observer design for singular systems with unknown inputs abstract the problem of designing unknown input observers which is especially meaningful for singular systems because differentiation terms exist in the inputs in the solution is discussed the conditions for such observers existing and an algorithm for designing the observers under these conditions are obtained and a numerical example is presented

statistical approach to normalization of feature vectors and clustering of mixed datasets normalization of feature vectors of datasets is widely used in a number of fields of data mining in particular in cluster analysis where it is used to prevent features with large numerical values

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

design of a class of luenberger observers for descriptor systems a new approach to the design of a class of luenberger observers for descriptor systems is presented the design is performed in a straightforward manner the conditions required in the design are given in terms of the original system matrices the relationship among these and other well known conditions is highlighted

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

observer design for descriptor systems two detectability notions are introduced for the most general class of linear descriptor systems necessary and sufficient conditions are derived for the detectabilities and for the existence of observers depending on existence conditions satisfied the observers can be either in the familiar luenberger form or in a generalized form i e in descriptor form as the original system itself is a unified procedure is provided for the design of observers in both forms

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

efficient least angle regression for identification of linear in the parameters models least angle regression as a promising model selection method differentiates itself from conventional stepwise and stagewise methods in that it is neither too greedy nor too slow it is closely related to l1 norm optimization which has the advantage of low prediction variance through sacrificing part of model bias property in order to enhance model generalization capability in this paper we propose an efficient least angle regression algorithm for model selection for a large class of linear in the parameters models with the purpose of accelerating the model selection process the entire algorithm works completely in a recursive manner where the correlations between model terms and residuals the evolving directions and other pertinent variables are derived explicitly and updated successively at every subset selection step the model coefficients are only computed when the algorithm finishes the direct involvement of matrix inversions is thereby relieved a detailed computational complexity analysis indicates that the proposed algorithm possesses significant computational efficiency compared with the original approach where the well known efficient cholesky decomposition is involved in solving least angle regression three artificial and real world examples are employed to demonstrate the effectiveness efficiency and numerical stability of the proposed algorithm

on the reduced order causal observer for generalized control systems abstract this paper considers the problem of a causal state observer for a square singular system necessary and sufficient conditions for solvability of the problem have been established the approach of this study leads to a new insight concerning synthesis of a causal observer and establishes new results with regard to the structure and the dimension of a reduced order observer for singular systems

random search with k prototypes algorithm for clustering mixed datasets a new algorithm to cluster datasets with mixed numerical and categorical values is presented the algorithm called rankpro random search with k prototypes algorithm combines the advantages of a recently introduced population based optimization algorithm called the bees algorithm ba and k prototypes algorithm the ba works with elite and good solutions and continues to look for other possible extrema solutions keeping the number of testing points constant however the improvement of promising solutions by the ba may be time consuming because it is based on random neighbourhood search on the other hand an application of the k prototypes algorithm to a promising solution may be very effective because it improves the solution at each iteration the rankpro algorithm balances two objectives it explores the search space effectively owing to random selection of new solutions and improves promising solutions fast owing to employment of the k prototypes algorithm the efficiency of the new algorithm is demonstrated by clustering several datasets it is shown that in the majority of the considered datasets when the average number of iterations that the k prototypes algorithm needs to converge is over 10 the rankpro algorithm is more efficient than the k prototypes algorithm

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

brief paper jump modes analysis and observer design for discrete singular systems this paper is concerned with the reconstructability and detectability of the jump modes of discrete singular systems the notion of jump modes can be considered as the discrete counterpart of the impulsive modes of continuous singular systems which has been extensively studied the effect of jump modes on the existence of reduced order normal luenberger observers for discrete singular systems is investigated necessary and sufficient conditions for the existence of normal luenberger observers and the corresponding observer design procedure are obtained

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

experiments on elastomechanical wave functions in chaotic plates and their statistical features we measure the amplitude of the elastomechanical displacement at a fine grid of points on a free plate having the shape of a sinai stadium the obtained displacement field formally corresponds to a wave function in a quantum system while the distribution of the squared amplitudes agrees with the prediction of random matrix theory rmt there is a strong deviation of the spatial correlator from the standard prediction for quantum chaotic systems we show that this is due to the presence of two modes leading to a beating phenomenon we construct a proper extension of the spatial correlator within the framework of rmt

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

semiclassical spatial correlations in chaotic wave functions we study the spatial autocorrelation of energy eigenfunctions psi n q corresponding to classically chaotic systems in the semiclassical regime our analysis is based on the weyl wigner formalism for the spectral average c epsilon q q e of psi n q psi n q defined as the average over eigenstates within an energy window epsilon centered at e in this framework c epsilon is the fourier transform in the momentum space of the spectral wigner function w x e epsilon our study reveals the chord structure that c epsilon inherits from the spectral wigner function showing the interplay between the size of the spectral average window and the spatial separation scale we discuss under which conditions is it possible to define a local system independent regime for c epsilon in doing so we derive an expression that bridges the existing formulas in the literature and find expressions for c epsilon q q e valid for any separation size q q

geometric design techniques for observers in singular systems abstract we present geometric design techniques for observer design in generalized state space or singular systems the unknown input conditioned invariant subspaces are defined for singular systems and are related to the solution of a generalized lyapunov equation the approach emphasizes solution procedures based on the lyapunov equation and a singular system structure algorithm this paper extends the singular system observer theory both by providing a geometric basis for it and showing the best that may be accomplished if the system is not observable

design of observers for time varying discrete time descriptor systems in this note linear discrete time time varying descriptor systems i e control systems of the form ek 1 xk 1 akxk bkuk yk ckxk are considered where ek 1 and ak 1 are constant not necessarily square matrices the problem is to design a reduced order observer for the system using matrix generalized inverses and singular value decomposition a procedure is developed which either leads to a routine observer design problem involving an ordinary discrete time state equation or proves that no observer exists for the given descriptor system the method of this note is partly based on an earlier paper of the authors on observer design for continuous time time invariant systems of the form e dx dt ax bu y cx where e and a are square matrices el tohami et al 1983 and makes use of some results of luenberger on discrete time time varying descriptor systems luenberger 1977

observers for descriptor systems abstract the design of observers is investigated for the estimation of the descriptor vector x t of descriptor systems of the form ex t ax t bu t y t cx t such systems are divided into two categories non causal impulsive systems and causal impulse free systems for non causal systems an observer of order rank e is proposed and termed a full order observer since the dynamical order of a descriptor system is regarded as rank e for causal systems it is shown that the number of independent outputs m does not exceed rank e and an observer of order rank e m is introduced and termed a minimal order observer the approach is based on the transformation of the system into particular design coordinates via a singular value decomposition of the matrix e illustrative examples are included

random matrix elements and eigenfunctions in chaotic systems the expected root mean square value of a matrix element a ensuremath alpha ensuremath beta in a classically chaotic system where a is a smooth ensuremath elzxh independent function of the coordinates and momenta and ensuremath alpha and ensuremath beta label different energy eigenstates has been evaluated in the literature in two different ways by treating the energy eigenfunctions as gaussian random variables and averaging a ensuremath alpha ensuremath beta 2 over them and by relating a ensuremath alpha ensuremath beta 2 to the classical time correlation function of a we show that these two methods give the same answer only if berry s formula for the spatial correlations in the energy eigenfunctions which is based on a microcanonical density in phase space is modified at large separations in a manner that we previously proposed

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

a reduced order observer for descriptor systems abstract in this paper a reducer order observer is analyzed for so called generalized state space or descriptor systems based on the staircase form for generalized state space models a recursive algorithm is presented to construct a reduced order observer for a given observable descriptor system in the algorithm only the eigenfrequencies of the observer have to be specified

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

design of observers for descriptor systems using a descriptor standard form methods are developed for constructing luenberger type observers for linear time invariant descriptor systems based on the descriptor standard form derived here by means of a simple matrix calculus full order observers reduced order observers and function observers of a general type are proposed some necessary and sufficient conditions for the function observers to estimate linear functions of the state are discussed then the so called separation property of combined observer controller systems is established

statistical properties of high lying chaotic eigenstates we study the statistical properties of the high lying chaotic eigenstates 200000 and above which are deep within the semiclassical regime the system we are analysing is the billiard system inside the region defined by the quadratic complex conformal map of the unit disk as introduced by robnik 1983 we are using heller s method of plane wave decomposition of the numerical eigenfunctions and perform extensive statistical analysis with the following conclusions i the local average probability density is in excellent agreement with the microcanonical assumption and all statistical properties are also in excellent agreement with the gaussian random model ii the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with berry s 1977 prediction iii although the scars of unstable classical periodic orbits in such an ergodic regime are expected to exist so far we have not found any around the 200000th state other than a scar like feature resembling the whispering gallery modes

gaussian random eigenfunctions and spatial correlations in quantum dots we show that spatial correlations in wave functions of quantum dots obtained earlier by averaging over a random potential via supermatrix techniques can be computed with much less effort by making use of berry s conjecture that the energy eigenfunctions in a quantized chaotic system are gaussian random variables furthermore in the case of a time reversal invariant system we find a greatly simplified though equivalent formula for these correlations textcopyright 1996 the american physical society

analysis of impulsive modes and luenberger observers for descriptor systems abstract in this paper the distinction between impulsive modes that can be generated by inputs and impulsive modes that can only be generated by initial conditions is investigated for linear time invariant descriptor systems it reveals that the input generatable impulsive modes are very crucial to the existence of normal luenberger observers of the systems based on this observation an earlier known concept called strongly detectability for regular descriptor systems is extended to irregular case that is shown to be of much clearer physical meanings in classical sense than a recently proposed equivalent notion of causal detectability does thus similar to the regular case strong detectability is also a sufficient and necessary condition for the existence of normal luenberger observers of irregular descriptor systems

how chaotic is the stadium billiard a semiclassical analysis the impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described semiclassically by the gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits i show that this belief is erroneous the gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the green s function semiclassical contributions to the trace show an dependent transition from hard chaos to integrable behaviour for trajectories approaching the bouncing ball orbits a whole region in phase space surrounding the marginal stable family acts semiclassically like a stable island with boundaries being explicitly dependent the localized bouncing ball states found in the billiard derive from this semiclassically stable island the bouncing ball orbits themselves however do not contribute to individual eigenvalues in the spectrum an ebk like quantization of the regular bouncing ball eigenstates in the stadium can be derived the stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics the behaviour is generically found at the border of classically stable islands in systems with a mixed phase space structure

design of a minimal order observer for singular systems this paper considers the design of an observer for generalized state space systems using the concept of generalized inverses in contrast to the luenberger observer theory a new method is proposed which does not presuppose the observer structure it is shown that under certain conditions it is possible to construct a minimal order observer for this class of systems illustrative examples are included

comments on on the robust popov criterion for interval lur e systems this paper supplements the remarks made in a paper by dahleh et al ieee trans automat contr vol 38 p 1400 5 1993 regarding the popov sector of lur e systems with interval plants a numerical example is provided to show that strict inequalities hold among three sectors those obtained by the theorem of the paper by applying popov theorem to each member system and by applying that to each of the systems with kharitonov plants this counterexample negates a tempting conjecture on strong robust popov criterion

an approach for pole assignment in singular systems pole assignment in a singular system edx dt ax bu is discussed it is shown that the problem of assigning the roots of det se a bf by applying a proportional feedback u fx r in a given singular system is equivalent to the problem of pole assignment of an appropriate regular system an immediate application of this result is that procedures and computational algorithms that were originally developed for assigning eigenvalues in regular systems become useful tools for pole assignment in singular systems the approach provides a useful tool for the combined problem of eliminating impulsive behavior and stabilizing a singular system