controllability of linear single input positive discrete time systems linear positive systems are utilized in various scientific areas such as economic modelling behavioural science stochastic processes etc a recent study has focused on some sufficient conditions for the controllability of these systems a full characterization is provided of the controllability for discrete time single input positive systems necessary and sufficient conditions are derived using a digraph theoretic approach the main results show that strong constraints are to be imposed on the digraph associated with the pair a b to ensure the controllability of the system

perturbation of the svd in the presence of small singular values abstract this paper gives svd perturbation bounds and expansions that are of use when an m n m n matrix a has small singular values the first part of the paper gives subspace bounds that are closely related to those of wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace in the second part first and second order approximations are given for perturbed singular values the subspace bounds are used to show that all approximations retain accuracy when applied to small singular values the paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation

minimality and local state decompositions of a nonlinear state space realization using energy functions in this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state space realization of a formal power series is minimal specifically it is shown that positivity of these functions in addition to a stability requirement and a few technical conditions implies minimality using the nonlinear analogue of the kalman decomposition connections are then established between minimality singular value functions balanced realizations and various notions of reachability and observability for nonlinear systems

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

faa di bruno hopf algebra of the output feedback group for multivariable fliess operators abstract 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

a note on discrete time positive realizations abstract a discrete time positive linear system has the property that any nonnegative initial state and any nonnegative input produces a nonnegative trajectory in the state space and output for all time such systems are r n reachable if every state of the nonnegative orthant r n is reachable from origin with a nonnegative input function in a finite time interval a simple necessary and sufficient condition for r n realizability i e for the existence of a nonnegative realization that is also r n reachable is stated the realization procedure is straightforward

local realizations of nonlinear causal operators we give necessary and sufficient conditions for a causal operator to have a local realization of class c k k 1 cdots infty omega of the form dot x f x u quad y h x we show that two minimal local realizations of the same response map are locally difleomorphic

computation of convergence bounds for volterra series of linear analytic single input systems in this paper the volterra series decomposition of a class of single input time invariant systems analytic in state and affine in input is analyzed input to state convergence results are obtained for several typical norms l 0 t l r as well as exponentially weighted norms from the standard recursive construction of volterra kernels new estimates of the kernel norms are derived the singular inversion theorem is then used to obtain the main result of the paper namely an easily computable bound of the convergence radius guaranteed error bounds for the truncated series are also provided the relevance of the method is illustrated in several examples

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

positive realizations of linear systems abstract this paper deals with the realization problem of linear systems in which a sign pattern is imposed on the realization a b c t this problem arises in the context of positive linear systems i e systems in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative using a unitary framework for continuous time and discrete time systems we derive necessary and sufficient conditions for positive realizability by means of convex analysis moreover the complete solution in the case of a transfer function of mcmillan degree n 3 is obtained

non causal fliess operators and their shuffle algebra fliess operators as a class of non linear operators have been well studied in several respects they have a well developed realization theory and convenient representations in terms of directed infinite products of exponential lie series their interconnection as subsystems has been studied as has their relationship to rational systems they find applications in such diverse areas as discretization methods for controls systems optimal control neural network analysis and the numerical solution of stochastic differential equations one issue concerning fliess operators however that has received little attention is their possible generalization to the non causal case examples of such operators appear implicitly in the literature addressing hilbert adjoints of causal non linear operators and system inversion for the purpose of output tracking but a general systematic treatment of the subject has not appeared in this paper a non causal extension of a fliess operator is developed with the primary focus being on local convergence continuity the associated shuffle algebra and computing adjoint operators

on the existence of a positive realization the positive realization problem for linear systems is to find conditions for a given transfer function with nonnegative impulse response to have a realization such that the resulting system is a positive system recently it has been shown that under a mild assumption on the long term behaviour of the impulse response this problem is related to the maximum modulus poles only in this paper necessary and sufficient conditions for positive realizability of discrete time systems are given they show that also nondominant poles play a role in the most general case positive realizability conditions for the continuous time case are also given

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

a faa di bruno hopf algebra for a group of fliess operators with applications to feedback a faa 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

a unified approach to generating series for mixed cascades of analytic nonlinear input output systems the primary goal is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input output systems in particular it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of analytic integral operators fliess operators and analytic functions provided that the composite system is well defined and resides in one of these two classes in each case the radius of convergence of the interconnection is computed

existence and uniqueness of realizations of nonlinear systems necessary and sufficient conditions for the existence of either analytic or smooth symmetric realizations of nonlinear input output maps are given it is also shown that two minimal realizations are diffeomorphic

optimal estimation and rank detection for sparse spiked covariance matrices this paper considers a sparse spiked covariance matrix model in the high dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection the optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices we also establish the minimax rate under the spectral norm for estimating the principal subspace the primary object of interest in principal component analysis in addition the optimal rate for the rank detection boundary is obtained this result also resolves the gap in a recent paper by berthet and rigollet ann stat 41 4 1780 1815 2013 where the special case of rank one is considered

nonnegative linear systems in the behavioral approach the autonomous case nonnegative linear systems which have traditionally been investigated within the state space framework are in this paper introduced and analyzed by means of the behavioral approach starting from certain definitions and results presented in a recent paper j w nieuwenhuis systems control lett 1 1982 283 we have explored the general autonomous case by deriving an extended set of necessary and sufficient conditions for an autonomous behavior to be nonnegative in the scalar case in particular necessary conditions for nonnegativity which refer to the set of zeros of any polynomial involved in the kernel description of the behavior have been provided

nonlinear system identification for multivariable control via discrete time chen fliess series abstract the nonlinear system identification problem is solved for a multivariable nonlinear input output system that can be represented in terms of a chen fliess functional expansion the problem is formulated in terms of a regression which is linear in the parameters and has a nonlinear regressor an inductive implementation of the nonlinear regressor is developed using the underlying noncommutative algebraic and combinatorial structures the method is demonstrated in an adaptive control application involving a two input two output lotka volterra dynamical system

discrete time approximations of fliess operators a convenient way to represent a nonlinear input output system in control theory is via a chen fliess functional expansion or fliess operator the general goal of this paper is to describe how to approximate fliess operators with iterated sums and to provide accurate error estimates for two different scenarios one where the series coefficients are growing at a local convergence rate and the other where they are growing at a global convergence rate in each case it is shown that the error estimates are asymptotically achievable for certain worst case inputs the paper then focuses on the special case where the operators are rational i e they have rational generating series and thus are realizable in terms of bilinear ordinary differential state equations in particular it is shown that a discretization of the state equation via a kind of euler approximation coincides exactly with the discrete time fliess operator approximator of the continuous time rational operator

bilinear system interconnections and generating series of weighted petri nets 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

controllability and reachability criteria for discrete time positive systems discrete time positive systems have been the object of widespread interest in the literature in the last decade particular attention has been devoted to the analysis and characterization of the different notions of reachability and controllability which in this context make their appearance in this paper the properties of ordinary and essential controllability and reachability are analysed based on a graph theoretic approach we provide practical criteria to verify whether a given positive system f g is endowed with these properties together with canonical forms for describing reachable controllable pairs

fliess operators on lp spaces convergence and continuity 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 coe3cients in their series representations insure uniform and absolute convergence provided every input is uniformly bounded by some 4xed 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 su3cient conditions for convergence and continuity when the admissible inputs are from a ball in lp t0 t 0 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 coe3cients of a fliess operator have a state space representation it is shown that the state space model will always locally realize the correspondinginput output map on lp t0 t 0 t for su3ciently small t 0 if certain well posedness conditions are satis4ed then the state space model will globally realized the input output mapping for unbounded t when the coe3cients satisfy the global growth condition c 2002 elsevier science b v all rights reserved

graph theoretic characterization of fixed modes in centralized and decentralized control a closed loop system is represented by a weighted directed graph which exhibits all couplings of the system in a very clear and simple manner the coefficients of the characteristic polynomial of the closed loop system may be obtained with the aid of cycle families in this graph based on this result a criterion for the existence of fixed modes is derived the multiplicities of fixed modes may also be determined graph theoretically the consequences of structural constraints on the output feedback pattern become evident decentralized feedback is discussed as a special case

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

controllability of multi input positive discrete time systems linear discrete time positive systems arise in various fields of science such as economic modelling behavioural science stochastic processes etc recently some authors have examined the problem of the controllability of such systems for the single input case this paper provides necessary and sufficient conditions for characterizing the controllability properties of multi input systems the results here developed show that only a narrow class of these systems enjoys the properties of reachability and controllability