# FAST APPROXIMATION OF THE H∞ NORM VIA OPTIMIZATION OVER SPECTRAL VALUE SETS∗

@inproceedings{Guglielmi2012FASTAO, title={FAST APPROXIMATION OF THE H∞ NORM VIA OPTIMIZATION OVER SPECTRAL VALUE SETS∗}, author={Nicola Guglielmi and Mert Gurbuzbalaban and Michael L. Overton}, year={2012} }

The H∞ norm of a transfer matrix function for a control system is the reciprocal of the largest value of ε such that the associated ε-spectral value set is contained in the stability region for the dynamical system (the left half-plane in the continuous-time case and the unit disk in the discrete-time case). After deriving some fundamental properties of spectral value sets, particularly the intricate relationship between the singular vectors of the transfer matrix and the eigenvectors of the…

## 45 Citations

### CALCULATING THE H ∞ -NORM USING THE IMPLICIT DETERMINANT METHOD

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- 2013

. We propose a fast algorithm to calculate the H ∞ -norm of a transfer matrix. The method builds on a well-known relationship between singular values of the transfer function and pure imaginary…

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An improved spectral value set based algorithm is presented using a novel hybrid expansion-contraction scheme that guarantees convergence to a stationary point of the optimization problem without incurring breakdown and completes the entire test set 3 to 18 times faster, depending on which optimizations are enabled.

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- 2014

This paper exploits the relationship between the H ∞ -norm and the structured complex stability radius of a corresponding matrix pencil and uses a new fast iterative scheme based on certain rank-1 perturbations of a matrix pencil to compute the structured stability radius.

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- 2015

This paper presents a novel approach to approximate real stability radii, particularly well-suited for large sparse matrices, and requires the computation of the rightmost eigenvalue of a sequence of matrices given by the sum of the original matrix A and a low-rank one.

### LARGE-SCALE COMPUTATION OF L∞-NORMS BY A GREEDY

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- 2017

We are concerned with the computation of the L∞-norm for an L∞-function of the form H(s) = C(s)D(s)−1B(s), where the middle factor is the inverse of a meromorphic matrix-valued function, and C(s),…

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- Computer Science
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Two new algorithms with practical application to the problem of designing controllers for linear dynamical systems with input and output are considered: a new spectral value set based algorithm called hybrid expansion-contraction intended for approximating the H∞ norm and a new BFGS SQP based optimization method for nonsmooth, nonconvex constrained optimization motivated by multi-objective controller design.

### Certifying Global Optimality for the L-Infinity-Norm Computation of Large-Scale Descriptor Systems

- Computer Science
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An algorithm is designed that determines whether a given value is less than the L∞-norm of the transfer function under consideration, and that does not require user input other than the system matrices, and is tested without any parameter adaptation on a benchmark collection of large-scale systems.

### On Linear-Quadratic Optimal Control and Robustness of Differential-Algebraic Systems

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This thesis considers the linear-quadratic optimal control problem for differentialalgebraic equations. In this first part we present a complete theoretical analysis of this problem. The basis is a…

### Approximation of stability radii for large-scale dissipative Hamiltonian systems

- MathematicsAdv. Comput. Math.
- 2020

The proposed frameworks are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake and derive subspaces yielding a Hermite interpolation property between the full and projected problems.

### Subspace Methods for Computing the Pseudospectral Abscissa and the Stability Radius

- Computer ScienceSIAM J. Matrix Anal. Appl.
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This paper proposes to combine a linearly converging iterative method for computing the pseudospectral abscissa and its variants with subspace acceleration, and observes local quadratic convergence and proves local superlinear convergence of the resulting subspace methods.

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