is coming into the input of the controller. Also a limiter is used at the input of

Fig. 5 ANFIS desing for crane

Fig. 6 Varification of the ANFIS model with factual data


198

controller to prevent the distance error not to digress the desired limitations. The parameters of PIλDµ were taken as Kp = 4, Kd = 4 Ki = 1, λ = 0,9, µ = 0,9.

Results and Discussion

For simulation a trolley and pendulum based model was used. The simulations were conducted for a 1000 kg of crane mass (M=1000 kg), a 100 kg of load mass (m=100 kg) , with a rope length of 5 m (l =5 m.), under the gravity g = 9.81 m/s2.

For different target displacement the performances for trolley position (x), swing angle (θ) and control

force (N) were observed and these performances were compared with the performances of PID & Fractional Order PID (PIλDµ) controllers. And also to see the performance of crane for different load mass, we increased the mass of load to 250 kg and executed the same simulation. The first simulation were conducted for: X= 10 m and m=100 kg. To see the performance of the designed Adaptive Neuro Fuzzy Controller, a control surface of

Errore and Velocity,x output Controlleru === �., is obtained. The second simulation were conducted for: X=20 m and m=100 kg.

Fig. 7 (a) Displacement error, (b) velocity and (c) angular velocity


199

We executed the computer simulations of adaptive neuro fuzzy controller designed for overhead crane for two target displacement to verify the performance of the proposed ANFIS controller. And we compare the performance of adaptive neuro fuzzy controller with that of PID & PIλDµ controllers in terms of trolley position (x), swing angle (θ) and control force.

For the first simulation experiment the target displacement was taken as 10 m. For this displacement the obtained trolley position is shown in Fig. 9. As can be seen in this figure, the trolley reached the 10 m target in about 6 s with adaptive neuro-fuzzy controller. In the same figure it can be seen that, the trolley reached the target in obout 9 s with PID controller and in about 12 s with fractional

order PID (PIλDµ) controller. Sway angles of this experiment are shown in Fig. 10. For adaptive neuro-fuzzy controller the sway angle was reduced to 0 in about 6 s. Sway angle of PID controller was reduced to 0 in about 6 s and that of fractional order PID controller was reduced to 0 in about 8 s. The control forces were compared in Fig. 11. The smallest force is applied with adaptive neuro-fuzzy controller.

The same experiment was repeated for a longer target displacement, 20 m. And the obtained trolley position figures were shown in Fig. 13. Although the target displacement increase, the trolley was reached to the target in the same time with adaptive neuro-fuzzy controller. There was a small increasing in time of transportation with PID controller and fractional

Fig. 8 Fractional order PID desing for crane

Fig. 9The trolley position for different type of controllers (PID&PIλDµ&ANFIS) for 10 m target displacement

Fig. 10 The swing angles for different type of controllers (PID&PIλDµ&ANFIS) for 10 m target displacement


200

order PID (PIλDµ) controller. From this simulation it can be said that in crane systems, which controlled with adaptive neuro-fuzzy controller, for longer displacement the load is transported in short time. Swing angles were shown in Fig. 14. For adaptive

neuro-fuzzy controller the sway angle was reduced to 0 in 6 s. For this magnitude of displacement it can be considered to a good performance. Control forces were shown in Fig. 15. The force for adaptive neuro-fuzzy controller was between the reasonable limits.

Fig. 11 The control forces for different type of controllers (PID&PIλDµ&ANFIS) for 10 m target displacement

Fig. 12 Control surface for 10 m target displacement

Fig. 13 The trolley position for different type controllers (PID&PIλDµ&ANFIS) for 20 m target displacement

Fig. 14 The swing angles for different type of controllers (PID&PIλDµ&ANFIS) for 20 m target displacement

Fig. 15 The control forces for different type of controllers (PID&PIλDµ&ANFIS) for 20 m target displacement

Fig. 16 Control surface for 20 m target displacement


201

Fig. 17 For 10 m and 250 kg (a) trolleyposition, (b) swing angle, (c) control forces and (d) control surface

Fig. 18 For 20 m and 250 kg (a) Trolley position (b) Swing angle (c) Control forces (d) Control surface


202

For both simulations, the control surfaces, Figs 12 and 16, are also obtained for error (e), trolley velocity

( x� ) and control force (U), to see the effect of θ� (angular veloity) on the movement. As seen on the surfaces, at the begining of movement the values of force, velocity and error were high. As crane, whose movement follows the control surface, come near to the target; force, velocity and error reduced to 0.

Because of the effect of θ� sometimes there were some drift of movement of crane from the control surface.

The above simulations were repeated for same displacements with a 250 kg load to test the performance for different mass of load. The obtained results were given in Figs 17 and 18. As seen in these figures, although the mass of load increased, the best performance again was obtained with adaptive neuro-fuzzy controller.

Conclusions In this paper, an adaptive neuro-fuzzy controller is

proposed for controlling of overhead cranes. We designed a controller for crane with adaptive neuro-fuzzy inference systems (ANFIS) for controlling to transport the load safely, rapidly, correctly, and without swing. We also executed overhead crane system simulation with other controllers; PID, and fractional order PID (PIλDµ). The performance of adaptive neuro-fuzzy controller is compared with the performances of PID and PIλDµ controllers in terms of trolley positions, swing angles and control forces. The results of obtained simulations showed that adaptive neuro-fuzzy controller had better performances.

Nomenclature ai, bi, ci= parameter for the membership function

Ai, Bi= fuzzy sets in parameter form

αta D = fractional calculus operator

E= error

F= output of Sugeno fuzzy inference system

Fx= horizontal force (N)

G= gravitational acceleration (9.81 m/s2)

Gc= controller

Gp= plant

Kcr= critical gain

Kd= derivative constant

Ki= ıntegral constant

Kp= proportional constant

L= delay

L= length of hoisting rope (m)

M= moment

m= payload mass (kg)

M= trolley mass (kg)

Oi= output of the i.th node in the j.th layer

Pcr= critical period

T= time constant

T= torque

U= input force (N)

wcg= gain crossover frequency

wh= high frequency

wi= weight of neuron

wl= low frequency

X= trolley position (m)

Λ= order of integration

θ�� = sway angle acceleration (rad/s2)

θ�= sway angle velocity (rad/s)

Θ= sway angle (rad)

x�� = trolley acceleration (m/s2)

x� = trolley velocity (m/s)

: ,ii BA µµ =node function φm= phase margin

Г= euler’s gamma function

ANFIS= adaptive neuro-fuzzy ınference system

PID= proportional-ıntegral-derivative

PIλDµ= proportional-fractional order ıntegral-fractional order derivative (fractional order PID)

References

1 Manson G A, Optim Control Appl Methods, 3 (1982) 115-120.

2 Karihaloo B L & Parbery R D, J Optim Theory Appl, 36(3) (1982) 409-417.

3 Sakawa Y & Shinde Y, Int Fed Autom Control, 18(3) (1982) 257-266.

4 Ridout A J, New feedback control system for overhead cranes, Electric Energy Conf, Adelaide, Australia, 1987, 135-140.

5 Lee H H, Cho S K & Cho J S, A new anti-swing control of overhead cranes, Proc ASI’97, 1997, 137-142.

6 Ohnishi E, Tsuboi I & Egusa T, Automatic control of an overhead crane, IFAC World Congress, Kyoto, Japan, 1981, 1885-1890.

7 Moustafa A & Ebeid A, ASME J Dyn Syst, Meas, Control,

110 (1988) 266-271.

8 Hurteau R & Desantis R, Microprocessor-based adaptive control of a crane system, Proc 22nd IEEE Conf on Decision

and Control, including The Symp on Adaptive Processes, New York, 2 (1983) 944–947.

9 Omar H M & Nayfeh A H, Gantry Cranes Gain Scheduling Control with Friction Compensation 39th Ann Tech Meet of


203

the Society of Engineering Science, Penn State University, PA, October 13-16, 2002.

10 Marttinen A, Salminen R & Virkkunen J, Adaptive pole placement control of a pilot crane, Proc 11th Triennial World

Congress of the International Federation of Automatic

Control, Tallin, USSR, 2 (1990) 313-318. 11 Moustafa K A & Emara-Shabaik H, Int J Syst Sci, 23(7)

(1992)1167-1177. 12 Yasunobu S & Hasegawa T, Control Theory Adv Technol,

2(3) (1986) 419-432. 13 Nalley M J & Trabia M B, Dyn Syst Control, 1 (1994)

389-398.

14 Suzuki Y, Yamada S & Fujikawa H, Anti-Swing Control of the container Crane by Fuzzy Control, Proc IEEE, 1993, 230-235.

15 Benhidjeb A & Gissinger G L, Control Eng Pract, 3(12)(1995) 1687-1696.

16 Monje C A, Vinagre B M, Chen Y Q, Feliu V, Lanusse P & Sabatier J, Proposals for fractional PIλDµ tuning, First IFAC

Workshop on Fractional Differentiation and its Applications,

Bordeaux, 2004.

17 Vale´rio D & Costa J S, Signal Process, 86 (2006) 2771-2784.

anfis & pi d controller design and comparison for overhead ... · nonlinear elements of the...

Documents