All pass system

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Description

This procedural benchmark generates an all-pass SISO system based on . For an all-pass system, the transfer function has the property $g(s)g(-s) = \sigma^2$, $\sigma > 0$, or (equivalently) the controllability and observability Gramians are quasi inverse to each other: $W_C W_O = \sigma I$, which means this system has a singular Hankel singular value of multiplicity of the system's order. The system matrices are constructing based on the scheme:

$\begin{align} A &= \begin{pmatrix} a_{1,1} & -\alpha_1 \\ \alpha_1 & 0 & -\alpha_2 \\ & \ddots & \ddots & \ddots \\ & & \alpha_{N-1} & 0 & -\alpha_{N-1} \end{pmatrix}, \\ B &= \begin{pmatrix} b_1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \\ C &= \begin{pmatrix} s_1 b_1 & 0 & \dots & 0 \end{pmatrix}, \\ D &= -s_1 \sigma. \end{align}$

We choose $s_1 \in \{-1,1\}$, to be $s_1 \equiv -1$, as this makes the system state-space-anti-symmetric. Furthermore, $b_1 = 1$ and $\sigma_1 = 1$, which makes $a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}$.

Data

This benchmark is procedural and the state dimensions can be chosen. Use the following MATLAB code to generate a random system as described above:

function [A,B,C,D] = allpass(N)
% allpass (all-pass system)
% by Christian Himpe, 2020
% released under BSD 2-Clause License
%*

    A = gallery('tridiag',N,-1,0,1);
    A(1,1) = -0.5;
    B = sparse(1,1,1,N,1);
    C = -B';
    D = 1;
end

The function call requires one argument; the number of states $N$. The return value consists of four matrices; the system matrix $A$, the input matrix $B$, the output matrix $C$, and the feed-through matrix $D$.

[A,B,C,D] = allpass(N);

Dimensions

$\begin{array}{rcl} \dot{x}(t) &=& Ax(t) + Bu(t) \\ y(t) &=& Cx(t) + Du(t) \end{array}$

System dimensions:

$A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times 1}$, $C \in \mathbb{R}^{1 \times N}$, $D \in \mathbb{R}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, All-Pass System. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/All_pass_system

@MISC{morwiki_allpass,
  author =       {{The MORwiki Community}},
  title =        {All-Pass System},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {http://modelreduction.org/index.php/All_pass_system},
  year =         {2020}
}