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According to Matlab documentation:

sigma plots the singular values of the frequency response of a dynamic system.

bode creates a Bode plot of the frequency response of a dynamic system.

I am reasonably familiar with Bode plots and dynamic systems but I don't understand what the singular values of the system are or how they are calculated. Are they related to the magnitudes of the system (which can be outputted by bode)?

For reference, I have a discrete state-space system which I am attempting to model as a reduced order system. One of the papers I have been using to accomplish this mentions using the sigma function to compare the two models. The paper is 'Dynamic Mode Decomposition with Control' by J.L.Proctor.

Any help with understanding singular values or this sigma function would be appreciated.

Joe
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2 Answers2

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For a MIMO system $y(s) = G(s)d(s)$, with $m$ inputs and $l$ outputs. Consider a fixed frequency $\omega$ where $G(j\omega)$ is a constant $l \times m$ complex matrix. For the sake of simplicity the matrix $G(j\omega)$ is written as $G$.

In short, the singular value decomposition (SVD) states that any matrix $G$ may be decomposed into an input rotation $V$, a scaling matrix $\Sigma$ and an output rotation $U$

$$G = U \Sigma V^H$$ where $V^H$ denotes the conjugate transpose or Hermitian transpose of matrix $V$. Furthermore,

  • $U$ is an $l \times l$ unitary matrix of output singular vectors, $u_i$;
  • $V$ is an $m \times m$ unitary matrix of input singular vectors, $v_i$;
  • $\Sigma$ is an $l \times m$ diagonal matrix with $k = \min\lbrace l,m\rbrace$ non-negative singular values, $\sigma_i$.

The gain of matrix $G$ in the $i$'th direction (from input $v_i$ to output $u_i$) is given by the singular value $\sigma_i$.

Hence, for a SISO system, the singular value $\sigma$ describes the amplification from the input to the output on frequency $\omega$. This information can also be obtained from the Bode magnitude response.

That is why the MATLAB documentation of the function sigma states:

The singular value response of a SISO system is identical to its Bode magnitude response.

useless-machine
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There is a relationship between the $\mathcal{H}_{\infty}$-norm and the supremum of the singular value of the system's transfer function, I probably read about this in the book:

LMIs in Control Systems: Analysis, Design and Applications By Guang-Ren Duan, Hai-Hua Yu CRC Press

The analysis of Bode plots, especially in SISO cases, makes it easier to compare the norms between different control systems. If you're designing a control system by minimizing the $\mathcal{H}_{\infty}$ norm to attenuate disturbances, it's a good idea to check both the Bode plots of the transfer function and those of the singular values.

Marconi Oliveira de Almeida
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