strong detectability and observers conditions are given for the existence of observers that estimate unmeasured outputs on the basis of partial information on the input and the state the concepts of strong detectability and strong observability introduced before in the literature for discrete time systems only are defined and studied for continuous time systems it is shown that there are two different concepts of strong detectability which coincide for discrete time systems algebraic conditions for either concept are given it is shown that these concepts are intimately related to the existence of strong observers i e observers that only use the output of the plant

an entropy based measure for comparing distributions of complexity abstract this paper is part of a series addressing the empirical statistical distribution of the diversity of complexity within and amongst complex systems here we consider the problem of measuring the diversity of complexity in a system given its ordered range of complexity types i and their probability of occurrence p i with the understanding that larger values of i mean a higher degree of complexity to address this problem we introduce a new complexity measure called case based entropy c c a modification of the shannon wiener entropy measure h the utility of this measure is that unlike current complexity measures which focus on the macroscopic complexity of a single system c c can be used to empirically identify and measure the distribution of the diversity of complexity within and across multiple natural and human made systems as well as the diversity contribution of complexity of any part of a system relative to the total range of ordered complexity types

brief paper zeros of structured linear systems this paper provides a detailed study of transmission zeros in the context of linear structured systems for such systems the entries of the state space model matrices are supposed to be either fixed zeros or free parameters a directed graph can be associated to such systems and structural properties can be studied i e properties which are valid for almost all values of the parameters we show that there exists two kinds of transmission zeros the parameter dependent ones and the constant ones which can only occur in zero moreover the constant transmission zeros can be directly obtained on the associated graph in terms of particular paths and cycles

uniqueness of matrix square roots and an application abstract let a m n c let a denote the spectrum of a and f a the field of values of a it is shown that if a 0 then a has a unique square root b m n c with b in the open right complex half plane this result and lyapunov s theorem are then applied to prove that if f a 0 then a has a unique square root with positive definite hermitian part we will also answer affirmatively an open question about the existence of a real square root b m n r for a m n r with f a 0 where the field of values of b is in the open right half plane

the computation of the principal square roots of centrosymmetric h matrices in this paper two algorithms for computing the principal square root of a centrosymmetric h matrix with positive diagonal entries are proposed it is showed that our algorithms ensure significant savings in computational costs as compared to the case of an arbitrary h matrix a with positive diagonal entries

fixed modes in parametrized systems several necessary graph theoretic conditions are presented for a parametrized system nut to have structurally fixed modus under output feedback of some arbitrarily specified pattern the criteria are formulated by the aid of a new graph which represents the couplings inherent in the system as well as the admissible feedback pattern for the special class of structured systems the conditions turn out to be also sufficient

partial state observability recovering for linear systems by additional sensor implementation this paper deals with the problem of additional sensor location in order to recover the observability of any given part of the state for structured linear systems the proposed method is based on a graph theoretic approach and assumes only the knowledge of the system s structure we first provide new graphical necessary and sufficient conditions for the generic partial observability then we study the location of additional sensors in order to satisfy the latter conditions we provide necessary and sufficient requirements to be satisfied by these additional sensors and all their possible locations the proposed solution is simple to implement because it is based on well known algorithms usually used for finding successors and predecessors of vertex subsets or on computation of maximal linkings in a digraph all the used algorithms have polynomial complexity orders

hadamard inverses square roots and products of almost semidefinite matrices abstract let a aij be an n n symmetric matrix with all positive entries and just one positive eigenvalue bapat proved then that the hadamard inverse of a given by a 1 1 a ij is positive semidefinite we show that if moreover a is invertible then a 1 is positive definite we use this result to obtain a simple proof that with the same hypotheses on a except that all the diagonal entries of a are zero the hadamard square root of a given by a 1 2 a ij 1 2 has just one positive eigenvalue and is invertible finally we show that if a is any positive semidefinite matrix and b is almost positive definite and invertible then a b 1 e t b 1 e a

computing the square roots of matrices with central symmetry for computing square roots of a nonsingular matrix a which are functions of a two well known fast and stable algorithms which are based on the schur decomposition of a were proposed by bjork and hammarling a bjork s hammarling a schur method for the square root of a matrix linear alg appl 52 53 1983 127 140 for square roots of general complex matrices and by higham n j higham computing real square roots of a real matrix linear alg appl 88 89 1987 405 430 for real square roots of real matrices in this paper we further consider the computation of the square roots of matrices with central symmetry we first investigate the structure of the square roots of these matrices and then develop several algorithms for computing the square roots we show that our algorithms ensure significant savings in computational costs as compared to the use of standard algorithms for arbitrary matrices

survey generic properties and control of linear structured systems a survey in this survey paper we consider linear structured systems in state space form where a linear system is structured when each entry of its matrices like a b c and d is either a fixed zero or a free parameter the location of the fixed zeros in these matrices constitutes the structure of the system indeed a lot of man made physical systems which admit a linear model are structured a structured system is representative of a class of linear systems in the usual sense it is of interest to investigate properties of structured systems which are true for almost any value of the free parameters therefore also called generic properties interestingly a lot of classical properties of linear systems can be studied in terms of genericity moreover these generic properties can in general be checked by means of directed graphs that can be associated to a structured system in a natural way we review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms by properties we mean here system specific properties like controllability the finite and infinite zero structure and so on as well as solvability issues of certain classical control problems like disturbance rejection input output decoupling and so on in this paper we do not try to be exhaustive but instead by a selection of results we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool we emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only moreover the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well known polynomially bounded combinatorial techniques for larger systems

the generic number of invariant zeros of a structured linear system in this paper we consider a structured linear system represented by means of a directed graph we present a graph theoretic method to compute the generic number of invariant zeros of the corresponding rosenbrock matrix the method is based on a fundamental decomposition of the directed graph representing the structured system the generic number of invariant zeros is important in generic versions of problems involving controllability and pole placement

renyi entropy and complexity measure for skew gaussian distributions and related families abstract in this paper we provide the renyi entropy and complexity measure for a novel flexible class of skew gaussian distributions and their related families as a characteristic form of the skew gaussian shannon entropy we give closed expressions considering a more general class of closed skew gaussian distributions and the weighted moments estimation method in addition closed expressions of renyi entropy are presented for extended skew gaussian and truncated skew gaussian distributions finally additional inequalities for skew gaussian and extended skew gaussian renyi and shannon entropies are reported

the computation of the square roots of circulant matrices abstract in the first part of this paper we investigate the reduced forms of circulant matrices and quasi skew circulant matrices by using their properties we present two efficient algorithms to compute the square roots of circulant matrices and quasi skew circulant matrices respectively those methods are faster than the traditional algorithm which is based on the schur decomposition in the second part we further consider circulant h matrices with positive diagonal entries and develop two algorithms for computing their principal square roots those two algorithms have the common advantage that is they only need matrix matrix multiplications in their iterative sequences an operation which can be done very efficiently on modern high performance computers

structural invertibility of transfer functions in this note we study certain properties of a transfer function in terms of the structure of the state space representation a b c specifically the rank of the transfer function and the degree of a decoupled system which can be obtained by applying a left decoupler are characterized in terms of the structure of a b c

the iterative methods for centrosymmetric matrices in this paper we give some new centrosymmetric splittings of a centrosymmetric matrix a and obtain corresponding iterative methods for a linear system ax b these iterative schemes are particularly based on the centrosymmetric property of a and aimed at reduction of the cost of computation and storage

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

observer design for general linear time invariant systems it is noted that for both explicit and implicit linear systems several observer design problems including system inversion for systems with or without unknown inputs can all be formulated as a single problem involving descriptor systems the method used for solving this problem is based on recursively applying the reduction operation which consists of identifying and discarding equations that do not provide useful information for the estimation problem the reduction algorithm produces a reduced dimension system for which standard observers can be constructed the method presented can be implemented using numerically robust algorithms one such algorithm is presented

analysis of linear structured systems using a primal dual algorithm in this paper we give a polynomial algorithm to compute the infinite structure of a structured system a directed graph is associated with such structured systems the infinite zero orders can be computed on the associated graph via the determination of the minimal length of vertex disjoint input output paths this search corresponds to a minimum cost flow determination on an appropriate directed graph the proposed algorithm is based on the primal dual algorithm linked to linear programming this polynomial algorithm is one of the most efficient for this type of problem moreover it allows an iterative determination of the generic infinite structure which is a key tool for solving numerous control problems

on almost invariant subspaces of structural systems and decentralized control graph theoretic conditions are obtained for a structured system to have the property that the supremal l sub p almost invariant controllability subspace is generically the entire state space and that the infimal sup l sub p almost conditional invariant complementary observability subspace is generically the zero subspace the conditions are used to determine stabilizability of structured interconnected systems by means of decentralized feedback control although the obtained graph theoretic conditions are conservative they are considered satisfactory to the extent that the benefits of easy testability involving only binary calculations outweigh the conservativeness in the results

a statistical measure of complexity with nonextensive entropy abstract a statistical measure of complexity utilising the concept of entropy or information is proposed our way in this study is to use a nonextensive entropy instead of an extensive additive shannon entropy in the definition but can be characterised as a difference between the qth order renyi entropy and the second one furthermore we devise a conditional joint and mutual complexity measure as a coherent possibility the behavior of the measure for the logistic map shows that it is more sensitive to nonextensivity at the transition point ac 3 8284 than any other values when 0

hamiltonian square roots of skew hamiltonian matrices revisited abstract recently h fassbender et al linear algebra appl 287 1999 125 proved the following theorem every real skew hamiltonian matrix w has a real hamiltonian square root h i e h 2 w we prove an analog of this theorem for complex matrices our approach may be of independent interest namely we use the polar decomposition of a nonsingular operator acting in a space with the symplectic inner product

positive integer powers of complex skew symmetric circulant matrices abstract in j rimas on computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices i applied mathematics and computation 165 2005 137 141 j rimas on computing of arbitrary positive integer powers for one type of even order symmetric circulant matrices i applied mathematics and computation 172 2006 86 90 rimas derived a general expression for the entries of the qth power q n of the n n real symmetric circulant matrix circ n 0 1 0 0 0 0 1 for all n n in this paper we present an extension of that interesting work deriving a similar expression for the entries of the positive integer powers of any complex symmetric circulant matrix

structure at infinity of structured descriptor systems and its applications the generic structure at infinity of the transfer matrix of a descriptor system is investigated under the physically reasonable assumption that the coefficients in the equations are classified into independent physical parameters and dimensionless constants for such a structured descriptor system the generic structure at infinity is characterized in terms of an independent assignment or weighted matroid intersection problem this leads to necessary and sufficient conditions for the generic solvability of the exact matching problem for a singular descriptor system and for the generic solvability of the disturbance decoupling problem for a nonsingular descriptor system the obtained conditions can be checked by efficient matroid theoretic algorithms

some results on symmetric circulant matrices and on symmetric centrosymmetric matrices we derive a readily computable sufficient condition for the existence of a nonnegative symmetric circulant matrix having a prescribed spectrum moreover we prove that any set 1 2 n is the spectrum of real symmetric centrosymmetric matrices s1 and s2 such that for s1 an eigenvector corresponding to 1 is the all ones vector and for s2 an eigenvector corresponding to 1 is the vector with components 1 and 1 alternately the proof is constructive then we derive an improved condition on k k 1n such that s1 1e where e is a stochastic symmetric centrosymmetric matrix finally we propose an algorithm to compute the eigenvalues of some real symmetric centrosymmetric matrices all the numerical procedures are based on the use of the fast fourier transform

simultaneous decoupling and disturbance rejection a structural approach abstract the combined problem of disturbance rejection and decoupling for linear systems is considered new necessary and sufficient conditions are derived for the existence of a state feedback controller that rejects the disturbances and simultaneously decouples the closed loop system these conditions are simple intuitively appealing and are expressed in terms of system structural invariants such as infinite zero orders then we consider the disturbance rejection with simultaneous decoupling problem for structured linear systems necessary and sufficient conditions for the generic solvability of this problem are provided they are expressed in terms of input output path length comparisons on the system associated graph

on the hss iteration methods for positive definite toeplitz linear systems we study the hss iteration method for large sparse non hermitian positive definite toeplitz linear systems which first appears in bai golub and ng s paper published in 2003 z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 and hss stands for the hermitian and skew hermitian splitting of the coefficient matrix a in this note we use the hss iteration method based on a special case of the hss splitting where the symmetric part h 12 a a t is a centrosymmetric matrix and the skew symmetric part s 12 a a t is a skew centrosymmetric matrix for a given toeplitz matrix hence fast methods are available for computing the two half steps involved in the hss and ihss iteration methods some numerical results illustrate their effectiveness

output feedback disturbance decoupling graph interpretation for structured systems abstract in this paper we consider first the output feedback disturbance decoupling problem for linear systems we derive a set of new necessary or sufficient conditions for the solvability of this problem these conditions are expressed in terms of system structural invariants such as infinite zero orders and essential orders then we consider the output feedback disturbance decoupling problem for structured linear systems necessary or sufficient conditions for the generic solvability of this problem are provided they are expressed in terms of input output path length comparisons on the system associated graph

brief paper state and input observability recovering by additional sensor implementation a graph theoretic approach this paper deals with the problem of additional sensor location in order to recover the state and input observability for structured linear systems the proposed method is based on a graph theoretic approach and assumes only the knowledge of the system s structure it allows one to provide the minimal number of the required sensors and either their pertinent location or a necessary and sufficient condition which allows one to check if a sensor location is adequate or not we obtain a sensor placement procedure based on classical and well known graph theory algorithms which have polynomial complexity orders

structurally fixed modes a notion of structurally fixed modes is introduced in the framework of decentralized control systems to identify the modes that cannot be shifted by decentralized feedback regardless of the numerical values of system parameters in this way a qualitative characterization of decentralized fixed modes is provided which is suitable for delineating decentrally stabilizable systems the characterization leads naturally to the notion of fixed modes with respect to arbitrary feedback structure constraints

brief paper graph theoretic characterization of fixed modes in frequency domain we examine a characterization of fixed modes in a frequency domain framework based on the graphical representation of systems with multiple poles we present a graph theoretic characterization of the characteristic polynomials of closed loop systems under arbitrary structural constraints based on this characterization fixed modes are characterized using the weight products of 1 factors of the weighted directed graphs associated with closed loop systems it is also shown that the well known characterization of fixed modes in terms of transmission zeros can be derived from the proposed graph theoretic characterization