a new algorithm for constructing approximate transformations for nonlinear systems an algorithm is provided for constructing an approximating transformation for a feedback linearizable system and for a system that is not feedback linearizable instead of approximating the given nonlinear system the authors approximate the transformation in a proper sense they assume that the lead variables of a transformation have a certain form with unknown parameters substitute these variables into the required equations and determine the unknown parameters through linear algebraic equations there is a tremendous savings in terms of number of equations and unknowns and it is sometimes possible to find exact or best possible transformations in a surprisingly small number of steps

strict feedforward control systems linearisability and convergent normal forms this article discusses the feedback equivalence of multi inputs feedforward control systems via smooth resp analytic feedback transformations we first address the state resp feedback linearisation problem and provide easily computable algorithms that yield explicit state resp feedback linearising coordinates for systems in strict feedforward form the application of the algorithms does not require checking the commutativity resp involutivity of the distributions associated with the system and the algorithms fail after few steps if the system is not linearisable in the latter case the algorithms are extended to provide coordinate systems bringing the system into a normal form which is a smooth resp analytic counterpart of kang s formal normal form illustrative examples for both the linearisation and convergent normal form include the vertical take off and landing aircraft the multi vehicle wireless testbed among others

approximated stable inversion for nonlinear systems with nonhyperbolic internal dynamics a technique to achieve output tracking for nonminimum phase nonlinear systems with nonhyperbolic internal dynamics is presented the present paper integrates stable inversion techniques that achieve exact tracking with approximation techniques that modify the internal dynamics to circumvent the nonhyperbolicity of the internal dynamics this nonhyperbolicity is an obstruction to applying presently available stable inversion techniques the theory is developed for nonlinear systems and the method is applied to a two cart with inverted pendulum example

stabilization of uncertain systems via linear control this note considers the problem of stabilizing a linear dynamical system whose state equation includes a time varying uncertain parameter vector q cdot given the dynamics dot x t a q t x t b q t u t and a bounding set q for the values q t the objective is to choose a control law u t p x t guaranteeing uniform asymptotic stability for all admissible variations of q cdot our results differ from previous work in one fundamental way that is we show that when working with linear controllers it is possible to dispense with all assumptions on b cdot which have been made by previous authors e g see 1 9 this elimination of hypotheses on b cdot is accomplished roughly as follows the system sigma underline underline delta a q b q is shown to be equivalent to another system sigma underline underline delta a q b as far as stabilization is concerned since b is a constant matrix independent of q the desired result is readily obtained

a direct method for the construction of nonlinear discrete time observer with linearizable error dynamics we provide a direct method for the design of nonlinear discrete time observers by a construction of a change of variables an explicit expression of the change of variables is given some simulations for chaotic systems such as lozi system and he spl acute non system are provided to illustrate the proposed method

observability quadratic normal form for discrete time systems this note deals with quadratic observability normal form for nonlinear discrete time single input single output siso system first of all the main concept of quadratic equivalence with respect to the observability property is introduced for discrete time systems subsequently normal form structure for discrete time system is developed for system with unobservable linear approximation in one direction finally the effect of the so called resonant terms on the observer design and synchronization of chaotic systems is pointed out in an illustrative example

nonlinear discrete time observer design with linearizable error dynamics a necessary and sufficient condition for the existence of a discrete time nonlinear observer with linearizable error dynamics is provided the result can be applied to any real analytic nonlinear system whose linear part is observable the necessary and sufficient condition is the solvability of a nonlinear functional equation furthermore the well known siegel s theorem on the linearizability of a mapping is naturally reproduced in a corollary the proposed observer design method is constructive and can be applied approximately to any sufficiently smooth linearly observable system yielding a local observer with approximately linear error dynamics

control problems in super articulated mechanical systems control of super articulated mechanical systems sams controlled mechanical system in which the dimension of the configuration space exceeds the dimension of the control input space is studied as a starting point a graphical characterization of general sams is developed in which a so called control flow diagram cfd is constructed to represent the interaction forces among the degrees of freedom of the system three types of structure namely the chain structure the tree structure and isolated points are identified degree of complexity of the system is defined based on the structure of the system a systematic backstepping type controller design is proposed for systems with a chain structure to achieve global asymptotic stability and trajectory tracking such a controller design itself illustrates the sufficient conditions for the existence of smooth feedback control such that the systems with a chain structure can be globally asymptotically stabilized the connection of the authors control algorithm with existing results is discussed

nonlinear control via uniform system approximation abstract we introduce the notion of uniform system approximation and use it to solve the following problem given a single input nonlinear system f g x f x g x u that is linearly controllable on a manifold e of equilibrium points find vector fields f g such that f g and f g agree to first order on e and f g is input to state linearizable on a neighborhood of e the solution proceeds by constructing a linearizable uniform system approximation and yields as a by product a feedback control law that can be used to guarantee stable operation of the given system in a neighborhood of e this includes for example smooth transitions between operating points that are not necessarily close to each other a simple example is given to illustrate the approach

higher order approximate feedback linearization about a manifold for multi input systems considers the problem of approximately feedback linearizing a multi input nonlinear system around the equilibrium manifold spl epsi while making the error terms be of highest order on spl epsi necessary and sufficient conditions are given for approximately feedback linearizing the system around spl epsi with error terms of order spl rho on spl epsi a simple multi input example is given to illustrate the higher order approximate feedback linearization procedure

normal forms and approximated feedback linearization in discrete time abstract the paper discusses approximated feedback linearization of nonlinear discrete time dynamics which are controllable in first approximation and introduces two types of normal forms the study is set in the context of differential difference representations of discrete time dynamics proposed in monaco normand cyrot in normand cyrot ed perspectives in control a tribute to ioan dore landau springer londres 1998 pp 191 205

input output pseudolinearization for nonlinear systems an input output pseudolinearized system has the property that its family of linearizations about constant operating points has input output behavior that is independent of the operating point we develop a necessary and sufficient condition for existence of a state feedback and state coordinate change that transforms a given system into a pseudo normal form that is input output pseudolinearized these ideas are applied to obtain new approaches to approximate tracking and approximate inversion of nonlinear systems

sampled data adaptive control of a class of continuous nonlinear systems the problem is addressed of deriving sampled data control laws for a class of nonlinear continuous systems the concept of approximate state feedback linearizing control is defined for sampled data models arising from the sampling of continuous time systems such control laws are then designed for both non adaptive and adaptive cases and a detailed performance analysis is performed

feedback control of analytic nonlinear systems by extended linearization for mufti input mufti output analytic nonlinear systems a design method based on the family of linearizations of the system parameterized by the family of constant operating points is considered using the cauchy kowalewski theorem it is shown that nonlinear state feedback control laws and output observer feedback control laws exist such that the eigenvalues of the family of closed loop linearizations have specified analytically scheduled values with respect to the family of closed loop operating points

feedback linearization families for nonlinear systems characterizations are developed for parameterized linear dynamic feedback control laws that can arise when linearizing a nonlinear dynamic feedback control law for a specified nonlinear system about a family of constant operating points such characterizations are important in applying the recently developed extended linearization design approach to various types of control problems to illustrate the input output decoupling problem is considered

approximate linearization by state feedback and coordinate change a nonlinear system can always be approximated to first order by linear systems it has been shown by jakubczyk and respondek 4 and hunt and su 3 that certain nonlinear systems are the exact transforms of linear systems under nonlinear state coordinate change and nonlinear state feedback in this paper we give necessary and sufficient conditions for a nonlinear system to be approximated to higher order by the transform of a linear system the use of this technique in the design of nonlinear compensators has been suggested recently by the author 6

nonlinear discrete time control of systems with a naimark sacker bifurcation in this paper we study the problem of the stabilization of nonlinear control system with one complex uncontrollable mode we find normal forms and quadratic invariants then we compute center manifolds and use bifurcation theory to synthesize quadratic controllers

on a nonlinear multivariable servomechanism problem abstract multivariable nonlinear plants with both measured and unmeasured vector disturbance signals are considered output feedback dynamic control laws are designed for the problem of tracking a constant or slowly varying reference input while rejecting constant or slowly varying disturbances

a new method for linearizing non linear systems the pseudolinearization consider the single input non linear system x f x u it has been shown that under mild conditions there exist transformations z t x v s x u such that in the z space the linear tangent model is independent of the operating point a constructive procedure to obtain the above transformations is described this new method is computationally attractive and effective in many practical cases

sampled data observer error linearization the effects of time sampling on the solvability conditions for the observer linearization design methodology are investigated it is shown that the class of systems for which this design method can be applied for an open set of sampling times is quite small in particular when the dimension of the state space is two it consists only of those systems that are state equivalent to a linear system the practical implication is that digital implementations of this methodology will have to be approximate

on diagonal singularities of the green functions for schrodinger operators we investigate the behavior of the green functions of schrodinger operators near the diagonal the only nontrivial cases where the on diagonal singularities are nonzero and do not depend on the spectral parameter are two and three dimensions in the case of two dimensions we show that the singularity is independent of both the scalar and the gauge potentials in dimension three we obtain conditions for preserving the singularity under perturbations by nonregular potentials some examples illustrating dependence of the singularity on general scalar and gauge potentials are presented

the global existence of nonlinear observers with linear error dynamics a topological point of view in this paper we show that under suitable assumptions there exists a global homeomorphism 1 of rn which maps a nonlinear system x f x x 0 x0 y h x onto a linear system with output injection z az y z 0 x0 thus an observer for state x can be directly constructed as x f x y h x which is a generalized version of luenberger observer an important feature of the obtained result is that there is no need to find the corresponding change of coordinates explicitly which is different from current various existing approaches

feedback classification of multi input nonlinear control systems we study the feedback group action on multi input nonlinear control systems with uncontrollable mode we follow slightly an approach proposed in kang and krener w kang and a j krener siam j control optim 30 1992 pp 1319 1337 which consists of analyzing the system and the feedback group step by step we construct a normal form which generalizes on one hand the results obtained in the single input case and on the other hand those recently obtained by the same author in the controllable case we illustrate our results by studying the caltech multi vehicle wireless testbed mvwt and the prototype of planar vertical takeoff and landing aircraft pvtol we also study the notion of bifurcation of controllability for systems with one nonzero uncontrollable mode we first show that the equilibria for those systems is a p dimensional submanifold p equals number of inputs provided that one term in their normal form is nonzero we show that these systems are linearly controllable hence stabilizable at any nearby equilibrium point of the origin

an approximate nonlinear observer with polynomial coordinate transformation maps the state equivalence condition to the nonlinear observer form is stated in terms of the exactness of one form using the definition of the approximate closedness we relax the equivalence condition thereby obtaining a necessary and sufficient condition for the approximate observer error linearizability this approximate observer theory extends the realm of applications another great advantage of this approximation method is that the coordinate transformation is directly obtainable as vectors of polynomials while it is in general not possible with the existing method

a stability property in this note we obtain a stability property not for a point of equilibrium but for a certain family of asymptotically stable equilibria of a dynamical system with an input we show that this property depends not only on the initial condition but also on the time derivative of the input function we suggest possible applications of this result to some control problems

on input output linearization of discrete time nonlinear systems abstract we consider a nonlinear discrete time system of the form x t 1 f x t u t y t h x t where x r n u r m y r q and f and h are analytic necessary and sufficient conditions for local input output linearizability are given we show that these conditions are also sufficient for a formal solution to the global input output linearization problem finally we show that zeros at infinity of can be obtained by the structure algorithm for locally input output linearizable systems

on weak relative degree and the mcgehee transformation points where jumps in relative degree occur are presented as singularities in the vector field of the internal dynamics of systems that are candidates for input output linearization via nonlinear feedback for a certain class of vector fields it is shown that the point at which the relative degree is weak can be transformed to a saddle equilibrium point at the origin

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

controllability distributions and systems approximations a geometric approach given a nonlinear system we determine a relation at an equilibrium between controllability distributions defined for a nonlinear system and a taylor series approximation of it the value of such a relation is appreciated if we recall that the solvability conditions as well as the solutions to some control synthesis problems can be stated in terms of geometric concepts like controlled invariant controllability distributions the relation between these distributions at the equilibrium will help us to decide when the solvability conditions of this kind of problems are equivalent for the nonlinear system and its approximation

exact and approximate feedback linearization without the linear controllability assumption the feedback linearization problem of nonlinear control systems has been solved in the literature under the assumption that the nonlinear system is linearly controllable in this paper the assumption of linear controllability is removed necessary and sufficient conditions are given through lie orbital symmetries thus giving a geometric characterization of the problem in the analytic case both the exact and the approximate linearization problems are considered in the analytic case

global convergence of nonlinear cascade flows with morse bott zero dynamics abstract we derive a sufficient condition that a flow captures the dynamics on an invariant submanifold this leads to a refinement of the lasalle invariance principle as a consequence we generalize a well known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set this includes the case where a globally asymptotically stable system is coupled to a morse bott flow

approximate noninteracting control with stability for nonlinear systems a notion of approximate noninteracting control with stability for nonlinear systems is introduced and the design of both static and dynamic state feedback control laws that achieve this property for a given plant is considered this approach to the problem is based on the notion of extended linearization and begins with an extension of the linearization approach pointwise to all constant operating points in an open neighborhood of the nominal under natural assumptions it is shown that obstructions to extract noninteracting control with internal stability or input state output stability can be circumvented in the context of approximate noninteraction

fixed period of temporary immunity after run of anti malicious software on computer nodes abstract sirs epidemic model has been developed with a fixed period of temporary immunity following temporary recovery from the infection of malicious objects in place of an exponentially distributed period of temporary immunity when a node is recovered from the infected class it recovers temporarily acquiring temporary immunity with probability p 0 p 1 and dies from the attack of malicious object with probability 1 p temporary immunity is observed in the computer network when anti malicious software is run after a node gets affected by a malicious object s the model consists of a set of integro differential equations the stability of the results is stated in terms of threshold parameter it has been observed that the endemic equilibrium for this model may be unstable thus giving an example of a generalization which leads to new possibilities for the behavior of the model numerically it has been verified that the endemic equilibrium is not asymptotically stable for all parameter values

a homotopy algorithm for approximating geometric distributions by integrable systems in the geometric theory of nonlinear control systems the notion of a distribution and the dual notion of codistribution play a central role many results in nonlinear control theory require certain distributions to be integrable distributions and codistributions are not generically integrable and moreover the integrability property is not likely to persist under small perturbations of the system therefore it is natural to consider the problem of approximating a given codistribution by an integrable codistribution and to determine to what extent such an approximation may be used for obtaining approximate solutions to various problems in control theory in this note we concentrate on the purely mathematical problem of approximating a given codistribution by an integrable codistribution we present an algorithm for approximating an m dimensional nonintegrable codistribution by an integrable one using a homotopy approach the method yields an approximating codistribution that agrees with the original codistribution on an m dimensional submanifold e 0 of r n

asymptotic stabilization of a generic class of three dimensional homogeneous quadratic systems abstract we give necessary and sufficient conditions for the asymptotic stabilizability of a generic class of single input three dimensional homogeneous quadratic systems

further results on the observer design problem abstract in this paper the observer design problem for nonlinear systems is considered sufficient lyapunov like conditions are presented for the existence of a nonlinear observer the theory we develop considerably improves and extends the results of our recent work 14

nonlinearizable single input control systems do not admit stationary symmetries the main result of the paper states that almost any analytic single input control system which is truly nonlinear that is not feedback linearizable with controllable linearization at an equilibrium point does not admit any symmetry preserving that point by almost any system we mean that we exclude a small class of odd systems that admit just one nontrivial symmetry conjugated to minus identity the obtained results are based on a recent classification of nonlinear single input systems under formal feedback we also describe symmetries of feedback linearizable systems

on stability properties of nonlinear systems with slowly varying inputs systems with slowly varying inputs are discussed as a special class of two time scale systems and singular perturbation results are seen as convenient tools to analyze their properties a lemma by f c hoppensteadt trans amer math soc vol 123 p 521 35 1966 for a parameterized family of systems is restated and applied to the analysis of systems with slowly varying inputs

gauge theories on manifolds with boundary the boundary value problem for laplace type operators acting on smooth sections of a vector bundle over a compact riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied the condition of strong ellipticity of this boundary value problem is formulated the resolvent kernel and the heat kernel in the leading approximation are explicitly constructed as a result the previous work in the literature on heat kernel asymptotics is shown to be a particular case of a more general structure for a bosonic gauge theory on a compact riemannian manifold with smooth boundary the problem of obtaining a gauge field operator of laplace type is studied jointly with local and gauge invariant boundary conditions which should lead to a strongly elliptic boundary value problem the scheme is extended to fermionic gauge theories by means of local and gauge invariant projectors after deriving a general condition for the validity of strong ellipticity for gauge theories it is proved that for euclidean yang mills theory and rarita schwinger fields all the above conditions can be satisfied for euclidean quantum gravity however this property no longer holds i e the corresponding boundary value problem is not strongly elliptic some non standard local formulae for the leading asymptotics of the heat kernel diagonal are also obtained it is shown that due to the lack of strong ellipticity the heat kernel diagonal is non integrable near the boundary

dynamic gain scheduled control and extended linearisation extensions explicit formulae and stability a controller designed for linearisations at various trim operating points of a nonlinear system using linear approaches is not necessarily performing well or stable once scheduled with a state under dynamic conditions the key idea of using this scheduled control law design is to retain states close to the current usually dynamically varying operating point dynamic gain scheduling dgs is a technique aimed to resolve this controller scheduling issue for rapidly changing dynamics and states it entails scheduling the control law gains with a fast varying state variable rather than with a slowly varying state it has been successfully applied to the aircraft system models allowing also for nested loops the aim of this paper is to extend dgs to a more general setting allowing for more complex multi input dynamics incorporating the more simple static scheduling within the same controller hence given a linear design suitable transformations are provided which allow fast scheduling of multi variable co

stabilization of single input nonlinear systems using piecewise constant controllers the purpose of this note is to present a new approach for stabilizing single input control affine systems for which the robust relative degree equals the dimension of the state space in contrast to the method used in approximate feedback linearization we employ piecewise constant controllers which allows us to account for higher order terms

on the pseudo linearization problem for nonlinear systems abstract based on well known results for linear systems an alternate treatment of tthe pseudo linearization problem of reboulet and champetier is given sufficient conditions for pseudo linearization are obtained in the general case and these are shown to be satisfied under mild hypothesis by systems with one or two inputs advantages of the new approach include the simplicity of the derivation and a more explicit representation for a pseudo linearizing transformation

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

quadratic forms and approximate feed back linearization in discrete time quadratic feedback linearization is discussed for discrete time systems i e approximate feedback linearization up to the second order with respect to input and state variables after introducing in discrete time two equivalence classes invariant under quadratic static state feedback and diffeomorphism two canonical quadratic representations are proposed then it is shown that any nonlinear discrete time dynamics which are controllable in their first approximations can be linearized up to the second order under dynamic feedback the results stated here can be considered as the general discrete time analogues of the continuous time ones kang and krener 1992

pseudolinearization and nonlinear optimal control for a class of infinite horizon nonlinear optimal control problems the pseudo linearization transformation is used to investigate a new type of approximate solution for the hamilton jacobi equation and corresponding approximate optimal control law

scattering on compact manifolds with infinitely thin horns the quantum mechanical scattering on a compact manifold with semi axes attached to the manifold hedgehog shaped manifold is considered the complete description of the spectral structure of schrodinger operators on such a manifold is done the proof of existence and uniqueness of scattering states is presented an explicit form for the scattering matrix is obtained and unitarity of this matrix is proven it is shown that the positive part of the spectrum of the schrodinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached half line moreover the positive part of the spectrum of the initial schrodinger operator is fully determined by the conductance properties of an electronic device consisting of the initial manifold and two wires attached to it

nonlinear control via approximate input output linearization the ball and beam example approximate input output linearization of nonlinear systems which fail to have a well defined relative degree is studied for such systems a method for constructing approximate systems that are input output linearizable is provided the analysis presented is motivated through its application to a common undergraduate control laboratory experiment the ball and beam where it is shown to be more effective for trajectory tracking than the standard jacobian linearization

approximate linearization of nonlinear control systems abstract the paper address the problem of approximate linearization of a nonlinear control system by state feedback or dynamic feedback the main result proved is that if the fast k d relative degrees are equal to each other then the input output response of a nonlinear control system can be linearized to degree k any system having linear d relative degree can be linearized to any degree by dynamic feedbacks one example of signal tracking using dynamic feedback linearization method to improve the performance is also given

on the control of linear systems whose coefficients are functions of parameters the stabilization and regulation of linear discrete time systems whose coefficients depend on one or more parameters is studied for linear systems whose coefficients are continuous functions of real or complex parameters respectively analytic or rational functions of real parameters it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous analytic rational functions of the parameters closed form expressions for a collection of stabilizing gains are given in terms of the reachability gramian for systems which are stabilizable for all values of the parameters it is shown that continuous analytic rational stabilizing gains can be computed from a finite time solution to a riccati difference equation whose coefficients are functions of the parameters these results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters

feedback control of nonlinear systems by extended linearization for single input multiple output nonlinear systems we consider a design method based on the family of linearizations of the system parameterized by constant operating points nonlinear state feedback control laws and observer state feedback control laws are designed such that the eigenvalues of the family of linearized closed loop systems are placed at specified values that are locally invariant with respect to the closed loop operating point the method is illustrated by application to the problem of automatically balancing an inverted pendulum

right coprime factorizations and stabilization for nonlinear systems the problem of constructing right coprime factorizations which are based on the graph of an input output map instead of the bezout identity of nonlinear input output maps is considered the map is assumed to arise from a state variable realization with a fixed initial state the main result is that the existence of a stabilizing state feedback implies the existence of a right coprime factorization for the map the technique is illustrated by application to nonlinear systems which are affine in the control and have a controllable linear past and nonlinear systems which are feedback linearizable a notion of input output stability that requires a bound on the magnitude of the input signals is introduced methods of constructing such bounds are developed for a locally feedback linearizable system the problem of input bounds is transferred to the equivalent linear system this leads to a technique that allows state and input constraints for a feedback linearizable system to be mapped to the equivalent linear system

a local stabilizing control scheme using an approximate feedback linearization we present a method of stabilizing single input approximately feedback linearizable systems in the generic situation where the linearizing feedback and coordinate transformation map are not directly available in obtaining a stabilizing feedback we utilize a technique of an approximate feedback linearization which is based on an approximate integrating factor calculated from a recursive algorithm we construct a feedback controller for the approximately linearized system so that the system is asymptotically stable in a neighborhood of the origin where the one to one relation holds with the original coordinate system

observer based control law for a class of non linear systems control and observation laws are considered for a class of non linear systems an algebraic procedure is proposed to design approximate observer based controllers this procedure is similar to that used for linear time invariant systems the paper is completed by an example based on elastically jointed robots

state and feedback linearizations of single input control systems abstract in this paper we address the problem of state resp feedback linearization of nonlinear single input control systems using state resp feedback coordinate transformations although necessary and sufficient geometric conditions have been provided in the early eighties the problems of finding the state resp feedback linearizing coordinates are subject to solving systems of partial differential equations we will provide here a solution to the two problems by defining algorithms allowing to compute explicitly the linearizing state resp feedback coordinates for any nonlinear control system that is indeed linearizable resp feedback linearizable each algorithm is performed using a maximum of n 1 steps n being the dimension of the system and they are made possible by explicitly solving the flow box or straightening theorem we illustrate with several examples borrowed from the literature

some numerical aspects of approximate linearization of single input non linear systems we characterize approximate linearization to order m of a single input non linear system in terms of the existence of an integrating factor to order m 2 which is equivalent to the well known condition that a system of vector fields be order m 1 involutive we present a constructive method to obtain the required transformation for approximate linearization which results in substantial computational savings

dynamical behavior of computer virus on internet abstract in this paper we presented a computer virus model using an sirs model and the threshold value r 0 determining whether the disease dies out is obtained if r 0 is less than one the disease free equilibrium is globally asymptotically stable by using the time delay as a bifurcation parameter the local stability and hopf bifurcation for the endemic state is investigated numerical results demonstrate that the system has periodic solution when time delay is larger than a critical values the obtained results may provide some new insight to prevent the computer virus

withdrawn optimal control in a novel computer virus spread model this article has been withdrawn at the request of the corresponding author the publisher apologizes for any inconvenience this may cause the full elsevier policy on article withdrawal can be found at http www elsevier com locate withdrawalpolicy

immersion and immersion by nonsingular feedback of a discrete time nonlinear system into a linear system an analytic discrete time nonlinear system is considered necessary and sufficient conditions for immersion of the system into a linear system are given local immersion is defined by nonsingular feedback into a linear system and given necessary and sufficient conditions for this problem it is shown that a similar approach can also be applied to continuous time affine nonlinear systems

parametric stabilization is easy a polynomially parametrized family of continuous time controllable linear systems is always stabilizabte by polynomially parametrized feedback

a neural approach for control of nonlinear systems with feedback linearization several schemes for feedback linearization using neural networks have been investigated and compared then an approach to design a neurocontroller in the sense of feedback linearization is introduced the contents include 1 full input output linearization when a system has relative degree n 2 partial input output linearization when a system has relative degree r r n and 3 approximate linearization when the involutivity condition does not hold corresponding programs and examples are given to illustrate the proposed methodology

brief paper a velocity algorithm for the implementation of gain scheduled controllers a new method is proposed to implement gain scheduled controllers for nonlinear plants given a family of linear feedback controllers designed for linearizations of a nonlinear plant about constant operating points a nonlinear gain scheduled controller is derived that preserves the input output properties of the linear closed loop systems locally about each equilibrium point the key procedures in the proposed method are to provide integral action at the inputs to the plant and differentiate some of the measured outputs before they are fed back to the scheduled controller for a fairly general class of systems the nonlinear gain scheduled controllers are easy to obtain and their structure is similar to that of the original linear controllers

approximation by regular input output maps the authors study approximate dynamic input output linearization in the sense of j hauser et al 1989 for multiinput multioutput mimo nonlinear systems which exhibit singularities in the dynamic extension process

feedback classification of nonlinear single input control systems with controllable linearization normal forms canonical forms and invariants we study the feedback group action on single input nonlinear control systems we follow an approach of kang and krener based on analyzing step by step the action of homogeneous transformations on the homogeneous part of the same degree of the system we construct a dual normal form and dual invariants with respect to those obtained by kang we also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms resp their dual canonical forms coincide we give an explicit construction of transformations bringing the system to its normal dual normal canonical and dual canonical forms we illustrate our results by simple examples on mathbb r 3 and mathbb r 4

green functions of higher order differential operators the green functions of the partial differential operators of even order acting on smooth sections of a vector bundle over a riemannian manifold are investigated via the heat kernel methods we study the resolvent of a special class of higher order operators formed by the products of second order operators of laplace type defined with the help of a unique riemannian metric but with different bundle connections and potential terms the asymptotic expansion of the green functions near the diagonal is studied in detail in any dimension as a by product a simple criterion for the validity of the huygens principle is obtained it is shown that all the singularities as well as the non analytic regular parts of the green functions of such high order operators are expressed in terms of the usual heat kernel coefficients ak for a special laplace type second order operator

an algorithm for orbital feedback linearization of single input control affine systems feedback linearization is an effective design and analysis tool used in the study of nonlinear control systems however cases arise where linearizability conditions cannot be met for systems that are not linearizable by classical techniques orbital feedback linearization or feedback linearization by state dependent time scaling has been proposed to relax these conditions unfortunately approaches proposed to date have led to conditions that tend to be more difficult to check then conditions for state feedback linearization in this paper necessary and sufficient conditions for orbital feedback linearizability is presented for a class of single input nonlinear systems the conditions are simple and can be checked directly from the data of the problem using an exterior calculus approach a simple algorithm is developed to compute state dependent time scaling that yield state feedback linearizable systems it is shown that orbital feedback linearizability generalizes the concept state feedback linearizability to deal with locally weakly accessible control systems

on approximate feedback equivalence of affine control systems as a generalization of some ideas of krener 1984 a b we introduce problems of k l and k r feedback equivalence of affine systems and relate them to the action of so called k l and k r feedback groups then using some elementary properties of jets of smooth maps we establish necessary conditions under which the groups act on a manifold of systems with open orbits this result allows us to determine a range of dimensions n m k r where n is the dimension of the state space and m the number of input within which the problems are ill posed in particular the result applies to problems of k r reduction of affine systems i e of finding a feedback transforming the kth order taylor polynomial of an affine system into the rth order one r k

analysis and control of hopf bifurcations in this paper control systems with two uncontrollable modes on the imaginary axis are studied the main contributions include the local orientation control of periodic solutions and center manifolds the quadratic normal form of systems with two imaginary uncontrollable modes the stabilization of the hopf bifurcation by state feedback and the quadratic invariants that characterize the nonlinearity of a system and its hopf bifurcation

nonlinear observer design in the siegel domain we extend the method of kazantzis and kravaris systems control lett 34 1998 pp 241 247 for the design of an observer to a larger class of nonlinear systems the extended method is applicable to any real analytic observable nonlinear system it is based on the solution of a first order singular nonlinear pde this solution yields a change of state coordinates which linearizes the error dynamics under very general conditions the existence and uniqueness of the solution is proved lyapunov s auxiliary theorem and siegel s theorem are obtained as corollaries the technique is constructive and yields a method for constructing approximate solutions

survey approximate linearization via feedback an overview fostered by a growing interest in nonlinear control theory and catalyzed by the discovery in the early 1980s of the exact conditions under which a nonlinear plant can be linearized by static state feedback and coordinate transformation in the last decades there has been a rapid increase of interest in the search for approximate solutions to the problem of linearizing nonlinear systems by state or output feedback main reason for that is the limited applicability of the rigorous methods and the complexity sensitivity and design difficulties of the exact linearizing compensators if any in the present paper the literature on the subject is reviewed and organized in what is believed to be a new and consistent perspective recent works especially in the area of data based techniques are in fact described and related whenever possible to fundamental results previously obtained by model based differential geometric methods this is expected to bring modern system linearization methods closer to the needs of practicing control engineers and to stimulate further research eventually able to fill visible gaps in this direction