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 , of states
and outputs
.
Optionally, a symmetric system can be enforced with the parameter
.
For reproducibility, the random number generator seed can be controlled by the parameter
.
The return value consists of three matrices; the system matrix
, the input matrix
and the output matrix
.
[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:
System dimensions:
,
,
.
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
- ↑ S.C. Smith, J. Fisher, "On generating random systems: a gramian approach", Proceedings of the American Control Conference, 3: 2743--2748, 2003.
- ↑ 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.