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Formal power series products appear in nonlinear control theory when systems modeled by Chen–Fliess series are interconnected to form new systems. In fields like adaptive control and learning systems, the coefficients of these formal power series are estimated sequentially with real-time data. The main goal is to prove the continuity and analyticity of such products with respect to several natural (locally convex) topologies on spaces of locally convergent formal power series in order to establish foundational properties behind these technologies. In addition, it is shown that a transformation group central to describing the output feedback connection is in fact an analytic Lie group in this setting with certain regularity properties. [ABSTRACT FROM AUTHOR]

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen–Fliess series and an additive static output feedback is applied. The first step is to consider the so-called Wiener-Fliess connection consisting of a Chen–Fliess series followed by a memoryless function. To explicitly compute the generating series, two Hopf algebras are needed: the existing output feedback Hopf algebra used to describe dynamic output feedback and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener–Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant. The convergence of the Wiener–Fliess composition product and the additive static feedback product are characterised. [ABSTRACT FROM AUTHOR]

Consider an input–output system where the output is the tracking error given some desired reference signal. It is natural to consider under what conditions the problem has an exact solution, that is, the tracking error is exactly the zero function. If the system has a well defined relative degree and the zero function is in the range of the input–output map, then it is well known that the system is locally left invertible, and thus, the problem has a unique exact solution. A system will fail to have relative degree when more than one exact solution exists. The general goal of this paper is to describe a decomposition of an input–output system having a Chen-Fliess series representation into a parallel product of subsystems in order to identify possible solutions to the problem of zeroing the output. For computational purposes, the focus is on systems whose generating series are polynomials. It is shown that the shuffle algebra on the set of generating polynomials is a unique factorization domain so that any polynomial can be uniquely factored modulo a permutation into its irreducible elements for the purpose of identifying the subsystems in a parallel product decomposition. This is achieved using the fact that this shuffle algebra is isomorphic to the symmetric algebra over the vector space spanned by Lyndon words. A specific algorithm for factoring generating polynomials into its irreducible factors is presented based on the Chen-Fox-Lyndon factorization of words. [ABSTRACT FROM AUTHOR]

Abstract The concept of relative degree plays an important role in nonlinear control theory. It provides, for example, a necessary and sufficient condition for the existence of a feedback linearizing control law for a single-input, single-output (SISO) input-affine nonlinear state space system. It also gives a sufficient condition under which a left inverse exists. In applications it is common for systems to be composed of smaller interconnected subsystems. It is known that various feedback structures preserve relative degree, but it is largely unknown how to compute the relative degree of interconnected systems using only their properties. So the goal of this paper is to determine the relative degree of two nonlinear control systems interconnected in a variety of different ways. A collection of illustrative examples is given. [ABSTRACT FROM AUTHOR]

A combinatorial interpretation is given of Devlin's word problem underlying the classical center problem of Poincaré for nonautonomous differential equations. It turns out that the canonical polynomials of Devlin are from the point of view of connected graded Hopf algebras the graded components of a Hopf algebra antipode applied to the formal power series of Ferfera. The link is made by applying control theory to the Abel equation, which can produce a center at the origin. The Abel equation is equivalent to an output feedback equation, and the Hopf algebra of output feedback is derived from the composition of iterated integrals rather than just the products of iterated integrals, which yields only the shuffle algebra. This means that the primary algebraic structure at play in Devlin's approach is actually not the shuffle algebra, but a Faà di Bruno type Hopf algebra, which is defined in terms of the unshuffle coproduct. [ABSTRACT FROM AUTHOR]

Given two nonlinear input–output systems written in terms of Chen–Fliess functional expansions, i.e., Fliess operators, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic setup is introduced using a new type of grading that has recently appeared in the literature. This grading also facilitates a fully recursive algorithm to compute the antipode of the Hopf algebra of the output feedback group, and thus, the corresponding feedback product can be computed much more efficiently. The final innovation is an improved convergence analysis of the antipode operation, namely, the radius of convergence of the antipode is computed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Theoretical tools are presented to analyze the relationship between the design choices for a class of NASA-inspired fault-tolerant, reconfigurable computer architectures and the tracking performance degradation of a digital flight control system implemented on such a platform while operating in a high-intensity radiated field (HIRF) environment. A HIRF experiment was conducted at the NASA Langley Research Center to validate the theory for a distributed Boeing 747 flight control system subject to HIRF upsets. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

Fliess operators, which are a type of functional series expansion, have been used to describe a broad class of nonlinear input–output maps driven by deterministic inputs. But in most applications, a system's inputs have noise components. This paper has three objectives. The first objective is to show that the notion of a Fliess operator can be generalized to admit a class of L 2-Itô stochastic input processes. The next objective is to show that they converge absolutely over an arbitrarily large but finite time interval when a certain coefficient growth condition is met. However, a significant number of systems fail to meet this condition. Thus, the final objective is to consider an interval of convergence having a random length so that a Fliess operator might converge under less restrictive growth conditions. [ABSTRACT FROM AUTHOR]

Abstract: A Faà di Bruno type Hopf algebra is developed for a group of integral operators known as Fliess operators, where operator composition is the group product. Such operators are normally written in terms of generating series over a noncommutative alphabet. Using a general series expansion for the antipode, an explicit formula for the generating series of the compositional inverse operator is derived. The result is applied to analytic nonlinear feedback systems to produce an explicit formula for the feedback product, that is, the generating series for the Fliess operator representation of the closed-loop system written in terms of the generating series of the Fliess operator component systems. This formula is employed to provide a proof that local convergence is preserved under feedback. [Copyright &y& Elsevier]

Abstract: This paper has three objectives. The first objective is to provide a simpler proof concerning a sufficient condition due to Ferfera under which bilinearity is preserved for cascade interconnections. The next objective is to provide a specific counterexample to show that this condition does not apply to the feedback connection. The final objective is to show that the well-known correspondence between rational series and formal power series recognized by weighted finite-state automata can be generalized to produce a correspondence between the generating series of cascaded and feedback connected bilinear systems and a class of weighted Petri nets. [Copyright &y& Elsevier]

This paper analyzes the output tracking performance of a Boeing 737 aircraft in closed loop with a Recoverable Computer System (RCS) operating in a high intensity neutron environment at the Los Alamos National Laboratory. A stochastic hybrid model, validated by simulated-neutron experiments done at the NASA Langley Research Center, is used to theoretically predict the outcome of the experiment. In particular, this model predicts and the data confirms that the recoverable digital flight control system, when functioning as designed, will provide accurate output tracking performance in the presence of neutron-induced single-event upsets (SEUs) at normal atmospheric levels. [ABSTRACT FROM AUTHOR]

Abstract: It is shown that the notion of balancing a state space realization about a stable, isolated equilibrium point can be generalized to systems possessing stable invariant regular submanifolds of arbitrary dimension, for example, a stable limit cycle. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents three results in singular value analysis of Hankel operators for nonlinear input–output systems. First, the notion of a Schmidt pair is defined for a nonlinear Hankel operator. This makes it possible to define a Hankel singular value function from a purely input–output point of view and without introducing a state space setting. However, if a state space realization is known to exist then a set of sufficient conditions is given for the existence of a Schmidt pair, and the state space provides a convenient representation of the corresponding singular value function. Finally, it is shown that in a specific coordinate frame it is possible to relate this new singular value function definition to the original state space notion used to describe nonlinear balanced realizations. [Copyright &y& Elsevier]

Fliess operators as input–output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory. In the classical analysis of these operators, certain growth conditions on the coefficients in their series representations insure uniform and absolute convergence, provided every input is uniformly bounded by some fixed upperbound. In some emerging applications, however, it is more natural to consider other classes of inputs. In this paper, Lp function spaces are considered. In particular, it is shown that the classic growth conditions also provide sufficient conditions for convergence and continuity when the admissible inputs are from a ball in Lp[t0,t0+T], where T is bounded and p⩾1. In addition, stronger global growth conditions are given that apply even for the case where T is unbounded. When the coefficients of a Fliess operator have a state space representation, it is shown that the state space model will always locally realize the corresponding input–output map on Lp[t0,t0+T] for sufficiently small T>0. If certain well-posedness conditions are satisfied then the state space model will globally realized the input–output mapping for unbounded T when the coefficients satisfy the global growth condition. [Copyright &y& Elsevier]

Presents information on a study which described a mathematical model for analyzing the stability effects of digital linear flight controllers subject to electromagnetic (EM) disturbances. Structure of the EM disturbance model; Components of EM disturbance model; Conclusions.

Focuses on a study that examined a motion detection circuit which is based on nerve membrane conduction. Background information on the motion detection model; Velocity estimation of neural network; Simulation of the study.

Consider a set of single-input, single-output nonlinear systems whose input–output maps are described only in terms of convergent Chen–Fliess series without any assumption that finite dimensional state space models are available. It is shown that any additive or multiplicative interconnection of such systems always has a Chen–Fliess series representation that can be computed explicitly in terms of iterated formal Lie derivatives. [ABSTRACT FROM AUTHOR]

This paper presents a symmetric Kronecker product analysis of independent and identically distributed jump linear systems to develop new, lower dimensional equations for the stability and performance analysis of this type of systems than what is currently available. In addition, new closed form expressions characterising multi-parameter relative sensitivity functions for performance metrics are introduced. The analysis technique is illustrated with a distributed fault-tolerant flight control example where the communication links are allowed to fail randomly. [ABSTRACT FROM AUTHOR]

In this work a combinatorial description is provided of a Faà di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given. [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]