brief paper identification and data driven model reduction of state space representations of lossless and dissipative systems from noise free data we illustrate procedures to identify a state space representation of a lossless or dissipative system from a given noise free trajectory important special cases are passive systems and bounded real systems computing a rank revealing factorization of a gramian like matrix constructed from the data a state sequence can be obtained the state space equations are then computed by solving a system of linear equations this idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced order model

matrices a such that as 1r ra with rk i abstract we study matrices a c n n such that a s 1 r r a where r k i n and s k are nonnegative integers with k 2 such matrices are called r s 1 k potent matrices the s 0 case corresponds to matrices such that a r a r 1 with r k i n and is studied using spectral properties of the matrix r for s 1 various characterizations of the class of r s 1 k potent matrices and relationships between these matrices and other classes of matrices are presented

passivity preserving model reduction via interpolation of spectral zeros abstract an algorithm is developed for passivity preserving model reduction of linear time invariant systems implementation schemes are described for both medium scale dense and large scale sparse applications the algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros these interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain block matrix which has the spectral zeros as its spectrum explicit interpolation is avoided and passivity of the reduced model is established instead through satisfaction of the necessary conditions of the positive real lemma it is also shown that this procedure indirectly solves the associated controllability and observability riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations from these a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable

synthesis of dissipative systems using quadratic differential forms part ii the problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative our formulation and solution of this problem is completely representation free the system dynamics are specified by a differential behavior and the performance is specified through a quadratic differential form we view control as interconnection a controller constrains a distinguished set of system variables the control variables the resulting behavior of the to be controlled variables is called the controlled behavior the constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior obtained by setting the control variables equal to zero and the plant behavior obtained by leaving the control variables unconstrained the main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications the essential requirement is a coupling condition an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior

a bilinear differential forms approach to parametric structured state space modelling we use one variable loewner techniques to compute polynomial parametric models for mimo systems from vector exponential data gathered at various points in the parameter space instrumental in our approach are the connections between vector exponential modelling via bilinear differential forms and the loewner framework

finite state incompressible infinite sequences in this paper we define and study finite state complexity of finite strings and infinite sequences as well as connections between these complexity notions to randomness and normality we show that the finite state complexity does not only depend on the codes for finite transducers but also on how the codes are mapped to transducers as a consequence we relate the finite state complexity to the plain kolmogorov complexity to the process complexity and to prefix free complexity working with prefix free sets of codes we characterise martin lof random sequences in terms of finite state complexity the weak power of finite transducers is compensated by the high complexity of enumeration of finite transducers we also prove that every finite state incompressible sequence is normal but the converse implication is not true these results also show that our definition of finite state incompressibility is stronger than all other known forms of finite automata based incompressibility in particular the notion related to finite automaton based betting systems introduced by schnorr and stimm the paper concludes with a discussion of open questions

on a matrix group constructed from an r s 1 k potent matrix abstract let r c n n be a k involutory matrix that is r k i n for some integer k 2 and let s be a nonnegative integer a matrix a c n n is called an r s 1 k potent matrix if a satisfies r a a s 1 r in this paper a matrix group corresponding to a fixed r s 1 k potent matrix is explicitly constructed and properties of this group are derived and investigated this group is then reconciled with the classical matrix group g a that is associated with a generalized group invertible matrix a

a subspace algorithm for balanced state space system identification in an earlier paper by moonen et at 1989 an algorithm has been introduced for identifying multivariable linear systems directly from input output data this new algorithm falls in the class of so called subspace methods and resembles impulse response based realization algorithms in this paper we extend this algorithm by incorporating a balancing step such that the identified model is always in balanced coordinates with this modification one obtains a data driven counterpart for kung s realization algorithm

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

the kolmogorov complexity of infinite words we present a brief survey of results on relations between the kolmogorov complexity of infinite strings and several measures of information content dimensions known from dimension theory information theory or fractal geometry special emphasis is placed on bounds on the complexity of strings in constructively given subsets of the cantor space finally we compare the kolmogorov complexity to the subword complexity of infinite strings

finite state dimension classical hausdorff dimension sometimes called fractal dimension was recently effectivized using gales betting strategies that generalize martingales thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary infinite sequences in this paper we use gales computed by multi account finite state gamblers to develop the finite state dimensions of sets of binary sequences and individual binary sequences the theorem of eggleston quart j math oxford ser 20 1949 31 36 relating hausdorff dimension to entropy is shown to hold for finite state dimension both in the space of all sequences and in the space of all rational sequences binary expansions of rational numbers every rational sequence has finite state dimension 0 but every rational number in 0 1 is the finite state dimension of a sequence in the low level complexity class ac0 our main theorem shows that the finite state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information lossless finite state compressors

on a disjoint idempotent decomposition for linear combinations produced from n commutative tripotent matrices abstract it has been established a 3 n term disjoint idempotent decomposition did for the linear combinations produced from n 2 commutative tripotent matrices their products and their products of power 2 at most the results obtained in this way generalize those in y tian a disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications linear multilinear algebra 59 2011 1237 1246 moreover an algorithm to get a did has been provided finally a numerical example has been given to exemplify the results

fast modes in the set of minimal dissipation trajectories abstract in this paper we study the set of trajectories satisfying both a given lti system s laws and also laws of the corresponding adjoint system in other words trajectories in the intersection of the system s behavior and that of the adjoint system this intersection has important system theoretic significance for example it is known that the trajectories in this intersection are the ones with minimal dissipation underlying the notion of adjoint is that of a power supply it is with respect to this supply rate that the trajectories in the intersection are known to be stationary in this paper we deal with half line solutions to the differential equations governing both the system and its adjoint analysis of half line solutions plays a central role for example in initial value problems and in well posedness studies of an interconnection we interpret the set of half line trajectories allowed by a system and its adjoint as an interconnection of these two systems and thus address issues about well posedness ill posedness of the interconnection we formulate necessary and sufficient conditions for this intersection to be autonomous for the case of an ill posed interconnection and resulting autonomous system we derive conditions for existence of initial conditions that lead to impulsive solutions in the states of the system we link our conditions with the strongly reachable and weakly unobservable subspaces of a state space system we show that absence of impulsive initial conditions is equivalent to the well known subspace iteration algorithms for these subspaces converging in one step

entropy rates and finite state dimension the effective fractal dimensions at the polynomial space level and above can all be equivalently defined as the c entropy rate where c is the class of languages corresponding to the level of effectivization for example pspace dimension is equivalent to the pspace entropy rate at lower levels of complexity the equivalence proofs break down in the polynomial time case the p entropy rate is a lower bound on the p dimension equality seems unlikely but separating the p entropy rate from p dimension would require proving p np we show that at the finite state level the opposite of the polynomial time case happens the reg entropy rate is an upper bound on the finite state dimension we also use the finite state genericity of ambos spies and busse automatic forcing and genericity on the diagonalization strength of finit automata in proc fourth int conf on discrete mathematics and theoretical computer science 2003 springer berlin pp 97 108 to separate finite state dimension from the reg entropy rate however we point out that a block entropy rate characterization of finite state dimension follows from the work of ziv and lempel compression of individual sequences via variable rate coding ieee trans inform theory 24 1978 530 536 on finite state compressibility and the compressibility characterization of finite state dimension by dai et al finite state dimension theoret comput sci 310 1 3 2004 1 33 as applications of the reg entropy rate upper bound and the block entropy rate characterization we prove that every regular language has finite state dimension 0 and that normality is equivalent to finite state dimension 1

the most powerful unfalsified model for data with missing values abstract the notion of the most powerful unfalsified model plays a key role in system identification since its introduction in the mid 80s many methods have been developed for its numerical computation all currently existing methods however assume that the given data is a complete trajectory of the system motivated by the practical issues of data corruption due to failing sensors transmission lines or storage devices we study the problem of computing the most powerful unfalsified model from data with missing values we do not make assumptions about the nature or pattern of the missing values apart from the basic one that they are a part of a trajectory of a linear time invariant system the identification problem with missing data is equivalent to a hankel structured low rank matrix completion problem the method proposed selects rank deficient complete submatrices of the incomplete hankel matrix under specified conditions the kernels of the submatrices form a nonminimal kernel representation of the data generating system the final step of the algorithm is reduction of the nonminimal kernel representation to a minimal one apart from its practical relevance in identification missing data is a useful concept in systems and control classic problems such as simulation filtering and tracking control can be viewed as missing data estimation problems for a given system the corresponding identification problems with missing data are data driven equivalents of the classical simulation filtering and tracking control problems

characterization and properties of matrices with k involutory symmetries we say that a matrix r cn n is k involutory if its minimal polynomial is xk 1 for some k 2 so rk 1 r 1 and the eigenvalues of r are 1 2 k 1 where e2 i k let 0 1 k 1 if r cm m a cm n s cn n and r and s are k involutory we say that a is r s symmetric if ras a we show that an r s symmetric matrix a can be usefully represented in terms of matrices fl cc l dl 0 l k 1 where cl and dl are respectively the dimensions of the l eigenspaces of r and s this continues a theme initiated in an earlier paper with the same title in which we assumed that 1 we say that a k involution is equidimensional with width d if all of its eigenspaces have dimension d we show that if r and s are equidimensional k involutions with widths d1 and d2 respectively then r s symmetric matrices are closely related to generalized circulants ras r r s 0k 1 where a0 a1 ak 1 cd1 d2 for this case our results are new even if 1 we also give an explicit formula for the moore penrose inverse of a unilevel block circulant as r r s 0k 1 for any 0 1 k 1 generalizing a result previously obtained for the case where gcd k 1

a characterization of solutions of the discrete time algebraic riccati equation based on quadratic difference forms this paper is concerned with a characterization of all symmetric solutions to the discrete time algebraic riccati equation dare dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper along the line of the continuous time results due to trentelman and rapisarda h l trentelman p rapisarda pick matrix conditions for sign definite solutions of the algebraic riccati equation siam j contr optim 40 3 2001 969 991 we show that the solvability of the dare is equivalent to a certain dissipativity of the associated discrete time state space system as a main result we characterize all unmixed solutions of the dare using the pick matrix obtained from the quadratic difference forms this characterization leads to a necessary and sufficient condition for the existence of a non negative definite solution it should be noted that when we study the dare and the dissipativity of the discrete time system there exist two difficulties which are not seen in the continuous time case one is the existence of a storage function which is not a quadratic function of state another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the dare due to the infinite generalized eigenvalues of the associated hamiltonian pencil one of the main contributions of this paper is to demonstrate how to resolve these difficulties

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

the dimensions of individual strings and sequences a constructive version of hausdorff dimension is developed using constructive supergales which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences this constructive dimension is used to assign every individual infinite binary sequence s a dimension which is a real number dim s in the interval 0 1 sequences that are random in the sense of martin lof have dimension 1 while sequences that are decidable 10 or 10 have dimension 0 it is shown that for every 20 computable real number in 0 1 there is a 20 sequence s such that dim s a discrete version of constructive dimension is also developed using termgales which are supergale like functions that bet on the terminations of finite binary strings as well as on their successive bits this discrete dimension is used to assign each individual string w a dimension which is a nonnegative real number dim w the dimension of a sequence is shown to be the limit inferior of the dimensions of its prefixes the kolmogorov complexity of a string is proven to be the product of its length and its dimension this gives a new characterization of algorithmic information and a new proof of mayordomo s recent theorem stating that the dimension of a sequence is the limit inferior of the average kolmogorov complexity of its first n bits every sequence that is random relative to any computable sequence of coin toss biases that converge to a real number in 0 1 is shown to have dimension h the binary entropy of

structure preserving model reduction of port hamiltonian systems by moment matching at infinity model reduction of port hamiltonian systems by means of the krylov methods is considered aiming at port hamiltonian structure preservation it is shown how to employ the arnoldi method for model reduction in a particular coordinate system in order to preserve not only a specific number of the markov parameters but also the port hamiltonian structure for the reduced order model furthermore it is shown how the lanczos method can be applied in a structure preserving manner to a subclass of port hamiltonian systems which is characterized by an algebraic condition in fact for the same subclass of port hamiltonian systems the arnoldi method and the lanczos method turn out to be equivalent in the sense of producing reduced order port hamiltonian models with the same transfer function

on and off line identification of linear state space models abstract a geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time invariant systems using possibly noisy input output i o measurements only as opposed to other mostly stochastic identification schemes no variance covariance information whatever is involved and only a limited number of i o data are required for the determination of the system matrices hence the algorithm can be best described and understood in the matrix formalism and consists of the following two steps first a state vector sequence is realized as the intersection of the row spaces of two block hankel matrices constructed with i o data then the system matrices are obtained at once from the least squares solution of a set of linear equations when dealing with noisy data this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition also the algorithm is easily applied to slowly time

resource bounded kolmogorov complexity revisited we take a fresh look at cd complexity where cdt x is the size of the smallest program that distinguishes x from all other strings in time t x we also look at cnd complexity a new nondeterministic variant of cd complexity and time bounded kolmogorov complexity denoted by c complexity we show several results relating time bounded c cd and cnd complexity and their applications to a variety of questions in computational complexity theory including the following showing how to approximate the size of a set using cd complexity without using the random string as needed in sipser s earlier proof of a similar result also we give a new simpler proof of this result of sipser s improving these bounds for almost all strings using extractors a proof of the valiant vazirani lemma directly from sipser s earlier cd lemma a relativized lower bound for cnd complexity exact characterizations of equivalences between c cd and cnd complexity showing that satisfying assignments of a satisfiable boolean formula can be enumerated in time polynomial in the size of the output if and only if a unique assignment can be found quickly this answers an open question of papadimitriou a new kolmogorov complexity based proof that bpp subseteq sigma 2 p new kolmogorov complexity based constructions of the following relativized worlds there exists an infinite set in p with no sparse infinite np subsets exp nexp but there exists a nexp machine whose accepting paths cannot be found in exponential time satisfying assignments cannot be found with nonadaptive queries to sat

relations between k s 1 potent matrices and different classes of complex matrices in this paper k s 1 potent matrices are considered a matrix a cn n is called k s 1 potent when kas 1k a where k is an involutory matrix and s 1 2 3 specifically k s 1 potent matrices are analyzed considering their relations to different classes of complex matrices these classes of matrices are s 1 generalized projectors k hermitian matrices normal matrices and matrices b cn n anti commuting with k or such that kb is involutory hermitian or normal in addition some new relations for k generalized centrosymmetric matrices have been derived

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

brief a stochastic realization algorithm via block lq decomposition in hilbert space a stochastic realization problem of a stationary stochastic process is re visited and a new stochastically balanced realization algorithm is derived in a hilbert space generated by second order stationary processes the present algorithm computes a stochastically balanced realization by means of the singular value decomposition of a weighted block hankel matrix derived by a block lq decomposition extension to a stochastic subspace identification method explains how the proposed abstract algorithm is implemented in system identification

an almost machine independent theory of program length complexity sophistication and induction abstract the purpose of this paper is to use a variant of program length complexity to formally define the structure of a binary string where the structure of an object is taken to mean the aggregate of its projectible properties

reprint of a bilinear differential forms approach to parametric structured state space modelling abstract we use one variable loewner techniques to compute polynomial parametric models for mimo systems from vector exponential data gathered at various points in the parameter space instrumental in our approach are the connections between vector exponential modelling via bilinear differential forms and the loewner framework

choice of state space basis in combined deterministic stochastic subspace identification this paper describes how the state space basis of models identified with subspace identification algorithms can be determined it is shown that this basis is determined by the input spectrum and by user defined input and output weightings through the connections between subspace identification and frequency weighted balancing the state space basis of the subspace identified models is shown to coincide with a frequency weighted balanced basis

when is a storage function a state function in discrete time the purpose of this paper is to investigate when a storage function is a state function in discrete time as shown by trentelman and willems systems control lett 32 1997 pp 249 259 siam j control optim 36 1998 pp 1703 1749 every storage function is a state function in continuous time at first glance the same claim seems to hold also in discrete time contrary to this expectation this is not true in general in fact the discrete time counterpart involves not only some different but also some more difficult issues compared with the continuous time case this paper addresses these issues exactly and shows that every nonnegative storage function is a state function of a supply rate with a linear time invariant dynamical system in discrete time

characterizations of k s 1 potent matrices and applications abstract recently situations where a matrix coincides with some of its powers have been studied this kind of matrices is related to the generalized inverse matrices on the other hand it is possible to introduce another class of matrices that involve an involutory matrix generalizing the well known idempotent matrix widely useful in many applications in this paper we introduce a new kind of matrices called k s 1 potent as an extension of the aforementioned ones first different properties of k s 1 potent matrices have been developed later the main result developed in this paper is the characterization of this kind of matrices from a spectral point of view in terms of powers of the matrix by means of the group inverse and via a block representation of a matrix of index 1 finally an application of the above results to study linear combinations of k s 1 potent matrices is derived

pseudo centrosymmetric matrices with applications to counting perfect matchings we consider square matrices a that commute with a fixed square matrix k both with entries in a field f not of characteristic 2 when k 2 i tao and yasuda defined a to be generalized centrosymmetric with respect to k when k 2 i we define a to be pseudo centrosymmetric with respect to k we show that the determinant of every even order pseudo centrosymmetric matrix is the sum of two squares over f as long as 1 is not a square in f when a pseudo centrosymmetric matrix a contains only integral entries and is pseudo centrosymmetric with respect to a matrix with rational entries the determinant of a is the sum of two integral squares this result when specialized to when k is the even order alternating exchange matrix applies to enumerative combinatorics using solely matrix based methods we reprove a weak form of jockusch s theorem for enumerating perfect matchings of 2 even symmetric graphs as a corollary we reprove that the number of domino tilings of regions known as aztec diamonds and aztec pillows is a sum of two integral squares

matrices that commute with a permutation matrix abstract let p be an n n permutation matrix and let p be the corresponding permutation let a be a matrix such that ap pa it is well known that when p is an n cycle a is permutation similar to a circulant matrix we present results for the band patterns in a and for the eigenstructure of a when p consists of several disjoint cycles these results depend on the greatest common divisors of pairs of cycle lengths

finite state complexity in this paper we develop a version of algorithmic information theory ait based on finite transducers instead of turing machines the complexity induced is called finite state complexity in spite of the fact that the universality theorem true for turing machines is false for finite transducers the invariance theorem holds true for finite state complexity we construct a class of finite state complexities based on various enumerations of the set of finite transducers in contrast with descriptional complexities plain prefix free from ait finite state complexity is computable and there is no a priori upper bound for the number of states used for minimal descriptions of arbitrary strings upper and lower bounds for the finite state complexity of arbitrary strings and for strings of particular types are given and incompressible strings are studied

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

discrete time average positivity and spectral factorization in a behavioral framework abstract quite recently willems and trentelman siam j control opt 36 1998 1703 1749 have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view in this paper we develop their discrete time version called quadratic difference forms and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories moreover we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms a numerical example will be given to show the validity of our result

the kolmogorov complexity of real numbers we consider for a real number the kolmogorov complexities of its expansions with respect to different bases in the paper it is shown that for usual and self delimiting kolmogorov complexity the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way that depends only on the relative information of one base with respect to the other more precisely we show that the complexity of the length l logrb prefix of the base r expansion of is the same up to an additive constant as the logrb fold complexity of the length l prefix of the base b expansion of then we consider the classes of reals of maximum and minimum complexity for maximally complex reals we use our result to derive a further complexity theoretic proof for the base independence of the randomness of real numbers finally we consider liouville numbers as a natural class of low complex real numbers

matrices a such that ra as 1r when rk i abstract this paper examines matrices a c n n such that ra a s 1 r where r k i the identity matrix and where s and k are nonnegative integers with k 2 spectral theory is used to characterize these matrices the cases s 0 and s 1 are considered separately since they are analyzed by different techniques

the inverse eigenvalue problem for a hermitian reflexive matrix and the optimization problem the inverse eigenvalue problem and the associated optimal approximation problem for hermitian reflexive matrices with respect to a normal k 1 potent matrix are considered first we study the existence of the solutions of the associated inverse eigenvalue problem and present an explicit form for them then when such a solution exists an expression for the solution to the corresponding optimal approximation problem is obtained

kolmogorov complexity and hausdorff dimension abstract in this paper various relationships between the kolmogorov complexity of infinite strings and measures of information content are given the general approach taken here is to bound the complexity of a maximally complex string in a given set of strings by the hausdorff dimension or the entropy of that set it turns out that hausdorff dimension yields lower bounds to the kolmogorov complexity whereas under certain recursiveness constraints on the structure of the respective sets their entropy yields upper bounds more detailed investigations result in a generalization of p martin lof s theorems on the kolmogorov complexity of random strings to maximally complex strings in regularly structured sets of infinite strings

the relation of the cca subspace method to a balanced reduction of an autoregressive model in this paper we consider an identification procedure called mest for multivariate time series based on ar modeling and stochastically balanced truncation and compare it with the cca subspace method the stochastically balancing of multivariate ar models is described using just linear algebraic operations i e no algebraic riccati equations need to be solved both identification procedures are formulated in a uniform manner and from these expressions we conclude that the only difference is that mest uses a covariance extension whereas cca is based on the sample covariances only finally it is shown that mest and cca are asymptotically equivalent

realization of lossless systems via constant matrix factorizations we study the realization problem for linear time invariant systems described by higher order differential equations which are j lossless with constant j our approach is based on the factorization of a constant matrix obtained in a straightforward way from the storage function of the system state equations are obtained directly from this factorization and from the original system representation

an example of a computable absolutely normal number the first example of an absolutely normal number was given by sierpinski in 1916 twenty years before the concept of computability was formalized in this note we give a recursive reformulation of sierpinski s construction which produces a computable absolutely normal number

a new result on passivity preserving model reduction abstract the problem of model reduction with preservation of passivity is investigated the approach is based on positive real interpolation and is inspired by the similarity between lowner and pick matrices the former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions it follows that interpolation of the original set of data together with an appropriately defined mirror image set of data yields automatically positive real interpolants subsequently we show how this result can be implemented using a krylov projection procedure the ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large scale systems

spectral study of r s 1 k and r s 1 k potent matrices abstract the r s 1 k and r s 1 k potent matrices have been studied in several recent papers we continue these investigations from a spectral point of view specifically a spectral study of r s 1 k potent matrices is developed using characterizations involving an associated matrix pencil a r the corresponding spectral study for r s 1 k potent matrices involves the pencil a r in order to present some properties the relevance of the projector i a a where a is the group inverse of a is highlighted in addition some applications and numerical examples are given particularly involving pauli matrices and the quaternions

canonical correlation analysis approximate covariance extension and identification of stationary time series abstract in this paper we analyze a class of state space identification algorithms for time series based on canonical correlation analysis in the light of recent results on stochastic systems theory in principle these so called subspace methods can be described as covariance estimation followed by stochastic realization the methods offer the major advantage of converting the nonlinear parameter estimation phase in traditional arma models identification into the solution of a riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity the reason for this is that an essential part of the problem is equivalent to the well known rational covariance extension problem therefore the usual deterministic arguments based on factorization of a hankel matrix are not valid for generic data something that is habitually overlooked in the literature we demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension unless some rather stringent assumptions are made which in general are not explicitly reported in this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory we review the basic ideas of stochastic realization theory in the context of identification discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation the model reduction method of desai and pal 1982 a realization approach to stochastic model reduction proc 1st decision and control conf pp 1105 1112 based on truncated balanced stochastic realizations is partially justified showing that the reduced system structure has a positive covariance sequence but is in general not balanced as a byproduct of this analysis we obtain a theorem prescribing conditions under which the subspace identification methods produce bona fide stochastic systems