# Solving Stable Generalized Lyapunov Equations for Hankel Singular Values Computation

### Vasile Sima

#### 2022

#### Abstract

Generalized Lyapunov equations are often encountered in systems theory, analysis and design of control systems, and in many applications, including balanced realization algorithms, procedures for reduced order models, or Newton methods for generalized algebraic Riccati equations. An important application is the computation of the Hankel singular values of a generalized dynamical system, whose behavior is defined by a regular matrix pencil. This application uses the controllability and observability Gramians of the system, given as the solutions of a pair of generalized Lyapunov equations. The left hand side of each of these equations follows from the other one by applying the (conjugate) transposition operator. If the system is stable, the solutions of both equations are non-negative definite, hence they can be obtained in a factorized form. But these theoretical results may not hold in numerical computations if the symmetry and non-negative definiteness are not preserved by a solver. The paper summarizes new related numerical algorithms for complex continuous- and discrete-time generalized systems. Such solvers are not yet available in the SLICOT Library or MATLAB. The developed solvers address the essential practical issues of reliability, accuracy, and efficiency.

Download#### Paper Citation

#### in Harvard Style

Sima V. (2022). **Solving Stable Generalized Lyapunov Equations for Hankel Singular Values Computation**. In *Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-989-758-585-2, pages 130-137. DOI: 10.5220/0011259900003271

#### in Bibtex Style

@conference{icinco22,

author={Vasile Sima},

title={Solving Stable Generalized Lyapunov Equations for Hankel Singular Values Computation},

booktitle={Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2022},

pages={130-137},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0011259900003271},

isbn={978-989-758-585-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - Solving Stable Generalized Lyapunov Equations for Hankel Singular Values Computation

SN - 978-989-758-585-2

AU - Sima V.

PY - 2022

SP - 130

EP - 137

DO - 10.5220/0011259900003271