convex inner approximations of nonconvex semialgebraic sets applied to fixed order controller design we describe an elementary algorithm to build convex inner approximations of nonconvex sets both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials even though no optimality guarantees can be given e g in terms of volume maximisation for bounded sets the algorithm is designed to preserve convex boundaries as much as possible while removing regions with concave boundaries in particular the algorithm leaves invariant a given convex set the algorithm is based on gloptipoly 3 a public domain matlab package solving nonconvex polynomial optimisation problems with the help of convex semidefinite programming optimisation over linear matrix inequalities or lmis we illustrate how the algorithm can be used to design fixed order controllers for linear systems following a polynomial approach

simple approximations of semialgebraic sets and their applications to control many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets that is as the intersection of sets described by means of polynomial inequalities important examples are for instance the solution set of linear matrix inequalities or the schur hurwitz stability domains these sets often have very complicated shapes non convex and even non connected which renders very difficult their manipulation it is therefore of considerable importance to find simple enough approximations of these sets able to capture their main characteristics while maintaining a low level of complexity for these reasons in the past years several convex approximations based for instance on hyperrect angles polytopes or ellipsoids have been proposed in this work we move a step further and propose possibly non convex approximations based on a small volume polynomial superlevel set of a single positive polynomial of given degree we show how these sets can be easily approximated by minimizing the l1 norm of the polynomial over the semialgebraic set subject to positivity constraints intuitively this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems from a computational viewpoint we design a hierarchy of linear matrix inequality problems to generate these approximations and we provide theoretically rigorous convergence results in the sense that the hierarchy of outer approximations converges in volume or equivalently almost everywhere and almost uniformly to the original set two main applications of the proposed approach are considered the first one aims at reconstruction approximation of sets from a finite number of samples in the second one we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set the efficiency of the proposed approach is demonstrated by different numerical examples

convex sets with semidefinite representation we provide a sufficient condition on a class of compact basic semialgebraic sets bf k subset mathbb r n for their convex hull co k to have a semidefinite representation sdr this sdr is explicitly expressed in terms of the polynomials g j that define k examples are provided we also provide an approximate sdr that is for every fixed epsilon 0 there is a convex set bf k epsilon such that rm co bf k subseteq bf k epsilon subseteq rm co bf k epsilon bf b where b is the unit ball of mathbb r n and bf k epsilon has an explicit sdr in terms of the g j s for convex and compact basic semi algebraic sets k defined by concave polynomials we provide a simpler explicit sdr when the nonnegative lagrangian l f associated with k and any linear f in mathbb r x is a sum of squares we also provide an approximate sdr specific to the convex case

sufficient and necessary conditions for semidefinite representability of convex hulls and sets a set s subseteq mathbb r n is called semidefinite programming sdp representable or semidefinite representable if s equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality lmi clearly if s is sdp representable then s must be convex and semialgebraic it is describable by conjunctions and disjunctions of polynomial equalities or inequalities this paper proves sufficient conditions and necessary conditions for sdp representability of convex sets and convex hulls by proposing a new approach to constructing sdp representations the contributions of this paper are i for bounded sdp representable sets w 1 dots w m we give an explicit construction of an sdp representation for mathrm conv cup k 1 mw k this provides a technique for building global sdp representations from the local ones ii for the sdp representability of a compact convex semialgebraic set s we prove sufficient the boundary partial s is nonsingular and positively curved while necessary is partial s has nonnegative curvature at each nonsingular point in terms of defining polynomials for s nonsingular boundary amounts to them having nonvanishing gradient at each point on partial s and the curvature condition can be expressed as their strict versus nonstrict quasi concavity at those points on partial s where they vanish the gaps between them are partial s having or not having singular points either of the gradient or of the curvature s positivity a sufficient condition bypassing the gaps is when some defining polynomials of s satisfy an algebraic condition called sos concavity iii for the sdp representability of the convex hull of a compact nonconvex semialgebraic set t we find that the critical object is partial c t the maximum subset of partial t contained in partial mathrm conv t we prove sufficient for sdp representability partial c t is nonsingular and positively curved and necessary is partial c t has nonnegative curvature at nonsingular points the gaps between our sufficient and necessary conditions are similar to case ii the positive definite lagrange hessian pdlh condition which meshes well with constructions is also discussed

empirical mode decomposition and correlation properties of long daily ozone records sreceived 16 november 2004 revised manuscript received 17 march 2005 published 27 may 2005d correlations for daily data of total ozone column are investigated by detrended fluctuation analysis sdfad the removal of annual periodicity does not result in a background free signal for the tropical station mauna loa in order to identify the remaining quasiperiodic constituent the relatively new method of empirical mode decomposition semdd is tested we found that the so called intrinsic mode functions do not represent real signal components of the ozone time series their amplitude modulation is very sensitive to local changes such as random data removal or smoothing tests on synthetic data further corroborate the limitations of decomposing quasiperiodic signals from noise with emd nevertheless the emd algorithm helps to identify dominating frequencies in the time series which allows to separate fluctuations from the remaining background we demonstrate that dfa analysis for the cleaned mauna loa record yields scaling comparable to a mid latitude station

subgraph centrality in complex networks we introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network smaller subgraphs are given more weight than larger ones which makes this measure appropriate for characterizing network motifs we show that the subgraph centrality c s i can be obtained mathematically from the spectra of the adjacency matrix of the network this measure is better able to discriminate the nodes of a network than alternate measures such as degree closeness betweenness and eigenvector centralities we study eight real world networks for which c s i displays useful and desirable properties such as clear ranking of nodes and scale free characteristics compared with the number of links per node the ranking introduced by c s i for the nodes in the protein interaction network of s cereviciae is more highly correlated with the lethality of individual proteins removed from the proteome

a note on walk entropies in graphs abstract the concept of walk entropy of a graph has been recently introduced in e estrada et al 2014 4 in that paper the authors formulated two conjectures about walk entropies in the present note we prove the first of these two conjectures and propose a stronger form of it

stochastic modeling of daily temperature fluctuations classical spectral hurst and detrended fluctuation analysis have been revealed asymptotic power law correlations for daily average temperature data for short time intervals however strong correlations characterize the dynamics that permits a satisfactory description of temperature changes as a low order linear autoregressive process dominating the texts on climate research here we propose a unifying stochastic model reproducing correlations for all time scales the concept is an extension of a first order autoregressive model with power law correlated noise the inclusion of a nonlinear atmospheric response function conveys the observed skew for the amplitude distribution of temperature fluctuations while stochastic models cannot help to understand the physics behind atmospheric processes they are capable to extract useful features promoting to benchmark physical models an example is shown possible applications for other systems of strong short range and asymptotic power law correlations are discussed

effect of nonlinear filters on detrended fluctuation analysis when investigating the dynamical properties of complex multiple component physical and physiological systems it is often the case that the measurable system s output does not directly represent the quantity we want to probe in order to understand the underlying mechanisms instead the output signal is often a linear or nonlinear function of the quantity of interest here we investigate how various linear and nonlinear transformations affect the correlation and scaling properties of a signal using the detrended fluctuation analysis dfa which has been shown to accurately quantify power law correlations in nonstationary signals specifically we study the effect of three types of transforms i linear y i a x i b ii nonlinear polynomial y i a x k i and iii nonlinear logarithmic y i log x i delta filters we compare the correlation and scaling properties of signals before and after the transform we find that linear filters do not change the correlation properties while the effect of nonlinear polynomial and logarithmic filters strongly depends on a the strength of correlations in the original signal b the power k of the polynomial filter and c the offset delta in the logarithmic filter we further apply the dfa method to investigate the apparent scaling of three analytic functions i exponential exp x a ii logarithmic log x a and iii power law x a lambda which are often encountered as trends in physical and biological processes while these three functions have different characteristics we find that there is a broad range of values for parameter a common for all three functions where the slope of the dfa curves is identical we further note that the dfa results obtained for a class of other analytic functions can be reduced to these three typical cases we systematically test the performance of the dfa method when estimating long range power law correlations in the output signals for different parameter values in the three types of filters and the three analytic functions we consider

walk entropy and walk regularity abstract a graph is said to be walk regular if for each l 1 every vertex is contained in the same number of closed walks of length l we construct a 24 vertex graph h 4 that is not walk regular yet has maximized walk entropy s v h 4 log 24 for some 0 this graph is a counterexample to a conjecture of benzi 2014 1 conjecture 3 1 we also show that there exist infinitely many temperatures 0 0 so that s v g 0 log n g if and only if a graph g is walk regular

nonuniversal atmospheric persistence different scaling of daily minimum and maximum temperatures an extensive investigation of 61 daily temperature records by means of detrended fluctuation analysis has revealed that the value of correlation exponent is not universal contrary to earlier claims furthermore statistically significant differences are found for daily minimum and maximum temperatures measured at the same station suggesting different degrees of long range correlations for the two extremes numerical tests on synthetic time series demonstrate that a correlated signal interrupted by uncorrelated segments exhibits an apparently lower exponent value over the usual time window of empirical data analysis in order to find statistical differences between the two daily extreme temperatures high frequency 10 min records were evaluated for two distant locations the results show that daily maxima characterize better the dynamic equilibrium state of the atmosphere than daily minima for both stations this provides a conceptual explanation why scaling analysis can yield different exponent values for minima and maxima

clustering of spectra and fractals of regular graphs we exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices the structure has a fractal threadlike appearance points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar zooming in reveals that this cluster splits into smaller segments filars labeled by the number of triangles in graphs further zooming in shows that the smaller filars split into subfilars labeled by the number of quadrangles in graphs etc we call this fractal structure discovered in a numerical experiment a multifilar structure we also provide a mathematical explanation of this phenomenon based on the ihara selberg trace formula and compute the coordinates and slopes of all filars in terms of bessel functions of the first kind

flat output characterization for linear systems using polynomial matrices this paper is devoted to the study of linear flat outputs for linear controllable time invariant systems in polynomial matrix form we characterize the transformations expressing the system variables in terms of a linear flat output and derivatives called defining matrices as the kernel of a polynomial matrix an application to trajectory planning is presented showing the usefulness of the present characterization

appropriateness of correlated first order auto regressive processes for modeling daily temperature records abstract the present study investigates linear and volatile nonlinear correlations of first order auto regressive process with uncorrelated ar 1 and long range correlated car 1 gaussian innovations as a function of the process parameter in the light of recent findings a kiraly i m janosi phys rev e 65 2002 0511021 we discuss the choice of car 1 in modeling daily temperature records we demonstrate that while car 1 is able to capture linear correlations it is unable to capture nonlinear volatile correlations in daily temperature records

continuous time linear predictive control and flatness a module theoretic setting with examples a general flatness based framework for linear continuous time predictive control is presented the mathematical setting which is valid for multivariable systems is provided by the algebraic theory of modules where a controllable system corresponds to a finitely generated free module over a principal ideal ring any basis of this free module is a flat output which yields an easy calculation of the predicted trajectory this formalism permits one to handle non minimum phase systems system constraints and to deal with additive perturbations three concrete case studies namely a dc motor a flexible system and a cement mill are analysed and simulations are given these examples are written in such a way that any reader who is not familiar with module theory may nevertheless grasp the proposed control strategy

approximate volume and integration for basic semialgebraic sets given a basic compact semialgebraic set mathbf k subset mathbb r n we introduce a methodology that generates a sequence converging to the volume of mathbf k this sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on mathbf k can be approximated as closely as desired which permits the approximation of the integral on mathbf k of any given polynomial the extension to integration against some weight functions is also provided finally some numerical issues associated with the algorithms involved are briefly discussed

ellipsoidal approximation of the stability domain of a polynomial the stability domain in the space of coefficients of a polynomial is a nonconvex set in general in this note we propose a new convex ellipsoidal inner approximation of this set derived via optimization over linear matrix inequalities as a by product we obtain new simple sufficient conditions for stability that may prove useful in robust control design

effect of the signal filtering on detrended fluctuation analysis abstract detrended fluctuation analysis dfa is an effective method to accurately quantify long term correlations embedded in a nonstationary time series in this paper we study the effect of signal filtering of a signal on the dfa method the theoretical and simulated results show that the signal filtering will affect the range of scale in dfa moreover this effect is different for fractal gaussian noise series and fractal brown movement series our study is meaningful for improving accuracy and efficiency of dfa method in theory and practice

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems necessary and sufficient conditions for convexity and strong convexity respectively of connected sublevel sets that are defined by finitely many real valued c 1 1 maps are presented a novel characterization of strongly convex sets in terms of the so called local quadratic support is proved the results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous time nonlinear systems to be strongly convex an application of these conditions is a novel method to overapproximate attainable sets when strong convexity is present

superroughening by linear growth equations with spatiotemporally correlated noise we give an extensive study on a class of interfacial superroughening processes with finite lateral system size in 1 1 dimensions described by linear growth equations with spatiotemporally power law decaying correlated noise since some of these processes have an extremely long relaxation time we first develop a very efficient method capable of simulating the interface morphology of these growth processes even in very late time we numerically observe that this class of superrough growth processes indeed gradually develops macroscopic structures with the lateral size comparable to the lateral system size through the rigorous analytical study of the equal time height difference correlation function the different time height difference correlation function and the local width we explicitly evaluate not only the leading anomalous dynamic scaling term but also all the subleading anomalous dynamic scaling terms which dominate over the ordinary dynamic scaling term moreover the relation between the macroscopic structure formation and anomalous interfacial roughening of the superrough growth processes is analytically investigated in detail

maximum walk entropy implies walk regularity the notion of walk entropy sv g sv g for a graph g at the inverse temperature was put forward recently by estrada et al 2014 7 it was further proved by benzi 1 that a graph is walk regular if and only if its walk entropy is maximum for all temperatures i i where i is a set of real numbers containing at least an accumulation point benzi 1 conjectured that walk regularity can be characterized by the walk entropy if and only if there is a 0 0 such that sv g sv g is maximum here we prove that a graph is walk regular if and only if the view the mathml sourcesv g 1 lnn we also prove that if the graph is regular but not walk regular view the mathml sourcesv g 0 0 and view the mathml sourcelim 0sv g lnn lim sv g if the graph is not regular then view the mathml sourcesv g lnn for every 0 0 for some 0 0

positive polynomials and robust stabilization with fixed order controllers recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial an application to the design of fixed order controllers robustly stabilizing a linear system subject to polytopic uncertainty is then proposed based on linear matrix inequality optimization the key ingredient in the design procedure resides in the choice of the central polynomial several numerical examples illustrate the relevance of the approach

walk entropies in graphs abstract entropies based on walks in graphs and in their line graphs are defined they are based on the summation over diagonal and off diagonal elements of the exponential of the adjacency matrix known as the network communicability the walk entropies are strongly related to the walk regularity of graphs and line graphs they are not biased by the graph size and have significantly better correlation with the inverse participation ratio of the eigenmodes of the adjacency matrix than other graph entropies a homogeneous weighting of the edges of the graph is used to simulate the effects of the temperature over the entropies defined in particular the walk entropy of graphs is non monotonic for regular but non walk regular graphs in contrast to non regular graphs

estimating the estrada index abstract let g be a graph on n vertices and let 1 2 n be its eigenvalues the estrada index of g is a recently introduced graph invariant defined as ee i 1 n e i we establish lower and upper bounds for ee in terms of the number of vertices and number of edges also some inequalities between ee and the energy of g are obtained

use and misuse of some hurst parameter estimators applied to stationary and non stationary financial time series abstract the detection of long range dependence lrd is an important task in time series analysis lrd is often summarized by the well known hurst parameter or exponent h 0 1 which can be estimated by a number of methods some of these techniques are designed to be applied to signals behaving as a stationary fractional gaussian noise fgn whereas others imply that the analyzed time series behave as a non stationary fractional brownian motion fbm moreover some estimators do not yield the hurst parameter but indexes related to h and ranging outside the unit interval therefore the fgn or fbm nature of the studied time series has to be preliminarily analyzed before applying any estimation method and the relationships between h and the indexes resulting from the analyses have to be taken into account to obtain coherent results since fgn like series represent the increments of fbm like processes and both the signals are characterized by the same h value by definition estimators designed for fgn like series can be applied to fbm like sequences after preventive differentiation and conversely estimators designed for fbm like processes can be applied to fgn like series after preventive integration the signal characterization is particularly important when h is estimated on financial time series because the returns represent the first difference of price time series which are often assumed to behave like self affine sequences the analysis of simulated fgn and fbm time series shows that all the considered methods yield comparable h values when properly applied the reanalysis of several market price time series already studied in the literature points out that a correct application of the estimators supported by a preventive signal classification yields homogeneous h values allowing for a useful cross validation of results reported in different works moreover some conclusions reported in the literature about the anti persistence of some financial series are shown to be incorrect because of the inappropriate application of the estimation methods

robust pole assignment via reflection coefficients of polynomials a robust version of the output controller design for discrete time systems is introduced instead of a single stable point a stable polytope or simplex is preselected in the closed loop characteristic polynomial coefficients space a constructive procedure for generating stable polytopes is given starting from the unit hypercube of reflection coefficients of monic polynomials this procedure is quite straightforward because for a special family of polynomials the linear cover of so called reflection vectors is stable the roots placement of reflection vectors is studied if a stable target simplex is preselected then the robust output controller design task is solved by quadratic programming approach

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability

on canonical forms for linear constant systems let f 0 h be a reachable system we introduce the notion of a nice selection and that of a minimal nice selection of column vectors of the reachability matrix g fg these notions lead to a unified treatment and a parametrization of all the and all the t canonical forms of f 0 h which are sometimes called first and second control canonical forms this global approach shows that the and the canonical forms are local coordinates of the quasi projeetive algebraic variety of all reachable systems of fixed dimension modulo basis change in the state space more over the canonical forms of f g h we mentioned above are in bijective correspondence with the so called and and matrix fraction representations of its transfer function finally we investigate a canonical form for proper rational matrix fractions which turns out to be a matrix fraction representation