on the generalized dimensions for the fourier spectrum of the thue morse sequence we present explicit relations for the generalized dimensions of the spectral measure of the thue morse symbolic sequence for positive integer values of q each is expressed through the eigenvalue of the corresponding matrix it is also demonstrated how these dimensions can be recovered from the products of the values of the autocorrelation function

brief paper interval observer design for consistency checks of nonlinear continuous time systems this paper deals with fault detection for nonlinear continuous time systems a procedure based on interval analysis is proposed to build a guaranteed qlpv quasi linear parameter varying approximation of the nonlinear model the interval qlpv approximation makes it possible to derive two point observers which estimate respectively the lower and the upper bound of the state vector using cooperativity theory a set guaranteed to contain the actual value of the residual is then designed the modelling uncertainties and measurement errors are taken into account at the design stage the proposed methodology is illustrated through numerical simulations

robustness of luenberger observers linear systems stabilized via non linear control abstract given a dynamical system whose state equations include time varying uncertain parameters it is often desirable to design a state feedback controller leading to uniform asymptotic stability of a given equilibrium point if however the controller operates on some estimate of the state instead of the true state itself it is of interest to know whether the desired stability will be preserved e g suppose that the measured output is processed by a luenberger observer this paper concentrates on the scenario above and in addition our analysis permits the controller to be non linear as a first step inequalities are developed which have implications on the system s robustness that is when the uncertain parameters satisfy these inequalities it becomes possible to separately design controller and observer this amounts to an extension of the classical separation theorem to the case when the controller is non linear it is also of interest to note that the approach given here guarantees stability for some non zero range of admissible parameter variations this is achieved by introducing a certain tuning parameter into the lyapunov function which is used to assure the stability of the combined plant observer controller system

on unscented kalman filtering for state estimation of continuous time nonlinear systems this paper considers the application of the unscented kalman filter ukf to continuous time filtering problems where both the state and measurement processes are modeled as stochastic differential equations the mean and covariance differential equations which result in the continuous time limit of the ukf are derived the continuous discrete ukf is derived as a special case of the continuous time filter when the continuous time prediction equations are combined with the update step of the discrete time ukf the filter equations are also transformed into sigma point differential equations which can be interpreted as matrix square root versions of the filter equations

necessary and sufficient conditions for quadratic stability and stabilizability of uncertain linear time varying systems in this paper we consider linear time varying systems subject to norm bounded one block structured uncertainties first given an unforced system we find necessary and sufficient conditions for quadratic stability then considered a controlled system we provide necessary and sufficient conditions for quadratic stabilizability both for the state and the output feedback case

towards a semiclassical theory of the quantum baker s map abstract we develop semiclassical 0 treatments of the quantum baker s map paying special attention to a the discrete and finite nature of its hilbert space and b the asymptotic analysis of the true quantum objects themselves new techniques are described at the theoretical level exact symbolic decomposition of the propagator and at the numerical level color graphics display of operators in the kirkwood representation both of which are of general interest for other quantized chaotic maps periodic orbit theory is tested numerically and we find that log corrections need to be included before an accurate calculation of the spectrum from periodic orbits can be achieved however the asymptotic analysis of the traces of the propagator reveals unexpected departures from standard semiclassical theory those anomalies which we compute explicitly in a few cases are traced ultimately to the discontinuity of the map and to the compactness of the phase space the analysis is far from complete but the results point towards the necessity of careful assessment of semiclassical folklore when applied to chaotic maps

exact eigenfunctions for a quantised map exact eigenfunctions for quantised linear maps on a torus are constructed a connection with the number of periodic orbits of the classical map is established it is shown that in the semiclassical limit the eigenfunctions become more and more random i e their correlation function approaches a delta function

asymptotically stable interval observers for planar systems with complex poles in some parametric domains the problem of designing an exponentially stable interval observer for an exponentially stable two dimensional time invariant linear system is open we show that in some cases no linear time invariant change of coordinates can help to determine an exponentially stable interval observer next we solve the problem by constructing interval observers of a new type which have as key feature the property of being time varying this new design is applied to the chaotic chua s system

interval state estimation for uncertain polytopic systems the aim of this article is the state estimation of uncertain polytopic dynamic systems the parametric uncertainties affecting the system are time varying unknown and bounded with known bounds the objective is to determine the state estimates consisting in the smallest interval containing the real state value caused by the parametric uncertainties this set will be characterized by the lower and upper bounds of the state trajectory given the uncertainty bounds the set can be computed by a direct simulation of the system but a more accurate estimation is obtained with a luenberger type observer fed with the system measurements the proposed observer is designed to minimize the interval width of the estimates the observer gains are obtained by solving an optimization problem under lmi constraints the efficiency of the proposed approach is illustrated by numerical examples

multifractal analysis of the spectral measure of the thue morse sequence a periodic orbit approach the fourier spectral density of the thue morse sequence is reinterpreted as the invariant measure of a stochastic dynamical system based on this fact its generalized r nyi dimension and f statistics are calculated with high precision by cycle expansions of spectral determinant and dynamical zeta function q at integer values of q are also computed in an operator scheme and the asymptotic result in the large q limit is derived

detailed balance as a quantum group symmetry of kraus operators a unital completely positive map governing the time evolution of a quantum system is usually called a quantum channel and it can be represented by a tuple of operators which are then referred to as the kraus operators of the channel we look at states of the system whose correlations with respect to the channel have a certain symmetry then we show that detailed balance holds if the kraus operators satisfy a very interesting algebraic relation which plays an important role in the representation theory of any compact quantum group

an unknown input interval observer for lpv systems under l2 gain and l gain criteria abstract this paper investigates the design of an unknown input interval observer for linear parameter varying systems the proposed method provides a solution to the decoupling conditions under l 2 gain and l gain performance level it is shown that the design of the unknown input interval observer parameters can be formulated as a sdp problem the major advantages with regard to other solutions is that first it does not require the explicit knowledge of a time varying state transformation for the estimation error dynamics to satisfy the meztler property rather it is shown how such a property can be formulated as a part of the sdp optimization process second decoupling and l 2 l constraints are formulated on the thickness of the interval length in spite of half the interval length the proposed methodology is illustrated on a numerical academic example

unified canonical forms for matrices over a differential ring abstract linear differential equations with variable coefficients of the vector form i x a t x and the scalar form ii y n n t y n 1 2 t y 1 t y 0 can be studied as operators on a differential module over a differential ring using this differential algebraic structure and a classical result on differential operator factorizations developed by cauchy and floquet a new variable eigenvalue theory and an associated set of matrix canonical forms are developed in this paper for matrices over a differential ring in particular eight new spectral and modal matrix canonical forms are developed that include as special cases the classical companion jordan diagonal and generalized vandermonde canonical forms as traditionally used for matrices over a number ring field therefore these new matrix canonical forms can be viewed as unifications of these classical ones two additional canonical matrices that perform order reductions of a cauchy floquet factorization of a linear differential operator are also introduced general and explicit formulas for obtaining the new canonical forms are derived in terms of the new variable eigenvalues the results obtained here have immediate applications in diverse engineering and scientific fields such as physics control systems communications and electrical networks in which linear differential equations i and ii are used as mathematical models

positive observers for linear positive systems and their implications a treatment of the positive observation problem for positive systems is provided for the continuous and discrete time cases the proposed observers are positive that is they ensure that the estimates are nonnegative at any time moreover necessary and sufficient conditions for the existence of such positive observers are formulated in terms of linear programming also we show how positive observers can provide guaranteed bounds on the real states of uncertain positive systems in addition some implications are discussed especially it is shown that it is not possible to stabilise an unstable system using these positive observers finally the applicability of the proposed interval estimation approach is shown for an illustrative application from pharmacokinetics

brief paper near optimal interval observers bundle for uncertain bioreactors in this paper we design an interval observer for the estimation of unmeasured variables of uncertain bioreactors the observer is based on a bounded error observer as proposed in lemesle v gouze j l 2005 hybrid bounded error observers for uncertain bioreactor models bioprocess and biosystems engineering 27 311 318 that makes use of a loose approximation of the bacterial kinetics we first show how to generate guaranteed upper and lower bounds on the state provided that known intervals for the initial condition and the uncertainties are available these so called framers depend on a tuning gain they can be run in parallel and the envelope provides the best estimate an optimality criterion is introduced leading to the definition of an optimal observer we show that this criterion provides directly a gain set containing the best framers the method is applied to the estimation of the total biomass of an industrial wastewater treatment plant demonstrating its efficiency

technical communique nonlinear bounded error state estimation of continuous time systems this paper presents a first study on the application of interval analysis and consistency techniques to state estimation of continuous time systems described by nonlinear ordinary differential equations the approach is presented in a bounded error context and the resulting methodology is illustrated by an example

generalized quantum baker maps as perturbations of a simple kernel we present a broad family of quantum baker maps that generalize the proposal of schack and caves to any even hilbert space with arbitrary boundary conditions we identify a structure common to all maps consisting of a simple kernel perturbed by diffraction effects this essential baker s map has a different semiclassical limit and can be diagonalized analytically for hilbert spaces spanned by qubits in all cases this kernel provides an accurate approximation to the spectral properties char22 eigenvalues and eigenfunctions char22 of all the different quantizations

multifractal analysis in reciprocal space and the nature of the fourier transform of self similar structures the authors propose to use multifractal analysis in reciprocal space as a tool to characterise in a statistical and global sense the nature of the fourier transform of geometrical models for atomic structures this approach is especially adequate for shedding some new light on a class of structures introduced recently which exhibit singular scattering using the language of measure theory the fourier intensity of these models is presumably singular continuous and therefore represents an intermediate type of order between periodic or quasiperiodic structures characterised by bragg peaks atomic fourier transform and amorphous structures which exhibit diffuse scattering absolutely continuous fourier transform this general approach is illustrated in several examples of self similar one dimensional sequences and structures generated by substitutions a special emphasis is put on the relationship between the nature of the fourier intensity of these models and the f alpha spectrum obtained by multifractal analysis in reciprocal space

stable interval observers in bbc for linear systems with time varying input bounds this technical note deals with the design of stable interval observers and estimators for continuous time linear dynamic systems under uncertain initial states and uncertain inputs enclosed within time varying zonotopic bounds no monotony assumption such as cooperativity is required in the vector field the interval observer stability directly derives from the stability of the observer state matrix where any poles real or complex single or multiple are handled in the same way

multifractal eigenstates of quantum chaos and the thue morse sequence we analyze certain eigenstates of the quantum baker s map and demonstrate using the walsh hadamard transform the emergence of the ubiquitous thue morse sequence a simple sequence that is at the border between quasiperiodicity and chaos and hence is a good paradigm for quantum chaotic states we show a family of states that are also simply related to the thue morse sequence and are strongly scarred by short periodic orbits and their homoclinic excursions we give approximate expressions for these states and provide evidence that these and other generic states are multifractal

interval observers for linear time invariant systems with disturbances it is shown that for any time invariant exponentially stable linear system with additive disturbances time varying exponentially stable interval observers can be constructed the technique of construction relies on the jordan canonical form that any real matrix admits and on time varying changes of coordinates for elementary jordan blocks which lead to cooperative linear systems the approach is applied to detectable linear systems

periodic orbit basis for the quantum baker map a set of quantum states dynamically related to the classical periodic orbits of a chaotic map is used as a basis in which the description of the eigenstates of its quantum version is greatly simplified this set can be improved with the inclusion of short time propagation along the stable and unstable manifolds of the periodic orbits resulting in a construction similar to the scar functions of vergini j phys a 33 4709 2000 the average participation ratio is used to quantify the quality of the basis

design of reduced order interval observers for lti systems with bounded time varying disturbances a parametric approach this paper investigates the design problem of reduced order interval observers for linear time invariant systems with bounded time varying disturbances although there are many excellent design met

interval state estimation for a class of nonlinear systems the goal of this technical note is to design interval observers for a class of nonlinear continuous time systems the first part of this work shows that it is usually possible to design an interval observer for linear systems by means of linear time invariant changes of coordinates even if the system is not cooperative this result is extended to a class of nonlinear systems using partial exact linearisations the proposed observers guarantee to enclose the set of system states that is consistent with the model the disturbances and the measurement noise moreover it is only assumed that the measurement noise and the disturbances are bounded without any additional information such as stationarity uncorrelation or type of distribution the proposed observer is illustrated through numerical simulations

estimation of solutions of observable nonlinear systems with disturbances a family of continuous time observable nonlinear systems with input and output is considered a new technique of estimation of the state variables is proposed it relies on the use of past values of the output as done to construct some observers which converge in finite time and on a recent technical result pertaining to the theory of the monotone systems it applies to systems with additive disturbances and disturbances in the output the nonlinear terms are not supposed to be globally lipschitz but it is requested that they depend only on the input and output variables

shuffling cards factoring numbers and the quantum baker s map it is pointed out that an exactly solvable permutation operator viewed as the quantization of cyclic shifts is useful in constructing a basis in which to study the quantum baker s map a paradigm system of quantum chaos in the basis of this operator the eigenfunctions of the quantum baker s map are compressed by factors of around five or more we show explicitly its connection to an operator that is closely related to the usual quantum baker s map this permutation operator has interesting connections to the art of shuffling cards as well as to the quantum factoring algorithm of shor via the quantum order finding one hence we point out that this well known quantum algorithm makes crucial use of a quantum chaotic operator or at least one that is close to the quantization of the left shift a closeness that we also explore quantitatively

interval state observer for nonlinear time varying systems this paper is devoted to the design of interval observers for linear time varying ltv systems and a class of nonlinear time varying systems in the output canonical form an interval observer design is feasible if it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable it is shown that under some mild conditions the cooperativity of an ltv system can be ensured by a static linear transformation of coordinates the efficiency of the proposed approach is demonstrated through numerical simulations

an easy design for interval observers this paper deals with the problem of interval estimation the proposed approach is based on the use of successive derivatives in continuous time or forward measurements in discrete time in an alg

stability and l 1 l1 to l 1 l1 performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays abstractsolutions to the interval observation problem for delayed impulsive and switched systems with l1 performance are provided the approach is based on first obtaining stability and l1 l1 to l1

an effective method to interval observer design for time varying systems an interval observer for linear time varying ltv systems is proposed in this paper usually the design of such observers is based on monotone systems theory monotone properties are hard to satisfy in many situations to overcome this issue in a recent work it has been shown that under some restrictive conditions the cooperativity of an ltv system can be ensured by a static linear transformation of coordinates however a constructive method for the construction of the transformation matrix and the observer gain making the observation error dynamics positive and stable is still missing and remains an open problem in this paper a constructive approach to obtain a time varying change of coordinates ensuring the cooperativity of the observer error in the new coordinates is provided the efficiency of the proposed approach is shown through computer simulations

using the hadamard and related transforms for simplifying the spectrum of the quantum baker s map we rationalize the somewhat surprising efficacy of the hadamard transform in simplifying the eigenstates of the quantum baker s map a paradigmatic model of quantum chaos this allows us to construct closely related but new transforms that do significantly better thus nearly solving many states of the quantum baker s map these transforms which combine the standard fourier and hadamard transforms in an interesting manner are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit flip parity and possess bit reversal time reversal symmetry