measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

eigenvalue based characterization and test for positive realness of scalar transfer functions an eigenvalue based characterization of positive realness of transfer functions of a single input single output time invariant linear system is derived based on this characterization we propose an efficient computational procedure to determine if a given transfer function is positive real the input for this eigenvalue based test is any given not necessarily minimal state space representation of the linear system the test only involves standard matrix computations such as computing eigenvalues of a matrix or a matrix pencil results of numerical experiments are reported

criteria for copositive matrices abstract some finite criteria for copositive copositive plus and strictly copositive matrices are proposed and compared with existing determinantal tests the basic mathematical tool is principal pivoting

indefinite kac moody algebras of classical type the first author is supported in pat by nsf under grant dms 9025111 the 3rd author is supported in part by nsa msp grant mda 90 h 4039

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

a procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

polynomially parameter dependent lyapunov functions for robust stability of polytopic systems an lmi approach in this note robust stability of state space models with respect to real parametric uncertainty is considered specifically a new class of parameter dependent quadratic lyapunov functions for establishing stability of a polytope of matrices is introduced i e the homogeneous polynomially parameter dependent quadratic lyapunov functions hpd qlfs the choice of this class which contains parameter dependent quadratic lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form is motivated by the property that a polytope of matrices is stable if and only there exists an hpd qlf the main result of the note is a sufficient condition for determining the sought hpd qlf which amounts to solving linear matrix inequalities lmis derived via the complete square matricial representation csmr of homogeneous matricial forms and the lyapunov matrix equation numerical examples are provided to demonstrate the effectiveness of the proposed approach

brief paper state space solution to weight optimization problem in h loop shaping control this paper proposes a state space solution to weight optimization problem in loop shaping control a pointwise in frequency weight optimization framework is transformed into convex optimization searches that are independent of frequency the introduced optimization problem therefore avoids gridding of the frequency space and consequently the inaccuracies attributed to fitting transfer functions to magnitude data in this optimization problem the order of the weight is specified a priori thus facilitating the synthesis of low order controllers which is desirable from an implementation perspective the proposed solution algorithm simultaneously synthesizes a robust stabilizing controller and a loop shaping weight pre compensator that maximize the robust stability margin subject to constraints on the performance and the singular values of the weight three numerical examples are given to illustrate the effectiveness of the technique

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

brief paper an alternative kalman yakubovich popov lemma and some extensions this paper introduces an alternative formulation of the kalman yakubovich popov kyp lemma relating an infinite dimensional frequency domain inequality fdi to a pair of finite dimensional linear matrix inequalities lmi it is shown that this new formulation encompasses previous generalizations of the kyp lemma which hold in the case the coefficient matrix of the fdi does not depend on frequency in addition it allows the coefficient matrix of the frequency domain inequality to vary affinely with the frequency parameter one application of this results is illustrated in an example of computing upper bounds to the structured singular value with frequency dependent scalings

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

brief paper linear matrix inequality tests for frequency domain inequalities with affine multipliers this paper revisits an alternative formulation of the kalman yakubovich popov kyp lemma relating an infinite dimensional frequency domain inequality fdi to a pair of finite dimensional linear matrix inequalities lmi it is shown that this alternative formulation allows a certain class of the coefficient matrix of the fdi to be frequency dependent without introducing conservatism the construction provided in the present paper is helpful in other problems where system augmentation is commonly used such as those involving rational or polynomial multipliers for stability and performance analysis

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

stability multipliers and spl mu upper bounds connections and implications for numerical verification of frequency domain conditions the paper shows the equivalence between the following two approaches for obtaining sufficient conditions for the robust stability of systems with structured uncertainties 1 apply the classical absolute stability theory with multipliers and 2 use modern spl mu theory specifically the spl mu upper bound obtained by fan et al 1991 in particular the relationship between the stability multipliers used in absolute stability theory and the scaling matrices used in the cited reference is explicitly characterized the development hinges on the derivation of certain properties of a parametrized family of complex linear matrix inequalities a result of independent interest the derivation also suggests a general computational framework for checking the feasibility of a broad class of frequency dependent conditions based on which bisection schemes can be devised to reliably compute several quantities of interest for robust control

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

deformed kac moody algebras and their representations abstract a class of lie algebras g a associated to generalized cartan matrices a is studied the lie algebras g a have much simpler structure than kac moody algebras but have the same root spaces with g a in particular g a has an abelian subalgebra of half size we show that g a has a non degenerate invariant symmetric bilinear form if and only if a is symmetrizable g x 1 g x 2 if and only if the gcms x 1 and x 2 are the same up to a permutation of rows and columns we study the lowest respectively highest weight verma module v respectively v over g a and obtain the necessary and sufficient conditions for v to be irreducible and also find its maximal proper submodule when v is reducible then using graded dual module of v we deduce the necessary and sufficient conditions for v to be irreducible

a parameterization of parahermitian matrix functions and its application to a state space solution for μ analysis abstract in this paper an economic parameterization for positive parahermitian matrix functions is introduced and applied to the analysis framework wherein we propose a new state space optimization problem for finding the required d scales among the four state space matrices to be used to realize the optimal d scale a and b are chosen via a laguerre parameterization whereas the other two state space matrices c and d are obtained by spectral factorization after solving a convex optimization problem formulated in an lmi framework the obtained d scale satisfies the commuting property with the uncertainty structure the proposed economic parameterization yields advantages in terms of less computational time and less number of decision variables and also the proposed state space optimization framework gives a frequency independent solution algorithm in state space variables for the required d scales two numerical examples are used to demonstrate the effectiveness of the proposed algorithm

propagation of periodic and chaotic action potential trains along nerve fibers abstract in this paper we report the findings that the action potential trains transmitted along the nerve fibers are encoded not only periodically but also chaotically first spontaneous action potentials along a single fiber of injured sciatic nerves in the anesthetized rat were recorded then the data were divided into two groups and analyzed with different methods phase space representation spectral analysis and the calculation of correlation dimension were used for the first group of data sampled with constant frequency due to the serious influence of the measurement noise no reliable conclusion can be drawn from them for the second group of data of the interspike intervals isi which seem to convey more rich and important information nonlinear forecasting method the surrogate data and the plot of isi n 1 vs isi n were used in the analysis good results have been obtained which confirm with those from the cell model the largest lyapunov exponent lle were calculated not only to further support our findings of chaos but also to quantitatively determine the degree of chaos

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

periodic orbits as the skeleton classical and quantum chaos a description of a low dimensional deterministic chaotic system in terms of unstable periodic orbits cycles is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos comparable to the familiar perturbation expansions for nearly integrable systems the infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a selberg product form dominated by the short cycles and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits methods for implementing this computation for finite subshift dynamics are introduced

on the combinatorial and algebraic complexity of quantifier elimination in this paper a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given this algorithm improves the complexity of the asymptotically fastest algorithm for this problem known to this data a new feature of this algorithm is that the role of the algebraic part the dependence on the degrees of the imput polynomials and the combinatorial part the dependence on the number of polynomials are sparated another new feature is that the degrees of the polynomials in the equivalent quantifier free formula that is output are independent of the number of input polynomials as special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields and also for solving the existential problem in the first order theory over real closed fields are obtained

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

classification of finite dimensional solvable lie algebras with nondegenerate invariant bilinear forms abstract in this paper finite dimensional solvable lie algebras with nondegenerate invariant bilinear forms are classified the classification is based on the investigation of the minimal generating sets which are proved to be one of the three types paired type isolated type and mixed type

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

generalized kyp lemma unified frequency domain inequalities with design applications the celebrated kalman yakubovic spl caron popov kyp lemma establishes the equivalence between a frequency domain inequality fdi and a linear matrix inequality and has played one of the most fundamental roles in systems and control theory this paper first develops a necessary and sufficient condition for an s procedure to be lossless and uses the result to generalize the kyp lemma in two aspects the frequency range and the class of systems and to unify various existing versions by a single theorem in particular our result covers fdis in finite frequency intervals for both continuous discrete time settings as opposed to the standard infinite frequency range the class of systems for which fdis are considered is no longer constrained to be proper and nonproper transfer functions including polynomials can also be treated we study implications of this generalization and develop a proper interface between the basic result and various engineering applications specifically it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain phase properties in several frequency ranges the method is illustrated by numerical design examples of digital filters and proportional integral derivative controllers

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

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

on the complexity of purely complex spl mu computation and related problems in multidimensional systems in this paper the following robust control problems are shown to be np hard given a purely complex uncertainty structure spl delta determine if 1 spl mu sub spl delta m 1 for a given rational matrix m 2 spl par m spl middot spl par sub spl mu 1 for a given rational transfer matrix m s and 3 inf q spl isin h sup spl infin spl par f t q spl par sub spl mu 1 for a given linear fractional transformation f t q with rational coefficients in other words purely complex spl mu computation analysis and synthesis problems are np hard it is also shown that checking stability and computing the h sup spl infin norm of a multidimensional system are np hard problems therefore it is rather unlikely to find nonconservative polynomial time algorithms for solving the problems in complete generality