Author's Accepted Manuscript

Fuzzy PID supervised online ANFIS based

speed controller For Brushless dc motor
K. Premkumar, B.V. Manikandan

www.elsevier.com/locate/neucom

PII: S0925-2312(15)00053-3
DOI: http://dx.doi.org/10.1016/j.neucom.2015.01.032
Reference: NEUCOM15070

To appear in: Neurocomputing

Received date: 7 May 2014

Revised date: 27 November 2014
Accepted date: 16 January 2015

Cite this article as: K. Premkumar, B.V. Manikandan, Fuzzy PID supervised
online ANFIS based speed controller For Brushless dc motor, Neurocomputing,
http://dx.doi.org/10.1016/j.neucom.2015.01.032

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 Fuzzy PID supervised online ANFIS based speed controller for
Brushless dc motor
K.Premkumara and B.V.Manikandanb
a
Department of Electrical and Electronics Engineering, Pandian Saraswathi Yadav Engineering College, Sivagangai
-630561, Tamilnadu state, India
b
Department of Electrical and Electronics Engineering, Mepco Schlenk Engineering college, Sivakasi – 626 005,
Tamilnadu state, India
a
Corresponding author: prem.kamaraj@gmail.com

Abstract: In this paper, two different speed controllers i.e., fuzzy online gain tuned anti wind up
Proportional Integral and Derivative (PID) controller and fuzzy PID supervised online ANFIS
controller for the speed control of brushless dc motor have been proposed. The control system
parameters such as rise time, settling time, peak time, recovery time, peak overshoot and
undershoot of speed response of the brushless dc motor with the proposed controllers have been
compared with already published controllers such as anti wind up PID controller, fuzzy PID
controller, offline ANFIS controller, PID supervised online ANFIS controller and On-line
Recursive least square – Error back propagation algorithm based ANFIS controller. In order to
validate the effectiveness of the proposed controllers, the brushless dc motor is operated under
constant load condition, varying load conditions and varying set speed conditions. The
simulation results under MATLAB environment have predicted better performance with fuzzy
PID supervised online ANFIS controller under all operating conditions of the drive.

Keywords: Brushless dc motor, PID controller, anti wind up PID controller, fuzzy PID
controller, offline ANFIS, online ANFIS.

1. Introduction

Speed regulation is an important aspect in the field of brushless dc motor drive for
precise speed and position control applications. For the enhancement of brushless dc motor
performance, different controllers have been developed [1-4]. Proportional Integral (PI) based
speed controller has been implemented for Brushless DC motor in [1] but, the controller has
produced large settling time, rise time and more oscillations in the speed response. In [2], an off-
line least-squares approximation method has been developed for identifying BLDC motor
parameters and it has resulted in unacceptable level of tolerance which cannot be acceptable for
precise speed control applications. In [3], optimization of PI coefficients of speed controller for a
PMBLDC motor using Genetic Algorithm (GA) has been developed. The speed response has
larger overshoot in transient period and more fluctuation during the steady state period. A
brushless dc motor drive system incorporated with proportional integral speed control loop has
been implemented in [4], but it has resulted in larger steady state error.

An adaptive PID neural network controller has been developed and Particle Swarm
Optimization (PSO) algorithm was used for initializing the weight of the neural network and
improved gradient descent algorithm was used for adjusting the parameter of the PID neural
network [5]. The disadvantage of this method is that PSO algorithm takes long time for
initializing the weight of the PID neural network. A comparative analysis of PSO and BFO
methods for the tuning of PID controller for the speed control of a BLDC motor has been
proposed in [6]. But, during sudden load disturbances, speed response has larger undershoot and
larger steady state error. A comparative analysis of PI, Anti wind up PI and fuzzy Logic
controllers have been developed for brushless dc motor [7]. During load disturbance, the speed
response has larger undershoot and steady state error with anti wind up PI control.

Adaptive fuzzy PID controller using multi objective PSO reinforcement evolutionary
algorithm has been developed for automobile suspension system. The adaptive mechanism was
formed by using parallel combination of PID controller and fuzzy logic controller. The controller
effectively controls the system response, but it exhibits large steady state error [8]. Different
types of fractional order hybrid fuzzy PID structure have been developed for the lag dominant,
balanced lag and delay dominant system. Recommended structure for each system with specified
operating conditions was suggested. Conversely, the controller has reduced the control system
performance [9]. The Adaptive tuning method for the classical PID controller has been
developed for nonlinear process system. The PID controller has been cascaded with a fuzzy
predictor, where the controller gains are adjusted online based on the predictions of a fuzzy
predictor, but this controller produced larger overshoot and noise in the output response [10].

Fuzzy tuned PID controller was developed for Brushless dc motor. Simulations were
carried out for sudden load disturbance and step change in input. The effectiveness of controller
was compared with the conventional PID controller, the controller produced ±50 percentage
overshoot and large settling time in the speed response [11]. In [12], particle swarm optimization
algorithm has been applied for PID control of electrical dc drive system. From the simulated and
experimental results, it was observed that speed response has large overshoot and more
oscillation in the steady state. Also, there were uncertainty problems due to load variations. In
[13], comparative analysis between fuzzy logic and PID sliding mode fuzzy logic controller has
been made for brushless dc motor. However, the controller produced indecision problem due to
different loading conditions. In [14], GA and PSO optimized fractional order fuzzy PID
controller has been developed for nonlinear process with time delay and unstable process with
time delay. Effectiveness of this controller was compared with fuzzy PID controller. The
controller has imprecision due to load disturbances as outlined in [15]. Online tuning of fuzzy
PID controllers via rule weighing based on normalized acceleration has been developed for the
second order system. This method effectively controlled the system but, the system output has
larger overshoot and undershoots during sudden load disturbance [16]. Self tuning of PID type
fuzzy logic controller has been developed for dc motor system. The system output was compared
with conventional PI controller and this control mechanism outperforms the PI controller, but the
output has ±10 percentage of overshoot [17].

ANFIS based automatic generation control has been developed for multi area power
system. The effectiveness of controller was compared with the integral controller. The ANFIS
controller enhanced the steady state response but degraded the transient response [18]. Hybrid
based controller with Neuro Fuzzy and PI controller has been developed for brushless dc motor.
The Hybrid controller has sluggish response during transient period and also, has uncertainty
problem due to load variations [19]. Comparative analysis between PI controller, fuzzy tuned
PID controller, Fuzzy variable structure controller and ANFIS controller based speed control of

2
the brushless dc motor has been explained in [20]. The main drawback with the developed
ANFIS controllers cited in [18-20] is that the controllers have been trained in off line mode.

Fuzzy critic supervised Neuro-fuzzy controller was developed for brushless dc motor
[21]. Fuzzy critic algorithms utilize the control action of proportional derivative controller and it
has modified the output layer of the Neuro-fuzzy logic controller. Gain tuning of the PD
controller has significant effect on the control system performance. Online learning of Neuro-
fuzzy controller using sliding mode algorithm has been explained in [22]. Conventional PID and
PD controller was paralleled with Neuro-Fuzzy controller. This controller also has effect on
control system performance with tuning of PID controller. In [23], online mixed learning method
of recursive least square and error back propagation trained Neuro fuzzy system with evolving
clustering has been applied for the machinery condition monitoring applications.

Most of the research seldom concentrated on control system performance parameter such
as rise time, settling time, recovery time, steady state error, undershoot and overshoot. In this
paper, all the above notified control system parameters are measured for the proposed controllers
namely, fuzzy online gain tuned anti wind up PID controller and fuzzy PID supervised learning
of online ANFIS controller and compared with the already published controllers namely anti
wind up PID controller, fuzzy PID controller, offline ANFIS controller, PID supervised online
ANFIS controller and On-line Recursive least square – Error back propagation based ANFIS
controller notified in [7, 14, 20,22 and 23]. Organization of the paper is as follows: State space
model of the brushless dc motor presented in section 2. Structure of the speed control of
brushless dc motor is described in section 3 and proposed controllers are explained in section 4.
Section 5 discusses the simulation results and comments about the results discussion is
summarized in section 6. Concluding remarks are outlined in section 7.

2. State space modeling of brushless dc motor

The BLDC motor has three stator windings and permanent magnets on the rotor. The
mathematical state space representation of variables of the BLDC motor can be described by the
following equation in (1),
 −R 
L−M 0 0 0 0  1 
L−M 0 0 0  −1 
  L−M 0 0 
 ia   0 −R  
0 0 0   ia   1  
i     0 0 0 0  Va   −1
L−M
  ib     0 0  ea 
d   
L−M
 Vb  +   
b

 ic  =  0
−R  L−M
0   ic  +  1  −1   b 
0 0 e
dt    L−M   0 0 0 0  Vc  
0 0
ωr   −B  ωr   L−M  
 L−M 
 ec 
 θ r   0 0 0 0  θ   −1  TL  
 J  r   0 0 0  0 0 0 
 J   
 P  
 0 0 0 
 0 0 0 0  0 0 0 0 
 2 

(1)
where Va, Vb and Vc denotes stator phase voltages of the BLDC motor in volts and R represents
stator winding resistance in ohms. Phase currents of the motor are represented by ia, ib and ic in
amps. The self inductance of the motor winding is represented by L and the mutual inductances
between stator windings are denoted by M in Henry. ea, eb and ec denotes the trapezoidal back-
EMF of each phase in volts. P is the number of poles in the rotor and θr is the rotor position of

3
the rotor in radians. J, B, ωr and TL denotes the moment of inertia, frictional coefficient, angular
velocity and load torque of the motor respectively. The electromechanical torque is expressed in
equation (2) as,

d ωr
Te = J + Bωr + TL (2)
dt
The equation for instantaneous electrical Torque is given in equation (3) and also the relationship
between angular velocity and rotor position.

Te =
( eaia + ebib + ec ic ) and ωr =
dθ r
(3)
ωr dt

3. Structure of Speed control of Brushless DC Motor

The closed loop speed control system of brushless dc motor is taken from [20]. The
overall speed control system is modeled by using MATLAB/Simulink tool box. Figure 1 shows
the overall Simulink model of speed control of brushless dc motor. The Simulink model consist
of three phase voltage source PWM inverter, three phase BLDC motor, speed controller,
Switching logic and PWM generator and motor measurement blocks.

4
 Fig.1 Simulink model of speed control of brushless dc motor
4. Proposed controllers for Brushless DC Motor
Two types of controllers are proposed for the speed control of brushless dc motor, i.e.,
fuzzy online gain tuned anti wind up PID controller and fuzzy PID supervised online ANFIS
controller.

4.1 Fuzzy online gain tuned anti wind up PID controller

The PID controller does not have output saturation limiters and hence potential
conditions for the wind up phenomenon exist. In order to overcome this difficulty, a maximum
integrator output value will be kept within limit and a strategy is adapted for this condition,
which is known as anti-windup phenomenon. The main goal of anti windup PID control scheme
is to avoid the over value in the integrator thereby the integration output will be kept within a
limited range.

The back calculation anti wind up PID controller is commonly accepted for speed control
of brushless dc motor [7]. Unfortunately, many of the back calculation anti wind up PID loops
that are in process are in need of continual monitoring and tuning of gains, i.e., Kp, Ki, Kd and
Kc, since they can easily become improperly tuned or become constant. Consequently, the
control system parameters such as rise time, settling time and steady state error of the speed

5
response of the brushless dc motor will undergo drastic change with change in operating
conditions of the brushless dc drive. To overcome this drawback and for improving the control
system parameters, online tuning of gains in the anti wind up PID controller is proposed and this
is considered to be vital. It is expected that there will be considerable improvement in the
controller performance under all operating conditions of the brushless dc drive due to online
gains tuning.

The advantage of the fuzzy inference system has been proved in controlling nonlinear,
complex, time-varying dynamic processes in real world problems. It can be used as an online
gain tuner for the back calculation anti wind up PID controller. The proposed control scheme
uses mamdani fuzzy inference system as a gain tuning-tool for the back calculation anti wind up
PID controller. The structure of this controller is shown in figure 2 and in order to obtain the
variable gain, fuzzy logic is employed for tuning the controller gain in online mode under all
operating conditions.

Fig. 2 Structure of fuzzy online gain tuned anti wind up PID controller

Fuzzy online gain tuner is modeled by mamdani fuzzy inference system. Figure 3 shows
the structure of fuzzy online gain tuner for anti wind up PID controller. It has two inputs i.e.,
speed error (e) and the rate of change of error (∆e) and four outputs (Kp, Ki, Kd and Kc). Each
input has five membership functions with bell shape norm and outputs have five membership
functions with Gaussian norm.

6
 Fig.3 Proposed structure of fuzzy online gain tuner

Distribution of input membership function is shown in figure 4. The universal bell shaped
membership function is expressed by equation (4) as,

1
f ( x; a, b, c ) = 2b
x−c
1+
a
(4)

Fig.4 Distribution of input membership functions for error (rpm) and rate of change of error
(rev/s2)

7
 The universal bell shaped function depends on three parameters a, b and c where
‘a’ denotes the half width, ‘b’ controls the slopes at the intersecting points and ‘c’ determines the
centre of the corresponding membership function. The input range for error and rate of change
error are from -1500 to 1500 and distributed with five bell shaped membership functions denoted
by Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS) and Positive Big
(PB).

Similarly each output is distributed with five Gaussian shaped membership functions.
Gaussian membership function has been used to ease the design task. The Gaussian membership
function has been widely used with mamdani fuzzy inference system and has shown good
performance in many applications. Besides, the use of Gaussian membership function ensures
that the whole output space is covered by fuzzy rules and avoids the zero firing strength
problems. The generalized Gaussian function expressed by the equation (5) as,
2
/ 2σ 2
f ( x; σ , c ) = e − ( x − c )
(5)

where c and σ represents the center and width of the membership function. Five Gaussian
membership functions are denoted by Small (S), Medium (M), Big (B), Very Big (VB), Very-
Very Big (VVB). The range of proportional gain is from 0 to 5, integral gain ranges from -1 to 1,
derivative gain ranges from -1 to 0 and back calculation gain ranges from -1 to 1. Totally 25
rules are created for fuzzy online gain tuner and few rules are described in equation (6) as,

rule 1: if e is NB and ∆e is NB then ( Kp is VVB )( Ki is VVB )( Kd is S )( Kc is VVB )

rule 25: if e is PB and ∆e is PB then ( Kp is S )( Ki is S )( Kd isVB ) ( Kc is S )
(6)
Table. I Rule base of fuzzy online gain tuner

Rate of change of error (∆e) Rate of change of error (∆e)

Speed Error (e)

Speed Error (e)

Kp NB NS Z PS PB Ki NB NS Z PS PB
NB VVB VVB VVB VB B NB VVB VB B B B
NS VVB VVB VB VB B NS VVB VB VB B M
Z VB VB B PS PS Z B M S S S
PS B M M M M PS B B M S S
PB M M S S S PB M S S S S
Rate of change of error (∆e) Rate of change of error (∆e)
Speed Error (e)

Speed Error (e)

Kd NB NS Z PS PB Kc NB NS Z PS PB
NB S S M M VB NB VVB VB B B B
NS S S S S M NS VVB VB VB B M
Z S S S M VB Z B M S S S
PS M M M VB S PS B B M S S
PB S S S M VB PB M S S B S

8
 The overall rule base for fuzzy online gain tuner is shown in Table I. The Fuzzy system
utilizes the centroid defuzzification method. Centroid defuzzification provides the center of area
under the curve. This is the most commonly used technique and it is very accurate. It is described
by the equation (7) as,

gainkp , ki ,kd and kc =

∫µ A ( x ) x dx
∫µ A ( x ) dx
(7)

The output of the fuzzy online gain tuner is then multiplied with anti wind up PID
controller and it provides the control signal (Uc) to the system. The simulink model of this
controller is shown in figure 5.

Fig. 5 Simulink model of fuzzy online gain tuned anti wind up PID controller

4.2 Fuzzy PID supervised online ANFIS controller

Supervised learning techniques are more powerful in machine learning than unsupervised
techniques because the availability of labeled training data provides clear criteria for model
optimization [22]. Supervised learning of ANFIS structure can be formed using off line operation
and online operation. Both operation have two type of learning i.e., structure learning and
parameter learning. Structure learning is primary to extract the fuzzy logic rules from the input
data with tuning of fuzzy partitions for the input and output spaces. Then, the parameter learning
adjusts the parameters of each rule. These two phases are completed sequentially in off line
operation. In the first phase, input data is partitioned and in second phase, premises and
consequent parameter of the network is updated using gradient descent method and recursive
least square method respectively.

9
 Main drawback of this sequential learning scheme of off-line operation is that, it requires
large quantity of representative data collection in advance and also independent realization of the
structure and parameter learning usually takes lot of time. To conquer these problems and for
faster learning, online operation has been introduced to perform the structure and parameter
learning phases concurrently. To enhance the performance still further, fuzzy PID supervised
online learning of ANFIS controller is proposed in this paper.

The proposed Fuzzy PID Supervised online Adaptive Neuro Fuzzy Inference System
combines the merits of fuzzy ART, neural network and fuzzy inference system. Moreover, it
performs the structure and parameter learning phases simultaneously. The structure learning is
based on the partition of input space by using different partitioning methods such as grid
partitioning, tree partitioning and scatter partitioning. In grid partitioning, size of fuzzy rules
generated grows exponentially. Tree partitioning relieves the above problem but it needs more
membership functions for each input to define fuzzy regions and these membership functions do
not usually bear clear linguistic meanings. The scatter partitioning is usually dictated by the
desired input-output data pairs and makes it hard to estimate the overall mapping directly from
the consequent of each rule’s output. In order to overcome the drawbacks of partitioning
methods, fuzzy ART clustering method is proposed for structure learning of ANFIS speed
controller. In addition, fuzzy ART has faster convergence rate i.e., it requires much less training
iterations. Also it has high online learning capability property i.e., fuzzy ART can learn a new
pattern without having to retain the network. The parameter learning is based on the fuzzy PID
adaptation control law.

The structure of fuzzy PID supervised learning of ANFIS controller is shown in figure 6
and its simulink model is shown in figure 7.

Fig. 6 Structure of fuzzy PID supervised online ANFIS controller

10
 Fig. 7 Simulink model of fuzzy PID supervised online ANFIS controller

Typical fuzzy PID controller is created with a fuzzy PD controller with an integrator and
a summation unit at the output [14]. It has two inputs that are error (e) and rate of change of error
(∆e) and one output that is the supervised output (UF). The equation for fuzzy PID controller is
expressed in equation (8) as,

U F = αU + β ∫U (8)

where α is the proportional gain, β is the integral gain and U is the output of the fuzzy PD
controller. The Fuzzy PD controller has two inputs that are error (e) and rate of change of error
(∆e) and one output (U) and it is shown in figure 8. The inputs are distributed with seven
triangular membership functions and output is distributed with forty-nine constant values.

Fig. 8 Structure of fuzzy PD controller

11
 Fuzzy inference system is modeled by zero order Takagi–Sugeno (T-S) fuzzy system. In
realistic control applications, the triangular membership function is generally selected for
representing fuzzy sets in T-S fuzzy inference system. Because, in term of real-time
requirements by the inference engine, their parametric, functional description of membership
function can be easily obtained, stored with minimal use of memory and can be manipulated
efficiently. The triangular membership function is described by the equation (9) as,
 0, x≤0
 x−a
 j
, a j ≤ x ≤ bj
 b j − a j
f ( x, a, b, c ) = 
 cj − x , b ≤ x ≤ c
 c j − bj j j


 0, cj ≤ x
(9)

where a and c locate the feet of the triangle and the parameter b locates the peak. The distribution
of membership functions for the error and the rate of change of error are shown in figure 9(a) and
9(b). The input range for the error is from -500 to 1500 and membership function denote by
A,B,C,D,E,F and G. The range for the rate of change of error is from -1*108 to 3.824*104 and
membership function denoted by A1, B1, C1, D1, E1, F1 and G1. The range of output is from -
1.584*104 to 1.142*104. The distribution of output is shown in figure 10.

Fig. 9(a) Distribution of membership function for error in rpm

12
 Fig. 9(b) Distribution of membership function for rate of change of error in rev/s2

Fig. 10 Distribution of output of the fuzzy PD controller

Totally 49 rules are created for fuzzy PD controller and few rules are described in
equation (10). The overall fuzzy rule is shown in Table II. The set of rules can be separated into
three clusters to realize the logic of incorporating the rule base as in Table II. Cluster 1: In this
cluster of rules, error (e) and its rate of change of error (∆e) have very small positive or negative
values or equal to zero. This implies that the process output has strayed off slightly from the set
point but is still close to it. Thus small values of control signals (S) are required to correct these
small deviations and these rules mainly relate to the steady state performance of the process.
Cluster 2: For this group, error (e) is positive big and its rate of change of error (∆e) is negative
big or error (e) is negative big and rate of change of error (∆e) is positive big, suggesting that the
process output is far below or far away from the set point. To bring the process output towards
the set point, the controller applies an appropriate positive control signal (P) to speed up or slow
down the approach towards the set point. Cluster 3: In this cluster the error is negative large or
medium, implying that the process output is significantly above the set point. Also the fractional
derivative of the error is negative implying that the process output is moving away from the set

13
point. Hence the controller applies a negative control signal (N) to speed up or slow down the
approach towards the set point. Weighted average defuzzification method is used for converting
fuzzy set into crisp set.
rule1: if e is A and ∆e is A1thenU is P


rule14: if e is B and ∆eis G1thenU is N


rule 28: if e is D and ∆eis G1thenU is P


rule 35: if eis E and ∆e is D1thenU is S


rule 49: if e is G and ∆e is G1 thenU is P
(10)

Table. II Rule base for fuzzy PD controller

Rate of Change of Error (∆e)
U A1 B1 C1 D1 E1 F1 G1
A P P S S S S S Cluster 1
Speed Error (e)

B P P S S S S S Cluster 2
C P P S S S S S Cluster 3
D N N S S S S S
E N N S S S S S
F N N N N P P P
G N N N P P P P

The output of fuzzy PD controller is processed by proportional integral controller and it

will provide supervised output for the online ANFIS controller. Error between supervised output
and online ANFIS controller output is expressed by the equation (11) as,
EA = U F − UC
(11)

Next, the process of applying online learning algorithm to identify ANFIS parameters has
been discussed. The ANFIS system, as the name suggests, is an adaptive Neuro-fuzzy inference
machine. It has two structures that are artificial neural network and fuzzy inference system. The
advantages of neural networks are: it has better learning capacity, generalization capacity and
robustness in relation to disturbances. The disadvantages of the neural networks are: impossible
interpretation of the functionality and difficulty in determining the number of layers and number
of neurons. The advantages of the fuzzy systems are: capacity to represent inherent uncertainties
of the human knowledge with linguistic variables, simple interaction of the expert of the domain
with the engineer designer of the system, easy interpretation of the results because of the natural

14
rules representation, easy extension of the base of knowledge through the addition of new rules
and robustness in relation of the possible disturbances in the system. The disadvantages fuzzy
system are: incapable to generalize, or either, it only answers to what is written in its rule base,
it’s not robust in relation to the topological changes of the system, such changes would demand
alterations in the rule base and it depends on the existence of an expert to determine the
inference logical rules. ANFIS combines the advantages of both artificial neural networks and
classic fuzzy systems. It also eliminates the shortcomings of the neural network and fuzzy logic
system and it works as a universal approximator [20].

The Fuzzy ART- ANFIS controller consist of six layers and shown in figure 11. They
employ two algorithms for parameter learning (i.e. Recursive Least Square and error - back
propagation) and one algorithm for automatic structure learning (i.e. fuzzy-ART). Fuzzy ART
implements fuzzy logic into ART pattern recognition, thus enhancing generalizability. An
optional feature of fuzzy ART is complement coding, a means of incorporating the absence of
features into pattern classifications, which goes a long way towards preventing inefficient and
unnecessary category proliferation [24-25].

Fig.11 Architecture of Fuzzy ART- ANFIS Network

Layer 1 is known as the input normalization layer. In this layer, ANFIS-ART uses the
technique of complement coding from fuzzy-ART to normalize the input training data.
Complement coding helps in avoiding the problem of category proliferation when using fuzzy-
ART for data clustering [24].

Layer 2 is known as input fuzzification layer. The nodes belonging to this layer are called
input-term nodes and each represents a term of an input-linguistic variable and functions as a 1-
D membership function. Fuzziness of the trapezoidal membership function is regularized.
Premises parameters or non-linear parameters adjust the shape and the location of the

15
membership function. Those parameters are adjusted during the training mode of operation by
the error back-propagation algorithm. These premises parameters or nonlinear parameters are
updated at each iteration i.e., after each input-output pair is received during training and the
instantaneous error function is minimized. For each input-output training data pair, the ANFIS
operates in the forward pass in order to calculate the current output. Afterwards, starting from
the output layer, and moving backwards, the error back-propagation executes to calculate the
derivatives for each node at every layer of the network. At the end of each iteration, non-linear
parameter of the input membership function is updated.

Layer 3 is known as fuzzy AND operation layer. Each node in this layer performs a
fuzzy-AND operation. T-norm operator of the algebraic product is selected and it will results in
each node’s output being the product of all of its inputs. The output of each node in this layer
represents the firing strength or the activation value of the corresponding fuzzy rule and the
number of fuzzy rules will be equal to the number of input term nodes. The latter is common for
all the input variables. Therefore, each fuzzy rule may be assigned an index equal to the
corresponding index of the input term node, which is common for each input linguistic variable.

Layer 4 is known as the normalization of each rule firing strength layer. The output of the
kth node is the firing strength of each rule divided by the total sum of the activation values of all
the fuzzy rules. This results in the normalization of the activation value for each fuzzy rule.

Layer 5 is known as a linear rule consequence parameter layer. Each node k in this layer
is accompanied by a set of adjustable parameters and implements the linear function. Adjustable
parameters are called consequent parameters or linear parameters of the ANFIS system and they
are adjusted by the Recursive Least Square algorithm. For the Online supervised ANFIS
controller, the inputs and output parameters are considered to be e, ∆e and UC. The output is
expressed in equation (12) as,
f ( e ( m ) , ∆e ( m ) ) d ( m ) = U C ( m )
(12)

where e(m) and ∆e(m) are controller input vectors, f is the known function of the inputs and
d(m) is the unknown parameter to be estimated. In order to identify the unknown parameter
d(m), input-output training data is required for the target system and it is obtained from the
fuzzy PID supervised control algorithm and expressed in set of ‘t’ linear equation given in (13)
as,
f t ( e ( m ) , ∆e ( m ) ) d ( m ) = U F ( m )
(13)

By the application of recursive least square algorithm, the consequent or linear parameter of the
online ANFIS controller is updated in the layer 5.

Layer 6 is known as output layer. This layer consists of one and only node that creates the
network’s output as the algebraic sum of the node’s inputs.

Table. III Specifications of BLDC Motor drive

16
 Specifications Value
Rated Voltage (Volts) 470
Rated Current (Amps) 50
Rated Speed (rpm) 1500
Stator phase resistance, R (ohm) 3
Stator phase inductance, L (H) 0.001
Flux linkage established by magnets, λ (V-sec) 0.175
Voltage Constant, Kb (V/rpm) 0.1466
Torque Constant, Kt (N-m / A) 1.4
Moment of Inertia, J (kg-m2/rad) 0.0008
Friction factor, B (N-m/(rad/sec)) 0.001
Pole pairs, P 4

5. Simulation results and discussions

Speed response for constant load condition, varying load conditions and varying set
speed conditions are analyzed for the considered brushless dc motor. Control system
performance parameters such as rise time, settling time, recovery time, steady state error,
overshoot and undershoot are obtained for the proposed controllers and compared with anti wind
up PID controller, fuzzy PID controller, offline ANFIS controller, PID supervised ANFIS
controller and On-line Recursive least square – Error back propagation algorithm based ANFIS
controller. The specifications of the BLDC motor are taken from [20] and it is shown in Table
III.

5.1. Result for Constant Load Condition

In this section, simulation results of the speed response of brushless dc motor under no
load and full load conditions are presented. Figure 12 (a) shows the speed response curve for no
load condition with a set speed of 1500 rpm and the control system parameters are presented in
Table IV. The speed response is separated into two periods i.e., transient period and steady state
period. During transient period, the peak overshoot of speed response is 2.5008 % and 2.1150 %
for anti wind up PID controller and fuzzy PID controller respectively. For offline ANFIS
controller, the peak overshoot is 1.5789 %, it is 1.2457% for PID supervised ANFIS controller
and 0.7460% overshoot for online recursive least square – error back propagation based ANFIS
controller. For fuzzy online gain tuned anti wind up PID controller, it is 0.2133 % and therefore
it is better than the controllers notified above. But with fuzzy PID supervised online ANFIS
controller, the peak overshoot is 0.1103% only and it is the best controller.

During steady state period, the set speed is attained in short time of 0.036 sec for fuzzy
PID supervised online ANFIS controller and it is comparatively higher for other controllers.
Steady state error is comparatively lower (0.133% ) for fuzzy online gain tuned anti wind up
PID controller and it gets almost eliminated (0.15 rpm or 0.01 %) for fuzzy PID supervised
online ANFIS controller. From the comparison results, it evident that, fuzzy online gain tuned
anti wind up PID controller is the better controller and fuzzy PID supervised online ANFIS
controller is the best controller.

17
 Figure 12(b) shows the speed response curve for full load conditions (25 Nm). Control
system parameters are provided in Table V. From this plot and Table V, rise time for the
proposed controllers i.e., fuzzy online gain tuned anti wind up PID controller and fuzzy PID
supervised online ANFIS controller, it is only 0.03 sec but other controllers has larger rise time
i.e., greater than 0.04 sec. Also, set speed of 1500 rpm is attained quickly with fuzzy PID
supervised online ANFIS controller. The steady state error is also in favor of the proposed
controllers, i.e., 0.166% for fuzzy online gain tuned anti wind up PID controller and 0.066% for
fuzzy PID supervised online ANFIS controller. It is clear that, the proposed controllers
outperform the other controllers in all aspects. Fuzzy PID supervised online ANFIS controller is
the best of all considered controllers.

Fig 12 (a). Speed response of brushless dc motor under no load condition (0 Nm) with set speed
of 1500 rpm

Table. IV Control system parameters for no load condition

Control system parameters
Steady
Peak Peak % Steady
Controllers Rise time Peak time Settling State
value Overshoot State
(sec) (sec) Time(sec) Error
(rpm) (%) error
(rpm)
AW PID 0.0314 0.0436 1537.5 2.5008 0.0446 17 1.133
Fuzzy PID 0.0300 0.0415 1531.7 2.1150 0.0420 10.5 0.7
Offline
0.0300 0.0411 1523.7 1.5789 0.0384 4 0.266
ANFIS
PID
0.0300 0.0410 1518.7 1.2457 0.0384 3.5 0.233
+

18
 ANFIS
Online
RLS-BP- 0.0300 0.403 1511.2 0.746 0.045 2.5 0.166
ANFIS
Fuzzy
0.0300 0.0420 1503.2 0.2133 0.043 2 0.133
AW PID
Fuzzy
PID + 0.0300 0.0403 1501.7 0.1103 0.036 0.15 0.01
ANFIS

Fig 12 (b). Speed response of brushless dc motor under full load condition (25 Nm) with set
speed of 1500 rpm

Table. V Control system parameters for full load condition

Control system parameters
Steady
Peak Peak % Steady
Controllers Rise time Peak time Settling State
value Overshoot State
(sec) (sec) Time(sec) Error
(rpm) (%) error
(rpm)
AW PID 0.0449 0.4343 1527.9 1.8633 0.0578 16 1.066
Fuzzy PID 0.0418 0.0573 1518.7 1.2490 0.0538 10.8 0.72
Offline
0.0418 .0569 1509.5 0.6351 0.0538 3 0.2
ANFIS
PID
+ 0.0453 0.7662 1500 0 0.0584 5.5 0.36
ANFIS

19
 Online
RLS-BP- 0.044 - - - 0.05 8.5 0.566
ANFIS
Fuzzy
0.0300 0.0405 1500 0 0.042 2.5 0.166
AW PID
Fuzzy
PID + 0.0300 0.0403 1501.7 0.1103 0.036 1 0.066
ANFIS

5.2. Result for Varying Load Condition

In most of the industrial applications, the drive is always subjected to varying load
conditions. In order to ascertain the superiority of the proposed controller, the closed loop system
of the brushless dc motor is operated with sudden change in load conditions. The speed
responses for varying load conditions are described in this section for two cases. For case A,
speed is set at 1500 rpm and load is varied from no load (0 Nm) to full load (25 Nm) at 0.4 sec.
In case B, speed is set at 1500 rpm and load is varied from full load (25 Nm) to no load (0 Nm)
at 0.4 sec. Figure 13 (a) shows the speed response curves for case A.

At the time of load change, very low speed drop is witnessed for fuzzy online gain tuned
anti wind up PID controller and lowest speed drop for fuzzy PID supervised online ANFIS
controller. Also, with these controllers, the overshoot is zero. When any sudden disturbance in
operating condition occurs, the system will take time to return to the set speed and this is termed
as recovery time. Recovery time is measured for all controllers and it is 0.42 sec for fuzzy online
gain tuned anti wind up PID controller and 0.41 sec for fuzzy PID supervised online ANFIS
controller. For the other controllers the recovery time is larger.

Figure 13(b) shows the speed response of brushless dc motor with load change from full
load to no load and control system parameter for both cases are provided in Table VI. When
sudden load rejection occurs, speed will get increased and rise in speed is witnessed for all
controllers except proposed controllers. For realistic load variation conditions, fuzzy online gain
tuned anti wind up PID controller is performing better and clearly fuzzy PID supervised online
ANFIS controller is the best controller than the other considered controllers.

20
Fig 13 (a). Speed response of brushless dc motor with load variation from no load (0 Nm) to full
load (25 Nm)

Fig 13 (b). Speed response of brushless dc motor with load variation from full load (25 Nm) to
no load (0 Nm)

Table. VI control system parameters for varying load condition

Controllers Load Control system parameters

21
 conditions Steady
Peak Peak
Peak time Recovery State % Steady
value Overshoot
(sec) Time(sec) Error State error
(rpm) (%)
(rpm)
Case A 0.4417 1519.5 1.3025 0.46 10 0.666
AW PID
Case B 0.5313 1533.5 2.2304 0.6 21.5 1.433
Case A 0.4010 1516.5 1.1255 0.44 10.5 0.7
Fuzzy PID
Case B 0.4517 1521.9 1.4587 0.445 9.5 0.633
Offline Case A 0.4002 1501.8 0.120 0.46 2.8 0.186
ANFIS Case B 0.4028 1507.2 0.4821 0.45 3.5 0.233
PID Case A 0.7948 1500 0 0.44 5.5 0.366
+
Case B 0.7752 1503.5 0.2363 0.48 2 0.133
ANFIS
Online Case A - - - 0.43 6.5 0.433
RLS-BP-
Case B 0.430 1503 0.2 0.45 2.5 0.166
ANFIS
Fuzzy AW Case A - - - 0.42 1.5 0.1
PID Case B 0.404 1502 0.1333 0.42 1 0.066
Fuzzy PID Case A 0.4001 1500 0 0.41 1 0.066
+
Case B 0.7939 1500.2 0.0144 0.405 0.16 0.0106
ANFIS

5.3. Result for Varying Set Speed Condition

In process industries, the set speed of the drive will be changed as per the requirement of
processes. In order to validate the effectiveness of the proposed controllers under varying set
speed conditions, two varying set speed operating conditions are assumed and simulated. In case
A, set speed is varied from 1500 rpm to 1000 rpm and in case B, set speed is varied from 1000
rpm to 1500 rpm. For both cases, the load is set at no load. Figure 14 (a) shows the speed
response for case A and figure 14 (b) shows the speed response for case B.

Fig 14 (a). Speed response of brushless dc motor with set speed variation from 1500 rpm to
1000 rpm

22
Fig 14 (b). Speed response of brushless dc motor with set speed variation from 1000 rpm to 1500
rpm

The control system parameters for both cases are presented in Table VII. The system
exhibits oscillatory response for all other controllers except the proposed fuzzy online gain tuned
anti wind up PID controller and fuzzy PID supervised online ANFIS controller.

Table. VII Control system parameters for varying set speed condition
Control system parameters
Set Speed Peak Peak
Controllers Recovery Steady State % Steady
conditions undershoot Overshoot
Time(sec) Error (rpm) State error
(%) (%)
Case A 0.1 - 0.495 15.5 1.55
AW PID
Case B - 2.133 0.48 9.5 0.633
Case A 0.35 - 0.47 10.3 1.03
Fuzzy PID
Case B - 2.033 0.475 10.8 0.72
Offline Case A 0.6 - 0.47 6 0.6
ANFIS Case B - 1.58 0.465 4.5 0.3
PID Case A 1.6 - 0.48 20 2
+
Case B - 1.23 0.46 1.5 0.1
ANFIS
Online RLS- Case A 0 - 0.43 4 0.4
BP-ANFIS Case B - 1.58 0.45 3 0.133
Fuzzy AW Case A 0.1 - 0.44 2.5 0.25
PID Case B - 0.33 0.425 3 0.100
Fuzzy PID Case A 0 - 0.415 1 0.1
+
Case B - 0.32 0.421 0.2 0.013
ANFIS

23
Also, these controllers have produced less steady state error than the other controllers. Steady
state error is 0.25% for case A and 0.1% for case B of fuzzy online gain tuned anti wind up PID
controller and it is 0.1% for case A and 0.013% for case B of fuzzy PID supervised online
ANFIS controller. The proposed controllers have outperformed the other considered controllers.
In the controllers proposed for brushless dc motor, fuzzy PID supervised online ANFIS
controller has shown superior performance under all operating conditions. Fuzzy online gain
tuned anti wind up PID controller is the second best controller in all aspects.

6. Comments on Results and Discussion

All considered controllers are able to track the set speed correctly. Performance under
sudden load variations is not satisfactory for anti wind up PID, fuzzy PID and offline ANFIS
controllers. Again, anti wind up PID, fuzzy PID and offline ANFIS controllers are not able to
provide complete damping as oscillations are witnessed in the speed response curves. Remaining
controllers are able to damp out the oscillation in speed response. Overall performance attributes
are provided in Table VIII for comparing the performance of all controllers at ease. From the
summary of results it can be concluded that, wind up phenomenon elimination and complete
reduction of steady state error are the special features of the two proposed controllers. Almost all
controllers have shown improvement in settling time. Reduction of peak overshoot and
undershoot is the additional advantage claimed by fuzzy PID supervised online ANFIS
controller. Considering the control capabilities, it is very easy to single out the proposed fuzzy
PID supervised online ANFIS controller as the most versatile controller. The other proposed
controller, i.e., fuzzy online gain tuned anti wind up PID controller is the next best controller.

Table VIII. Comparative performance analysis of considered controllers

Controllers
Online Fuzzy PID
Sl. PID Fuzzy AW
Control Objectives AW Fuzzy Offline RLS- +
No + PID
PID PID ANFIS BP ANFIS
ANFIS (proposed)
ANFIS (proposed)
Set speed tracking
1 ● ● ● ● ● ● ●
ability
Sudden Load
2 disturbance ● ● ● ●
rejection
Damping of
3 ● ● ● ●
Oscillations
Wind up
4 phenomenon ● ●
elimination
Steady State Error
5 ● ●
minimization
Reduction of
6 Overshoot/ ●
Undershoot

24
 Improved Settling
7 ● ● ● ● ● ● ●
Time
Comments on overall performance Better Best

Considering the two proposed controllers alone for comparison, for constant and varying
load conditions, fuzzy PID supervised online ANFIS controller performs better than the fuzzy
online gain tuned anti wind up PID controller. It is evident from the values obtained and shown
in Tables IV, V and VI for vital parameters such as overshoot, settling time and steady state
error. The control system parameters under varying speed conditions which are shown in Table
VII are also in favor of fuzzy supervised online ANFIS controller. The important positive point
about the two proposed controllers is that the wind up phenomenon is effectively rejected by
them.

7. Conclusion
Two effective controllers have been presented for the speed control of brushless dc
motor. The control system parameters are obtained for the proposed controllers and compared
with already published modern controllers such as anti wind up PID controller, fuzzy PID
controller, offline ANFIS controller, PID supervised ANFIS controller and On-line Recursive
least square – Error back propagation algorithm based ANFIS controller. In order to test the
effectiveness of the proposed controllers under realistic operating environment, various operating
conditions such as constant load, varying load and varying set speed conditions are considered
and the performances are observed. From the parameters considered for comparison, it has been
ascertained that, the fuzzy PID supervised online ANFIS controller clearly outperforms the other
controllers under all considered operating conditions of the brushless dc motor and it is the best
of all. The other proposed controller, Fuzzy online gain tuned anti wind up PID controller is
considered to be second best controller. Since the two controllers have been rigorously tested
under varying operating conditions, it can be readily implemented for speed control of brushless
motor.
The efficiency of proposed controllers relies on fuzzy membership function selection,
fuzzy rules and input and output scaling factor of the controller. This might be the limitation of
the proposed controllers. Certain optimization algorithms may be applied for fuzzy membership
function selection, tuning of fuzzy rule and input-output scaling factor tuning to achieve effective
results under different circumstances and this may be reserved as scope for future work.

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27
Research Highlights

1. Fuzzy online gain tuned anti wind up PID and Fuzzy PID supervised online ANFIS based

speed controllers are proposed for Brushless DC Motor.

2. Simulation has been performed and analyzed for varying speed and load conditions.

3. The proposed controller effectively eliminates the problem of wind up phenomenon and

also improves the control system performance in all operating conditions of the brushless

dc motor.

Graphical Abstract

28
 Fuzzy PID Supervised Online ANFIS Controller

Fuzzy Online Gain Tuned Anti Wind up PID Controller

29
 Author Biography

K.Premkumar received the B.E. degree from Anna University C Chennai,

hennai, Tamilnadu, India,
in 2005, and M.E. Degree from Anna University Chennai, Tamilnadu, India, in 2007, all in
faculty of electrical and electronics engineering. At present
present, he e is doing Ph.D. in Anna
University Chennai, Tamilnadu, India. Also he is working as Assistant Professor in the
Department of Electrical and Electronics Engineering of Pa Pandian
dian Saraswathi Yadav
Engineering College, Sivagangai, Tamilnadu, India. His current research interests include
design of speed and current control
controller is based on PID controller, fuzzy logic controller, ANFIS controller
and CANFIS controller for the special electrical machines.

B.V.Manikandan obtained his B.E. degree in Electrical and Electronics Engineering during
1990 and M.E. degree in Power SystemSystemss Engineering during 1992 from Madurai Kamaraj
University. He obtained his Ph.D. degree from Anna University, Chennai in the year 2010.
His special fields of interest include power system restructuring issues, FACTS controllers,
Special machines and Drives & Controls. Presently, he is working as Professor in the
Electrical and Electronics Engineering department of Mepco Schlenk Engineering College,
Sivakasi, and Tamilnadu, India.

30
 Table. 1 Rule base of fuzzy online gain tuner.

Rate of change of error (∆e) Rate of change of error (∆e)

Kp NB NS Z PS PB Ki NB NS Z PS PB
Speed NB VVB VVB VVB VB B Speed NB VVB VB B B B
Error NS VVB VVB VB VB B Error NS VVB VB VB B M
(e) Z VB VB B PS PS (e) Z B M S S S
PS B M M M M PS B B M S S
PB M M S S S PB M S S S S
Rate of change of error (∆e) Rate of change of error (∆e)
Kd NB NS Z PS PB Kc NB NS Z PS PB
Speed NB S S M M VB Speed NB VVB VB B B B
Error NS S S S S M Error NS VVB VB VB B M
(e) Z S S S M VB (e) Z B M S S S
PS M M M VB S PS B B M S S
PB S S S M VB PB M S S B S

Table 2 Rule base for fuzzy PD controller.

Rate of Change of Error (∆e)

U A1 B1 C1 D1 E1 F1 G1
A P P S S S S S Cluster 1
Speed Error (e)

B P P S S S S S Cluster 2
C P P S S S S S Cluster 3
D N N S S S S S
E N N S S S S S
F N N N N P P P
G N N N P P P P

31
 Table. 3 Specifications of BLDC Motor drive.
Specifications Value
Rated Voltage (Volts) 470
Rated Current (Amps) 50
Rated Speed (rpm) 1500
Stator phase resistance, R (ohm) 3
Stator phase inductance, L (H) 0.001
Flux linkage established by magnets, λ (V-sec) 0.175
Voltage Constant, Kb (V/rpm) 0.1466
Torque Constant, Kt (N-m / A) 1.4
Moment of Inertia, J (kg-m2/rad) 0.0008
Friction factor, B (N-m/(rad/sec)) 0.001
Pole pairs, P 4

Table. 4 Control system parameters for no load condition

Control system parameters
Steady
Peak Peak % Steady
Controllers Rise time Peak time Settling State
value Overshoot State
(sec) (sec) Time(sec) Error
(rpm) (%) error
(rpm)
AW PID 0.0314 0.0436 1537.5 2.5008 0.0446 17 1.133
Fuzzy PID 0.0300 0.0415 1531.7 2.1150 0.0420 10.5 0.7
Offline
0.0300 0.0411 1523.7 1.5789 0.0384 4 0.266
ANFIS
PID +
0.0300 0.0410 1518.7 1.2457 0.0384 3.5 0.233
ANFIS
Online
RLS-BP- 0.0300 0.403 1511.2 0.746 0.045 2.5 0.166
ANFIS
Fuzzy
0.0300 0.0420 1503.2 0.2133 0.043 2 0.133
AW PID
Fuzzy
PID + 0.0300 0.0403 1501.7 0.1103 0.036 0.15 0.01
ANFIS

Table. 5 Control system parameters for full load condition.

Control system parameters
Controllers
Rise time Peak time Peak Peak Settling Steady % Steady
32
 (sec) (sec) value Overshoot Time(sec) State State
(rpm) (%) Error error
(rpm)
AW PID 0.0449 0.4343 1527.9 1.8633 0.0578 16 1.066
Fuzzy PID 0.0418 0.0573 1518.7 1.2490 0.0538 10.8 0.72
Offline
0.0418 .0569 1509.5 0.6351 0.0538 3 0.2
ANFIS
PID +
0.0453 0.7662 1500 0 0.0584 5.5 0.36
ANFIS
Online
RLS-BP- 0.044 - - - 0.05 8.5 0.566
ANFIS
Fuzzy
0.0300 0.0405 1500 0 0.042 2.5 0.166
AW PID
Fuzzy
PID + 0.0300 0.0403 1501.7 0.1103 0.036 1 0.066
ANFIS

Table. VI control system parameters for varying load condition

Control system parameters

Steady
Load Peak Peak
Controllers Peak time Recovery State % Steady
conditions value Overshoot
(sec) Time(sec) Error State error
(rpm) (%)
(rpm)
Case 1 0.4417 1519.5 1.3025 0.46 10 0.666
AW PID
Case 2 0.5313 1533.5 2.2304 0.6 21.5 1.433
Case 1 0.4010 1516.5 1.1255 0.44 10.5 0.7
Fuzzy PID
Case 2 0.4517 1521.9 1.4587 0.445 9.5 0.633
Offline Case 1 0.4002 1501.8 0.120 0.46 2.8 0.186
ANFIS Case 2 0.4028 1507.2 0.4821 0.45 3.5 0.233
PID Case 1 0.7948 1500 0 0.44 5.5 0.366
+
Case 2 0.7752 1503.5 0.2363 0.48 2 0.133
ANFIS
Online Case 1 - - - 0.43 6.5 0.433
RLS-BP-
Case 2 0.430 1503 0.2 0.45 2.5 0.166
ANFIS
Fuzzy AW Case 1 - - - 0.42 1.5 0.1
PID Case 2 0.404 1502 0.1333 0.42 1 0.066
Fuzzy PID Case 1 0.4001 1500 0 0.41 1 0.066
+
Case 2 0.7939 1500.2 0.0144 0.405 0.16 0.0106
ANFIS
Table. VII Control system parameters for varying set speed condition
Control system parameters
Set Speed Peak Peak
Controllers Recovery Steady State % Steady
conditions undershoot Overshoot
Time(sec) Error (rpm) State error
(%) (%)
AW PID Case 1 0.1 - 0.495 15.5 1.55

33
 Case 2 - 2.133 0.48 9.5 0.633
Case 1 0.35 - 0.47 10.3 1.03
Fuzzy PID
Case 2 - 2.033 0.475 10.8 0.72
Offline Case 1 0.6 - 0.47 6 0.6
ANFIS Case 2 - 1.58 0.465 4.5 0.3
PID Case 1 1.6 - 0.48 20 2
+
Case 2 - 1.23 0.46 1.5 0.1
ANFIS
Online RLS- Case 1 0 - 0.43 4 0.4
BP-ANFIS Case 2 - 1.58 0.45 3 0.133
Fuzzy AW Case 1 0.1 - 0.44 2.5 0.25
PID Case 2 - 0.33 0.425 3 0.100
Fuzzy PID Case 1 0 - 0.415 1 0.1
+
Case 2 - 0.32 0.421 0.2 0.013
ANFIS

Table VIII. Comparative performance analysis of considered controllers

Controllers
Online Fuzzy PID
Sl. PID Fuzzy AW
Control Objectives AW Fuzzy Offline RLS- +
No + PID
PID PID ANFIS BP ANFIS
ANFIS (proposed)
ANFIS (proposed)
Set speed tracking
1 ● ● ● ● ● ● ●
ability
Sudden Load
2 disturbance ● ● ● ●
rejection
Damping of
3 ● ● ● ●
Oscillations
Wind up
4 phenomenon ● ●
elimination
Steady State Error
5 ● ●
minimization
Reduction of
6 Overshoot/ ●
Undershoot
Improved Settling
7 ● ● ● ● ● ● ●
Time
Comments on overall performance Better Best

34