Finally, obtain the accurate or approximate solution of x, which is based on a strict mathematical optimization problem: min[[parallel]y - [PHI][[PSI].sup.T]x[parallel].sub.2] + [lambda][[parallel][[psi].sup.T]x[parallel].sub.0], where [[parallel] [parallel].sub.2] and [[parallel] [parallel].sub.0] is the

2-norm and 0-norm, respectively.

Furthermore, error

2-norm of first-order natural frequency is also shown in Table 6.

If the matrix B is full rank, then [parallel]([I.sub.n] - [BB.sup.+])E[parallel] [less than or equal to] [parallel]E[parallel] where the operator [parallel]*[parallel] denotes the

2-norm of the matrix *.

(Necessarily, d = 0 because the result is a norm and therefore positive homogeneous.) In the special case where \\ * \\ [DELTA] and \\ * \\ [DELTA] are both the

2-norm, then [A.sub.i], [b.sub.i], [B.sub.i], and [K.sub.i] are given by (2.3) in Example 2.2.

where {[??]} is scaled by its

2-norm because [V] is a unit orthogonal matrix.

where [parallel] x [parallel] denotes the

2-norm and [parallel][([[??].sub.r]).sub.max][parallel] = [parallel][y.sub.d] - [y.sub.0][parallel]/[tau] is the maximum value of [[??].sub.r] which is obtained from (3), where [DELTA]y = [parallel][y.sub.d] - [y.sub.0][parallel] is the difference between initial and desired values of the output.

Usually, matrix infinite norm is larger than matrix

2-norm; thus the result established in Theorem 5 can be rewritten by matrix

2-norm form further.

Hence, the

2-Norm of the estimation lingering of [z.sub.a] is

where [[phi].sub.u] is the time length center, ||[s.sub.p]|| is the

2-norm of [s.sub.p], and [[phi].sub.u] can be gotten by:

The iteration stops when the Euclidean norm (

2-norm) of the residual vector is reduced by [10.sup.-14].

In order to assess the quality of the calibration process, it was used the

2-norm condition number of the first matrix in Equation (2) as:

The theory of probabilistic normed spaces was initiated and developed in [1], [17], [31], [32] and further it was extended to random/probabilistic

2-normed spaces [8] by using the concept of

2-norm [7].