generalized controller canonical form for linear and nonlinear dynamics a generalized controller canonical form for linear and nonlinear dynamics is proposed it is obtained using the theorem of the primitive element from differential algebra the derivation of the controller form does not apply to multivariable constant linear systems with the transformations all nonlinear dynamics can be exactly linearized by dynamic feedback the main departure from standard theory is that transformations may depend on input derivatives once differential algebraic tools are introduced the proofs of the results are easy

a numerical model for the boltzmann equation with applications to micro flows given an integer lattice email protected r d we define g as the orthogonal group leaving l invariant starting from a basic kinetic model on g we construct a collision operator on l which keeps all the essential features of the classical boltzmann collision operator for a particular 3d lattice we demonstrate the suitability of this discrete model for the numerical simulation of rarefied flows for several examples e g in the context of micro flows we find a good qualitative and quantitative agreement of our simulation results with test data

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

on minimal compensators for decoupling control abstract the general construction of minimal regular compensators for decoupling a nonlinear system is given by means of vector subspaces which are generated by differential vectors of outputs and states and the minimal dimension of a decoupling compensator is expressed in intrinsic characterization the equivalence relation of all minimal regular compensators is established though these compensators may have different dynamic behavior from each other

the differential field associated to a general analytic nonlinear dynamical system a rigorous and detailed construction of the differential field k sub sigma u y associated in a differential algebraic context to a nonlinear system sigma described in state space form with analytic coefficients is presented the field k sub sigma u y represents an input output description of sigma and its study is of key importance in problems concerning the input output behavior of sigma technically the basic step of the construction of k sub sigma u y consists in proving that the differential ideal associated to the state evolution equation of sigma is prime

global stabilization of a class of invertible mimo nonlinear systems in this paper we consider the class of input affine mimo nonlinear systems which are assumed to be invertible and input output linearizable this class includes as a trivial special case the class of mimo systems possessing a well defined vector relative degree it is shown that if a system in this class is strongly minimum phase in a sense specified in the paper it is globally stabilizable via partial state feedback or semiglobally stabilizable via dynamic output feedback the result substantially broadens the class of nonlinear mimo systems for which global semiglobal stabilization via limited state information is known to be possible

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

local stabilization of minimum phase nonlinear systems abstract in this note we briefly review the notion of a nonlinear minimum phase system i e a system which when constrained in such a way as to produce zero output evolves with an asymptotically stable dynamics the main purpose of the paper is to show that any minimum phase nonlinear system can always be locally stabilized by smooth state feedback

structure at infinity revisited abstract the structure at infinity of an ordinary differential control system is a finite sequence of increasing integers ending with the differential output rank of the system namely the number of outputs that can be given as arbitrary functions of time when the inputs are unknown its definition and construction originally done for linear systems has been extended to affine nonlinear systems and used in order to study dynamic decoupling or model matching it essentially relies on a state representation the purpose of this paper is to make a critical examination of this concept and to modify it in order to avoid the state representation at the same time we extend it to nonlinear partial differential control systems by exhibiting a link with formal integrability a highly important concept in the formal theory of systems of partial differential equations that cannot be handled by means of a transfer matrix approach many explicit examples will illustrate the main results and the possibility to use computer algebra techniques will be pointed out

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

block decoupling of non linear systems by static measurement feedback in this paper the block input output decoupling problem for non linear systems is stated and solved by static feedback of measured variables only in contrast with standard solutions which assume that the full state is available for feedback

zeros at infinity for affine nonlinear control systems a definition of zeros at infinity for affine nonlinear control systems is proposed the definition is local which means that we exclude certain singularities we argue the reasonableness of our definition by showing its relevance to the problem of nonlinear decoupling in particular we give a necessary and sufficient condition for the solvability of the general regular decoupling problem for affine systems in terms of the zeros at infinity

invertibility of switched nonlinear systems this article addresses the invertibility problem for switched nonlinear systems affine in controls the problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state we extend the concept of switch singular pairs introduced recently to nonlinear systems and develop a formula for checking if the given state and output form a switch singular pair a necessary and sufficient condition for the invertibility of switched nonlinear systems is given which requires the invertibility of individual subsystems and the nonexistence of switch singular pairs when all the subsystems are invertible we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs detailed examples are included to illustrate these newly developed concepts

minimality of dynamic input output decoupling for nonlinear systems in this note we study the strong dynamic input output decoupling problem for nonlinear systems using an algebraic theory for nonlinear control systems we obtain for a dynamic input output decouplable nonlinear system a compensator of minimal dimension that solves the decoupling problem

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

nonlinear strong model matching the problem of matching a given input output behavior for systems described by general nonlinear differential equations is considered it is shown that by appropriately modifying the zero dynamics algorithm it is possible to obtain a simple necessary and sufficient condition for the solvability of the model matching problem which requires that the initial state be on an appropriate submanifold of the state space another condition necessary and sufficient for the solvability of the strong model matching problem is proposed this last condition is then related to an equality of a list of integers which under some regularity assumptions coincide with the algebraic structures at infinity of the process and of a composition of the process and the model the relation between these conditions and the equality of the algebraic structures at infinity of the process and the model is established

parameterization of decoupling control laws for affine nonlinear systems the problem of parameterizing all the decoupling feedbacks is investigated and applied to the design of a stable noninteracting control for affine nonlinear systems an algebraic study of controlled invariance is presented it covers a new criterion for controlled invariance parameterization and novel properties of the friend set and computational aspects of the maximal controlled invariant distribution this analysis is used to derive falb wolovich like conditions for general decoupling including disturbance decoupling and noninteracting control problems it is shown that to determine all the decoupling feedbacks one usually needs infinitely many sets of controlled invariant or controllability distributions that can arise as solutions to the problems it is also shown that under certain conditions stable noninteracting design based on first parameterizing the decoupling feedbacks is still a feasible approach in the sense that a feedback can be chosen from the friend set of the maximal solution for the exponentially stable and noninteracting design problem

almost disturbance decoupling for single input single output nonlinear systems the almost disturbance decoupling problem for nonlinear single input single output systems is addressed by using singular perturbation methods and high gain feedback sufficient conditions and the explicit high gain nonlinear state feedback in solvable cases are given they generalize both previous almost results for linear systems and exact ones for nonlinear systems the necessity of the conditions is discussed in particular an example is given where the main structural condition is not satisfied and the high gain control designed on the basis of linear approximations fails to achieve almost disturbance decoupling for the original system

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

global state and feedback equivalence of nonlinear systems the problem of finding global state space transformations and global feedback of the form u t x t to transform a given nonlinear system xdot f x g x u to a controllable linear system on rn or on an open subset of rn is considered here we give a complete set of differential geometric conditions which are equivalent to the existence of a solution to the above problem

use of observers for the inversion of nonlinear maps abstract in this paper we deal with the problem of the inversion of nonlinear maps the first contribution of the paper is the definition in a formal framework of some problems related to map inversion which are already present in the literature for instance in the robotics area and in observer theory the second and the main contribution consists in the solution of such problems by means of observers for which some structures are proposed for the state estimation of a time varying nonlinear system associated with the given map and reference trajectory

stable inversion for nonlinear systems abstract there have been two recent major developments in output tracking for nonlinear systems and our first main contribution is to relate these under appropriate assumptions we show that the bounded solution of the partial differential equation of isidori and byrnes for each trajectory of an exosystem must be given by an integral representation formula of devasia chen and paden under restrictive hypotheses devasia chen and paden develop a picard process that converges to the solution of the integral equation this solution to the integral equation is also a bounded solution to a dynamical equation driven by the desired outputs in aircraft applications our nonlinear systems are perturbations of pure feedback systems with outputs and we find a solution to the stable inversion problem i e finding bounded controls and bounded state trajectories in response to bounded output signals in two steps the first step ignores the perturbation error and computes the major part of the desired control and corresponding state trajectory the second step computes the remaining control and states by finding a noncausal and stable solution to an error driven dynamical equation in other words the method of devasia chen and paden is applied to an error system and not to the full system this two step procedure is the second main contribution of this paper throughout this paper it is assumed that our nonlinear systems have vector relative degree

on implicit systems of differential equations abstract this article considers implicit systems of differential equations the implicit systems that are considered are given by polynomial relations on the coordinates of the indeterminate function and the coordinates of the time derivative of the indeterminate function for such implicit systems of differential equations we are concerned with computing algebraic constraints such that on the algebraic variety determined by the constraint equations the original implicit system of differential equations has an explicit representation our approach is algebraic although there have been a number of articles that approach implicit differential equations algebraically all such approaches have relied heavily on linear algebra the approach of this article is different we have no linearity requirements at all instead we rely on algebraic geometry in particular we use birational mappings to produce an explicit system the methods developed in this article are easily implemented using various computer algebra 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 consistency result for a discrete velocity model of the boltzmann equation in this paper we study the link between a certain class of discrete velocity models dvms and the boltzmann equation those models possess an infinite number of velocities laying on a regular grid of step h in r3 our aim is to prove that it is possible to construct models consistent with the boltzmann equation i e such that the discrete collision term can be seen as an approximation of the collision integral of the boltzmann equation

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

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

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

a solution to the generalized problem of nonlinear map inversion abstract this paper deals with the generalization of some problems related to map inversion solutions of such problems are proposed by means of observers for which a structure is proposed

on the sliding mode control of multivariable nonlinear systems sliding mode control of multivariable nonlinear systems is addressed from the perspective of linear differential algebra for a special but large class of linearizable systems known as differentially flat systems essential orders and differential flatness are shown to be a relevant concept and a sensible requirement respectively associated with the possibilities of designing static or dynamical sliding mode multivariable feedback regulators for nonlinear systems

persistent identification of systems with unmodeled dynamics and exogenous disturbances in this paper a novel framework of system identification is introduced to capture the hybrid features of systems subject to both deterministic unmodeled dynamics and stochastic observation disturbances using the concepts of persistent identification control oriented system modeling and stochastic analysis we investigate the central issues of irreducible identification errors and time complexity in such identification problems upper and lower bounds on errors and speed of persistent identification are obtained the error bounds are expressed as functions of observation lengths sizes of unmodeled dynamics and probability distributions of disturbances asymptotic normality and complexity lower bounds are investigated when periodic inputs and ls estimation are applied generic features of asymptotic normality are further explored to extend the asymptotic lower bounds to a wider range of signals and identification mappings

brief paper asymptotic rejection of asymmetric periodic disturbances in output feedback nonlinear systems this paper deals with asymptotic rejection of periodic disturbances which may have asymmetric basic wave patterns this class of disturbances covers asymmetric wave forms in the half period such as alternating sawtooth wave form some disturbances which are generated from nonlinear oscillation such as van de pol oscillators as well as disturbances with symmetric half period wave forms such as sinusoidal disturbances and triangular disturbances etc the systems considered in this paper can be transformed to the nonlinear output feedback form the amplitude and phase of the disturbances are unknown the novel concept of integral phase shift is introduced together with the newly introduced half period integration operator to investigate the invariant properties of asymmetric periodic disturbances they are used for the estimation of unknown disturbances in the systems together with observer design techniques to deal with nonlinearity the proposed control design with the disturbance estimation asymptotically rejects the unknown disturbance and ensures the overall stability of the system

discrete velocity models for mixtures without nonphysical collision invariants an important aspect of constructing discrete velocity models dvms for the boltzmann equation is to obtain the right number of collision invariants it is a well known fact that dvms can also have extra collision invariants so called spurious collision invariants in plus to the physical ones a dvm with only physical collision invariants and so without spurious ones is called normal for binary mixtures also the concept of supernormal dvms was introduced meaning that in addition to the dvm being normal the restriction of the dvm to any single species also is normal here we introduce generalizations of this concept to dvms for multicomponent mixtures we also present some general algorithms for constructing such models and give some concrete examples of such constructions one of our main results is that for any given number of species and any given rational mass ratios we can construct a supernormal dvm the dvms are constructed in such a way that for half space problems as the milne and kramers problems but also nonlinear ones we obtain similar structures as for the classical discrete boltzmann equation for one species and therefore we can apply obtained results for the classical boltzmann equation

output feedback disturbance decoupling in nonlinear systems the problem of disturbance decoupling by means of output feedback is addressed the notion of conditioned invariance is reconsidered in a linear algebraic framework using this geometric notion a necessary and sufficient condition respectively necessary condition is obtained for solving the disturbance decoupling problem by quasi static output feedback respectively the disturbance decoupling problem by dynamic output feedback dddpo the necessary conditions for dddpo which are derived are weaker than the existing ones a sufficient condition for the disturbance decoupling problem by static output feedback is given as well

normal high order discrete velocity models of the boltzmann equation abstract this paper aims at approximations of the collision operator in the boltzmann equation the developed framework guarantees the normality of the approximation which means correct collision invariants h theorem and equilibrium solutions it fits into the discrete velocity model framework is given in such a way that it is understandable with undergraduate level mathematics and can be used to construct approximations with arbitrary high convergence orders at last we give an example alongside a numerical verification here the convergence orders range up to 3 2 and the time complexity is given by 3 1 2 4 2 3 in 2 3 dimensions

a stopping rule for least squares identification a stopping rule for least squares identification is developed the stopping rule is determined by the construction of a confidence ellipsoid for any predetermined estimation error epsilon 0 if the iterates are inside of an ellipsoidal confidence region with volume less than or equal to epsilon sup r then the recursive online algorithm will be terminated with high probability

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

equivalence of nonlinear systems to input output prime forms the problem of transforming nonlinear control systems into input output prime forms is dealt with using state space static state feedback and also output space transformations necessary and sufficient geometric conditions for the solvability of this problem are obtained the results obtained generalize well known results both on feedback linearization as well as input output decoupling of nonlinear systems it turns out that from a computational point of view the output space transformation is the crucial step that is performed by constructing rectifying coordinates for a nested sequence of distributions on the output manifold

an augmented ud identification algorithm an augmented ud identification audi algorithm for system identification is developed by rearranging the data vectors and augmenting the covariance matrix of bierman s ud factorization algorithm the structure of the augmented information covariance matrix is particularly easy to interpret and it is shown that the audi algorithm is a direct extension of the familiar recursive least squares rls algorithm the proposed algorithm permits simultaneous identification of the model parameters plus loss functions for all orders from 1 to n at each time step with approximately the same calculation effort as th order rls this provides a basis for simultaneous model order and parameter identification so that problems due to over and under estimation of model can be avoided based on its least squares properties numerical robustness theoretical basis and the fact that it simultaneously estimates multiple models the proposed audi algorithm is recommended for use in place of rls and bierman s ud factorizatio

output input stability implies feedback stabilization abstract we study the recently introduced notion of output input stability which is a robust variant of the minimum phase property for general smooth nonlinear control systems this paper develops the theory of output input stability in the multi input multi output setting we show that output input stability is a combination of two system properties one related to detectability and the other to left invertibility for systems affine in controls we derive a necessary and sufficient condition for output input stability which relies on a global version of the nonlinear structure algorithm this condition leads naturally to a globally asymptotically stabilizing state feedback strategy for affine output input stable systems

canonical decomposition of non linear systems the construction of a canonical or normal representation for non linear systems similar to the morse canonical form in the linear case is obtained the problem considered consists in particular of studying the existence of generalized transformations that achieve a desired block decoupling such transformations generalize the notions of state transformation state feedback and output injection a sufficient condition is given in terms of a separability condition which in turn is completely characterized by using differential forms and their properties

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

feedback linearization of discrete time systems abstract we give necessary and sufficient conditions for a nonlinear discrete time system to be locally linearizable by a change of state coordinates and a feedback the conditions are expressed in terms of controllability distributions associated to the system

on the construction of static stabilizers and static output trackers for dynamically linearizable systems related results and applications it is shown that the property of dynamic linearizability to be understood as linearizability by means of the dynamic extension algorithm implies the existence of static possibly time varying control laws yielding asymptotic output tracking with arbitrary speed of convergence and asymptotic stabilization with a computable bound on the region of attraction similar results hold for systems which are only input output linearizable by means of dynamic state feedback provided that the inverse dynamics possess certain stability properties applications of these results to the problem of regional stabilization of a vtol aircraft is considered together with the tracking problem for a class of flexible joints robots moreover a novel parameterization for flexible joint robots is also proposed

recursive information forgetting with augmented ud identification fundamental analysis of information forgetting schemes in recursive identification shows that they can be classified into two basic approaches relative and absolute forgetting the augmented ud identification audi algorithm developed by the authors reveals that all the information pertinent to identification and information forgetting is contained in a single matrix called the augmented information matrix aim the tracking ability of the recursive algorithms is determined by the relative weight that is put on to the new information contained in the regressor compared with the information already contained in the aim this relative weight can be easily controlled by the simple diagonal loss function matrix produced by the audi method this simplifies the interpretation of both new and existing information forgetting approaches and also provides helpful guidelines for design

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

an intrinsic sufficient condition for regular decoupling abstract this paper introduces an intrinsic regularity property namely regular invertibility which requires that a sequence of vector spaces has constant dimension after a certain order thus allowing some degeneracies in the beginning it corresponds to the system and its tangent having the same defect using the constant rank theorem this property is shown to be sufficient condition for regular decoupling

discrete kinetic models in the fluid dynamic limit we investigate discrete kinetic models in the fluid dynamic limit described by the euler system and the navier stokes correction obtained by the chapman enskog procedure we show why reliable small systems can be expected only for small mach numbers and derive a calculus for designing models for given prandtl numbers

convergence rate of least squares identification and adaptive control for stochastic systems the strong consistency and the convergence rate of least squares identification for the multidimensional armax model are established under some decaying excitation conditions which are satisfied if both input and output do not grow too fast and the attenuating excitation technique is applied the parameter identification results are applied to adaptive control systems with a quadratic loss function the rate of convergence of the loss function to its minimum is also obtained

stabilization by output feedback of multivariable invertible nonlinear systems in this paper the problem of global stabilization of a rather general class of mimo nonlinear systems is addressed the systems considered in the paper are invertible have a trivial zero dynamics and possess a normal form in which certain multipliers are functions of the state vector of a special kind while special structural dependence of such multipliers on the components state vector has been exploited before in the context of achieving stabilization via full state feedback though the novelty in the approach of this paper is that a peculiar structure is identified which happens to be intimately related to the property of uniform complete observability thus making the design of observers possible and to the property of uniform invertibility relations never established before in the literature as a result for this class of mimo nonlinear systems a dynamic output feedback law can be designed yielding semiglobal and even global under appropriate assumptions asymptotic stability

discrete velocity models without nonphysical invariants models for mixtures of discrete velocity gases have been recently introduced by the authors and have produced unexpected results particularly with regard to the possible existence of nonphysical collision invariants here we discuss a method to construct models without spurious invariants the method can be extended to very general models including polyatomic gases chemical reactions etc

necessary conditions for asymptotic tracking in nonlinear systems in the literature it has been shown that if a single input single output analytic nonlinear plant 1 has a well defined relative degree and 2 is minimum phase it is possible to achieve asymptotic tracking for an open set of output trajectories containing the origin in c sup n 0 spl infin the space of n times continuously differentiable functions taking values in r when either of these sufficient conditions is not met various authors have investigated approximate analytic solutions discontinuous solutions and solutions for restricted sets of trajectories in this paper it is shown that conditions 1 and 2 are necessary for the existence of an analytic compensator which yields asymptotic tracking for an open set of output trajectories analogous results are established for multi input multi output systems

the nonlinear interactor and its application to input output decoupling the solvability of the input output decoupling problem via pure dynamic state feedback a special type of dynamic compensator without zeros at infinity is investigated a necessary and sufficient condition for the solvability of this problem is established in terms of a nonlinear interactor a new concept in the theory of nonlinear control

global stabilization of partially linear composite systems a linear stabilizable nonlinear asymptotically stable cascade system is globally stabilizable by smooth dynamic state feedback if a the linear subsystem is right invertible and weakly minimum phase and b the only linear variables entering the nonlinear subsystem are the output and the zero dynamics corresponding to this output both of these conditions are coordinate free and there is freedom of choice for the linear output variable this result generalizes several earlier sufficient conditions for stabilizability moreover the weak minimum phase condition for the linear subsystem cannot be relaxed unless a growth restriction is imposed on the nonlinear subsystem

generalized normal form and stabilization of non linear systems this paper considers the normal form of non linear control systems first we propose a generalized relative degree relative degree vector for non linear single respectively multiple input control system which is called the point relative degree respectively point relative degree vector for the systems without output the concepts of essential relative degree respectively essential relative degree vector and the essential point relative degree respectively essential point relative degree vector are defined unlike the classical definition which requires regularity the point relative degree vector is always well defined using these new concepts the generalized normal form is obtained its relationship with the jacobian linearization is investigated using it a straightforward computation algorithm is provided to achieve the generalized normal form based on the generalized normal form we prove that with an additional condition if the zero dynamics is stable the overall system is stabil

a new approach to the structure at infinity of nonlinear systems abstract zeros at infinity for nonlinear systems are defined by using differential algebra this approach can be viewed as a nonlinear generalization of silverman s structure algorithm the connection with the problem of inverting input output systems is briefly described

trading the stability of finite zeros for global stabilization of nonlinear cascade systems analyzes the stabilizability properties of nonlinear cascades in which a nonminimum phase linear system is interconnected through its output to a stable nonlinear system it is shown that the instability of the zeros of the linear system can be traded with the stability of the nonlinear system up to a limit fixed by the growth properties of the cascade interconnection term below this limit global stabilization is achieved by smooth static state feedback beyond this limit various examples illustrate that controllability of the cascade may be lost making it impossible to achieve large regions of attractions

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

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

immersion and invariance adaptive control of linear multivariable systems abstract we show in this paper that it is possible to globally adaptively stabilize linear multivariable systems with reduced prior knowledge of the high frequency gain in particular we relax the restrictive nongeneric symmetry condition usually required to solve this problem instrumental for the establishment of our result is the use of the new immersion and invariance approach to adaptive control recently proposed in the literature the controllers obtained with this technique are not certainty equivalent though smooth and without projections or overparameterizations and the resulting lyapunov functions contain cross terms between the plant states and the parameter errors

on the solvability of the regulator equations for a class of nonlinear systems regulator equations arise in studying the output regulation problem for nonlinear systems the solvability of the regulator equations is the necessary condition for that of the output regulation problem it has been shown under various assumptions that the solvability of the regulator equations can be reduced to that of a center manifold equation defined by the zero dynamics of a composite system consisting of the plant and exosystem in this paper we further show that for a quite general class of nonlinear systems the solvability of the regulator equations can be reduced to that of a center manifold equation if the relative degree of the composite system at the origin exists

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

a canonical dynamic extension for noninteraction with stability for affine nonlinear square systems we study the problem of noninteracting control with stability via dynamic feedback for continuous affine nonlinear square systems given a system which may not be decouplable via static state feedback but is decouplable via dynamic feedback a canonical dynamic extension is constructed we show that every dynamic extension contains this canonical one hence the mix dynamics as defined by wagner of the canonical extension is contained in every other dynamic extension following wagner s result the stability of this mix dynamics is necessary for noninteraction with internal stability the results in this paper show that under some regularity assumptions in the investigation of noninteraction with stability one can assume without loss of generality that the system is decouplable via static state feedback

output input stability and minimum phase nonlinear systems this paper introduces and studies the notion of output input stability which represents a variant of the minimum phase property for general smooth nonlinear control systems the definition of output input stability does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics in the spirit of the input to state stability philosophy it requires the state and input of the system to be bounded by a suitable function of the output and derivatives of the output modulo a decaying term depending on initial conditions the class of output input stable systems thus defined includes all affine systems in global normal form whose internal dynamics are input to state stable and also all left invertible linear systems whose transmission zeros have negative real parts as an application we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control

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

a polynomial approach to discrete time non linear system controllability this paper uses a polynomial approach to present a necessary and sufficient condition for local controllability of single input single output siso discrete time non linear systems the condition is presented in terms of common factors of a non commutative polynomial expression this extends the result on controllability of linear systems and the recent result on controllability of continuous time non linear systems to expose controllability of discrete time non linear systems in the input output framework the controllability condition yields a finite step computing procedure for examining controllability of discrete time input output non linear systems

a unifying framework for global regulation via nonlinear output feedback from iss to iiss this paper presents a unifying framework for the problem of robust global regulation via output feedback for nonlinear systems with integral input to state stable inverse dynamics subject to possibly unknown control direction the contribution of the paper is two fold firstly we consider the problem of global regulation instead of global asymptotic stabilization gas for systems with generalized dynamic uncertainties it is shown by an elementary example that gas is not solvable using conventional smooth output feedback secondly we reduce the stability requirements for the disturbance and demand relaxed assumptions for the system using our framework most of the known classes of output feedback form systems are broadened in several directions unmeasured states and unknown parameters can appear nonlinearly restrictive matching and growth assumptions are removed the dynamic uncertainty satisfies the weaker condition of sontag s integral input to state stability and the sign of high frequency gain may be unknown a constructive strategy is proposed to design a dynamic output feedback control law that drives the state to the origin while keeping all other closed loop signals bounded