A new approach for calculating the real stability radius

We present a new fast algorithm to compute the real stability radius with respect to the open left half plane which is an important problem in many engineering applications. The method is based on a well-known formula for the real stability radius and the correspondence of singular values of a transfer function to pure imaginary eigenvalues of a three-parameter Hamiltonian matrix eigenvalue problem. We then apply the implicit determinant method, used previously by the authors to compute the complex stability radius, to find the critical point corresponding to the desired singular value. This corresponds to a two-dimensional Jordan block for a pure imaginary eigenvalue in the parameter dependent Hamiltonian matrix. Numerical results showing quadratic convergence of the algorithm are given.

[1]  Stephen P. Boyd,et al.  A bisection method for computing the H∞ norm of a transfer matrix and related problems , 1989, Math. Control. Signals Syst..

[2]  A. Griewank,et al.  Characterization and Computation of Generalized Turning Points , 1984 .

[3]  D. Hinrichsen,et al.  Stability radius for structured perturbations and the algebraic Riccati equation , 1986 .

[4]  A. Spence,et al.  The computation of Jordan blocks in parameter-dependent matrices , 2014 .

[5]  Willy Govaerts.  Stable solvers and block elimination for bordered systems , 1991 .

[6]  Michael L. Overton,et al.  Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix , 2011, SIAM J. Matrix Anal. Appl..

[7]  Paul Van Dooren,et al.  A fast algorithm to compute the real structured stability radius , 1996 .

[8]  Mert Gürbüzbalaban,et al.  Theory and methods for problems arising in robust stability, optimization and quantization , 2012 .

[9]  C. Loan.  How Near is a Stable Matrix to an Unstable Matrix , 1984 .

[10]  Daniel Kressner,et al.  Algorithm 854: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices II , 2005, TOMS.

[11]  Gene H. Golub,et al.  Matrix computations , 1983 .

[12]  Charles R. Johnson,et al.  Linear algebra and its role in systems theory , 1985 .

[13]  Richard O. Akinola,et al.  The calculation of the distance to a nearby defective matrix , 2012, Numer. Linear Algebra Appl..

[14]  Daniel Kressner,et al.  On the computation of structured singular values and pseudospectra , 2010, Syst. Control. Lett..

[15]  P. Dooren.  Robustness measures and level set methods , 2006 .

[16]  Edward J. Davison,et al.  A formula for computation of the real stability radius , 1995, Autom..

[17]  Alastair Spence,et al.  Photonic band structure calculations using nonlinear eigenvalue techniques , 2005 .

[18]  S. Boyd,et al.  A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L ∞ -norm , 1990 .

[19]  L. Trefethen.  Spectra and pseudospectra , 2005 .

[20]  Daniel Kressner,et al.  Subspace Methods for Computing the Pseudospectral Abscissa and the Stability Radius , 2014, SIAM J. Matrix Anal. Appl..

[21]  G. Alistair Watson,et al.  An Algorithm for Computing the Distance to Instability , 1998, SIAM J. Matrix Anal. Appl..

[22]  R. Byers.  A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices , 1988 .

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

[24]  P. Benner,et al.  New Hamiltonian Eigensolvers with Applications in Control , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[25]  Alastair Spence,et al.  A Newton-based method for the calculation of the distance to instability , 2011 .

[26]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[27]  L. Trefethen,et al.  Spectra and pseudospectra : the behavior of nonnormal matrices and operators , 2005 .