All pass system: Difference between revisions

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Description

This procedural benchmark generates an all-pass SISO system based on ^[1]. For an all-pass system, the transfer function has the property $g (s) g (- s) = σ^{2}$ , $σ > 0$ , or (equivalently) the controllability and observability Gramians are quasi inverse to each other: $W_{C} W_{O} = σ I$ , 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{aligned} A & = (\begin{matrix} a_{1, 1} & - α_{1} \\ α_{1} & 0 & - α_{2} \\ α_{2} & 0 & ⋱ \\ ⋱ & ⋱ & - α_{N - 1} \\ α_{N - 1} & 0 \end{matrix}), \\ B & = (\begin{matrix} b_{1} \\ 0 \\ ⋮ \\ 0 \end{matrix}), \\ C & = (\begin{matrix} s_{1} b_{1} & 0 & \dots & 0 \end{matrix}), \\ D & = - s_{1} σ . \end{aligned}

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 $σ = 1$ , which makes $a_{1, 1} = - \frac{b_{1}^{2}}{2 σ} = - \frac{1}{2}$ .

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{aligned} \dot{x} (t) & = A x (t) + B u (t) \\ y (t) & = C x (t) + D u (t) \end{aligned}

System dimensions:

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

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 =          {https://modelreduction.org/morwiki/index.php/All_pass_system},
  year =         {2020}
}

References

↑ R.J. Ober. "Asymptotically Stable All-Pass Transfer Functions: Canonical Form, Parametrization and Realization", IFAC Proceedings Volumes, 20(5): 181--185, 1987.

[Obe87-1] R.J. Ober. "Asymptotically Stable All-Pass Transfer Functions: Canonical Form, Parametrization and Realization", IFAC Proceedings Volumes, 20(5): 181--185, 1987.

[1]

@@ Line 1: / Line 1: @@
-{{preliminary}} <!-- Do not remove -->
 [[Category:benchmark]]
 [[Category:procedural]]
 [[Category:linear]]
+[[Category:time invariant]]
 [[Category:first differential order]]
-[[Category:time invariant]]
+[[Category:Sparse]]
 [[Category:SISO]]
@@ Line 12: / Line 11: @@
 This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.
-For an all-pass system, the trasfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
+For an all-pass system, the transfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
-or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>.
+or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,
-The system matrices are constructing based on the scheme:
+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:
 :<math>
 \begin{align}
-A &= \begin{pmatrix} a_{1,1} & -\alpha_1 \\
+A &=
-                      \alpha_1 & 0 & -\alpha_2 \\
+\begin{pmatrix}
-                       & \ddots & \ddots & \ddots \\
+  a_{1,1} & -\alpha_1 \\
-                       & & \alpha_{N-1} & 0 & -\alpha_{N-1} \end{pmatrix}, \\
+  \alpha_1 & 0 & -\alpha_2 \\
-B &= \begin{pmatrix} b_1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \\
+  & \alpha_2 & 0 & \ddots \\
-C &= \begin{pmatrix} s_1 b_1 & 0 & \dots & 0 \end{pmatrix}, \\
+  & & \ddots & \ddots & -\alpha_{N-1} \\
+  & & & \alpha_{N-1} & 0
+\end{pmatrix}, \\
+B &=
+\begin{pmatrix}
+  b_1 \\
+\\
+  \vdots \\
+\end{pmatrix}, \\
+C &=
+\begin{pmatrix}
+  s_1 b_1 & 0 & \cdots & 0
+\end{pmatrix}, \\
 D &= -s_1 \sigma.
 \end{align}
 </math>
-We choose <math>s_1 \in \{-1,1\}</math>, to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.
+We choose <math>s_1 \in \{-1,1\}</math> to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.
-Furthermore, <math>b_1 = 1</math> and <math>\sigma_1 = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.
+Furthermore, <math>b_1 = 1</math> and <math>\sigma = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.
 ==Data==
@@ Line 37: / Line 50: @@
 <!-- TODO add unbalancing transformation -->
+:<syntaxhighlight lang="matlab">
-<div class="thumbinner" style="width:20%;text-align:left;"><!--[[Media:allpass.m|-->
+function [A,B,C,D] = allpass(n)
-<source lang="matlab">
-function [A,B,C,D] = allpass(N)
 % allpass (all-pass system)
 % by Christian Himpe, 2020
@@ Line 47: / Line 57: @@
 %*
-     A = gallery('tridiag',N,-1,0,1);
+     A = gallery('tridiag',n,-1,0,1);
      A(1,1) = -0.5;
-     B = sparse(1,1,1,N,1);
+     B = sparse(1,1,1,n,1);
      C = -B';
      D = 1;
 end
-</source>
+</syntaxhighlight>
-<!--]]--></div>
-The function call requires one argument; the number of states <math>N</math>.
+The function call requires one argument; the number of states <math>n</math>.
 The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.
-:<source lang="matlab">
+:<syntaxhighlight lang="matlab">
-[A,B,C,D] = allpass(N);
+[A,B,C,D] = allpass(n);
-</source>
+</syntaxhighlight>
+An equivalent [https://www.python.org/ Python] code is
+:<syntaxhighlight lang="python">
+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
+</syntaxhighlight>
 ==Dimensions==
 :<math>
-\begin{array}{rcl}
+\begin{align}
-\dot{x}(t) &=& Ax(t) + Bu(t) \\
+\dot{x}(t) &= Ax(t) + Bu(t) \\
-y(t) &=& Cx(t) + Du(t)
+y(t) &= Cx(t) + Du(t)
-\end{array}
+\end{align}
 </math>
 System dimensions:
-<math>A \in \mathbb{R}^{N \times N}</math>,
+<math>A \in \mathbb{R}^{n \times n}</math>,
-<math>B \in \mathbb{R}^{N \times 1}</math>,
+<math>B \in \mathbb{R}^{n \times 1}</math>,
-<math>C \in \mathbb{R}^{1 \times N}</math>,
+<math>C \in \mathbb{R}^{1 \times n}</math>,
 <math>D \in \mathbb{R}</math>.
@@ Line 85: / Line 110: @@
 * For the benchmark itself and its data:
-::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/All_pass_system
+::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/All_pass_system
   @MISC{morwiki_allpass,
@@ Line 91: / Line 116: @@
     title =        {All-Pass System},
     howpublished = {{MORwiki} -- Model Order Reduction Wiki},
-    url =          <nowiki>{http://modelreduction.org/index.php/All_pass_system}</nowiki>,
+    url =          <nowiki>{https://modelreduction.org/morwiki/index.php/All_pass_system}</nowiki>,
     year =         {2020}
   }