Describing Function Analysis | MATLAB Tutorial

Describing Function Analysis | MATLAB Tutorial

Introduction

The describing function analysis is an approximate technique for studying non-linear systems. The basic idea of the method is to approximate the non-linear systems. The basic idea is of the method is to approximate the nonlinear components in nonlinear control system by linear equivalents and then use frequency domain techniques to analyze the resulting system. The accuracy of the describing function analysis increases with the order of the system.

The describing function method of a nonlinear system is defined to be the complex ratio of amplitudes and phase angle between fundamental harmonic components of output to input sinusoid.

Mathematically,

Where,

N = describing function,

X = amplitude of input sinusoid,

Y = amplitude of fundamental harmonic component of output,

φ1 = phase shift of the fundamental harmonic component of output.

So in the describing function analysis, we assume that only the fundamental harmonic component of the output. Since the higher harmonics in the output of a non linear system are often of smaller amplitude than the amplitude of fundamental harmonic component.

 

The block diagram of a nonlinear system is as follows:

Describing Function for Saturation Non-Linearity

Follow the given steps to do describing function analysis of a non-linear system.

1.  Open Simulink Model: Go to New > Simulink Model.

2.  Now, we will be designing a relay control system for the saturation non-linearity. The components being used are: Sine Wave, Mux, Sum, Transfer Fcn, Saturation, Constant, Scope. You can search these terms by typing these names in the search box of the Simulink Library Browser.

3.  Make the following connection.

The coefficients of the first transfer function are: Numerator: [1 0] ; Denominator : [ 1 4 5]. The coefficients of the second transfer function are: Numerator: [1 3]; Denominator: [2 4 1]. These values can be changed by double clicking on the transfer function box.

4.  Simulate the connection and double click on the scope to obtain the following output:

5.  Replace the saturation nonlinearity with the dead zone nonlinearity in the above connection as follows:

6.  Simulate the above connection and double click on the scope to obtain the following output:

Conclusion

A relay control system was analyzed with the help of describing function analysis for deadzone and saturation non linearity in Simulink.


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