Optimization of Fuzzy Controller Parameter by Using A Firefly Algorithm

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Volume 02, No. 3, March 2016

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Optimization of Fuzzy Controller Parameter by Using

A Firefly Algorithm

Soumya Chauhan

Department of Electrical Engineering, Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat (Haryana) India

ABSTRACT:

Firefly Algorithm (FA) is one of the latest nature-inspired algorithms. Finding the best global optimal point, speed up the convergence by tuning the FA parameters and the ability of dealing with multimodality are some merits of FA. In this paper, FA is applied to tune optimal fuzzy parameters for FLC which are used in plant for liquid level and temperature control. MATLAB/ Simulink program has been used to achieve the optimal parameters of the membership functions. The results show clearly that the optimized FLC using FA has better performance compared to manually adjustments of the system parameters for different datasets.

KEYWORDS: Firefly algorithm, Fuzzy logic controller, Particle swarm optimization,

1. INTRODUCTION

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solutions for a given optimization problem. There exist a diverse range of algorithms for optimization.

Firefly Algorithm (FA) is one of the latest nature-inspired algorithms. Finding the best global optimal point, speed up the convergence by tuning the FA parameters and the ability of dealing with multimodality are some merits of FA[39]. Fireflies cover the search space by random and information based steps which present two random search and local search to the algorithm respectively which conveys exploration and exploitation in firefly algorithm [33]. But, important point in firefly algorithm is keeping balance between exploration and exploitation. In standard firefly algorithm, controlling parameters of these two kinds of searches are initialized at the beginning and these values don‘t change until end of the searching process and this cannot indicate proper balance between these two kinds of searches considering different conditions that algorithm will be faced in different steps of solving problem. Therefore, we have used a fuzzy controller for tuning the parameters which control balance between random search and local search dynamically in each step of solving problem (with considering progress of solving problem). This method increases performance of the firefly algorithm in finding optimal solution by keeping balance between these two kinds of searches. We have used the proposed method for optimization of fuzzy controller parameters for a temperature and water level system.

In this paper, fuzzy controller (Mamdani type) to control of liquid level and temperature in tanks are used. We present optimization of Mamdani-type fuzzy regulator parameters using FA. MATLAB software is used for designing and simulating. This paper is organized as follows. In Section2, system model is presented. In Section3, the controller design based on Mamdani system is given. In Section4, Simulink model is presented. In Section5,FA is described. Finally, the results, conclusion and future direction are illustrated.

II. GENERAL FORMULATION OF THE SYSTEM MODEL OF LIQUID LEVEL AND TEMPERATURE

1. System Model for Level

The tank system [1-2] is shown in Fig. 1

Fig. 1: Schematic diagram of coupled tank system

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= (1)

= - (2)

Where:

F =steady-state liquid flow rate, c /sec.

F = out flow rate from first tank, c /sec.

= out flow rate from second tank, cm3/sec.

And = coefficients, c /sec.

( = level first tank, cm. = level second tank, cm).

=the cross sectional area for first tank, c .

=the cross sectional area for second tank, c

2. SYSTEM MODEL FOR TEMPERATURE [29]

Fig. 2 depicts the circuit configuration for a simple Triac circuit where the primary power control device is the Triac whose gate firing device is the Diac (Skvarenina, 2002). The Diac is a fixed break-over voltage device compared to the Triac that permits a variable firing angle through its gate terminal. Compared to the Silicon Controlled rectifier (SCR) which is also gate controlled, the Triac permits device firing on the positive and negative half-cycles of the AC waveform.

Fig 2: Triac circuit

III. DESIGNING OF FUZZY LOGIC CONTROLLER

A. Fuzzy Logic Controller Review [1-2]

The basic fuzzy structure is shown in fig 3. The development of FLC consists following steps-

a) Specify the range of controlled variable and manipulated variables;

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c) Determine the rule base to specify control action; d) Application of suitable defuzzification method.

The number of necessary fuzzy sets and their ranges were designed based upon the experience gained on the process. The standard fuzzy set consists of three stages: Fuzzification, Decision- Making Logic and Defuzzification.

Fig 3: Block Diagram of basic structure of fuzzy control system.

The controller in present work is Mamdani based one. It uses a rule based in linguistic terms. There are two inputs: error in liquid level and rate of change of liquid level and one output parameter: the inlet valve control angle in level control. Triangular membership functions are selected to fuzzify the inputs and output variables. The membership functions with five linguistic values for error (e) and change in error (de) is used and is shown in fig.4, 5.

Fig 4: Membership function for error Fig 5: Membership function for change in error

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Fig 6: Membership function for output (rotation)

2. HAND-TUNED MEMBERSHIP FUNCTION FOR TEMERATURE

Fig 7: Membership function for error Fig8: Membership function

forchange-in-error

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IV. SIMULINK BLOCK DIAGRAM DESCRIPTION

1. For Level

Simulink model for liquid level control and Fuzzy Logic Controller and by using program MatlabR2013. Based on the dynamic equations (1) and (2) a Simulink block diagram. Fig.2 showing the nonlinear model of the plant can be formed successively .Fig.3 shows the subsystem the Simulink block diagram of the nonlinear model of the plant. For pumping liquid is pumping capacity (F = 588 c \sec). Fig.4 showing the subsystem of stepper motor.

Fig.10 showing subsystem of flowrate using stepper motor and valve control and Fig.11 shows the subsystem 1 of valve control.

Fig 10: Simulink for Water level Control System with FLC

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Fig 12: Subsystem of stepper motor

Fig: Subsystem of flow rate using stepper motor and valve control

Fig 13: Subsystem 1 of valve control

2. For Temperature

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(t)]. In this the fuzzy sets e (t) (Temperature error) and de (t) (Change in temperature error) acts as the input. And the output is a firing angle which lies in the interval of [ ]. Here

through fuzzy output, get the firing angle which triggers the triac. Then fuzzy is further connected with triac circuit and through this firing angle calculate the value of TRAIC voltage (load voltage) from this formula(1).

V=f ( (1)

V=triac voltage

f ( = function of firing angle

Through this triac voltage, we have to calculate (Rms voltage) from TRIAC circuit.

Then, the value of will be put in the formula of power which is shown in formula (2)

P= (2)

In this formula, the value of R will be taken as a constant value. And the changes

according to the firing angle. The value of power will put in this formula (3) to get rise in temperature ( =100 ) which is shown in formula (3)

(3)

, P= power, t= time, density=1,

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Fig 15: Subsystem 1 of AC source and measuring unit

Fig 16: Subsystem 2 of measuring unit

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Fig 18: subsystem 4 of triac

Fig 19: Subsystem 5 of rise in temperature

V. FIREFLY OPTIMZATION ALGORITHM (FA)

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tropic fireflies can even synchronise their flashes, thus forming emerging biological self-organized behaviour. We know that the light intensity at a particular distance r from the light source obeys the inverse square law. That is to say, the light which becomes weaker and

Fig 20: Flowchart of Fa

Weaker as the distance increases. These two combined factors makes most fireflies visual to a limit distance, usually several hundred meters at night, which is good enough for fireflies to communicate. The flashing light can be formulated in such a way that it is associated with the objective function to be optimized, which makes it possible to formulate new optimization algorithms. Use the following three idealized rules:[39]

1) All the fireflies are unisex so it means that one firefly is attracted to other fireflies irrespective of their sex.

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brightness both decrease as their distance increases. If there no one brighter than other firefly, it will move randomly.

3) The brightness of a firefly is determined by the view of the objective function.

Input: f(X), X= ( )

m, // user defined constant

= f(X)

Output ;

For i=1: m

= initial _solution ( )

End

While (termination requirements are not met) do min argmi (f( )

fori= 1: m do for j 1 : m do

if j 1 : m do

if f( then

distance ( , )

attractiveness ( , )

(1- ). + + α. (random ()-1/2)

end if end j endi

+ α. (random ()-1/2)

end

Fig 21: Pseudo codes of FA

A. DESIGN ISSUES FOR CONSTRUCTION OF mfs [38]

Following design points are considered while constructing rule base and sfs for FAFLC:

1. Choice of Input Variables: We considered only 2 input variables, E(n) and R(n), each with 5 mfs and thus total number of rules & output mfs being 25 and 9 resp.

2. Choice of Input And Output Mfs: Exponential type of mfs (such as sigmoidal, Gaussian etc.) were avoided as their infinitely long tails add to noise only. For outermost fuzzy sets on UOD, particularly suited are non-exponential type zmf and smf (requiring only two defining parameters, each) and for the rest, triangular mfs, trimf, (requiring 3 parameters, each) were adopted. Defining equations are:

zmf(x;a,b) = 1 for x<=a,

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= 2((x-b)/(b-a))2 for (a+b)/2 <= x <= b

= 0 for x >=b.

where parameters a and b locate extremes of sloped portion of curve.

smf(x;a,b) = 1 – zmf(x;a,b)

trimf(x;a,b) = max(min((x-a)/(b-a), (c-x)/(c-b)), 0)

where a,b,c respectively locate start, peak and end points.

B. DESIGN OF ADAPTIVE FITNESS FUNCTION

A performance criterion such as ISE is although quite general but has disadvantage of being unable to distinguish between persisting errors (even though well within tolerance band) and initial sluggishness. To cover more facets of time-response, a synthetic, more comprehensive fitness-function as follows is adopted:

Fitness-function= (w1+w2+w3+w4)/ (w1*ISE +w2*Mp+w3*tr+ w4*ts) (1) where w1,…,w4 are user-settable weights (free-parameters) to lay emphasis on different facets of response. Design of such a comprehensive fitness function needs much more attention in case FA-FLC is to operate at SPC. We considered following points in its design:

i) In synthesizing a comprehensive fitness function for a FA meant to tune an FLC operating at SPC, one cannot be oblivious of numerical properties/ scale of different factors constituting the fitness function. For instance, at higher SPCs, ISE can get numerically too dominant over Mp, tr and ts, so static chosen weights can mean that ISE is getting the tilt in its favour, totally eclipsing other performance indices. In other words, firefly good on all other accounts (viz. Mp, tr and ts) will keep on getting superseded by firefly poor on these accounts but dominant merely on account of ISE. We, therefore, felt a need to adapt fitness function by choosing weight w1 as follows: w1 = 0.2/(SPC)2.

ii) Values of w3 and w4 were held constant at 0.5 and 2 respectively, as these indices are relatively independent of SPC; latter weight is kept much higher as ts is generally more crucial than tr.

iii) w2 is adapted as follows depending upon whether Mp is within tolerance band (normal weight) or Mp exceeds it (heavily de-weighted):

If (Mp>0.05*abs(SPC)),

w2 =0.1/abs(SPC); else w2 = 0.5/sqrt(abs(SPC)

VI. EXPERIMENTAL RESULTS

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Fig 28: Response of Level control with FLC

Fig 5.4 shows the response of fuzzy controller of water level of tank 2 with load disturbance at t = 5 sec under simulation environment

Fig 29: Response of water level of tank 2 with load disturbance

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Fig 32: Membership function of Output

Fig 33: Response of water level of tank 2 with FA-FLC

FAFLC: Evolved mfs shown in Fig and correspond to following bestf firefly:

[-60, -36, -60, -36, 0, -36, 0, 24, 0, 24, 59.4, 24, 59.4, -1200, -720, -1200, -720, 0 ,-720,0, 480, 0,480, 1188, 480, 1188, 0,480.0,0,480.0,864,480.0,864,1104, 864, 1104, 1200, 1104, 1200, 1320, 1200, 1320, 1824, 1320, 1824, 2064, 1824, 2064, 2376, 2064, 2376]

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The greatest reductions in settling time has occurred which is a proof to great ability of FA for finding global search of a problem. After optimization of controller membership function, the main obvious changes can be interpreted as a significant lowering in rise time as shown in figure

FAFLC: Evolved mfs shown in Fig and correspond to following bestf firefly:

[-100, -60, -100, -60, 0, -60, 0, 40, 0, 40, 99, 40, 99, -2000, -1800, -2000, -1800, 0, -1800, 0, 1200, 0, 1200, 2970, 1200, 2970, 0, 1020, 0, 1020, 2160, 1020, 2160, 2760, 2160, 2760, 3000, 2760, 3000, 3300, 3000, 3300, 4560, 3300, 4560, 5160, 4560, 5160, 5940, 5160, 5940]

Table 1: Comparison between FA-FLC and FLC for temperature

FA-FLC FLC

Rise time ( ) 0.0355 0.1421

Settling time 0.5211 0.582

Overshoot 4.115 4.775

Table 2: Comparison between FA-FLC and FLC for water level

FA-FLC FLC

Rise time 0.3572 0.3760

Settling time 1.15 1.245

Table 3: Values of FA Parameters

Number of optimization variables 19

Number of alpha 0.55

Number of firefly (n) 30

Number of delta 0.97

Number of dimension(d) 51

Number of beta 0.20

Number of gamma 1

TABLE 4: Bounds for mfs of input and output variables and for input and output scaling factors of FLC

Note 1: (a,b)= points of zmf of smf (a,b,c) =points of trimf

Note 2; m =Multiplying Factor. Its value is SPC for E(n); 20*SPC for R(n) & 50*SPC for u(n)

lb ub

Bounds for mfs of inputs E(n) and R(n)

MN

SN

ZE

SP

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V. CONCLUSION AND FUTURE DIRECTION

With a large number of parameters (51 points of mfs for a two-inputs and one-output FLC), FLC design truly represents a search problem in high-dimensional, multi-modal hyperspace typically suited for FA. In this work, simulation experimental results on the control of level and temperature using firefly algorithm have been provided and it has been demonstrated that FA is an efficient method. FA-FLC outperforms a hand-tuned FLC.

Following new directions can be pursued: (a) adapting FLC for temperature and level control systems subjected to wide range of set-points. (b) adapting control parameters of FA.

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