Numerical solution of stable generalized complex Lyapunov equations

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 (E, A), with E nonsingular. This application uses the controllability and observability Gramians of the system, given as the solutions of a pair of related generalized Lyapunov equations. For a stable system, the solutions of both equations are non-negative definite. The paper summarizes the numerical algorithms for complex continuous- and discrete-time generalized systems. Such solvers are not yet available in the SLICOT Library or MATLAB toolboxes, but could be an important addition. The developed solvers address the essential practical issues of reliability, accuracy, and efficiency.