1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 3 where Sxx, Re denotes the real portion of Sxx, Sxx, Im the imaginary portion of Sxx, Lmat, Re the upper triangular matrix for the Cholesky factorization of Sxx, Re, andLmat, Imthe upper triangular matrix for the Cholesky factorization of Sxx, Im. Lmat, Re is formed directly fromLvec, Re. Lmat, Imis found using the following expressions to maintain the positive definiteness of Sxx, Im: Lmat,Im = ⎡ ⎢⎢ ⎢⎢ ⎣ 0 Lmat,Im1,2 Lmat,Im1,3 . . . Lmat,Im1,2 0 0 Lmat,Im2,3 . . . . . . . . . . . . 0 . . . Lmat,Imn−1,2 0 0 0 0 0 ⎤ ⎥⎥ ⎥⎥ ⎦ + ⎡ ⎢⎢ ⎢⎣ 1 0 . . . 0 0 1 . . . 0 . . . 0 . . . 0 . . . 0 . . . 1 ⎤ ⎥⎥ ⎥⎦ (1.6) where Lvec,Im = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ Lmat,Im1,2 Lmat,Im1,3 Lmat,Im2,3 . . . Lmat,Im1,2 . . . Lmat,Imn−1,2 ⎫ ⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎭ (1.7) andLmat,Imi,p denotes one of the upper triangular terms in the i th rowandpth columnof Lmat, Im taken directly fromLvec, Im, shown in Eq. (1.7). Once Sxx, Im is formed using Eq. (1.5), the diagonal entries resulting from the addition of the identity matrix in the definition of Lmat, Im are removed. The complete optimization process can be broken down as follows: 1 Make an initial guess for the complete Lmatrix in the form of Lvec, dv 2 Extract real and imaginary parts from the current guess vector, Lvec, dv 3 Form triangular matrices, Lmat, Re and Lmat, Im, from current Lvec, dv 4 FormSxx, Re and Sxx, Im fromLmat, Re andLmat, Im using Eq. (1.5) 5 Form complete input matrixSxx as sumof Sxx, Re and Sxx, Im 6 Solve forward MIMO control problem, defined with Eq. (1.1), to get response from this optimization iteration 7 Compute one of several scalar objective functions to compare current response to the field data where the objective functions used in this paper are explored in Sect. 1.3. In general, the Sxx formed using the method outlined above will be Hermitian positive definite, but numerically it could become slightly nonpositive definite. Therefore, a small correction can be applied to ensure the resultingSxx is Hermitian positive definite: Sxx =Sxx +Sxx H−SxxD SxxD = Sxx 0 if i =p otherwise (1.8) whereSxxDdenotes the matrix of just the diagonal elements of Sxx, i the row index of a matrix element, andpthe column index of a matrix element. As an additional guarantee of positive definiteness, the eigenvalues of Sxx are adjusted to numerically enforce the constraint, if needed. The process outlined in Steps 1–7 above are repeated until an acceptable solution is obtained based on the parameters used with the optimization algorithms implemented with DAKOTA, an optimization software created at the Sandia National Laboratories, for each frequency step within the frequency range of interest [6].
RkJQdWJsaXNoZXIy MTMzNzEzMQ==