Before setting the optimal objective function, the observability matrix [W.sub.obs] and the controllability matrix [W.sub.con] are, respectively, obtained by the expression between Gramian matrix and the system modal energy .
where W is Gramian matrix, its value is [W.sub.obs] or [W.sub.con], [sigma]([[lambda].sub.i]) is standard deviation for the eigenvalues [[lambda].sub.i] of Gramian matrix W, [2n[square root of det(W)] is the geometric mean of the eigenvalues, its physical significance is the volume of the ellipse, n is the coefficient of freedom degree, [sigm]([[lambda].sub.i]) is the location mainly avoiding both great and small eigenvalues, and trace(W) is the output energy of the actuator.
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.
Nicholson, "On a fundamental property of the cross- Gramian matrix
," IEEE Transactions on Circuits and Systems, vol.
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.