Anonymous
×
Create a new article
Write your page title here:
We currently have 105 articles on MOR Wiki. Type your article name above or click on one of the titles below and start writing!



MOR Wiki

Inverse Lyapunov Procedure

Revision as of 16:53, 6 March 2018 by Himpe (talk | contribs) (Added missing section and various fixes)


Description

The Inverse Lyapunov Procedure (ilp) is a synthetic random linear system generator. It is based on reversing the Balanced Truncation procedure and was developed in [1], where a description of the algorithm is given. In aggregate form, for randomly generated controllability and observability gramians, a balancing transformation is computed. The balanced gramian is the basis for an associated state-space system, which is determined by solving a Lyapunov equation and then unbalanced. A central point is the solution of the Lyapunov equations for the system matrix instead of the gramian matrix. This is feasable due to the symmetric (semi-)positive definiteness of the gramians and the requirement for a stable system, yet with a non-unique solution.

Inverse Sylvester Procedure

A variant of the Inverse Lyapunov Procedure is the inverse Sylvester procedure [2].

Data

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

function [A,B,C] = ilp(M,N,Q,s,r)
% ilp (inverse lyapunov procedure)
% by Christian Himpe, 2013--2018
% released under BSD 2-Clause License
%*
  if(nargin==5)
    rand('seed',r);
    randn('seed',r);
  end;

% Gramian Eigenvalues
  WC = exp(0.5*rand(N,1));
  WO = exp(0.5*rand(N,1));

% Gramian Eigenvectors
  [P,S,R] = svd(randn(N));

% Balancing Transformation
  WC = P*diag(sqrt(WC))*P';
  WO = R*diag(sqrt(WO))*R';
  [U,D,V] = svd(WC*WO);

% Input and Output
  B = randn(N,M);

  if(nargin>=4 && s~=0)
    C = B';
  else
    C = randn(Q,N);
  end

% Scale Output Matrix
  BB = sum(B.*B,2);  % = diag(B*B')
  CC = sum(C.*C,1)'; % = diag(C'*C)
  C = bsxfun(@times,C,sqrt(BB./CC)');

% Solve System Matrix
  A = -sylvester(D,D,B*B');

% Unbalance System
  A = V*A*U';
  B = V*B;
  C = C*U';

end

The function call requires three parameters; the number of inputs M, of states N and outputs Q. Optionally, a symmetric system can be enforced with the parameter s \neq 0. For reproducibility, the random number generator seed can be controlled by the parameter r \in \mathbb{N}. The return value consists of three matrices; the system matrix A, the input matrix B and the output matrix C.

[A,B,C] = ilp(M,N,Q,s,r);

A variant of the above code using empirical Gramians instead of a matrix equation solution can be found at http://gramian.de/utils/ilp.m , which may yield preferable results.

Dimensions

System structure:


\begin{align}
\dot{x}(t) &= Ax(t) + Bu(t) \\
y(t) &= Cx(t)
\end{align}

System dimensions:

A \in \mathbb{R}^{N \times N}, B \in \mathbb{R}^{N \times M}, C \in \mathbb{R}^{Q \times N}.


Citation

To cite this benchmark, use the following references:

  • For the benchmark itself and its data:
The MORwiki Community. Inverse Lyapunov Procedure. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Inverse_Lyapunov_Procedure
   @MISC{morwiki-invlyapproc,
    author = {The {MORwiki} Community},
    title = {Inverse Lyapunov Procedure},
    howpublished = {{MORwiki} -- Model Order Reduction Wiki},
    url = {http://modelreduction.org/index.php/Inverse Lyapunov Procedure},
    year = {2018}
   }
  • For the background on the benchmark:
    @INPROCEEDINGS{SmiF03,
     author =       {Smith, S.~C. and Fisher, J.},
     title =        {On generating random systems: a gramian approach},
     booktitle =    {Proc. Am. Control. Conf.},
     volume =       3,
     pages =        {2743--2748},
     year =         2003,
     doi =          {10.1109/ACC.2003.1243494}
    }


References

  1. S.C. Smith, J. Fisher, "On generating random systems: a gramian approach", Proceedings of the American Control Conference, 3: 2743--2748, 2003.
  2. C. Himpe, M. Ohlberger, "Cross-Gramian-Based Model Reduction: A Comparison", In: Model Reduction of Parametrized Systems, Modeling, Simulation and Applications, vol. 17: 271--283, 2017.

Contact

Christian Himpe