high gain feedback in non linear control systems abstract high gain state and output feedback are investigated for non linear control systems with a single additive input by using singular perturbation techniques classical approximation results tihonov like theorems in singular perturbation theory are extended to non linear control systems by defining a composite additive control strategy a control dependent fast equilibrium manifold and non linear change of coordinates those tools and an appropriate change of coordinates show that high gain state feedback and variable structure control systems can be equivalently used for approximate non linearity compensation in feedback linearizable systems next the effect of high gain output feedback is shown to be related to the strong invertibility property and the relative order of invertibility for strongly invertible systems the slow reduced subsystem coincides with the dynamics of the inverse system when zero input is applied and with the unobservable dynamics when a certain input output feedback linear

discretization behaviors of equivalent control based sliding mode control systems in this note discretization behaviors of the equivalent control based sliding mode control smc systems are studied some inherent dynamical properties of the discretized second order systems are first explored upper bounds for the system steady states are established the system s steady state behaviors are discussed the analysis for the second order systems is then extended to higher order systems simulations are presented to verify the theoretical results

robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems we present a patch based equilibrated flux recovery procedure for the conforming finite element approximation to diffusion problems the recovered flux is computed as the solution to a local constraint free minimization problem on each patch the approach is valid for higher order conforming elements in both two and three dimensions the resulting estimator admits guaranteed reliability and the robust local efficiency is proved under the quasi monotonicity condition of the diffusion coefficient numerical experiments are given to confirm the theoretical 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

adaptive finite element methods for elliptic equations with non smooth coefficients nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3 et sa discretisation par elements finis le but de cet article est de demontrer des estimations d erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients

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

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

a note on constant free a posteriori error estimates in this note we look at constant free a posteriori error estimates from a different perspective we show that they can be interpreted as an alternative way of expressing the residual of a finite element approximation and thus fit into the same framework as other a posteriori error estimates such as residual error indicators our approach also reveals that when applied to singularly perturbed reaction diffusion or convection diffusion problems constant free a posteriori error estimates will not be fully robust unless extra measures are taken

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

flux recovery and a posteriori error estimators conforming elements for scalar elliptic equations in this paper we first study two flux recovery procedures for the conforming finite element approximation to general second order elliptic partial differential equations one is accurate in a weighted l 2 norm studied in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 for linear elements and the other is accurate in a weighted h mathrm div norm up to the accuracy of the current finite element approximation for the l 2 recovered flux we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual based estimator based on the h mathrm div recovered flux we introduce two a posteriori error estimators one estimator may be regarded as an extension of the recovery based estimator studied in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 to higher order conforming elements the global reliability and the local efficiency bounds for this estimator are established provided that the underlying problem is neither convection nor reaction dominant the other is proved to be exact locally and globally on any given mesh with no regularity assumptions with respect to a norm depending on the underlying problem numerical results on test problems for these estimators are also presented

superconvergent derivative recovery for lagrange triangular elements of degree p on unstructured grids in this paper we develop a postprocessing derivative recovery scheme for the finite element solution u h on general unstructured but shape regular triangulations in the case of continuous piecewise polynomials of degree p geq 1 by applying the global l 2 projection q h and a smoothing operator s h the recovered p th derivatives s h m q h partial p u h superconverge to the exact derivatives partial p u based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the p 1 st derivatives approximated by partial s h m q h partial p u h we provide several numerical examples illustrating the effectiveness of our schemes we also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

a new integrable toda related three particle system a study of a three particle system that possesses in addition to the energy function two nonlinear constants of motion is presented herein the system is obtained by modification of the lagrangian of the toda lattice the explicit expression of the two integrals is obtained and it is proven that this new three particle lagrangian is integrable finally the possible generalization to the case of n particles is discussed

semi global stabilization of partially linear composite systems via feedback of the state of the linear part abstract we consider the problem of semi global stabilization of a class of partially linear composite systems we show by explicit construction of the control laws that a cascade of linear stabilizable and nonlinear asymptotically stable subsystems is semi globally stabilizable by a dynamic feedback of the state of the linear subsystem if a the linear subsystem is right invertible and has all its invariant zeros in the closed left half s plane and b the only linear variables entering the nonlinear subsystem are the output of the linear subsystem our work generalizes previous results by c i byrnes and a isidori 1991 h j sussmann and p v kokotovic 1991 and a r teel 1992

a class of integrable hamiltonian systems whose solutions are perhaps all completely periodic we show that the dynamical system characterized by the complex equations of motion qj i qj k 1 k jnqjqkf qj qk j 1 n with f x x x is hamiltonian and integrable and we conjecture that all its solutions qj t j 1 n are completely periodic with a period that is a finite integral multiple of t 2 here n is an arbitrary positive integer is an arbitrary nonvanishing real constant y y is the weierstrass function with arbitrary semiperiods and are two arbitrary constants special cases are f x 2 coth x 1 r2 sinh2 x f x 2 coth x f x 2 sinh x f x 2 x 1 2x2 and of course f x 2 x these findings as well as the conjecture which is shown to be true in some of these special cases are based on the possibility to recast these equations of motion in the modified lax form l i l l m with l and m appropriate n n matrix functions of the n dynamical variables qj and of their time derivatives qj

integrable three particle systems hidden symmetries and deformations of the calogero moser system the properties of some three particle systems are studied using as an approach the theory of generalized or hidden symmetries it is proven that the existence of a family that possesses in addition to the energy function two nonlinear constants of motion in involution this family of integrable systems is obtained by considering deformations of the lagrangian of the calogero moser system finally the generalization of the results to the general case of n particles is discussed

a necessary condition for local asymptotic stability of discrete time nonlinear systems with exogenous disturbance local asymptotic stability of nonlinear systems with real parametric uncertainty or disturbance is one of the important problems in the control systems literature in this paper we investigate the problem of asymptotic stability for discrete time nonlinear systems with time varying disturbance we assume that the disturbance vector is generated by an exosystem which is neutrally stable thus the disturbances that we consider include both constant and periodic signals for this class of nonlinear systems with time varying disturbance we derive a necessary condition for local asymptotic stability of equilibria as corollaries of our general result we deduce the necessary condition obtained by sundarapandian 1 for discrete time nonlinear systems with constant real parametric uncertainty and the necessary condition obtained by lin and byrnes 2 for discrete time nonlinear autonomous systems we illustrate our result with several examples

a hybrid a posteriori error estimator for conforming finite element approximations abstract this paper introduces a hybrid a posteriori error estimator for the conforming finite element method which may be regarded as a combination of the explicit residual and the improved zz error estimators with comparable cost the hybrid estimator is more accurate than the residual estimator it is shown that the hybrid estimator is reliable on all meshes unlike estimators of the zz type moreover the reliability constant is independent of the jump of the diffusion coefficients for elliptic interface problems under the monotonicity assumption of the coefficients finally numerical examples confirm the robustness of the estimator with respect to coefficient jumps and also better effectivity index compared to the residual estimator

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

robust equilibrated residual error estimator for diffusion problems conforming elements this paper analyzes an equilibrated residual a posteriori error estimator for the diffusion problem the estimator which is a modification of those in d braess and j schoberl math comput 77 2008 pp 651 672 r verfurth siam j numer anal 47 2009 pp 3180 3194 is based on the prager synge identity and on a local recovery of an equilibrated flux numerical results for an interface test problem show that the modification is necessary for the robustness of the estimator when the distribution of diffusion coefficients is local quasi monotone it is shown theoretically that the estimator is robust with respect to the size of jumps

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

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

brief paper zoh discretization effect on single input sliding mode control systems with matched uncertainties in this paper discretization behaviors of equivalent control based sliding mode control systems with matched uncertainties are studied upper bounds for system steady states are established some inherent dynamical periodic properties of the systems subject to matched constant and periodic uncertainties are explored simulations are presented to verify the theoretical results

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

equilibrated residual error estimates are p robust abstract equilibrated residual error estimators applied to high order finite elements are analyzed the estimators provide always a true upper bound for the energy error we prove that also the efficiency estimate is robust with respect to the polynomial degrees a complete theory is provided for tensor product elements in the case of simplicial elements the theorem was first based on a conjecture but the conjecture was quite recently proved by costabel and mcintosh m costabel a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains irmar preprint 08 40 rennes august 2008 numerical evidence for the p robustness is provided

normal form for linear systems with respect to its vector relative degree for multi input multi output mimo linear systems with existing vector relative degree a normal form is constructed this normal form is not only structural simple but allows to characterize the system s zero dynamics for the design of feedback controllers a characterization of the zero dynamics in terms of the normal form is given

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

periodic orbits from spl delta modulation of stable linear systems the spl delta modulated control of a single input discrete time linear stable system is investigated the modulation direction is given by c sup t x where c spl isin r sup n spl bsol 0 is a given otherwise arbitrary vector we obtain necessary and sufficient conditions for the existence of periodic points of a finite order some concrete results about the existence of a certain order of periodic points are also derived we also study the relationship between certain polyhedra and the periodicity of the spl delta modulated orbit

passivity and absolute stabilization of a class of discrete time nonlinear systems abstract by using the theory of discrete time passive systems and the concept of feedback equivalence we present an approach toward deriving sufficient conditions for the global stabilization of discrete time nonlinear systems in the form x k 1 tf x k g x k u k the central feature of the approach is that a unifying set of stability criteria can be obtained and the stability results recover the previous stabilization schemes proposed in the literature

an efficient method for locating and computing periodic orbits of nonlinear mappings abstract the accurate computation of periodic orbits of nonlinear mappings and the precise knowledge of their properties are very important for studying the behavior of many dynamical systems of physical interest in this paper we present an efficient numerical method for locating and computing to any desired accuracy periodic orbits stable unstable and complex of any period the method described here is based on the topological degree of the mapping and is particularly useful since the only computable information required is the algebraic signs of the components of the mapping this method always converges rapidly to a periodic orbit independently of the initial guess and is particularly useful when the mapping has many periodic orbits stable and unstable close to each other all of which are desired for the application we illustrate this method first on a two dimensional quadratic mapping used in the study of beam dynamics in particle accelerators to compute rapidly and accurately its periodic orbits of periods p 1 5 16 144 1296 10368 and then obtain periodic orbits of its four dimensional complex version for periods which also reach up to the thousands

output feedback stabilization the stabilization of control systems using output feedback is addressed necessary and sufficient lyapunov conditions are established generalizing the artstein s stabilization theorem for linear control systems a necessary and sufficient lyapunov condition is provided for stabilization by means of a linear output feedback law

dynamical behaviors of an euler discretized sliding mode control systems the dynamical behaviors of an euler discretized sliding mode control smc systems based on equivalent control strategy are studied a periodic 2 orbit in steady state is found for the switching function of the euler discretized smc systems the time steps for the switching function to converge toward the periodic 2 orbit are obtained when the discretized smc system is stable it further shows that the system states of the smc systems will also enter into some periodic 2 orbits and these periodic 2 orbits are characterized by explicit analytic expressions finally simulation examples are given to illustrate the theoretical results

recovery based error estimator for interface problems conforming linear elements this paper studies a new recovery based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems instead of recovering the gradient in the continuous finite element space the flux is recovered through a weighted l 2 projection onto h mathrm div conforming finite element spaces the resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results this paper also proposes an adaptive finite element method based on either the recovery based estimators or the edge estimator through local mesh refinement and establishes its convergence in particular it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps

the lax representation for an integrable class of relativistic dynamical systems we exhibit the lax pair for the class of relativistic dynamical systems recently introduced by ruijsenaars and schneider whose equations of motion read whereb is an arbitrary constant and the weierstrass elliptic function

on normal forms of nonlinear systems affine in control the nonlinear equivalences of both finite and infinite zero structures of linear systems have been well understood for single input single output systems and have found many applications in nonlinear control theory the extensions of these notions to multiple input multiple output systems have proven to be highly sophisticated in this paper we propose constructive algorithms for decomposing a nonlinear system that is affine in control these algorithms require modest assumptions on the system and apply to general multiple input multiple output systems that do not necessarily have the same number of inputs and outputs they lead to various normal form representations and reveal the structure at infinity the zero dynamics and the invertibility properties all of which represent nonlinear equivalences of relevant linear system structural properties

δ modulated feedback in discretization of sliding mode control modulated feedback control introduces periodicity the global attracting property of the periodic points is established for a simple scalar discrete time system under d modulated feedback attracting regions of the periodic points are also characterized when the discretization effects of the equivalent control based sliding mode control systems are studied we show that the zero order hold discretization gives rise to d modulation in the sliding mode direction the global attracting property of d modulated feedback offers a vivid illustration of the way sliding is achieved interestingly we find that a zoh discretization scheme of the equivalent control based sliding mode control system with relative degree one results in only 2 periodic orbits

euler s discretization of single input sliding mode control systems in this note the dynamical behaviors of euler s discretization of the equivalent control based sliding mode control smc systems are studied we show that if the discretized smc system is asymptotically stable then every trajectory converges to a period 2 cycle bounds for the steady states are derived which allow one to estimate the maximum chattering amplitude for a given value of the time step the analysis and results are verified with several computer simulations

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

regulation of relaxed static stability aircraft the authors formulate and solve a regulator problem for nonlinear parameter dependent dynamics it is shown that the problem is solvable except at parameter values associated with bifurcation of the equilibrium equations and that such bifurcations are inherently linked to the system zero dynamics these results are applied to the study of the regulation of the longitudinal dynamics of aircraft it is shown how bifurcation points arise in these problems and why they affect solvability of the regulator problem the relationships between bifurcation system zeros and dynamic and static stability are illustrated

asymptotic model matching for nonlinear systems this paper investigates the problem of designing a compensating control law for a square invertible nonlinear plant so that the response of the closed loop system asymptotically matches that of a prescribed driven nonlinear model a set of necessary conditions for achieving asymptotic model matching is established one of these involves the stability properties for subdynamics which are common to every model matching closed loop these subdynamics called the fixed dynamics are intrinsically characterized based on these results a new set of sufficient conditions for achieving asymptotic model matching is given the interrelation between the minimum phase and vector relative degree properties of a plant and a matchable model are studied

applications of singular perturbation techniques to control problems this paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years the first three sections are devoted to the standard model and its time scale stability and controllability properties the next two sections deal with linear quadratic optimal control and one with cheap near singular control then the composite control and trajectory optimization are considered in two sections and stochastic control in one section the last section returns to the problem of modeling this time in the context of large scale systems the bibliography contains more than 250 titles

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

a necessary condition for local asymptotic stability of nonlinear systems with exogenous disturbance local asymptotic stability of nonlinear systems with real parametric uncertainty or disturbance is one of the important problems in control systems literature in this paper we investigate this problem for nonlinear systems with time varying disturbance we assume that the disturbance vector is generated by an exosystem which is neutrally stable thus the disturbances that we consider include both constant and periodic signals for this class of nonlinear systems with time varying disturbance we derive a necessary condition for local asymptotic stability of equilibria as corollaries of our general result we deduce the necessary condition obtained by byrnes and sundarapandian 1 for nonlinear systems with constant real parametric uncertainty and the necessary condition obtained by brockett 2 for nonlinear autonomous systems we illustrate our result with several examples

interpolation in multidimensions a convenient finite dimensional matrix representation of the partial differential operators and some applications an extension to multidimensions of a generalized notion of lagrangian interpolation is used to introduce finite dimensional matrix representations of the partial differential operators this makes it possible to extend to a multidimensional environment various results which were obtained in the past by exploiting such a representation in a one dimensional context some such applications are outlined the construction of remarkable matrices convenient techniques to solve numerically eigenvalue problems and differential equations in multidimensions and the manufacture of solvable many body dynamical systems in multidimensional spaces some instances of such systems are exhibited

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

solvable nonrelativistic classical n body problems on the line i several solvable n body problems on the line are exhibited they are characterized by equations of motion of newtonian type mjxj fj j 1 n with the forces fj given as explicit functions of the particle coordinates xk and their velocities xk k 1 n the forces fj generally also depend on some free parameters so that in each case there is actually a class of solvable models in this paper the emphasis is on translation invariant models and on values of n sufficiently small few body problems to allow the display of the solution of the initial value problem in completely explicit form

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

stabilization of a class of nonlinear systems by a smooth feedback control we consider the stabilizability problem for systems of the form x f x ub in the neighborhood of an equilibrium point of f x first by means of center manifold theory a lower order system is introduced if this system is stabilizable then so is the original system results on the stabilization of the low order system are presented no resort is taken to liapunov theory the relation between stabilizability and controllability is discussed

zero dynamics of sampled nonlinear systems abstract the behaviour of the zero dynamics of a discrete time system obtained when sampling a continuous time nonlinear system is studied we give a result which generalizes the known behaviour of the zeros of a sampled linear system for small time intervals

output tracking design of affine nonlinear plant via variable structure system an output tracking design for affine nonlinear systems via a variable structure system is proposed the structure algorithm is used to determine if the considered system is left invertible it is shown that if such a condition is satisfied then a sliding mode controller can be designed to achieve output tracking the decouplability condition in conventional approaches is now replaced by the weaker left invertibility condition stable tracking is guaranteed if the internal mode is bounded input bounded state stable simulation results are included to demonstrate the design procedure