Nevanlinna-Pick Interpolation Research Articles

Robustness Result for Linear Continuous-Time SISO Systems

This paper contains results on stability robustness for continuous-time SISO systems comprising a plant with H∞ norm bounded uncertainty in the numerator and a feedback controller with integral action. An upper bound on the H∞ norm of the minimal destabilising perturbation is obtained. Also the class of nominal plants for which this robustness bound can be achieved with a linear time invariant controller is distinguished. Membership of this class is equivalent to the positive semidefiniteness of a Pick matrix, whose dimension is the number of unstable plant poles. A design example is also included.

Bitangential structured interpolation theory

In this paper we present a generalization of the classical bitangential Nevanlinna-Pick theory in which one bounds not the norm of the interpolating functions but their structured singular value This work was motivated by some problems arising in robust control of systems with structured uncertainty. This approach is based on the commutant lifting theory of Sz.-Nagy and Foias (1968) and extends previous work of Bercovici, Foias and Tannenbaum (1990) on structured matrix Nevanlinna-Pick interpolation.

The trisecant identity and operator theory

For the special case of a Riemann surface which arises as the double of a planar domainR, the trisecant identity has a natural interpretation as a relation among reproducing kernels for subspaces of the Hardy spaceH2(R). This relation and Riemann's theorem on the vanishing of the theta function is applied to Nevanlinna-Pick interpolation onR.

Nevanlinna–Pick Type Interpolation in a Dual Algebra

The duality approach of Sarason, Abrahamse, Ball, and others is extended to obtain a verion of Nevanlinna–Pick interpolation for a weak star closed algebra of operators on Hilbert space. In this context factorization is the polar decomposition of a positive trace class operator. Several concrete Nevanlinna–Pick type theorems are seen to follow from the main result and various versions of the Wold decomposition.

双一次変換パラメータをもつ補間コントローラ

A stabilization problem is considered for linear time-invariant systems interpolated with two nominal plants, where the controller is an interpolation of stabilizing controllers for the nominal plants. So far, a linear function is used as the interpolation, and the linearity restricts the class of stabilizable plants. Employing a bilinear function as the interpolation leads to some enlargement of the freedom of the controller design, although the cost of the implementation remains to be almost same as the linear case. Therefore, we consider the stabilization problem, where both the plant and the controller are interpolated bilinearly.In the SISO case, a necessary and sufficient condition for the stabilization of the interpolated plants is derived based on the Nevanlinna-Pick interpolation problem. Furthermore, an LMI description of the condition gives a method to decide a feasible bilinear interpolation. In the MIMO case, a sufficient condition is formed as a strict positive realness condition, and the design method is also proposed with LMI conditions in the state-space form. The effectiveness of the proposed methods is shown with some examples in both cases.

Solutions to the combined sensitivity and complementary sensitivity problem in control systems

In a recent paper, a set of necessary and sufficient conditions for the existence of solutions to the combined sensitivity and complementary sensitivity (CSCS) problem in control systems was given. However, a practical method for constructing solutions to this problem was not presented. The computation procedure that was proposed involves constructing a sequence of rational functions to approximate an irrational function and may result in complicated high-order controllers. In this paper, we consider the synthesis aspect of the CSCS problem and derive an algorithm for constructing controllers that solve our CSCS problem in a simple manner. Moreover, an error in the closed-form solution of the Nevanlinna-Pick interpolation problem is revealed.

On the solution of Nevanlinna Pick problem with selfadjoint extensions of symmetric linear relations in Hilbert space

The representation of Nevanlinna Pick Problem is well known, see [7], [8] and [11]. The aim of this paper is to find the necessary and sufficient condition for the solution of Nevanlinna Pick Problem and to show that there is a one‐to‐one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of symmetric linear relation in Hilbert space. Finally, we define the resolvent matrix which gives the solutions of the Nevanlinna Pick Problem.

On the Operator-Valued Nevanlinna-Pick Problem

In this note we construct a specific Schur-Nevanlinna type algorithm that will be used in describing more precisely the solutions of the operator-valued Nevanlinna-Pick problem with the so-called method of Weyl circles. We also present an approach to the Pick criterion in the operator-valued setting.

Classes of pairs of meromorphic matrix valued functions generated by nonnegative kernels and associated Nevanlinna-Pick problems

Families of pairs of matrix-valued meromorphic functions $$\mathcal{P}$$ (ρ,P) depending on two parameters ϱ andP are introduced. They are the projective analogues of classes of functions studied in [1] and include as special cases the projective Schur, Nevanlinna and Caratheodory classes. A two sided Nevanlinna-Pick interpolation problem is defined and solved in $$\mathcal{P}$$ (ρ,P), using the fundamental matrix inequality method.

Worst case system identification in H/sub ∞/: validation of a priori information, essentially optimal algorithms, and error bounds

Resolves several important open issues pertaining to a worst-case control-oriented system identification problem known as identification in H/sub /spl infin//. First, a method is presented for developing confidence that certain a priori information available for identification is not invalid. This method requires the solution of a certain nondifferentiable convex program. Second, an essentially optimal identification algorithm is constructed. This algorithm is (worst-case strongly) optimal to within a factor of two. Finally, new upper and lower bounds on the optimal identification error are derived and used to estimate the identification error associated with the given algorithm. Interestingly, the development of each of these results draws heavily upon the classical Nevanlinna-Pick interpolation theory. As such, the authors' results establish a clear link between the areas of system identification and optimal interpolation theory. Both the formulation and techniques in this paper are applicable to problems where the frequency data available for identification may essentially be arbitrarily distributed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stability margin optimization via interpolation and conformal mappings

Many interesting problems of robust stabilization can be solved using the complex function theoretic approach of Khargonekar and Tannenbaum (1985). This method is based on the construction of a conformal mapping which reduces the original robust stabilization problem to a Nevanlinna-Pick interpolation problem. The main difficulty of this approach lies in the construction the conformal mapping. The authors present a new numerical algorithm to construct such conformal mappings. Then they demonstrate its effectiveness in the synthesis of optimal gain-phase margin controllers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A lower bound for limiting time delay for closed-loop stability of an arbitrary SISO plant

This correspondence is concerned with the time delay margin for closed-loop stability of a single-input single-output plant with time delay uncertainty. Since no restriction is placed on the stabilizing controller, the limiting value of time delay depends on the plant transfer function only. The approach, using a modified form of classical Nevanlinna-Pick interpolation theory, provides a lower bound for the limiting time delay given the plant open-loop poles and zeros only.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On maximum entropy interpolants and maximum determinant completions of associated Pick matrices

In this paper we shall show that “the value” of the maximum ω-entropy solution of an interpolation problem of the Nevanlinna-Pick type at the point ω maximizes the determinant of the solution of an associated matrix completion problem. This serves to show that the solutions of two distinct extremal problems coincide.

On the tangential interpolation problem for 𝐻₂ functions

The aim of this paper is to solve a matrix-valued version of the Nevanlinna-Pick interpolation problem for H 2 {H_2} functions. We reduce this problem to a Nevanlinna-Pick interpolation problem for Schur functions and obtain a linear fractional transformation which describes the set of all solutions.

Nevanlinna-Pick interpolation problem for two frequency scale systems

The Nevanlinna-Pick interpolation problem for a parametric set of data is considered. Sufficient conditions are obtained that guarantees a two frequency scale solution. Decomposition of the problem into two Nevanlinna-Pick interpolation problems with smaller data size is presented. An algorithm is developed which computes the solution as the result of combining the solutions for these two smaller problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Active Versus Passive Vibration Absorbers

Passivity based control has several advantages, e.g., robustness and simplicity. However, the passivity constraint causes performance degradations. Specifically, the degree to which passively controlled absorber systems yield a flat response around the nominal absorber frequency is limited. An inherent flatness bound is derived using Nevanlinna-Pick boundary interpolation theory. For active vibration absorbers, this limitation cancels, and magnitude response can be made arbitrarily flat.

Nehari and Nevanlinna-Pick Problems and Holomorphic Extensions in the Polydisk in Terms of Restricted BMO

A new class BMOr is introduced, motivated by the following extension of the Nehari theorem to d ≥ 1: The norm of a bounded (big) Hankel form B in the d-dimensional torus satisfies || φ|| ≤ || B|| ≤ √ d || φ || where the norm of φ, the antianalytic symbol of B, is taken in BMOr. This extension in turn provides solutions to Nevanlinna-Pick type interpolation problems in the polydisk, with control of the BMOr norm. Necessary and sufficient conditions for the existence of H 2 ∩ BMOr extensions of analytic functions defined on a zero variety V are given, with control of the norm, under general conditions on V.

Computing matrix-valued Nevanlinna-Pick interpolation

We describe a computational method, known as the Nevanlinna algorithm, for the matrix-valued Nevanlinna-Pick interpolation. The original interpolation problem formulated using the Carathéodory class of matrix-valued rational functions is first converted to an equivalent setting using the Schur class of rational functions. As a result, the necessary and sufficient Pick's condition for the interpolation becomes consistent with the scalar-valued formulation, so that some efficient techniques developed for the scalar-valued interpolation can be employed or modified for the matrix-valued case. We give a brief, yet sufficiently clear, derivation and a detailed arithmetic complexity analysis for the algorithm. We show that an n-point matrix-valued Nevanlinna-Pick interpolation using the new algorithm requires approximately 95 nm 3 complex arithmetic operations, where m is the matrix dimension.

Robust stabilization: some extensions of the gain margin maximization problem

Considers robust stabilization for SISO systems with linear dependence on a real uncertain parameter which may occur either in the numerator or denominator of the plant transfer function. The authors show necessary and sufficient conditions for the existence of robust controllers. Their structure and parameterization can be obtained from the corresponding Nevanlinna-Pick interpolation problems. This work extends the previously known case of an uncertain gain parameter. >

On a Class of Functions Analytic in a Half Disk and an Associated Interpolation Problem

In this paper we introduce a class of functions analytic in a half disk and for which the Nevanlinna-Pick interpolation problem has a solution in terms of a linear fractional transformation. The method used to solve this problem is that of the fundamental matrix inequality, suitably adapted to the present situation.