representations of control systems in the fliess algebra and in the algebra of nonlinear power moments in this paper we analyze a representation of affine nonlinear systems in the form of series of nonlinear power moments a difference between such a representation and the representation as the fliess series is caused by a difference in the structure of corresponding algebras it leads to a difference between the conditions of realizability in the form of n dimensional affine systems for these two types of series

a bound on the number of integrators needed to linearize a control system this paper considers the problem of dynamic feedback linearization for nonlinear control systems for a restricted class of dynamic compensators that correspond to adding chains of integrators to the inputs we give an upper bound on the order of the compensator that needs to be considered in the case of 2 input systems we show that this bound is sharp

trajectory generation for the n trailer problem using goursat normal form develops the machinery of exterior differential forms more particularly the goursat normal form for a pfaffian system for solving nonholonomic motion planning problems i e motion planning for systems with nonintegrable velocity constraints the authors use this technique to solve the problem of steering a mobile robot with n trailers the authors present an algorithm for finding a family of transformations which will convert the system of rolling constraints on the wheels of the robot with n trailers into the goursat canonical form two of these transformations are studied in detail the goursat normal form for exterior differential systems is dual to the so called chained form for vector fields that has been studied previously consequently the authors are able to give the state feedback law and change of coordinates to convert the n trailer system into chained form three methods for planning trajectories for chained form systems using sinusoids piecewise constants and polynomials as inputs are presented the motion planning strategy is therefore to first convert the n trailer system into goursat form use this to find the chained form coordinates plan a path for the corresponding chained form system and then transform the resulting trajectory back into the original coordinates simulations and frames of movie animations of the n trailer system for parallel parking and backing into a loading dock using this strategy are included

using action based hierarchies for real time diagnosis abstract an intelligent agent diagnoses perceived problems so that it can respond to them appropriately basically the agent performs a series of tests whose results discriminate among competing hypotheses given a specific diagnosis the agent performs the associated action using the traditional information theoretic heuristic to order diagnostic tests in a decision tree the agent can maximize the information obtained from each successive test and thereby minimize the average time number of tests required to complete a diagnosis and perform the appropriate action however in real time domains even the optimal sequence of tests cannot always be performed in the time available nonetheless the agent must respond for agents operating in real time domains we propose an alternative action based approach in which a each node in the diagnosis tree is augmented to include an ordered set of actions each of which has positive utility for all of its children in the tree and b the tree is structured to maximize the expected utility of the action available at each node upon perceiving a problem the agent works its way through the tree performing tests that discriminate among successively smaller subsets of potential faults when a deadline occurs the agent performs the best available action associated with the most specific node it has reached so far although the action based approach does not minimize the time required to complete a specific diagnosis it provides positive utility responses with step wise improvements in expected utility throughout the diagnosis process we present theoretical and empirical results contrasting the advantages and disadvantages of the information theoretic and action based approaches

control of flat systems by quasi static feedback of generalized states generalized states the state representations of which depend on time derivatives of the input are naturally used in the control design for flat systems quasi static feedback of these generalized states which involves time derivatives of the closedloop input is shown to allow control of all flat systems by feedback linearization of the tracking error dynamics this is then used in a method proposed for the analysis and design by subsystems of certain flat series connections the results are illustrated on a gantry crane example

a linear algebraic framework for dynamic feedback linearization to any accessible nonlinear system we associate a so called infinitesimal brunovsky form this gives an algebraic criterion for strong accessibility as well as a generalization of kronecker controllability indices an output function which defines a right invertible system without zero dynamics is shown to exist if and only if the basis of the brunovsky form can be transformed into a system of exact differential forms this is equivalent to the system being differentially flat and hence constitutes a necessary and sufficient condition for dynamic feedback linearizability

brief paper on state representations of time varying nonlinear systems this work considers a nonlinear time varying system described by a state representation with input u and state x a given set of functions v which is not necessarily the original input u of the system is the new input candidate the main result provides necessary and sufficient conditions for the existence of a local classical state space representation with input v these conditions rely on integrability tests that are based on a derived flag as a byproduct one obtains a sufficient condition of differential flatness of nonlinear systems

fliess series a generalization of the ree s theorem and an algebraic approach to a homogeneous approximation problem in this paper an algebraic technique originated by m fliess is developed and applied to a homogeneous approximation problem for nonlinear affine control systems the central role is played by a generalization of the r ree s theorem on a connection of lie elements and the shuffle product operation in a free associative algebra

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

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

a theory of diagnosis from first principles suppose one is given a description of a system together with an observation of the system s behaviour which conflicts with the way the system is meant to behave the diagnostic problem is to determine those components of the system which when assumed to be functioning abnormally will explain the discrepancy between the observed and correct system behaviour we propose a general theory for this problem the theory requires only that the system be described in a suitable logic moreover there are many such suitable logics e g first order temporal dynamic etc as a result the theory accommodates diagnostic reasoning in a wide variety of practical settings including digital and analogue circuits medicine and database updates the theory leads to an algorithm for computing all diagnoses and to various results concerning principles of measurement for discriminating among competing diagnoses finally the theory reveals close connections between diagnostic reasoning and nonmonotonic reasoning

classical electrical engineering questions in the light of fliess s differential algebraic framework of non linear control systems this paper addresses two important questions of electrical engineering in the differential algebraic framework introduced by fliess this lead to an intrinsic formulation of the well known park transformation and to a generalized state representation of stand alone electrical generators a mon maitre en hommage respecteux pour son jubile

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

extended luenberger observer for non linear multivariable systems for non linear multiple input multiple output systems xdot f x u y h x u nonlinear observers are designed using a transformation into the non linear observer canonical form and an extended linearization the differential equation of observer error in canonical coordinates is linearized about the reconstructed trajectory and dimensioned by eigenvalue assignment with reference to the extended kalman filter algorithm this non linear observer design is called the extended luenberger observer this observer design is possible for all sufficiently smooth and locally observable systems in comparison with single output systems the non linear observer design can be essentially simplified using the degrees of freedom available in the case of multiple outputs

dynamic decoupling for right invertible nonlinear systems abstract it is shown that every system with more inputs than outputs that is right invertible in the sense of 5 can be decoupled via a dynamic state feedback that can be computed using a given algorithm this result generalizes the one of 2 and completes the results of 3

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

decoupling with dynamic compensation for strong invertible affine non linear systems abstract in this paper we tackle for affine non linear systems the morgan problem i e the scalar input scalar output decoupling problem for square systems with dynamic compensation the result provided here theorem 3 1 generalizes that one previously given by wang 1970 for linear systems

a remark on nonlinear accessibility conditions and infinite prolongations the lie rank condition for strong nonlinear accessibility is interpreted via the differential geometry of jets and prolongations of infinite order it yields to a new lie algebraic criterion and to the consideration of first integrals which apply to nonlinear systems in quite general form the latter characterization in particular is valid without distinguishing between the system variables and is independent of time scalings weak accessibility is also considered

a condition for dynamic feedback linearization of control affine nonlinear systems in this paper a necessary and sufficient condition for dynamic feedback linearizability by endogenous feedback of control affine nonlinear systems is presented the condition is based on a modified derived flag of a pfaffian system associated with a control system which takes into account the presence of a precompensator it is shown that the generators associated with this modified derived flag must satisfy a number of congruences the congruences ensure that the conditions of the gs algorithm are satisfied for the prolongated system and thus provide an algorithm to calculate the required feedback and state space transformations a tower structure is identified with respect to a set of indices corresponding to a set of generalized kronecker indices using these conditions a bound related to the extent of precompensation required for each input channel is obtained it is shown how linearizing outputs can be identified by studying the derived flag of the pfaffian system associated with a nonlinear system

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

differential flatness and absolute equivalence of nonlinear control systems this paper presents a formulation of differential flatness a concept originally introduced by fliess levine martin and rouchon in terms of absolute equivalence between exterior differential systems systems that are differentially flat have several useful properties that can be exploited to generate effective control strategies for nonlinear systems the original definition of flatness was given in the context of differential algebra and required that all mappings be meromorphic functions the formulation of flatness presented here does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems in particular it is shown that under regularity assumptions and in the case of single input control systems i e codimension 2 pfaffian systems a system is differentially flat if and only if it is feedback linearizable via static state feedback in higher codimensions our approach does not allow one to prove that feedback linearizability about an equilibrium point and flatness are equivalent one must be careful with the role of time as well as the use of prolongations that may not be realizable as dynamic feedback in a control setting applications of differential flatness to nonlinear control systems and open questions are also discussed

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

existence of realizations of nonlinear analytic input output maps we are interested in the existence of realizations of nonlinear input output maps when they are defined for positive times and when the set of control values is finite first we give examples that point out the nature of the problems which may appear in the attempt to find global results in realization theory then we state a necessary and sufficient condition for the existence of realizations and we give a sketch of the proof

feedback stabilization of control systems described by a class of nonlinear differential algebraic equations a theoretical framework for the stabilization of con trol systems defined by a class of nonlinear differential alge braic equations is established assumptions are stated which guarantee that the nonlinear differential algebraic equations can be described by a nonlinear control system defined on a smooth manifold a procedure for obtaining a local state realization is developed conditions for local stabilization of a single equilibrium solution including one set of conditions which can easily be checked using standard computations are indicated

technical communique some remarks on static feedback linearization for time varying systems this work summarizes some results about static state feedback linearization for time varying systems three different necessary and sufficient conditions are stated in this paper the first condition is the one by sluis w m 1993 a necessary condition for dynamic feedback linearization systems control letters 21 277 283 the second and the third are the generalizations of known results due respectively to aranda bricaire e moog c h pomet j b 1995 a linear algebraic framework for dynamic feedback linearization ieee transactions on automatic control 40 127 132 and to jakubczyk b respondek w 1980 on linearization of control systems bulletin del academie polonaise des sciences serie des sciences mathematiques 28 517 522 the proofs of the second and third conditions are established by showing the equivalence between these three conditions the results are re stated in the infinite dimensional geometric approach of fliess m levine j martin p rouchon p 1999 a lie backlund approach to equivalence and flatness of nonlinear systems ieee transactions on automatic control 44 5 922 937

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

dynamic input output decoupling of nonlinear control systems the problem of dynamic input output decoupling of nonlinear control systems is studied based on an analytic algorithm the authors obtain necessary and sufficient conditions for the solvability of this problem the solution of the problem is constructed by applying a series of simple precompensations and linking maps some interesting connections with other approaches in nonlinear control theory are discussed the authors also give a few simple examples to illustrate the methods used

flatness based control of a nonlinear chemical reactor model the nonlinear model of a continuous stirred tank reactor is shown to be flat the flatness permits the design of suitable trajectories on the basis of the explicit stationary solution and the tracking of these trajectories asymptotically using quasi static state feedback linearization a nonlinear local observer with a time varying gain is designed allowing the realisation of the state feedback in the case of partial measurement of the state simulation results illustrate the tracking behavior of the closed loop with the observer

on state representations of nonlinear implicit systems this work considers a semi implicit system that is a pair s y where s is an explicit system described by a state representation where x t n and u t m which is subject to a set of algebraic constraints y t h t x t u t 0 where y t l an input candidate is a set of functions v v 1 v s which may depend on time t on x and on u and its derivatives up to a finite order the problem of finding a local proper state representation z g t z v with input v for the implicit system is studied in this article the main result shows necessary and sufficient conditions for the solution of this problem under mild assumptions on the class of admissible state representations of these solvability conditions rely on an integrability test that is computed from the explicit system s the approach of this article is the infinite dimensional differential geometric setting of fliess levine martin and rouchon 1999 a lie backlund approach to equivalence and flatnes

sufficient conditions for dynamic state feedback linearization sufficient conditions are given for the existence of a dynamic state feedback compensator for a multi input nonlinear system such that the closed loop system is transformable into a linear controllable one by an extended state space change of coordinates an example shows that the conditions are not necessary necessary conditions are also given which are shown to be sufficient when the number of states minus the number of controls is equal to one several examples illustrate how the sufficient conditions obtained lead to the design of the dynamic compensator

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

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

generalized state variable representation for a simplified crane description recent theoretical advances in nonlinear systems theory insist on the relevance of dynamics where control derivatives appear we discuss here a simplified model of a crane which exhibits such a description

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

numerical accuracy of a certain class of iterative methods for solving linear system abstract one of the most important problem for solving the linear system ax b by using the iterative methods is to use a good stopping criterion and to determine the common significant digits between each corresponding components of computed solution and exact solution in this paper for a certain class of iterative methods we propose a way to determine the number of common significant digits of x m and x where x m and x are computed solution at iteration m and exact solution respectively by using the cadna library which allows us to estimate the round off error effect on any computed result we also propose a good stopping criterion which is able to stop the process as soon as a satisfactory informatical solution is obtained numerical examples are used to show the good numerical properties

rank invariants of nonlinear systems a linear algebraic framework for the analysis of rank properties of nonlinear systems is introduced this framework gives a high level interpretation of several existing algorithms built around the recursive computation of certain algebraic ranks associated with right invertibility left invertibility and dynamic decoupling furthermore it can be used to establish links between these algorithms and the differential algebraic approach as well as to solve some static and dynamic noninteracting control problems

absolute equivalence and dynamic feedback linearization abstract elie cartan s results on absolute equivalence of differential systems are shown to imply a generalization of a recent result on dynamic feedback linearization of scalar control systems

the behavior of conjugate gradient algorithms on a multivector processor with a hierarchical memory abstract in this paper an analysis of some of the tradeoffs involved in the design and efficient implementation of conjugate gradient based algorithms for a multivector processor with a two level memory hierarchy is presented and supplemented by experimental results obtained on an alliant fx 8 the algorithms considered consist of the classical conjugate gradient method preconditioning techniques that are well suited for parallel computers such as polynomial preconditioners and several versions of the incomplete cholesky preconditioners as well as the reduced system approach for linear systems arising from the 5 point finite difference discretization of 2 d self adjoint elliptic p d e s the analysis shows that conjugate gradient methods do not perform as well as algorithms for dense matrix computations on the considered architecture due to lack of data locality by using the reduced system approach however a significant decrease in time could be obtained

a modified algorithm for invertibility in nonlinear systems the algorithm of hirschorn 1 for the construction of inverses of nonlinear systems is modified and new sufficient conditions for invertibility and inverse system for a larger class of systems are obtained for the systems which do not satisfy the invertibility conditions a characterization of input space on which the input output map is injective is derived

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

tracking in nonlinear differential algebraic control systems with applications to constrained robot systems we consider the design of a feedback control law for control systems described by a class of nonlinear differential algebraic equations so that certain desired outputs track given reference inputs the nonlinear differential algebraic control system being considered is not in state variable form assumptions are introduced and a procedure is developed such that an equivalent state realization of the control system described by nonlinear differential algebraic equations is expressed in a familiar normal form a nonlinear feedback control law is then proposed which ensures under appropriate assumptions that the tracking error in the closed loop differential algebraic system approaches zero exponentially applications to simultaneous contact force and position tracking in constrained robot systems with rigid joints constrained robot systems with joint flexibility and constrained robot systems with significant actuator dynamics are discussed

the geometry controllability and flatness property of the n bar system we propose a kinematic model of a system moving in an m 1 dimensional euclidean space and consisting of n rigid bars attached successively to each other and subject to the nonholonomic constraints that the instantaneous velocity of the source point of each bar is parallel to that bar we prove that the associated control system is controllable and feedback equivalent to the m chained form around any regular configuration as a consequence we deduce that the n bar system is flat and show that the cartesian position of the source point of the last from the top bar is a flat output the n bar system is a natural generalisation of the n trailer system and we provide a comparison of flatness properties of both systems

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability

relative flatness and flatness of implicit systems in this work we define the concept of relative flatness of a system with respect to a subsystem the subsystem associated to a set of outputs of a system is constructed and called here output subsystem it is shown that the relative flatness of a system with respect to the output subsystem implies the flatness of the corresponding implicit system obtained by setting these outputs to zero a sufficient condition of relative flatness based on a relative derived flag is presented based on these results a sufficient condition for the flatness of a class of nonlinear implicit systems is obtained