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Direct approximation of fractional order systems as a reduced integer/fractional-order model by genetic algorithm

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Abstract

In this paper, a new method is proposed for the reduced-order model approximation of commensurate/incommensurate fractional order (FO) systems. For integer order approximation, the model order is determined via Hankel singular values of the original system; while the order of FO approximations is determined via optimization. Unknown parameters of the reduced model are obtained by minimizing a fitness function via the genetic algorithm (GA). This fitness function is the weighted sum of differences of Integral Square Error (ISE), steady-state errors, maximum overshoots, and ISE of the magnitude of the frequency response of the FO system and the reduced-order model. Therefore, both time and frequency domain characteristics of the system considered in obtaining the reduced-order model. The stability criteria of the reduced-order systems were obtained in various cases and added to the cost function as constraints. Three fractional order systems were approximated by the proposed method and their properties were compared with famous approximation methods to show the out-performance of the proposed method.

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Abbreviations

A, B, C, D :

State space model matrices of system

\( a_{i} \), \( b_{j} \) :

Coefficients of fractional order Transfer Function (TF)

\( c_{i} \), \( d_{i} \) :

Unknown coefficients of reduced TF

\( e_{ss} \) :

Steady state error

\( E_{i} , M_{i} \) :

Parent chromosomes

K 0 :

Gain of fractional commensurate TF

M :

Number of terms in numerator of fractional TF

M :

Matrix of initial population of GA

N :

Number of terms in denominator of fractional TF

O i :

Recombined parents

Q :

Order of fractional derivative

R :

Ratio parameter in GA

R :

Order of reduced model

\( t_{f} \) :

Final time

U :

Input vector

V :

Population size in GA

\( w_{1} ,w_{2} ,w_{3} ,w_{4} \) :

Weighting coefficients

W :

Number of unknown parameters in GA

X :

State vector

Y :

Output vector

\( y_{f} \) :

Step response of fractional order system

\( y_{r} \) :

Step response of reduced order model

\( \alpha \) :

Commensurate fractional order

\( \alpha_{i} ,\beta_{j} \) :

Orders of fractional TF

\( \varGamma \) :

Gamma function

\( \lambda \) :

Eigenvalues of system

\( \nu_{i} \) :

Positive integer number

\( \omega \) :

Frequency range

\( D_{t}^{q} \) :

Fractional derivative operator

f :

Continuous function

\( G_{commensurate} \) :

Commensurate fractional order TF

\( G_{f} \) :

General fractional order TF

\( G_{incommensurate} \) :

Incommensurate fractional TF

\( G_{r} \) :

Reduced integer order TF

\( Gr_{commensurate} \) :

Reduced fractional order commensurate TF

\( Gr_{incommensurate} \) :

Reduced fractional order incommensurate TF

\( J_{\text{integer}} \) :

Constrained fitness function for reduced integer model

\( J_{\text{commensurate}} \) :

Constrained fitness function for reduced commensurate fractional model

\( J_{\text{incommensurate}} \) :

Constrained fitness function for reduced incommensurate fractional model

J’ :

Unconstrained fitness function

L:

Laplace operator

mag :

Magnitude of frequency response

N :

The set of natural number

OS:

Overshoot

\( R^{ + } \) :

Positive real numbers

Z :

The set of integer numbers

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Authors and Affiliations

  1. Control Engineering Department, Imam Khomeini International University, Qazvin, Iran

    Hasan Nasiri Soloklo & Nooshin Bigdeli

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  1. Hasan Nasiri Soloklo
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  2. Nooshin Bigdeli
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Correspondence to Hasan Nasiri Soloklo.

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Soloklo, H.N., Bigdeli, N. Direct approximation of fractional order systems as a reduced integer/fractional-order model by genetic algorithm. Sādhanā 45, 277 (2020). https://doi.org/10.1007/s12046-020-01503-1

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