Anisotropic $\epsilon$-optimal model reduction for linear discrete time-invariant system

We consider an $\epsilon$-optimal model reduction problem for a linear discrete time-invariant system, where the anisotropic norm of reduction error transfer function is used as a performance criterion. For solving the main problem, we state and solve an auxiliary problem of $\mathcal H_2$ $\epsilon$-optimal reduction of a weighted linear discrete time system. A sufficient optimality condition defining a solution to the anisotropic $\epsilon$-optimal model reduction problem has the form of a system of cross-coupled nonlinear matrix algebraic equations including a Riccati equation, four Lyapunov equations, and five special-type nonlinear equations. The proposed approach to solving the problem ensures stability of the reduced model without any additional technical assumptions. The reduced-order model approximates the steady-state behavior of the full-order system.