Identification of microwave filters by analytic and rational approximation

Abstract

In this paper, an original approach to frequency identification is explained and demonstrated through an application in the domain of microwave filters. This approach splits in two stages: a stable and causal model of high degree is first computed from the data (completion stage); then, model reduction is performed to get a rational low order model. In the first stage the most is made of the data taking into account the expected behavior of the filter. A reduced order model is then computed by rational H 2 approximation. A new and efficient method has been developed, improved over the years and implemented to solve this problem. It heavily relies on the underlying Hilbert space structure and on a nice parametrization of the optimization set. This approach guarantees the stability of the MIMO approximant of prescribed McMillan degree.

Minimax principle and lower bounds in H2-rational approximation

“…In applied sciences, problem RAB(n) and weighted variants thereof are of great importance to model time series as well as to identify linear dynamical systems, 2 e.g. in modal analysis of mechanical structures or in frequency analysis of microwave devices [22,24,27,35,30]. The importance of the L 2 norm in this context stems from its statistical interpretation as a variance.…”

“…Our purpose here is not to discuss these techniques, nor to compare them with dedicated optimization algorithms [13,30], but rather to stress a link between the value of RAB(n) and the minimization of linearized errors.…”

“…Special interest attaches to the case where the approximated function extends holomorphically on one side of the curve. In connection with system identification and control, such issues typically arise on the line or the circle where they make contact with extremal problems in Hardy spaces [4,15,33,29,30,32,43]. Our model curve in this paper will be the circle, though everything translates easily to the line.…”

Model-Free Closed-Loop Stability Analysis: A Linear Functional Approach

“…The projection described here has been used before to perform stable interpolation and extrapolation of FRF data [22]. With slight modifications, it can be turned into a full-blown stability analysis.…”

Constrained L2-Approximation by Polynomials on Subsets of the Circle

“…Indeed, on the line, constrained extremal problems for analytic functions naturally arise in Engineering when studying deconvolution issues, in particular those pertaining to system identification and design. This motivation is stressed in [12,4,5,19,2], whose results are effectively used today to identify microwave devices [1,14]. More precisely, recall that a linear time-invariant dynamical system is just a convolution operator, hence the Fourier-Laplace transform of its output is that of its input times the Fourier-Laplace transform of its kernel.…”