stability of invariant maximal semidefinite subspaces ii applications self adjoint rational matrix functions algebraic riccati equations abstract the stability of various factorizations of self adjoint rational matrix functions and matrix polynomials as well as of hermitian solutions of symmetric matrix algebraic riccati equations is studied in the first part of this paper results on stability of certain classes of invariant subspaces of a matrix which is self adjoint in an indefinite inner product were obtained these results serve as the main tools in the investigation

on self adjoint matrix polynomials with constant signature abstract this paper mainly concerns the class of self adjoint matrix polynomials with constant signature we define the order of neutrality for a regular self adjoint matrix polynomial l suppose that the leading coefficient of l is nonsingular then it is proved that l is of constant signature if and only if nl is even and the order of neutrality of l is equal to nl 2 where n is the degree l is the size of the polynomial l a similar result is obtained for regular self adjoint matrix polynomials we give an answer to an open problem concerning symmetric factorizations of self adjoint matrix polynomials of constant signature let l be a self adjoint matrix polynomial of even degree and constant signature and with a nonsingular leading coefficient an if all the elementary divisors of l are linear then l m gl a n m with m a monic matrix polynomial in particular this shows that generically a self adjoint matrix polynomial of even degree and constant signature admits such a factorization the proof of this factorization result is based on a result concerning invariant maximal neutral subspaces for a matrix which is self adjoint in an indefinite inner product the latter result also applies to other situations e g to factorizations of rational matrix functions with constant signature and to the existence of hermitian solutions for a class of algebraic riccati equations

hermitian solutions of the discrete algebraic riccati equation hermitian solutions of the discrete algebraic riccati equation play an important role in the least squares optimal control problem for discrete linear systems in this paper we describe the set of hermitian solutions in various ways in terms of factorizations of rational matrix functions which take hermitian values on the unit circle in terms of certain invariant subspaces of a matrix which is unitary in an indefinite scalar product and in terms of all invariant subspaces of a certain matrix these results are inspired by known results for the algebraic riccati equation arising in the least squares optimal control problem for continuous linear systems

stability of invariant maximal semidefinite subspaces i abstract we study stability in classes of subspaces which are invariant under a self adjoint matrix in an indefinite inner product and have various maximality and semidefiniteness properties with respect to this indefinite inner product descriptions of all subspaces in such a class for which these properties are stable in one or another way are obtained

dissipative matrices and invariant maximal semidefinite subspaces abstract an explicit construction of invariant maximal semidefinite subspaces for a dissipative matrix in an indefinite scalar product is given necessary and sufficient conditions for uniqueness of such a subspace are given these subspaces play an important role in the explicit construction of pseudo canonical factorizations for certain rational matrix valued functions

attack detection in sensor network target localization systems with quantized data we consider a sensor network focused on target localization where sensors measure the signal strength emitted from the target each measurement is quantized to one bit and sent to the fusion center a general attack is considered at some sensors that attempts to cause the fusion center to produce an inaccurate estimation of the target location the attack is a combination of man in the middle hacking and spoofing attacks that can effectively change both signals going into and coming out of the sensor nodes in a realistic manner we show that the essential effect of attacks is to alter the naive estimate of the distance between the target and each attacked sensor which ignores the existence of attacks to a different extent giving rise to a geometric inconsistency among the attacked and unattacked sensors with the help of two secure sensors a class of detectors are proposed to detect the attacked sensors by scrutinizing the existence of the geometric inconsistency we show that the false alarm and miss probabilities of the proposed detectors decrease exponentially as the number of measurement samples increases which implies that with sufficient measurement samples the proposed detectors can identify the attacked and unattacked sensors with any required accuracy numerical results show that compared to the cases where all sensors are employed without detecting attacks or only the secure sensors are employed the localization performance can be significantly improved if we employ the secure sensors and the sensors which are declared as unattacked by the proposed detector

basic classes of matrices with respect to quaternionic indefinite inner product spaces basic classes of matrices or linear transformations in finite dimensional quaternionic vector spaces with a regular indefinite inner product are introduced and studied the classes include plus matrices selfadjoint skew adjoint and unitary matrices in particular results are proved concerning extensions of invariant semidefinite subspaces canonical form for unitary matrices is developed and subsequently applied to stability of periodic hamiltonian systems

attacks on sensor network parameter estimation with quantization performance and asymptotically optimum processing byzantine attacks on sensor systems estimating the value of an unknown deterministic parameter based on quantized observations are studied initially asymptotically optimum processing is investigated for the family of attacks that would pass bad data detectors at the sensors bad data detectors check that the sensor data fits the models employed by the estimation approach when it assumes no attack is present it is then shown that it is possible to identify the attacked sensors under stated assumptions with perfect accuracy as the number of observations k from each sensor tends to infinity if the number of sensors tends to infinity while having a finite k it is shown that the attacked sensors can be identified with a given accuracy that can be set by k allowing considerable design flexibility next the performance of any estimation approach employed by the sensor system under any general attack is described for cases where any number of observations and sensors are employed a classification for these general attacks which categorizes them according to the information available to the attacking entity is introduced solving optimization problems over these attack classes leads to expressions which describe the performance of any specific estimation algorithm under the most devastating attacks with full information and the generally less effective information free attacks constraints are considered to account for some attack detection performed by the sensor system

a supercritical sobolev type inequality in higher order sobolev spaces and related higher order elliptic problems abstract a sobolev type embedding for radially symmetric functions on the unit ball b in r n n 3 into the variable exponent lebesgue space l 2 x b 2 2 n n 2 0 is known due to j m do o b ruf and p ubilla namely the inequality sup b u x 2 x d x u h 0 rad 1 b u l 2 b 1 holds in this work we generalize the above inequality for higher order sobolev spaces of radially symmetric functions on b namely the embedding h 0 rad m b l 2 m x b with 2 m n 2 2 m 2 n n 2 m and 0 holds questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied to illustrate the finding an application to a boundary value problem on balls driven by polyharmonic operators is presented this is the first in a set of our works concerning functional inequalities in the supercritical regime

minimal square spectral factors abstract recently there has been renewed interest in the problem of spectral factorization and in particular the problem of parametrizing all square minimal spectral factors of a given spectrum for instance let us mention two recent papers in this journal one of which dealing with more computational aspects of this problem clements 1993 the other giving a parametrization under additional constraints ferrante et al 1993 this paper is motivated in a large part by ferrante et al 1993 in fact we shall show that a parametrization similar to the one given there can be achieved without one of the additional constraints imposed in ferrante et al 1993 a secondary aim of this paper is to give an overview of several parametrizations available in the literature

hyperplane based vector quantization for distributed estimation in wireless sensor networks this paper considers distributed estimation of a vector parameter in the presence of zero mean additive multivariate gaussian noise in wireless sensor networks due to stringent power and bandwidth constraints vector quantization is performed at each sensor to convert its local noisy vector observation into one bit of information which is then forwarded to a fusion center where a final estimate of the vector parameter is obtained within such a context this paper focuses on a class of hyperplane based vector quantizers which linearly convert the observation vector into a scalar by using a compression vector and then carry out a scalar quantization it is shown that the key of the vector quantization design is to find a compression vector for each sensor under the framework of the cramer rao bound crb analysis the compression vector design problem is formulated as an optimization problem that minimizes the trace of the crb matrix such an optimization problem is extensively studied in particular an efficient iterative algorithm is developed for the general case along with optimal and near optimal solutions for some specific but important noise scenarios performance analysis and simulation results are carried out to illustrate the effectiveness of the proposed scheme

bandwidth constrained distributed estimation for wireless sensor networks part i gaussian case we study deterministic mean location parameter estimation when only quantized versions of the original observations are available due to bandwidth constraints when the dynamic range of the parameter is small or comparable with the noise variance we introduce a class of maximum likelihood estimators that require transmitting just one bit per sensor to achieve an estimation variance close to that of the clairvoyant sample mean estimator when the dynamic range is comparable or larger than the noise standard deviation we show that an optimum quantization step exists to achieve the best possible variance for a given bandwidth constraint we will also establish that in certain cases the sample mean estimator formed by quantized observations is preferable for complexity reasons we finally touch upon algorithm implementation issues and guarantee that all the numerical maximizations required by the proposed estimators are concave

unitary solutions of a class of algebraic riccati equations and factorization abstract we consider the algebraic riccati equation xd 1 x xd 2 d 3 x d 1 0 where d 2 and d 3 are self adjoint we are interested in contractive and unitary solutions x of this equation stability properties of such solutions under perturbations of the coefficients are studied further the connection between unitary solutions of such equations and symmetric factorizations of nonnegative rational matrix functions is investigated

asymptotically optimum distributed estimation in the presence of attacks distributed estimation of a deterministic mean shift parameter in additive zero mean noise is studied when using quantized data in the presence of byzantine attacks several subsets of sensors are assumed to be tampered with by adversaries using different attacks such that the compromised sensors transmit fictitious data first we consider the task of identifying and categorizing the attacked sensors into different groups according to distinct types of attacks it is shown that increasing the number k of time samples at each sensor and enlarging the size n of the sensor network can both ameliorate the identification and categorization but to different extents as k rightarrow infty the attacked sensors can be perfectly identified and categorized while with finite but sufficiently large k as n rightarrow infty it can be shown that the fusion center can also ascertain the number of attacks and obtain an approximate categorization with a sufficiently small percentage of sensors that are misclassified next in order to improve the estimation performance by utilizing the attacked observations we consider joint estimation of the statistical description of the attacks and the parameter to be estimated after the sensors have been well categorized when using the same quantization approach successfully employed without attacks it can be shown that the corresponding fisher information matrix fim is singular to overcome this a time variant quantization approach is proposed which will provide a nonsingular fim provided that k geq 2 furthermore the fim is employed to provide necessary and sufficient conditions under which utilizing the compromised sensors in the proposed fashion will lead to better estimation performance when compared to approaches where the compromised sensors are ignored

a fundamental limitation on maximum parameter dimension for accurate estimation with quantized data it is revealed that there is a link between the quantization approach employed and the dimension of the vector parameter which can be accurately estimated by a quantized estimation system a critical quantity called inestimable dimension for quantized data idqd is introduced which does not depend on the quantization regions and the statistical models of the observations but instead depends only on the number of sensors and on the precision of the vector quantizers employed by the system it is shown that the idqd describes a quantization induced fundamental limitation on the estimation capabilities of the system to be specific if the dimension of the desired vector parameter is larger than the idqd of the quantized estimation system then the fisher information matrix for estimating the desired vector parameter is singular and moreover there exist infinitely many nonidentifiable vector parameter points in the vector parameter space furthermore it is shown that under some common assumptions on the statistical models of the observations and the quantization system a smaller idqd can be obtained which can specify an even more limiting quantization induced fundamental limitation on the estimation capabilities of the system

functional forms of optimum spoofing attacks for vector parameter estimation in quantized sensor networks estimation of an unknown deterministic vector from quantized sensor data is considered in the presence of spoofing attacks which alter the data presented to several sensors contrary to the previous work a generalized attack model is employed which manipulates the data using transformations with arbitrary functional forms determined by some attack parameters whose values are unknown to the attacked system for the first time necessary and sufficient conditions are provided under which the transformations provide a guaranteed attack performance in terms of cramer rao bound crb regardless of the processing the estimation system employs thus defining a highly desirable attack interestingly these conditions imply that for any such attack when the attacked sensors can be perfectly identified by the estimation system either the fisher information matrix fim for jointly estimating the desired and attack parameters is singular or that the attacked system is unable to improve the crb for the desired vector parameter through this joint estimation even though the joint fim is nonsingular it is shown that it is always possible to construct such a highly desirable attack by properly employing a sufficiently large dimension attack vector parameter relative to the number of quantization levels employed which was not observed previously to illustrate the theory in a concrete way we also provide some numerical results which corroborate that under the highly desirable attack attacked data are not useful in reducing the crb

universal decentralized estimation in a bandwidth constrained sensor network consider a situation where a set of distributed sensors and a fusion center wish to cooperate to estimate an unknown parameter over a bounded interval u u each sensor collects one noise corrupted sample performs a local estimation and transmits a message to the fusion center while the latter combines the received messages to produce a final estimate this correspondence investigates optimal local estimation and final fusion schemes under the constraint that the communication from each sensor to the fusion center must be a one bit message such a binary message constraint is well motivated by the bandwidth limitation of the communication links fusion center and by the limited power budget of local sensors in the absence of bandwidth constraint and assuming the noises are bounded to the interval u u additive independent but otherwise unknown the classical estimation theory suggests that a total of o u2 epsi2 sensors are necessary and sufficient in order for the sensors and the fusion center to jointly estimate the unknown parameter within epsi root mean squared error mse it is shown in this correspondence that the same remains true even with the binary message constraint furthermore the optimal decentralized estimation scheme suggests allocating 1 2 of the sensors to estimate the first bit of the unknown parameter 1 4 of the sensors to estimate the second bit and so on

stable invariant lagrangian subspaces factorization of symmetric rational matrix functions and other applications several applications of earlier results by the authors concerning various notions of stability of invariant lagrangian subspaces are studied the applications include stability of symmetric minimal factorizations of real symmetric rational matrix functions stability of factorizations of certain classes of symmetric matrix polynomials and stably well posed matricial boundary value problems with symmetries an interpretation of many stability results in terms of equilibria of lie group actions on subspaces is given as well

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

simple forms and invariant subspaces of h expansive matrices in this paper we consider a simple form for pairs of matrices a h where h is a real symmetric invertible matrix and a is a real h expansive matrix that is atha h is positive semidefinite for such pairs we use the simple form to give an explicit construction of real a invariant maximal h semidefinite subspaces

nonnegative solutions of algebraic riccati equations abstract nonnegative hermitian solutions of various types of continuous and discrete algebraic riccati equations are studied the hamiltonian is considered with respect to two different indefinite scalar products for the set of nonnegative solutions the order structure and the topology of the set and the stability of solutions is treated for general hermitian solutions a method to compute the inertia is given although most attention is payed to the classical types arising from lq optimal control theory the case where the quadratic term has an indefinite coefficient is studied as well

target location estimation in sensor networks with quantized data a signal intensity based maximum likelihood ml target location estimator that uses quantized data is proposed for wireless sensor networks wsns the signal intensity received at local sensors is assumed to be inversely proportional to the square of the distance from the target the ml estimator and its corresponding crameacuter rao lower bound crlb are derived simulation results show that this estimator is much more accurate than the heuristic weighted average methods and it can reach the crlb even with a relatively small amount of data in addition the optimal design method for quantization thresholds as well as two heuristic design methods are presented the heuristic design methods which require minimum prior information about the system prove to be very robust under various situations

super resolution delay doppler estimation for ofdm passive radar in this paper we consider the problem of joint delay doppler estimation of moving targets in a passive radar that makes use of orthogonal frequency division multiplexing communication signals a compressed sensing algorithm is proposed to achieve supper resolution and better accuracy using both the atomic norm and the ell 1 norm the atomic norm is used to manifest the signal sparsity in the continuous domain unlike previous works that assume the demodulation to be error free we explicitly introduce the demodulation error signal whose sparsity is imposed by the ell 1 norm on this basis the delays and doppler frequencies are estimated by solving a semidefinite program sdp which is convex we also develop an iterative method for solving this sdp via the alternating direction method of multipliers where each iteration involves closed form computation simulation results are presented to illustrate the high performance of the proposed algorithm

decentralized estimation with dependent gaussian observations this paper considers decentralized estimation with correlated noises under the bayesian framework for a tandem network with correlated additive gaussian noises we establish that threshold quantizers on local observations are optimal in the sense of maximizing fisher information at the fusion center this is true despite the fact that subsequent estimators may differ at the fusion center depending on the statistical distribution of the parameter to be estimated in addition it is always beneficial to have the better sensor i e the one with higher signal to noise ratio serve as the fusion center in a tandem network finally we identify different correlation regimes in terms of their impact on the estimation performance these include the well known case where negatively correlated noise benefits estimation performance as it facilitates noise cancellation as well as two distinct regimes with positively correlated noises