W. Steven Gray

Generating Series for Interconnected Analytic Nonlinear Systems

Given two analytic nonlinear input-output systems represented as Fliess operators, four system interconnections are considered in a unified setting: the parallel connection, product connection, cascade connection, and feedback connection. In each case, the corresponding generating series is produced and conditions for the convergence of the corresponding Fliess operator are given. In the process, an existing notion of a composition product for formal power series has its set of known properties significantly expanded. In addition, the notion of a feedback product for formal power series is shown to be well defined in a broad context, and its basic properties are characterized.

Observability functions for linear and nonlinear systems

A generalization of the zero-state observability function is considered for nonlinear systems. The linear time-invariant case is considered as an application in model reduction problems.

Nonlinear Hilbert adjoints: properties and applications to Hankel singular value analysis

The notion of an adjoint operator for a nonlinear mapping has few interpretations in the literam. In this paper a new nonlinear Hilbert adjoint operator is proposed. It is shown to unite several existing concepts and provides an essential tool for singular value analysis of nonlinear Hankel operators.

Energy Functions and Algebraic Gramians for Bilinear Systems

Abstract In linear system theory, Gramian matrices help quantify the input-tostate and state-to-output interactions in a state space model. In this paper two different existing generalizations of this idea are examined for the case of a stable bilinear system. One method is based on using the L 2 norm to measure signal sizes and leads to the notion of the controllability and observability functions, which are referred to collectively as energy functions. The other method is motivated by introducing algebraic generalizations of the linear system Lyapunov equations, the solutions of which are called the algebraic Gramians. While these generalizations are distinct, in this paper some new relationships between the two approaches are presented.

Hamiltonian realizations of nonlinear adjoint operators

This paper addresses the issue of state-space realizations for nonlinear adjoint operators. In particular, the relationships between nonlinear Hilbert adjoint operators, Hamiltonian extensions and port-controlled Hamiltonian systems are established. Then, characterizations of the adjoints of controllability, observability and Hankel operators are derived from this analysis. The state-space realizations of such adjoint operators provide new insights on singular value analysis and duality issues in nonlinear control systems theory. Finally, a duality between the controllability and observability energy functions is proved.

On the nonuniqueness of singular value functions and balanced nonlinear realizations

The notion of balanced realizations for nonlinear state space model reduction was first introduced in 1993. Analogous to the linear case, the so called singular value functions of a system describe the relative importance of each state component from an input-output point of view. In this paper it is shown that the usual procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case. Specifically, it appears that the singular value functions as currently defined are dependent on a particular factorization of the observability function. It is shown by example that in a fixed coordinate frame this factorization is not unique, and thus other distinct definitions for the singular value functions and balanced realizations are possible. One method relating singular value functions from different factorizations is presented.

Discrete Time Nonlinear Balancing

Abstract This paper extends earlier work on balanced realizations for smooth nonlinear systems to the discrete time setting. The main idea behind the global balanced structure, if it exists, is that the linearized system along a nominal solution is balanced in the usual linear time-varying sense. Hence the global nonlinear balanced system is defined as the one that closes the commutative diagram between linearization and balancing.

A Geometric Approach to the Minimum Sensitivity Design Problem

In this paper a Riemannian geometric framework is given for the minimum sensitivity design problem and its solution using a natural optimization criterion. The theory is then applied to the case of linear systems to generate a class of minimum sensitivity realizations related to the so-called balanced realizations. In particular, conditions are given which are applicable in the cases of fixed point and floating point implementations.

Left inversion of analytic nonlinear SISO systems via formal power series methods

Given a single-input, single-output (SISO) system, F , and a function y in the range of F , the left inversion problem is to determine an input u such that y = F u . The goal of this paper is to provide an exact and explicit analytical solution to this problem in the case where F is an analytic mapping in the sense that it has a convergent Chen-Fliess functional expansion, and y is a real analytic function. In particular, it will be shown that given a certain condition on the generating series c of F , a corresponding unique analytic u can always be determined via operations on formal power series. The condition on c turns out to be equivalent to having a well-defined relative degree when F has an input-affine analytic state space realization with finite dimension. But the method is applicable even when F does not have such a realization. The technique is demonstrated on four examples, including a continuous stirred chemical reactor.

ON THE RADIUS OF CONVERGENCE OF INTERCONNECTED ANALYTIC NONLINEAR INPUT-OUTPUT SYSTEMS ∗

A complete analysis is presented of the radii of convergence of the parallel, product, cascade and feedback interconnections of analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, which are indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions for the coefficients of the generating series for the subsystems, the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent. In the process of deriving the radius of convergence for the feedback connection, it is shown definitively that local convergence is preserved under feedback. This had been an open problem in the literature until recently.