complexity of stability and controllability of elementary hybrid systems in this paper we consider simple classes of nonlinear systems and prove that basic questions related to their stability and controllability are either undecidable or computationally intractable np hard as a special case we consider a class of hybrid systems in which the state space is partitioned into two halfspaces and the dynamics in each halfspace correspond to a different linear system

realisation and estimation of piecewise linear output error models piecewise linear systems in input output form can have different switching schedules in this article two categories instant and delayed switching are analysed even though a general piecewise linear state space model cannot be converted into input output form it is shown that it is possible to find state space models representing instant and delayed switching in addition a prediction error minimisation pem method for piecewise linear output error predictors is derived and it is concluded that the instant switching model candidate is not necessarily the most suitable for the parameter estimation procedure

balanced topological field theories we describe a class of topological field theories called balanced topological field theories these theories are associated to moduli problems with vanishing virtual dimension and calculate the euler character of various moduli spaces we show that these theories are closely related to the geometry and equivariant cohomology of iterated superspaces that carry two differentials we find the most general action for these theories which turns out to define morse theory on field space we illustrate the constructions with numerous examples finally we relate these theories to topological sigma models twisted using an isometry of the target space

a clustering technique for the identification of piecewise affine systems we propose a new technique for the identification of discrete time hybrid systems in the piecewise affine pwa form this problem can be formulated as the reconstruction of a possibly discontinuous pwa map with a multi dimensional domain in order to achieve our goal we provide an algorithm that exploits the combined use of clustering linear identification and pattern recognition techniques this allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures moreover the clustering step used for classifying the datapoints is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

an algorithm for the n lyapunov exponents of an n dimensional unknown dynamical system abstract an algorithm for estimating lyapunov exponents of an unknown dynamical system is designed the algorithm estimates not only the largest but all lyapunov exponents of the unknown system the estimation is carried out by a multivariate feedforward network estimation technique we focus our attention on deterministic as well as noisy system estimation the performance of the algorithm is very satisfactory in the presence of noise as well as with limited number of observations

a multilayer neural network with piecewise linear structure and back propagation learning a multilayer neural network which is given a two layer piecewise linear structure for every cascaded section is proposed the neural networks have nonlinear elements that are neither sigmoidal nor of a signum type each nonlinear element is an absolute value operator it is almost everywhere differentiable which makes back propagation feasible in a digital setting both the feedforward signal propagation and the backward coefficient update rules belong to the class of regular iterative algorithms this form of neural network specializes in functional approximation and is anticipated to have applications in control communications and pattern recognition

the asymptotic distribution of nonparametric estimates of the lyapunov exponent for stochastic time series this paper derives the asymptotic distribution of a smoothing based estimator of the lyapunov exponent for a stochastic time series under two general scenarios in the first case we are able to establish root t consistency and asymptotic normality while in the second case which is more relevant for chaotic processes we are only able to establish asymptotic normality at a slower rate of convergence we provide consistent confidence intervals for both cases we apply our procedures to simulated data

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

a bounded error approach to piecewise affine system identification this paper proposes a three stage procedure for parametric identification of piecewise affine autoregressive exogenous pwarx models the first stage simultaneously classifies the data points and estimates the number of submodels and the corresponding parameters by solving the partition into a minimum number of feasible subsystems min pfs problem for a suitable set of linear complementary inequalities derived from data second a refinement procedure reduces misclassifications and improves parameter estimates the third stage determines a polyhedral partition of the regressor set via two class or multiclass linear separation techniques as a main feature the algorithm imposes that the identification error is bounded by a quantity spl delta such a bound is a useful tuning parameter to trade off between quality of fit and model complexity the performance of the proposed pwa system identification procedure is demonstrated via numerical examples and on experimental data from an electronic component placement process in a pick and place machine

superconformal invariance and the geography of four manifolds the correlation functions of supersymmetric gauge theories on a four manifold x can sometimes be expressed in terms of topological invariants of x we show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four manifold topology in particular in the example of gauge group su 2 with one doublet hypermultiplet we derive a theorem relating classical topological invariants such as the euler character and signature to sum rules for seiberg witten invariants a short account of this paper can be found in 1

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

realization theory for linear hybrid systems the paper develops realization theory for linear hybrid systems i e hybrid systems in continuous time without guards whose continuous dynamics is determined by linear control systems and whose discrete dynamics is determined by a finite state automaton we will formulate necessary and sufficient conditions for the existence of a realization we will show that minimality is equivalent to observability and span reachability and that minimal systems are isomorphic in turn observability and span reachability can be characterized by rank conditions

supersymmetric yang mills theory on a four manifold by exploiting standard facts about n 1 and n 2 supersymmetric yang mills theory the donaldson invariants of four manifolds that admit a kahler metric can be computed the results are in agreement with available mathematical computations and provide a powerful check on the standard claims about supersymmetric yang mills theory

a statistical framework for testing chaotic dynamics via lyapunov exponents abstract one common criticism of the existing lyapunov exponents algorithms is the absence of a distributional theory which provides a framework for the statistical hypothesis testing for the calculated lyapunov exponents this paper presents a methodology to calculate the empirical distributions of lyapunov exponents by using a bootstrapping technique the methodology of the paper provides a formal test of chaos under the null hypothesis the numerical examples show that the method works well with small data sets

new resampling method to assess the accuracy of the maximal lyapunov exponent estimation abstract in this paper we propose a novel method to assess standard errors and confidence intervals for the maximal lyapunov exponent estimated on time continuous chaotic systems the method is based on resampling the original series by means of spline interpolation in the time domain in such a way new time series of increased size are obtained and the sample distribution of the estimators is constructed the method is explained and tested on the basis of computer simulations both for clean and noisy series we give evidence that the distribution of the maximal lyapunov exponent calculated by this method fairly agrees with the one obtained by true series with different initial conditions an empirical criterion for the choice of the parameters of the resampling is also suggested

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

the bootstrap and lyapunov exponents in deterministic chaos inasmuch as lyapunov exponents provide a necessary condition for chaos in a dynamical system confidence bounds on estimated lyapunov exponents are of great interest estimates derived either from observations or from numerical integrations are limited to trajectories of finite length and it is the uncertainties in the distribution of these finite time lyapunov exponents which are of interest within this context a bootstrap algorithm for quantifying sampling uncertainties is shown to be inappropriate for multiplicative ergodic statistics of deterministic chaos this result remains unchanged in the presence of observational noise as originally proposed the algorithm is also inappropriate for general nonlinear stochastic processes a modified version is presented which may prove of value in the case of stochastic dynamics a new approach towards quantifying the minimum duration of observations required to estimate global lyapunov exponents is suggested and is explored in a companion paper

an attractor in a solar time series a time series analysis of observed solar radio pulsations suggests that there must be a low dimensional attractor the power spectrum cannot be interpreted as a superposition of periodic components estimates of the maximum lyapunov exponent and of the kolmogorov entropy give some indications for a deterministic chaos in order to study the limitations inherent in small data samples we include data from the lorenz model and an artificial noise record consequences for the physical modelling of the pulsation event are discussed

euler number of instanton moduli space and seiberg witten invariants we show that a partition function of topological twisted n 4 yang mills theory is given by seiberg witten invariants on a riemannian four manifolds under the condition that the sum of the euler number and the signature of the four manifolds vanishes the partition function is the sum of the euler number of instanton moduli space when it is possible to apply the vanishing theorem also we obtain a relation of the euler number labeled by the instanton number k with seiberg witten invariants all calculations in this article are done without assuming duality

testing chaotic dynamics via lyapunov exponents abstract this paper presents a bootstrap based test statistic for testing the presence of chaotic dynamics from data by using the lyapunov exponents in particular a one sided test statistic in gencay s gencay physica d 89 1996 261 266 framework is designed and its small sample properties are tested on the henon map the numerical examples show that the test statistic has desirable small sample properties

identification of piecewise affine systems via mixed integer programming this paper addresses the problem of identification of hybrid dynamical systems by focusing the attention on hinging hyperplanes and wiener piecewise affine autoregressive exogenous models in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron in particular we provide algorithms based on mixed integer linear or quadratic programming which are guaranteed to converge to a global optimum for the special case where the estimation data only seldom switches between the different submodels we also suggest a way of trading off between optimality and complexity by using a change detection approach

realization theory of discrete time linear switched systems the paper presents realization theory of discrete time linear switched systems we present necessary and sufficient conditions for an input output map to admit a discrete time linear switched system realization in addition we present a characterization of minimality of discrete time linear switched systems in terms of reachability and observability further we prove that minimal realizations are unique up to isomorphism we also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state space representation from input output data the paper uses the theory of rational formal power series in non commutative variables

the orientation of yang mills moduli spaces and 4 manifold topology si x est une 4 variete lisse orientee fermee dont la forme intersection q h 2 x z torsion z est definie negative alors la forme est equivalente sur les entiers a la forme standard 1 1 1 soit x une 4 variete lisse orientee fermee avec les trois proprietes suivantes 1 h 1 x z n a pas de 2 torsion 2 la forme intersection q sur h 2 x torsion a une partie positive de rang 1 ou 2 3 la forme intersection est paire alors q est equivalente sur les entiers a une des formes 01 10 01 10 01 10

hinging hyperplanes for regression classification and function approximation a hinge function y h x consists of two hyperplanes continuously joined together at a hinge in regression prediction classification pattern recognition and noiseless function approximation use of sums of hinge functions gives a powerful and efficient alternative to neural networks with computation times several orders of magnitude less than is obtained by fitting neural networks with a comparable number of parameters a simple and effective method for finding good hinges is presented

on the input output representation of piecewise affine state space models this paper addresses the conversion of discrete time piecewise affine pwa state space models into input output form necessary and sufficient conditions for the existence of equivalent input output representations of a given pwa state space model are derived connections to the observability properties of pwa models are investigated under a technical assumption it is shown that every finite time observable pwa model admits an equivalent input output representation when an equivalent input output model exists a constructive procedure is presented to derive its equations several examples illustrate the proposed results

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

identification of piecewise affine systems based on statistical clustering technique this paper is concerned with the identification of a class of piecewise affine systems called a piecewise affine autoregressive exogenous pwarx model the pwarx model is composed of arx sub models each of which corresponds to a polyhedral region of the regression space under the temporary assumption that the number of sub models is known a priori the input output data are collected into several clusters by using a statistical clustering algorithm we utilize support vector classifiers to estimate the boundary hyperplane between two adjacent regions in the regression space in each cluster the parameter vector of the sub model is obtained by the least squares method it turns out that the present statistical clustering approach enables us to estimate the number of sub models based on the information criteria such as caic and mdl the estimate of the number of sub models is performed by applying the identification procedure several times to the same data set after having fixed the number of sub models to different values finally we verify the applicability of the present identification method through a numerical example of a hammerstein model

a statistical approach to estimate the lyapunov spectrum in disc brake squeal abstract the estimation of squeal propensity of a brake system from the prediction of unstable vibration modes using the linear complex eigenvalue analysis cea in the frequency domain has its fair share of successes and failures while the cea is almost standard practice for the automotive industry time domain methods and the estimation of lyapunov spectra have not received much attention in brake squeal analyses one reason is the challenge in estimating the true lyapunov exponents and their discrimination against spurious ones in experimental data a novel method based on the application of the eckmann ruelle matrices is proposed here to estimate lyapunov exponents by using noise in a statistical procedure it is validated with respect to parameter variations and dimension estimates by counting the number of non overlapping confidence intervals for lyapunov exponent distributions obtained by moving a window of increasing size over bootstrapped same length estimates of an observation function a dispersion measure s width is calculated and fed into a bayesian beta binomial model results obtained using this method for benchmark models of white and pink noise as well as the classical henon map indicate that true lyapunov exponents can be isolated from spurious ones with high confidence the method is then applied to accelerometer and microphone data obtained from brake squeal tests estimated lyapunov exponents indicate that the pad s out of plane vibration behaves quasi periodically on the brink to chaos while the microphone s squeal signal remains periodic

application of bootstrap to detecting chaos in financial time series abstract a moving blocks bootstrap procedure is used to investigate the dynamics of nominal exchange rates and the return rates of the us dollar against the polish zloty the problem if these financial time series exhibit chaotic behavior is undertaken a possibility of detecting the presence of a positive lyapunov exponent is studied

estimating the lyapunov exponent of a chaotic system with nonparametric regression abstract we discuss procedures based on nonparametric regression for estimating the dominant lyapunov exponent 1 from time series data generated by a nonlinear autoregressive system with additive noise for systems with bounded fluctuations 1 0 is the defining feature of chaos thus our procedures can be used to examine time series data for evidence of chaotic dynamics we show that a consistent estimator of the partial derivatives of the autoregression function can be used to obtain a consistent estimator of 1 the rate of convergence we establish is quite slow a better rate of convergence is derived heuristically and supported by simulations simulation results from several implementations one local thin plate splines and three global neural nets radial basis functions and projection pursuit are presented for two deterministic chaotic systems local splines and neural nets yield accurate estimates of the lyapunov exponent however the spline method is sensitive to the choice of the emb

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

noncommutative cohomological field theory and gms soliton we show that it is possible to construct a quantum field theory that is invariant under the translation of the noncommutative parameter this is realized in a noncommutative cohomological field theory as an example a noncommutative cohomological scalar field theory is constructed and its partition function is calculated the partition function is the euler number of gopakumar minwalla and strominger gms soliton space