brief paper a new concept of invariance for saturated systems in this paper the notions of sns invariance and sns domain of attraction are introduced the sns domain of attraction serves as an estimation of the domain of attraction of a saturated system it is shown that in the case of single input saturated systems any contractive set is contained in the sns domain of attraction another important characteristic of the sns domain of attraction is that it contains any estimation obtained by means of a linear difference inclusion of the saturated system a simple algorithm that converges to the sns domain of attraction is presented an illustrative example is given

probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation we present a methodology to perform the identification and validation of complex uncertain dynamical systems using experimental data for which uncertainties are taken into account by using the nonparametric probabilistic approach such a probabilistic model of uncertainties allows both model uncertainties and parameter uncertainties to be addressed by using only a small number of unknown identification parameters consequently the optimization problem which has to be solved in order to identify the unknown identification parameters from experiments is feasible two formulations are proposed the first one is the mean square method for which a usual differentiable objective function and an unusual non differentiable objective function are proposed the second one is the maximum likelihood method coupling with a statistical reduction which leads us to a considerable improvement of the method three applications with experimental validations are presented in the area of structural vibrations and vibroacoustics

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

proper orthogonal decomposition pod of a class of vibroimpact oscillations abstract this study employs the method of proper orthogonal decomposition pod also known as the karhunen loeve k l method to extract dominant coherent structures modes that approximate the system behavior from time series data these mode shapes can be used in a galerkin reconstruction process to obtain lower dimensional models for the structural systems under consideration the mode shapes and the energies obtained by this process can also be used to predict certain kinds of periodic or non periodic non linear motions the k l method has been applied successfully to fluid dynamical thermal processes and signal processing however only a handful of works exist in the area of vibration analysis and structural mechanics an extensive k l analysis is performed numerically for two vibroimpacting systems a beam and a rotor lower dimensional models are created and used to study non linear energy transmission from low to higher k l modes extensive reconstructions are also performed to prove the efficacy of the k l method to provide accurate low dimensional dynamical models in addition experimental investigations are presented for the case of the overhung impacting rotor and qualitative comparisons with the theoretical analysis are presented

a nonparametric probabilistic approach for quantifying uncertainties in low dimensional and high dimensional nonlinear models summary a nonparametric probabilistic approach for modeling uncertainties in projection based nonlinear reduced order models is presented when experimental data are available this approach can also quantify uncertainties in the associated high dimensional models the main underlying idea is twofold first to substitute the deterministic reduced order basis rob with a stochastic counterpart second to construct the probability measure of the stochastic reduced order basis srob on a subset of a compact stiefel manifold in order to preserve some important properties of a rob the stochastic modeling is performed so that the probability distribution of the constructed srob depends on a small number of hyperparameters these are determined by solving a reduced order statistical inverse problem the mathematical properties of this novel approach for quantifying model uncertainties are analyzed through theoretical developments and numerical simulations its potential is demonstrated through several example problems from computational structural dynamics copyright 2016 john wiley sons ltd

uncertainty based experimental validation of nonlinear reduced order models abstract this paper presents the first results of a combined experimental computational investigation focused on the validation of reduced order models of geometrically nonlinear structures in the presence of uncertainty the validation approach considered here is based on the premise that the model is valid if the experimental results can be considered as random sample responses of the stochastic system of which the reduced order model is the mean for the situation considered here the power spectra of the experiments should lie within the 2nd and 98th percentiles of the response forming the uncertainty band of the stochastic model nominally clamped clamped beams are considered to demonstrate the entire process the construction of two mean reduced order models and their stochastic counterparts are first performed then the validation effort is carried out by comparing experimentally obtained power spectra and their corresponding computational uncertainty bands this process leads for both reduced order models to a very good representation of the important upper envelope 98th percentile of the experimental data but a less good fit of the lower envelope 2nd percentile

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

post buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation the paper presents a complete experimental validation of an advanced computational methodology adapted to the nonlinear post buckling analysis of geometrically nonlinear structures in presence of uncertainty a mean nonlinear reduced order computational model is first obtained using an adapted projection basis the stochastic nonlinear computational model is then constructed as a function of a scalar dispersion parameter which has to be identified with respect to the nonlinear static experimental response of a very thin cylindrical shell submitted to a static shear load the identified stochastic computational model is finally used for predicting the nonlinear dynamical post buckling behavior of the structure submitted to a stochastic ground motion

critical slowing down in local dynamics simulations the author reports some universal and non universal aspects of the critical dynamics of the three dimensional ising model obtained in extensive monte carlo finite size simulations the author shows that the time dependence of the magnetization of finite lattices is composed of two kinds of fluctuations at the critical point i phase fluctuations from one metastable minimum of the free energy to the other dominating the long time behaviour of the magnetization ii critical fluctuations inside each minimum decaying on a comparatively short timescale both kinds of fluctuations show up the same critical exponent z 2 10 or 0 02 which is also in excellent agreement with the exponent z of energy fluctuations

a state space error estimate for pod deim nonlinear model reduction this paper derives state space error bounds for the solutions of reduced systems constructed using proper orthogonal decomposition pod together with the discrete empirical interpolation method deim recently developed for nonlinear dynamical systems siam j sci comput 32 2010 pp 2737 2764 the resulting error estimates are shown to be proportional to the sums of the singular values corresponding to neglected pod basis vectors both in galerkin projection of the reduced system and in the deim approximation of the nonlinear term the analysis is particularly relevant to ode systems arising from spatial discretizations of parabolic pdes the derivation clearly identifies where the parabolicity is crucial it also explains how the deim approximation error involving the nonlinear term comes into play

stochastic reduced order models for uncertain geometrically nonlinear dynamical systems a general methodology is presented for the consideration of both parameter and model uncertainty in the determination of the response of geometrically nonlinear structural dynamic systems the approach is rooted in the availability of reduced order models of these nonlinear systems with a deterministic basis extracted from a reference model the mean model uncertainty both from parameters and model is introduced by randomizing the coefficients of the reduced order model in a manner that guarantees the physical appropriateness of every realization of the reduced order model i e while maintaining the fundamental properties of symmetry and positive definiteness of every such reduced order model this randomization is achieved not by postulating a specific joint statistical distribution of the reduced order model coefficients but rather by deriving this distribution through the principle of maximization of the entropy constrained to satisfy the necessary symmetry and positive definiteness properties several desirable features of this approach are that the uncertainty can be characterized by a single measure of dispersion affects all coefficients of the reduced order model and is computationally easily achieved the reduced order modeling strategy and this stochastic modeling of its coefficients are presented in details and several applications to a beam undergoing large displacement are presented these applications demonstrate the appropriateness and computational efficiency of the method to the broad class of uncertain geometrically nonlinear dynamic systems

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

brief nonlinear model predictive control with polytopic invariant sets ellipsoidal invariant sets have been widely used as target sets in model predictive control mpc these sets can be computed by constructing appropriate linear difference inclusions together with additional constraints to ensure that the ellipsoid lies within a given inclusion polytope the choice of inclusion polytope has a significant effect on the size of the target ellipsoid but the optimal inclusion polytope cannot in general be computed systematically this paper shows that use of polytopic invariant sets overcomes this difficulty allowing larger stabilizable sets without loss of performance in the interests of online efficiency consideration is focused on interpolation based mpc

buckling analysis of cylindrical shells with cutouts including random boundary and geometric imperfections abstract in this paper the effect of random geometric imperfections on the critical load of isotropic thin walled cylindrical shells under axial compression with rectangular cutouts is presented second moment characteristics of geometric imperfections are estimated by data of available measurements a simulation procedure based on the karhunen loeve expansion is applied for generating realizations of geometric imperfections nonlinear static finite element analyses are carried out for the calculation of the response statistics of the critical load of the cylindrical shells cumulative distribution functions of the critical load as obtained by direct monte carlo simulation are presented furthermore the individual and combined effects of random boundary and geometric imperfections on the limit loads of isotropic thin walled cylindrical shells under axial compression are also treated again second moment characteristics of these imperfections are estimated by data of available measurements of imperfections a simulation procedure also based on the karhunen loeve expansion is applied for generating realizations of both boundary and geometric imperfections a nonlinear static finite element analysis using the general purpose code stags c c rankin f a brogan w a loden h d cabiness stags structural analysis of general shells user manual lmsc p032594 version 3 0 lockheed martin missiles and space co inc palo alto ca usa 1998 is carried out for the calculation of the buckling response of the cylindrical shells finally the cumulative distribution functions of the limit load using direct monte carlo simulation are shown

convex models and interval analysis method to predict the effect of uncertain but bounded parameters on the buckling of composite structures a new non probabilistic set theoretical models that is interval analysis method is developed in this study to predict the variability or uncertainty in buckling loads of composite structures resulting from the unavoidable scatter in structural parameters by mathematically analyzing and proving the width of the upper and lower bounds on the critical buckling loads calculated by the interval analysis method is sharper than those that are obtained by convex models method moreover the interval analysis method has less computational cost than convex models method comparison between convex models and interval analysis method is performanced through numerical examples for the specific cases considered the results of numerical examples indicate that uncertainties in structural parameters have significant effect on the critical buckling loads of composite structures

brief paper on the computation of convex robust control invariant sets for nonlinear systems in this paper we provide a method to compute robust control invariant sets for nonlinear discrete time systems a simple criterion to evaluate if a convex set in state space is a robust control invariant set for a nonlinear uncertain system is presented the criterion is employed to design an algorithm for computing a polytopic robust control invariant set the method is based on the properties of dc functions i e functions which can be expressed as the difference of two convex functions since the elements of a wide class of nonlinear functions have dc representation or at least admit an arbitrarily close approximation the method is quite general the algorithm requires relatively low computational resources

acceleration of uncertainty propagation through lagrange multipliers in partitioned stochastic method abstract a partitioned stochastic method psm is proposed for the solution of static structural mechanics problems with uncertainties whose solution vectors are the displacements for each partition and lagrange multipliers along with the partition interfaces the proposed partitioned stochastic method employs three stochastic basis selection steps an arbitrary initial choice of displacement random bases a set of conjugate bases for the lagrange multipliers and finally modification of the displacement bases affected by those of the lagrange multipliers the present psm thus propagates the uncertainties instantly across partitioned substructures resulting in an improved rate of convergence numerical experiments illustrate the proposed psm outperforms a conventional partitioned solution method for structural mechanics problems

critical dynamics a consequence of a random stepwise growth of clusters critical dynamics of correlated particles here glauber dynamics of singly flipping ising spins is explained by a random stepwise growth and contraction of clusters as follows at equilibrium a cluster of size s is described by its length l a random walk like path connecting a sequence of neighbor spins the length scales as l s where constitutes a new static critical exponent we assume that on the average the random growth of a cluster from zero to size s and length ls requires a sequence of l2 spin flips this gives for dynamic critical exponent z 2 where and are the usual static exponents exact results at dimension d 1 and 4 and simulation results at d 2 and 3 support the theory

buckling analysis of imperfect shells with stochastic non gaussian material and thickness properties in this paper the effect of material and thickness spatial variation on the buckling load of isotropic shells with random initial geometric imperfections is investigated to this purpose a random spatial variability of the elastic modulus as well as of the thickness of the shell is introduced in addition to the random initial geometric deviations of the shell structure from its perfect geometry the main novelty of this paper compared to previous works is that a non gaussian assumption is made for the distribution of the two aforementioned uncertain parameters i e the modulus of elasticity and the shell thickness which are described by two dimensional uni variate 2d 1v homogeneous non gaussian stochastic fields the initial geometric imperfections are described as a 2d 1v gaussian non homogeneous stochastic field with properties derived from corresponding experimental measurements numerical examples are presented focusing on the influence of the non gaussian assumption on the variability of the buckling load which is calculated by means of the monte carlo simulation method it is shown that the choice of the marginal probability distribution for the description of the material and thickness variability is crucial since it affects significantly the statistics of the buckling load of imperfection sensitive shell type structures

uncertain buckling its past present and future an authoritative review on stochastic buckling of structures was written by amazigo some quarter century ago the present review summarizes some of the developments which took place since then a brief overview of the effect of uncertainty in the initial geometric imperfections elastic moduli applied forces and thickness variation is given for the benefit of the thoughtful reader the review is of critical nature

simulation of time exponent in 2d glauber kinetic ising model abstract simulations of the magnetization relaxation of large square lattices at the curie point give a dynamical exponent near 2 2 somewhat higher than expected

a cluster model of critical dynamics abstract critical relaxation exponent z is described by a reversible growth of clusters particles of a cluster are ordered into connected sequences such that a stepwise variation of their length describes relaxation we derive z z where is a scaling exponent relating cluster size to connected length at equilibrium available from simulation of ising clusters furthermore a flory like equation gives and hence z z in d 2 and 3 theoretical z an simulation aided z agree reasonably well with each other and with direct simulations of z in d 4 e z agrees with the exact rg

invariant sets computation for convex difference inclusions systems in this paper we introduce the convex difference inclusion cdi systems as a modeling framework useful to analyze set theory and invariance related issues for nonlinear and uncertain systems the dynamics of a cdi system is given by a set valued map whose values are convex compact subsets of the space and are determined by convex bounding functions necessary and sufficient boundary type conditions for invariance and contractiveness characteristic of the linear systems are given for the cdi systems lyapunov functions are proved to be induced by contractive sets for cdi systems as in the linear context a computational procedure for obtaining polytopic invariant and contractive sets for nonlinear systems based on the properties of the cdi systems is presented

a computational procedure to estimate the stochastic dynamic response of large non linear fe models abstract an algorithm for the computation of the stochastic non stationary non linear response of large fe models is presented in stochastic analysis the response is described by the mean and covariance function and possibly by the probability distribution of the non linear response to estimate the covariance matrix the method of equivalent statistical linearization is applied for linearizing all non linear elements the large fully populated and symmetric covariance matrix of dimension 2 n is described by the so called karhunen loeve expansion representation which allows one to employ a feasible description in this context an efficient procedure is suggested to determine the minimal number of karhunen loeve vectors necessary to assure a sufficiently accurate representation this method in fact allows one to employ any available deterministic integration scheme to compute the karhunen loeve vectors to increase the computational efficiency modal coordinates are used to represent the linear sub system special attention is given to the effects of mode truncation which are generally not negligible for large fe systems moreover the suggested approach has no limitation w r t the size of the model in terms of degrees of freedom dofs or the number of non linear elements the feasibility of the proposed procedure is demonstrated in the numerical examples where the methodology is applied to an office building with approximately 25 50 and 100 thousand dofs containing hysteretic damping elements

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

remarks on the efficiency of pod for model reduction in non linear dynamics of continuous elastic systems the first objective of this paper is to analyse the efficiency of the reduced models constructed using the proper orthogonal decomposition pod basis and the lin basis in non linear dynamics for continuous elastic systems the pod basis is the hilbertian basis constructed with the pod method while the lin basis is the hilbertian basis derived from the generalized continuous eigenvalue problem associated with the underlying conservative part of the continuous elastic system and usually called the eigenmodes of vibration the efficiency of the pod basis or the lin basis is related to the rate of convergence in the frequency domain of the solution constructed with the reduced model with respect to its dimension a basis will be more efficient than another if the reduced order solution of the galerkin projection converges to the solution of the dynamical system more rapidly than the reduced order solution of the other

uncertainty quantification in computational linear structural dynamics for viscoelastic composite structures this paper deals with the analysis of a stochastic reduced order computational model in computational linear dynamics for linear viscoelastic composite structures in the presence of uncertainties the computational framework proposed is based on a recent theoretical work that allows for constructing the stochastic reduced order model using the nonparametric probabilistic approach in the frequency domain the generalized damping matrix and the generalized stiffness matrix of the stochastic computational reduced order model are random matrices due to the causality of the dynamical system these two frequency dependent random matrices are statistically dependent and are linked by a compatibility equation induced by the causality of the system involving a hilbert transform the computational aspects related to the nonparametric stochastic modeling of the reduced stiffness matrix and the reduced damping matrix that are frequency dependent random matrices are presented a dedicated numerical approach is developed for obtaining an efficient computation of the cauchy principal value integrals involved in those equations for which an integration over a broad frequency domain is required a computational analysis of the propagation of uncertainties is conducted for a composite viscoelastic structure in the frequency range it is shown that the uncertainties on the damping matrix have a strong influence on the observed statistical dispersion of the stiffness matrix