necessary and sufficient small gain conditions for integral input to state stable systems a lyapunov perspective this paper is concerned with conditions for the stability of interconnected nonlinear systems consisting of integral input to state stable iiss systems with external inputs the treatment of iiss and input to state stable iss systems is unified both necessary conditions and sufficient conditions are investigated using a lyapunov formulation in the presence of model uncertainty this paper proves that for the stability of the interconnected system at least one subsystem is necessarily iss which is a stronger stability property in the set of iiss the necessity of a small gain type property is also demonstrated this paper proposes a common form of smooth lyapunov functions which can establish the iiss and the iss of the interconnection comprising iiss and iss subsystems whenever the small gain type condition is satisfied the result covers situations more general than the earlier study and removes technical conditions assumed in the previous literature global asymptotic stability is discussed as a special case

exponential stability of impulsive systems with application to uncertain sampled data systems we establish exponential stability of nonlinear time varying impulsive systems by employing lyapunov functions with discontinuity at the impulse times our stability conditions have the property that when specialized to linear impulsive systems the stability tests can be formulated as linear matrix inequalities lmis then we consider lti uncertain sampled data systems in which there are two sources of uncertainty the values of the process parameters can be unknown while satisfying a polytopic condition and the sampling intervals can be uncertain and variable we model such systems as linear impulsive systems and we apply our theorem to the analysis and state feedback stabilization we find a positive constant which determines an upper bound on the sampling intervals for which the stability of the closed loop is guaranteed the control design lmis also provide controller gains that can be used to stabilize the process we also consider sampled data systems with constant sampling intervals and provide results that are less conservative than the ones obtained for variable sampling intervals

robust input to state stability for hybrid systems this work addresses input to state stability iss for hybrid dynamical systems which combine continuous time dynamics on a flow set and discrete time dynamics on a jump set the main result entails equivalent characterizations of iss when the right hand side of the differential equation for the continuous time dynamics is locally lipschitz but the set of admissible derivative values generated by considering all possible input values is not necessarily convex under some mild assumptions we show that existence of an iss lyapunov function is equivalent to various types of robust iss for these hybrid systems as an application we demonstrate how a hybrid system framework can be used to study some stability properties for continuous time systems in particular we use the derived equivalent characterization results of iss for hybrid systems to recover and generalize lyapunov and asymptotic characterizations of input output to state stability for continuous time systems

brief paper input to state stability of switched systems and switching adaptive control in this paper we prove that a switched nonlinear system has several useful input to state stable iss type properties under average dwell time switching signals if each constituent dynamical system is iss this extends available results for switched linear systems we apply our result to stabilization of uncertain nonlinear systems via switching supervisory control and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide iss properties with respect to the estimation errors detailed illustrative examples are included

stability analysis for stochastic hybrid systems this survey addresses stability analysis for stochastic hybrid systems shs which are dynamical systems that combine continuous change and instantaneous change and that also include random effects we re emphasize the common features found in most of the models that have appeared in the literature which include stochastic switched systems markov jump systems impulsive stochastic systems switching diffusions stochastic impulsive systems driven by renewal processes diffusions driven by levy processes piecewise deterministic markov processes general stochastic hybrid systems and stochastic hybrid inclusions then we review many of the stability concepts that have been studied including lyapunov stability lagrange stability asymptotic stability and recurrence next we detail lyapunov based sufficient conditions for these properties and additional relaxations of lyapunov conditions many other aspects of stability theory for shs like converse lyapunov theorems and robustness theory are not fully developed hence we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces

input output to state stability and state norm estimators for switched nonlinear systems in this paper the concepts of input output to state stability ioss and state norm estimators are considered for switched nonlinear systems under average dwell time switching signals we show that when the average dwell time is large enough a switched system is ioss if all of its constituent subsystems are ioss moreover under the same conditions a non switched state norm estimator exists for the switched system furthermore if some of the constituent subsystems are not ioss we show that still ioss can be established for the switched system if the activation time of the non ioss subsystems is not too big again under the same conditions a state norm estimator exists for the switched system however in this case the state norm estimator is a switched system itself consisting of two subsystems we show that this state norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for

brief paper input to state stability for networked control systems via an improved impulsive system approach this paper presents a novel impulsive system approach to input to state stability iss analysis of networked control systems ncss with time varying sampling intervals and delays this approach is based upon the new idea that an ncs can be viewed as an interconnected hybrid system composed of an impulsive subsystem and an input delay subsystem a new type of time varying discontinuous lyapunov krasovskii functional which makes full use of the information on the piecewise constant input and the bounds of the network delays is introduced to analyze the iss property of ncss linear matrix inequality based sufficient conditions are derived for iss of ncss with respect to external disturbances when applied to the approximate tracking problem for ncss the derived iss result provides bounds on the steady state tracking error numerical examples are provided to show the efficiency of the proposed approach

leader following consensus of a class of stochastic delayed multi agent systems with partial mixed impulses in this paper the exponential leader following consensus problem is investigated for a class of nonlinear stochastic networked multi agent systems with partial mixed impulses and unknown time varying but bounded delays the main feature of partial mixed impulses is that time varying impulses are not only composed of synchronizing and desynchronizing impulses simultaneously but they are also injected into a fraction of nodes in multi agent systems three kinds of partial mixed impulses are proposed fixed partial mixed impulses periodic partial mixed impulses and try once discard like partial mixed impulses by means of the lyapunov function theory and the comparison principle conditions are derived for ensuring global exponential leader following consensus under the presented three kinds of partial mixed impulses simulations of leader following consensus of robotic systems are provided to validate the effectiveness of the proposed results and to show the advantages of the proposed partial mixed impulses

input to state stability of nonlinear impulsive systems we prove that impulsive systems which possess an input to state stable iss lyapunov function are iss for time sequences satisfying the fixed dwell time condition if an iss lyapunov function is the exponential one we provide a stronger result which guarantees uniform iss of the whole system over sequences satisfying the generalized average dwell time condition then we prove two small gain theorems that provide a construction of an iss lyapunov function for an interconnection of impulsive systems if the iss lyapunov functions for subsystems are known the construction of local iss lyapunov functions via the linearization method is provided relations between small gain and dwell time conditions as well as between different types of dwell time conditions are also investigated although our results are novel already in the context of finite dimensional systems we prove them for systems based on differential equations in banach spaces that makes obtained results considerably more general

composition of interactions in relativistic quantum theory a scheme of composition of interactions which is the generalisation of sokolov s method 1975 of packing operators is proposed this scheme can be used in all forms of relativistic quantum dynamics in contrast to the formulation of coester and polyzou 1982 the author makes no use of the properties of multichannel wave operators but uses only the properties of the poincare group unitary representations a solution to the problem of composition of interactions in the instant form is given and it is shown that this solution agrees with that obtained by coester and polyzou

the sparse matrix transform for covariance estimation and analysis of high dimensional signals covariance estimation for high dimensional signals is a classically difficult problem in statistical signal analysis and machine learning in this paper we propose a maximum likelihood ml approach to covariance estimation which employs a novel non linear sparsity constraint more specifically the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform smt the smt is formed by a product of pairwise coordinate rotations known as givens rotations using this framework the covariance can be efficiently estimated using greedy optimization of the log likelihood function and the number of givens rotations can be efficiently computed using a cross validation procedure the resulting estimator is generally positive definite and well conditioned even when the sample size is limited experiments on a combination of simulated data standard hyperspectral data and face image sets show that the smt based covariance estimates are consistently more accurate than both traditional shrinkage estimates and recently proposed graphical lasso estimates for a variety of different classes and sample sizes an important property of the new covariance estimate is that it naturally yields a fast implementation of the estimated eigen transformation using the smt representation in fact the smt can be viewed as a generalization of the classical fast fourier transform fft in that it uses butterflies to represent an orthonormal transform however unlike the fft the smt can be used for fast eigen signal analysis of general non stationary signals

bayesian estimation of karhunen loeve expansions a random subspace approach one of the most widely used procedures for dimensionality reduction of high dimensional data is principal component analysis pca more broadly low dimensional stochastic representation of random fields with finite variance is provided via the well known karhunen loeve expansion kle the kle is analogous to a fourier series expansion for a random process where the goal is to find an orthogonal transformation for the data such that the projection of the data onto this orthogonal subspace is optimal in the l 2 sense i e which minimizes the mean square error in practice this orthogonal transformation is determined by performing an svd singular value decomposition on the sample covariance matrix or on the data matrix itself sampling error is typically ignored when quantifying the principal components or equivalently basis functions of the kle furthermore it is exacerbated when the sample size is much smaller than the dimension of the random field in this paper we introduce a bayesian kle procedure allowing one to obtain a probabilistic model on the principal components which can account for inaccuracies due to limited sample size the probabilistic model is built via bayesian inference from which the posterior becomes the matrix bingham density over the space of orthonormal matrices we use a modified gibbs sampling procedure to sample on this space and then build probabilistic karhunen loeve expansions over random subspaces to obtain a set of low dimensional surrogates of the stochastic process we illustrate this probabilistic procedure with a finite dimensional stochastic process inspired by brownian motion

input to state stability of interconnected hybrid systems we consider the interconnections of arbitrary topology of a finite number of iss hybrid systems and study whether the iss property is maintained for the overall system we show that if the small gain condition is satisfied then the whole network is iss and show how a non smooth iss lyapunov function can be explicitly constructed in this case

characterizations of input to state stability for hybrid systems abstract we study input to state stability iss for a broad class of hybrid systems which are combinations of a differential equation on a constraint set and a difference equation on another constraint set for this class of hybrid systems we establish the equivalence of iss nonuniform iss and the existence of a smooth iss lyapunov function by additionally assuming that the right hand side of the differential equation has a convex property with respect to inputs moreover we demonstrate by examples that the equivalence may fail when such a convexity assumption is removed

brief paper input to state stability and integral input to state stability of nonlinear impulsive systems with delays this paper is concerned with analyzing input to state stability iss and integral iss iiss for nonlinear impulsive systems with delays razumikhin type theorems are established which guarantee iss iiss for delayed impulsive systems with external input affecting both the continuous dynamics and the discrete dynamics it is shown that when the delayed continuous dynamics are iss iiss but the discrete dynamics governing the impulses are not the iss iiss property of the impulsive system can be retained if the length of the impulsive interval is large enough conversely when the delayed continuous dynamics are not iss iiss but the discrete dynamics governing the impulses are the impulsive system can achieve iss iiss if the sum of the length of the impulsive interval and the time delay is small enough in particular when one of the delayed continuous dynamics and the discrete dynamics are iss iiss and the others are stable for the zero input the impulsive system can keep iss iiss no matter how often the impulses occur our proposed results are evaluated using two illustrative examples to show their effectiveness

on input to state stability of stochastic retarded systems with markovian switching this note develops a razumikhin type theorem on pth moment input to state stability of hybrid stochastic retarded systems also known as stochastic retarded systems with markovian switching which is an improvement of an existing result an application to hybrid stochastic delay systems verifies the effectiveness of the improved result

input to state stability of impulsive and switching hybrid systems with time delay this paper investigates input to state stability iss and integral input to state stability iiss of impulsive and switching hybrid systems with time delay using the method of multiple lyapunov krasovskii functionals it is shown that even if all the subsystems governing the continuous dynamics in the absence of impulses are not iss iiss impulses can successfully stabilize the system in the iss iiss sense provided that there are no overly long intervals between impulses i e the impulsive and switching signal satisfies a dwell time upper bound condition moreover these impulsive iss iiss stabilization results can be applied to systems with arbitrarily large time delays conversely in the case when all the subsystems governing the continuous dynamics are iss iiss in the absence of impulses the iss iiss properties can be retained if the impulses and switching do not occur too frequently i e the impulsive and switching signal satisfies a dwell time lower bound condition several illustrative examples are presented with their numerical simulations to demonstrate the main results

a stabilized pod model for turbulent flows over a range of reynolds numbers optimal parameter sampling and constrained projection abstract we present a reduced basis technique for long time integration of parametrized incompressible turbulent flows the new contributions are threefold first we propose a constrained galerkin formulation that corrects the standard galerkin statement by incorporating prior information about the long time attractor for explicit and semi implicit time discretizations our statement reads as a constrained quadratic programming problem where the objective function is the euclidean norm of the error in the reduced galerkin algebraic formulation while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the n term expansion second we propose an a posteriori error indicator which corresponds to the dual norm of the residual associated with the time averaged momentum equation we demonstrate that the error indicator is highly correlated with the error in mean flow prediction and can be efficiently computed through an offline online strategy third we propose a greedy algorithm for the construction of an approximation space procedure valid over a range of parameters the greedy is informed by the a posteriori error indicator developed in this paper we illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two dimensional turbulent lid driven cavity flow on the reynolds number

some group theoretical aspects of the so0 1 4 invariant theory some properties of unitary representations of the de sitter group so0 1 4 are considered in the case of spin zero and positivity of the mass operator these representations involve a decomposition which ensures the fulfilment of the cluster separability condition and allows for the conventional interpretation of the de sitter analogues of momentum and boost operators the decomposition of the tensor product of representations belonging to the principal series contains the representations of the principal series with all masses under natural assumptions concerning the interaction operators there are no bound states in the theory which is due to the universal de sitter antigravity

the problem of interactions in de sitter invariant theories the structure of representations describing systems of free particles in the theory with the invariance group is investigated the property of the particles to be free means as usual that the representation describing a many particle system is the tensor product of the corresponding single particle representations i e no interaction is introduced it is shown that the mass operator contains only continuous spectrum in the interval and such representations are unitarily equivalent to ones describing interactions gravitational electromagnetic etc this means that there are no bound states in the theory and the hilbert space of the many particle system contains a subspace of states with the following property the action of free representation operators on these states is manifested in the form of different interactions possible consequences of the results are discussed

spectral representation and reduced order modeling of the dynamics of stochastic reaction networks via adaptive data partitioning dynamical analysis tools are well established for deterministic models however for many biochemical phenomena in cells the molecule count is low leading to stochastic behavior that causes deterministic macroscale reaction models to fail the main mathematical framework representing these phenomena is based on jump markov processes that model the underlying stochastic reaction network conventional dynamical analysis tools do not readily generalize to the stochastic setting due to nondifferentiability and absence of explicit state evolution equations we developed a reduced order methodology for dynamical analysis that relies on the karhunen loeve decomposition and polynomial chaos expansions the methodology relies on adaptive data partitioning to obtain an accurate representation of the stochastic process especially in the case of multimodal behavior as a result a mixture model is obtained that represents the reduced order dynamics of the system the schlogl model is used as a prototype bistable process that exhibits time scale separation and leads to multimodality in the reduced order model

brief paper a unified synchronization criterion for impulsive dynamical networks this paper focuses on the problem of globally exponential synchronization of impulsive dynamical networks two types of impulses are considered synchronizing impulses and desynchronizing impulses in previous literature all of the results are devoted to investigating these two kinds of impulses separately thus a natural question arises is there any unified synchronization criterion which is simultaneously effective for synchronizing impulses and desynchronizing impulses in this paper a unified synchronization criterion is derived for directed impulsive dynamical networks by proposing a concept named average impulsive interval the derived criterion is theoretically and numerically proved to be less conservative than existing results numerical examples including scale free and small world structures are given to show that our results are applicable to large scale networks

stability analysis of a class of hybrid stochastic retarded systems under asynchronous switching the stability of a class of hybrid stochastic retarded systems hsrss with an asynchronous switching controller is investigated in this model the controller design relies on the observed jumping parameters which are however delayed and thus can not be measured in real time precisely this delayed time interval referred to as the asynchronous switching interval is markovian and dependent on the actual switching signal the sufficient conditions under which the system is either stochastically asymptotic stable or input to state stable are obtained by applying the extended razumikhin type theorem to the asynchronous switching interval these results are less conservative as it is only required that the designed lyapunov function is non decreasing it is shown that the stability of the considered system can be guaranteed by a sufficiently small mode transition rate of the underlying markov process which is a conclusion similar to that in asynchronous deterministic switched systems the effectiveness and correctness of the obtained results are finally verified by a numerical example

input to state stability of networked control systems a new class of lyapunov uniformly globally asymptotically stable ugas protocols in networked control systems ncs is considered it is shown that if the controller is designed without taking into account the network so that it yields input to state stability iss with respect to external disturbances not necessarily with respect to the error that will come from the network implementation then the same controller will achieve semi global practical iss for the ncs when implemented via the network with a lyapunov ugas protocol moreover the iss gain is preserved the adjustable parameter with respect to which semi global practical iss is achieved is the maximal allowable transfer interval mati between transmission times

integral input to state stability for hybrid delayed systems with unstable continuous dynamics this paper considers integral input to state stability iiss for a class of hybrid time delay systems discrete dynamics includes impulsive and switching signals and continuous dynamics is not necessarily stable based on multiple lyapunov krasovskii functionals a dwell time bound is explicitly given to guarantee iiss of the hybrid delayed system compared with existing results on related problems the obtained stability criteria can be applied to a larger class of hybrid delayed systems moreover the obtained dwell time bound is less conservative than existing ones at last an example related to networked control systems ncss is provided to illustrate the effectiveness of the proposed result

robust stability of networks of iiss systems construction of sum type lyapunov functions this paper gives a solution to the problem of verifying stability of networks consisting of integral input to state stable iiss subsystems the iiss small gain theorem developed recently has been restricted to interconnections of two subsystems for large scale systems stability criteria relying only on gain type information that were previously developed address only input to state stable iss subsystems to address the stability problem involving iiss subsystems interconnected in general structure this paper shows how to construct lyapunov functions of the network by means of a sum of nonlinearly rescaled individual lyapunov functions of subsystems under an appropriate small gain condition

input to state stability of impulsive stochastic delayed systems under linear assumptions in this paper the input to state stability iss integral iss iiss and stochastic iss siss are investigated for impulsive stochastic delayed systems by means of the lyapunov krasovskii function and the average impulsive interval approach the conditions for iss type properties are derived under linear assumptions respectively for destabilizing and stabilizing impulses it is shown that if the continuous stochastic delayed system is iss and the impulsive effects are destabilizing then the hybrid system is iss with respect to a lower bound of the average impulsive interval moreover it is unveiled that if the continuous stochastic delayed system is not iss the impulsive effects can successfully stabilize the system for a given upper bound of the average impulsive interval an improved comparison principle is developed for impulsive stochastic delayed systems which facilitates the derivations of our results for iss iiss siss an example of networked control systems is provided to illustrate the effectiveness of the proposed results

technical communique infinite horizon h2 h control for stochastic systems with markovian jumps this paper studies robust h 2 h control problem for systems subjected to multiplicative noise and markovian parameter jumps a necessary sufficient condition for the existence of h 2 h control is presented by means of two coupled algebraic riccati equations respectively finally a suboptimal h 2 h controller design algorithm is also obtained by solving a convex optimization problem

stochastic input to state stability of switched stochastic nonlinear systems in this paper global asymptotic stability in probability gasip and stochastic input to state stability siss for nonswitched stochastic nonlinear nssnl systems and switched stochastic nonlinear ssnl systems are investigated for the study of gasip the definition which we considered is not the usual notion of asymptotic stability in probability stability in probability plus attractivity in probability it can depict the properties of the system quantitatively correspondingly based on this definition some sufficient conditions are provided for nssnl systems and ssnl systems furthermore the definition of siss is introduced and corresponding criteria are provided for nssnl systems and ssnl systems in the proof of the above results to overcome the difficulties coming with the appearance of switching and the stochastic property at the same time we generalize the past comparison principle and fully use the properties of the functions which we constructed in terms of the average dwell time of the switching laws a sufficient siss condition is obtained for ssnl systems finally some examples are provided to demonstrate the applicability of our results

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

connections between razumikhin type theorems and the iss nonlinear small gain theorem a razumikhin type theorem that guarantees input to state stability for functional differential equations with disturbances is established using the nonlinear small gain theorem the result is used to show that input to state stabilizability for nonlinear finite dimensional control systems is robust in an appropriate sense to small time delays at the input also relaxed razumikhin type conditions guaranteeing global asymptotic stability for differential difference equations are given

lyapunov conditions for input to state stability of impulsive systems this paper introduces appropriate concepts of input to state stability iss and integral iss for impulsive systems i e dynamical systems that evolve according to ordinary differential equations most of the time but occasionally exhibit discontinuities or impulses we provide a set of lyapunov based sufficient conditions for establishing these iss properties when the continuous dynamics are iss but the discrete dynamics that govern the impulses are not the impulses should not occur too frequently which is formalized in terms of an average dwell time adt condition conversely when the impulse dynamics are iss but the continuous dynamics are not there must not be overly long intervals between impulses which is formalized in terms of a novel reverse adt condition we also investigate the cases where i both the continuous and discrete dynamics are iss and ii one of these is iss and the other only marginally stable for the zero input while sharing a common lyapunov function in the former case we obtain a stronger notion of iss for which a necessary and sufficient lyapunov characterization is available the use of the tools developed herein is illustrated through examples from a micro electro mechanical system mems oscillator and a problem of remote estimation over a communication network