a matroid theoretic approach to structurally fixed modes of control systems the structurally fixed modes for a decentralized control system described in the descriptor form f dot bf x a bf x b bf u h bf y c bf x are investigated under a physically reasonable assumption that the coefficients in the equations are classified into independent physical parameters and dimensionless fixed constants a necessary and sufficient condition for the existence of structurally fixed modes is given in matroid theoretic terms the condition can be tested by an efficient algorithm for the independent flow problem the combinatorial canonical form of a layered mixed matrix plays a central role in deriving the condition

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

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

the pencil se a and controllability observability for generalized linear systems a geometric approach in the first part of the paper we study the pencil se a by using some geometric concepts given by bernhard siam j control optim 20 1982 pp 612 6331 and some new ones which are introduced here such concepts are then used to obtain an alternative characterization for the finite and infinite zero structure of the pencil and to construct a polynomial basis for ker se a necessary and sufficient conditions for the rows or and the columns of the pencil to be linearly independent over the ring of the polynomials are also given the main geometric properties of a regular pencil are presented including the identification of the subspace in which the impulsive response of the autonomous generalized linear system e dot x ax takes place in the second part we consider the generalized linear system e dot x ax bu y cx and we give necessary and sufficient conditions for the infinite zeros of the regular pencil se a to be controllable and observable

note on a graph theoretic criterion for structural output controllability the structural output controllability of a dynamical system is studied it is known that the maximum size of linking in an associated dynamic graph provides a good upper bound on the dynamic graph and it was believed that the bound is exact this conjecture is disproved and a new efficiently computed lower bound is presented

group decision making with a fuzzy linguistic majority abstract the determination of solutions in group decision making is considered the point of departure is a collection of individual fuzzy preference relations a solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation as opposed to conventional approaches in which a crisp threshold type majority rule is used we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier e g most much more than 50 etc a calculus of linguistically quantified propositions is applied using the fuzzy majority various solution concepts are derived mainly of the type of core minimax opposition set and consensus winner

a noncommutative analogue of d xk kxk 1 abstract the paper concerns alternating powers of a hilbert space let k be defined by k a x1 xk ax1 axk it is proved that the norm of the linear map d k a depends only upon a and is assumed at the identity

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

graph theoretic approach to symbolic analysis of linear descriptor systems abstract continuous descriptor systems e x ax bu y cx where e is a possibly singular matrix are symbolically analyzed by means of digraphs starting with four different digraph characterizations of square matrices and determinants the author favors the cauchy coates interpretation then an appropriate digraph representation of the matrix pencil se a is given which is followed by a digraph interpretation of det se a and the transfer function matrix c se a 1 b next a graph theoretic procedure is derived to reveal a possibly hidden factorizability of the determinant det se a this is very important for large scale systems finally as an application of the derived results an electrical network is analyzed symbolically

some recent results in combinatorial approaches to dynamical systems abstract this paper is a unified presentation of recent results obtained by the author on the structural analysis of dynamical systems by matroid theoretic combinatorial methods the first part consists of the results on combinatorial and algorithmic characterizations of control theoretic generic properties the analysis is based on a physically reasonable mathematical model which assumes that the coefficients in the equations are classified into independent physical parameters and dimensionless fixed constants the concept of mixed matrix and its canonical form turn out to be convenient mathematical tools the second part describes a combinatorial analogue of the dynamical system theory developed in the matroid theoretic framework the matrices a and b in the state space equations ax bu are replaced by bimatroids or linking systems

managing non homogeneous information in group decision making within the frame of decision aid literature decision making problems with multiple sources of information have drawn the attention of researchers from a wide spectrum of disciplines in decision situations with multiple individuals each one has his own knowledge on the alternatives of the decision problem the use of information assessed in different domains is not a seldom situation this non homogeneous information can be represented as values belonging to domains with different nature as linguistic numerical and interval valued or can be values assessed in label sets with different granularity multi granular linguistic information decision processes for solving these problems are composed by two phases aggregation and exploitation the main problem to deal with non homogeneous contexts is how to aggregate the information assessed in these contexts in this paper taking as base the 2 tuple fuzzy linguistic representation model we shall develop an aggregation process for dealing with non homogeneous contexts in first place we shall develop an aggregation process for combining numerical interval valued and linguistic information afterwards we shall propose different extensions of this process to deal with contexts in which can appear other type of information as intuitionistic fuzzy sets or multi granular linguistic information

generic controllability theorems for descriptor systems this paper presents an extension of the structural controllability theorem to the case of structured descriptor systems sds a major difficulty in this problem is the loss of independence among the entries of the system matrices this difficulty is resolved by introducing the idea of column structured matrices csm first the paper defines and characterizes c and r controllability for descriptor systems then the generic controllability theorem is proved extending the structural controllability theorem to column structured systems css finally based on this result necessary conditions and sufficient conditions for generic r and c controllability are obtained the result gives a complete characterization of generic controllability for sds with nonsingular e and a partial solution for the case of singular e

quantum analysis non commutative differential and integral calculi a new scheme of quantum analysis namely a non commutative calculus of operator derivatives and integrals is introduced this treats differentiation of an operator valued function with respect to the relevant operator in a banach space in this new scheme operator derivatives are expressed in terms of the relevant operator and its inner derivation explicitly derivatives of hyperoperators are also defined some possible applications of the present calculus to quantum statistical physics are briefly discussed

variation of grassman powers and spectra the operator norm of the derivative of the map which takes a finite dimensional linear operator to its kth grassman power the kth compound is evaluated this leads to a bound for the distance between the grassman powers of two operators as an important application a bound for the distance between the eigenvalues of two operators is obtained

variation of induced linear operators abstract let v be an n dimensional inner product space let be an irreducible character of the symmetric group s m and let v be the symmetry class of tensors associated with it let a be a linear operator on v and let k a be the operator it induces on v we obtain an explicit expression for the norm of the derivative of the map a k a in terms of the singular values of a two special cases of this problem antisymmetric and symmetric tensor products have been studied earlier and our results reduce to the earlier ones in these cases

variation of symmetric tensor powers and permanents the norm of the derivative of the symmetric tensor power of an operator is evaluated exactly a bound for the distance between the permanents of two matrices is obtained as a consequence

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

on structural controllability theorem the structural controllability theorem was given first by lin for the single input case by the graph theoretic approach and several authors extended this to the multiinput case by the matrix algebra in this correspondence a graph theoretic interpretation of the condition that the structured matrix a b has generically full rank is derived from well known theorem in the graph theory and this leads to a quite simple proof of the structural controllability theorem

derivations derivatives and chain rules for a smooth function f on the space of bounded operators in a hilbert space we obtain formulas for the nth order commutator f a x x x in terms of the frechet derivatives dmf a we illustrate the use of these formulas in obtaining bounds for norms of generalised commutators f a x xf b and their higher order analogues

eigensets and power products of a bimatroid abstract we consider a bimatroid linking system which has a natural one to one corre spondence between the row set and the column set such bimatroid can be regarded as a combinatorial counterpart of a matrix which is subject to similarity transformations for such a bimatroid we can consider the power product introduce the notion of an eigenset which is a combinatorial analogue of an eigenvector and define the jordan type in particular the maximum size of an eigenset is characterized by the rank of the power products

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

derivatives for antisymmetric tensor powers and perturbation bounds abstract in an earlier paper r bhatia t jain higher order derivatives and perturbation bounds for determinants linear algebra appl 431 2009 2102 2108 we gave formulas for derivatives of all orders for the map that takes a matrix to its determinant in this paper we continue that work and find expressions for the derivatives of all orders for the antisymmetric tensor powers and for the coefficients of the characteristic polynomial we then evaluate norms of these derivatives and use them to obtain perturbation bounds

on the gap between the structural controllability of time varying and time invariant systems structural controllability of time invariant and time varying systems when the input control sequences have a restricted length k is compared the dimensions of controllable space coincide in the following three special cases the input sequences have length k 2 the input sequences have k n where n is the size of the system i e the ultimate controllability is the same in both cases and for every length of input sequences provided that the system has a single input only it is proved that there may appear a gap for every input length k such that 2

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

refined study on structural controllability of descriptor systems by means of matroids for a dynamical system described in the descriptor form fd bf x dt a bf x b bf u where the coefficients are classified into generic parameters and fixed constants the structural controllability is investigated under a physically reasonable assumption that can be justified by the dimensional analysis a necessary and sufficient condition for the structural controllability is given in matroid theoretic terms the condition can be tested by efficient algorithms for the matroid union intersection problem

hierarchical decomposition of symmetric discrete systems by matroid and group theories an algebraic method is proposed for the hierarchical decomposition of large scale group symmetric discrete systems into partially ordered subsystems it aims at extracting substructures and hierarchy for such systems as electrical networks and truss structures

a controllability condensed form and a state feedback pole assignment algorithm for descriptor systems a direct algorithm for the pole assignment problem of a time invariant linear multivariable descriptor system with linear state feedback is presented the system is assumed to be completely controllable for both finite and infinite poles the algorithm is based on the controllability condensed form which is derived