Quenching and Suppression of Limit Cycles in 3x3 Nonlinear Systems
KARTIK CHANDRA PATRA1*, ASUTOSH PATNAIK2
1,2Department of Electrical Engineering
C. V. Raman Global University
Bhubaneswar, Odisha, PIN - 752054
INDIA
*Corresponding Author
*ORCID ID: 0000-0002-4693-4883
Abstract: - For several decades, the importance and weight-age of prediction of nonlinear self-sustained
oscillations or Limit Cycles (LC) and their quenching by signal stabilization have been discussed which is
confined to Single Input and Single Output (SISO) system. However, for the last five to six decades, the analysis
of 2x2 Multi Input and Multi Output (MIMO) Nonlinear Systems gained importance in which a lot of literature
available. In recent days few literatures are available which addresses the exhibition of LC and their
quenching/suppression in 3x3 MIMO Nonlinear systems. Poor performances in many cases like Load Frequency
Control (LFC) in multi area power system, speed and position control in robotics, automation industry and other
occasions have been observed which draws attention of Researchers. The complexity involved, in implicit non-
memory type and memory type nonlinearities, it is extremely difficult to formulate the problem in particular for
3x3 systems. Under this circumstance, the harmonic linearization/ harmonic balance reduces the complexity
considerably. Still the analytical expressions are so complex which loses the insight into the problem particularly
for memory type nonlinearity in 3x3 system. Hence in the present work a novel graphical method has been
developed for prediction of limit cycling oscillations in a 3x3 nonlinear system. The quenching of such LC using
signal stabilization technique using deterministic (Sinusoidal) and random (Gaussian) signals has been explored.
Suppression LC using pole placement technique through arbitrary selection and optimal selection of feedback
Gain Matrix K with complete state controllability condition and Riccati Equation respectively. The method is
made further simpler assuming a 3x3 system exhibits the LC predominantly at a single frequency, which
facilitates clear insight into the problem and its solution.
The proposed techniques are well illustrated with example and validated/substantiated by digital simulation (a
developed program using MATLAB codes) and use of SIMULINK Tool Box of MATLAB software.
The Signal stabilization with Random (Gaussian) Signals and Suppression LC with optimal selection of state
feedback matrix K using Riccati Equation for 3x3 nonlinear systems have never been discussed elsewhere and
hence it claims originality and novelty.
The present work has the brighter future scope of:
i. Adapting the Techniques like Signal Stabilization and Suppression LC for 3x3 or higher dimensional
nonlinear systems through an exhaustive analysis.
ii. Analytical/Mathematical method may also be developed for signal stabilization using both deterministic
and random signals based on Dual Input Describing function (DIDF) and Random Input Describing Function
(RIDF) respectively.
iii. The phenomena of Synchronization and De-synchronization can be observed/identified analytically
using Incremental Input Describing Function (IDF), which can also be validated by digital simulations.
Key-Words: - Limit Cycles, Describing function, 3x3 non-linear systems, Pole placement technique, Suppression
limit cycle, signal stabilization.
Received: March 11, 2024. Revised: August 28, 2024. Accepted: September 21, 2024. Published: November 19, 2024.
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1 Introduction
It has been observed for years long the importance
and weightage on self-sustained oscillations or
nonlinear oscillations or limit cycles (LC) [1], [2],
[3], [4], [5].
For last several decades, the analysis of 2x2
multivariable nonlinear systems drawn attention of
the researchers and good number of literature is
available [6], [7], [8], [9], [10], [11], [12], [13], [14],
[15], [16], [17], [18], [19], [20], [21], [22], [23], [24],
[25], [26], [27], [28], [29], [30], [31], [32], [33], [33],
[34], [35], [36], [37], [38], [39], [40], [41], [42], [43],
[44], [45], [46], cover this area of research. The
prediction of LC in 2 x 2 system, in means of
increasing the reliability of the describing function
(DF) are well established [4], [5], [10], [13], [16],
[23], [47], [48] and others used harmonic
linearization/ harmonic balance, [13], [27], [31],
[49].
In the event of existence of limit cycling
oscillations, the possibility, of quenching the
sustained oscillations using the method of signal
stabilization has been investigated, [5], [28], [29],
[47], [48], in 2X2 nonlinear systems with non-
memory type nonlinear elements and in memory type
nonlinear elements in [37] using deterministic signals
whereas the same has been addressed with Gaussian
signals [52].
Prediction and suppression of limit cycling
oscillations in 2 x 2 memory type nonlinear systems
using arbitrary pole placement has been discussed in
[30], [41], [42], [50].
The present work follows the dynamics of general
3X3 nonlinear systems shown in Fig. 2, Fig.3 [6],
which is an equivalent representation of the general
multivariable system of Fig.1[26].
Having realized the importance of
quenching/suppression of limit cycle oscillations the
present work first establishes the exhibitions of limit
cycles in 3X3 nonlinear systems following the
similar procedure as depicted/illustrated [6].
2. Graphical Method of prediction of
LC in a general 3x3 Nonlinear Systems
In order to avoid complexity, involved in this
structure a graphical method is developed for
prediction of limit cycles in 3x3 nonlinear systems
[6], [53].
2.1 Graphical Method
Consider a system of Fig.1, a class of 3x3
nonlinear systems for simplicity it is assumed that the
whole 3x3 system exhibits the LC predominantly of
a single frequency sinusoid and harmonic
linearization/harmonic balance leading to use of
describing function methods have been opted.
The normalized phase diagrams [44] are drawn
for 3x3 systems with three combinations such as:
Combination 1: For subsystems S1, S2 & S3: C1
(+ve), C2 (-ve) and C3 (+ve)
Combination 2: For subsystems S3, S2 & S1: C2
(+ve), C3 (-ve) and C1 (+ve).
Combination 3: For subsystems S1, S3 & S2: C3
(+ve), C1 (-ve) and C2 (+ve).
Example: Used for illustration of procedures of
Normalized phase diagrams.
The linear elements are represented by
; ; =
and Nonlinear elements are taken, Ideal
relays as shown in Fig.2.
Fig. 1: A class of 3x3 multivariable nonlinear systems
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Fig. 2: All Ideal Relays
Assuming harmonic linearization these
nonlinear elements can be equivalently
represented by their describing functions which
are real functions in these two examples and do
not contribute any phase angles to the system.
Hence the phase angles of the system are due to
linear functions, G1(s), G2(s), G3(s) which are
complex functions of complex variable s, the
Laplace operator. It may be noted that for
frequency response, input is sinusoidal and
outputs are steady state values considered, so
that s (Laplace Operator) is replaced by j
[6].
X1, X2 & X3 are the amplitudes of respective
sinusoidal inputs to the nonlinear elements. C1,
C2 & C3 are the amplitudes of sinusoidal output
of subsystems S1, S2 & S3 respectively. G1, G2 &
G3 are the magnitudes/absolute values of linear
elements represented by their transfer functions
of subsystems S1, S2 & S3 respectively and N1,
N2 & N3 are the magnitudes/absolute values of
linear elements represented by their describing
functions of subsystems S1, S2 & S3 respectively.
θL1 = Arg. ( (jω)) = -90 - (ω):
θL2 = Arg. ( (jω)) = -90 - ( ):
θL3 = Arg. ( (jω)) = -90 - ( ):
N2=(11-3 ) ±
…..(1),
N1= N2+ (2),
= (3),
With reference to Fig. 3(a),
= = ….(4),
For a fixed value of ω the combinations of
subsystems 1, 2, and 3, Normalised Phase
Diagrams are shown in Figure 3(a), (b), and (c)
respectively. However, any one of these
combinations can be used for the determination
of limit cycling conditions and the related
quantities of interest.
FIG.3 (a): Normalised phase diagram with C1, C2 & C3 for
the combination 1, where C1 (+ve), C2 (-ve) and C3 (+ve).
FIG.3 (b): Normalised phase diagram with C1, C2 & C3 for
the combination 2, where C2 (+ve), C3 (-ve) and C1 (+ve).
FIG.3 (c): Normalised phase diagram with C1, C2 & C3 for
the combination 3, C3 (+ve), C1 (-ve) and C2 (+ve).
With reference to a normalized phase diagram [44],
the phase representing X2 would lie along a straight
line drawn at an angle θL2 with the phase C2 (C2 = -
R1). The intersections of this straight line with the
circle drawn with respect to θL1 would represent
possible self-oscillations. The concept has been
extended for 3 x 3 as:
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(i) Consider Fig. 3(a) the phase representing X2 and
X3 would lie along straight lines drawn at angles θL2
and ϴL3 with the phase C2 (C2 = - R1) and C3 (C3=R1)
respectively. The intersections of these straight lines
with the circle drawn with respect to θL1 would
represent possible self-oscillations.
(ii) Consider Fig. 3(b), the phase representing X3 and
X1 would lie along straight lines drawn at angles θL3
and θL1 with the phase C3 (C3= - R2) and C1 (C1=R2)
respectively. The intersections of these straight lines
with the circle drawn with respect to θL2 would
represent possible self-oscillations.
(iii) Consider Fig. 3(c) the phase X1 and X2 would lie
along straight lines drawn at angles θL1 and θL2 with
phase C1(C1=-R3) and C2(C2 = R3) respectively. The
intersections of these straight lines with the circle
drawn with respect to θL3 would represent possible
self-oscillations.
Table 1: Shows the θL1, θL2, θL3, r (radius), and the
intersection points of the straight lines and circle for
combination 1 corresponding to the example. It may
be noted that Table 1: Contains obtained from
Eqn.3 and Eqn.4 are matched at a limit cycling
frequency.
2.2 Digital Simulation
The Example is revisited:
A program has been developed [6] with the use of
MATLAB code for digital simulation.
The equivalent canonical form of Fig. 1 for the
example is shown in Fig. 4(a) and digital
representation is shown in Fig. 4(b) respectively.
Numerical results obtained from different
methods are compared in Table 2 for the example.
The results/images for the example obtained from
digital simulation (using the developed program) and
that of obtained using SIMULINK Toolbox of
MATLAB software are shown in Fig. 5 for
comparison.
Fig. 4(a): Equivalent Canonical form of Fig.1 for the
Example
Fig. 4(b): The Digital representation of Fig.1 for the
Example
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Table 1: Shows the θL1, θL2, θL3, r (radius), and the intersection points of the straight lines and circles for
combination 1 corresponding to the Example (with reference to Fig. 3(a)).
θL1
θL2
θL3
r
X1/X
2
from
eqn.
3
X1/X2
from eqn. 4
Normalized Phase
Diagrams
Remark
0.600
-151.93
-98.531
-106.7
-0.55257
-
-
No intersection
of straight lines
and circle
0.650
-156.05
-99.23
-108
0.58256
-
-
No intersection
of straight lines
and circle
0.700
-159.98
-99.926
-109.29
-2.128
-
-
No intersection
of straight lines
and circle
0.701
-160.06
-99.94
-109.32
-3.1323
1.0
1.02
(matched
)
The intersection
of st. lines &
circle found:
Confirms the
occurrence of
limit cycles
=0.701,
C1 = OD2 = 6
C2 = 1
C3 = 1
X1=BD2=6.08
X2=AD2=6.08
X3=B’D2=
6.32
0.750
-163.74
-100.62
-110.56
-1.3583
-
-
No intersection
of straight lines
and circle
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Fig. 5: Results/Images from digital simulation and
SIMULINK for C1, C2, C3, X1, X2 and X3 of the Example
(relay type nonlinearities).
Table 2 Results obtained using different methods
corresponding to Ideal Relay Example
Sl.
No
Methods
C1
C2
C3
X1
X2
X3
1
Graphical
6.0
1.0
1.0
6.08
6.08
6.32
0.701
2
Digital
Simulation
(developed
program)
4.83
0.74
0.95
4.72
4.91
5.23
0.70
3
Using
SIMULINK
TOOL BOX
OF MATLAB
5.95
1.01
0.96
4.84
5.12
5.62
0.70
3. Signal Stabilization in 3x3
Nonlinear System
The System exhibits limit cycles (LC) in the
autonomous state, the possibility of quenching
the LC by injecting a suitable high frequency
signal, preferably, at least 10 times of the limit
cycling frequency.
3.1 Using Deterministic Signal
The forced oscillations can be realised by
feeding deterministic or random signals of high
frequency, at least greater than 10 times the limit
cycling frequency at any one /all input points of the
subsystems S1, S2, S3.
If the amplitude B of the high frequency signal
is gradually increased, the system would exhibit
complex oscillations before the synchronization
takes place. On the reverse operation, if the
amplitude B is gradually reduced at certain value of
B the self-oscillations i.e. the Limit cycle would
reappear and the system would exhibit complex
oscillations again which can be called the de-
synchronisation. The phenomena of synchronization
and de-synchronization can be observed / identified
analytically using Incremental Input Describing
function (IDF) [44].
However, the forced oscillation can also be
analysed using the Equivalent Gain/Dual input
Describing Function (DIDF) [44] in case of a
deterministic forcing signal in particular with a
sinusoidal signal. Taking the second option i.e. all
three inputs are same as B sinft at 3 input points U1,
U2, & U3, shown in Fig.6. Amplitude B is gradually
increased, the frequency of self-
oscillation, s would gradually change, the system
will synchronize to forcing frequency i.e. the self-
oscillation would be quenched and the system would
exhibit forced oscillations at frequency f.
Fig 6: Equivalent System of Fig. 1 for forced oscillations
(Signal Stabilization) with deterministic signal for the
Example.
The results/images from digital simulation for signal
stabilization with deterministic (sinusoidal signal)
for the Example shown in Fig.7.
Fig. 7: Forced Oscillations by Signal Stabilization with
deterministic signal for the Example Forcing Signal U =
5sinf t (f = 7.5 rad / sec)
30 40 50 60
-6
-4
-2
0
2
4
6
8
C1 = 5.95, w = 0.69
Result / Image from Simulink Application
Result / Image from Digital Simulation
C1 = 4.51, w = 0.70
(a)
30 40 50 60
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
(c)
Result / Image from Digital Simulation
C2 = 0.74, w = 0.70
Result / Image from Simulink Application
C2 = 1.01, w = 0.69
30 40 50 60
-1.0
-0.5
0.0
0.5
1.0
(e)
Result / Image from Digital Simulation
C3 = 0.94, w = 0.70
Result / Image from Simulink Application
C3 = 0.95, w = 0.69
30 40 50 60
-6
-4
-2
0
2
4
6
(b)
Result / Image from Digital Simulation
X1 = 4.84, w = 0.70
Result / Image from Simulink Application
X1 = 4.72, w = 0.70
30 40 50 60
-6
-4
-2
0
2
4
6
(d)
Result / Image from Digital Simulation
X2 = 5.12, w = 0.70
Result / Image from Simulink Application
X2 = 4.91, w = 0.70
30 40 50 60
-6
-4
-2
0
2
4
6
(f)
Result / Image from Digital Simulation
X3 = 5.24, w = 0.70
Result / Image from Simulink Application
X3 = 5.62, w = 0.70
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3.2 Using Gaussian signal:
The forced oscillation is analysed with Equivalent
Gain (similar to DIDF - Random Input Describing
Function) (RIDF) in case of the Random Signals in
particular with Gaussian Signals [52].
Consider the Example. The system is exhibiting
LC under autonomous state, a Gaussian signal with
specified mean and variance is injected at U1, U2 &
U3 of subsystems for stabilizing the system /
quenching the self-sustained oscillations. At a
suitable value of mean () and variance (), the self-
sustained oscillations are vanished/the system is
synchronised to high frequency forcing input.
The results/images are shown in Fig. 8, which is
obtained from digital simulation by signal
stabilization with Gaussian signals for the Example
replacing B sin ft with a suitable random signals in
Fig.7.
Fig. 8: Forced Oscillations by signal stabilization with
Gaussian Signal of mean 50 and variance 0.05 for the
Example
4. Suppression of limit cycle in 3x3
nonlinear system using pole placement
technique
The System of the Example exhibits Limit Cycles
which can be suppressed by pole placement
technique [46]. The closed loop poles or Eigen values
of the closed loop systems can be placed at the
desired location through state feedback using an
appropriate feedback gain matrix K [k1, k2, k3].
Necessary and sufficient condition for arbitrary pole
placement is that the system be completely state
controllable [46]. This can also be done by optimal
selection of feedback gain matrix K using Riccati
Equation [46].
4.1 Suppression of Limit Cycles in 3x3
Nonlinear system using arbitrary Pole
Placement by state feedback:
Pole placement technique by state feedback is
done by determining the Eigen values or poles of the
system. These Eigen values cause the limit cycles in
the system, and as the complete removal of these self-
oscillations may not be possible, the location of the
poles must be changed from its original position so
as to bring about suppression of the limit cycle. The
most general multivariable nonlinear system [53] is
shown in Fig. 9 (a). For existence of limit cycles, an
autonomous system (input U=0) Fig. 9(a) can be
represented in simplified form as shown in Fig. 9(b).
Making use of the first harmonic linearization of the
nonlinear elements, the matrix equation for the
system of Fig. 9(b) can be expressed as
X = -HC, where C = GN(x) X. Hence,
X = -HGN(x) = AX (5)
Where, A = -HGN(x)
Fig. 9(a): Block diagram representation of a most general
nonlinear multivariable system
Fig. 9(b): Equivalent of the system of Fig. 9 (a) with input
U= 0
Realizing Eqn. (5) as a transformation of the vector
X onto itself, it is noted that for a limit cycle to exist
the following two conditions should be satisfied, [6],
[53]:
(i) For every non-trivial solution of X, the matrix
A must has an Eigen value λ equal to unity, and
(ii) The Eigen vector of “A” corresponding to this
unity Eigen value must be coincident with X.
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4.1.1: Arbitrary Pole Placement for
suppression of limit cycles in the Example
with all ideal relays
In order to suppress the limit cycles, arbitrary pole
placements may be possible if the system is
completely state controllable [46].
The controllability matrix
(6)
Where,
;
;
From Table 1 for the Example,
X1= 6.08, X2= 6.08, X3 =6.32
N1(X1) = = = 0.419; N2(X2) = =
= 0.314, N3(X3) = = = 0.202
= = 1.913
= = = 0.351
= 0.673
On substitution of the numerical values:
= -0.419 X 1.913 = - 0.802,
= - 0.314 X 0.351 = -0.110,
= -0.202 X 0.673 = -0.136
; AB =
= ;
B = =
Hence S = = 0.0215≠0 (The
system is completely state controllable)
Hence arbitrary pole placement is possible [46]
= (7)
The system under autonomous state is represented as
shown in Fig. 10.
.
Fig. 10: A system with state feedback
Consider Fig. 10:
The control law u = -KX (8)
Where K= [ ] is the feedback matrix.
Replacing K in Eqn. (7) by Eqn. (8), we get,
= (A-BK) X (9)
Substituting the values of A, B and K, we get: The
Characteristic Equation as
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Hence
=
=
= + ( + + + )+ (2 +2
+2 + + +
+ )+(4 +
+=0 (10) (Ch.
Equation)
On substitution of the values of , , , and
, in Eqn. (10), we get,
+ (0.136+0.11+0.802+ +
0.177+0.218+0.030+ + +
}+(0.048+0.522+ =0 Or
+ (1.048+ + 0.136
(0.57 x0.03)=0
(11)
If the poles are selected arbitrarily at
respectively, the
characteristic equation becomes:
( ( ( = +4 +5 +2=0
(12)
Comparing Eq. (12) with Eq. (11), and equating the
coefficients of like powers of we get:
4 = 1.048 (13)
2 = (0.57 x 0.03), whence … (14)
5 = (0.425 x 0.136 + x 0.136 x 0.91)
5= (0.425 0.136 + x 0.136
x 0.91) or
5=(0.425 + x 0.136 ), whence
(15)
Hence K = =
(16)
From Eqn. (9), (A – BK) = A1, with shifted poles for
Example 1. Or
A1 = =
(17)
The images/response in the
autonomous state obtained from digital simulation
for A1 of Example, are shown in Fig.11.
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Fig. 11: Suppression of Limit Cycles by State Feedback
with arbitrarily selection of feedback gain matrix for the
Example.
4.1.2 Optimal Selection of Feedback gain
Matrix using Riccati Equation for Example 1
The Riccati Equation is A′P+PA- B′P+Q=0
(18)
And K = Feedback gain matrix = B′P (19)
Assuming R = 1, B= , Q =
Let P= , considering P to be
symmetric matrix: = ,
Hence P =
A′ P =
=
(20)
PA=
=
(21)
B′P= ,
=
=
=
=
(22)
On substitution of numerical values, Eqn. 20 can be
written as
---- (23)
On substitution of numerical values, Eqn. 21 can be
written as
---- (24)
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On substitution of these values of Eqns. (22), (23),
(24) and the assumed value of Q in Riccati Eqn. 18
yields:
(-1.604 -1.604 )+1-p213 = 0
(25)
(-0.912 +0.802 -0.802 -0.11 +0.11 -
p13 p23=0 (26)
(-0.938 +0.802 p23-0.802 p33+0.136 p11-0.136 p12-
p13 p33=0 (27)
(-0.22 p12-0.22 p22+0.22 p23- p223 = 0
(28)
(-0.11 p13-0.246 p23+0.11 p33+0.136 p12-0.136 p22- p23
p33=0 (29)
(0.272 p13-0.272 p23-0.272 p33- p33 p23) = 0
(30)
Further, subtracting Eqn. (29) from Eqn. (30), we get,
0.382 p13 – 0.026 p23 – 0.382 p33 – 0.136 + 0.136
p22 (31)
The solution of these simultaneous Eqns.
(26),(27),(28),(29),(30) & (31) yields :
= -116.68, = -110.48, =6.58, = -
93.24, p23 = -6.58, p33 = 0
From Eqn. (19), K = B′P = 1
Or =
Or =
= ,
Whence, = 6.58, = -6.58 and = 0
(32)
Hence, A – BK =
A2=
Fig. 12: Suppression of Limit Cycles by State Feedback
with optimal selection of feedback gain matrix for the
Example.
On substitution of numerical values for Example 1,
A2 becomes:
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A2=
=
(33)
The images/responses C = and = in the
autonomous state, obtained from digital simulation
for the Example, are shown in Fig. 12.
5 Conclusion
In today’s scenario, nonlinear self-sustained
oscillations or Limit Cycles are the basic feature of
instability. The existence /exhibition of such
phenomena limit the performance of most of the
physical systems such as the speed and position
control in robotics, automation industry in particular.
Quenching/Complete extinction of such LC has been
a severe headache among the researchers for several
decades. There are some methods, seen in the
available literature which suggests the solution to this
problem occurring in SISO or 2x2 systems. However,
our present work explores the solution for 3x3
systems in the event of existence of LC problem and
establishes the result graphically & validated by
digital simulation. The novelty of the work claims in:
(i) Quenching of LC exhibited in nonlinear systems
by Signal Stabilization with deterministic as well as
random (Gaussian) signals, (ii) Suppression of limit
cycles in 3x3 nonlinear systems by Pole Placement
using State feedback with arbitrary selection as well
as optimal selection of feedback gain matrix K.
More importantly the poles of such 3x3 systems
are shifted or placed suitably by State feedback so
that the system do not exhibit limit cycles. This pole
placement is done either by arbitrary selection
satisfying the complete state controllability condition
or by optimal selection of feedback gain matrix K
using Riccati equation which has not been attempted
elsewhere.
The present work has the brighter future scope of
adopting the techniques like signal stabilization [44]
and suppression of limit cycles [46] in the event of
the existence of limit cycling oscillations for 3x3
higher dimensional systems through an exhaustive
analysis.
Analytical/Mathematical procedures may also be
developed for signal stabilization using both
deterministic and random signals applying DIDF and
RIDF respectively.
Backlash is one of the nonlinearities commonly
occurring in physical systems which are an inherent
characteristic of Governor, more popularly used for
load frequency control (LFC) in power systems. The
LFC shows poor performance due to the backlash
characteristic of the governor. Similarly, the backlash
characteristic limits the performance of speed and
position control in the robotics, automation industry.
The poor performance of LFC, speed and position
control in robotics and in automation industries are
happening since these systems exhibit limit cycles
due to their backlash type of nonlinear
characteristics. The proposed method of suppression
of L.C. can be extended and developed for backlash
type nonlinearity in 3x3 systems and used to
completely eliminate the limit cycle to mitigate such
problems.
The phenomena of synchronization and de-
synchronization can be observed/identified
analytically using Incremental Input Describing
function (IDF).
Acknowledgement:
The Authors wish to thank the C.V Raman Global
University, Bhubaneswar – 752054, Odisha, India for
providing the computer facilities for carrying out the
research and preparation of this paper.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Kartik Chandra Patra has formulated the problem,
methodology of analysis adopted and algorithm of
computation presented.
Asutosh Patnaik has made the validation of the
results using the geometric tools and SIMULINK
toolbox of MATLAB software.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The C.V. Raman Global University has provided all
computer facilities with relevant software for the research
work and also for the preparation of the paper.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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