Anisotropy-based analysis for case of nonzero initial condition

Well-known $\mathcal{H}_2$-norm- and $\mathcal{H}_\infty$-norm-theories allow construction of optimal regulators that minimize external disturbances' influence on the linear time-idependent system output. They are based on quality criteria of $\mathcal{H}_2$- and $\mathcal{H}_\infty$-norms of closed-loop transfer functions. In anisotropy theory, a notion of anisotropy norm is introduced. Usually in the context of the anisotropy-based robust performance analysis stochastic systems with zero initial condition are investigated. In this paper we extend this analysis and consider a linear discrete time invariant system under random disturbances and with the nonzero initial condition. In accordance with the basic postulates of the anisotropy-based control theory, the disturbance attenuation capabilities of system are quantified by the anisotropic generalized gain which is defined as the largest root mean square gain of the system with respect to a random input and the nonzero initial condition, anisotropy of which is bounded by a given nonnegative parameter $a$. A numerical example is considered.