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

robust control of cascaded and individually feedback linearizable nonlinear systems abstract feedback linearization technique is applied to design robust controllers for a class of nonlinear uncertain systems every system in the class is a series connection of finite number of nonlinear subsystems which are individually feedback linearizable and have significant but matched nonlinear functional uncertainties the key difference between this work and existing robust results is that the matching conditions and the existence of feedback linearization transformation are only required individually for each subsystem a constructive robust control design procedure is proposed for the unique control in the overall system the procedure is conceptually simple and contains three steps first perform input output feedback linearization for all individual subsystems second design fictitious robust controllers for inputs of each subsystem as if the subsystem had an independent control finally a mapping is developed which maps the individual fictitious controls recursively into the unique control of the overall system since neither the matching conditions nor feedback linearization are required for the overall system this becomes the broadest class of nonlinear uncertain or known systems for which a robust regular tracking control can be constructed using feedback linearization method global stability of the overall system is always guaranteed

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

on the linearization via a restricted class of dynamic feedback we deal with the problem of linearization of nonlinear systems using a restricted class of dynamic compensator which is composed of chains of integrators followed by static feedback in the vector field formulation we present an upper bound on the number of integrators to linearize nonlinear systems and show that the bound is sharp when n 2 spl ges m spl ges 2 this results in the checkable necessary and sufficient conditions

remarks on linearization of discrete time autonomous systems and nonlinear observer design abstract this paper presents necessary and sufficient conditions under which a discrete time autonomous system with outputs is locally state equivalent to an observable linear system or a system in the nonlinear observer form krener and isidori 1983 in particular an open problem raised in lee and nam 1991 namely the observer linearization problem is solved for a nonlinear system which may not be invertible i e the mapping f may not be a local diffeomorphism as a consequence the nonlinear observer design problem is solved by means of exact linearization techniques

a model reference adaptive control scheme for pure feedback nonlinear systems a model reference adaptive control mrac scheme is presented for nonlinear systems in a pure feedback canonical form with unknown parameters the present of parameter uncertainty in the system causes imperfect linearization i e it introduces nonlinear additive terms in the transformed coordinates under some mild technical assumptions global convergence of the output error is established for all initial estimates of the parameter vector lying in an open neighborhood of the true parameters in the parameter space

singular pdes and the single step formulation of feedback linearization with pole placement the present work proposes a new formulation and approach to the problem of feedback linearization with pole placement the problem under consideration is not treated within the context of geometric exact feedback linearization where restrictive conditions arise from a two step design method transformation of the original nonlinear system into a linear one in controllable canonical form with an external reference input and the subsequent employment of linear pole placement techniques in the present work the problem is formulated in a single step using tools from singular pde theory in particular the mathematical formulation of the problem is realized via a system of first order quasi linear singular pdes and a rather general set of necessary and sufficient conditions for solvability is derived by using lyapunov s auxiliary theorem the solution to the system of singular pdes is locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law computed through the solution of the system of singular pdes both feedback linearization and pole placement design objectives may be accomplished in a single step effectively overcoming the restrictions of the other approaches by bypassing the intermediate step of transforming the original system into a linear controllable one with an external reference input

state feedback control of non linear systems a design method for state feedback controllers for single input non linear systems is proposed the method makes use of the transformations of the non linear system into controllable like canonical forms the resulting non linear state feedback is designed in such a way that the eigenvalues of the linearized closed loop model are invariant with respect to any constant operating point the method constitutes an alternative approach to the design methodology recently proposed by baumann and rugh also a review of different transformation methods for non linear systems is presented an example and simulation results of different control strategies are provided to illustrate the design technique

linearization of discrete time nonlinear systems and a canonical structure linearizable nonlinear discrete time systems are described as a composition of diffeomorphisms linearizability of a general nonlinear system is obtained as an extension of linearizability of the system expressed in terms of the diffeomorphisms

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

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

an input output characterization for linearization by feedback for a stationary initially relaxed single input single output bilinear realizable system that is described by a sequence of regular transfer functions the relative degrees of the individual transfer functions are defined in a manner similar to the linear case these relative degrees are used to characterize the region where linear input output behavior can be obtained by state feedback in a bilinear realization of the system

generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes abstract in this paper we study the error in the approximation of a convex function obtained via a one parameter family of approximation schemes which we refer to as barycentric approximation schemes for a given finite set of pairwise distinct points x n x i i 0 n in r d the barycentric approximation of a convex function f is of the form b f x i 0 n i x f x i where i i 0 n is a set of barycentric coordinates with respect to the point set x n the main content of this paper is two fold the first goal is to derive sharp upper and lower bounds on all barycentric coordinates over the convex polytope conv x n the second objective of the paper is to exploit the convexity assumption heavily and establish a number of upper and lower pointwise bounds on the approximation error for approximating arbitrary convex functions these bounds depend solely on computable quantities related to the data values of the function the largest and smallest barycentric coordinates for convex twice continuously differentiable functions we derive an optimal error estimate we show that the delaunay triangulation gives access to efficient algorithms for computing optimal barycentric approximation finally numerical examples are used to show the success of the method

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

derivative computations for the log likelihood function this note improves on a method for computing derivatives of the log likelihood function used in the identification of discrete linear time invariant gaussian systems a formula for the derivative is obtained in terms of the derivatives of the matrices used to define the system under identification

the effect sampling on linear equivalence and feedback linearization we investigate the effect of sampling on linearization for continuous time systems it is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change also it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant even if this is known to be linearizable with continuous time feedback for n 2 we show under the assumption of completeness of adfg that if the discretized system is lineariable by state coordinate change and feedback then the continuous time affine complete analytic system is linearizable by state coordinate change only also we suggest a method of proof when n 3

linearization of discrete time systems abstract we characterize the equivalence of single input single output discrete time nonlinear systems to linear ones via a state coordinate change and with or without feedback four cases are distinguished by allowing or disallowing feedback as well as by including the output map or not the interdependence of these problems is analyzed an important feature that distinguishes these discrete time problems from the corresponding problem in continuous time is that the state coordinate transformation is here directly computable as a higher composition of the system and output maps finally certain connections are made with the continuous time case

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

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

a simple algorithm for computing canonical forms abstract it is well known that all linear time invariant controllable systems can be transformed to brunovsky canonical form by a transformation consisting only of coordinate changes and linear feedback however the actual procedures for doing this have tended to be overly complex the technique intorduced here is envisioned as an on line procedure and is inspired by george meyer s tangent model for nonlinear systems the process utilizes meyer s block triangular form as an intermediate step in going to brunovsky form the method also involves orthogonal matrices thus eliminating the need for the computation of matrix inverses in addition the kronecker indices can be computed as a by product of this transformation so it is not necessary to know them in advance

on the extension of the korobov s class of linearizable triangular systems by nonlinear control systems of the class c1 abstract we give the complete description of nonlinear control systems of the class c 1 with one dimensional control which are feedback linearizable by means of a change of the control of the class c 1 and a change of variables of the class c 2 as well as the control systems which are linearizable by means of a change of variables only the both results are originated from the approach proposed by v i korobov in 1973 for the linearization of triangular control systems

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

a corrective feedback design for nonlinear systems with fast actuators recent two time scale results can be derived from a geometric framework which allows further extensions and computational improvements in this note the two time scale behavior of singularly perturbed systems is exploited to design slow and fast controls and to combine them into a composite control as an illustration we present a corrective design to compensate for fast actuator dynamics modeled as singular perturbations

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

a new method for evaluating the log likelihood gradient the hessian and the fisher information matrix for linear dynamic systems a method is presented for evaluating the log likelihood gradient score the hessian and the fisher information matrix of the parameters of linear dynamic stochastic systems the method incorporates the optimal kalman smoothing equations and is therefore ideal for simultaneous state estimation and parameter identification the result can be used for efficient implementation of gradient based algorithms for maximum likelihood identification of the unknown system parameters and for assessing the mean square estimation accuracy

a functional equations approach to nonlinear discrete time feedback stabilization through pole placement abstract the present work proposes a new approach to the nonlinear discrete time feedback stabilization problem with pole placement the problem s formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived using tools from functional equations theory one can prove that the solution to the above system of nonlinear functional equations is locally analytic and an easily programmable series solution method can be developed under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete time state feedback control law that are both computed through the solution of the system of nonlinear functional equations the feedback stabilization with pole placement design objective can be attained under rather general conditions the key idea of the proposed single step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach however since the proposed method does not involve an external reference input it cannot meet other control objectives such as trajectory tracking and model matching

the immersion under feedback of a multidimensional discrete time non linear system into a linear system this paper deals with the problem of finding conditions which ensure the reproducibility of the input output behaviour of a linear analytic discrete time system by means of a linear discrete time system on this basis the problem of modifying the system by feedback in order for it to enjoy such a property is posed and solved

deterministic feedback linearization girsanov transformations and finite dimensional filters the problem of extending state feedback linearization methods of deterministic control theory to stochastic systems is addressed for stratonovitch stochastic differential equations with smooth vector fields necessary and geffucient geometric conditions for local and global linearization by diffeomorphism and absolutely continuous change of probability law are obtained using the interpretation of girsanov transformations as state feedback on brownian motions for stochastic systems with single with single input or onde dimensional brownian motion and single output or one dimensional observation process necessary and sufficient geometric conditions to transform the duncan mortensen zakai dmz equation of filtering into that of an affine prime system are obtained as well as interpretation of gauge transformation as girsanov change of probability law

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 external linearization of nonlinear systems via feedback this note presents necessary conditions and sufficient conditions for an affine nonlinear system to be globally feedback equivalent to a controllable linear system over an open subset v of rn when v equals rn necessary and sufficient conditions are obtained

sharp error estimates for interpolatory approximation on convex polytopes let mathfrak p be a convex polytope in the d dimensional euclidean space we consider an interpolation of a function f at the vertices of mathfrak p and compare it with the interpolation of f and its derivative at a fixed point y in mathfrak p the two methods may be seen as multivariate analogues of an interpolation by secants and tangents respectively for twice continuously differentiable functions we establish sharp error estimates with respect to a generalized lp norm for 1 le p le infty the case p 1 is of special interest since it provides analogues of the midpoint rule and the trapezoidal rule for approximate integration over the polytope mathfrak p in the case where mathfrak p is a simplex and p 1 this investigation covers recent results by s waldron siam j numer anal 35 1998 pp 1191 1200 and by m stampfle j approx theory 103 2000 pp 78 90

feedforward and feedback linearization of non linear systems with disturbances the problem of transforming non linear systems with input disturbances to their linear equivalents is investigated a characterization of the class of non linear systems with disturbances that are equivalent to controllable quasi linear systems with disturbances z az bv z d is given the necessary transformations state space coordinate changes and control input space coordinate changes by non linear feedforward and feedback are presented and their implications for control system synthesis are discussed finally a numerical example is given

global exact quadratization of continuous time nonlinear control systems we consider a class denoted by cal q of the nonlinear control systems which can be densely represented as a subsystem of a certain kind of quadratic system namely a quadratic target we say that a system in cal q undergoes a globally exact quadratization here globally adds up to a slight extension of the notion of c infty immersion of systems namely a dense immersion which amounts to saying that it is defined on the whole manifold of the system states except possibly a zero measure set it is proven that the class cal q includes all systems characterized by vector fields whose components are analytic integral closed form functions icffs the result is first proven for algebraic system functions by means of a constructive proof and next extended up to analytic icffs for nonanalytic icffs a weaker result is proven as well also the case of a partially observed system is considered as well as the internal structure of every quadratic representation which is proven to be alwa

reduced gradient computation in prediction error identification one way to design practical identification algorithms using the prediction error method is to find a computationally efficient expression for the gradient of the prediction error cost function here results on adjoint equations are used to derive an efficient gradient expression which reduces the computational burden of evaluating it

feedback equivalence of control systems abstract cartan s method of equivalence is applied to the problem of equivalence of 2 state systems with scalar control under feedback the differential invariants produced by the method completely characterize equivalence classes the phenomenon of linearizability is associated with the presence of infinite lie pseudogroups the generic non linearizable case has a geometrically defined variational problem which yields a time optimal closed loop control

on time scaling for nonlinear systems application to linearization in this note we propose a method to analyze systems in a time scale which is varied depending on the state such as dt d tau s x where t and are the actual time scale and that of new one respectively and s x is the function which we call time scaling function analysis of the system in the new time scale enables us to investigate the intrinsic structure of the system a linearization problem in the new time scale is formulated as wide sense feedback equivalence and is solved it is also shown that the time scaling function which makes the system linear is derived as the solution of differential equations

stabilization and feedback equivalence to linear coupled oscillators sufficient conditions for the system xdot f x u t g x x r2n to be stabilized via a smooth state feedback are given it is also shown that certain lie algebraic conditions are necessary and sufficient for the system to be feedback equivalent to a system of controlled linear coupled oscillators which is feedback stabilizable and which is proposed as a reference model for stabilization problems two overlapping sets of sufficient conditions which guarantee the existence and the explicit construction of stabilizing smooth state feedback controls are therefore given and their connections are discussed

control systems in neural networks with local memory abstract based on a new paradigm of neural networks consisting of neurons with local memory nnlm we discuss the representation of a control system by neural networks using this representation the basic issues of complete controllability and observability for the system are addressed a separation principle of learning and control is presented for nnlm the result shows that the weights of the network will not affect its dynamics some results about local linearization via a regular static feedback and nonlinear transformation are also given

on approximation of linear functionals over convex functions recently there has been renewed interest in the problem of finding under and over estimations on the set of convex functions to a given non negative linear functional that is approximations which estimate always below or above the functional over a family of convex functions the most important example of such an approximation problem is given by the multidimensional versions of the midpoint rectangle rule and the trapezoidal rule which provide under and over estimations to the true value of the integral on the set of convex functions also known as the hermite hadamard inequality in this paper we introduce a general method of constructing new families of under over estimators on the set of convex functions for a general class of linear functionals in particular under the regularity condition namely the functions belonging to c 2 not necessarily convex we will show that the error estimations based on such estimators are always controlled by the error associated with using the quadratic function the result is also extended to the class of lipschitz functions we also propose a modified approximation technique to derive a general class of under over estimators with better error estimates

symmetrization procedures and convexity in centrally symmetric polytopes univariate symmetrization technique has many good properties in this paper we adopt the high dimensional viewpoint and propose a new symmetrization procedure in arbitrary convex polytopes of r d with central symmetry moreover the obtained results are used to extend to the arbitrary centrally symmetric polytopes the well known hermite hadamard inequality for convex functions

receding horizon control and discontinuous state feedback stabilization this paper addresses three aspects of receding horizon control in discrete time 1 feedback stabilization of general nonlinear systems with receding horizon control 2 the generation of stabili

a lyapunov approach for robust nonlinear state feedback synthesis the authors apply the input output linearization approach for nonlinear state feedback synthesis the model uncertainty under consideration is a class of state model perturbations that satisfy appropriate matching conditions the controller design uses a lyapunov based approach to guarantee uniform ultimate boundedness of the states and the output

a slow manifold approach to linear equivalents of nonlinear singularly perturbed systems abstract the problem of external feedback linearization of a nonlinear system into a linear controllable system has received considerable attention during the past few years in this paper the concept of slow manifold theory is applied to a class of nonlinear singularly perturbed systems to achieve external linearization for the full system via reduced order decompositions into exact slow and fast subsystems approximations to the exact state feedback and the control space transformation are obtained in particular by a specific choice of the manifold the same state space coordinate transformation can be utilized to approximate the exact solution the notion of weakly and strongly linearizable systems are introduced it is shown that if the system is weakly linearizable the state space coordinate transformation and the state feedback results in a high gain feedback control finally the notion of input output linearization and parasitic decoupling problem is introduced

some comments on global linearization of nonlinear systems it is shown that even in the case of a single input system on r 2 it will be rather difficult to find conditions sufficient to guarantee existence of a global linearization and that local conditions alone will definitely not be enough a class of examples which is fairly typical and it is hoped useful is discussed

convexity results and sharp error estimates in approximate multivariate integration jjan interesting property of the midpoint rule and the trapezoidal rule which is expressed by the so called hermite hadamard inequalities is that they provide one sided approximations to the integral of a convex function we establish multivariate analogues of the hermite hadamard inequalities and obtain access to multivariate integration formulae via convexity in analogy to the univariate case in particular for simplices of arbitrary dimension we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases the first family also includes a multivariate analogue of a maclaurin formula and of the two point gaussian quadrature formula the second family includes a multivariate analogue of a formula by p c hammer and of simpson s rule in both families we trace out those formulae which satisfy a hermite hadamard inequality as an immediate consequence of the latter we obtain sharp error estimates for twice continuously differentiable functions

losslessness feedback equivalence and the global stabilization of discrete time nonlinear systems in this paper a necessary and sufficient condition for a nonlinear system of the form spl sigma given by x k 1 f x k g x k u k y k h x k j x k u k to be lossless is given and it is shown that a lossless system can be globally asymptotically stabilized by output feedback if and only if the system is zero state observable then we investigate conditions under which spl sigma can be rendered lossless via smooth state feedback in particular we show that this is possible if and only if the system in question has relative degree spl lcub 0 0 spl rcub and has lossless zero dynamics under suitable controllability like rank conditions we prove that nonlinear systems having relative degree spl lcub 0 0 spl rcub and lossless zero dynamics can be globally stabilized by smooth state feedback as a consequence we obtain sufficient conditions for a class of cascaded systems to be globally stabilizable the global stabilization problem of the nonlinear system spl sigma without output is also investigated in this paper by means of feedback equivalence some of the results are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear

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

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

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

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

state observers and state feedback controllers for a class of non linear systems a design methodology for observers and controllers for a class of single input single output non linear systems is developed a non linear observer form is defined and a corresponding observer using non linear observer gains is specified the resulting error dynamics which are non linear are asymptotically stable for a proper choice of those gains and bounded inputs and outputs a non linear controller form is defined and a corresponding controller which results in linear closed loop dynamics with arbitrary eigenvalue placement is devised transformations from the class of interest to the defined forms are derived in the observer case the transformation to the observer form and the non linear entries in the state matrix of that form can be determined separately leading to a system of linear partial differential equations a major simplification in the controller case a simple well known system of linear partial differential equations is obtained the efficacy of the developed methodology is sho

the differential geometric structure of general mechanical systems from the lagrangian point of view several differential geometric points of view on analytical mechanics of systems with a finite number of degrees of freedom are developed in generality emphasizing cartan s calculus of differential forms and ehresmann s theory of jet spaces the classical theory of lagrange s equations with external forces and constraints holonomic or nonholonomic is put into an invariant and coordinate free form the relation between this lagrangian and the hamiltonian symplectic approach which is that most extensively used in the contemporary mathematical physics literature is also developed

adaptive observers for single output nonlinear systems an adaptive version of the nonlinear observer obtained by a j krener et al 1983 is presented this version involves the cancellation of nonlinear terms by output injection as an intermediate step necessary and sufficient conditions are given for transforming a nonlinear system by state space change of coordinates into the special adaptive observer form that was used by y bastin et al 1988 to design adaptive observers

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

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

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

multivariate approximation by a combination of modified taylor polynomials we study an approximation of a multivariate function f by an operator of the form i 1n r f xi x i x where 1 n are certain basis functions and r f xi x are modified taylor polynomials of degree r expanded at xi the modification is such that the operator has highest degree of algebraic precision in the univariate case this operator was investigated by xuli multinode higher order expansions of a function j approx theory 124 2003 242 253 special attention is given to the case where the basis functions are a partition of unity of linear precision for this setting we establish two types of sharp error estimates in the two dimensional case we show that this operator gives access to certain classical interpolation operators of the finite element method in the case where 1 n are multvariate bernstein polynomials we establish an asymptotic representation for the error as n

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

an extended direct scheme for robust adaptive nonlinear control abstract the proposed adaptive scheme achieves regulation for a class of nonlinear systems with unknown constant parameters and unmodeled dynamics the scheme does not employ overparametrization and does not restrict the class of nonlinearities by any growth conditions instead the dependence on the unknown parameters is restricted by an extended matching condition which however is satisfied in many systems of practical importance such as most types of electric motors

on the approximation of strongly convex functions by an upper or lower operator the aim of this paper is to find a convenient and practical method to approximate a given real valued function of multiple variables by linear operators which approximate all strongly convex functions from above or from below our main contribution is to use this additional knowledge to derive sharp error estimates for continuously differentiable functions with lipschitz continuous gradients more precisely we show that the error estimates based on such operators are always controlled by the lipschitz constants of the gradients the convexity parameter of the strong convexity and the error associated with using the quadratic function see theorems 3 1 and 3 3 moreover assuming the function we want to approximate is also strongly convex we establish sharp upper as well as lower refined bounds for the error estimates see corollaries 3 2 and 3 4 as an application we define and study a class of linear operators on an arbitrary polytope which approximate strongly convex functions from above finally we present a numerical example illustrating the proposed method

approximate and local linearizability of nonlinear discrete time systems we consider a single input non linear discrete time system of the form where x n u and f x u n 1 n is a c n valued function necessary and sufficient conditions for approximate linearizability are given for we also give necessary and sufficient conditions for local linearizability finally we present analogous results for multi input non linear discrete time systems

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

approximations of differentiable convex functions on arbitrary convex polytopes abstract let x n x i i 0 n be a given set of n 1 pairwise distinct points in r d called nodes or sample points let p conv x n let f be a convex function with lipschitz continuous gradient on p and i i 0 n be a set of barycentric coordinates with respect to the point set x n we analyze the error estimate between f and its barycentric approximation b n f x i 0 n i x f x i x p and present the best possible pointwise error estimates of f additionally we describe the optimal barycentric coordinates that provide the best operator b n for approximating f by b n f we show that the set of linear finite element barycentric coordinates generated by the delaunay triangulation gives access to efficient algorithms for computing optimal approximations finally numerical examples are used to show the success of the method

feedback control of linear systems by multirate pulse width modulation in this note sampled data methods for modeling and control design of pulse width modulated systems are examined both single rate and multirate approaches are considered local controllability and observability analyses of the sampled data models are performed and feedback linearizability conditions for the sampled data models are derived multirate nonlinear state and output feedback designs are presented that permit asymptotic regulation or tracking

feedback linearization of sampled data systems it is shown that feedback linearizability of a continuous time system can be destroyed through the introduction of the usual sample and hold devices a method whereby a feedback linearizable discrete time model can be recovered is given

asymptotic linearization on uncertain systems by variable structure control abstract a nonlinear control system with uncertain dynamics described by a single ordinary differential equation with scalar control and available state is given based on knowledge of upper and lower estimates of the unknown dynamics we show that the system can be transformed to an asymptotically equivalent linear model arbitrarily fixed described by a single linear ordinary differential equation of the same order by using an explicitly described discontinuous feedback control law of variable structure type the linear behaviour is obtained up to an arbitrarily fast exponentially decaying error term

canonical forms of decouplable and controllable linear systems in this paper we present the canonical forms of decouplable and controllable linear systems some of the non zero elements in gilbert s canonically decoupled structure which remain undetermined are explicitly determined by a complete set of invariances our result constitutes the key step to characterization of the whole class of nonlinear systems that are feedback equivalent to decouplable and controllable linear systems

new cubature formulas and hermite hadamard type inequalities using integrals over some hyperplanes in the d dimensional hyper rectangle abstract this paper focuses on the problem of approximating a definite integral of a given function f when rather than its values at some points a number of integrals of f over certain hyperplane sections of a d dimensional hyper rectangle c d are only available we develop several families of integration formulas all of which are a weighted sum of integrals over some hyperplane sections of c d and which contain in a special case of our result multivariate analogs of the midpoint rule the trapezoidal rule and simpson s rule basic properties of these families are derived in particular we show that they satisfy a multivariate version of hermite hadamard inequality this latter does not require the classical convexity assumption but it has weakened by a different kind of generalized convexity as an immediate consequence of this inequality we derive sharp and explicit error estimates for twice continuously differentiable functions more precisely we present explicit expressions of the best constants which appear in the error estimates for the new multivariate versions of trapezoidal midpoint and hammer s quadrature formulas it is shown that as in the univariate case the constant of the error in the trapezoidal cubature formula is twice as large as that for the midpoint cubature formula and the constant in the latter is also twice as large as for the new multivariate version of hammer s quadrature formula numerical examples are given comparing these cubature formulas among themselves and with uniform and non uniform centroidal voronoi cubatures of the standard form which use the values of the integrand at certain points

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