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

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 presents a new structure-preserving balancing technique for skew-Hamiltonian/Hamiltonian 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 with difficult examples.

[1]  C. Loan,et al.  A Schur decomposition for Hamiltonian matrices , 1981 .

[2]  Peter Benner,et al.  Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils , 2002, SIAM J. Matrix Anal. Appl..

[3]  M. Steinbuch,et al.  A fast algorithm to computer the H ∞ -norm of a transfer function matrix , 1990 .

[4]  Peter Benner,et al.  CAREX - A Collection of Benchmark Examples for Continuous-Time Algebraic Riccati Equations (Version , 1999 .

[5]  Vasile Sima,et al.  ${\cal L}_{\infty}$-Norm Computation for Continuous-Time Descriptor Systems Using Structured Matrix Pencils , 2012, IEEE Transactions on Automatic Control.

[6]  Daniel Kressner,et al.  Numerical Methods for General and Structured Eigenvalue Problems , 2005, Lecture Notes in Computational Science and Engineering.

[7]  Vasile Sima,et al.  Algorithms for Linear-Quadratic Optimization , 2021 .

[8]  Vasile Sima,et al.  Solving SLICOT benchmarks for continuous-time algebraic Riccati equations by Hamiltonian solvers , 2015, 2015 19th International Conference on System Theory, Control and Computing (ICSTCC).

[9]  Vasile Sima,et al.  Pitfalls when solving eigenproblems with applications in control engineering , 2015, 2015 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO).

[10]  Subroutines for the Solution of Skew-Hamiltonian / Hamiltonian Eigenproblems – Part II : Implementation and Numerical Results , 2013 .

[11]  C. Loan.  A Symplectic Method for Approximating All the Eigenvalues of a Hamiltonian Matrix , 1982 .

[12]  M. Voigt,et al.  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 , 2013 .

[13]  Vasile Sima,et al.  Robust and Efficient Algorithms for L∞ -Norm Computation for Descriptor Systems , 2012, ROCOND.

[14]  Peter Benner,et al.  Symplectic Balancing of Hamiltonian Matrices , 2000, SIAM J. Sci. Comput..

[15]  Robert C. Ward,et al.  Balancing the Generalized Eigenvalue Problem , 1981 .

[16]  V. Mehrmann,et al.  Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications , 2005 .

[17]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.