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On Pseudo-state Matrices of Fractional-Order Multi-frequency Oscillators: A Routh Array Based Oscillatory Criterion and Harmonic Preserving Transformations

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Abstract

This paper deals with investigating the properties of pseudo-state matrices in multi-order dynamic systems which generate steady-state oscillations combined of some sinusoidal components with different frequencies (multi-frequency steady state oscillations). In this investigation, a Routh array based criterion is found which should be satisfied by the pseudo-state matrix of a multi-order multi-frequency oscillatory system. In design of multi-frequency oscillators, this criterion can be rewritten as a system of monomial equations whose variables are the tunable elements of the pseudo-state matrix of the oscillator. Furthermore, a class of transformations on the pseudo-state matrix of a multi-order multi-frequency oscillator is obtained which can guarantee the preservation of the multi-frequency feature in the dynamic structure constructed based on the transformed pseudo-state matrix.

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Notes

  1. The Householder matrix \(H \in {\mathbb{R}}^{n \times n}\) constructed from the vector \(v \in {\mathbb{R}}^{n \times 1}\) is defined as \(H = I - \frac{{2vv^{T} }}{{v^{T} v}}\) [3].

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  1. Electrical Engineering Department, Sharif University of Technology, Tehran, Iran

    Mohammad Saleh Tavazoei

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Tavazoei, M.S. On Pseudo-state Matrices of Fractional-Order Multi-frequency Oscillators: A Routh Array Based Oscillatory Criterion and Harmonic Preserving Transformations. Circuits Syst Signal Process 42, 5714–5724 (2023). https://doi.org/10.1007/s00034-023-02378-3

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