Lyapunov Generalized Controllability

Available identifiers

MaRDI QIDQ6674145

generalized Lyapunov equation used in the stability analysis of linear dynamical systems to determine the controllability

Description

For a numerical solution, one can resort to iterative schemes, which requires the solution of a standard Lyapunov equation in each step. As an alternative, one may use the biconjugate gradient method (with preconditioner) as suggested by Tobias Breiten from TU Graz, Austria, now TU Berlin.
For the solvability of generalized Lyapunov equations, there are two requirements: (1) stability condition for A and (2) suitable upper bound for the norm of N.

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Defining Formula: $A{W}_c+W_c{A}^{*}+\sum _k{N}_k{W}_c{N}_k^{*}+B{B}^{*}=0$

$A$ symbol represents Control System Matrix A

$B$ symbol represents Control System Matrix B

$N$ symbol represents Control System Matrix N

$W_c$ symbol represents Controllability Gramian (Generalized)

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