This proposed method is developed from the energy approach based on the controllability Gramian matrix of the linearized system.
Matrices A and B of the above system (17) are controllable if and only if the controllability Gramian matrix [G.sub.c](T) on horizon T defined as (21) has full rank n and is positive definite.
We remark that, by using the Gramian matrix G defined by
Further, using 2n x 2n Gramian matrix, we generalized the result to Theorem 2.
The DoFs of [D.sub.s] and [B.sup.a.sub.s] are connected to those of [E.sub.s] and [H.sub.s] by the Galerkin's discrete Hodges and the
Gramian matrix. By using the sparse approximate inverse of the
Gramian matrix, the resultant eigensystem involve sparse matrices only, which can be easily solved with conventional eigensolvers.
Using Claim 1, we observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [sigma](M) denotes the spectrum of the Gramian matrix M as an operator on [l.sup.2].
In this case, the Gramian matrix M is a convolution operator, having some sequence a as kernel.
The N x N Gramian matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to plays a crucial role in the recovery of missing samples; its elements are the h-periodic functions
In the recovery of missing samples the structure of the mixed Gramian matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plays a crucial role.