Distance Problems for Linear Dynamical Systems

This chapter is concerned with distance problems for linear time-invariant differential and differential-algebraic equations. Such problems can be formulated as distance problems for matrices and pencils. In the first part, we discuss characterizations of the distance of a regular matrix pencil to the set of singular matrix pencils. The second part focuses on the distance of a stable matrix or pencil to the set of unstable matrices or pencils. We present a survey of numerical procedures to compute or estimate these distances by taking into account some of the historical developments as well as the state of the art.

[1]  Heike Faßbender,et al.  Breaking Van Loan’s Curse: A Quest forStructure-Preserving Algorithms for Dense Structured Eigenvalue Problems , 2015 .

[2]  Ricardo Riaza,et al.  Differential-Algebraic Systems: Analytical Aspects and Circuit Applications , 2008 .

[3]  Emre Mengi,et al.  Numerical Optimization of Eigenvalues of Hermitian Matrix Functions , 2011, SIAM J. Matrix Anal. Appl..

[4]  L. Trefethen,et al.  Spectra and Pseudospectra , 2020 .

[5]  Nicola Guglielmi,et al.  Approximating real stability radii , 2015 .

[6]  Do Duc Thuan,et al.  Stability radius of implicit dynamic equations with constant coefficients on time scales , 2011, Syst. Control. Lett..

[7]  Dirk Pflüger,et al.  Lecture Notes in Computational Science and Engineering , 2010 .

[8]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[9]  Volker Mehrmann,et al.  Differential-Algebraic Equations: Analysis and Numerical Solution , 2006 .

[10]  Matthias Voigt,et al.  On Linear-Quadratic Optimal Control and Robustness of Differential-Algebraic Systems , 2015 .

[11]  Timo Reis,et al.  Surveys in Differential-Algebraic Equations III , 2015 .

[12]  D. Hinrichsen,et al.  Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness , 2010 .

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

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

[15]  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 .

[16]  Wen-Wei Lin,et al.  Palindromic Eigenvalue Problems in Applications , 2015 .

[17]  Alastair Spence,et al.  A new approach for calculating the real stability radius , 2014 .

[18]  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..

[19]  Michael Karow,et al.  μ-Values and Spectral Value Sets for Linear Perturbation Classes Defined by a Scalar Product , 2011, SIAM J. Matrix Anal. Appl..

[20]  Diederich Hinrichsen,et al.  Modelling, state space analysis, stability and robustness , 2005 .

[21]  D. Hinrichsen,et al.  Stability radii of linear systems , 1986 .

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

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

[24]  Paul Van Dooren,et al.  Calculating the HINFINITY-norm Using the Implicit Determinant Method , 2014, SIAM J. Matrix Anal. Appl..

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

[26]  M. Voigt,et al.  A Structured Pseudospectral Method for H-infinity-Norm Computation of Large-Scale Descriptor Systems , 2012 .

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

[28]  R. Decarlo,et al.  Computing the distance to an uncontrollable system , 1991 .

[29]  Daniel Kressner,et al.  Low rank differential equations for Hamiltonian matrix nearness problems , 2015, Numerische Mathematik.

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

[31]  Wim Michiels,et al.  An Iterative Method for Computing the Pseudospectral Abscissa for a Class of Nonlinear Eigenvalue Problems , 2012, SIAM J. Sci. Comput..

[32]  Stephan Trenn,et al.  Solution Concepts for Linear DAEs: A Survey , 2013 .

[33]  Nicola Guglielmi,et al.  Computing the Structured Pseudospectrum of a Toeplitz Matrix and Its Extreme Points , 2012, SIAM J. Matrix Anal. Appl..

[34]  Volker Mehrmann,et al.  On the distance to singularity via low rank perturbations , 2015 .

[35]  Peter Benner,et al.  A structured pseudospectral method for $$\mathcal {H}_\infty $$H∞-norm computation of large-scale descriptor systems , 2013, Math. Control. Signals Syst..

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

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

[38]  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..

[39]  Nicola Guglielmi,et al.  Differential Equations for Roaming Pseudospectra: Paths to Extremal Points and Boundary Tracking , 2011, SIAM J. Numer. Anal..

[40]  Nicola Guglielmi,et al.  Low-Rank Dynamics for Computing Extremal Points of Real Pseudospectra , 2013, SIAM J. Matrix Anal. Appl..

[41]  Michael L. Overton,et al.  Fast Approximation of the HINFINITY Norm via Optimization over Spectral Value Sets , 2013, SIAM J. Matrix Anal. Appl..

[42]  N.J. Higham,et al.  The sensitivity of computational control problems , 2004, IEEE Control Systems.

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

[44]  Ivan Markovsky,et al.  Software for weighted structured low-rank approximation , 2014, J. Comput. Appl. Math..

[45]  Volker Mehrmann,et al.  Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form , 2005 .

[46]  Paul Van Dooren,et al.  Convergence of the calculation of Hoo norms and related questions , 1998 .

[47]  Tim Mitchell,et al.  Robust and Efficient Methods for Approximation and Optimization of Stability Measures , 2014 .

[48]  Ivan Markovsky,et al.  Low Rank Approximation - Algorithms, Implementation, Applications , 2018, Communications and Control Engineering.

[49]  Peter Benner,et al.  Dimension Reduction of Large-Scale Systems , 2005 .

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

[51]  D. Hinrichsen,et al.  Real and Complex Stability Radii: A Survey , 1990 .

[52]  Daniel Boley.  The algebraic structure of pencils and block Toeplitz matrices , 1998 .

[53]  Nicola Guglielmi,et al.  Erratum/Addendum: Differential Equations for Roaming Pseudospectra: Paths to Extremal Points and Boundary Tracking , 2012, SIAM J. Numer. Anal..

[54]  Vu Hoang Linh,et al.  Robust Stability of Differential-Algebraic Equations , 2013 .

[55]  Do Duc Thuan,et al.  Spectrum-Based Robust Stability Analysis of Linear Delay Differential-Algebraic Equations , 2015 .

[56]  Adrian S. Lewis,et al.  Alternating Projections on Manifolds , 2008, Math. Oper. Res..

[57]  Daniel Kressner,et al.  Computing Extremal Points of Symplectic Pseudospectra and Solving Symplectic Matrix Nearness Problems , 2014, SIAM J. Matrix Anal. Appl..

[58]  J. Demmel.  On condition numbers and the distance to the nearest ill-posed problem , 2015 .

[59]  D. Hinrichsen,et al.  Optimization problems in the robustness analysis of linear state space systems , 1989 .

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

[61]  Peter Benner,et al.  Robust numerical methods for robust control , 2004 .

[62]  S. Bora,et al.  Structured Eigenvalue Perturbation Theory , 2015 .

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

[64]  P. Rentrop,et al.  Differential-Algebraic Equations , 2006 .

[65]  Daniel Kressner.  Finding the distance to instability of a large sparse matrix , 2006, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control.

[66]  M. Overton,et al.  NORM VIA HYBRID EXPANSION-CONTRACTION USING SPECTRAL VALUE SETS , 2014 .

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

[68]  Peter Benner,et al.  Algorithm 800: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices. I: the square-reduced method , 2000, TOMS.

[69]  Volker Mehrmann,et al.  Where is the nearest non-regular pencil? , 1998 .

[70]  Timo Reis,et al.  Circuit synthesis of passive descriptor systems—a modified nodal approach , 2010 .