Computational Methods in Systems and Control Theory

MATLAB Codes for Computing the H_∞-/L_∞-Norm for Large-Scale Descriptor Systems

MATLAB implementation of various algorithms for the computation of the H_∞-/L_∞-norm for large-scale descriptor systems. Based on the computation of dominant poles two optimization methods were implemented to determine the norm value. Both implementations were tested with MATLAB 2012a under Linux and should work with a reasonably current version of MATLAB.

Method 1: Computation of the H_∞-Norm via Optimization over Structured Pseudospectra

This algorithm is based on the relation between the H_∞-norm and the structured complex stability radius of a transfer function. A nested iteration is used. In the inner iteration, the rightmost point of a structured ε-pseudospectrum is computed for a fixed ε. In the outer iteration, ε is updated via Newton steps to determine the value of ε for which the structured ε-pseudospectrum touches the imaginary axis.

Structured pseudospectra with most dominant poles (black crosses)

An inner iteration with intermediate iterates (black circles)

Author

• Matthias Voigt, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany.

Downloads

• MATLAB code: HINORM_v1.0.zip
• Test examples: testexamples.zip

License and Usage

This software is published under the GNU General Public License, version 3. It is research code and there is no warranty for correctness of numerical results. This software uses the MATLAB implementation of the SAMDP algorithm (samdp.m) by Joost Rommes, which underlies own conditions. If you use this code for your own work, please cite the publication stated below.

Related Software

• HINFNORM: Fast approximation of the H_∞-norm via optimization over spectral value sets (by Mert Gürbüzbalaban and Tim Mitchell);
• PSAPSR: Fast algorithms for the approximation of the pseudospectral abscissa and the pseudospectral radius of a matrix (by Nicola Guglielmi and Michael Overton);
• subspace_pspa: Subspace methods for the computation of the pseudospectral abscissa (by Daniel Kressner and Bart Vandereycken).

Reference

• P. Benner and M. Voigt. A Structured Pseudospectral Method for H_∞-Norm Computation of Large-Scale Descriptor Systems. Math. Control Signals Systems, 2013.

Method 2: Computation of the L_∞-Norm via Optimization on Level Sets

This algorithm is an extension of the well-known Bruinsma/Steinbuch algorithm to large-scale problems. Using the dominant poles of the transfer function, shifts for a structure-preserving iterative eigensolver for even eigenvalue problems (even IRA) are computed. The obtained imaginary eigenvalues can now be used to determine level sets that contain the optimal frequency.

Plot of a transfer function with computed norm value (red circle)

Plot of the level sets for every iteration

Authors

• Ryan Lowe, Queens University, Ontario, Canada (main author);
• Matthias Voigt, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany.

Downloads

• MATLAB code: LINORM_v1.0.zip
• Test examples: testexamples.zip

License and Usage

This software is published under the GNU General Public License, version 3. It is research code and there is no warranty for correctness of numerical results. This software uses the MATLAB implementations of the SAMDP algorithm (samdp.m) by Joost Rommes and the even IRA (even_ira.m) by Volker Mehrmann, Valeria Simoncini, and Christian Schröder, which underly own conditions. If you use this code for your own work, please cite the publications stated below.

References

• R. Lowe and M. Voigt. L_∞-Norm Computation for Large-Scale Descriptor Systems Using Structured Iterative Eigensolvers. Preprint MPIMD/13-20, MPI Magdeburg, 2013.
• P. Benner, V. Sima, and M. Voigt. L_∞-Norm Computation for Continuous-Time Descriptor Systems Using Structured Matrix Pencils. IEEE Trans. Automat. Control, 57(1):233-238, 2012.