This paper discusses transformations between bases used in closed-loop learning control experiments. The goal is to transform to a basis in which the number of control parameters is minimized and in which the parameters act independently. We demonstrate a simple procedure for testing whether a unitary linear transformation (i.e., a rotation amongst the control variables) is sufficient to reduce the search problem to a set of globally independent variables. This concept is demonstrated with closed-loop molecular fragmentation experiments utilizing shaped, ultrafast laser pulses.
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Two matrices that commute share the same eigenvectors. The eigenvectors can be used to form a transformation matrix that diagonalizes the matrices.
A spectral cutoff frequency of was used for our two-dimensional low pass box filter.
The width of the smoothed Heaviside function centered at frequency is (10%–90% of the step rise).
