The lower triangular Toeplitz matrix of Markov parameters is the finite time, time domain equivalent of the z-transfer function of digital systems. Its properties are therefore of interest, and this paper discovers previously unknown properties. The singular value decomposition (SVD) of the Toeplitz matrix contains the matrix of the singular values, and two orthonormal matrices of input and output singular vectors. An understanding of these three matrices in terms of system frequency response has been available for the limiting case when the Toeplitz matrix tends to infinity. Here we obtain an understanding of what happens to the SVD as the dimension is reduced. A system identification algorithm uses the entries in the input/output singular vectors as samples of a continuous time linear system under free response, with the result that such identified dynamics represent the data. Numerical examples for several different system orders are used to show the characteristics of identified eigenvalues embedded in the singular vectors. Each singular vector contains at least a pure frequency that is close to but not the same as the frequency when the dimension tends to infinity. Other eigenvalues help the output singular vectors to end at zero, and the input singular vectors to start at zero, a property of the finite time matrix. Singular vector dynamics in various Iterative Learning Control (ILC) laws are discussed here and numerically investigated, and the dynamics are relevant to basis function ILC, to finite time low pass filtering design, and to new versions of stable inverse theory, among other applications within the theory of control.
Identification of the Dynamics in the Singular Vectors of the System Toeplitz Matrix of Markov Parameters
J Astronaut Sci
The Journal of the Astronautical Sciences ; 69 , 4 ; 1115-1148
2022-08-01
34 pages
Aufsatz (Zeitschrift)
Elektronische Ressource
Englisch
Toeplitz matrix , Frequency response , Finite time frequency response , Stable inverse , Singular vector dynamics , Iterative learning control , Partial system inverse Engineering , Aerospace Technology and Astronautics , Mathematical Applications in the Physical Sciences , Space Sciences (including Extraterrestrial Physics, Space Exploration and Astronautics)
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