Self-Tuning fuzzy controller for sun-tracker system using Gray Wolf Optimization (GWO) technique

Journal Pre-proof Self-Tuning fuzzy controller for sun-tracker system using Gray Wolf Optimization (GWO) technique Sandeep Tripathi, Ashish Shrivastava, Kartick C Jana

PII: DOI: Reference:

S0019-0578(20)30012-4 https://doi.org/10.1016/j.isatra.2020.01.012 ISATRA 3453

To appear in:

ISA Transactions

Received date : 1 December 2018 Revised date : 10 December 2019 Accepted date : 7 January 2020 Please cite this article as: S. Tripathi, A. Shrivastava and K.C. Jana, Self-Tuning fuzzy controller for sun-tracker system using Gray Wolf Optimization (GWO) technique. ISA Transactions (2020), doi: https://doi.org/10.1016/j.isatra.2020.01.012. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier Ltd on behalf of ISA.

Journal Pre-proof

pro of

Self-Tuning Fuzzy Controller for Sun-Tracker System using Gray Wolf Optimization (GWO) Technique Sandeep Tripathi1, Ashish Shrivastava2 and Kartick C Jana1 Indian Institute of Technology (ISM) Dhanbad, Bihar, India 2 Manipal University Jaipur, Rajasthan, India

Jo

urn a

lP

re-

1

Journal Pre-proof

Self-Tuning Fuzzy Controller for SunTracker System using Gray Wolf Optimization (GWO) Technique current from the PV cells and generates up to 25% more power than the permanently fixed or tilted system [3-4]. In past few years, a lot of researchers have proposed their work in the field of PV cells for maximum optimizing power using different controllers, such as Proportional Integral Derivative (PID), Fuzzy PID and Self-Tuning Fuzzy-PID (STF-PID), which are widely used because of their simple, inexpensive, and easy to design structure [1,2,5,6]. The conventional PID controller uses Ziegler Nicolas (ZN) tuning, which is out dated. The auto tune PID controller is not applicable for practical time varying uncertainties or nonlinear system. Robust control(H∞) [7] and linear quadratic regulator (LQR) have been developed for robust system performance to control external disturbances. But these types of controllers have difficulty in model accuracy and optimal controller design [8]. The Artificial intelligence (AI) based controller and combination of AI with conventional controllers, like fuzzy controller, PID Fuzzy [9-11], Genetic algorithm, hybrid algorithm [12-13] and multi objective genetic algorithm [14] have been suggested and widely used for modeling and controlling such type of time varying problems. The Fuzzy controller, which works on knowledge based reasoning for processing of inaccurate information to exact algorithm, is used for diverse applications; however, it has its own limitation in terms of tuning the scaling factor [15-16].In addition to this, several meta-heuristic algorithms, such as simulated annealing [17], Artificial Bee Colony for fractional PID controller [18], Ant Colony [19], Gravitational Search Algorithm [20], Particle Swarm Optimization (PSO) [21-22] and Modified PSO [23], have been used by researchers for parameter optimization problems. Solving optimization problems, specific to fuzzy control parameter, tuning is complex due to objective functions and local minima trapping [24-26]. To solve such problems, nature-inspired optimization (NIO) algorithms can be used to minimize these objective functions. In 2014, Mirjali et al. [27] introduced Gray Wolf Optimization (GWO), which is a population based algorithm inspired by Canadian wolfs. It mimics their hunting behavior and shows good response for the uni-model and multi-model systems [28]. Other population based methods, Whale Optimization Technique (WOT) proposed by Mirjali in 2016 [29] and Adaptive Global Whale Optimization (AGWO) [30] proposed by Indrajit N et al., are much similar to GWO that is adopted as the methodology for exploration and exploitation in optimization of problems. In this paper, the above mentioned nature inspired optimization algorithms have been applied on TSK fuzzy controller with the

lP

re-

increasing very rapidly worldwide and to fulfill this requirement, solar energy is one of the most viable solution as renewable energy source. Photovoltaic (PV) cell based sun-tracker system (STS) produces maximum current when sunlight vertically incident on its surface. Hence, there is a need of optimized continuous axis position control of STS to achieve maximum output current. This task can be done on the basis of the fuzzy control system. Usually, in the traditional fuzzy control system (FCS), tuning of designed fuzzy parameter is done by trial and error method. However, this type of FCS parameter tuning approach may or may not give optimal solution. Thus, in presented work, an optimal tuning technique with Takagi, Sugeno and Kang (TSK) fuzzy controller (TFC) using Gray Wolf Optimization (GWO) for STS has been proposed. In order to validate the proposed work, different objective functions have been employed to carry out fuzzy controller parameter optimization. A comparative analysis has been performed on the basis of three parameters: settling time, maximum-overshoot and optimal fuzzy parameter on different constrain set. The results obtained with the GWO optimization algorithm were also compared with other popular population algorithms, i.e. Whale Optimization Technique (WOT) and Particle Swarm Optimization (PSO) algorithms.

pro of

Abstract—The demand of electric power consumption is

Index Terms—Sun-tracker system, TSK-Fuzzy Controller, Gray Wolf Optimization, Particle Swarm Optimization, Whale Optimization.

I. INTRODUCTION

Jo

urn a

In recent years due to the conventional energy resource crisis, environmental pollution and their non-renewable nature attracted the attention of researchers and modern scientists to search for alternate renewable energy resource (RER). The RER searching is a solution not only for limited conventional energy resources but also for pollutant gas emission, which creates major environment problems. To overcome aforementioned problems, a Photovoltaic (PV) cell can be used for possible aspect due to its abundance, easy availability and its provision of providing clean and green energy. This type of renewable energy is also used in zero emission type vehicle (ZEV) design to limit emission of pollutants, such as hydrocarbon, carbon mono oxide, and nitrogen oxide [1-2]. The amount of energy generated by PV will be maximum and proportional to its current flow when sun light falls vertically on its surface. In this type of vehicle design, the sun-tracker system (STS) incorporated with PV cells is connected in such a way that maximum illumination of sun light incident on its surface. The sun-tracker system with the help of servo motor continuously changes its sun-tracking axis (STA) in accordance with the direction of light rays to extract maximum amount of

Journal Pre-proof

vertically to produce zero error voltage and then the DC motor stops. In this system, one-axis tracking has been considered and the angle ( ) between sun tracker axis and solar axis has to be minimized. The desired and reference angular position of sun tracker and solar axis with respect to reference line is  r and  0 , respectively, as shown in Fig.2. The coordinate position, sun ray’s alignment on PV cells and error discriminator of suntracker system is shown in Fig.3. Solar Axis

pro of

aim of reducing fuzzy tuning parameters and minimize error objective function for sun-tracker system. This paper discusses two different contributions for maximum power extraction from STS: (i) TSK-fuzzy controller designed for STS, which uses GWO, PSO and WOT algorithms for fuzzy tuning parameter optimization; (ii) For different objective function a comparative study has been performed for GWO, PSO, WOT on different constrain set. Section II describes the STS model with different parameters and system transfer function. Section III discusses about the fuzzy controller and the approach for tuning of TSK FCs for the STS. Section IV deals with the GWO, PSO and WOT based optimization techniques for the different type of fitness functions. Section V shows the comparison of results and subsequently conclusion is discussed in the last section of this article.



II. SUN-TRACKER SYSTEM

Sun rays

Output gear centre

α

Fig.2: Sun Tracker coordinate system

The main purpose of control system is to minimize error α (t) to zero, i.e.,  (t ) =  r (t ) −  0 (t )

(1) In Fig.1 and Fig.2, it has been clearly demonstrated that when sun-tracker axis is completely aligned with the sun axis, then  (t ) becomes zero, the current flow through PV cells, i a ( t ) = ib ( t ) ,will become maximum. When sun-tracker axis

ia

Cell A

Output gear 1/n

Cell A and Cell B mounted on space vehicle

Jo

Solar axis

Ө0

Cell B

ib

Reference Line

lP

urn a

Error discriminator

r

o

re-

The schematic diagram of STS used in this paper is shown in Fig.1 [31]. A pair of PV cells, A and B, is used as the current source that generates output when sun light falls on it. The output current is proportional to the intensity of illumination falling on it. When the sun light is vertically incident on each cell then both the cells produce equal current and there is no error voltage. But, when the light does not fall vertically on each cell then both the cells produce different currents and subsequently the voltage error signal is generated. This error signal is fed to the controller and the servo amplifier magnifies this voltage signal which is fed to the DC servo motor drive system. The motor drive system rotates the PV cell and aligns it

Sun Tracker Axis

Өm

α

Sun Tracker axis

Rf

R

eo R

Controller

u

Servo Amplifier K

ea

M

et T

Fig 1: Schematic diagram of Sun-Tracker system (STS) [31]

DC servo motor for tracking max. efficiency Tachometer

Journal Pre-proof

is not aligned completely with respect to solar axis, then,  (t ) ≠ 0 and  (t ) depends upon as follows: ao =

co =

L

+ R tan  ( t )

2

L 2

(2)

− R tan  ( t )

The tachometer voltage output related to tachometer constant C t and motor angular velocity  m ( t ) is expressed as: et (t ) = − m (t ).C t =  m .C t (6) Through gear ration (1/n), the relation between output gear angular position  o (t ) and position of motor  m ( t ) is defined:  o (t ) =

1 n

(7)

 m (t )

The modelling of DC servo motor is as follows:

pro of

a

E bm (t ) =  mot ( t ). K bm

(9) (10)

Tmot ( t ) = K im i ( t )

R tan 



Tmot ( t ) = J

b

L/2

(8)

E a (t ) = ra ia (t ) + E bm (t )

Cell A

d  mot ( t ) dt

(11)

+ B  mot ( t )

Where, ra is armature resistance , i a is armature current, E a is

o

voltage source, E bm is back emf voltage, K bm is back emf constant,  mot is angular velocity of motor, Tmot is motor torque, K im is motor torque constant, J is rotor movement, B is frictional constant. After combining all the above equations from 1 to 11 and putting numerical parameters of the system from Table I, the state space model of sun-tracker system will be represented as:

c L



Cell B d

R

re-

Sun Rays

Table I SUN-TRACKER SYSTEM NUMERICAL PARAMETERS

Parameters

where, sun rays width inclination for cell A is a o and for cell B is c o .

When the flow of current i a (t ) is directly proportional to a o

ia ( t ) = I + ib ( t ) = I −

2 RI L 2 RI L

tan  ( t ) tan  ( t )

urn a

and ib (t ) is directly proportional to c o , then:

Hence, from the above equation, the error discriminator of the system is: ia ( t ) − ib ( t ) =

4 RI L

tan  ( t )

Or ia (t ) − ib (t ) = M s tan  (t ) Ms =

4 RI

(3)

Jo

where,

is error discriminator constant.

L

Now, the relation between PV cell output current i a ( t ) , ib ( t ) and tracker system operational amplifier is expressed as: eo ( t ) = −[ ia ( t ) − ib (t )] R f (4) The servo amplifier output of system is defined as: (5) Where, K is servo amplifier gain, e ( t ) is operational amplifier output voltage, e ( t ) is tachometer output voltage. e a ( t ) = − K [ eo ( t ) − et ( t )]

o

t

Operational-Amp Gain (Rf )

10000 

Resistance of armature (Ra )

6.25 

Back EMF (Kb )

0.0125 V/rad/sec

Torque Constant (Ki )

0.0125Newton-meter/Amp

Moment of inertia (J)

10-06 Kilogram/meter2

Error constant (Ks )

0.1 Amp/radian

Viscous Friction Coefficient (B)

0

Gear Ratio (n)

800

Gain of Servo Amplifier (K)

1

lP

Fig. 3: Sun-tracker system error discriminator diagram

Values

.   0 x  1=  .  0  x 2  

y ( t ) = 1

1 

0  x1 ( t )    + k −1    x 2 ( t )    c    c 0  x1 ( t )

 u ,  in 

(12)

x 2 (t ) 

T

where,  c is time constant, x1 ( t ) is angular position, x2 (t ) is angular speed, k is proportional gain, uin is input applied on STS, y (t ) is output and T is matrix transpose. From the state space equation 12, neglecting the static nonlinearity and simplifying the process model in terms of the desired position output, the open loop transfer function (OLTF) of STS is represented as:

Journal Pre-proof

(13)

where, K is process gain amplifier and discriminator.

Ms

Di

1

is error

III. FUZZY LOGIC CONTROLLER The fuzzy logic controller (FLC) is a convenient tool for designing the non-linear controller that is based on organized skilled knowledge, which is in the form of rule based behavior. Several forms of the FLC studied in literature [7,10] and its fundamental schematic diagram is shown in Fig.4. Here, FLC is the fuzzy controller, P is the system, r is the reference input, y is the control output, u is the control signal and e = r − y is the control error. r

e

+

u FLC

y P

, i 2 , ....in

 Z

is constant corresponding to rule R

L i1 , i 2 ,

...i

. n

3. Fuzzy implication and product inference. 4. Weighted-average defuzzifier. In this paper, two input variable, such as error e(t) and increment in input error Δe(t), and one output Δu(t) have been considered for fuzzy controller design as given in equations1517. e ( t ) = e ( t ).T (15) 1

1

(16)

 e ( t ) =  e1 ( t ).T2

pro of

G p ( s ) = M s . K/  s ( + s ) 

 e1 ( t ) =

d

dt

Where,

(17)

( e1 ( t ))

e1 ( t )

is error input,

T1

is error input tuning factor,

is change in error, T is change in error tuning factor, and Δu(t) is output of fuzzy controller, which is defined as: (18)  u ( t ) = [ e ( t ).S  +  e ( t ).S ]  e1 ( t )

2

¥

_

where,

S  = [ S e (t enext − t e int ) / 2]

S ¥ = [ S ed (t ednext − t ed int ) / 2]

where,

S e = S1, 2 ,3, 4 ,5

and

re-

Fig. 4: Fuzzy control system block diagram structure

e1(t)

e(t)

T1

Δu(t)

T3

u(t)

urn a

TSK-Fuzzy Δe1(t) T2

similarly, t ednext is final point of change in error membership , t ed int is initial point of change in error membership. In TSK fuzzy, for input error and increment in input error, triangular membership function is equally distributed in the range of -1 to 1. For the defuzzification average weighted method has been used. The input error, increment of input error and output error membership function are shown in Fig. 6. The sum and product operation of input-output membership function with applied rule base are given in Table II.

Error

if

xk

Jo

n

ik

is B then Y

= Di

(14) where, i = 1, 2, ..M , k = 1, 2, ... m is input fuzzy variable, Y is output variable and M is number of membership function of kth input. B  U , i = 1, 2...M where, k=1,2...m is linguistic term i

1, 2 ,

i3...... i n

k

k =1

k

1

, i 2 ,...in

k

ik

k

characterized

k

k

by

k

fuzzy

membership

are error and change in

TABLE II TSK FUZZY CONTROLLER RULE BASE [9]

From Fig.5, let us assume that fuzzy systems are MISO mappings f : P  R → Z  R ,where P = P1 .P2 ...Pn  Rn is the input space and Z ⊂ R is the output space [9]. The TSK model consists of four principal components: 1. A fuzzifier that consists of normal, complete and consistent fuzzy sets. 2. A complete fuzzy rule base form: RL i

S¥

error membership, t e int is initial point of error membership .

Error dot BN SN

Fig.5: TSK Fuzzy Logic Controller Structure [9]

n

, S  and

error membership parameter respectively. t enext is final point of

Δe(t)

d/dt

S ed = S 6 , 7 ,8 ,9

lP

For the calculation of output value and processing time, Takagi-Sugeno-Kang (TSK) type fuzzy is more suitable than Mamdani type fuzzy. The classification of fuzzy controller is based on input and output membership function structure and generally two to three input-output structure are most common [5]. In this article, two input one output, i.e., multiple input single output (MISO), type fuzzy controller have been taken which is shown in Fig.5.

function 

ik Bk

( xk )

.

Z

SP

BP

BN

BN

BN

BN

SN

Z

SN

BN

BN

SN

Z

SP

Z

BN

SN

Z

SP

BP

SP

SN

Z

SP

BP

BP

BP

Z

SP

BP

BP

BP

From the above rule base, the output of the fuzzy controller is:  u ( t ) = ή.[ e ( t ).S  +  e ( t ).S ] (19) ¥

and u(t) = T .u (t ) where, T1, T2 and T3 are fuzzy tuning parameter and ή is 0< ή<1. In this paper ή has been taken as 0.5 interval. The fuzzy membership linguistic variables taken as ‘BN’, ‘SN’, 3

Journal Pre-proof

that fuzzy controller gives optimized output of the system. ρ= [T1 T2 T3]T

(a)

4.1 GRAY WOLF OPTIMIZATION TECHNIQUE The Gray wolf optimization (GWO), inspired by Canadian gray wolves hunting behaviors, is a population based algorithm. This technique was proposed by Mirjali et al. [27]. In decreasing order of dominance, there are four categories of these wolves: Alpha (α), Beta (β), Delta (δ) and Omega (ω). It also starts with random population like Particle Swarm Optimization. In every iteration, Alpha, Beta and Delta wolves update their position with respect to the position of the prey. This updating continues till the distance between the prey and predator wolf stops moving or satisfactory result is met. In modeling of these wolves, α is the best solution. Every other wolf follows according to their dominance. The hunting is predominantly guided by α and β and then guided by δ, which is followed by ω. The GWO algorithm that solves the objective function consists of the following steps: Step 1. The gray wolf population are initially generated. The generated population represented by n dimensional search space for M agent positions. For the iteration, initially it initialized from k=0, and set to them k max .

re-

(20)

optimization function and  is angular error position.

pro of

‘Z’, ‘SP’ and ‘BP’ are defined as ‘Big Negative’, ‘Slightly Negative’, ‘Zero’, ‘Slightly Positive’ and ‘Big Positive’, respectively. According to equations 18 and 19, for tuning of fuzzy controller [10, 13], the centroid and base location of FCS is used in every considered case to vary input-output membership functions. For obtaining the optimal value of the gain scaling factor T1, 2 ,3 [9], it needs to be chosen in such a way

(b)

Pj ( k ) = [P j (k).......... P j (k)............. P j (k)] , j  { ,  ,  } 1

f

n

T

where, k = 1........ k max , k is the current number of iteration.

k max is max. iteration, and P ( k ), P ( k ) , P ( k ) is the vector

urn a

lP

solutions. Step 2. Each population member’s performance is evaluated on FLC based simulation. The member performance evaluation leads to controller objective function value, which is used for GWO based optimization using Pi ( k ) =  , i = 1...... M Step 3. The population members best three solutions acquired i.e., P ( k ), P ( k ) , P ( k ) using . i = j ..... M

(c) Fig. 6: (a) Error, (b) Error dot, (c) Output of TSK Fuzzy membership

J ( )

(22)

 (t ,  ) dt

(23)

 t .  (t ,  ) dt

(24)



J (  ) = IAE =



J (  ) = MSE =

(21)

(t ,  ) dt

J (  ) = ISE =

J (  ) = ITAE =

Jo

  D

1 t

2

 t

2

(t ,  )dt

i = j ..... M

(26)

i = j ..... M

To design an optimal fuzzy controller, its tuning parameter play an important role for desired output. Using ‘ρ’ notation as shown in equation 20 for tuning parameter the objective function is defined as: * =

J(P ( k )) = min { J (Pi ( k )), Pi ( k )  D p / P (k)}}, J(P ( k )) = min { J (Pi ( k )), Pi ( k )  D p / P (k), P (k)}},

IV. OPTIMIZATION TECHNIQUE

min

J(P ( k )) = min { J (Pi ( k )), Pi ( k )  D p },

(25)

where,  * is optimal solution of controller parameter (optimal value of  , D  is the constrain set of  , J (  ) is the

The above equation has the condition for the result J(P ( k ))  J(P ( k ))  J(P ( k ))

Step 4. The search vector coefficients are taken using bellow equation. 1

f

n

T

R j ( k ) = [r j ( k )............ r j ( k ).......... r j ( k )]

S j ( k ) = [s j ( k )............ s j ( k ).......... s j ( k )] , j  { ,  ,  }, 1

f

n

T

(27)

With

r j (k) = r ( k )(2 q − 1), s j ( k ) = 2 q , j   ,  ,  }, f

f

f

f

f

Where q f is uniformly random number distribution in the range of 0  q f  1, f = 1... n , and vector coefficient r f ( k ) decreases from 2 to 0 in searching process. f

r (k) = 2[1 − (k − 1) / (k max − 1)], f = 1........ n

Step 5. The search coefficient agents are allowed to find their new position by Pi ( k + 1) using bellow equation.

Journal Pre-proof

V j ( k ) = S j ( k ) P j ( k ) − Pi ( k ) , i = 1......... M, j  { ,  ,  } (28) i

f

f

f

factors ( T1, 2 ,3 ) are obtained using different optimization

techniques with different objective functions. Four standard error objective functions, from equations 22 to j j1 jf jm T P ( k ) = [p ( k ).......... p ( k ).......... x ( k )] , j  { ,  ,  }. 25, have been used for optimization. For the fuzzy control and tuning approach the swarm size is 60, no. of maximum jf f f i P ( k ) = p j ( k ) − r j ( k ) v j ( k ), f = 1....... n, i = 1...... M, j  { ,  ,  } iterations is 40, number of attempts is 3, random starting point and 0.01 step size have been set for all the optimization the updated Pi ( k + 1) vector solution will obtained by algorithms i.e. WOT, PSO and GWO. Using equation 22-25 as    Pi ( k + 1) = (P ( k ) + (P ( k ) + (P ( k )) / 3, i = 1....... M . a fitness functions, the optimal scaling factors of fuzzy tuning Step 6. The updated solution from above equation validated for parameter( T1, 2 ,3 )and fitness value (J) are shown in Table III. It TSK fuzzy controller tuning parameter ρ = Pi ( k + 1) is clearly visible that fitness value (J) is having lowest Step 7. The algorithm repeated from step 2, until iteration k optimized value, while considering ITAE as a fitness function reached from initial iteration to its maximum limit. as compared to others, i.e., IAE, MSE and ISE for all the Step 8. In last step algorithm stopped and find best solution individual optimization techniques. Hence, ITAE is selected for using error minimizationon different constrain set. The system step  * = arg min J (Pi ( k max )) response using different fitness function for GWO, PSO and i =1...... M The GWO optimization with three variables (n = 3) that belong WOT is shown in Fig. 7(a)-7(c). to the controller parameter vector

pro of

By taking notation P j ( k ) for  ,  ,  ,updated solution

 = [ 1 ,  2 ,  3 ]T = [T1 , T2 , T3 ]T

V. SIMULATION RESULTS AND DISCUSSION

urn a

lP

re-

The STS is represented schematically in Fig.1. The mathematical model of Sun-Tracker system and its input output relation is given by equation 13. Due to integration of system model with inertia characteristics, the system represented by the transfer function in equation 13 is unstable. For such type of system controlling, two-input Sugeno-type fuzzy controller is used. The Fuzzy controller implementation is based on its Error, Error dot membership function as shown in Fig. 6(a)6(b). All simulations have been performed on MATLAB/Simulink platform. For controlling the STS axis position, primarily the fuzzy input output membership function is uniformly distributed in the range of [-1, 1] and then optimal fuzzy tuning scaling

Fig. 7 (a): Step response of STS for IAE as objective function on [0-10] boundary for GWO, PSO, WOT optimization

TABLE III

OPTIMAL VALUE USING DIFFERENT OBJECTIVE FUNCTIONS

Cost Function IAE [0-10] GWO

ISE[0-10] MSE[0-10] ITAE[0-10] IAE[0-10]

WOT

ISE[0-10] MSE[0-10]

Jo

PSO

T1

T2

T3

Optimal solution (J)

1.11622

10

1.9363

0.025074

1.11794

10

1.90653

0.014473

1.10193

10

1.8524

0.01456

1.10137

10

1.84525

0.00155

1.5483

9.7952

1.8122

0.0335861

2.2453

9.2934

1.3129

0.01764

2.2034

9.2934

1.3261

0.015761

ITAE[0-10]

1.0764

10

0.1.4968

0.002836

IAE[0-10]

1.1459

8.3734

1.648

0.0381

ISE[0-10]

1.9168

9.0595

1.8779

0.022237

MSE[0-10]

1.0018

8.6076

1.7683

0.016151

ITAE[0-10]

1.10714

10

1.90247

0.0013556

Journal Pre-proof

TABLE IV PARAMETER OPTIMIZATION FOR ITAE WITH 60 POPULATION SIZE AND 40 ITERATION FOR GWO, PSO, WOT ON DIFFERENT FEASIBLE SPACE

ITAE

WOT

ITAE

PSO

ITAE

T1

T2

T3

[0-10]

1.10137

10

1.84525

Of error Minimization value 0.00155

[0-100]

1.006364

100

8.5436

[0-10]

1.10714

10

[0-100]

1.00108

100

[0-10]

1.0764

10

[0-100]

1.1115

100

Max Overshoot

0.5

0

5.579e-05

0.041

0

1.90247

0.0013556

0.4

0

8.6981

5.622e-05

0.05

1.02

0.1.4968

0.002836

0.7

0

8.0893

6.066e-05

0.055

1.01

Fig. 8(a)-8(b) shows system output response, Fig. 8(c)- 8(d) shows minimization of objective function and Table IV gives the GWO, PSO and WOT tuning parameter, optimal solution and settling time on different constrain set. From Table IV, when objective function ITAE is chosen in the range of [0-10], WOT gives better optimized value T1 than GWO and PSO while for [0-100] range GWO performance is superior than other two techniques. The response of GWO, WOT and PSO on different constrain set, when ITAE is considered as objective function is shown in Fig.8(a) and Fig.8(b). It can be observed from Fig.8(a), Fig.8(b) and Table IV that the performance of GWO comes out to be better than PSO and WTO with respect to error minimization functions as well as output response.

urn a

lP

Fig. 7 (b): Step response of STS for ISE as objective function on [0-10] boundary for GWO, PSO, WOT optimization

Settling Time

pro of

GWO

Boundary

re-

Cost Function

Fig. 7 (c): Step response of STS for MSE as objective function on [0-10] Boundary for GWO, PSO, WOT optimization

Fig. 8 (b): Step response of GWO, PSO and WOT for ITAE as objective function on [0-100] constrain set

Jo

From figure 7(a) while IAE considered as objective function the STS system settling time is 0.6 sec for GWO, 0.75 sec for PSO and 0.8 sec for WOT which shows GWO gives better output as compared to WOT and PSO. Similarly, for ISE and MES, GWO shows better response as shown in Fig. 7 (b)-7(c) respectively.

Fig. 8 (a): Step response of GWO, PSO and WOT for ITAE as objective function on [0-10] constrain set

Fig. 8 (c): ITAE of GWO, PSO, WOT [0-10]

Journal Pre-proof TABLE V PARAMETER OPTIMIZATION FOR ITAE & ITAU WITH 60 POPULATION SIZE AND 40 ITERATION FOR GWO, PSO, WOT ON DIFFERENT FEASIBLE SPACE

Objective Function GWO ITAE & ITAU

WOT

T1

T2

T3

[0-10]

1.00046

10

1.65283

Of error Minimization value 0.0041871

[0-100]

1.3255

96.52

3.95

[0-10]

3.6929

10

[0-100]

1.7009

96.35

[0-10]

1.00025

10

[0-100]

1.38905

94.6904

Settling Time

Max Overshoot

0.5

0

0.0003358

0.01

0

1.1143

0.004653

2

0

0.01582

0.04590

0.55

0

ITAE & ITAU

ITAE & ITAU

pro of

PSO

Boundary

0.3385

0.008414

0.59

0

3.81096

0.0003356

0.125

0

TABLE VI

PARAMETER OPTIMIZATION FOR ISE & ISU WITH 60 POPULATION SIZE AND 40 ITERATION FOR GWO, PSO, WOT ON DIFFERENT FEASIBLE SPACE

ISE & ISU

PSO

ISE & ISU

WOT

ISE & ISU

T1

T2

[0-10] [0-100] [0-10] [0-100] [0-10] [0-100]

1.00958 1.64964 1.2389 1.8433 1.008 1.7447

10 89.801 10 99.97 10 94.89

T3

Of error Minimization value

Settling Time

Max Overshoot

1.31137 1.429 1.0920 1.6075 1.2118 1.5079

0.02798 0.015075 0.02786 0.015075 0.02869 0.0150

0.65 0.19 0.8 0.24 0.65 0.24

0 1.02 0 1.08 0 1.03

lP

GWO

Boundary

re-

Objective Function

For better analysis, minimization of error magnitude and the magnitude of system input (which is controller output) are also considered as objective function is shown in equations 29 to 31. 



urn a

J = min imize (  S e ( t ) tdt + 

T

(29)

u( t ) tdt )

0



(30)

J = min imize (  Se (t ) dt +  T u (t ) dt ) 2

2

0

0





(31)

J = min imize (  S e (t ) dt +  T u(t ) dt 0

0

Where, S and T are weighing factors, e(t) is error magnitude of controller input, u(t) is magnitude of system input signal. In this manuscript Equation 29 is noted as ITAE & ITAU, equation 30 is ISE & ISU and equation 31 is IAE & IAU respectively. In order to better visualize the effect of the weighing factors, ‘S

Fig. 8 (d): ITAE of GWO, PSO, WOT [0-100]

Jo

0

Table VII

PARAMETER OPTIMIZATION FOR IAE & IAU WITH 60 POPULATION SIZE AND 40 ITERATION FOR GWO, PSO, WOT ON DIFFERENT FEASIBLE SPACE

Objective Function GWO

IAE & IAU

PSO

IAE & IAU

WOT

IAE & IAU

Boundary

T1

T2

T3

Of error Minimization value

Settling Time

Max. Overshoot

[0-10]

1.00284

10

1.41908

0.056002

0.75

0

[0-100]

1.839995

100

3.1534

0.02287

0.12

0

[0-10]

1.0116

10

1.4524

0.05598

0.75

0

[0-100]

1.8257

100

3.0426

0.02287

0.14

0

[0-10]

1.005

10

1.40

0.05585

0.8

0

[0-100]

1.85043

100

2.7607

0.023055

0.15

0

Journal Pre-proof

and T’, on the optimal scaling factors ( T1, 2 ,3 ) and the

pro of

performance of the closed-loop system, weighing factor is selected as 0.5. From Table V, while taking equation 29 as objective function, i.e., ITAE & ITAU, on different constrain set, the optimized value of GWO gives better dynamic performance in terms of error minimization and settling time as compared to WOT and PSO as shown in Fig.9 (a)-9 (d).

Fig. 9 (c): ITAE & ITAU of GWO, PSO, WOT [0-10]

lP

re-

Fig. 9(a): Step response of GWO, PSO and WOT for ITAE & ITAU as objective functions on [ 0-10] search domain

Fig. 9 (d): ITAE & ITAU of GWO, PSO, WOT [0-100]

Fig. 9(b): Step response of GWO, PSO and WOT for ITAE & ITAU as objective functions on [ 0-100] search domain

Jo

urn a

Moreover, when equation 30 is considered as an objective function, i.e., ISE & ISU, the objective function minimization of GWO and PSO are almost similar in [0-10] & [0-100] constrain set but GWO gives improved settling time for [0-10] & [0-100] constrain set. Considering the system settling time, the GWO and WOT in [0-10] boundary range is almost similar as shown in Fig.10(a)-(b), but in [0-100] boundary range GWO gives better response. The comparison table of WOT, PSO and GWO on different parameters while equation 43 is considered as objective function is given in table VI. The objective function minimization for PSO, WOT and GWO is shown in Fig.10(c)-(d).

Fig. 10 (a): Step response of STS for ISE & ISU as objective function for [0-10] boundary on GWO, PSO, WOT optimization

Journal Pre-proof

urn a

lP

Fig. 10 (c): ISE & ISU of GWO, PSO, WOT [0-10]

Fig. 11 (a): Step response of STS for IAE & IAU as objective function for [0-10] boundary on GWO, PSO, WOT optimization

re-

Fig. 10 (b): Step response of STS for ISE & ISU as objective function for [0100] boundary on GWO, PSO, WOT optimization

pro of

Thus, it is observed that GWO is considered the best optimization solution for tuning of fuzzy parameters pertaining to Sun-Tracker System.

Fig. 11 (b): Step response of STS for IAE & IAU as objective function for [0-100] boundary on GWO, PSO, WOT optimization

Fig. 11 (c): IAE & IAU of GWO, PSO, WOT [0-10]

Fig. 10 (d): ISE & ISU of GWO, PSO, WOT [0-100]

Jo

Similarly, when equation 30 is taken as an objective function for GWO, PSO and WOT, GWO [0.056002, 0.02287], PSO [0.05598, 0.02287] and WOT [0.05585, 0.023055] minimize value on [0-10 & 0-100] constrain set respectively. Furthermore, settling time of system using PSO [0.75, 0.14] sec, GWO [0.75, 0.12] sec, WOT [ 0.8, 0.15] sec for [0-10 & 0100] constrain set. The comparison table of WOT, PSO and GWO on different parameter with equation 31 being considered as objective function is shown in table VII. The step response of system while IAE & IAU taken as objective function on [010] and [0-100] constrain set is shown in Fig. 11(a)-11(b). Fig. 11(c)-11(d) show the objective function error minimization of GWO, PSO, and WOT on different constrain set.

Fig. 11 (d): IAE & IAU of GWO, PSO, WOT [0-100]

Journal Pre-proof

CONCLUSION In this article a performance comparison of GWO, PSO and WOT algorithms has been done. Also the optimal values of Fuzzy controller tuning parameters, error minimization and stability performance have been computed for STS. The design and implementation of TSK fuzzy with variable membership functions has been used to test different objective functions, i.e., MSE, IAE, ISE & ITAE, ITAE & ITAU, ISE & ISU and IAE & IAU in optimization process. The obtained results authenticate that GWO based optimization technique is a good choice to find optimal solution of fuzzy tuning parameter and error minimization on different constrain set for Sun-Tracker System. Moreover, as the constrain set varies, it creates critical impact on the performance of GWO, PSO and WOT. The research clearly states that GWO gives best fuzzy tuning parameters and error minimization on different constrain set [010 & 0-100]. It also provides better result in context of overshoot and settling time, as compared to PSO and WOT. It has also been observed that ITAE & ITAU objective function gives better result for STS as compared to ISE & ISU and IAE & IAE objective function.

[15]

REFERENCES

[24]

[4]

[5] [6] [7] [8] [9]

[10] [11] [12]

[13] [14]

[17]

pro of

[18]

[19] [20] [21] [22]

re-

[23]

[25] [26]

lP

[3]

urn a

[2]

Husain, I. “Electric and hybrid vehicles, design fundamentals,” Second Edition, CRC Press, Taylor and Francis, New York, USA, 2011. Yadav, A.K., Gaur P., Jha, S.K., Gupta, J.R.P., and Mittal A.P., “Optimal speed control of hybrid electric vehicles,” Journal of Power Electronics, vol. 11, no. 4, pp. 390-392, Jul.2011. S. K. Jha, A. K. Yadav, Prerna Gaur, H. Parthsarathy, and J.R.P. Gupta, “Robust and Optimal Control Analysis of Sun Seeker System,” Journal of Control Engineering and Applied Informatics, vol.16, no.1, pp. 70-79, 2014. Rastogi, S., S. K. Jha, and R. Oberoi, “Robust Stability Analysis of Sun Seeker System byArguon's Theorem,” in Proc. of Fifth International Conference on Advanced Computing and Communication Technologies (ACCT), Haryana, India, Feb. 2015. Md. Shahidul Islam, Md. Ebrahim Khalil, Md. Alif Al Shefaet, “Tuning of PID Controller for A Sun Seeker System Using Fuzzy Logic,” Asian Journal of Applied Science and Engineering, vol.5, no.1,2016. Kealy, T. O'Dwyer, A. “Analytical ISE Calculation and Optimum Control System Design,” in Proc. Irish Signals and Systems Conference, University of Limerick, Ireland, 2003. Tripathi S.K., Panday H, Gaur P, “Robust control of Inverted Pendulum using fuzzy logic controller,” in Proc. of IEEE Students Conference on Engineering and Systems (SCES) Allahabad, India, Apr. 2013, pp 1-6. Vishal, Jyoti Ohri, “GA tuned LQR and PID controller for aircraft pitch control,” in Proc. IEEE 6thIndia International Conference on Power Electronics (IICPE), Kurukshetra, India, Dec. 2014, pp. 1-6. A. Noshadi, J. Shi, W. S. Lee, P. Shi, A. Kalam, “Optimal PID-type fuzzy logic controller for a multi-input multi-output active magnetic bearing system", Neural Comput. Appl., DOI: 10.1007/s00521-015-1996-7, 2015. Woo ZW, Chung HY, Lin JJ, “A PID type fuzzy controller with selftuning scaling Factors,” Fuzzy Sets and System., vol. 115, no.2, pp.321326, Oct. 2000. Ahn, K.K.; Truong, D.Q. “Online tuning fuzzy PID controller using robust extended Kalman Filter,”. J. Process Control vol.19, no.6, pp.1011–1023, Jun. 2009. Sandeep Kumar Tripathi, Mohit Dev Srivastava, Babita Singh, Priyanshu Dixit, “Genetic Algorithm based H∞ Controller for Inverted Pendulum,” in Proc. Eighth International Conference on Image and Signal Processing (ICISP), Bangalore, India Jul. 2014, pp.25-27. R. K. Mudi and N. R. Pal, “A robust self-tuning scheme for PI- and PDtype fuzzy Controllers,” IEEE Trans. Fuzzy Systems , vol. 7, no.1,pp.216, Feb. 1999. Neath, M.J.; Swain, A.K.; Madawalaand, U.K.; Thrimawithana, D.J., “An optimal PID controller for a bidirectional inductive power transfer system

Jo

[1]

[16]

using multiobjective genetic algorithm,” IEEE Trans. Power Electron., vol. 29, no.3, pp.1523-1531, May 2013. Rajni Jain a, N. Sivakumaran b , T.K. Radhakrishnan,” Design of self tuning fuzzy controllers for nonlinear systems,” Expert Systems with Applications, vol.38, no.4, pp.4466-4476, Oct. 2010. Anil Kr. Yadav, Prerna Gaur, Sandeep Tripathi, “Design and Control of an Intelligent Electronic throttle Control System,” in Proc. IEEE international Conference on Energy Economics and Environment (ICEEE) , Noida, India,Mar. 2015,pp.1-5. S. Kirkpatrick, C.D. Gelatto, M.P. Vecchi, “Optimization by simulated annealing,” Science New Series, vol.200, no.4598, pp-671–680, May 1983. Ameya Anil Kesarkar& N. Selvaganesan “Tuning of optimal fractionalorder PID controller using an artificial bee colony algorithm,” Systems Science & Control Engineering, An Open Access Journl Taylor & Francis, vol.3, no.1, pp. 99-105, Dec. 2014. M. Dorigo, V. Maniezzo, A. Colorni, The ant system: optimization by a colony of cooperating agents, IEEE Trans. Systems, Man, and Cybernetics, vol. 26, no.1, pp. 29–41, 1996. EsmatRashedi, Hossein Nezamabadi-pour, SaeidSaryazdi, “GSA: A Gravitational Search Algorithm,” Information Sciences, vol.179, no.13, pp. 2232–2248, Jun. 2009. T.H. Kim, I. Maruta, T. Sugie, “Robust PID controller tuning based on the constrained particle swarm optimization,” Automatica, vol. 44, no. 4, pp.1104–1110, Apr. 2008. J. Kennedy, R. C. Eberhart, Particle swarm optimization, in Proc. IEEE Int. Conf. Neural Networks, Perth, Australia, Nov.1995, pp. 1942–1948. Shi Y, Eberhart R, “A modified particle swarm optimizer,” in Proc of evolutionary computation,” pp 69–73,1998. Radu-Emil Precup, Radu- Codrut David and Emil M. Petriu, “Grey Wolf Optimizer Algorithm-Based Tuning of Fuzzy Control Systems With Reduced Parametric Sensitivity,” IEEE Trans. Ind. Electronics, vol.64, no.1, pp. 527-534, Jan.2017. Gupta, E., &Saxena, A. “Grey wolf optimizer based regulator design for automatic generation control of interconnected power system,”. Cogent Engineering, vol.3, no.1, Mar. 2016. S. Kamyab, A. Bahrololoum “Designing of rule base for a TSK- fuzzy system using bacterial foraging optimization algorithm (BFOA),” in Proc.4th International Conference of Cognitive Science (ICCS), Feb. 2012, pp.176-183. S. Mirjalili, S. M. Mirjalili , and A. Lewis , “ Grey wolf optimizer ,”Adv. Eng. Software, vol.69 , pp. 46-61, Mar. 2014. Seyedali Mirjalili, Shahrzad Saremi, Seyed Mohammad Mirjalili,Leandro dos S. Coelho, “Multi-objective grey wolf optimizer: A novel algorithm for multi-criterion Optimization,” Expert Systems with Applications,vol.47, no.1, pp. 106-119, Apr. 2016. Seyedali Mirjalili, Andrew Lewis, “The Whale optimization Algorithm”, Advance in engineering Software, vol.95, pp 51-67,May 2016. Indrajit N. Trivedi, Jangir Pradeep, Jangir Narottam, Kumar Arvind, Ladumor Dilip, “ A Novel Adaptive Whale Optimization Algorithm for Global Optimization,” Indian Journal of Science & Technology, vol.9, no. 38, Oct. 2016. Kuo, B.C. and Golnaraghi, F., “Automatic control system,” John Wiley and Sons (Asia) Pvt. Ltd, Eighth Edition., 2003.

[27] [28]

[29] [30]

[31]

Journal Pre-proof Highlights for Review

pro of

re-

•

lP

•

urn a

•

Optimal tuning technique for TSK fuzzy controller parameters using Gray Wolf optimization (GWO) for Sun Tracker System has been described. Minimization of objective function, represented as weighted sum of control error and output sensitivity of fuzzy controller with GWO is proposed. A comparative analysis of GWO, Particle Swarm Optimization (PSO) and Whale optimization technique (WOT) on different constrain set for optimal TSK fuzzy parameter estimation for STS have been analyzed. Using GWO technique, maximum power extraction of STS with respect to control performance indices have been improved.

Jo

•

Journal Pre-proof

Dear Editor-in-chief, This is to inform you that there is no conflict of interest related to any authors associated with the submitted manuscript.

Jo

urn a

lP

re-

pro of

With regards, Sandeep Tripathi, Ashish Shrivastava and K C Jana