the t k diagnosability of star graph networks the t k diagnosis is a diagnostic strategy at system level that can significantly enhance the system s self diagnosing capability it can detect up to t faulty processors or nodes units which might include at most k misdiagnosed processors where k is typically a small number somani and peleg 1996 claimed that an n dimensional star graph denoted s n a well studied interconnection model for multiprocessor systems is k 1 n 3k 2 k diagnosable recently chen and liu 2012 found counterexamples for the diagnosability obtained in without further pursuing the cause of the flawed result in this paper we provide a new complete proof that an mbi n dimensional star graph is actually k 1 n 3k 1 k diagnosable where 1 leq k leq 3 and investigate the reason that caused the flawed result in based on our newly obtained fault tolerance properties we will also outline an o n log n diagnostic algorithm n n is the number of nodes in s n to locate all up to k 1 n 3k 1 faulty processors among which at most k 1 leq k leq 3 fault free processors might be wrongly diagnosed as faulty

the pessimistic diagnosabilities of some general regular graphs the pessimistic diagnosis strategy is a classic strategy based on the pmc model a system is t t diagnosable if provided the number of faulty processors is bounded by t all faulty processors can be isolated within a set of size at most t with at most one fault free node mistaken as a faulty one the pessimistic diagnosability of a system g denoted by t p g is the maximal number of faulty processors so that the system g is t t diagnosable in this paper we study the pessimistic diagnosabilities of some general k regular k connected graphs g n the main result t p g n 2 k 2 g under some conditions is obtained where g is the maximum number of common neighbors between any two adjacent vertices in g n as applications of the main result every pessimistic diagnosability of many famous networks including some known results such as the alternating group networks an n the k ary n cubes q n k the star graphs s n the matching composition networks g g 1 g 2 m and the alternating group graphs ag n are obtained we study the pessimistic diagnosabilities of some general regular graphs under the pmc model the exact value of t p g n is obtained as applications the pessimistic diagnosability of an n q n k s n some mcn and ag n etc are obtained

positive invariant sets for fault tolerant multisensor control schemes this article deals with fault tolerant multisensor control schemes for systems with linear dynamics positive invariance is a common analysis and control design tool for systems affected by bounded constraints and disturbances this article revisits the construction of e approximations of minimal robust positive invariant sets for linear systems upon contractive set iterations the cases of switching between different sets of disturbances and the inclusion of a predefined region of the state space are treated in detail all these results are used in multisensor control schemes which have to deal with specific problems originated by the switching between different estimators and by the presence of faults in some of the sensors the construction of positive invariant sets for different operating regimes provides in this context effective fault detection information within the same framework global stability of the switching strategies can be assured if the invariant sets topology allows the exclusive se

diagnosability of star graphs with missing edges in this paper we study the system diagnosis on an n dimensional star under the comparison model following the concept of local diagnosability 3 the strong local diagnosability property 7 is discussed this property describes the equivalence of the local diagnosability of a node and its degree we prove that an n dimensional star has this property and it keeps this strong property even if there exist n 3 missing edges in it

the t k diagnosability of the bc graphs processor fault diagnosis takes an important role in fault tolerant computing on multiprocessor systems there are two classical diagnosis strategies the precise strategy and the pessimistic strategy both of which are based on the well known pmc diagnostic model nevertheless the degree of diagnosability of the system is limited under these two strategies a better method called the t k diagnosis strategy is proposed by somani and peleg in which the identified fault set is allowed to contain at most k fault free processors using this diagnosis strategy the degree of diagnosability of the hypercube increases greatly as the number of the fault free processors in the fault set increases we study the t k diagnosability of so called bc graphs that include hypercubes crossed cubes mobius cubes and twisted cubes etc we show that any n dimensional bc graph is t n k k diagnosable when n spl ges 4 and 0 spl les k spl les n where t n k k 1 n 1 2 k 1 k 2 1 therefore the crossed cube the mobius cube and the twisted cube all have the same t k diagnosability as the hypercube as a result the algorithms developed for diagnosis on the hypercube may also be used to diagnose multiprocessor systems whose network topologies are based on bc graphs

lee distance and topological properties of k ary n cubes in this paper we consider various topological properties of a k ary n cube q sub n sup k using lee distance we feel that lee distance is a natural metric for defining and studying a q sub n sup k after defining a q sub n sup k graph using lee distance we show how to find all disjoint paths between any two nodes given a sequence of radix k numbers a function mapping the sequence to a gray code sequence is presented and this function is used to generate a hamiltonian cycle embedding the graph of a mesh and the graph of a binary hypercube into the graph of a q sub n sup k is considered using a k ary gray code we show the embedding of a k n sub 1 spl times k n sub 2 spl times spl times k n sub m dimensional mesh into a q sub n sup k where n spl sigma sub i l sup m n sub i then using a single digit 4 ary reflective gray code we demonstrate embedding a q sub n into q sub n 2 sup 4 we look at how lee distance may be applied to the problem of resource placement in a q sub n sup k by using a lee distance error correcting code although the results in this paper are only preliminary lee distance error correcting codes have not been applied previously to this problem finally we consider how lee distance can be applied to message routing and single node broadcasting in a q sub n sup k in this section we present two single node broadcasting algorithms that are optimal when single port and multi port i o is used

active fault isolation a duality based approach via convex programming this paper presents the mathematical conditions and the associated design methodology of an active fault diagnosis technique for continuous time linear systems given a set of faults known a priori the system is modeled by a finite family of linear time invariant systems accounting for one healthy and several faulty configurations by assuming bounded disturbances and using a residual generator an invariant set and its projection in the residual space i e its limit set are computed for each system configuration each limit set related to a single system configuration is parameterized with respect to the system input thanks to this design active fault isolation can be guaranteed by the computation of a test input either constant or periodic such that the limit sets associated with different system configurations are separated and the residual converges toward one limit set only in order to alleviate the complexity of the explicit computation of the limit set an implicit dual representation is adopted leading to efficient procedures based on quadratic programming for computing the test input the developed methodology offers a competent continuous time solution to the optimization based computation of the test input via hahn banach duality simulation examples illustrate the application of the proposed active fault diagnosis methods and its efficiency in providing a solution even in relatively large state dimensional problems

fault diagnosability of data center networks abstract a kind of generalization of diagnosability for a network g is g good neighbor diagnosability which is denoted by t g g let g g be the r g connectivity lin et al 2016 17 and xu et al 2017 29 gave the same problem independently that the relationship between the r g connectivity g g and t g g of a general graph g needs to be studied in the future in this paper this open problem is solved for general regular graphs we firstly establish the relationship of g g and t g g and obtain that t g g g g g under some conditions secondly we obtain the g good neighbor diagnosability of data center network d k n which are t g d k n g 1 k 1 n g for 1 g n 1 under the pmc model and the mm model respectively furthermore we show that d k n is tightly super n k 1 connected for n 2 and k 2 and we also prove that the largest connected component of the survival graph contains almost all of the remaining vertices in d k n when n 2 k 2 vertices removed

conditional diagnosability of alternating group graphs let an be the alternating group of degree n with n 3 set s 1 2 i 1 i 2 3 i n the alternating group graph denoted by agn is defined as the cayley graph on an with respect to s jwo et al networks 23 1993 315 326 introduced alternating group graph agn as an interconnection network topology for computing systems conditional diagnosability a new measure of diagnosability introduced by lai et al ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of agn is 4 for n 4 and 6n 18 for each n 5

remarks on two integral operators and numerical methods for csie in this paper the author extends the mapping properties of some singular integral operators in zygmund spaces equipped with uniform norm as a by product quadrature methods for solving csie having indices 0 and 1 are proposed their stability and convergence are proved and error estimates in zygmund norm are given some numerical tests are also shown

bounds and invariant sets for a class of discrete time switching systems with perturbations we present a novel method to compute componentwise ultimate bounds and invariant regions for a class of switching discrete time linear systems with perturbation bounds that may depend nonlinearly on a delayed state the method has the important advantage that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise this componentwise method does not employ a standard norm for bounding either the perturbation or state vectors and thus may avoid conservativeness due to different perturbation or state vector components having substantially different bounds we also establish the relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic lyapunov function we illustrate the application of our method via numerical examples including the fault tolerance analysis of the feedback control of a winding machine

the extra connectivity of bubble sort star graphs connectivity plays an important role in measuring the fault tolerance of a multiprocessor system in the case of vertices failures extra connectivity is an important indicator of a network s ability for diagnosis and fault tolerance in this paper we analyze the fault tolerance ability of the bubble sort star network denoted by b s n a well known interconnection network proposed for multiprocessor systems and establish the g extra connectivity for 1 g 3

continuous time probabilistic ultimate bounds and invariant sets the concepts of ultimate bounds and invariant sets play a key role in several control theory problems as they replace the notion of asymptotic stability in the presence of unknown disturbances however when the disturbances are unbounded as in the case of gaussian white noise no ultimate bounds nor invariant sets can in general be found to overcome this limitation we introduced in previous work the notions of probabilistic ultimate bound pub and probabilistic invariant set pis for discrete time systems this article extends the notions of pub and pis to continuous time systems studying their main properties and providing tools for their calculation in addition the use of these concepts in robust control design by covariance assignment is presented

brief paper feedback and invariance under uncertainty via set iterates we examine discrete time control systems under non parametric disturbances sets which a given control feedback makes invariant under the disturbance are analyzed via lifting the feedback operation to the space of sets properties of being an attractor of the disturbed dynamics and being a minimal invariant set are derived from the corresponding notions of the set dynamics yielding in turn useful characterizations and error estimates for numerical algorithms which detect the minimal invariant sets concrete numerics for some examples of practical feedback rules are offered

sufficient conditions for k restricted edge connected graphs for a connected graph g v e an edge set s e is a k restricted edge cut if g s is disconnected and every component of g s has at least k vertices the k restricted edge connectivity of g denoted by k g is defined as the cardinality of a minimum k restricted edge cut let k g min x x x k g x is connected where x v x g is maximally k restricted edge connected k optimal for short if k g k g the k restricted edge connectivity is more refined network reliability indices than edge connectivity in this paper let k 2 be an integer and let g be a graph of order g at least 2k satisfying n u n v 2 k 2 for all pairs u v of nonadjacent vertices if for each triangle t there exists at least one vertex v v t such that d v g 2 k 1 then g is k optimal

set theoretic approach to fault tolerant control of linear parameter varying systems with sensor reintegration abstractin this paper we present a fault tolerant control scheme for linear parameter varying systems that utilises multiple sensor switching to compensate for sensor faults the closed loop scheme consists of an estimator based feedback tracking controller and sensor estimate switching strategy which allows for the reintegration of previously faulty sensors the switching mechanism tracks the transitions from faulty to healthy behaviour by means of set separation and pre computed transition times the sensor estimate pairings are then reconfigured based on available healthy sensors under the proposed scheme preservation of closed loop system boundedness is guaranteed for a wide range of sensor fault situations an example is presented to illustrate the performance of the fault tolerant control strategy

on the computation of set induced control lyapunov functions for continuous time systems the paper presents a computation oriented method for characterizing and obtaining local control lyapunov functions induced by particular star shaped nonconvex sets for continuous time nonlinear systems with bounded inputs for a given set the necessary and sufficient conditions for the induced function to be a nonconvex local control lyapunov function are provided the related convex problems for computing the exact region in which the function is decreasing and the optimal control input are presented the results are applied to the brockett integrator

robust time optimal control of constrained linear systems a version of dynamic programming which computes level sets of the value function rather than the value function set itself is used to design robust non linear controllers for linear discrete time dynamical systems subject to hard constraints on controls and states the controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time for the robust regulator problem the control invariant terminal set is a neighborhood preferably small of the origin for the robust tracking problem the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero two non linear controllers which utilize the level sets of the value function are described the first requires the controller to solve on line a modest linear program whose dimension is approximately the same as that of the control variable the second decomposes each level set into a set of simplices a piecewise linear control law affine in each simplex is then constructed

on robust stability and set invariance of switched linear parameter varying systems in this paper we address the problem of robust stability and set invariance of switched linear parameter varying lpv systems affected by bounded disturbances a switched lpv system is a collection of single lpv subsystems with a switching rule that decides which subsystem is active at each time this framework is useful for lpv systems with large parameter variation range for which it may be difficult to assess or guarantee stability over the whole parameter range our first contribution is to derive a result for a single lpv system that extends and generalises previous work for switched systems our second and main contribution is to apply this seed result in the derivation of dwell time type conditions on the switched lpv switching rule which ensure robust closed loop stability and set invariance of the trajectories of the switched lpv system across the whole parameter range as an application of the results we consider the problem of reference tracking for lpv systems using switched lpv state fe

k restricted edge connectivity for some interconnection networks abstract k restricted edge connectivity is an important parameter in measuring the reliability and fault tolerance of large interconnection networks in this paper we present two families of graphs similar with the networks proposed by chen et al y c chen j j m tan l h hsu s s kao super connectivity and super edge connectivity for some interconnection networks applied mathematics and computation 140 2003 245 254 and study the k k 2 3 restricted edge connectivity of these graphs in particular as the applications of our results the k k 2 3 restricted edge connectivity of the recursive circulant graphs and the n ary k cubes is given

the g extra connectivity and diagnosability of crossed cubes abstract connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network g in 1996 fabrega and fiol proposed the g extra connectivity of g in 2016 zhang et al proposed the g extra diagnosability of g that requires every component of g s has at least g 1 vertices the g extra connectivity of g is necessary for g extra diagnosability of g in this paper we show that the g extra connectivity of the crossed cube cq n is n g 1 1 2 g g 3 for n 5 0 g n 2 and the g extra diagnosability of cq n is n 1 2 g g 1 under the pmc model for n 5 0 g n 2 and the mm model for n 7 0 g n 2

extraconnectivity of k ary n cube networks given a graph g and a non negative integer g the g extraconnectivity of g is the minimum cardinality of a set of vertices in g if such a set exists whose deletion disconnects g and leaves every remaining component with more than g vertices this study shows that the 2 extraconnectivity of a k ary n cube q n k for k 4 and n 5 is equal to 6n 5

an interval approach to compute invariant sets this paper proposes an original interval based method to compute an outer approximation of all invariant sets such as limit cycles of a continuous time nonlinear dynamic system which are included inside a prior set of the state space contrary to all other existing approaches our method has the following properties first it is guaranteed a solution cannot be lost second it is applicable to a large class of systems without any specific assumption such as the knowledge of a lyapunov function or any partial linearity and third there is no need to integrate the system

a fast algorithm for solving the generalized airfoil equation abstract the present paper is devoted to the quadrature method for cauchy type singular integral equations on 1 1 with an additional logarithmic singularity of the kernel an algorithm being based on this method is given whose realization only requires a computational work of o n ln n operations in order to determine n unknown coefficients of the approximate solution furthermore in a scale of weighted sobolev like spaces on 1 1 having analogous properties like the classical sobolev spaces on the unit circle the order of convergence is optimal

further remarks on asymptotic stability and set invariance for linear delay difference equations we studied the existence of positively invariant sets for linear delay difference equations in particular we regarded two strong stability notions robust with respect to delay parameter asymptotic stability for the discrete time case and delay independent stability for the continuous time case the correlation between these stability concepts is also considered furthermore for the delay difference equations with two delay parameters we provided a computationally efficient numerical routine which is necessary to guarantee the existence of contractive sets of lyapunov razumikhin type this condition also appears to be necessary and sufficient for the delay independent stability and sufficient for the robust asymptotic stability

fault tolerance and diagnosability of burnt pancake networks under the comparison model processor fault diagnosis plays an important role in measuring the reliability of a multiprocessor system and the diagnosabilities of many well known multiprocessor systems have been investigated the conditional diagnosability has been widely accepted as a new measure of diagnosability by assuming an additional condition that any faulty set can t contain all the neighbors of any node in a multiprocessor system in this paper we explore algebraic and combinatorial properties of burnt pancake networks and investigate the structural vulnerability as well as super and extra connectivities furthermore we show that the classic diagnosability and the conditional diagnosability of n dimensional burnt pancake network bp n n 4 are n and 3 n 4 respectively algebraic and combinatorial properties of burnt pancake networks are established the structural vulnerability as well as super and extra connectivities are explored the classic and conditional diagnosabilities under the comparison model are derived

on g extra connectivity of folded hypercubes let g be a connected graph and g a non negative integer the g extra connectivity of g is the minimum cardinality of a set of vertices in g if it exists whose deletion disconnects g and leaves each remaining component with more than g vertices in several recent publications the g extra connectivity of an n dimensional folded hypercube was determined for g 3 and some specific n see for example chang tsai and hsieh 2014 4 in this paper an extension of the above results to all 0 g n 1 and n 7 is presented