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Artificial Intelligence Techniques for
Electrical Engineering Systems
AITEES-2022
Order Reduction of Continuous Time Linear Interval Systems
Using Whale Optimization Algorithm (AITEES2022-020)
G Ramesh
Assistant Professor, Department of EEE, Gudlavalleru
M. Siva Kumar
Professor, Department of EEE, Gudlavalleru
B. Dasu
Associate Professor, Department of EEE, Gudlavalleru
R. Srinivasa Rao
Professor, JNTUK, Kakinada
AITEES 2022, 6th -7th May, 2022 @ SRGEC , Gudlavalleru
International Conference
on

Contents:
• Abstract
• Introduction
• Problem statement
• Implementation of WOA
• Application of proposed method
• Comparison with other method
• Conclusion
• References
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Abstract:
 This paper presents nature-inspired meta-heuristic optimization
algorithm, called Whale Optimization Algorithm (WOA), which
mimics the social behaviour of humpback whales. The algorithm has
inspired by the bubble-net hunting strategy.
 In this paper, WOA has been used for getting Reduced Order
Interval Model (ROIM) from higher order linear continuous time
interval system.
 In this proposed method, the reduced order model denominator and
numerator polynomials are obtained based on minimization of cost
function of Integral Squared Error (ISE) by using WOA.
 Optimization results proved that the WOA algorithm is very
competitive compared to the state-of-art meta-heuristic algorithms
as well as conventional methods.
 The WOA has found to be simple, easy in implementation and
provides the optimal solution.
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Introduction:
 Scientists and engineers have often confronted with the analysis, design
and synthesis of real world problems. The first step in such studies is that
development of a 'mathematical model' which could be a substitute for the
real life problem.
 Whenever a physical system has represented by a mathematical model it
may yield a transfer function of very high order.
 The analysis of higher order system has one of the most important subjects.
 Available methods for analysis and design may become cumbersome when
applied to a system of higher order.
 At this juncture, application of order reduction methods has inevitable to
have less computational effort and process time. Since recent years much
research work has been reported in international literature.
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Introduction:
 Most of the industrial processes can be modeled as Linear Time
Invariant (LTI) systems, in spite of this fact, that their real behavior
of the processes has oftentimes different, much more complicated.
 The motivation is evident – owing to this, the transfer functions can
be used for description of such systems and subsequently also the
control theory of linear systems, which is very well-developed, can
be applied.
 However, an effort to create the simple enough model almost always
leads to the origin of uncertainty. Their emergence often consists in
neglect of “less important properties”, especially from the realms of
fast dynamic effects, nonlinearities or time-variant behaviors of the
plant.
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Introduction:
 Approximation of higher order systems to lower order models
facilitates simulation and design over the complex models.
 There has been a tremendous growth in the research area of model
order reduction methods, resulting in a development of variety of
techniques.
 Also, engineering and science designs involve uncertainty specified
in a number of ways, as convex or fuzzy descriptions to varying
degree, inclining a must study to estimate the upper and lower
bounds of the systems for the proper examination.
 So, systems with constant coefficients but uncertain within finite
range, is classified under interval systems.
 Over the years, analysis, stability and transient behavior of interval
systems have attracted the attention of researchers.
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Problem statement:
 Consider an asymptotically stable higher order interval system as:
…….(1)
 Where and are
lower and upper bounds for numerator and denominator interval
polynomial parameters respectively.
 It is required to obtain kth Reduced Order Interval Model (ROIM)
Rk( s, u, v) using the proposed reduction procedure, defined as:
…….(2)
 It is desired to reduce order of the system represented in (1) into (2).
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Problem statement:
 The Higher Order Interval System represented in equation (1) can be
represented as four fixed parameter Kharitonov’s transfer functions [10].
They are given as:
…….(3)
 After obtaining the parameters from the algorithm the Kth order and Ith
fixed parameter reduced order model is obtained as follows:
…….(4)
 This procedure has applied to all four Kharitonov transfer function and the
reduced order interval model has constructed with the coefficients of
numerator and denominator polynomials, using the following equation
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Implementation of WOA:
 Whale Optimization Algorithm (WOA) is a meta-heuristics
algorithm. The inspiration for WOA comes from social behavior and
the bubble-net hunting of humpback whales in oceans.
 The Humpback whales have a one-of-a-kind hunting mechanism
known as the bubble-net feeding method. Humpback whales prefer
to hunt school of krill or small fishes close to the surface.
 The step-wise procedure for the implementation of WOA is given
below.
 Step1: Exploration Phase (Searching Model): The search agent
(humpback whale) looks for the best solution (the prey) randomly
based on the position of each agent. Search agent position will be
updated during this phase by using a randomly selected search agent
rather than the best search agent.
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Implementation of WOA:
 Step 2: Encircling Prey: Current best candidate solution is assumed
to be closes to target prey and other solutions update their position
towards the best agent
…….(10)
…….(11)
 Step 3: Bubble-net attacking method (exploitation phase): In order
to mathematically model the bubble-net behavior of humpback
whales, two approaches are follows as:
i) Shrinking encircling mechanism:
ii) Spiral updating position:
 The mathematical model behind the humpback whale’s swimming
style around the prey using a shrinking circle and also following a
spiral-shaped path at the same time:
 ……(12)
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Application of proposed method:
 Consider an asymptotically stable Higher Order Interval System [4]
 Using the procedure given in (3), the four-fixed parameter
Kharitonov transfer functions are obtained as follows
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 The second order reduced models developed by minimizing ISE
using reduction technique, is given by
 Then the reduced order interval model can be constructed using eq.
(5) and is given by
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Fig. 2 Step Response of Lower Bound Fig. 3 Step Response of Upper Bound
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Comparison with other method :
 To show the effectiveness of the proposed method is applied to
numerical example and compared with methods [Gamma Delta [2],
Mixed Method [3] Mixed Evolutionary Method [5] presented in
literature.
i) The second order reduced interval model is obtained using Method in
[Gamma Delta [2], given by
ii) The second order reduced interval model is obtained using Method
in [Mixed Method [3], given by
iii) The second order reduced interval model is obtained using Method
in [Mixed Evolutionary Method [5], given by
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Comparison with other method :
Fig. 4: Step Response Comparison of Upper Bound Fig. 5: Step Response Comparison of Lower Bound
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Conclusion:
 In this proposed method, the reduced order interval model
denominator and numerator coefficients have been obtained by
minimization of a cost function ISE between higher order interval
system and reduced order interval model.
 It has been observed that the transient and steady state response of
reduced order interval model obtained by the proposed method are
closely matched.
 The proposed method uses a new optimization technique based on
nature inspired meta-heuristic from the social behavior and the
bubble-net hunting of humpback whales in oceans in solving
complex problems.
 This method uses single parameter in tuning which in turn reduces
computational time hence make it simple and easy in
implementation.
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References:
[1] Vijaya Anand N, Siva Kumar M and Srinivasa Rao R, A novel order reduction procedure
for linear time invariant interval systems using SGO algorithm, International Journal of
Engineering &Technology, vol.7, no. 8, (2018) 118-122.
[2]B Bandyopadhyay, Avinash Upadhye and Osman Ismail (1997), 𝛾 − 𝛿Routh
Approximation for Interval Systems, IEEE Trans. On Automatic Control, vol.42, no. 8,
pp. 1126-1130.
[3] N Selvaganesan (2007), Mixed Method of Model Reduction for Uncertain Systems,
Serbian Journal of Electrical Engineering, vol.4, no. 1, pp. 1-12.
[4] B. Bandyopadhyay, O. Ismail, and R. Gorez, „Routh Pade Approximation for Interval
Systems‟, IEEE Trans. Autom. Control, 39, 2454–2456, 1994.
[5] Devender kumar saini and Dr. Rajendra prasad (2010), Mixed evolutionary techniques to
reduce order of linear interval systems using generlized routh array, International Journal
of Engineering science and technology, vol. 2, no. 10, pp. 5197-5205.
[6] B. Bandyopadhyay, γ-δ Routh Approximations for Interval Systems‟, IEEE Trans.
Autom. Control, 42, 1127-1130, 1997.
[7] Y. Dolgin, and E. Zeheb, „On Routh Pade Model Reduction of Interval Systems‟, IEEE
Trans. Autom. Control, 48 (9), 1610–1612, 2003.
[8] O. Ismail, and B. Bandyopadhyay, „Model Order Reduction of Linear Interval Systems
Using Pade Approximation‟, IEEE International symposium on circuit and systems,
1995.
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Thank you
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