Arthur E. Frazho

The commutant lifting approach to interpolation problems

I. Analysis of the Caratheodory Interpolation Problem.- II. Analysis of the Caratheodory Interpolation Problem for Positive-Real Functions.- III. Schur Numbers, Geophysics and Inverse Scattering Problems.- IV. Contractive Expansions on Euclidian and Hilbert Space.- V. Contractive One Step Intertwining Liftings.- VI. Isometric and Unitary Dilations.- VII. The Commutant Lifting Theorem.- VIII. Geometric Applications of the Commutant lifting Theorem.- IX. H? Optimization and Functional Models.- X. Some Classical Interpolation Problems.- XI. Interpolation as a Momentum Problem.- XII. Numerical Algorithms for H? Optimization in Control Theory.- XIII. Inverse Scattering Algorithms for the Commutant Lifting Theorem.- XIV. The Schur Representation.- XV. A Geometric Approach to Positive Definite Sequences.- XVI. Positive Definite Block Matrices.- XVII. A Physical Basis for the Layered Medium Model.- References.- Notation.

Metric constrained interpolation, commutant lifting, and systems

Part 1 Interpolation and time-invariant system: interpolation problems for time-valued functions proofs using the commutant lifting theorem time invariant systems central commutant lifting central state space solutions parametization of intertwinning and its applications applications to control systems. Part 2 Nonstationary interpolation and time-varying systems nonstationary interpolation theorems nonstationary systems and point evaluation reduction techniques - from nonstationary to stationary and vice versa proofs of the nonstationary interpolation theorems by reduction to the stationary case a general completion theorem applications of the three chains completion theorem to interpolation parameterization of all solutions of the three chains completion problem.

Models for noncommuting operators

Abstract This paper develops a model theory for a pair of noncommuting operators. Using backward shift operators on a Fock space Rotas Theorem is generalized, i.e., it is shown that any two bounded operators on a Hilbert space are simultaneously similar to part of a pair of backward shift operators on a Fock space. These shift operators and the Fock space framework are also used to develop a dilation theory for two noncommuting operators.

Linear Systems and Control : An Operator Perspective

Systems and control stability Lyapunov theory least squares observability controllability controllable and observable realizations realization theory state feedback state estimators output feedback controllers zeros and constant output tracking linear quadratic regulators H analysis H control.

Complements to models for noncommuting operators

Abstract This paper develops a dilation theory for two noncommutative operators. Many of the results and techniques in dilation theory for one operator are extended to this setting. This is done by extending the Schaffer construction for one operator to the Fock space setting. A new Wold decomposition for two isometrics with orthogonal range is obtained. This Wold decomposition and the Schaffer construction is used to extend the Sz.-Nagy-Foias lifting theorem and the characteristic function theory to the noncommutative setting. The characteristic function is operator valued and defined in a Fock space.

A geometric approach to the maximum likelihood spectral estimator for sinusoids in noise

The problem of estimating sinusoids that have been corrupted by additive stationary noise is addressed. It is shown how the Naimark dilation for the data correlation sequence can be used to provide additional insight into some fundamental results on orthogonal polynomials and to give a new interpretation of the maximum-likelihood (ML) spectral estimator. >

Application of alternating convex projection methods for computation of positive Toeplitz matrices

Uses alternating convex projection techniques to compute the closest positive definite Toeplitz matrix that satisfies certain inequality constraints to a specified symmetric matrix. Some applications to signal processing and control problems are discussed. >

Relaxation of metric constrained interpolation and a new lifting theorem

In this paper a new lifting interpolation problem is introduced and an explicit solution is given. The result includes the commutant lifting theorem as well as its generalizations in [27] and [2]. The main theorem yields explicit solutions to new natural variants of most of the metric constrained interpolation problems treated in [9]. It is also shown that via an infinite dimensional enlargement of the underlying geometric structure a solution of the new lifting problem can be obtained from the commutant lifting theorem. However, the new setup presented obtained from the commutant lifting theorem. However, the new setup presented in this paper appears to be better suited to deal with interpolations problems from systems and control theory than the commutant lifting theorem.

An Operator Perspective on Signals and Systems

Preface.- I Basic Operator Theory.- 1 The Wold Decomposition.- 2 Toeplitz and Laurent Operators.- 3 Inner and Outer Functions.- 4 Rational Inner and Outer Functions.- 5 The Naimark Representation.- 6 The Rational Case.- II Finite Section Techniques.- 7 The Levinson Algorithm and Factorization.- 8 Isometric Representations and Factorization.- 9 Signal Processing.- III Riccati Methods.- 10 Riccati Equations and Factorization.- 11 Kalman and Wiener Filtering.- IV Interpolation Theory.- 13 Contractive Nevanlinna-Pick Interpolation.- V Appendices.- 14 A Review of State Space.- 15 The Levinson Algorithm.- Index.

Alternating convex projection methods for discrete-time covariance control design

The problem of designing a controller for a linear, discretetime system is formulated as a problem of designing an appropriate plant-state covariance matrix. Closed-loop stability and multiple-output performance constraints are expressed geometrically as requirements that the covariance matrix lies in the intersection of some specified closed, convex sets in the space of symmetric matrices. We solve a covariance feasibility problem to determine the existence and compute a covariance matrix to satisty assignability and output-norm performance constraints. In addition, we can treat a covariance optimization problem to construct an assignable covariance matrix which satisfies output performance constraints and is as close as possible to a given desired covariance. We can also treat inconsistent constraints, where we look for an assignable covariance which best approximates desired but unachievable output performance objectives; we call this the infeasible covariance optimization problem. All these problems are of a convex nature, and alternating convex projection methods are proposed to solve them, exploiting the geometric formulation of the problem. To this end, analytical expressions for the projections onto the covariance assignability and the output covariance inequality constraint sets are derived. Finally, the problem of designing low-order dynamic controllers using alternating projections is discussed, and a numerical technique using alternating projections is suggested for a solution, although convergence of the algorithm is not guaranteed in this case. A control design example for a fighter aircraft model illustrates the method.