All pass system

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 single Hankel singular value of multiplicity of the system's order. The system matrices are constructed based on the scheme:

$\begin{array}{rl}A& =\left(\begin{array}{cc}{a}_{1,1}& -{\alpha }_1\\ {\alpha }_1& 0& -{\alpha }_2\\ & {\alpha }_2& 0& \ddots \\ & & \ddots & \ddots & -{\alpha }_{N-1}\\ & & & {\alpha }_{N-1}& 0\end{array}\right),\\ B& =\left(\begin{array}{c}{b}_1\\ 0\\ ⋮\\ 0\end{array}\right),\\ C& =\left(\begin{array}{cccc}{s}_1{b}_1& 0& \cdots & 0\end{array}\right),\\ D& =-s_1\sigma .\end{array}$

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 $\sigma =1$, 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);

An equivalent Python code is

from scipy.sparse import diags, lil_matrix

def allpass(n):
    A = diags([-1, 0, 1], offsets=[-1, 0, 1], shape=(n, n), format='lil')
    A[0, 0] = -0.5
    A = A.tocsc()
    B = lil_matrix((n, 1))
    B[0, 0] = 1
    B = B.tocsc()
    C = -B.T
    D = 1
    return A, B, C, D

Dimensions

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

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}
}