a general and exact method for determining limit cycles in decentralized relay systems an exact method for evaluating the periods and stability of limit cycles in decentralized relay systems is presented the method can cope with relays having dead zones and hystereses the underlying idea is to convert the continuous decentralized relay system under a limit cycle to an equivalent fictitious sampled data system with synchronous samplers consequently regular sampled data tools can be applied in a straightforward manner to derive closed form necessary conditions as well as stability conditions and other properties of such systems this paper extends and generalises recent results obtained by the authors

parameter tuning of second order sliding mode controllers for linear plants with dynamic actuators tuning of second order sliding mode control 2 smc algorithms in linear systems with dynamic actuators is considered by means of a describing function df approach it is investigated how the parameters of a 2 smc algorithm the so called generalized sub optimal algorithm affect the frequency and the magnitude of the limit cycles that occur when the overall relative degree of the plant plus the actuator is three or more explicit formulas are given that allow for setting the parameters of the algorithm to obtain a periodic solution with the prescribed characteristics by means of simulation examples we show that the estimated chattering parameters are in good agreement with the actual ones we also show that the proposed design procedure can also be useful when the local linearization of a nonlinear dynamics is sufficiently accurate

on frequency domain criterion of finite time convergence of second order sliding mode control algorithms transient processes in the systems controlled by the twisting second order sliding mode sosm control algorithm and certain generic sosm given by the describing function are analyzed in the frequency domain the analysis is based on the approximate describing function method the relationship between the frequency response nyquist plot of the plant the shape of the negative reciprocal describing function of the controller and the transient process convergence rate is investigated a simple criterion of the existence of finite time convergence is proposed it is shown that the convergence rate in a system controlled by a sosm controller depends on the angle between the high frequency asymptote of the nyquist plot of the plant and the low amplitude asymptote of the negative reciprocal of the describing function of the controller which is named the phase deficit

perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods we derive a correct first order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted most previous perturbative methods give incorrect results for this case even to lowest order because of the complicated discontinuous boundary conditions on the electric field at such an interface our final expression is simply a surface integral over the material interface of the continuous field components from the unperturbed structure the derivation is based on a localized coordinate transformation technique which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking the limit in which the coordinate perturbation is infinitesimally localized around the boundary not only is our result potentially useful in evaluating boundary perturbations e g from fabrication imperfections in highly anisotropic media such as many metamaterials but it also has a direct application in numerical electromagnetism in particular we show how it leads to a subpixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes

chattering avoidance by second order sliding mode control relying on the possibility of generating a second order sliding motion by using as control the first derivative of the control signal instead of the actual control a new solution to the problem of chattering elimination in variable structure control vsc is presented such a solution inspired by the classical bang bang optimal control strategy is first depicted and expressed in terms of a control algorithm by introducing a suitable auxiliary problem involving a second order uncertain system with unavailable velocity then the applicability of the algorithm is extended via suitable modifications to the case of nonlinear systems with uncertainties of more general types the proposed algorithm does not require the use of observers and differential inequalities and can be applied in practice by exploiting such commercial components as peak detectors or other approximated methods to evaluate the change of the sign of the derivative of the quantity accounting for the distance to the sliding manifold

hosm observer for a class of non minimum phase causal nonlinear mimo systems a higher order sliding mode observer is proposed for asymptotic identification of the full state vector and the vector of unknown inputs for mimo nonlinear causal systems with unstable internal dynamics the problem is addressed via consistent application of exact higher order sliding mode differentiators in conjunction with the method of stable system center a numerical example illustrates the performance of the proposed algorithm

perturbation theory for maxwell s equations with shifting material boundaries perturbation theory permits the analytic study of small changes on known solutions and is especially useful in electromagnetism for understanding weak interactions and imperfections standard perturbation theory techniques however have difficulties when applied to maxwell s equations for small shifts in dielectric interfaces especially in high index contrast three dimensional systems due to the discontinous field boundary conditions in fact the usual methods fail even to predict the lowest order behavior by considering a sharp boundary as a limit of anisotropically smoothed systems we are able to derive a correct first order perturbation theory and mode coupling constants involving only surface integrals of the unperturbed fields over the perturbed interface in addition we discuss further considerations that arise for higher order perturbative methods in electromagnetism

generalization of gao s reaching law for higher relative degree sliding variables in this paper a generalized discrete time reaching law for sliding variables with arbitrary relative degree is proposed the reaching law is based on gao s strategy and ensures higher order switching type sliding motion which is a novel concept in the field of discrete time plants the controller synthesis process is simplified by performing a specific system state transformation it is demonstrated that the width of the layer to which the representative point of the system is driven decreases as the relative degree of the sliding variable gets bigger it has further been proven that achieving a narrower quasi sliding mode band width is reflected in reduced state error finally the reaching phase elimination for variables with arbitrary relative degree is considered this is obtained by implementing a time varying sliding surface in a finite number of initial time instants uniform ultimate boundedness of all state variables is guaranteed

design of continuous twisting algorithm abstract for the double integrator with matched lipschitz disturbances we propose a continuous homogeneous controller providing finite time stability of the origin the disturbance is compensated exactly in finite time using a discontinuous function through an integral action since the controller is dynamic the closed loop is a third order system that achieves a third order sliding mode in the steady state the stability and robustness properties of the controller are proven using a smooth and homogeneous strict lyapunov function lf in a first stage the gains of the controller and the lf are designed using a method based on polya s theorem in a second stage the controller s gains are adjusted through a sum of squares representation of the lf

unknown input and state estimation for unobservable systems the concept of strong detectability and its relation with the concept of invariant zeros is reviewed for strongly detectable systems which includes the strongly observable systems it is proposed a hierarchical design of a robust observer whose trajectories converge to those of the original state vector furthermore it is shown that neither left invertibility is a sufficient condition nor strong detectability is a necessary condition to estimate the unknown inputs it is shown that the necessary and sufficient condition for estimating the unknown inputs is that the set of the invariant zeros that do not belong to the set of unobservable modes be within the interior of the left half plane of the complex space this shows that the unknown inputs could be estimated even if it is impossible to estimate the entire state vector of the system two numerical examples illustrate the effectiveness of the proposed estimation schemes

sliding order and sliding accuracy in sliding mode control the synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered usually what is called sliding mode is employed in such synthesis this sliding mode is characterized in practice by a high frequency switching of the control it turns out that the deviation of the system from its prescribed constraints sliding accuracy is proportional to the switching time delay a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced these algorithms feature a bounded control continuously depending on time with discontinuities only in the control derivative it is also shown that the sliding accuracy is proportional to the square of the switching time delay

generating self excited oscillations for underactuated mechanical systems via two relay controller a tool for the design of a periodic motion in an underactuated mechanical system via generating a self excited oscillation of a desired amplitude and frequency by means of the variable structure control is proposed first an approximate approach based on the describing function method is given which requires that the mechanical plant should be a linear low pass filter the hypothesis that usually holds when the oscillations are relatively fast the method based on the locus of a perturbed relay systems provides an exact model of the oscillations when the plant is linear finally the poincare map s design provides the value of the controller parameters ensuring the locally orbitally stable periodic motions for an arbitrary mechanical plant the proposed approach is shown by the controller design and experiments on the furuta pendulum

accelerated twisting algorithm the proposed modified twisting controller provides for the exact output regulation of uncertain dynamic systems with relative degree 2 the convergence rate is prescribed in advance the fixed time convergence is obtained as a particular case if the transient time is uniformly bounded an output feedback version of the controller is presented as well

parameter identification via modified twisting algorithm a method for identification of any even number of parameters of the transfer function from the test of the process which involves application of the modified twisting algorithm is proposed equations for determining the unknown parameters can be written separately for the magnitude and the argument of the transfer function that simplifies the task of the identification as a result the problem can be reduced to the iterative solution of a system of algebraic equations

a continuous asymptotic tracking control strategy for uncertain nonlinear systems in this note we present a new continuous control mechanism that compensates for uncertainty in a class of high order multiple input multiple output nonlinear systems the control strategy is based on limited assumptions on the structure of the system nonlinearities a new lyapunov based stability argument is employed to prove semiglobal asymptotic tracking

analysis of chattering in systems with second order sliding modes a systematic approach to the chattering analysis in systems with second order sliding modes is developed the neglected actuator dynamics is considered to be the main cause of chattering in real systems the magnitude of oscillations in nonlinear systems with unmodeled fast nonlinear actuators driven by second order sliding mode control generalized suboptimal 2 smc g so algorithms is evaluated sufficient conditions for the existence of orbitally stable periodic motions are found in terms of the properties of corresponding poincare maps for linear systems driven by 2 smc g so algorithms analysis tools based on the frequency domain methods are developed the first of these techniques is based on the describing function method and provides for a simple approximate approach to evaluate the frequency and the amplitude of possible periodic motions the second technique represents a modified tsypkin s method and provides for a relatively simple theoretically exact approach to evaluate the periodic motion parameters examples of analysis and simulation results are given throughout this paper

extremum seeking control with second order sliding mode the extremum seeking control esc approach with a first order sliding mode has been proposed for searching a setpoint by making a sliding mode occur where a performance index tracks a reference signal to reach its extremum the system dynamics is assumed to be much faster than the parameter updating rate determined by the esc rule and thus is omitted in the stability analysis for the esc system with a first order sliding mode as a result only a static optimization can be obtained in this paper an esc approach with a second order sliding mode is proposed with the consideration of the system dynamics such that the performance index can be optimized dynamically even if the system dynamics is not faster than the parameter updating rate for the extremum seeking the asymptotic second order sliding mode relay control algorithm is implemented to guarantee the asymptotic convergence to the second order sliding mode without using the derivative of the performance index simulation results show that the second

an improved second order sliding mode control scheme robust against the measurement noise in this note the second order sliding mode control problem is tackled by explicitly taking into account the presence of measurement error with unknown upper bound spl delta under sensible assumptions regarding the uncertain dynamics of a broad class of nonlinear plants a new switching controller is proposed guaranteeing a sliding accuracy of order o spl delta simulations highlight the robustness and good performance of the proposed approach and confirm the expected precision order

tracking control of the uncertain heat and wave equation via power fractional and sliding mode techniques in the present paper preliminary results towards the generalization to the infinite dimensional setting of some well known robust finite dimensional control algorithms are illustrated more specifically we deal with the tracking problem for some classes of linear uncertain infinite dimensional systems evolving in hilbert spaces we design distributed variable structure stabilizers that are shown to be effective in the presence of external disturbances the main focus of the present paper is on the rejection of nonvanishing external disturbances the generalization to the infinite dimensional setting of the well known finite dimensional controllers namely the power fractional controller s p bhat and d s bernstein siam j control optim 38 2000 pp 751 766 and two second order sliding mode control algorithms the twisting and supertwisting algorithms l fridman and a levant higher order sliding modes as a natural phenomenon in control theory in robust control via variable structure and lyapunov techniques springer verlag berlin 1996 pp 107 133 a levant internat j control 58 1993 pp 1247 1263 is the main contribution of the present investigation first the distributed twisting control algorithm is developed to address the asymptotic state tracking of the perturbed wave equation next the finite time state tracking of the unperturbed heat equation is provided by means of a distributed power fractional controller finally the distributed supertwisting controller is suggested to address the asymptotic state tracking of the heat equation in spite of the presence of persistent disturbances constructive proofs of stability are developed via the lyapunov functional technique which leads to simple tuning rules for the controller parameters simulation results are discussed to verify the effectiveness of the proposed schemes

analysis of second order sliding mode algorithms in the frequency domain a frequency domain analysis of the second order sliding mode algorithms particularly of the twisting algorithm is carried out in the frequency domain with the use of the describing function method and tsypkin s approach it is shown that in the presence of an actuator the transient process may converge to a periodic motion parameters of this periodic motion are analyzed a comparison of the periodic solutions in the systems with higher order sliding mode controllers and the oscillations that occur in classical sliding mode systems with actuators is done

reaching law for dsmc systems with relative degree 2 switching variable abstractthis paper presents the first attempt to design a reaching law based discrete time sliding mode controller with a relative degree 2 switching variable the current value of this variable is only affected by the control signal and disturbance generated two time instants ago it is demonstrated that the new reaching law based strategy offers a smaller quasi sliding mode band width when compared to a similar control scheme with relative degree 1 switching variable this in turn leads to reduced system output error in the control system proposed in the paper

singularly perturbed analysis of chattering in relay control systems for sliding mode control systems with fast actuators sufficient conditions for the exponential decreasing of the amplitude of chattering and unlimited growth of frequency are found the connection between the stability of actuators and the stability of the plant on the one hand and the stability of the sliding mode system as the whole on the other hand is investigated the algorithm for correction of sliding mode equations is suggested for taking into account the presence of fast actuators

super twisting algorithm in presence of time and state dependent perturbations abstractin this paper generalized super twisting algorithm is applied to a class of systems whose perturbation and uncertain control coefficient are time and state dependent a non smooth strict lyapunov function is used to obtain the conditions for the global finite time stability and to estimate the time of convergence

l functions with large analytic rank and abelian varieties with large algebraic rank over function fields the goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of l functions allows one to produce many examples of l functions over function fields vanishing to high order at the center point of their functional equation conjectures of birch and swinnerton dyer bloch and beilinson relate the orders of vanishing of some of these l functions to mordell weil groups and other groups of algebraic cycles for certain abelian varieties of high analytic rank we are also able to prove the conjecture of birch and swinnerton dyer thus establishing the existence of large mordell weil groups in those cases in the rest of this section we state the main results of the paper

reduced order observer in the sliding mode control of nonlinear nonaffine systems the note considers the variable structure control of nonlinear known nonaffine systems when the state vector is not completely available and the use of observers is required the strategy of introducing integrators in the input channel is exploited to enlarge the class of tractable control systems a new reduced order observer is proposed and conditions are found under which it is proven the convergence to the unique ideal solution of both system and observer the control problem is solved by forcing a sliding regime for the observer while satisfying an exponential stability criterion for the observation error state equation

second order sliding fuzzy interval type 2 control for an uncertain system with real application a new second order sliding mode type 2 fuzzy controller for nonlinear uncertain perturbed systems is developed in this paper to overcome the constraint on the knowledge of the system model we have used local models that are related to some operating points to synthesize a type 2 nominal fuzzy global model the control is based on the super twisting algorithm which is among second order sliding mode controls smcs moreover two adaptive fuzzy type 2 systems have been introduced to generate the two super twisting signals to avoid both the chattering and the constraint on the knowledge of disturbances and uncertainties upper bounds these adaptive fuzzy type 2 systems are adjusted online by two adaptation laws that are deduced from the stability analysis in the lyapunov sense it has only one input i e the sliding surface and one output i e the optimal values of the gains control which are hard to compute with the original algorithm many results of the one link manipulator are obtained first by the simulation in order to compare the performances of the proposed method with that given by levant and then in a real time application in order to confirm the efficiency of the proposed approach the experimentation and simulation are done for the tracking control problem

performance analysis of second order sliding mode control systems with fast actuators the robustness of the second order sliding mode control 2 smc algorithm known as the generalized sub optimal algorithm is analyzed with respect to the cascade introduction of a linear fast actuator it is shown that if the actuator dynamics are sufficiently fast then the system trajectories converge to an invariant set that includes the second order sliding domain s s 0 it is also shown that the size of the invariant set has quadratic dependence with respect to the actuator small parameter mu for the sliding variable s and linear dependence for its derivative s this means that in the steady state the system trajectories converge to an invariant domain described by the following conditions s les o mu2 and s les o mu a simulation example is given to confirm the proposed

universal single input single output siso sliding mode controllers with finite time convergence an universal controller is constructed formulated in input output terms only which causes the output of any uncertain smooth single input single output siso minimum phase dynamic system with known relative degree to vanish in finite time this allows exact tracking of arbitrary real time smooth signals only one parameter is to be adjusted since the approach is based on higher order finite time convergence sliding modes the control can be made arbitrarily smooth providing for the arbitrarily high tracking accuracy order with respect to the sampling step

arithmetic of abelian varieties in artin schreier extensions we study abelian varieties defined over function fields of curves in positive characteristic p focusing on their arithmetic within the system of artin schreier extensions first we prove that the l function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor second we develop an artin schreier variant of a construction of berger this yields a new class of jacobians over function fields for which the birch and swinnerton dyer con jecture holds third we give a formula for the rank of the mordell weil groups of these jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between jacobians of auxiliary artin schreier curves we illustrate these theorems by computing the rank for explicit examples of jaco bians of arbitrary dimension g exhibiting jacobians with bounded rank and others with unbounded rank in the tower of artin schreier extensions fi nally we compute the mordell weil lattices of an isotrivial elliptic curve and a family of non isotrivial elliptic curves the latter exhibits an exotic phenom enon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields our methods also yield new results about supersingular factors of jacobians of artin schreier curves msc2000 primary 11g10 11g40 14g05 secondary 11g05 11g30 14h25 14j20 14k15

sliding sector control using new equivalent sector control this paper presents a variable structure control design method for linear and nonlinear continuous time systems with a special type of sliding sector using a new equivalent sector control the pr s

control of nonlinear variable structure systems etude des systemes de commande decrits par des equations d etat extension de la theorie des systemes de commande discontinue a structure variable a quelques cas non lineaires

exponential stabilization of the uncertain wave equation via distributed dynamic input extension the wave equation is studied under the effect of a persistent external perturbation a dynamical distributed controller is suggested based on an infinite dimensional generalization of second order sliding mode control techniques that provides the exponential attainment of a sufficiently smooth arbitrary reference of the state trajectory the control system comprises of both feedforward and feedback parts the latter being a discontinuous term directly connected to the plant input through a dynamical filter that augments the system state smoothing out the discontinuity of the feedback control loop as a result a continuous input is applied to the plant a constructive lyapunov based proof of convergence of the proposed control algorithm is carried out and supporting numerical results are presented

analysis of chattering in continuous sliding mode controllers an analysis of two most popular continuous sliding mode algorithms the power fractional sliding mode algorithm and a second order sliding mode algorithm known as the super twisting is carried out in the frequency domain with the use of the describing function method it is shown that in the presence of an actuator the transient process converges to a periodic motion parameters of this periodic motion are analyzed a few examples are considered to illustrate the obtained results

higher order super twisting for perturbed chains of integrators in this paper we present a generalization of the super twisting algorithm for perturbed chains of integrators of arbitrary order this higher order super twisting host controller is homogeneous with respect to a family of dilations and is continuous it is built as a dynamic controller with respect to the state variable of the chain of integrators and the convergence analysis is performed by the use of a homogeneous strict lyapunov function which is explicitly constructed the effectiveness of the controller is finally illustrated with simulations for a chain of integrators of order four first pure then perturbed where we compare the performances of two host controllers

adaptive robust controller based on integral sliding mode concept abstractthis paper proposes for a class of uncertain nonlinear systems an adaptive controller based on adaptive second order sliding mode control and integral sliding mode control concepts the adaptation strategy solves the problem of gain tuning and has the advantage of chattering reduction moreover limited information about perturbation and uncertainties has to be known the control is composed of two parts an adaptive one whose objective is to reject the perturbation and system uncertainties whereas the second one is chosen such as the nominal part of the system is stabilised in zero to illustrate the effectiveness of the proposed approach an application on an academic example is shown with simulation results

adaptive continuous twisting algorithm abstractin this paper an adaptive continuous twisting algorithm acta is presented for double integrator acta produces a continuous control signal ensuring finite time convergence of the states to zero moreover the control signal generated by acta compensates the lipschitz perturbation in finite time i e its value converges to the opposite value of the perturbation acta also keeps its convergence properties even in the case that the upper bound of the derivative of the perturbation exists but it is unknown

observation and output adaptive tracking for a class of nonlinear non minimum phase systems abstractin this paper the output tracking problem for a class of systems with unstable zero dynamics is addressed the state is assumed not measurable the output of the dynamical system to be controlled has to track a signal which is the sum of a known number of sinusoids with unknown frequencies amplitudes and phases the non minimum phase nature of the considered systems prevents the direct tracking by standard sliding mode methods which are known to generate unstable behaviours of the internal dynamics the proposed method relies on the availability of a flat output and its time derivatives which are functions of the unavailable state therefore a nonlinear observer is needed due to the uncertainty in the frequencies and in the parameters defining the relationship between the output of the system and the flat states adaptive indirect methods are applied

an averaging approach to chattering the singularly perturbed relay control systems sprcs as mathematical models of chattering in the small neighborhood of the switching surface in sliding mode systems are examined sufficient conditions for existence and stability of fast periodic solutions to the sprcs are found it is shown that the slow motions in such sprcs are approximately described by equations derived from equations for the slow variables of sprcs by averaging along fast periodic motions it is shown that in the general case when the equations of a plant contain relay control nonlinearly the averaged equations do not coincide with the equivalent control equations or with the filippov s definition 1988 for the sliding motions in the reduced system however in the linear case they coincide

brief digital second order sliding mode control for uncertain nonlinear systems in this note we analyze the discrete time implementation of a second order sliding mode control 2 smc scheme the treatment is detailed for a simple class of feedback linearizable nonlinear systems expressed in the brunowsky normal form first it is shown that the direct discretization of a continuous time 2 smc scheme guarantees the finite time attainment of a motion in an o t 2 boundary layer of the sliding manifold t being the sampling period then a suitable iterative learning procedure that leads to the asymptotic reduction of the boundary layer to o t 3 is proposed simulation results are reported at the end of the paper

generating self excited oscillations via two relay controller a tool for the design of a self excited oscillation of a desired amplitude and frequency in linear plants by means of the variable structure control is proposed an approximate approach based on the describing function method is given which requires that the mechanical plant should be a low pass filter the hypothesis that usually holds when the oscillations are relatively fast the proposed approach is demonstrated via the controller design and experiments on the furuta pendulum

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods

dielectric profile variations in high index contrast waveguides coupled mode theory and perturbation expansions perturbation theory formulation of maxwell s equations gives a theoretically elegant and computationally efficient way of describing small imperfections and weak interactions in electromagnetic systems it is generally appreciated that due to the discontinuous field boundary conditions in the systems employing high dielectric contrast profiles standard perturbation formulations fails when applied to the problem of shifted material boundaries in this paper we developed coupled mode and perturbation theory formulations for treating generic perturbations of a waveguide cross section based on hamiltonian formulation of maxwell equations in curvilinear coordinates we show that our formulation is accurate beyond the first order and rapidly converges to an exact result when used in a coupled mode theory framework even for the high index contrast discontinuous dielectric profiles among others our formulation allows for an efficient numerical evaluation of such quantities as deterministic pmd and change in the gv and gvd of a mode due to generic profile distortion in a waveguide of arbitrary cross section to our knowledge this is the first time perturbation and coupled mode theories are developed to deal with arbitrary profile variations in high index contrast waveguides