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

a general class of sliding surface for sliding mode control in the conventional sliding mode control the relative degree of the chosen sliding surface is usually one this paper addresses a general class of sliding surface whose relative degree is no longer restricted to one we show that when the relative degree is more than two it is inevitable that there will exist a limit cycle for the nonlinear sign function and for the case of relative degree two the system possesses poor phase margin however these disadvantages can be remedied remarkably by smoothing out the control nonlinearity in a suitable boundary layer with a saturation function that ensures asymptotic stability of the overall system

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

an output feedback stabilization scheme using the filtered coordinate transformation an output feedback stabilizing controller is proposed for a class of nonlinear systems that are diffeomorphic to nonlinear observer form which contains uncertainty in constructing the controller a filtered coordinate transformation is utilized so as to make the vector field with input constant some riccati type inequality criteria are derived for the exponential convergence of an observer the filter and the system

brief paper output feedback variable structure control for linear systems with uncertainties and disturbances this paper proposes a dynamic output feedback variable structure controller for linear mimo systems with mismatched and matched norm bounded uncertainties and matched nonlinear disturbances the proposed controller consists of nonlinear and linear parts similar to the unit vector type of controller the nonlinear part of the new controller takes care only of matched uncertainties and disturbances and on the other hand the linear part with full dynamics completely handles mismatched uncertainties designing such a linear part leads to use a lyapunov function associated with the full states which achieves the global stability against the mismatched uncertainties the resulting criteria are furthermore converted into solvable ones by using the so called cone complementary linearization algorithm for bi convex problems

boundary problems for diffusion in a fluctuating potential abstract using a newly developed formalism adapted for diffusion in the presence of partially absorbing boundaries we treat the overdamped motion of a particle in a fluctuating potential and in a restricted space the potential is assumed to be linear with a slope switching between two possible values the switching process is taken as the standard markov alternating process we first construct the free dynamics propagator no boundaries which controls the evolution of the joint probability for both the particle s position and the state of the potential secondly at a given spatial point we affix a two channel boundary a given channel absorbs the trajectories arriving at the boundary with the potential in a given state we demonstrate the asymmetry of the filling process for the individual channels and we discuss their asymptotic occupation depending on the mean switching frequency finally we study the motion in a domain restricted by two boundaries of the above two channel type and the corresponding first passage time the simple problem setting allows for an explicit analysis of the correlation between the particle s motion and the potential switching process

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

topological galois theory abstract we introduce an abstract topos theoretic framework for building galois type theories in a variety of different mathematical contexts such theories are obtained from representations of certain atomic two valued toposes as toposes of continuous actions of a topological group our framework extends grothendieck s theory of galois categories and allows to build galois type equivalences in new contexts such as for example graph theory and finite group theory

sites whose topoi are the smooth representations of locally prodiscrete monoids abstract we define a class of sites such that the associated topos is equivalent to the category of smooth sets representations of some locally prodiscrete monoids to be defined examples of locally prodiscrete monoids include profinite groups and finite adele valued points of algebraic groups this is a generalization of the fact that the topos associated with the etale site of a scheme is equivalent to the category of sets with continuous action by the etale fundamental group we then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group

inner outer factorization for nonlinear noninvertible systems this paper considers inner outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor an application of the main result to control of nonminimum phase nonlinear systems is indicated

output feedback multivariable global variable gain super twisting algorithm a novel output feedback super twisting algorithm design is proposed for uncertain multivariable plants in order to allow global finite time exact tracking in the presence of a rather general class of matched nonlinear disturbances which may depend on measured and or unmeasured states the algorithm is designed with variable gains this represents a significant extension of a recently proposed nondecoupled multivariable super twisting algorithm based on state feedback with fixed gains to the case of output feedback with variable gains

technical communique a remark on passivity based and discontinuous control of uncertain nonlinear systems we address the problem of robust stabilisation of nonlinear systems affected by time varying uniformly bounded in time affine perturbations our approach relies on the combination of sliding mode techniques and passivity based control roughly speaking we show that under suitable conditions the sliding mode variable can be chosen as the passive output of the perturbed system then we show how to construct a controller which guarantees the global uniform convergence of the plant s outputs towards a time varying desired reference even in the presence of permanently exciting time varying disturbances we illustrate our result on the tracking control of the van der pol oscillator

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

global robust stabilization of nonlinear cascaded systems this paper gives a sufficient condition for a class of nonlinear cascaded systems to be globally stabilizable via state feedback in the presence of uncertainty which does not necessarily satisfy the so called matching condition the authors result is an extension of the former stabilization results which treated systems without uncertainty in the sense that the uncertainty is taken into account

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

a tannakian context for galois theory abstract strong similarities have been long observed between the galois and tannaka theories of the representation of groups in this paper we construct an explicit neutral tannakian context for the galois theory of atomic topoi and prove equivalence for the fundamental theorem since the theorem is known for the galois context this yields a proof of the fundamental recognition theorem for a new tannakian context this example is different from the additive cases or their generalization for which the theorem is known to hold and for which the unit of the tensor product is always an object of finite presentation which is not the case in our context

feedback stabilization of planar nonlinear systems abstract the stabilization of second order control systems by smooth state feedback is addressed a reduction procedure proposed in 8 is applied in order to reduce the problem to the stabilization of a first order control system by a c 2 feedback function generic sufficient conditions are given for the solution of the last problem and by the reduction procedure of the first problem as well counter examples show that the conditions are not necessary and which kind of pathologies may arise

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

ergodicity of dissipative differential equations subject to random impulses differential equations subject to random impulses are studied randomness is introduced both through the time between impulses which is distributed exponentially and through the sign of the impulses which are fixed in amplitude and orientation such models are particular instances of piecewise deterministic markov processes and they arise naturally in the study of a number of physical phenomena particularly impacting systems the underlying deterministic semigroup is assumed to be dissipative and a general theorem which establishes the existence of invariant measures for the randomly forced problem is proved further structure is then added to the deterministic semigroup which enables the proof of ergodic theorems characteristic functions are used for the case when the deterministic component forms a damped linear problem and irreducibility measures are employed for the study of a randomly forced damped double well nonlinear oscillator with a gradient structure

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

escape statistics for systems driven by dichotomous noise ii the imperfect pitchfork bifurcation as a case study we use the general results for the escape probabilities and mean exit times obtained in an accompanying paper to analyze in detail a nonlinear system presenting an imperfect subcritical pitchfork bifurcation we redraw the bifurcation diagram to show the effect of the noise to avoid spurious results we introduce the concept ofextinction level as the minimum possible value for the system and discuss its effect on the bifurcation diagram

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

brief variable structure output feedback control design for a class of uncertain dynamic systems in this paper we consider the problem of designing a variable structure output feedback control vsofc law for a class of uncertain systems with mismatched uncertainty in the state matrix we derive existence conditions of linear sliding surfaces in terms of constrained linear matrix inequalities lmis and we give vsofc strategies we also give a simple design procedure and parameterizations of the linear switching surfaces together with solvability conditions for the given constrained lmis finally we give a design example in order to show the effectiveness of our method

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

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

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

peaking free variable structure control of uncertain linear systems based on a high gain observer an output feedback model reference variable structure controller based on a high gain observer hgo is proposed and analyzed for single input single output siso linear plants with relative degree greater than one the control law is generated using the hgo signals only to drive the sign function of the variable structure control component while the sign function gain also called modulation as well as the other components of the control signal are generated using signals from state variable filters which do not require high gain and are free of peaking this scheme achieves global exponential stability with respect to a small residual set and does not generate peaking in the control signal

on variable structure output feedback controllers for uncertain dynamic systems a class of variable structure output feedback controllers for uncertain dynamic systems with bounded uncertainties is proposed no statistical information about the uncertain elements is assumed a variable structure systems approach and a geometric approach to the analysis and synthesis of system zeros are employed in the synthesis of the proposed controllers

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

localic galois theory abstract in proposition i of memoire sur les conditions de resolubilite des equations par radicaux galois considers the splitting field a of a polynomial with rational coefficients given any intermediate extension x he proves that the action of the galois group gal a q on the set of morphisms x a is transitive and that x is the fixed field of its galois group gal a x in this article we first state and prove a dual categorical formulation of these statements which turns out to be a theorem about atomic sites with a representable point these abstract developments correspond exactly to classical galois theory we then consider the same situation but with the point no longer representable it determines a proobject and it becomes tautologically prorepresentable mutatis mutandis we state and prove the prorepresentable version of the categorical formulation of galois theorems in this case the classical group of automorphisms has to be replaced by the localic group of automorphisms these developments form the content of a theory that we call localic galois theory from our results it immediately follows the theorem a topos with a point is connected atomic if and only if it is the classifying topos of a localic group and this group can be taken to be the locale of automorphisms of the point the hard implication in this equivalence was first proved in print in joyal and tierney mem amer math soc 151 1984 theorem 1 section 3 chapter viii and it follows from a characterization of atomic topoi in terms of open maps and from a theory of descent for morphisms of topoi and locales our proof of this theorem is completely independent of descent theory and of any other result in joyal and tierney 1984 the theorem follows as a straightforward consequence of our direct generalization of the fundamental results of galois we explain and give the necessary definitions to understand this paper without previous knowledge of the theory of locales other than this our principal results theorems 2 2 and 2 3 require only elementary notions of category theory

output feedback stabilization of uncertain plants a variable structure systems approach the problem of output stabilization for uncertain s1so systems is considered using structural transformations uncertain systems can change to the form convenient for output feedback design synthesis of observer based variable structure control for asymptotical stabilization or uniform ultimate boundedness of the closed loop system is provided examples are considered and simulation results are given

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

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

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

on variable structure output feedback controllers in a recent work by zak hui 1993 a sliding mode controller for mimo systems using static output feedback was proposed very nice geometric conditions for how to design sliding surfaces were given however there are two restrictive assumptions in it one is that the uncertainties in the system must be bounded by a known function of outputs which excludes some possible uncertainties in the a matrix if the system is described by the triple a b c the other one requires a matrix equality to be held which may also be very difficult to satisfy in many systems in this paper we propose a modification of the sliding mode controller for a class of single input single output siso systems which can eliminate the above mentioned limitations and under certain conditions guarantee global closed loop stability hence the range of applicability of zak hui s method can be greatly broadened

stabilization of minimum phase nonlinear systems by dynamic output feedback in this paper the stabilization problem of minimum phase nonlinear systems by dynamic output feedback is investigated it is shown that if the zero dynamics of an affine nonlinear system is asymptotically stable according to the kth approximation then the system is stabilizable by dynamic output feedback the proof is constructive which provides a method to design the stabilizer the result is applied to investigating the stabilization problem of the rotating stall in axial flow compressors

a transformed lure problem for sliding mode control and chattering reduction the sliding mode control problem is investigated with the frequency domain approach the authors show that the sliding mode control can be transformed into a lure problem with this formulation it is shown that the conventional sliding mode control with a nonlinear sign function can only guarantee to make the overall system stable but not asymptotically stable the authors also consider a practical situation where the sign nonlinearity may contain a hysteresis loop for this situation they show that even if the hysteresis loop is very small there still exists a limit cycle however chattering phenomenon caused by the limit cycle behavior can be eliminated by a certain class of nonlinearities

adaptive stabilization of a class of uncertain nonlinear systems a differential inclusion approach abstract within the framework of controlled differential inclusions a stabilizing strategy is developed for a class of uncertain nonlinear dynamical systems the strategy has a linear plus discontinuous output feedback structure with bounded adaptive scalar gain

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

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

output feedback stabilization using variable structure control we consider the stabilization of a class of multivariable nonlinear systems about an equilibrium point at the origin using variable structure output feedback control in particular the system can be transformed into a normal form with no zero dynamics a robust high grain observer is used to estimate the state variables while rejecting the effect of disturbance a globally bounded discontinuous variable structure controller is designed to compensate for modelling error we show that the controller can stabilize the closed loop system and does not suffer from the peaking phenomenon that exists in previous designs

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