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

locally f g invariant distributions conditions are derived for the existence of f g invariant distributions in nonlinear systems and a procedure for the construction of the corresponding feedback control law is given

differential geometric methods in variable structure control this article presents a differential geometric approach for the design of sliding modes in non linear variable structure feedback systems coordinate free characterizations of local existence conditions for sliding regimes and a geometric reformulation of some of its most salient features are presented the approach uses basic notions from differential geometry involving vector fields distributions and 1 forms both single and multiple input cases are treated with some illustrative examples

implicit discrete time systems and accessibility an intrinsic description for dynamic systems whose evolution along discrete time is governed by nonlinear implicit difference equations in one independent variable and zero order algebraic equations is presented by means of differential geometrical methods where systems are associated with appropriate geometric objects reflecting their dynamics dynamic systems given in implicit form have the peculiarity that they may contain so called hidden restrictions a normal form is presented which is characterized by the circumstances that there are no further restrictions in addition it is illustrated that such a normal form allows for an equivalent system representation in explicit form based on the geometric picture of implicit discrete time systems the qualitative property of accessibility along a fixed trajectory is discussed by applying symmetry groups of discrete time systems and studying invariants of these groups a formal approach is provided that allows us to gather local accessibility criteria successively which can be tested by computer algebra several examples illustrate the results

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

controllability distributions for nonlinear control systems the purpose of this paper is to relate the notion of controlled invariance to a notion of controllability for nonlinear control systems by means of the linear ideas of a subsystem and a quotient system we derive a very structured description of nonlinear systems

single experiment observability decomposition of discrete time analytic systems this paper addresses the single experiment observability decomposition of discrete time analytic systems unlike the continuous time case there exist systems which cannot be decomposed into observable and unobservable subsystems due to the fact that the observable space is not integrable in this paper a necessary and sufficient condition for integrability of observable space is given as a corollary of this condition it is proven that if the system is reversible the observability decomposition can be always achieved moreover integrability of observable space is also addressed for delta domain models of non uniformly sampled systems

measurement feedback disturbance decoupling in discrete time nonlinear systems the paper studies the disturbance decoupling problem by the dynamic measurement feedback for discrete time nonlinear control systems to address the problem the algebraic approach called the algebra of functions is applied which allows the system description also depend on non differentiable functions a necessary and sufficient condition is given in terms of controlled and h f invariant functions also algorithms are derived which find invariant functions and the required feedback the algorithms are implemented in mathematica software which is made available over the internet

elastic plastic unilateral contact problems for slackened systems abstract the paper deals with the mathematical modelling of a class of unilateral elastic plastic contact problems the problem is posed in the framework of the so called slackened systems the considerations are confined to small strains and piecewise linear approximations of the yield surface and the clearance surface an incremental formulation is proposed in the form of a linear complementarity problem lcp the formulation allows one to analyse a structural system which can be subjected to nonproportional loading paths the included numerical results illustrate the application of this formulation to an elastic plastic beam that comes into contact with a unilateral elastic plastic foundation

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

convergence analysis of the extended kalman filter used as an observer for nonlinear deterministic discrete time systems in this paper convergence analysis of the extended kalman filter ekf when used as an observer for nonlinear deterministic discrete time systems is presented based on a new formulation of the first order linearization technique sufficient conditions to ensure local asymptotic convergence are established furthermore it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified ekf significantly the efficiency of this approach compared to the classical version of the ekf is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete time systems

invariant codistributions and the feedforward form for discrete time nonlinear systems the goal of this paper is two fold first given an arbitrary n dimensional discrete time nonlinear dynamical system necessary and sufficient conditions for the existence of a one dimensional invariant codistribution are obtained second it is shown that the previous conditions can be used iteratively to obtain a nested sequence of n invariant codistributions with the properties that each codistribution contains the previous one and the last one coincides with the cotangent bundle of the state manifold as a byproduct necessary and sufficient conditions are obtained for a discrete time nonlinear dynamical system to be equivalent to the so called feedforward form

non linear non interacting control with stability in discrete time a dynamic solution the geometric approach proposed in califano et al nonlinear noninteracting control with stability in discrete time a geometric approach int j control 75 1 pp 11 22 2002 for non linear non interacting control with stability via static state feedback in discrete time is now used to find a dynamic solution for systems characterized by an invertible drift geometric necessary and sufficient conditions are given a constructive procedure for the computation of the dynamic compensator solving the problem is proposed

computer analysis of slackened systems under variable loads the paper deals with a particular class of slackened elastic plastic skeletal structures where one parameter contact and yield conditions are introduced the mathematical model of this problem represents a discrete form of kuhn tucker s conditions and is equivalent to dual quadratic programming problems qpps the uniqueness of the solution to this problem is guaranteed with exception of some particular cases where the structure converts into a mechanism and the solution corresponds to dual linear programming problems lpps the authors propose another approach to solving the problem the uniqueness of the solutions allows a pure elastic analysis procedure which satisfies some additional constraints imposed on the objective function to be used the numerical solution is obtained by means of an iterative procedure using the trial and error method where the physical energetic meaning of the problem is utilized

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

observability of autonomous discrete time non linear systems a geometric approach in this paper the observability of autonomous discrete time systems is studied from a purely differential geometric point of view as with continuous time systems this approach leads to a local canonical form for an observable system a proposal for the generalization of an invariant subspace is made

nonlinear offset free model predictive control this paper addresses offset free reference tracking of asymptotically constant reference signals using model predictive control existing results for linear models are extended to general nonlinear models the core of the proposed method employs a disturbance model and an observer to estimate its state typical disturbance models are shown and the implications of using them are discussed conditions are given for which this setup eliminates the tracking error asymptotically basically we prove that error free output estimation and error free nominal tracking imply offset free model predictive control

non linear non interacting control with stability in discrete time a geometric framework the aim of the paper is to set the non linear non interacting control problem with stability in discrete time in a correct geometric framework so providing a solution which recalls the continuous approach and to point out the peculiarities of discrete time dynamics geometric necessary and sufficient conditions for the solvability of the problem are given and a constructive method figures out the regular static state feedback solution it is shown that if the given dynamics is feedback equivalent to a decoupled system characterized by an invertible drift the computation of the distributions involved can be carried out in a simple and elegant way the main results are thus proved exhaustively in this case and then generalized to submersive systems

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

controlled invariance for nonlinear systems two worked examples in this note we present two worked examples of disturbance decoupling for nonlinear systems using the concept of controlled invariance which was recently generalized to nonlinear systems in the first example we explicitly construct a feedback which decouples a disturbance from the vertical components of the axes of a rotating rigid body while the second example deals with a particle in a potential field subject to a disturbance

a class of nonlinear filtering problems arising from drifting sensor gains this paper considers a state estimation problem where the nominal system is linear but the sensor has a time varying gain component giving rise to a bilinear output equation this is a general sensor self calibration problem and is of particular interest in the problem of estimating wafer thickness and etch rate during semiconductor manufacturing using reflectometry we explore the use of a least squares estimate for this nonlinear estimation problem and give several approximate recursive algorithms for practical realization stability results for these algorithms are also given simulation results for comparing the new algorithms with the extended kalman filter and iterated kalman filter are given

simple sufficient conditions for reachability of discrete time polynomial systems we consider reachability of discrete time polynomial systems the reachability of such systems can be interpreted as the surjectivity of the polynomial mappings defined by the systems for a polynomial mapping between suitable spaces of the same dimensions it has been shown that the injectivity of the mapping also implies the surjectivity by utilizing this fact we derive sufficient conditions for the reachability of the polynomial systems which essentially check the injectivity of the polynomial mappings

zero state observability and stability of discrete time nonlinear systems abstract this paper studies discrete time invariant autonomous systems x k 1 tf x k y k h x k we investigate the problem of when asymptotic stability of can be characterized by means of zero state observability of and square summability of the output function h x k the relationship among observability square summable series and stability of discrete time autonomous systems is exposed the main result developed in this paper unifies a series of stability theorems recently proposed in the literature and generalizes morse s lemma morse a s 1990 towards a unified theory of parameter adaptive control tunability ieee trans autom contr 35 1002 1012 in two directions from continuous time linear systems to their discrete counterparts and from linear systems to nonlinear systems

on the uniform observability of discrete time nonlinear systems in constructing an observer for a discrete time nonlinear system the system is commonly required to satisfy a certain kind of uniform observability condition that is the state should always be reconstructible from observation windows of a specific length irrespective of the values of the state and inputs in this technical note it is proved that this uniform requirement is unnecessary in the sense that if the initial state and inputs are on a compact set then the uniform observability is derived from its non uniform counterpart

conditioned invariant and locally conditioned invariant distributions abstract we define the concepts of conditioned invariant and locally conditioned invariant distributions for nonlinear system and show how they can be used to solve the problem of tracking an output signal of a nonlinear system in spite of unknown disturbances

controlled invariance for nonlinear systems necessary and sufficient conditions are derived for a b invariance called here controlled invariance for nonlinear systems dot x f x u the obtained results generalize and elucidate already known results about systems dot x a x sum min i 1 max m u i b i x a new and direct differential geometric interpretation of the concept of controlled invariance and the derived conditions is given

controllability of nonlinear discrete time systems a lie algebraic approach this paper presents a geometric study of controllability for discrete time nonlinear systems various accessibility properties are characterized in terms of lie algebras of vector fields some of the results obtained are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear

controlled invariance for affine control systems the paper gives a solution of the disturbance decoupling problem for non linear systems the main difference between this and related works in the field is the construction of a now state dependent basis for the inputs

brief paper on the observability of discrete time dynamic systems a geometric approach a coordinate free description for dynamic systems described by explicit nonlinear difference equations in one independent variable via a differential geometric framework is presented based on this covariant approach suitable geometric objects for discrete time dynamics are introduced especially the observability along a trajectory is discussed and transformations to normal forms are derived in addition the obtained local observability criteria can be checked by computer algebra algorithms some examples illustrate the proposed approach

reversible linear and nonlinear discrete time dynamics it is shown that any causal discrete time system can be realized with a reversible or invertible state variable representation this bridges the gap with continuous time where flows always satisfy this property and settles an old debate on the relevance of invertibility in nonlinear discrete dynamics as the result applies to the linear case it is shown how the classic approach should be slightly modified and several examples of the benefits that can be obtained are given difference algebra is employed for nonlinear systems it leads to a most elegant characterization of causality which enables reversibility to be demonstrated module theory suffices in the linear case which is supplemented by elementary matrix manipulations

generalized controlled invariance for discrete time nonlinear systems with an application to the dynamic disturbance decoupling problem in analogy with the continuous time case a general notion of controlled invariance with respect to quasi static state feedback is introduced for discrete time nonlinear systems which incorporates the earlier definition of controlled invariance with respect to regular static state feedback this new notion is used to derive a geometric solution to the dynamic disturbance decoupling problem the proposed solution is a natural generalization of the geometric solution to the static disturbance decoupling problem

observability at an initial state for polynomial systems we consider observability at an initial state for polynomial systems when testing for local observability for nonlinear systems the observability rank condition based on the inverse function theorem is commonly used however the rank condition is a sufficient condition and we cannot test for global observability using the rank condition in this paper first we derive necessary and sufficient conditions for global observability at an initial state for continuous time polynomial systems then necessary and sufficient conditions for local observability are derived from one of the global observability conditions using the localization of a polynomial ring using the same procedure we derive both global and local observability conditions for discrete time polynomial systems each condition is characterized by a finite set of equations since polynomial rings are noetherian finally examples demonstrate the proposed criteria for testing for observability

stochastic stability of the discrete time extended kalman filter the authors analyze the error behavior for the discrete time extended kalman filter for general nonlinear systems in a stochastic framework in particular it is shown that the estimation error remains bounded if the system satisfies the nonlinear observability rank condition and the initial estimation error as well as the disturbing noise terms are small enough this result is verified by numerical simulations for an example system

a lie theoretic approach to nonlinear discrete time controllability via ritt s formal differential groups the controllability of nonlinear discrete time systems is related to ritt s formal differential groups

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

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

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

further results on the uniform observability of discrete time nonlinear systems in this technical note the relation between notions of uniform observability of discrete time nonlinear systems based on injectivity of an observation map window the full rankness of its jacobian and a k function is investigated it is proven that a system is uniformly observable in the sense of injectivity of the observation map together with the full rankness of its jacobian if and only if it is so in some k function senses

invertibility of affine nonlinear control systems a geometric approach abstract the paper deals with the invertibility of multivariable non linear control systems by using the recently developed theory on controlled invariant and controllability distributions necessary and sufficient conditions for invertibility are derived

remarks on robustness analysis of nonlinear feedback systems an input output approach in our previous paper 1999 sec vii contains two examples of adaptive controllers examples 8 and 9 for which the nonlinear gap robustness margin is zero and which are destabilized by arbitrarily small perturbations in the gap metric the proof that the robustness margin is zero makes use of the fact that the parameter estimate spl theta t tends to infinity as t spl rarr spl infin when there is a constant input disturbance of size spl epsi 0 the purpose of this present note is a to complete the argument given in the above paper to prove the unboundedness of spl theta t and b to give an alternative proof that the convergence of the state x t spl rarr 0 is a consequence of the unboundedness of spl theta t in example 9

moving horizon state estimation for nonlinear discrete time systems new stability results and approximation schemes a moving horizon state estimation problem is addressed for a class of nonlinear discrete time systems with bounded noises acting on the system and measurement equations as the statistics of such disturbances and of the initial state are assumed to be unknown we use a generalized least squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding window strategy for the resulting estimator the existence of bounding sequences on the estimation error is proved in the absence of noises exponential convergence to zero is obtained moreover suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value the approximate solution can be determined either on line by directly minimizing the cost function or off line by using a nonlinear parameterized function simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended kalman filter

internally exponentially stable non linear discrete time non interacting control via static feedback the problem of discrete time block non interacting control with internal exponential stability is treated by identifying the linearized fixed modes common to every non interactive closed loop a necessary condition for achieving an exponentially stable solution to the problem is provided also through a special decomposition of the given system a sufficient condition is obtained owing to peculiarities intrinsic to discrete time non linear systems the development of the solution is not completely parallel to that of the solution for the class of continuous time systems

existence of an observation window of finite width for continuous time autonomous nonlinear systems in this note the relationship between notions of observability for continuous time nonlinear system related to distinguishability observability rank condition and k function has been investigated it is proved that an autonomous nonlinear system that is observable in both distinguishability and rank condition sense permits an observation window of finite width and it is possible to construct a k function related to observability for such system