An anisotropy-based boundedness criterion for time-invariant systems with multiplicative noises

This paper presents an anisotropy-based analysis of linear time-invariant systems with multiplicative noises. The system dynamics are described in the state space. The external disturbance belongs to the set of stationary sequences of random vectors with bounded mean anisotropy. The multiplicative noises are centered and have unit variance; the external disturbance and noises are mutually independent. We derive a boundedness criterion for the anisotropic norm in terms of Riccati-like inequalities using the bounded real lemma of the anisotropy-based theory. With a special change of variables, we reduce the analysis problem to a convex optimization problem with additional constraints. The existence of the latter's solution implies the bounded anisotropic norm of the system with multiplicative noises, and the minimal upper bound of the anisotropic norm can be obtained by solving this convex optimization problem.