on vanishing stability regions in nonlinear systems with high gain feedback when local stabilization of nonlinear systems is achieved by linear feedback the resulting stability region may vanish as the feedback gains increase this is demonstrated by examples in which neglected nonlinearities create an unstable limit cycle around an asymptotically stable equilibrium as the feedback gains tend to infinity the unstable limit cycle shrinks to the equilibrium if a feedback linearization design is applied the same instability mechanism may occur when the nonlinearities are not precisely known

feedback affinization of nonlinear control systems abstract in this paper conditions for the nonlinear control system x f x u x m u r m to have a nonlinear feedback control u x v v r m m m 0 such that the nonlinear system takes a form of an affine system x a x b x v are presented all results require algebraic operations and differentiation of functions only

controlled invariance for discrete time nonlinear systems with an application to the disturbance decoupling problems invariant distributions are defined for discrete time nonlinear control systems and necessary and sufficient conditions are given for their controlled invariance this extends to discrete time systems the basic tool which has been so important in solving the various synthesis problems for continuous time systems to indicate their utility in the discrete time setting they are used to locally solve the disturbance decoupling problem

the matching of a prescribed linear input output behavior in a nonlinear system in this paper we deal with the problem of finding a feedback under which the input output behavior of a nonlinear system becomes exactly the same as that of a specified linear model the solvability of this problem is shown to depend on a formal infinite zero structure associated with the system

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

towards a generic constructive nonlinear control design tool using relaxed control in this technical note we revisit a control design approach for general non affine nonlinear systems using relaxed control using the notion of relaxed input where the ordinary real valued control input is replaced by a measure valued control input we are able to manipulate the original system such that the resulting relaxed system is amenable to control we show the applicability of using relaxed input for transforming a non affine nonlinear system into an affine one such that the relaxed system has desirable control properties finally we present a practical implementation of relaxed input via ordinary control input which is based on the sampled data of the relaxed input

controllability distributions for nonlinear control systems the purpose of this paper is to relate the notion of controlled invariance to a notion of controllability for nonlinear control systems by means of the linear ideas of a subsystem and a quotient system we derive a very structured description of nonlinear systems

symmetry and the implementation of feedback linearization abstract we show that by making use of the symmetry group of the system one may find the linearizing feedback for any controllable linearizable system with p controls by solving independent completely integrable pfaffian systems whose dimensions are the multiplicities of the kronecker indices and carrying out purely algebraic calculations in particular if the kronecker indices are distinct this means solving p independent one dimensional completely integrable pfaffian systems

feedback equivalence for general control systems abstract we discuss the solution to the problem of local equivalence of control systems with n states and p controls in a neighbourhood of a generic point under the lie pseudo group of local time independent feedback transformations we have shown earlier that this problem is identical with the problem of simple equivalence of the time optimal variational problem here we indicate the way in which this identification may be used to obtain closed loop time critical controls for general systems we show that the classification of general nonlinear systems depends on the classification of all n p dimensional affine subspaces of the space of symmetric forms in p variables and that the case of control linear systems depends on the classification of all n p dimensional affine subspaces of the space of skew forms in p variables we show that in the latter case the g structure is the prolongation of one determined by a state space transformation group we give a complete list of normal forms for control linear systems in the case p n 1

a constructive condition for dynamic feedback linearization the paper deals with dynamic feedback linearization of continuous time affine systems a constructive procedure based on prolongations is proposed based on necessary geometric conditions the algorithm computes a set of prolongation indices

measurement feedback disturbance decoupling in discrete time nonlinear systems the paper studies the disturbance decoupling problem by the dynamic measurement feedback for discrete time nonlinear control systems to address the problem the algebraic approach called the algebra of functions is applied which allows the system description also depend on non differentiable functions a necessary and sufficient condition is given in terms of controlled and h f invariant functions also algorithms are derived which find invariant functions and the required feedback the algorithms are implemented in mathematica software which is made available over the internet

a time optimal solution for the path cover problem on cographs we show that the notoriously difficult problem of finding and reporting the smallest number of vertex disjoint paths that cover the vertices of a graph can be solved time and work optimally for cographs our result implies that for this class of graphs the task of finding a hamiltonian path can be solved time and work optimally in parallel it was open for more than 10 years to find a time and work optimal parallel solution for this important problem our contribution is to offer an optimal solution to this important problem we begin by showing that any algorithm that solves an instance of size n of the problem must take logn time on the crew even if an infinite number of processors are available we then go on to show that this time lower bound is tight by devising an erew algorithm that given an n vertex cograph g represented by its cotree finds and reports all the paths in a minimum path cover in o logn time using n logn processors

linearizability of systems of the class c1 with multi dimensional control abstract we give a complete description of nonlinear control systems of the class c 1 with multi dimensional control that are feedback linearizable by means of change of variables of the class c 2 and change of controls of the class c 1

local input output decoupling of discrete time non linear systems a local treatment of the restricted block input output decoupling problem is given the major tools employed are the invariant and locally controlled invariant distributions which have recently been extended to the discrete time domain

varying coefficient models we explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables general algorithms are presented for estimating the models flexibly and some examples are given this class of models ties together generalized additive models and dynamic generalized linear models into one common framework when applied to the proportional hazards model for survival data this approach provides a new way of modelling departures from the proportional hazards assumption

a geometric approach to the theory of 2 d systems the authors develop a geometric approach to the theory of 2 d two dimensional systems defining in a suitable way the notion of a sub 1 2 b sub 1 2 invariant subspaces and of controlled invariant subspaces such subspaces are shown to have good computational and feedback properties which make them useful in application in particular sufficient conditions and constructive procedures are obtained for the solutions of disturbance decoupling and model matching problems moreover it is shown that certain structural properties of a 2 d system can be described by means of a set of indexes defined in geometric terms and that such structural indexes can be used to reformulate the sufficient condition for model matching

observability of autonomous discrete time non linear systems a geometric approach in this paper the observability of autonomous discrete time systems is studied from a purely differential geometric point of view as with continuous time systems this approach leads to a local canonical form for an observable system a proposal for the generalization of an invariant subspace is made

reduced rank hazard regression for modelling non proportional hazards the cox proportional hazards model is the most common method to analyse survival data however the proportional hazards assumption might not hold the natural extension of the cox model is to introduce time varying effects of the covariates for some covariates such as surgical treatment non proportionality could be expected beforehand for some other covariates the non proportionality only becomes apparent if the follow up is long enough it is often observed that all covariates show similar decaying effects over time such behaviour could be explained by the popular gamma frailty model however the marginal effects of covariates in frailty models are not easy to interpret in this paper we propose the reduced rank model for time varying effects of covariates starting point is a cox model with p covariates and time varying effects modelled by q time functions constant included leading to a pxq structure matrix that contains the regression coefficients for all covariate by time function interactions by reducing the rank of this structure matrix a whole range of models is introduced from the very flexible full rank model identical to a cox model with time varying effects to the very rigid rank one model that mimics the structure of a gamma frailty model but is easier to interpret we illustrate these models with an application to ovarian cancer patients

a nonregular solution of the nonlinear dynamic disturbance decoupling problem with an application to a complete solution of the nonlinear model matching problem the nonregular dynamic disturbance decoupling problem for nonlinear control systems is introduced a local solution is given by means of a constructive algorithm that is based on singh s algorithm and the clamped dynamics algorithm further studied is the nonlinear model matching problem that is defined as follows given a nonlinear control system to be referred to as the plant and another nonlinear control system to be referred to as the model can a compensator for the plant be found in such a way that the input output behavior of the compensated plant matches that of the model by proving that the solvability of the nonlinear model matching problem is equivalent to the solvability of an associated nonregular dynamic disturbance decoupling problem a complete local solution of this problem can be established

on decomposition of nonlinear control systems in this paper the problem is studied when the system x i k 1 u t g t x x 0 x 0 defined on an n dimensional analytic manifold can be decomposed into either independent subsystems or a cascade of subsystems necessary and sufficient conditions for the existence of such local decompositions are given in terms of lie algebras generated by the system

on dynamic input output linearization of discrete time nonlinear systems this paper studies the problem of linearizing the input output map of an analytic discrete time nonlinear system locally around a given trajectory necessary and sufficient conditions are given for the existence of a regular dynamic state feedback control law under which the input dependent part of the response of a nonlinear system becomes linear in the input and independent of the initial state the proposed conditions are less restrictive than those obtained by lee and marcus for linearizing the input output map via a static state feedback instrumental in the problem solution is the inversion structure algorithm for a discrete time nonlinear system firstly the solvability conditions are expressed in terms of the inversion algorithm secondly the proof of the existence and construction of the dynamic state feedback compensator relies on this algorithm

the gs algorithm for exact linearization to brunovsky normal form an algorithm utilizing the minimal number of integrations for the exact linearization of nonlinear systems to brunovsky normal form under nonlinear feedback is presented the tools which are involved are based on classical constructions appearing in the theory of exterior differential systems

a relaxation of the gamma frailty burr model frailty models are used in univariate data to account for individual heterogeneity in the popular gamma frailty model the marginal hazard has the form of a burr model although the burr model is very useful and can offer insight on the data it is far from perfect the estimation of the covariate effects is linked to the baseline hazard and this makes the model coefficients hard to interpret at the same time the frailties are assumed constant over time while biological reasoning in some cases may indicate that frailties may be time dependent in this paper we present a relaxation of the burr model which is based on loosening the link between the estimation of the covariate effects and the baseline hazard this can be achieved by replacing the cumulative baseline hazard in the burr model by a set of time functions and the frailty variance by a vector of coefficients directly estimated from the data using a partial likelihood we illustrate the similarities of the model with the burr model and a further extension of the latter a model with an autoregressive stochastic process for the frailty we compare the models on simulated data sets with constant and time dependent frailties and show how the relaxed burr models performs on two different real data sets we show that the relaxed burr model serves as a good approximation to the burr model when the frailty is constant and furthermore it gives better results when the frailty is time dependent copyright 2006 john wiley sons ltd

observability and controllability for smooth nonlinear systems the definition of a smooth nonlinear system as proposed recently by willems is elaborated as a natural generalization of the more common definitions of a smooth nonlinear input output system mini

input output decoupling of hamiltonian systems the linear case in this note we give necessary and sufficient conditions for a linear hamiltonian system to be input output decouplable by hamiltonian feedback i e feedback that preserves the hamiltonian structure in a second paper we treat the same problem for nonlinear hamiltonian systems

controlled invariance for nonlinear systems necessary and sufficient conditions are derived for a b invariance called here controlled invariance for nonlinear systems dot x f x u the obtained results generalize and elucidate already known results about systems dot x a x sum min i 1 max m u i b i x a new and direct differential geometric interpretation of the concept of controlled invariance and the derived conditions is given

controlled invariance by static output feedback for nonlinear systems the paper deals with the notion of static output feedback for nonlinear systems necessary and sufficient conditions are derived for c a b invariance here called measured controlled invariance for nonlinear control systems

on dynamic feedback linearization abstract we address the problem of transforming a nonlinear multi input system into a linear controllable one via nonsingular dynamic feedback and extended state space diffeomorphism we show that in the single input case if a system is dynamic feedback linearizable it is also static feedback linearizable we then give sufficient conditions for a class of multi input systems to be dynamic feedback linearizable all three dimensional systems with two controls and a controllable linear approximation about the origin are shown to belong to such a class

characterization of allowable perturbations for robust decoupling of affine non linear systems for affine non linear systems we study the effects of uncertainties in systems vector fields on decoupled linear input output behaviour a characterization of an allowable class of perturbations which preserve the decoupling properties of the nominal system is given an immediate consequence of this characterization indicates that decoupling position control of rigid robotic manipulators is not robust under any perturbations in robot dynamics the result is applied to simple hamiltonian control systems to characterize the allowable perturbation of total energy function

a polynomial solution to the k fixed endpoint path cover problem on proper interval graphs we study a variant of the path cover problem namely the k fixed endpoint path cover problem or kpc for short given a graph g and a subset t of k vertices of v g a k fixed endpoint path cover of g with respect to t is a set of vertex disjoint paths p that covers the vertices of g such that the k vertices of t are all endpoints of the paths in p the kpc problem is to find a k fixed endpoint path cover of g of minimum cardinality note that if t is empty or equivalently k 0 the stated problem coincides with the classical path cover problem the kpc problem generalizes some path cover related problems such as the 1hp and 2hp problems which have been proved to be np complete note that the complexity status for both 1hp and 2hp problems on interval graphs remains an open question damaschke 1993 9 in this paper we show that the kpc problem can be solved in linear time on the class of proper interval graphs that is in o n m time on a proper interval graph on n vertices and m edges the proposed algorithm is simple requires linear space and also enables us to solve the 1hp and 2hp problems on proper interval graphs within the same time and space complexity

generalized output regulation problem for a class of nonlinear systems with nonautonomous exosystem the problem of the generalized output regulation is considered here in contrast to the classical output regulation problem where the references to be tracked and disturbances to be rejected are generated by an autonomous exogeneous system for the generalized output regulation an exogenous system is driven by an unknown external reference signal in such a way the class of reference or disturbance signals is expanded presented result solves the full measurement feedback case and is based on the well known notion of the regulator equation combined with the classical concept of the invariant distribution illustrative example including matlab simulink simulations is presented to demonstrate the suggested approach

dynamic surface control for a class of nonlinear systems a method is proposed for designing controllers with arbitrarily small tracking error for uncertain mismatched nonlinear systems in the strict feedback form this method is another synthetic input technique similar to backstepping and multiple surface control methods but with an important addition spl tau 1 low pass filters are included in the design where spl tau is the relative degree of the output to be controlled it is shown that these low pass filters allow a design where the model is not differentiated thus ending the complexity arising due to the explosion of terms that has made other methods difficult to implement in practice the backstepping approach while suffering from the problem of explosion of terms guarantees boundedness of tracking errors globally however the proposed approach while being simpler to implement can only guarantee boundedness of tracking error semiglobally when the nonlinearities in the system are non lipschitz

a necessary condition for dynamic feedback linearization abstract a necessary condition for dynamic feedback linearizability of control systems x dot f x u in which the controls enter nonlinearly is derived for each x the image of the function f x needs to allow a ruling i e a foliation by straight lines

on solvability of several affine system problems we examine seven problems and properties of single input single output affine control systems and classify the properties topologically as being structural generic or residual these properties are accessibility observability invertibility preservation of a given distribution and partial linearizability the mathematics involved relies upon the thorn transversality theorem

on the largest feedback linearizable subsystem abstract a feedback invariant set of integers is associated with any nonlinear multivariable system which is linear with respect to the inputs it is shown to be the set of controllability indices of the largest feedback linearizable subsystem i e the largest subsystem which can be made locally linear and controllable by means of nonsingular feedback transformations

an optimal path cover algorithm for cographs abstract the class of cographs or complement reducible graphs arises naturally in many different areas of applied mathematics and computer science in this paper we present an optimal algorithm for determining a minimum path cover for a cograph g in case g has a hamiltonian path cycle our algorithm exhibits the path cycle as well

dynamic cox modelling based on fractional polynomials time variations in gastric cancer prognosis the most popular model used for survival analysis is the proportional hazards regression model proposed by cox this is mainly due to its exceptional simplicity nevertheless the fundamental assumption of the cox model is the proportionality of the hazards for many applications however this assumption is doubtful proposals to extend the cox model for non proportional hazards to allow for dynamic effect structures usually either depend on prespecifications or require non standard estimation techniques and are thus not favoured in application moreover tests to verify the dynamic effect structures are not straightforward or lack omnibus power in this paper we propose a flexible method for modelling dynamic effects in survival data within the cox framework the method is based on fractional polynomials as introduced by royston and altman this allows for a transformation of the dynamic predictor which leads back to the conventional cox model and hence fitting is straightforward using standard estimation techniques in addition it offers the possibility of easily verifying the existence of time variation we describe a model selection algorithm which selects time varying effects only when evidence is given in the data we illustrate the properties of the approach in a simulation study and compare it with other methods in a survival study of gastric cancer patients we apply the approach to analyse dynamic structures in the effects of prognostic factors

linearization and input output decoupling for general nonlinear systems necessary and sufficient conditions are obtained for the linearization and input decoupling by state feedback of general nonlinear systems it is shown how these conditions can be derived from the already known conditions for affine nonlinear systems thereby also elucidating the existing theory for affine systems

cascade decomposition of nonlinear systems foliations and ideals of transitive lie algebras abstract we relate the cascade decomposition of nonlinear systems to the joint notions of foliations and ideals of transitive lie algebras

on normal forms of nonlinear systems affine in control the nonlinear equivalences of both finite and infinite zero structures of linear systems have been well understood for single input single output systems and have found many applications in nonlinear control theory the extensions of these notions to multiple input multiple output systems have proven to be highly sophisticated in this paper we propose constructive algorithms for decomposing a nonlinear system that is affine in control these algorithms require modest assumptions on the system and apply to general multiple input multiple output systems that do not necessarily have the same number of inputs and outputs they lead to various normal form representations and reveal the structure at infinity the zero dynamics and the invertibility properties all of which represent nonlinear equivalences of relevant linear system structural properties

invariant distributions for discrete time nonlinear systems the purpose of this paper is to introduce some notions of invariant distributions for the class of linear analytic discrete time systems and to give an algebraic characterization of them a brief final discussion motivates the usefulness of the notions introduced in the study of the geometric properties of the system

invariant functions of control linear systems abstract controllable autonomous systems in a fixed number of states and controls may be classified entirely into finitely many equivalence classes defined by time independent feedback 1 the equivalence class to which a linear system belongs may be determined by a computation of certain invariants attached to the system these constants are geometric features of i e they are independent of feedback diffeomorphisms and therefore describe what is truly intrinsic to the system the analog of the constant invariant in the nonlinear case is the invariant function these are curvature like objects which provide a measure of the nonlinearity of the system and serve as natural i e geometrically determined outputs in this paper we employ a technique from the method of equivalence of cartan to investigate the feasibility of constructing invariant functions of control linear systems we show that under suitable restrictions on the number of states and controls invariant functions may be obtained in the generic case after only one differentiation of the original system moreover these invariants are computed by the simultaneous solution of algebraic equations this provides a solution in principle of the problem of delineating feedback equivalence classes of nonlinear control systems

on the linear equivalents of nonlinear systems this paper studies feedback equivalence of nonlinear systems an equivalence relation is established for the systems x f x u a complete characterization of the class of nonlinear systems which are equivalent to controllable linear systems is obtained

global transformations of nonlinear systems recent results have established necessary and sufficient conditions for a nonlinear system of the form dot x t f x t u t g x t with f 0 0 to be locally equivalent in a neighborhood of the origin in rnto a controllable linear system we combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system in doing so we introduce a technique for constructing a transformation under the assumptions that g ldot f dotg ad n 1 f ldotg span an n dimensional space and that g ldot f ldot g ad n 2 f ldotg is an involutive set

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

extended lie brackets for nonlinear time delay systems in this note the extended lie bracket operator is introduced for the analysis and control of nonlinear time delay systems nltds this tool is used to characterize the integrability conditions of a given submodule the obtained results have two fundamental outcomes first they define the necessary and sufficient conditions under which a given set of nonlinear one forms in the n dimensional delayed variables x t x t sd with d constant but unknown are integrable thus generalizing the well known fundamental frobenius theorem to delay systems secondly they set the basis for the extension to this context of the geometric approach used for delay free systems the effectiveness of the results is shown by solving the problem of the equivalence of a nltds to an accessible linear time delay system ltds by bicausal change of coordinates

time dependent effects of fixed covariates in cox regression a nonparametric modification is proposed for cox s proportional hazards model cox 1972 journal of the royal statistical society series b 34 187 220 where the covariates are fixed but their effects are allowed to vary in time parameters are introduced for the covariate effects at the uncensored survival times estimates are obtained by maximizing the penalized partial log likelihood that arises if a penalty function of the first order differences of consecutive parameters is subtracted from the partial log likelihood the choice of the smoothing parameter which determines the weight of the penalty is based on akaike s information criterion the methods are illustrated in a set of ovarian cancer data and in a set of kidney transplantation data

a mathcal b invariant distributions and disturbance decoupling of nonlinear systems the concept of a b invariant subspaces has resulted in a unified approach to many of the basic structural properties of time invariant linear systems w m wonham lecture notes in economics and mathematical systems vol 101 springer verlag new york 1974 the purpose of this paper is to introduce the more general notion of a mathcal b invariant distributions on differentiable manifolds and to use this idea to study the disturbance decoupling problem for a class of nonlinear systems which evolve on real analytic manifolds

bisimilar control affine systems the notion of bisimulation plays a very important role in theoretical computer science where it provides several notions of equivalence between models of computation these equivalences are in turn used to simplify verification and synthesis for these models as well as to enable compositional reasoning in systems theory a similar notion is also of interest in order to develop modular verification and design tools for purely continuous or hybrid control systems in this paper we introduce two notions of bisimulation for nonlinear systems we present differential geometric characterizations of these notions and show that bisimilar systems of different dimensions are obtained by factoring out certain invariant distributions furthermore we also show that all bisimilar systems of different dimension are of this form

new results on f g invariance abstract the existing result for a nonsingular and involutive distribution to be f g invariant has a priori assumption that g is nonsingular this paper treats the problem of f g invariance for the case where g is singular necessary and sufficient conditions for a nonsingular and involutive distribution to be f g invariant are presented the constructive proof provides an algorithm for the construction of the feedback law

a special case of reduced rank models for identification and modelling of time varying effects in survival analysis flexible survival models are in need when modelling data from long term follow up studies in many cases the assumption of proportionality imposed by a cox model will not be valid instead a model that can identify time varying effects of fixed covariates can be used although there are several approaches that deal with this problem it is not always straightforward how to choose which covariates should be modelled having time varying effects and which not at the same time it is up to the researcher to define appropriate time functions that describe the dynamic pattern of the effects in this work we suggest a model that can deal with both fixed and time varying effects and uses simple hypotheses tests to distinguish which covariates do have dynamic effects the model is an extension of the parsimonious reduced rank model of rank 1 as such the number of parameters is kept low and thus a flexible set of time functions such as b splines can be used the basic theory is illustrated along with an efficient fitting algorithm the proposed method is applied to a dataset of breast cancer patients and compared with a multivariate fractional polynomials approach for modelling time varying effects copyright 2016 john wiley sons ltd

geometric tools for solving the fdi problem for linear periodic discrete time systems this paper studies the problem of detecting and isolating faults in linear periodic discrete time systems the aim is to design an observer based residual generator where each residual is sensitive to one fault whilst remaining insensitive to the other faults that can affect the system making use of the geometric tools and in particular of the outer observable subspace notion the fault detection and isolation fdi problem is formulated and necessary and solvability conditions are given an algorithmic procedure is described to determine the solution of the fdi problem

internally exponentially stable non linear discrete time non interacting control via static feedback the problem of discrete time block non interacting control with internal exponential stability is treated by identifying the linearized fixed modes common to every non interactive closed loop a necessary condition for achieving an exponentially stable solution to the problem is provided also through a special decomposition of the given system a sufficient condition is obtained owing to peculiarities intrinsic to discrete time non linear systems the development of the solution is not completely parallel to that of the solution for the class of continuous time systems