Nevanlinna-Pick Interpolation Research Articles

RANK OF THE DERIVATIVE OF THE PROJECTION TO SYMMETRIZED POLYDISC

The projection, also called the symmetrization mapping, from spectral ball to symmetrized polydisc is closely related to the spectral Nevanlinna-Pick interpolation problem. We prove that the rank of the derivative of the projection from the spectral unit ball to the symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take the derivative. Therefore, it explains why the corresponding lifting problem is easier when the matrix base-point is cyclic since it is a regular point of the symmetrization mapping in the differential sense.

Coherent equalization of linear quantum systems

This paper introduces a H∞-like methodology of coherent filtering for equalization of passive linear quantum systems to help mitigate degrading effects of quantum communication channels. For such systems, which include a wide range of linear quantum optical devices and signals, we seek to find a near optimal equalizing filter which is itself a passive quantum system. The problem amounts to solving an optimization problem subject to constraints dictated by the requirement for the equalizer to be physically realizable. By formulating these constraints in the frequency domain, we show that the problem admits a convex H∞-like formulation. This allows us to derive a set of suboptimal coherent equalizers using J-spectral factorization. An additional semidefinite relaxation combined with the Nevanlinna–Pick interpolation is shown to lead to a tractable algorithm for the design of a near optimal coherent equalizer.

Parameter and Topology Optimization of Structures in the Frequency Domain under Nevanlinna–Pick Interpolation Constraints

The design optimization of structures can be conducted in either the time domain or the frequency domain. The frequency domain approach is advantageous compared to its time domain counterpart, especially if the degree of freedom is large, the objectives and/or constraints are formulated in the frequency domain, or the structure is subject to random loading. In this paper, an attempt is undertaken to obtain feasible optimal solutions by implementing the Nevanlinna–Pick (NP) interpolation theory across multi-objective structural optimization problems in the frequency domain. The NP equations introduce a trade-off that originates from the interpolation theory for complex variables. According to the NP theory, a complex function cannot have an independent amplitude from its derivative at a certain frequency. Consequently, the frequency response of a physical system cannot be shaped arbitrarily at discrete frequencies. Our objectives within this paper include calculating the weight, natural frequency, fatigue life, frequency domain response, and its derivative. To illustrate our claims, sample parameter and topology optimization problems were formulated and solved, both with and without the NP constraints. It was found that the inclusion of NP constraints induced a considerable improvement in the optimal solutions, while also causing the convergence to the optimal solution to become smoother.

H ∞ interpolation constrained by Beurling–Sobolev norms

Abstract We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H 2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.

Worst Case Uncertainty Construction via Multifrequency Gain Maximization With Application to Flutter Control

In the analysis of uncertain systems, we often search for a worst case perturbation that drives the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_\infty $ </tex-math></inline-formula> gain of the system to the maximum over the set of allowable uncertainties. Employing the classical technique, an uncertainty sample is obtained, which, indeed, maximizes the gain but only at the single frequency where that maximum occurs. In contrast, this article considers a method to calculate a worst case perturbation that maximizes the gain of a system with mixed uncertainty at multiple frequencies simultaneously. This approach involves a nonlinear optimization that selects the worst case value of the uncertain parameters and the application of the boundary Nevanlinna–Pick interpolation to calculate the dynamic uncertainty sample. Such a perturbation can be used to augment Monte Carlo simulations of uncertain systems, especially if the system has multiple resonance frequencies. The worst case analysis of a flutter control system designed for a small flexible aircraft is provided to demonstrate the applicability of the proposed method.

Solvability of the Nevanlinna-Pick interpolation problem

A solvability theorem is proved for the Nevanlinna-Pick interpolation problem. Its extreme cases are Carathéodory's and Sсhur's criteria on the one hand (when all interpolation points coincide) and the Krein-Rekhtman theorem on the other (when the interpolation points are pairwise distinct). Bibliography: 19 titles.

Design of discrete-time matrix all-pass filters using subspace Nevanlinna pick interpolation

Unitary matrix-valued functions of frequency are matrix all-pass systems, since they preserve the norm of the input vector signals. Typically, such systems are represented and analyzed using their unitary-matrix valued frequency domain characteristics, although obtaining rational realizations for matrix all-pass systems enables compact representations and efficient implementations. However, an approach to obtain matrix all-pass filters that satisfy phase constraints at certain frequencies was hitherto unknown. In this paper, we present an interpolation strategy to obtain a rational matrix-valued transfer function from frequency domain constraints for discrete-time matrix all-pass systems. Using an extension of the Subspace Nevanlinna Pick Interpolation Problem (SNIP), we design a construction for discrete-time matrix all-pass systems that satisfy the desired phase characteristics. An innovation that enables this is the extension of the SNIP to the boundary case to obtain efficient time-domain implementations of matrix all-pass filters as matrix linear constant coefficient difference equations, facilitated by a rational (realizable) matrix transfer function. We also show that the derivative of matrix phase constraints, related to the group delay at the interpolating points, can be optimized to control the all-pass transfer matrices at the unspecified frequencies. Simulations show that the proposed technique for unitary matrix filter design performs as well as traditional DFT based interpolation approaches, including Geodesic interpolation and the popular Givens rotation based matrix parameterization.

A Hill-Pick matrix criterion for the Lyapunov order

The Lyapunov order appeared in the study of Nevanlinna-Pick interpolation for positive real odd functions with general (real) matrix points. For real or complex matrices A and B it is said that B Lyapunov dominates A ifH=H⁎,HA+A⁎H⩾0⇒HB+B⁎H⩾0. (In case A and B are real we usually restrict to real Hermitian matrices H, i.e., symmetric H.) Hence B Lyapunov dominates A if all Lyapunov solutions of A are also Lyapunov solutions of B. In this paper we restrict to the case that appears in the study of Nevanlinna-Pick interpolation, namely where B is in the bicommutant of A and where A is Lyapunov regular, meaning the eigenvalues λj of A satisfyλi+λ‾j≠0,i,j=1,…,n. In this case we provide a matrix criterion for Lyapunov dominance of A by B. The result relies on a class of ⁎-linear maps for which positivity and complete positivity coincide and a representation of ⁎-linear matrix maps going back to work of R.D. Hill. The matrix criterion asks that a certain matrix, which we call the Hill-Pick matrix, be positive semidefinite.

Multiplicative Property of the Series Used in the Nevanlinna–Pick Problem

We obtain a qualitatively new sufficient condition of nonnegativity for the coefficients of a power series inverse to a series with positive coefficients. In particular, we prove that the element-wise product of the power series retains this property. In particular, this generalizes the classical Hardy–Kaluza theorem on power series. These results are generalized for the case of multivariable power series. Results of this kind are applied in the Nevanlinna–Pick theory.

Convex invertible cones and Nevanlinna-Pick interpolation: The suboptimal case

Nevanlinna-Pick interpolation developed from a topic in classical complex analysis to a useful tool for solving various problems in control theory and electrical engineering. Over the years many extensions of the original problem were considered, including extensions to different function spaces, nonstationary problems, several variable settings and interpolation with matrix and operator points. Here we discuss a variation on Nevanlinna-Pick interpolation for positive real odd functions evaluated in real matrix points. This problem was studied by Cohen and Lewkowicz using convex invertible cones and the Lyapunov order, but was never fully resolved. In this paper we present a solution to this problem in a special case that we refer to as ‘suboptimal’ based on connections with the classical case. The solution requires a representation of linear matrix maps going back to R.D. Hill and an analysis of when positive linear matrix maps are completely positive, on which we reported in earlier work and which we will briefly review here.

Multiplicative property of series used in the Nevanlinna-Pick problem

In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna – Pick theory. For example, if function k(x, y) can be represented as power series Pn≥0 an(x¯y)n, an &gt; 0, and reciprocal function 1/k(x, y) can be represented as power series Pn≥0 bn(x¯y)n such that bn &lt; 0, n &gt; 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna – Pick property. The reproducing kernel 1/(1 − x¯y) of the classical Hardy space H2(D) is a prime example for our theorems.

Multiplier tests and subhomogeneity of multiplier algebras

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.

Indeterminacy conditions for the matrix Nevanlinna-Pick problem and rational matrix functions of the first and second kind

Indeterminacy conditions for the matrix Nevanlinna-Pick problem and rational matrix functions of the first and second kind

Small-signal model and control approach for a dual input boost converter with VMC

ABSTRACT The dual-input boost converter (DIBC) with voltage multiplier cell (VMC) and operating in continuous conduction mode (CCM) is feasible for applications with high voltage from low voltage renewable energy sources (RES). To achieve a high voltage gain, DIBC is operated with two VMC, and overlapping switch signals, i.e. at duty ratio higher than 0.5; these complicate the modelling and control design. The objective here is to achieve output voltage regulation under load and input voltage disturbances with the proposed feed-forward plus feedback (FFFB) control strategy for DIBC with two VMC. Nevanlinna–Pick interpolation method and MATLAB SISOtool are used to design the optimally stable feed-forward controller and feedback controller, respectively. The study of the non-minimum phase behaviour of the converter, followed by the control design, is done based on the derived averaged state-space small-signal transfer function models. The analytical model is validated using MATLAB and PSIM software and experimentally. A converter prototype of 250 W is built and experimentally tested; the control is realised using FPGA. The simulation and experimental results ensure that the converter with this proposed control design provides satisfactory voltage regulation.

Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows — generalizing a result of Ransford–White for the spectral unit ball — that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider [Formula: see text]-point and [Formula: see text]-point Pick–Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

Bergman spaces over noncommutative domains and commutant lifting

The goal of the present paper is to provide analogues of Sarason interpolation theorem in the Hardy algebra H∞(D) and Sz.Nagy-Foiaş commutant lifting theorem for contractions on Hilbert spaces in the setting of noncommutative Hardy spaces associated with noncommutative regular domains and varieties. This is accompanied by the study of multi-analytic operators with respect to the universal models associated with the regular domains (resp. varieties) and the study of multipliers of noncommutative Bergman spaces.As applications, we obtain Toeplitz-corona theorems for multi-analytic operators, commutant lifting in several variables where the liftings are in certain Schur classes, factorization of multi-analytic operators, and Nevanlinna-Pick interpolation results for multipliers of Bergman spaces over Reinhardt domains in Cn. In addition, we obtain Andô type dilations and inequalities on noncommutative bi-domains and varieties.

Interpolation by holomorphic maps from the disc to the tetrablock

The tetrablock is the setE={x∈C3:1−x1z−x2w+x3zw≠0whenever|z|≤1,|w|≤1}. The closure of E is denoted by E‾. A tetra-inner function is an analytic map x from the unit disc D to E‾ such that, for almost all points λ of the unit circle T,limr↑1⁡x(rλ) exists and lies in bE‾, where bE‾ denotes the distinguished boundary of E‾. There is a natural notion of degree of a rational tetra-inner function x; it is simply the topological degree of the continuous map x|T from T to bE‾. In this paper we give a prescription for the construction of a general rational tetra-inner function of degree n. The prescription exploits a known construction of the finite Blaschke products of given degree which satisfy some interpolation conditions with the aid of a Pick matrix formed from the interpolation data. It is known that if x=(x1,x2,x3) is a rational tetra-inner function of degree n, then x1x2−x3 either is identically 0 or has precisely n zeros in the closed unit disc D‾, counted with multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function x=(x1,x2,x3) consists of the points in D‾ for which x1x2−x3=0 and the values of x at these points.

Delay Margin Optimization for Systems with Internal Delayed Feedback

In this brief paper, controller design for delay margin optimization is considered for systems with internal feedback delays (systems with delays in the state variables). Similar to existing results on delay margin optimization for finite dimensional systems with I/O delays, it is shown that the problem considered can be solved by using Nevanlinna-Pick interpolation involving non-minimum phase zeros of the plant.

Effective noncommutative Nevanlinna-Pick interpolation in the row ball, and applications

We provide an effective single-matrix criterion, in terms of what we call the elementary Pick matrix, for the solvability of the noncommutative Nevanlinna-Pick interpolation problem in the row ball, and provide some applications. In particular we show that the so-called “column-row property” fails for the free semigroup algebras, in stark contrast to the analogous commutative case. Additional applications of the elementary Pick matrix include a local dilation theorem for matrix row contractions and interpolating sequences in the noncommutative setting. Finally we present some numerical results related to the failure of the column-row property.

Commutant Lifting and Nevanlinna–Pick Interpolation in Several Variables

This paper concerns a commutant lifting theorem and a Nevanlinna–Pick type interpolation result in the setting of multipliers from vector-valued Drury–Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in \({\mathbb {C}}^n\). The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.