on the simultaneous diagonal stability of a pair of positive linear systems in this paper we derive a necessary and sufficient condition for the existence of a diagonal common quadratic lyapunov function cqlf for a pair of positive linear time invariant lti systems

optimal entanglement witnesses from generalized reduction and robertson maps we provide a generalization of the reduction and robertson positive maps in matrix algebras they give rise to a new class of optimal entanglement witnesses their structural physical approximation is analyzed as a byproduct we provide new examples of ppt positive partial transpose entangled states

optimal entanglement criterion for mixed quantum states we develop a strong and computationally simple entanglement criterion the criterion is based on an elementary positive map phi which operates on state spaces with even dimension n or 4 it is shown that phi detects many entangled states with a positive partial transposition ppt and that it leads to a class of optimal entanglement witnesses this implies that there are no other witnesses which can detect more entangled ppt states the map phi yields a systematic method for the explicit construction of high dimensional manifolds of bound entangled states

a new criterion for indecomposability of positive maps we give a criterion for a positive mapping on the space of operators on a hilbert space to be indecomposable we show that this criterion can be applied to two families of positive maps these families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states

construction of 3 3 entangled edge states with positive partial transposes we construct a class of 3 3 entangled edge states with positive partial transposes using indecomposable positive linear maps this class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes

stronger than lyapunov notions of matrix stability or how flowers help solve problems in mathematical ecology abstract persistent patterns of interactions in a multi component system e g intra and inter species relations in a community of n interacting species may imply a number of formalizations as special stronger than lyapunov notions of matrix stability like d stability qualitative stability volterra lyapunov stability and others a variety of these notions each having a certain motivation with regard to uncertainties inherent in model applications constitute a hierarchical topology sometimes very intricate and not yet well understood in a formal space of real n n matrices as visible forms of this hierarchy matrix flowers are suggested where petals correspond to subsets of particular stability kinds whose visible inclusion intersection represent logical implication junction the flowers are constructed under a few simple conventions and in the absence of ready characterizations to draw a petal often poses a challenging mathematical problem whose solution may reveal a new biological knowledge of a general nature particular petals concern the topics of strong and weak interactions in a community key species in its structure diffusion instability in its spatial dynamics where the flower plays a heuristic role in formulating a new problem or and stimulating a new application

stability analysis of second order switched homogeneous systems we study the stability of second order switched homogeneous systems using the concept of generalized first integrals we explicitly characterize the most destabilizing switching law and construct a lyapunov function that yields an easily verifiable necessary and sufficient condition for asymptotic stability using the duality between stability analysis and control synthesis this also leads to a novel algorithm for designing a stabilizing switching controller

strong stability of discrete time systems the paper introduces a new notion of stability for internal state space autonomous system descriptions in discrete time referred to as strong stability which extends a parallel notion introduced in the continuous time case this is a stronger notion of stability compared to alternative definitions asymptotic lyapunov which prohibits systems described by natural coordinates to have overshooting responses for arbitrary initial conditions in state space three finer notions of strong stability are introduced and necessary and sufficient conditions are established for each one of them the class of discrete time systems for which strong and asymptotic stability coincide is characterized and links between the skewness of the eigen frame and the violation of strong stability property are obtained connections between the notions of strong stability in the continuous and discrete domains are briefly discussed finally strong stabilization problems under state and output feedback are studied the results of the paper are illustrated with a numerical example

cones of positive maps and their duality relations the structure of cones of positive and k positive maps acting on a finite dimensional hilbert space is investigated special emphasis is given to their duality relations to the sets of superpositive and k superpositive maps we characterize k positive and k superpositive maps with regard to their properties under taking compositions a number of results obtained for maps are also rephrased for the corresponding cones of block positive k block positive separable and k entangled operators due to the jamiolkowski choi isomorphism generalizations to a situation where no such simple isomorphism is available are also made employing the idea of mapping cones as a side result to our discussion we show that extreme entanglement witnesses which are optimal should be of special interest in entanglement studies

on cones and stability abstract lyapunov while studying the asymptotic stability of solutions of differential systems proved a theorem which yields a necessary and sufficient condition for stability of a matrix lyapunov s theorem was a breakthrough in the research of stability among others it links cones and stability in various directions in this survey we review some of the links between convexity and matrix stability mainly such links that are derived from lyapunov s theorem the article does not cover the wide variety of all such links but rather gives some of the flavor of this rich theory

a series of absolutely indecomposable positive maps in matrix algebras abstract we provide examples of positive maps in m n c n 4 which cannot be decomposed into a sum of a 2 positive map and a 2 copositive map

feedback controller norm optimization for linear time invariant descriptor systems with pole region constraint an algorithm is proposed to compute a state feedback gain matrix for a linear time invariant regular descriptor system which ensures that i the closed loop system is impulse free and ii all the finite poles are placed within a pre defined stability region in the complex plane the associated design flexibilities are exploited in minimizing upper bounds of the frobenius norm of respective gain matrices a class of linear matrix inequality lmi regions in the complex plane are chosen as constraints for the closed loop finite poles by representing a subset of the nonsingular matrices through solution of an lmi and linearizing the set of matrix inequalities arise in the regional pole finite poles assignment the associated optimizations are formulated as semidefinite programs the effectiveness of the developed algorithm is demonstrated through numerical examples significant reduction in the norm of feedback gain matrix is achieved in a two generator six bus power system

the structural physical approximation conjecture it was conjectured that the structural physical approximation spa of an optimal entanglement witness is separable or equivalently that the spa of an optimal positive map is entanglement breaking this conjecture was disproved first for indecomposable maps and more recently for decomposable maps the arguments in both cases are sketched along with important related results this review includes background material on topics including entanglement witnesses optimality duality of cones decomposability and the statement and motivation for the spa conjecture so that it should be accessible for a broad audience

constructing positive maps from block matrices positive maps are useful for detecting entanglement in quantum information theory any entangled state can be detected by some positive map in this paper the relation between positive block matrices and completely positive trace preserving maps is characterized consequently a new method for constructing decomposable maps from positive block matrices is derived in addition a method for constructing positive but not completely positive maps from hermitian block matrices is also obtained

robust pole placement in lmi regions discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality lmi regions a class of convex regions of the complex plane that embraces most practically useful stability regions the focus is on linear systems with static uncertainty on the state matrix for this class of uncertain systems the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary lmi regions the resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities lmis and cover both unstructured and parameter uncertainty these analysis results are then applied to the synthesis of dynamic output feedback controllers that robustly assign the closed loop poles in a prescribed lmi region with some conservatism this problem is again tractable via lmi optimization in addition robust pole placement can be combined with other control objectives such as h sub 2 or h sub spl infin performance to capture realistic sets of design specifications physically motivated examples demonstrate the effectiveness of this robust pole clustering technique

on completely copositive and decomposable linear transformations abstract characterization theorems and other results for the cone of completely copositive linear transformations are given further results including the cone of decomposable linear transformations and its dual follow

reduced order controller synthesis with regional pole constraint abstracta reduced order output feedback controller is designed for a linear time invariant system which guarantees that the closed loop poles are placed within some pre specified stability region in the complex plane a convex approximation of the non convex constraints is used to pose a sequence of semi definite programs which provide the lowest order proper controller satisfying the approximate constraints the proposed method is demonstrated on two practical controller design applications

partial pole placement with controller optimization an arbitrary subset n m of the n closed loop eigenvalues of an nth order continuous time single input linear time invariant system is to be placed using full state feedback at pre specified locations in the complex plane the remaining m closed loop eigenvalues can be placed anywhere inside a pre defined region in the complex plane this region constraint on the unspecified poles is translated into a linear matrix inequality constraint on the feedback gains through a convex inner approximation of the polynomial stability region the closed loop locations for these m eigenvalues are optimized to obtain a minimum norm feedback gain vector this reduces the controller effort leading to less expensive actuators required to be installed in the control system the proposed algorithm is illustrated on a linearized model of a 4 machine 2 area power system example

positive maps from irreducibly covariant operators in this paper we discuss positive maps induced by irreducibly covariant linear operators for finite groups the application of group theory methods allows deriving some new results of a different kind in particular a family of necessary conditions for positivity for such objects is derived stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map in the low dimensional cases for the permutation group s 3 and the quaternion group q the necessary and sufficient conditions are given together with the discussion on their decomposability in higher dimensions we present positive maps induced by a three dimensional irreducible representation of the permutation group s 4 and d dimensional representation of the monomial unitary group mu d in the latter case we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction we show that the generalised choi map can be obtained by considered covariant maps induced by the monomial unitary group as an additional result a novel interpretation of the fujiwara algolet conditions for positivity and complete positivity is presented finally in the end a new form of an irreducible representation of the symmetric group s n is constructed allowing us to simplify the form of certain choi jamiolkowski images derived for irreducible representations of the symmetric group

peres horodecki separability criterion for continuous variable systems the peres horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states the partial transpose operation admits in the continuous case a geometric interpretation as mirror reflection in phase space this recognition leads to uncertainty principles stronger than the traditional ones to be obeyed by all separable states for all bipartite gaussian states the peres horodecki criterion turns out to be a necessary and sufficient condition for separability

globally convergent algorithms for robust pole assignment by state feedback it is observed that an algorithm proposed in 1985 by kautsky nichols and van dooren knv amounts to maximizing at each iteration the determinant of the candidate closed loop eigenvector matrix x with respect to one of its columns with unit length constraint subject to the constraint that it remains an achievable closed loop eigenvector matrix this interpretation is used to prove convergence of the knv algorithm it is then shown that a more efficient algorithm is obtained if det x is concurrently maximized with respect to two columns of x and such a scheme is easily extended to the case where the eigenvalues to be assigned include complex conjugate pairs variations exploiting the availability of multiple processors are suggested convergence properties of the proposed algorithms are established their superiority is demonstrated numerically

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

brief positive polynomial matrices and improved lmi robustness conditions recently several new lmi conditions for stability of linear systems have been proposed introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non convex robust stability conditions in this paper we show that these improved lmi conditions can be derived with the help of some basic results on positive polynomial matrices the approach allows us to derive in a unifying way results in the state space and polynomial frameworks applications to robust stability analysis and robust stabilization of systems with multi linear parametric uncertainty are fully described

feedback norm minimisation with regional pole placement this article proposes a convex algorithm for minimising an upper bound of the state feedback gain matrix norm with regional pole placement for linear time invariant multi input systems the inherent non convexity in this optimisation is resolved by a combination of two separate approaches 1 an inner convex approximation of the polynomial matrix stability region due to henrion and 2 a novel convex parameterisation of column reduced matrix fraction system representations using a sequence of approximations enabled by the above two methods it is shown that the constraints on closed loop poles both pre specified exact locations and regional placement define linear matrix inequalities finally the effectiveness of the proposed algorithm is compared with similar pole placement algorithms through numerical examples

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

generation of mapping cones from small sets we answer in the affirmative a recently posed question that asked if there exists an untypical convex mapping cone i e one that does not arise from the transpose map and the cones of k positive and k superpositive maps we explicitly construct such a cone based on atomic positive maps our general technique is to consider the smallest convex mapping cone generated by a single map and we derive several results on such mapping cones we use this technique to also present several other examples of untypical mapping cones including a family of cones generated by spin factors we also provide a full characterization of mapping cones generated by single elements in the qubit case in terms of their typicality

a class of atomic positive linear maps in matrix algebras we find a large class of atomic positive linear maps between n dimensional matrix algebras and study their properties in connection with complete positivity 2 positivity and schwarz inequality

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

on positive linear maps between matrix algebras abstract there is a concrete example of a positive linear map from m 2 to m 4 which is not decomposable modification of this map gives an explicit counterexample to a conjecture of woronowicz on the strong kadison inequality

on structural physical approximations and entanglement breaking maps very recently a conjecture saying that the so called structural physical approximations spas to optimal positive maps optimal entanglement witnesses give entanglement breaking eb maps separable states has been posed korbicz et al 2008 phys rev a 78 062105 the main purpose of this contribution is to explore this subject first we extend the set of entanglement witnesses supporting the conjecture then we ask whether spas constructed from other than the depolarizing channel maps also lead to eb maps and show that in general this is not the case on the other hand we prove an interesting fact that for any positive map there exists an eb channel such that the spa of constructed with the aid of is again an eb channel finally we ask similar questions in the case of continuous variable systems we provide a simple way of constructing spa and prove that in the case of the transposition map it gives the eb channel

low order controller design for siso systems using coprime factors and lmi this paper develops a low order controller design method for linear continuous time invariant single input single output systems requiring only the solution of a convex optimization problem the technique integrates several well known results in control theory an important step is the use of coprime factors so that based on strictly positive real functions feedback stabilization using low order controllers becomes a zero placement problem which is convex from this result we develop algorithms to solve two optimal control problems

robust exact pole placement via an lmi based algorithm this technical note deals with the robust exact pole placement problem pole placement algorithms that guarantee a small variation of the assigned poles against possible perturbations the solution to this problem is related to the solvability of a sylvester like equation thus the main issue is to compute a well conditioned solution to this equation also the best candidate solution must possess the minimal condition number to reduce sensitivity to perturbation this problem is cast as a global optimization under linear matrix inequality constraints for which a numerical convergent algorithm is provided and compared with the most attractive methods in the literature

indecomposable positive maps in low dimensional matrix algebras abstract we classify a series of positive maps in low dimensional matrix algebras with respect to the degree of indecomposability and provide stronger indecomposable maps in m n we also present a basic result about the atomic property

a class of positive atomic maps we construct a new class of positive indecomposable maps in the algebra of d d complex matrices these maps are characterized by the weakest positivity property and for this reason they are called atomic this class provides a new rich family of atomic entanglement witnesses which define an important tool for investigating quantum entanglement it turns out that they are able to detect states with the weakest quantum entanglement

strong stability of internal system descriptions this article introduces a new notion of stability for internal autonomous system descriptions that is referred to as strong stability and characterises the case of avoiding overshoots for all initial conditions within a given sphere this is a stronger notion of stability compared with alternative definitions asymptotic lyapunov which allows the analysis and design of control systems described by natural coordinates to have no overshooting response for arbitrary initial conditions for the case of linear time invariant systems necessary and sufficient conditions for strong stability are established in terms of the negative definiteness of the symmetric part of the state matrix the invariance of strong stability under orthogonal transformations is established and this enables the characterisation of the property in terms of the invariants of the schur form of the state matrix some interesting relations between strong stability and special forms of coordinate frames are examined and links between th