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

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

brief freedom in coordinate transformation for exact linearization and its application to transient behavior improvement this paper deals with the use of extra freedom in the exact linearization technique firstly the freedom in choosing the coordinate transformation and nonlinear state feedback for the exact linearization is clarified secondly a controller design strategy for the transient behavior improvement in the presence of impulsive disturbances is shown finally numerical and experimental examples exhibit the effectiveness of this method

compensating drift vector fields with gradient vector fields for asymptotic submanifold stabilization we derive sufficient conditions on a drift vector field to let an asymptotically stable invariant submanifold of an input affine system without drift remain asymptotically stable for the system with drift in doing so we use the same feedback laws modulo control gain tuning such that no new feedback laws need to be designed for the system with drift our main assumption is that the vector field of the input affine system without drift assumes the form of a gradient vector field for given feedback laws we show how one can assess the performance of the system with drift only via knowledge about the system without drift finally we find that our results are relevant in synchronization problems and backstepping controllers for mechanical systems

high gain observers in the state and estimation of robots having elastic joints the paper deals with the state and parameter estimation of robots having elastic joints the estimation of the unknown parameters of the robot is reduced to the estimation of its state variables by a state space immersion the luenberger observer is used in the state estimation of the extended robot model the use of high gains is studied in the cancellation of nonlinearities in order to simplify the observer design the high gain induces a time scale separation between the robot model and the observer and therefore the singular perturbation theory can be used in the stability analysis of the estimate error dynamics in particular it is shown that the error dynamics reaches the stable equilibrium in a very fast transient ensuring that the slow dynamics of the observer is just that of the given robot

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

towards a classification of modular compactifications of mathcal m _ g n the moduli space of smooth curves admits a beautiful compactification mathcal m g n subset overline mathcal m g n by the moduli space of stable curves in this paper we undertake a systematic classification of alternate modular compactifications of mathcal m g n let mathcal u g n be the non separated moduli stack of all n pointed reduced connected complete one dimensional schemes of arithmetic genus g when g 0 mathcal u 0 n is irreducible and we classify all open proper substacks of mathcal u 0 n when g 1 mathcal u g n may not be irreducible but there is a unique irreducible component mathcal v g n subset mathcal u g n containing mathcal m g n we classify open proper substacks of mathcal v g n satisfying a certain stability condition

on linearization of nonlinear singularity perturbed systems this note considers external feedback linearization of nonlinear singularly perturbed systems when more accurate representation of some dynamic systems is taken into account the necessary condition for linearization may be violated in these cases the slow manifold theory is shown to provide a practical tool for solving this problem furthermore for a class of systems with fast actuators direct application of present methodology in the literature is shown to be valid provided the linear equivalent system contains fast dynamics as well

on the control of regularly e perturbed nonlinear systems abstract this paper deals with the control of regularly c perturbed nonlinear systems on the basis of a nominal control law i e designed with respect to the unperturbed plant an iterative procedure is proposed for achieving at a fixed order of approximation the requirements specified oh the dynamics of the feedback control system the sufficient conditions ensuring the applicability of the procedure are verified when dealing with feedback linearization problems some working examples developed in the paper illustrate the efficiency of such a design strategy

an efficient lyapunov function for discrete t s models observer design this paper deals with the design of a new observer synthesis for discrete takagi sugeno t s fuzzy models it is well established that quadratic synthesis for discrete t s models and or linear parameter varying systems can be outperformed easily via nonquadratic syntheses several lyapunov functions can be used nevertheless this paper shows that with a small change in the initial lyapunov function a better in the sense of solutions to the linear matrix inequality constraints problem lyapunov function can be reached this one can introduce very important improvements

a novel approach to control of nonminimum phase nonlinear systems we present an approach to control of nonminimum phase multiple input multiple output mimo nonlinear systems based on a singular perturbation like technique first we show that a mimo nonlinear system can be converted into a singularly perturbed system through magnitude and time scaling transformation in particular the proposed transformation technique can make the hidden or internal dynamics behave like a fast subsystem however the fast subsystem takes a weakly controllable form which implies that a small positive low gain parameter multiplies the control input in the fast subsystem for this reason the well known integral manifold approach is then utilized to decompose the resulting singularly perturbed system into two subsystems in separate timescales each of which is of a lower dimension and more importantly controllable in the sense that the control input will explicitly appear in the boundary layer system the proposed method is particularly useful to make the input output dynamic characteristics of a decouplable nonminimum phase nonlinear system decoupled and linear

global asymptotic stabilisation of non linear systems in this article the problem of global asymptotic stabilisation of non linear systems is considered first of all the global asymptotic stability of a class of non linear systems is considered it is proved that the class of non linear systems is globally asymptotically stable under certain appropriate conditions secondly based on the result obtained the global asymptotic stabilisation of a class of non linear systems is investigated and the feedback control laws which globally stabilise the class of systems are designed thirdly by the coordinate change and the results obtained the global stabilisation of the affine non linear systems are studied and then some sufficient conditions of stabilisation are presented finally the global asymptotic stabilisation of partially linear cascade non linear system is considered the sufficient conditions of global stabilisation are obtained and the corresponding feedback control laws which stabilise the non linear systems are designed in addition some

on the connection between nonlinear differential algebraic equations and singularly perturbed control systems in nonstandard form considers a class of control systems represented by nonlinear differential equations depending on a small parameter the systems are not in the standard singularly perturbed form and therefore one of the challenges is to show that the control systems do represent singularly perturbed two time scale systems assumptions are introduced which guarantee that an equivalent representation for the systems can be obtained in the standard singularly perturbed form thereby justifying the two time scale property the equations for the slow dynamics are characterized by a set of differential algebraic equations which have been studied previosly in the literature the fast dynamics are characterized by differential equations both the slow and the fast dynamics are easily derived and are defined in terms of variables that define the original control system control design for the class of systems being considered is studied using the composite control approach

non standard singularly perturbed control systems and differential algebraic equations we consider a class of linear control systems represented by equations depending on a small parameter but which are not in the standard singularly perturbed form one of the challenging system theoretic problem is to obtain an equivalent representation for the control system which if possible is in the standard singularly perturbed form assumptions are introduced which guarantee that an equivalent representation can be obtained which is in the standard singularly perturbed form thereby justifying the two time scale property the equations for the slow dynamics are characterized by a set of differential algebraic equations which are easily derived from the original equations by setting the parameter to zero the original assumptions guarantee the existence of solutions of the obtained differential algebraic equations the fast dynamics are characterized by different equations in terms of matrices that define the original control system control design for the system being considered is studied using th

singularly perturbed zero dynamics of nonlinear systems stability properties of zero dynamics are among the crucial input output properties of both linear and nonlinear systems unstable or nonminimum phase zero dynamics are a major obstacle to input output linearization and high gain designs an analysis of the effects of regular perturbations in system equations on zero dynamics shows that whenever a perturbation decreases the system s relative degree it manifests itself as a singular perturbation of zero dynamics conditions are given under which the zero dynamics evolve in two time scales characteristic of a standard singular perturbation form that allows a separate analysis of slow and fast parts of the zero dynamics the slow part is shown to be identical to the zero dynamics of the unperturbed system while the fast part represented by the so called boundary layer system describes the effects of perturbation

a geometric approach to nonlinear singularly perturbed control systems abstract applications of two time scale singular perturbation methods have been limited to the class of models appearing in a standard form many nonlinear control systems such as models of aircraft and robotic manipulators with flexible joints are two time scale systems but do not appear in the standard form the main result of this paper is a coordinate free characterization of time scales in terms of invariant manifolds which express conservation and equilibrium properties of the control system a procedure for finding slow and fast states is given previously developed slow fast composite control designs are thus made applicable to a wider class of nonlinear systems

on the state observation and output feedback problems for nonlinear uncertain dynamic systems abstract in this paper we examine the problems of state observation and state trajectory control by output feedback for the class of non linear systems in 3 11 we begin by modifying a known discontinuous variable structure type observer into a continuous type observer that guarantees the observation error is globally exponentially stable ges we then modify a known discontinuous variable structure type output feedback controller into continuous type output feedback controller that forces the state to the origin in the ges sense specific time varying bounds on the observation error and the state trajectory are also developed revealing how the corresponding observer or control parameters can be adjusted to improve the system performance

comparative study of non linear state observation techniques abstract this paper contains a comparative study of four techniques for observing the slates of non linear systems the first technique examined is inspired by bestle and zeitz and icrener and respondek in this method a non linear transformation is found that brings the system into a canonical form from where observer design is facilitated the second technique is attributable to thau in this method the error between the system s true state and the output of the observer is shown to be asymptotically convergent to zero provided that an additional assumption is valid the third technique is due to baumann and rugh in this method extended or pseudolinearization of the error differential equation about a family of equilibrium points results in an observer design such that the eigenvalues of the linearized error equation are locally invariant finally techniques from variable structure systems are utilized to design an observer that yields an exponentially decaying error like thau s observer but unl

on the dynamical sliding mode control of nonlinear systems the consequences of the differential algebraic approach in the sliding mode control of nonlinear single input single output systems are reviewed in tutorial fashion input dependent sliding surfaces possibly including time derivatives of the input signal are shown to arise naturally from elementary differential algebraic results pertaining to the fliess s generalized controller canonical forms of nonlinear systems this class of switching surfaces generally leads to chattering free dynamically synthesized sliding regimes in which the highest time derivative of the input signal undergoes all the bang bang type discontinuities examples illustrating the obtained results are also included

variable structure regulation of partially linearizable dynamics it is shown how the theory of decoupling and partial linearization of nonlinear affine systems can enhance the design of variable structure control systems and expand their range of applicability refined insights into the design of nonlinear switching surfaces and a new regular form are obtained application to an adjustable speed induction motor drive illustrates how this method allows stabilization of a periodic attractor

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

on static output feedback stabilization of systems with stable generalized zero dynamics in this note we define generalized zero dynamics introduce a generalized zero dynamics algorithm and give a sufficient condition which ensures stabilizability of linear systems with asymptotically stable generalized zero dynamics by static output feedback

some results in nonlinear output regulation and feedback stabilization the problems of nonlinear output regulation and feedback stabilization on compacta are studied high gain control laws are given to solve output regulation on compacta for a class of nonlinear systems for which exact tracking with internal stability may be impossible with some modification this result is applied to feedback stabilization on compacta for nonlinear systems whose zero dynamics may be globally unstable