Balancing Skew-Hamiltonian/Hamiltonian Pencils - With Applications in Control Engineering

@inproceedings{Sima2016BalancingSP,
  title={Balancing Skew-Hamiltonian/Hamiltonian Pencils - With Applications in Control Engineering},
  author={Vasile Sima},
  booktitle={ICINCO},
  year={2016}
}
  • V. Sima
  • Published in ICINCO 29 July 2016
  • Mathematics
Badly-scaled matrix pencils could reduce the reliability and accuracy of computed results for many numerical problems, including computation of eigenvalues and deflating subspaces, which are needed in many key procedures for optimal and H∞ control, model reduction, spectral factorization, and so on. Standard balancing techniques can improve the results in many cases, but there are situations when the solution of the scaled problem is much worse than that for the unscaled problem. This paper… 

Figures from this paper

Improved balancing for general and structured eigenvalue problems

  • V. SimaP. Benner
  • Computer Science
    2016 20th International Conference on System Theory, Control and Computing (ICSTCC)
  • 2016
TLDR
This paper presents an improved balancing technique for general or structured matrices and matrix pencils and illustrates its good performance in solving eigenvalue problems and algebraic Riccati equations for large sets of examples from well-known benchmark collections.

A Flexible Structured Solver for Continuous-time Algebraic Riccati Equations

TLDR
A flexible solver for continuous-time AREs that allows the user to choose among several structured solution approaches, orthogonalization methods, and balancing options and parameters is proposed.

References

SHOWING 1-10 OF 24 REFERENCES

Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils

We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems, $H_{\infty}$ optimization, multibody systems, and many other

Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications

TLDR
This work will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures.

Robust and Efficient Algorithms for L∞ -Norm Computation for Descriptor Systems

TLDR
This paper shows how one can achieve the computation of the ℒ ∞ -norm of transfer functions related to descriptor systems, both in the continuous- and discrete-time context by computing the eigenvalues of certain structured matrix pencils.

Pitfalls when solving eigenproblems with applications in control engineering

  • V. SimaP. Benner
  • Computer Science
    2015 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO)
  • 2015
TLDR
This paper investigates some numerical algorithms for the solution of common and structured eigenproblems, which have many applications in automatic control, but also in various areas of applied mathematics, physics, and computational chemistry.

A schur method for solving algebraic Riccati equations

  • A. Laub
  • Mathematics, Computer Science
    1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes
  • 1978
TLDR
A new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented, a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors thereby gaining substantial numerical advantages.

SLICOT Working Note 2013-3 MB 04 BV A FORTRAN 77 Subroutine to Compute the Eigenvectors Associated to the Purely Imaginary Eigenvalues of Skew-Hamiltonian / Hamiltonian Matrix Pencils

TLDR
A structure-preserving numerical algorithm for extracting the eigenvectors associated to the purely imaginary eigenvalues of skew-Hamiltonian/Hamiltonian matrix pencils is implemented and compared with the QZ algorithm.

Solving SLICOT benchmarks for continuous-time algebraic Riccati equations by Hamiltonian solvers

  • V. SimaP. Benner
  • Computer Science
    2015 19th International Conference on System Theory, Control and Computing (ICSTCC)
  • 2015
TLDR
A new solver based on the SLICot Library subroutines has been developed and tested on the CAREX benchmark collection included in SLICOT and shows similar or better accuracy in comparison with the state-of-the-art MATLAB solver.

CAREX - A Collection of Benchmark Examples for Continuous-Time Algebraic Riccati Equations (Version

A collection of benchmark examples is presented for the numerical solution of continuous-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new

Balancing the Generalized Eigenvalue Problem

  • R. Ward
  • Computer Science, Mathematics
  • 1981
TLDR
The three-step algorithm is specifically designed to precede the $QZ$-type algorithms, but improved performance is expected from most eigensystem solvers.