A numerical algorithm for Lyapunov equations

The Minkowski–Lyapunov equation for linear dynamics: Theoretical foundations

“…Nevertheless, despite numerous efforts on solving the Lyapunov equation within the class of quadratic functions (Bitmead, 1981;Gajić & Qureshi, 2008;Lu & Wachspress, 1991;Norman, 1989;Tian & Gu, 2008;Zhou, 2011) and numerous works on solving the polyhedral Lyapunov inequality (Blanchini & Miani, 2008;Kiendl et al, 1992;Lazar, 2010;Molchanov & Pyatnitskii, 1989;Polański, 1998), which can be traced back to as far as 1963 (Rosenbrock, 1963), to the best of the authors' knowledge, a characterization of the solution to the Lyapunov equation within the class of Minkowski functions is not available.…”

“…In particular, it is well known that the corresponding Lyapunov inequality can be equivalently expressed as a linear matrix inequality (Boyd, El Ghaoui, Feron, & Balakrishnan, 1994), while the associated Lyapunov equation takes the form of a linear matrix equation. The importance of the Lyapunov equation is also indicated by the numerous research efforts that have been made to develop algorithms for solving the Lyapunov equation within the class of quadratic functions, see, for instance, Bitmead (1981); Gajić and Qureshi (2008); Lu and Wachspress (1991); Norman (1989); Tian and Gu (2008); Zhou (2011) and the references therein. Another important class of Lyapunov functions is the class of polyhedral Lyapunov functions, e.g., weighted (1, ∞)-vector norms, see, for example, Molchanov and Pyatnitskii (1989); Kiendl, Adamy, and Stelzner (1992); Polański (1998); Blanchini and Miani (2008); Lazar (2010) and the references therein.…”

Gradient based iterative solutions for general linear matrix equations

“…But the method there is not suitable for solving general linear matrix equations in (1) in the next section, which include the Lyapunov equations, Sylvester equations as the special cases, e.g., [4].…”

Stability region estimation of statically unstable two wheeled mobile robots

“…The direct method of Lyapunov has been considered as clean and short method but for some systems it is not straight forward to find a Lyapunov function. Various methods proposed in literature for the construction of Lyapunov function were based on conventional [13] and numerical [14] approaches. The numerical methods have been applied in many applications [15] to avoid computational problems of conventional method.…”