van Sjl Stef Eijndhoven

Linear quadratic regulator problem with positive controls

In this paper, the Linear Quadratic Regulator Problem with a positivity constraint on the admissible control set is addressed. Necessary and sufficient conditions for optimality are presented in terms of inner products, projections on closed convex sets, Pontryagins maximum principle and dynamic programming. The main results are concerned with smoothness of the optimal control and the value function. The maximum principle will be extended to the infinite horizon case. Based on these analytical methods, we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem. The numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other.

New orthogonality relations for the Hermite polynomials and related Hilbert spaces

The Hermite polynomials give rise to orthonormal bases in Bagmann-like Hilbert spaces XA, 0 < A < 1, consisting of entire functions. These Hilbert spaces are related to the Gelfand-Shilov space S1212, viz. S1212 = ∪0 < A

Eigencurrent analysis of resonant behavior in finite antenna arrays

1 XA.

Spaces of type W, growth of Hermite coefficients, Wigner distribution, and Bargmann transform

Resonant behavior in a finite array that appears as (modulated) impedance or current-amplitude oscillations may limit the array bandwidth substantially. Therefore, simulations should predict such behavior. Recently, a new approach has been developed, called the eigencurrent approach, which can predict resonant behavior in finite arrays. A study of line arrays of either E- or H-plane-oriented strips and rings in free space and in half-spaces confirms our conclusion in earlier research that resonant behavior is caused by the excitation of one of the eigencurrents. The eigenvalue (or characteristic impedance) of this eigencurrent becomes small in comparison to the eigenvalues of the other eigencurrents that can exist on the array geometry. We demonstrate that the excitation of this eigencurrent results in an edge-diffracted wave propagating along the surface of the array, which may turn into a standing wave. In that case, the amplitudes and phases of the element impedances show the same standing-wave pattern as those of the excited eigencurrent. We demonstrate that the phase velocity of this wave is approximately equal to or slightly larger than the free-space velocity of light. Finally, we throw light on the relation between the excitation of eigencurrents with a small eigenvalue and the behavior of super directive arrays

An Eigencurrent Approach for the Analysis of Finite Antenna Arrays

Abstract In their famous monograph on generalized functions Gelfand and Shilov introduce the function spaces W M , W Ω , and W M Ω . These spaces consist of C∞-functions and holomorphic functions, respectively, with growth behaviour specified by suitable convex functions M and Ω. In this paper we study the spaces WMMx, where Mx denotes Youngs dual function corresponding to M. We characterize the Hermite expansion coefficients of the functions in WMMx, their Fourier transforms, Wigner distributions, and Bargmann transforms. In particular, we prove that WMMx = WM ∩ WMx.

IIR-based pure linear prediction

An accurate description of typical finite-array behavior such as edge effects and array resonances is essential in the design of various types of antennas. The analysis approach proposed in this paper is essentially based on the concept of eigencurrents and is capable of describing finite-array behavior. In the approach numerical simulation is carried out, first, by computing element eigencurrents from chosen expansion functions and, second, by expanding a limited set of array eigencurrents in terms of element eigencurrents that contribute to the mutual coupling in the array. Both types of eigencurrents are eigenfunctions of an impedance operator that relates the current to the excitation field. Highlighting both mathematical and physical features we describe the basic concepts of the approach, in particular the relation between eigenvalues and mutual coupling. We illustrate these features for uniform linear arrays of loops and dipoles, and demonstrate that the approach provides significant improvements in terms of computation time and memory use.

Equivalence of convolution systems in a behavioral framework

This paper considers general, pure linear prediction schemes, where the prediction of the input signal is based on IIR-filtered versions of the one-sample-delayed input signal. Properties of these schemes are discussed, in particular, the whitening property and the realization and stability of the synthesis filter. In contrast to warped linear prediction, the synthesis filter can be realized in a way similar to the analysis filter. Furthermore, we prove that, at least for a specific class of systems, input data windowing for the calculation of the optimal prediction coefficients guarantees the stability of the synthesis filters. By simulation we show that the proposed prediction scheme, using properly parameterized Laguerre or Kautz systems, shows a behavior similar to that of warped linear prediction.

Optimal parameters in modulated Laguerre series expansions

AbstractIn this paper the problem of system equivalence is tackled for a rather general class of linear time-invariant systems. We consider AR-systems described by linear continuous shift-invariant operators with finite memory, acting on Fréchet-signal spaces, containing the space {\cal E} ({\open R}) of infinitely differentiable functions on {\open R}. This class is in one–one correspondence with matrices of suitable sizes over the convolution algebra {\cal E}′ ({\open R}) of all compactly supported distributions. Using some deep results from the theory of Fréchet spaces, various necessary and sufficient conditions for system equivalence and system inclusion are formulated. It is shown that a surjectivity demand on the system defining convolution operator matrix is necessary and sufficient for being able to translate the problem of system equivalence into division properties over the convolution algebra {\cal E}′({\open R}). This surjectivity condition is guaranteed if the system defining matrix over {\cal E}′({\open R}) has a right-inverse over {\cal D}′({\open R}), the space of all Schwartz distributions.

Analyticity spaces of self adjoint operators subjected to perturbations with applications to Hankel invariant distribution spaces

The harmonically modulated Laguerre functions constitute an orthonormal basis in the Hilbert space of square-integrable functions on R/sup +/. This basis comprises three free parameters, namely a modulation, a scale and an order factor. In practice, we are interested in series expansions that are as compact as possible. The free parameters can be used as the means to obtain a compact series expansion for a given function. As the compactness criterion the first-order moment of the energy distribution in the transform domain is chosen. In that case, the optimum compactness parameters can be given in a simple analytic form depending on signal measurements only.

A measure theoretical Sobolev lemma

A new theory of generalized functions has been developed by one of the authors (de Graaf). In this theory the analyticity domain of each positive self-adjoint unbounded operator