how initial conditions affect generalization performance in large networks generalization is one of the most important problems in neural network research it is influenced by several factors in the network design such as network size weight decay factor and others we show here that the initial weight distribution for gradient decent training algorithms is one other factor that influences generalization the initial conditions guide the training algorithm to search particular places of the weight space for instance small initial weights tend to result in low complexity networks and therefore can effectively act as a regularization factor we propose a novel network complexity measure which is helpful in shedding insight into the phenomenon as well as in studying other aspects of generalization

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

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

chattering reduction and error convergence in the sliding mode control of a class of nonlinear systems to reduce chattering in sliding mode control a boundary layer around the switching surface is used and a continuous control is applied within the boundary the effects of various control laws within the boundary layer on chattering and error convergence in different systems are studied new functions for chattering reduction and error convergence inside the boundary layer are proposed which are discontinuous in magnitude but not in sign the internal model principle has been used to generalize the design for the class of nonlinear systems being considered

dynamic output feedback h control synthesis for discrete time t s fuzzy systems via switching fuzzy controllers this paper studies the problem of dynamic output feedback h control of discrete time takagi sugeno t s fuzzy systems via a switched dynamic parallel distributed compensation dpdc scheme the considered switched dpdc control structure is determined based on the values of membership functions some new properties of membership functions are obtained and exploited to derive sufficient conditions for designing switching dpdc controllers that guarantee the stability and h performances of closed loop nonlinear systems the design conditions are given in terms of solvability of a set of linear matrix inequalities lmis it is shown that the new approach provides results at least as good as the corresponding dpdc methods a numerical example is given to illustrate the effectiveness of the proposed approach

robust control with structure perturbations the problem of making a given stabilizing controller robust so that the closed loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant the problem is treated in the state space and transfer function domains in the state space domain a stability hypersphere is determined in the parameter space using lyapunov theory the radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere in the transfer function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial the design algorithms are illustrated by examples

fluctuations of finite time stability exponents in the standard map and the detection of small islands some statistical properties of finite time stability exponents in the standard map can be estimated analytically the mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants whereas the mean carries information about the strength of the interaction and only indirect information about dynamical correlations the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength in particular the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time scalings the average of the stability matrix inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter the usual accelerator modes and the small islands created through double saddle node bifurcations which come halfway between the positions in interaction strength of the usual accelerator modes are clearly visible in the variance whose time scaling is capable of detecting the presence of islands as small as 0 01 of the phase space we study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions we also discuss the nature of this local approximation in some detail

output feedback model reference sliding mode control of uncertain multivariable systems this note considers the robust output tracking problem using a model reference sliding mode controller for linear multivariable systems of relative degree one it is shown that the closed loop system is globally exponentially stable and the performance is insensitive to bounded input disturbances and parameter uncertainties the strategy is based on output feedback unit vector control to generate sliding mode the only required a priori information about the plant high frequency gain matrix k sub p is the knowledge of a matrix s sub p such that k sub p s sub p is hurwitz which relaxes the positive definiteness requirement usually needed by other methods

fuzzy adaptive output feedback control for mimo nonlinear systems in this paper two observer based adaptive fuzzy output feedback control schemes are presented for a class of uncertain continuous time multi input multi output mimo nonlinear dynamics systems whose states are not available within these schemes fuzzy logic systems are employed to approximate the plant s unknown nonlinear functions and then the state observer is designed for estimating the states of the plant upon which a fuzzy adaptive output feedback controller is firstly investigated in order to overcome the controller singularity problem and relax the requirement of bounding parameter values a second modified fuzzy adaptive output feedback controller is proposed by using a regularized inverse and a robustifying control term all parameter adaptive laws and robustifying control terms are derived based on lyapunov stability analysis so that convergence to zero of tracking errors and boundedness of all signals in the closed loop system can be guaranteed simulations performed on a two link robot manipulator illustrate the approach and exhibit its performance

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

anomalous convergence of lyapunov exponent estimates numerical experiments reveal that estimates of the lyapunov exponent for the logistic map ital x sub ital t 1 ital f ital x sub ital t 4 ital x sub ital t 1 minus ital x sub ital t are anomalously precise they are distributed with a standard deviation that scales as 1 ital n where ital n is the length of the trajectory not as 1 radical ital n the scaling expected from an informal interpretation of the central limit theorem we show that this anomalous convergence follows from the fact that the logistic map is conjugate to a constant slope map the lyapunov estimator is just one example of a chaotic walk we show that whether or not a general chaotic walk exhibits anomalously small variance depends only on the autocorrelation of the chaotic process

on multi input chattering free second order sliding mode control a solution to the problem of eliminating the chattering effect which is always associated with practical implementations of variable structure control is presented with reference to a class of uncertain multi input nonlinear systems the solution procedure relies on the application of an original control approach capable of enforcing a second order sliding mode i e a sliding regime on a surface s x t 0 in the system state space with s spl dot x t identically equal to zero a regime enforced by a control signal depending on s x t but directly acting only on s spl uml x t such an approach in its original formulation only applies to single input nonlinear systems with particular types of uncertainties in the present paper its validity is extended to multi input nonlinear systems characterized by uncertainties of more general nature covering a wide class of real processes

static output feedback controller design for fuzzy systems an ilmi approach this paper examines the problem of static output feedback control of a takagi sugeno ts fuzzy system the existence of a static output feedback control law is given in terms of the solvability of bilinear matrix inequalities an iterative algorithm based on the linear matrix inequality is proposed to compute the static output feedback gain to reduce the conservatism of the design the structural information of the membership function of the fuzzy rules is incorporated numerical examples are used to illustrate the validity of the methods

robust h control analysis and synthesis for takagi sugeno general uncertain fuzzy systems abstract in this paper robust h control for takagi sugeno general uncertain fuzzy systems where uncertainties come into all the system matrices is considered the main result is to establish equivalent relationships between quadratic stability and quadratic stability with h disturbance attenuation of general uncertain fuzzy systems and h control for fuzzy systems without uncertainty these relationships imply that quadratically stabilizing controllers and quadratically stabilizing controllers with h disturbance attenuation for general uncertain fuzzy systems can be obtained by designing h controllers for fuzzy systems without uncertainties we first give sufficient conditions for the h norm being less than a given number we then consider a general h problem with output feedback controllers and give a design method of h controllers and sufficient conditions which guarantee the required h performance of the closed loop system this design method can be applied to quadratic stabilizing controllers and quadratic stabilizing controllers with h disturbance attenuation for general uncertain fuzzy systems next we analyze quadratic stability and quadratic stability with h disturbance attenuation of general uncertain fuzzy systems and establish equivalent relationships to h control of fuzzy systems without uncertainties based on these relationships we design robust controllers for uncertain fuzzy systems finally examples are given to illustrate the theory

output feedback control of uncertain nonlinear systems using a self structuring adaptive fuzzy observer this paper describes the design of an output feedback controller based on an adaptive fuzzy observer for uncertain single input single output nonlinear dynamical systems we have focused on the realization of a minimal dynamic order for the adaptive fuzzy observer for this purpose we adopt an adaptive fuzzy observer in which no strictly positive real spr condition is needed and we combine a self structuring scheme with an on line estimation of fuzzy parameters by using this proposed scheme we can reduce the dynamic order of the adaptive output feedback fuzzy control system the lyapunov synthesis approach is used to guarantee a global uniform ultimate boundedness property of the state estimation error and tracking error as well as all other signals in the closed loop system no a priori knowledge of an upper limits on the uncertainties including optimal parameters and modeling errors is required the theoretical results are illustrated through simulation examples

approximation dimension reduction and nonconvex optimization using linear superpositions of gaussians this paper concerns neural network approaches to function approximation and optimization using linear superposition of gaussians or what are popularly known as radial basis function rbf networks the problem of function approximation is one of estimating an underlying function f given samples of the form y sub i x sub i i 1 2 n with y sub i f x sub i when the dimension of the input is high and the number of samples small estimation of the function becomes difficult due to the sparsity of samples in local regions the authors find that this problem of high dimensionality can be overcome to some extent by using linear transformations of the input in the gaussian kernels such transformations induce intrinsic dimension reduction and can be exploited for identifying key factors of the input and for the phase space reconstruction of dynamical systems without explicitly computing the dimension and delay they present a generalization that uses multiple linear projections onto scalars and successive rbf networks mlprbf that estimate the function based on these scaler values they derive some key properties of rbf networks that provide suitable grounds for implementing efficient search strategies for nonconvex optimization within the same framework

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

robust h fuzzy static output feedback control of t s fuzzy systems with parametric uncertainties this paper addresses robust h fuzzy static output feedback control problem for t s fuzzy systems with time varying norm bounded uncertainties sufficient conditions for synthesis of a fuzzy static output feedback controller for t s fuzzy systems are derived in terms of a set of linear matrix inequalities lmis in comparison with the existing literatures the proposed approach not only simplifies the design procedure but also achieves a better h performance three drawbacks existing in the previous papers such as coordinate transformation same output matrices and bmi problem have been eliminated the effectiveness of the proposed design method is demonstrated by an example for the control of a truck trailer system

output feedback stabilization of uncertain fuzzy systems using variable structure system approach based on the variable structure system theory an output feedback fuzzy control design method is developed for a class of uncertain nonlinear systems that can be represented by a t s takagi sugeno fuzzy model the uncertain fuzzy systems under consideration may have mismatched uncertainties in the state matrix as the local controller a compensator based sliding mode output feedback controller with a nonlinear discontinuous term is used in terms of linear matrix inequalities lmis a sufficient condition is derived for the existence of the fuzzy controller guaranteeing a stable sliding motion it is shown that the compensator gain and the switching surface parameter matrix can be characterized in terms of the solution of the lmi existence condition finally an lmi based design algorithm and numerical design examples are given

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

observer based fuzzy adaptive control for strict feedback nonlinear systems in this paper a new fuzzy adaptive control approach is developed for a class of siso strict feedback nonlinear systems with unmeasured states using fuzzy logic systems to approximate the unknown nonlinear functions a fuzzy adaptive observer is introduced for state estimation as well as system identification under the framework of the backstepping design fuzzy adaptive output feedback control is constructed recursively it is proven that the proposed fuzzy adaptive control approach guarantees the semi global boundedness property for all the signals and the tracking error to a small neighborhood of the origin simulation studies are included to illustrate the effectiveness of the proposed approach

state observation of nonlinear uncertain dynamical systems this note proposes new types of observers for nonlinear dynamical systems subjected to bounded nonlinearities or uncertainties the design of these observers utilizes techniques related to variable structure systems theory a measure for the rate at which the estimates converge to the actual states is derived

fuzzy observer based fuzzy control design for nonlinear systems with persistent bounded disturbances to date nonlinear l gain control problems have not been solved by the conventional control methods for the nonlinear dynamic systems with persistent bounded disturbances this study introduces a fuzzy control design to deal with the nonlinear l gain output feedback control problem first the takagi and sugeno t s fuzzy model is employed to approximate the nonlinear dynamic system next based on the fuzzy model a fuzzy observer based fuzzy controller is developed to minimize the upper bound of l gain of the closed loop system under some linear matrix inequality lmi constraints therefore the nonlinear l gain control problem is transformed into a suboptimal control problem i e to minimize the upper bound of the l gain of the closed loop system subject to some lmi constraints in this situation the nonlinear l gain output feedback control problem can be easily solved by a lmi based optimization method the proposed methods which efficiently attenuate the peak of perturbation due to persistent bounded disturbances extend the l gain control problems from linear dynamic systems to nonlinear dynamic systems

stabilization of uncertain fuzzy time delay systems via variable structure control approach in view of a recent new application of variable structure control vsc to the stabilization problem for takagi sugeno t s fuzzy models this paper aims to study the stabilization of uncertain fuzzy time delay systems in t s fuzzy model via vsc approach there are mainly two features in this paper one lies in the incorporation of time delays both smooth and nonsmooth delays in which case lyapunov functionals and razumikhin theorem are required to solve the stabilization problem the other feature is that not only matched uncertainties but also mismatched uncertainties in the state variables are considered as a sequence the contribution of this paper consists of various control schemes proposed for the vsc design and the present results are in terms of linear matrix inequalities lmis an illustrative example is given to show the effectiveness of our various results

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

temporal evolution of generalization during learning in linear networks we study generalization in a simple framework of feedforward linear networks with n inputs and n outputs trained from examples by gradient descent on the usual quadratic error function we derive analytical results on the behavior of the validation function corresponding to the lms error function calculated on a set of validation patterns we show that the behavior of the validation function depends critically on the initial conditions and on the characteristics of the noise under certain simple assumptions if the initial weights are sufficiently small the validation function has a unique minimum corresponding to an optimal stopping time for training for which simple bounds can be calculated there exists also situations where the validation function can have more complicated and somewhat unexpected behavior such as multiple local minima at most n of variable depth and long but finite plateau effects additional results and possible extensions are briefly discussed

how to design a fuzzy adaptive controller based on observers for uncertain affine nonlinear systems this paper focuses on the construction of a fuzzy adaptive output feedback control based on any observer high gain hg observer sliding mode like observer etc for a class of single input single output siso uncertain or ill defined affine nonlinear systems indeed the corrective term of the proposed observer involves a well defined design function which is shown to be satisfied by the commonly used hg based observers namely for the usual hg observers and the sliding mode observers together with their implementable versions the design of the underlying update law as well as the robust control term is based on an appropriate filtering of the output tracking error this particularly allows to overcome the output observation error filtering or the necessity of the famous strictly positive real spr condition

output feedback lmi tracking control conditions with h criterion for uncertain and disturbed t s models this work concerns the tracking problem of uncertain takagi sugeno t s continuous fuzzy model with external disturbances the objective is to get a model reference based output feedback tracking control law the control scheme is based on a pdc structure a fuzzy observer and a h performance to attenuate the external disturbances the stability of the whole closed loop model is investigated using the well known quadratic lyapunov function the key point of the proposed approaches is to achieve conditions under a lmi linear matrix inequalities formulation in the case of an uncertain and disturbed t s fuzzy model this formulation facilitates obtaining solutions through interior point optimization methods for some nonlinear output tracking control problems finally a simulation is provided on the well known inverted pendulum testbed to show the efficiency of the proposed approach

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

conditions of output stabilization for nonlinear models in the takagi sugeno s form this paper deals with the control law design with an observer for models in the takagi sugeno s ts form in the general case where the nonlinearities and the variables involved are not measurable the results presented in this paper give sufficient conditions of stabilization for the global loop i e ts model stabilized with a state feedback and a fuzzy observer conditions are given in the form of linear matrix inequalities lmi problem

output tracking control of mimo fuzzy nonlinear systems using variable structure control approach the output tracking control problem for nonlinear systems in the presence of both parameter perturbations and external disturbances is studied our approach is based on the takagi sugeno t s fuzzy modeling method and a variable structure control vsc technique therefore the systems considered are not and need not be in the triangular and parametric strict feedback form which are prevalent among adaptive model following control for nonlinear systems or in the normal form which pervades almost all existing results in neuro fuzzy model following control approach we first study the problem of stabilization of t s fuzzy systems by using a vsc technique a method for the design of a switching surface based on linear matrix inequalities is developed and a stabilizing controller based on a reaching law concept in the presence of both parameter perturbations and external disturbances is proposed then the method is extended to design controllers for output tracking of t s fuzzy nonlinear systems in two cases i e systems which possess the so called strong passive subsystems and strong stable zero dynamics respectively finally illustrative examples are presented to demonstrate the whole design procedure from the original nonlinear systems to their fuzzification and finally to the realization of the desired controllers simulation results show that the goal of output tracking can be achieved by the proposed controllers

no free lunch for early stopping we show that with a uniform prior on models having the same training error early stopping at some fixed training error above the training error minimum results in an increase in the expected generalization error

lmi based nonlinear fuzzy observer controller design for uncertain mimo nonlinear systems based on the variable structure system theory we develop a nonlinear fuzzy observer controller design method for a class of uncertain nonlinear systems that can be represented by a takagi sugeno fuzzy model we use a sliding mode observer and a sliding mode controller as the local state observer and the local controller in terms of linear matrix inequalities lmis we derive sufficient conditions for the existence of the fuzzy observer and the fuzzy controller we also derive lmi conditions for the existence of the fuzzy observer and the fuzzy controller guaranteeing the alpha stability or generalized h2 hinfin performance constraints we show that the observer and the controller can be independently designed i e the separation property holds finally we give simple lmi based design algorithms together with an example

learning from examples in large neural networks a statistical mechanical theory of learning from examples in layered networks at finite temperature is studied when the training error is a smooth function of continuously varying weights the generalization error falls off asymptotically as the inverse number of examples by analytical and numerical studies of single layer perceptrons we show that when the weights are discrete the generalization error can exhibit a discontinuous transition to perfect generalization for intermediate sizes of the example set the state of perfect generalization coexists with a metastable spin glass state