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

variable structure control of stochastic systems abstract in this paper the problem of variable structure control of stochastic svsc systems is proposed and the corresponding variable structure control strategies with complete and incomplete state information are established the concepts of stochastic sliding mode and sliding motion band are introduced which may be regarded as the basic characteristics of svsc systems robustness of svsc systems is considered and a general design procedure for svsc systems is presented the effectiveness of the proposed control method is shown by simulation results

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

analysis of a numerical dynamic programming algorithm applied to economic models in this paper we develop a discretized version of the dynamic programming algorithm and study its convergence and stability properties we show that the computed value function converges quadratically to the true value function and that the computed policy function converges linearly as the mesh size of the discretization converges to zero further the algorithm is stable we also discuss several aspects of the implementation of our procedures as applied to some commonly studied growth models

sliding mode control of linear systems with mismatched uncertainties abstract it is well known that sliding mode control is robust to matched uncertainties that lie in the range space of the input matrix however many systems are affected by mismatched uncertainties and yet do not enjoy the matching conditions it is the purpose of this paper to present a new dynamical approach of sliding variable formulation which when the system is in sliding mode can explicitly deal with mismatched uncertainties our first step is to treat certain states as inputs to a reduced order system and use adaptive techniques to design fictitious controllers for these inputs which can then tackle the mismatched uncertainties the second step is to use sliding control to realize the adaptive fictitious controllers the design is systematic modular and intuitively simple

convergence properties of the likelihood of computed dynamic models this paper studies the econometrics of computed dynamic models since these models generally lack a closed form solution their policy functions are approximated by numerical methods hence the researcher can only evaluate an approximated likelihood associated with the approximated policy function rather than the exact likelihood implied by the exact policy function what are the consequences for inference of the use of approximated likelihoods first we find conditions under which as the approximated policy function converges to the exact policy the approximated likelihood also converges to the exact likelihood second we show that second order approximation errors in the policy function which almost always are ignored by researchers have first order effects on the likelihood function third we discuss convergence of bayesian and classical estimates finally we propose to use a likelihood ratio test as a diagnostic device for problems derived from the use of approximated likelihoods

accuracy of simulations for stochastic dynamic models this paper is concerned with accuracy properties of simulations of approximate solutions for stochastic dynamic models our analysis rests upon a continuity property of invariant distributions and a generalized law of large numbers we then show that the statistics generated by any sufficiently good numerical approximation are arbitrarily close to the set of expected values of the model s invariant distributions also under a contractivity condition on the dynamics we establish error bounds these results are of further interest for the comparative study of stationary solutions and the estimation of structural dynamic models

design of fuzzy sliding mode control systems in this paper the design of fuzzy sliding mode control is discussed for a complex physical system represented by an aggregated fuzzy global model which compromises a set of linear models conditions for the fuzzy sliding mode control to stabilize the global fuzzy model are given simulations are presented to show the effectiveness of the control strategy

aftershocks in coherent noise models the decay pattern of aftershocks in the so called coherent noise models m e j newman and k sneppen phys rev e54 6226 1996 is studied in detail analytical and numerical results show that the probability to nd a large event at timet after an initial major event decreases as t for smallt with the exponent ranging from 0 to values well above 1 this is in contrast to sneppen und newman who stated that the exponent is about 1 independent of the microscopic details of the simulation numerical simulations of an extended model c wilke t martinetz phys rev e56 7128 1997 show that the power law is only a generic feature of the original dynamics and does not necessarily appear in a more general context finally the implications of the results to the modeling of earthquakes are discussed

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

technical communique an explicit formula of linear sliding surfaces for a class of uncertain dynamic systems with mismatched uncertainties in this note the problem of designing a variable structure control vsc law for uncertain systems where the uncertainty in the state model does not satisfy the matching condition is considered based on the lmi approach we develop a new design method of linear sliding surfaces which are linear to the state we give a sufficient condition for the existence of linear sliding surfaces in terms of lmis and we give an explicit formula of linear sliding surfaces guaranteeing the quadratic stability of the reduced order equivalent system dynamics restricted to the sliding surfaces we also give sufficient conditions for the existence of linear sliding surfaces insuring some pole clustering constraints thus one can easily design vsc laws for a class of uncertain systems with mismatched uncertainties which are difficult to deal with by using conventional vsc design methods finally we give a design example

stability analysis and bang bang sliding control of a class of single input bilinear systems introduces a bang bang sliding control of a class of single input bilinear systems the sliding function is chosen via the well known pole assignment method for linear time invariant systems importantly the bang bang sliding control generates a reaching and sliding region and a stable sliding region each expressed by a set of linear inequalities both regions comprise the equilibrium point shown to be asymptotically stable however the stability analysis is processed under the limitation that the system state should be initially located in the reaching and sliding region two numeric examples are used for demonstration

variable structure control using system decomposition an approach to the design of large variable structure systems subject to control bounds is introduced the method includes a switching hyperplane design based on generalized inverses and system decomposition to ensure reaching the hyperplane and achieving a sliding condition the control is switched between local equivalent control and bounded corrective control the design of the corrective control component is completed using system decomposition into smaller subsystems an estimate of the domain of attraction corresponding to the bounded control is obtained and used to select the appropriate controller bounds the method is illustrated using a fifth order numerical example

analysis and design of variable structure systems using a geometric approach multivariable variable structure systems in the sliding mode are studies using a geometric approach the properties of system order reduction and disturbance rejection are proved using projector theory new design methods for the choice of switching hyperplanes are derived for the closed loop eigenvalue eigenvector assignment problem

a unifying design of sliding mode and classical controllers a theory is given to unify output sliding mode control and classical control the idea is based on defining the sliding variable in such a way that once the system begins to slide the classical controller transfer function is realized this idea leads to the development of a hybrid sliding and classical controller which retains the merits of both types of controllers but eliminates their respective limitations the proposed method is robust and applies to non minimum phase single input single output siso systems no state measurement is required

estimation of sliding mode domains of uncertain variable structure systems with bounded controllers the problem of estimating regions of asymptotic stability ras of uncertain dynamical systems with bounded controllers and the sliding model requirement is discussed the notion of combining different lyapunov functions is utilized different lyapunov functions are found and for each lyapunov function an estimate of the ras is found different regions are combined in order to find an improved ras the combined region is not in general a convex region and cannot be found analytically by using just a single lyapunov function a class of uncertain linear time invariant multivariable control systems is considered and a transformation for decoupling the system into two subsystems is used the transformation simplifies the ras estimation problem stability domains are estimated for a linear variable structure control system with sliding mode performance in the case of a discontinuous bounded controller

unified scaling law in the coherent noise model abstract the waiting time distribution between successive events and the unified scaling law is studied using the coherent noise model it is shown that although this model generates uncorrelated event sizes and does not exhibit criticality it still provides the unified scaling law we argue the role of characteristic kink observed in the unified scaling law and the meaning of the parameter c used to fix the peak of the kink to unity our results indicate that the parameter c is indeed a physical quantity localizing the end of the linear tendency in the scaling law which corresponds to the completion of the dominance of correlated events in time

computation of the zeros and zero directions of linear multivariable systems a new geometric method of calculating multivariable system zeros and zero directions is presented by considering a particular choice of state feedback control law this control law has been motivated by the study of variable structure systems in the sliding mode the cases of singular and non singular cb have been treated separately

brief sliding mode controllers design for linear discrete time systems with matching perturbations a simple methodology for designing sliding mode controllers is proposed in this paper for a class of linear multi input discrete time systems with matching perturbations if the functions of perturbations are bounded the proposed controllers cannot only satisfy the discrete sliding condition but also drive the state trajectories toward a small bounded region whose bound is dependent on the magnitude of perturbations in order to stabilize the closed loop system the determination of designed parameter of controllers is discussed neither chattering phenomenon will occur nor the knowledge of upper bound of perturbations is required numerical example is also given for demonstrating the feasibility of the proposed control scheme

estimation of asymptotic stability region and sliding domain of uncertain variable structure systems with bounded controllers this work is concerned with the problem of estimating the asymptotic stability region and sliding domain of uncertain variable structure systems with bounded controllers the properties of the decoupling theorem and strictly positive real function are used to incorporate with the lyapunov stability to estimate the stability region and the attractive region of the sliding mode it will be shown that the special state transformation used in previous work is no longer needed obviously our result is directly related to the original system states

a modified variable structure controller abstract it has long been known that one of the difficulties with the conventional variable structure systems vss design is the chattering problem in this paper a modified variable structure controller vsc is proposed to alleviate this problem at the same time retaining the benefits achieved in conventional vss design namely fast response good robustness and disturbance rejection

dynamic output feedback variable structure control for system stabilization simplex type dynamic output feedback variable structure control for stabilizing multivariable linear time invariant plants is analysed two types of controllers are considered a compensator type and an observer type it is shown that while the compensator type controller stabilizes the plant only if the plant is minimum phase the observer type controller does not face the same restriction the design of low order less than the order of the plant observer type controllers is also considered

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

output feedback variable structure control design this paper describes a procedure for variable structure output feedback control vsofc for a class of multivariable linear time invariant systems the equivalent control method of utkin in the output feedback mode yields a reduced order system exhibiting output feedback equivalent dynamics using the kimura davison sufficient conditions for pole placement using output feedback a sufficient condition is derived to assign the eigenvalues of the reduced order system the observability and controllability of the reduced system is discussed two switching surface design algorithms are introduced based on both a static output feedback and on an eigenstructure assignment techniques a control design is discussed as well

an lmi based switching surface design method for a class of mismatched uncertain systems in this note we consider the problem of designing linear sliding surfaces for multivariable uncertain systems with mismatched uncertainties in the state matrix in terms of linear matrix inequalities we give a new invariance condition guaranteeing the existence of a linear switching surface such that the reduced order equivalent sliding mode dynamics restricted to the switching surface is not only stable but completely invariant to mismatched uncertainties we also give an explicit formula of such linear switching surfaces together with a design example

estimation of the asymptotic stability region of uncertain systems with bounded sliding mode controllers in this note the problem of estimating the asymptotic stability region asr of uncertain systems with bounded sliding mode controllers is considered to simplify the problem the authors use a state transformation and choose a suitable lyapunov function for the transformed control system using the lyapunov function the authors estimate the asr and show the exponential stability of the closed loop control system in the region also by an example the authors show that for a certain class of uncertain dynamical systems with bounded sliding mode controllers their method gives a more improved estimate of the asr

robust control of linear discrete time systems abstract given a linear discrete time system with additive disturbances a general methodology for designing a stable control algorithm is shown the approach is lyapunov based with certain liberty of tailoring the candidate function for a given problem comparison to the existing solutions based on a variable structure approach is given

robust hyperplane design in multivariable variable structure control systems the problem of designing a stable sliding mode giving robust performance of a variable structure control system is studied a set of specified eigenvalues is assigned to the closed loop feedback system during the sliding mode the design philosophy seeks to build on the known robustness and invariance properties associated with the use of discontinuous or continuous non linear controls by assigning the eigenvectors associated with the sliding mode eigenvalues in a manner that leads to a robust control scheme a previously developed canonical form for the hyperplane design problem is again employed the design technique centres on a recently published eigenvalue assignment algorithm and is illustrated by the inclusion of an example

new methods to design an integral variable structure controller this paper presents two methods to design a single input single output integral variable structure system once an overall transfer function is selected two simple ways are presented to determine the required switching surface and the control function the proposed methods not only avoid transforming the original plant into a companion form but also enable the resulting control system to achieve the asymptotic tracking for a step reference signal and disturbance rejection for a bounded disturbance signal simultaneously two examples are given to illustrate the effectiveness of the proposed methods

sliding mode design via quadratic performance optimization with pole clustering constraint we consider a novel method for designing the sliding mode that minimizes the quadratic performance while keeping a pole clustering constraint our approach is based on the manipulations of linear matrix inequalities lmis imposed by the design objectives for this purpose we newly propose lmi conditions for the quadratic performance optimization and the pole clustering problem respectively in a full order state then they are combined in the lmi framework that is typically devised for the sliding mode design in the convex form an effort is made to reduce the generic conservatism by allowing different lyapunov matrices in addition a class of polytope uncertain systems is addressed to illustrate the advantages of the proposed method

universal local versus unified global scaling laws in the statistics of seismicity the unified scaling law for earthquakes proposed by bak christensen danon and scanlon is shown to hold worldwide as well as for areas as diverse as japan new zealand spain or new madrid the scaling functions that account for the rescaled recurrence time probability densities show a power law behavior for long times with a universal exponent about minus 2 2 another decreasing power law governs short times but with an exponent that may change from one area to another this is in contrast with a local time homogenized version of bak et al s procedure which seems to present a universal scaling behavior

practical stabilization of nonlinear uncertain dynamic systems with bounded controllers we address the problem of practically stabilizing nonlinear and or uncertain continuous time dynamic systems when the norm of the control input is subject to a given fixed bound we consider a class of systems whose nominal part is linear and whose nonlinear uncertain part does not satisfy the matching condition in this treatment no statistical information regarding the uncertainties is needed only a norm bound on each nonlinear uncertain quantity is assumed we propose a bounded state feedback controller whose design incorporates elements from both variable structure control theory and the deterministic control methodology using the second method of lyapunov we obtain sliding domains regions of uniform ultimate boundedness and estimates of the domain of attraction for systems employing this controller the closed loop stability analysis as well as the design of the controller is facilitated by introducing a special coordinate transformation the approach is illustrated by a numerical example

convergence properties of policy iteration this paper analyzes asymptotic convergence properties of policy iteration in a class of stationary infinite horizon markovian decision problems that arise in optimal growth theory these problems have continuous state and control variables and must therefore be discretized in order to compute an approximate solution the discretization may render inapplicable known convergence results for policy iteration such as those of puterman and brumelle math oper res 4 1979 pp 60 69 under certain regularity conditions we prove that for piecewise linear interpolation policy iteration converges quadratically also under more general conditions we establish that convergence is superlinear we show how the constants involved in these convergence orders depend on the grid size of the discretization these theoretical results are illustrated with numerical experiments that compare the performance of policy iteration and the method of successive approximations

on stabilization of large scale control systems using variable structure systems theory a new approach to control design for large scale systems using variable structure systems vss theory is presented multilevel control concepts arc used to decompose a large control problem into a two level algorithm such that each subsystem is stabilized with local discontinuous controllers and higher level corrective control is designed to take into account the effect of interactions among the subsystems two numerical examples are discussed as illustrations

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

an adaptive variable structure model following control design for robot manipulators an adaptive variable structure model following control avsmfc design is proposed for trajectory tracking in a nonlinear robot system which ensures the stability of the intersection of the surfaces without necessarily stabilizing each individual one the approach avoids the difficulties linked to the strict positive realness requirement in traditional amfc by taking advantage of the inherent positive definiteness of the manipulator s inertia matrix and is easily extendible to a higher number of links the controller does not require any knowledge of a nonlinear robotic system and does not necessarily need the occurrence of a sliding mode at each individually stable discontinuity surface it thus greatly reduces the complexity of design in the closed loop the joint angles asymptotically converge to the reference trajectory with a prescribed transient response chattering is eliminated by restricting the state of the system to slide within a boundary layer rather than along the intersection of the surfaces

missing data in aftershock sequences explaining the deviations from scaling laws in this paper we extend the branching aftershock sequence model to study the role of missing data at short times and small amplitudes after a mainshock we apply this model which contains three parameters characterizing the missing data to the magnitude and temporal statistics of four aftershock sequences in california we find that the observed time dependent deviations of the frequency magnitude scaling from the gutenberg richter power law dependency can be described quantitatively by the model we also show that for the same set of parameters the model is able to explain quantitatively the observed magnitude dependent deviations of the temporal decay of aftershocks from omori s law in addition we show that the same sets of data can also reproduce quite well the various functional forms of the probability density functions of the return times between consecutive events with magnitudes above a prescribed threshold as well as the violation of scaling at short and intermediate time scales

accuracy of numerical solutions using the euler equation residuals in this paper we derive sorne asymptotic properties on the accuracy of numerical solutions we silow tilat the approximation error of the policy function is of the same order of magnitude as the size of the euler equation residuals moreover for bounding this approximation error tile most relevant parameters are the discount factor and the curvature of the return function these findings provide theoretical foundations for the construction of tests that can assess the performance of alternative computational methods

robust control synthesis for uncertain nonlinear dynamical systems this paper addresses the problem of robust output feedback controller design for uncertain and nonlinear dynamic systems first a robust state feedback nonlinear control law is synthesized this control strategy practically stabilizes the closed loop system then a state estimator for the nonlinear uncertain plant is designed and its performance analyzed finally the control law and the estimator are combined and the practical stability region is estimated the results are illustrated by their application to a benchmark problem for robust control design proposed by wie and bernstein 1990

a variable structure control with simple adaptation laws for upper bounds on the norm of the uncertainties in deterministic design of feedback controllers for uncertain dynamical systems upper bounds on the norm of the uncertainties are an important clue to guarantee the stability of the closed loop system sometimes however these upper bounds may not be easily obtained because of the complexity of structure of the uncertainties simple adaptation laws for upper bounds on the norm of the uncertainties are proposed and these adaptive upper bounds are used to design a variable structure control system that guarantees the asymptotic stability of uncertain dynamical systems

q exponential relaxation of the expected avalanche size in the coherent noise model abstract recently sarlis and christopoulos 2012 the threshold distribution function p t h r e s k x of the coherent noise model for infinite number of agents after the k th avalanche has been studied as a function of k and hence natural time an analytic expression of the expectation value e s k 1 for the size s k 1 of the next avalanche has been obtained in the case that the coherent stresses are exponentially distributed with an average value here by using a statistical ensemble of initially identical systems we investigate the relaxation of the average e s k 1 versus k for k values smaller than k max f the numerical results indicate that e s k 1 collapses to the q exponential tsallis 1988 as a function of k for larger k values the ensemble average can be effectively described by the time average threshold distribution function obtained by newman and sneppen 1996 an estimate k 0 f k max f of this transition is provided this ensemble of coherent noise models may be considered as a simple prototype following q exponential relaxation the resulting q values are compatible with those reported in the literature for the coherent noise model