new good quasi cyclic ternary and quaternary linear codes let n k d q codes be linear codes of length n dimension k and minimum hamming distance d over gf q the following quasi cyclic codes are constructed in this paper 44 11 20 3 55 11 26 3 66 11 32 3 48 12 21 3 60 12 28 3 56 13 24 3 65 13 29 3 56 14 23 3 60 15 23 3 64 16 25 3 36 9 19 4 90 9 55 4 99 9 61 4 30 10 14 4 50 10 27 4 55 10 30 4 33 11 15 4 44 11 22 4 55 11 29 4 36 12 16 4 48 12 23 4 60 12 31 4 all of these codes have established or exceed the respective lower bounds on the minimum distance given by brouwer

on k t subnormal covering codes the concept of a k t subnormal covering code is defined it is discussed how an amalgamated direct sumlike construction can be used to combine such codes the existence of optimal q n m 1 codes c is discussed such that by puncturing the first coordinate of c one obtains a code with q 1 subnorm 2

quasi cyclic codes over z sub 4 and some new binary codes previously linear codes over z sub 4 and quasi cyclic qc codes over fields have been shown to yield useful results in coding theory combining these two ideas we study z sub 4 qc codes and obtain new binary codes using the usual gray map among the new codes the lift of the famous golay code to z sub 4 produces a new binary code a 92 2 sup 24 28 code which is the best among all binary codes linear or nonlinear moreover we characterize cyclic codes corresponding to free modules in terms of their generator polynomials

linear codes with covering radius 2 and other new covering codes infinite families of linear binary codes with covering radius r 2 and minimum distance d 3 and d 4 are given using the constructed codes with d 3 r 2 families of covering codes with r 2 are obtained the parameters of many constructed codes with r

an explicit construction of 2 generator quasi twisted codes quasi twisted qt codes are a generalization of quasi cyclic qc codes based on consta cyclic simplex codes a new explicit construction of a family of 2 generator quasi twisted qt two weight codes is presented it is also shown that many codes in the family meet the griesmer bound and therefore are length optimal new distance optimal binary qc 195 8 96 210 8 104 and 240 8 120 codes and good ternary qc 208 6 135 and 221 6 144 codes are also obtained by the construction

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

control of linear switched systems with receding horizon modal information we provide an exact solution to two performance problems one of disturbance attenuation and one of windowed variance minimization subject to exponential stability considered are switched systems whose parameters come from a finite set and switch according to a language such as that specified by an automaton the controllers are path dependent having finite memory of past plant parameters and finite foreknowledge of future parameters exact convex synthesis conditions for each performance problem are expressed in terms of nested linear matrix inequalities the resulting semidefinite programming problem may be solved offline to arrive at a suitable controller a notion of path by path performance is introduced for each performance problem leading to improved system performance non regular switching languages are considered and the results are extended to these languages two simple physically motivated examples are given to demonstrate the application of these results

on input to state stability of discrete time switched nonlinear time varying systems in this paper input to state stability iss for discrete time switched nonlinear time varying sntv systems is investigated starting with discrete time nonlinear time varying ntv systems some improved sufficient conditions are proposed to verify the iss of systems by using the weak implication form iss wi iss lyapunov function weak dissipative form iss wd iss lyapunov function and interval descent technique then the results obtained are extended to study the iss of discrete time sntv systems several relaxed conditions are given by using piecewise wi iss and wd iss lyapunov functions minimum dwell time and infinite switching methods respectively comparing with the existing results the obtained conditions release the requirement on negative definiteness of the differences of piecewise lyapunov functions moreover all subsystems are allowed to be unstable in the case of infinite switching finally a numerical example is given to illustrate the theoretical results

nonconservative lifted convex conditions for stability of discrete time switched systems under minimum dwell time constraint in this note a novel conception called virtual clock which is defined by an artificial timer over a finite cycle is introduced for stability analysis of discrete time switched linear systems under minimum dwell time constraint two necessary and sufficient conditions associated with a virtual clock with a sufficient length are proposed to ensure the global uniform asymptotic stability of discrete time switched linear systems for the two nonconservative stability criteria the lifted version maintains the convexity in system matrices based on the lifted convex conditions the extensions to ell 2 gain computation and mathcal h infty control problems are presented in the sequel in particular a novel virtual clock dependent controller is designed which outperforms the traditional mode dependent and common gain controllers several numerical examples are provided to illustrate our theoretic results

a gel fand type spectral radius formula and stability of linear constrained switching systems abstract using ergodic theory in this paper we present a gel fand type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup s restricted to a subset that need not carry the algebraic structure of s this generalizes the berger wang formula using it as a tool we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one sided markov shift associated to s

unified stability criteria for slowly time varying and switched linear systems this paper presents a unified approach to formulating stability conditions for slowly time varying linear systems and switched linear systems the concept of total variation is generalized to the case of matrix valued functions using this generalized concept a result extending existing stability conditions for slowly time varying linear systems is derived as special cases of this result two sets of stability conditions are derived for switched linear systems which match known results in the literature a numerical example is included to further illustrate the application of the main result

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

stability of discrete time switching systems with constrained switching sequences we introduce a novel framework for the stability analysis of discrete time linear switching systems with switching sequences constrained by an automaton the key element of the framework is the algebraic concept of multinorm which associates a different norm per node of the automaton and allows to exactly characterize stability building upon this tool we develop the first arbitrarily accurate approximation schemes for estimating the constrained joint spectral radius that is the exponential growth rate of a switching system with constrained switching sequences more precisely given a relative accuracy r 0 the algorithms compute an estimate of within the range 1 r these algorithms amount to solve a well defined convex optimization program with known time complexity and whose size depends on the desired relative accuracy r 0

approximation of the constrained joint spectral radius via algebraic lifting this paper studies the linear constrained switching system which is a discrete time switched linear system whose switching sequences are constrained by a deterministic finite automaton the stability of a constrained switching system is characterized by its constrained joint spectral radius that is known to be difficult to compute or approximate using the semi tensor product of matrices the matrix form expression of a constrained switching system is shown to be equivalent to that of a lifted arbitrary switching system then the constrained joint generalized spectral radius of a constrained switching system is proved to be equal to the joint generalized spectral radius of its lifted arbitrary switching system which can be approximated by off the shelf algorithms

stabilizing switching signals for switched systems this article deals with stability of continuous time switched systems under constrained switching given a family of systems possibly containing unstable dynamics we characterize a new class of switching signals under which the switched system generated by it and the family of systems is globally asymptotically stable our characterization of such stabilizing switching signals involves the asymptotic frequency of switching the asymptotic fraction of activation of the constituent systems and the asymptotic densities of admissible transitions among them our techniques employ multiple lyapunov like functions and extend preceding results both in scope and applicability

graph lyapunov function for switching stabilization and distributed computation abstract this paper studies stabilization of discrete time switched linear systems slss using the notion of graph control lyapunov functions gclfs a gclf is a set of lyapunov functions defined on a weighted digraph where each lyapunov function is represented by a node in the digraph and there is a lyapunov inequality associated with each subgraph consisting of a node and its out neighbors the weight of a directed edge indicates the decay or growth rate of the lyapunov functions it is proved that an sls is switching stabilizable if and only if there exists a gclf the main benefits of gclfs are reduced computational cost and conservatism for stabilizability tests besides we show that the proposed gclf framework unifies several control lyapunov functions and the related stabilization theorems moreover we propose a distributed algorithm to evaluate the stabilizability with reduced computational costs by taking benefits of the graph structure of gclfs several examples are given to demonstrate the efficiency of the algorithm

six new binary quasi cyclic codes six new quasi cyclic codes are presented which improve the lower bounds on the minimum distance for a binary code a local exhaustive search is used to find these codes and many other quasi cyclic codes which attain the lower bounds

new quasi twisted quaternary linear codes let n k d sub q codes be linear codes of length n dimension k and minimum hamming distance d over gf q in this article 31 new quaternary q 4 codes are constructed which improve the respective lower bounds on minimum distance in brouwer s tables

uniform stabilization of discrete time switched and markovian jump linear systems the uniform stability of discrete time switched linear systems possibly with a strongly connected switching path constraint and the existence of finite path dependent dynamic output feedback controllers uniformly stabilizing such a system are both shown to be characterized by the existence of a finite dimensional feasible system of linear matrix inequalities this characterization is based on the observation that a linear time varying system is uniformly stable only if there exists a finite path dependent quadratic lyapunov function the synthesis of a uniformly stabilizing controller is done without conservatism by solving any feasible system of linear matrix inequalities among an increasing family of systems of linear matrix inequalities the result carries over to the almost sure uniform stabilization of markovian jump linear systems

optimal disturbance attenuation for discrete time switched and markovian jump linear systems an exact condition for uniform stabilization and disturbance attenuation for switched linear systems is given in the discrete time domain via the union of an increasing family of linear matrix inequality conditions associated with each markovian jump linear system is a switched linear system so we obtain a necessary and sufficient condition for almost sure uniform stabilization and disturbance attenuation for markovian jump linear systems as well the results lead to semidefinite programming based controller synthesis techniques from which optimal finite path dependent linear dynamic output feedback controllers arise naturally in particular under the notion of path by path optimal disturbance attenuation finite path dependent controllers can outperform the usual mode dependent ones

new ternary quasi cyclic codes with better minimum distances one of the most important problems of coding theory is to construct codes with best possible minimum distances in this article using the structure of quasi cyclic qc codes with a bch type bound given by lally and fitzpatrick see wcc 99 workshop on coding and cryptography paris france 1999 we have found 20 new ternary quasi cyclic codes with improved minimum distances

constructions for perfect mixed codes and other covering codes a construction for an infinite family of perfect mixed codes with covering radius 2 is presented these are the first known nontrivial perfect mixed codes with covering radius greater than 1 based on mixed codes constructions for binary covering codes that lead to a considerable improvement of upper bounds on the sizes of covering codes are presented these codes and some other codes can be obtained by the blockwise direct sum construction two infinite families of codes are of special interest they are quasi perfect nonlinear union of their disjoint translates covers the space and their density as covering codes is remarkably low

observability and controllability analysis of linear systems subject to data losses we provide algorithmically verifiable necessary and sufficient conditions for fundamental system theoretic properties of discrete time linear systems subject to data losses more precisely the systems in our modeling framework are subject to disruptions data losses in the feedback loop where the set of possible data loss sequences is captured by an automaton as such the results are applicable in the context of shared wireless communication networks and or embedded architectures where some information on the data loss behavior is available a priori we propose an algorithm for deciding observability or the absence of it for such systems and show how this algorithm can be used also to decide other properties including constructibility controllability reachability null controllability detectability and stabilizability by means of relations that we establish among these properties the main apparatus for our analysis is the celebrated skolem s theorem from linear algebra moreover we study the relation between the model adopted in this paper and a previously introduced model where instead of allowing dropouts in the feedback loop one allows for time varying delays

the berger wang formula for the markovian joint spectral radius abstract the berger wang formula establishes equality between the joint and generalized spectral radii of a set of matrices for matrix products whose multipliers are applied not arbitrarily but in accordance with some markovian law there are also known analogs of the joint and generalized spectral radii however the known proofs of the berger wang formula hardly can be directly applied in the case of markovian products of matrices since they essentially rely on the arbitrariness of appearance of different matrices in the related matrix products nevertheless as has been shown by x dai 1 the berger wang formula is valid for the case of markovian analogs of the joint and the generalized spectral radii too although the proof in this case heavily exploits the more involved techniques of multiplicative ergodic theory in the paper we propose a matrix theory construction allowing to deduce the markovian analog of the berger wang formula from the classical berger wang formula

stability and stabilizability of discrete time dual switching systems with application to sampled data systems in this paper stability and stabilizability of discrete time dual switching linear systems is investigated the switched systems under consideration have two switching variables one of them is stochastic described by an underlying markov chain the other one can be regarded either as a deterministic disturbance or as a control input leading to stability or stabilizability problems respectively for the considered class of systems sufficient conditions for mean square stability with or without control gain synthesis and mean square stabilizability are provided in terms of matrix inequalities when the stochastic switching is driven by an independent identically distributed sequence we establish simpler conditions without additional conservatism then it is shown how the proposed framework can be used to study aperiodic sampled data systems with stochastic computation times the results are illustrated on examples borrowed from the literature

stability analysis and control synthesis for switched systems a switched lyapunov function approach this paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete time domain the approach followed in this paper looks at the existence of a switched quadratic lyapunov function to check asymptotic stability of the switched system under consideration two different linear matrix inequality based conditions allow to check the existence of such a lyapunov function the first one is classical while the second is new and uses a slack variable which makes it useful for design problems these two conditions are proved to be equivalent for stability analysis investigating the static output feedback control problem we show that the second condition is in this case less conservative the reduction of the conservatism is illustrated by a numerical evaluation

on the accuracy of the ellipsoid norm approximation of the joint spectral radius the joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set this quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate we introduce in this paper an approximation hat rho that is based on ellipsoid norms that can be computed by convex optimization and that is such that the joint spectral radius belongs to the interval hat rho sqrt n hat rho where n is the dimension of the matrices we also provide a simple approximation for the special case where the entries of the matrices are non negative in this case the approximation is proved to be within a factor at most m m is the number of matrices of the exact value

new ternary linear codes let n k d q codes be linear codes of length n dimension k and minimum hamming distance d over gf q in this correspondence 18 codes are constructed which improve the known lower bounds on minimum distance

an ad hoc sor method abstract an ad hoc method is presented for determining a sequence of successive overrelaxation factors that although having no rigorous foundations at the moment appears to produce a highly competitive iteration scheme for several classes of difference equations a different is chosen for each difference equation

lie algebraic stability criteria for switched systems it was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable lie algebra possesses a quadratic common lyapunov function which implies that the corresponding switched linear system is exponentially stable for arbitrary switching in this paper we prove that the same properties hold under the weaker condition that the lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact lie group the corresponding local stability result for nonlinear switched systems is also established moreover we demonstrate that if a lie algebra fails to satisfy the above condition then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable relevant facts from the theory of lie algebras are collected at the end of the paper for easy reference

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

joint spectral radius and path complete graph lyapunov functions we introduce the framework of path complete graph lyapunov functions for approximation of the joint spectral radius the approach is based on the analysis of the underlying switched system via inequalities imposed among multiple lyapunov functions associated to a labeled directed graph inspired by concepts in automata theory and symbolic dynamics we define a class of graphs called path complete graphs and show that any such graph gives rise to a method for proving stability of the switched system this enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic common sum of squares path dependent quadratic and maximum minimum of quadratics lyapunov functions we compare the quality of approximation obtained by certain classes of path complete graphs including a family of dual graphs and all path complete graphs with two nodes on an alphabet of two matrices we derive approximation guarantees for several families of path complete graphs such as the de bruijn graphs this provides worst case performance bounds for path dependent quadratic lyapunov functions and a constructive converse lyapunov theorem for maximum minimum of quadratics lyapunov functions

some best rate 1 p and rate p 1 p systematic quasi cyclic codes over gf 3 and gf 4 the class of quasi cyclic qc codes has been proven to contain many good codes to date the known qc codes are primarily rate 1 p and p 1 p binary codes constructed from circulant matrices these results are extended to qc codes over gf 3 and gf 4 codes are constructed using integer linear programming and heuristic combinatorial optimization many of these attain the maximum possible minimum distance for any linear code with the same parameters and several improve the maximum known distances the link between power residue pr codes and qc codes is exploited as a means of constructing new qc codes and to initialize the search algorithm previously unknown minimum distances for nonbinary quadratic residue codes are given the minimum distances for the pr codes and the maximum known distances for the qc codes are tabulated

performance of certain iterative methods in solving implicit difference schemes for 2 d navier stokes equations the performance of three iterative methods local sor preconditioned generalized conjugate gradient methods such as orthomin 1 and the gcg ls method with preconditioning matrices derived by incomplete modified and unmodified pointwise factorizations of the matrix corresponding to implicit finite difference schemes for the 2 d navier stokes equations is compared the relative merits of these methods in terms of cpu time average number of iterations with respect to the reynolds number and discretization parameters are emphasized

dynamic output feedback control for positive roesser system under the switched and t s fuzzy rules abstract this paper is concerned with the dynamic output feedback controller design for the positive roesser type nonlinear system which is intrinsically characterized by the switched mechanism and subtly decomposed into the linear form under the takagi sugeno fuzzy rules firstly based on the co positive lyapunov function and the average dwell time method sufficient conditions are presented under which the resulting closed loop system is exponentially stable and has l 1 gain bound then explicit expressions are also given to derive the expected controller gain matrices with desired l 1 gain bound finally effectiveness of the proposed results is illustrated by two numerical examples

a new construction for covering codes a novel method for constructing normal binary nonlinear covering codes is presented the construction improves several upper bounds on k n r the minimum cardinality of a binary code of length n and covering radius r

on path complete lyapunov functions geometry and comparison we study optimization based criteria for the stability of switching systems known as path complete lyapunov functions and ask the question can we decide algorithmically when a criterion is less conservative than another our contribution is twofold first we show that a path complete lyapunov function which is a multiple lyapunov function by nature can always be expressed as a common lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it geometrically our results provide for each path complete criterion an implied invariant set second we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given path complete lyapunov functions