a formula for iterated derivatives along trajectories of nonlinear systems abstract the paper presents a formula for the n th order total derivative of the output function of a nonlinear systems in terms of derivatives of the inputs the formula is given in terms of noncommuting variables and the shuffle product and has potential applications to input output decoupling and inverses of nonlinear systems

learning overcomplete representations in an overcomplete basis the number of basis vectors is greater than the dimensionality of the input and the representation of an input is not a unique combination of basis vectors overcomplete representations have been advocated because they have greater robustness in the presence of noise can be sparser and can have greater flexibility in matching structure in the data overcomplete codes have also been proposed as a model of some of the response properties of neurons in primary visual cortex previous work has focused on finding the best representation of a signal using a fixed overcomplete basis or dictionary we present an algorithm for learning an overcomplete basis by viewing it as probabilistic model of the observed data we show that overcomplete bases can yield a better approximation of the underlying statistical distribution of the data and can thus lead to greater coding efficiency this can be viewed as a generalization of the technique of independent component analysis and provides a method for bayesian reconstruction of signals in the presence of noise and for blind source separation when there are more sources than mixtures

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

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

surface integrals in lower dimensions from higher order chern classes and a class of solutions in three dimensions it is shown that surface integrals involving gauge and higgs fields in both odd and even dimensions can be obtained by dimensional reduction from chern classes in higher dimensions for odd dimensions the physically useful dimensional reduction is characterized by mn en p sp and for even dimensions by mn en p s2 s2 1 2 p times the existence of a family of field configurations for which these surface integrals in three dimensions exist and are nonzero is presented this family includes a yang mills higgs ymh system with an algebra valued higgs field and one with the higgs field in the reducible representation adjoint scalar

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

robust adaptive control for a class of mimo nonlinear systems by state and output feedback in this technical note high dimensional integral lyapunov functions are introduced for a class of mimo nonlinear systems with unknown nonlinearities first adaptive state feedback control is presented based on the integral lyapunov function when only the output is measurable by using a high gain observer to estimate the derivative of the system output adaptive output feedback control is also derived the proposed control scheme provides a general approach to stabilize the mimo plant without any restrictive assumptions the control is continuous and ensures closed loop stability and convergence of the tracking error to a small residual set the size of the tracking error at steady state can be specified a priori and guaranteed by choosing the design parameters

essential orders and the non linear decoupling problem abstract the concept of essential orders has recently been introduced into linear system theory and has been shown to be useful in the solution of static and dynamic decoupling problems commauit et al 1986 de luca et al 1985 this paper presents a non trivial extension to the class of non linear affine systems of some of these definitions and results it is shown that these invariant integers give the minimal order for a regular dynamic compensator that decouples a right invertible system

multiple instantons representing higher order chern pontryagin classes ii this paper is a continuation of an earlier study on the generalized yang mills instantons over 4m dimensional spheres we will first present a discussion on the generalized yang mills equations the higher order chern pontryagin classes c 2m and the self dual or anti self dual equations we will then obtain some sharp asymptotic estimates for the self dual or anti self dual equations within the witten tchrakian framework which relates the integer value of c 2m to the number of vortices of the solution to a reduced 2 dimensional abelian higgs system over the poincare half plane we will prove that indeed for any integer n there exists a 2 n parameter family of the generalized self dual or anti self dual instantons realizing the topology c 2m n furthermore for the purpose of accommodating more general solutions we establish a removable singularity theorem which enables us to extend the solutions obtained on a 4m dimensional euclidean space with an integral bound to the holder continuous solutions on a 4m dimensional sphere

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

technical communique input output pseudolinearization on controlled invariant submanifolds the original notion of input output pseudolinearization on equilibrium submanifolds is extended to general controlled invariant submanifolds for nonlinear systems input output pseudolinearizing feedback laws render the input output behavior of linearizations about nominal trajectories in the submanifold trajectory independent and equal to that of a fixed linear time invariant system thereby facilitating the subsequent construction of control laws for stabilization and tracking sufficient existence conditions are derived that are weaker than the familiar relative degree conditions for nonlinear input output decoupling that have come to be associated with exact input output linearization

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

a finiteness criterion for nonlinear input output differential systems we solve the following problem which has been dealt with in several publications when does a system have the same input output behaviour as a system depending linearly on the state variables precise definitions of this question are supplied in the text a particular case of the problem is the equivalence of a system with a bilinear one at the end we illustrate our methods and the scope of our results by a brief study of several examples

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

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

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

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

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

noninteracting control with stability for a class of nonlinear systems abstract in this paper we address the problem of noninteracting control with stability for the class of nonlinear square systems for which noninteraction can be achieved without stability by means of invertible static state feedback the use of both static state feedback and dynamic state feedback is investigated we prove that in both cases the asymptotic stabilizability of certain subsystems is necessary to achieve noninteraction and stability we use this and some recent results to state a complete set of necessary and sufficient conditions in order to solve the problem

on the structure at infinity of linear square decoupled systems the purpose of this note is to demonstrate that a square invertible linear system c a b is decouplable by static state feedback if and only if its infinite zero orders are respectively equal to the infinite zero orders of the subsystems c i a b where c i denotes the i th row of c until now only the necessity of this condition has been established 1

hybrid adaptive control laws solving a path following problem for non holonomic mobile manipulators in this paper a general solution to the path following problem for mobile manipulators with non holonomic mobile platform has been presented new proposed control algorithms for mobile manipulators with fully known dynamics or with parametric uncertainty in the dynamics take into considerations the kinematics as well as the dynamics of the non holonomic mobile manipulator the convergence of the control algorithms is proved using the lasalle s invariance principle

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

effective decorrelation and space dimensionality reduction of multiscaling volatility abstract we consider an approach for modeling non stationary and non gaussian curves which has a natural impact on financial time series analysis due to the characteristic features of volatility processes provided that one can approximate the signal of interest in this case stock index returns with a greedy approximation scheme based on wavelet like functions an effective space dimensionality reduction of the problem can be found by a decomposition technique which selects the scales according to an energy based optimization scheme and finds the most informative sources of the underlying multiscaling volatility process

input linearization of nonlinear systems via pulse width control in this note it is shown that a general nonlinear system can be transformed to an affine nonlinear system by virtue of the use of pulsewidth control therefore the combined system from the pulse width control input to the nonlinear system output behaves as an affine linear in control system by this way a cumbersome nonlinear system model can be transformed to a simpler linear in control form without increasing the system dimension this in turn enables simpler control design furthermore using this methodology some control design methods developed only for affine systems can be adapted to general nonlinear systems

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

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

tracking control designs for both holonomic and non holonomic constrained mechanical systems a unified viewpoint a tracking control design is proposed for both holonomic and non holonomic constrained mechanical systems from a unified viewpoint it will be shown that according to the number of control inputs if the state variables are available and the output variables are chosen properly then the exponentially global tracking of positions and constraint forces on the constraint manifold is achieved for holonomic control systems by a static state feedback controller from the same point of view for non holonomic control systems the globally exponential tracking of positions is achieved by just a dynamic state feedback controller finally by means of much discussion and many simulations two typical non holonomic examples edge and a vertical wheel are given to illustrate the essential features of the proposed design method

local nonlinear model matching from linearity to nonlinearity the model matching problem consists of designing a compensator for a given system called the plant in such a way that the resulting input output behaviour matches that of a prespecified model in this paper a local solution of the nonlinear model matching problem is given for the case that the model is decouplable by static state feedback the main theorem states that under generic conditions on the plant the problem is solvable around an equilibrium point if and only if it is solvable for the linearization of plant and model the generic conditions are identified they naturally appear in the solution of the dynamic input output decoupling problem for the plant the theory is illustrated by means of two examples

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

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

robust output tracking for nonlinear systems with weakly non minimum phase this paper is concerned with the problem of designing a robust output tracking controller for mimo nonlinear systems with weakly non minimum phase based on our system formulation control plants with uncertainties and or with actuator dynamics fall into the class under consideration the controller design here is divided into two phases fast feedback control and slow feedback control so that a final composite control is obtained the former is chosen to stabilize the boundary layer system whereas the latter essentially handles the mismatched uncertainties after the system is reformulated under some mild assumptions it is shown that the overall states are bounded and the tracking errors converge to a residual set whose size is a class k function of e as e 0 the residual set shrinks to the origin an interesting application to a simplified aircraft model with fast actuator dynamics which turns out to be weakly non minimum phase is given the computer simulation has verified the expected satisfac

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

a sufficient condition for nonlinear noninteracting control with stability via dynamic state feedback a sufficient condition is given to solve the problem of achieving local noninteracting control with asymptotic stability via dynamic state feedback for a nonlinear affine system the condition which is automatically satisfied in the case of linear systems although only sufficient is helpful in overcoming some of the geometric obstructions pointed out in the literature an example in which the author assumptions are satisfied is given clearly showing how to construct a decoupling and stabilizing dynamic state feedback and how the condition can weaken the requirement that the p dynamics be locally asymptotically stable

on the synthesis of linear input output responses for nonlinear systems necessary and sufficient conditions are given for the existence of a feedback control law under which the input dependent part of the response of a given nonlinear system becomes linear in the input and independent of the initial state

linearization of discrete time systems by exogenous dynamic feedback it is shown that a discrete time system may be linearizable by exogenous dynamic feedback even if it cannot be linearized by endogenous feedback this property is completely unexpected and constitutes a fundamental difference with respect to the continuous time case the notion of exogenous linearizing output is introduced it is shown that existence of an exogenous linearizing output is a sufficient condition for dynamic linearizability necessary and sufficient conditions for the existence of an exogenous linearizing output are provided the results of the paper are obtained using transformal operator matrices the properties of such operators are studied the theory is applied to the exact discrete time model of a mobile robot showing that the above mentioned property concerns not only academic examples but also physical systems

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

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

a general approach to input output pseudolinearization for nonlinear systems a nonlinear system is said to be input output pseudolinearized if its family of linearizations about constant operating points has input output behavior that is independent of the particular operating point the problem of constructing static state feedback laws that achieve input output pseudolinearization for a general class of nonlinear systems is considered a generalization of the well known structure algorithm to the case of parameterized families of linear systems plays an important role

approximate decoupling and asymptotic tracking for mimo systems this paper presents an algorithm for approximate input output decoupling of nonlinear mimo systems that are either numerically ill posed or exhibit nearly singular behavior in the application of decoupling algorithms although the systems considered are regular so that the exact decoupling algorithms are applicable in this case they require inversion of an ill conditioned matrix and yield high gain feedback solutions that may result in actuator saturation and cancellation of high frequency zeros the approximate algorithms of this paper are numerically robust and provide solutions that do not cancel far off right half plane zeros this latter characteristic is especially valuable when some of the far off right half plane zeros are unstable the algorithms are inspired by and are generalizations of some examples in the flight control literature

a note on the invertibility of nonlinear input output differential systems abstract a new conceptual framework employing differential algebra is proposed for solving the long standing problem of inverting nonlinear input output differential systems simple characterizations of left and right invertibility are given which rest upon an invariant integer which for constant linear systems turns out to be the rank of the transfer matrix

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

a triangular canonical form for a class of 0 flat nonlinear systems this article proposes a triangular canonical form for a class of 0 flat nonlinear systems necessary and sufficient geometrical conditions are given in order to guarantee the existence of a local diffeomorphism to transform the studied nonlinear systems into the proposed 0 flat canonical form which enables us to compute the flat output as well

global approximate output tracking for nonlinear systems addresses the global output tracking problem for nonlinear systems with singular points for nonlinear systems which satisfy a suitable observability condition the authors identify a class of smooth output trajectories which the system can track using continuous open loop controls this class includes all output trajectories generated by smooth state feedback they then study the problem of approximate output tracking using discontinuous time varying feedback controllers given a smooth output trajectory for which exact tracking is possible the authors construct a discontinuous feedback controller which achieves robust tracking of the desired output trajectory in the face of perturbations finally it is shown that their results can be applied to the control of a chain system and some numerical results are presented to illustrate the performance of their controller

fixed modes and nonlinear noninteracting control with stability the existence of a fixed internal dynamics inherent to the noninteraction requirement is identified when unstable this represents an obstruction to the achievement of noninteracting control with stability by means of static state feedback remarkably whenever one leaves the class of linear systems this obstruction cannot in general be removed by dynamic state feedback

axially symmetric instantons in generalized yang mills theory in 4p dimensions axially symmetric gauge field configurations of a certain type are studied in generalized yang mills theory in 4p dimensions p 2 3 for all p finite action solutions of the generalized self duality equations with topological charge n 1 2 are found it is also shown that the self duality equations are overdetermined when other types of axial symmetry are imposed

data driven model free adaptive control for a class of mimo nonlinear discrete time systems in this paper a data driven model free adaptive control mfac approach is proposed based on a new dynamic linearization technique dlt with a novel concept called pseudo partial derivative for a class of general multiple input and multiple output nonlinear discrete time systems the dlt includes compact form dynamic linearization partial form dynamic linearization and full form dynamic linearization the main feature of the approach is that the controller design depends only on the measured input output data of the controlled plant analysis and extensive simulations have shown that mfac guarantees the bounded input bounded output stability and the tracking error convergence

a sufficient condition for non linear non interacting control with stability via dynamic state feedback block partitioned outputs a geometric sufficient condition is given for the first time to solve the problem of non linear block non interacting control with stability by means of dynamic state feedback this result extends a previous sufficient condition given by the same author battilotti 1991

q analogs of iu n and u n 1 the starting point is the nonsemisimple inhomogeneous lie algebra un i2n denoted also as iu n where i2n represents an abelian subalgebra in semidirect product with the homogeneous part u n this is realized by explicitly giving the matrix elements of the generators on a modified gelfand zetlin basis that allows representations of infinite dimensions the enveloping algebra is q lifted by introducing q brackets in the matrix elements giving uq iu n the deformation of the abelian structure of i2n is studied for q 1 some implications are pointed out the important invariants are constructed for arbitrary n the results are compared to the corresponding ones for jimbo s construction of uq u n 1 on a gelfeld zetlin basis finally the related construction of uq u n 1 is presented and discussed here uq su 1 1 the q analog of relativistic motion in a plane is analyzed in the context of this formalism

nonlinear morgan s problem case of p 1 inputs and p outputs noninvertible static state feedback solutions of the row by row decoupling problem are investigated for the class of nonlinear systems having one more input component than output components the analysis involves a combination of geometric and algebraic tools but notably avoids the use of regular or degenerate controllability distribution subspaces the main tools used to date for the class of linear systems the main idea is finding within the system a chain of integrators that may be attached to one of the inputs in order to appropriately modify the system s structure at infinity

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