On approximation of stable linear dynamical systems using Laguerre and Kautz functions

Approximation of stable linear dynamical systems by means of so-called Laguerre and Kautz functions, which are the Laplace transforms of a class of orthonormal exponentials, is studied. Since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials (times polynomials), it seems reasonable to use basis functions of the same type. Assuming that the transfer function of a system is bounded and analytic outside a given disc, it is shown that Laguerre basis functions are optimal in a mini-max sense. This result is extended to the “two-parameter” Kautz functions which can have complex poles, while the poles of Laguerre functions are restricted to the real axis. By conformai mapping techniques the “two-parameter” Kautz approximation problem is recast as two Laguerre approximation problems. Thus, the well-developed theory of Laguerre functions can be applied to analyze Kautz approximations. Unilateral shifts are used to further develop the connection between Laguerre functions and Kautz functions. Results on þ2 and þ∞ approximation using Kautz models are given. Furthermore, the weighted L2 Kautz approximation problem is shown to be equivalent to solving a block Toeplitz matrix equation.

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