Reachability sets and the generalized $H_2$-norm of a linear discrete descriptor system

The paper focuses on a linear discrete noncausal descriptor system on a finite horizon under consistent initial conditions and bounded external disturbances, i.e. a bounded $l_2$-norm. The notion of the generalized $H_2$-norm for a linear discrete descriptor system is introduced as the induced norm of the linear operator generated by the system under consideration. This norm is characterized in terms of difference projected generalized Lyapunov equation solutions. It is demonstrated that if the sum of the quadratic forms of the initial and final states and the sum of the quadratic forms of the disturbance over a finite time interval is bounded by a given value from above, the reachability set of this system is a time-varying ellipsoid whose matrix satisfies the difference projected generalized Lyapunov equation. It is established that the generalized $H_2$-norm of the system under non-zero initial conditions coincides with the value of the maximum half-axis of the reachability ellipsoidal set for a given output of the system. An example of a fourth-order descriptor system is provided as an illustration of the results. For this system a generalized $H_2$-norm is calculated and reachability sets are constructed. The paper demonstrates the results of numerical simulations and projections of reachability sets on the plane corresponding to the forward and backward subsystems.