dwell time based observer design for unknown input switched linear systems without requiring strong detectability of subsystems this paper investigates the state observer design of a class of unknown input switched linear systems via mode dependent dwell time switchings the distinguishing feature of the proposed method is that strong detectability condition of subsystems of the switched systems is unnecessarily required first a time varying coordinate transformation is introduced to design a suitable reduced order observer for each subsystem then computable sufficient conditions on the synthesis of the observers are proposed in the framework of a mode dependent dwell time technique since the observer of an individual subsystem cannot be designed due to unavailability of strong detectability condition of the subsystem the state of the switched system is estimated under the condition of confining the dwell time by a pair of upper and lower bounds restricting the growth of the lyapunov function of the active subsystem and forcing energy of the overall switched system to decrease at switching instants finally an example is presented to demonstrate the effectiveness of the proposed method

uncertainty investigations in nonlinear aeroelastic systems in this paper the stochastic collocation method scm is applied to investigate the nonlinear behavior of an aeroelastic system with uncertainties in the system parameter and the initial condition numerical case studies for problems with uncertainties are carried out in particular the performance of the scm is compared with solutions based on other computational techniques such as monte carlo simulation wiener chaos expansion and wavelet chaos expansion from the computational results we conclude that the scm is an effective tool to study a nonlinear aeroelastic system with random parameters

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

bayesian inference of nonlinear unsteady aerodynamics from aeroelastic limit cycle oscillations a bayesian model selection and parameter estimation algorithm is applied to investigate the influence of nonlinear and unsteady aerodynamic loads on the limit cycle oscillation lco of a pitching airfoil in the transitional reynolds number regime at small angles of attack laminar boundary layer trailing edge separation causes negative aerodynamic damping leading to the lco the fluid structure interaction of the rigid but elastically mounted airfoil and nonlinear unsteady aerodynamics is represented by two coupled nonlinear stochastic ordinary differential equations containing uncertain parameters and model approximation errors several plausible aerodynamic models with increasing complexity are proposed to describe the aeroelastic system leading to lco the likelihood in the posterior parameter probability density function pdf is available semi analytically using the extended kalman filter for the state estimation of the coupled nonlinear structural and unsteady aerodynamic model the posterior parameter pdf is sampled using a parallel and adaptive markov chain monte carlo mcmc algorithm the posterior probability of each model is estimated using the chib jeliazkov method that directly uses the posterior mcmc samples for evidence marginal likelihood computation the bayesian algorithm is validated through a numerical study and then applied to model the nonlinear unsteady aerodynamic loads using wind tunnel test data at various reynolds numbers

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

uncertainty quantification of limit cycle oscillations different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit cycle oscillation subject to parametric variability the aeroelastic system is that of a rigid airfoil supported by pitch and plunge structural coupling with nonlinearities in the component in pitch the nonlinearities are adjusted to permit the formation of a either a subcritical or supercritical branch of limit cycle oscillations uncertainties are specified in the cubic coefficient of the torsional spring and in the initial pitch angle of the airfoil stochastic projections of the time domain and cyclic equations governing system response are carried out leading to both intrusive and non intrusive computational formulations non intrusive formulations are examined using stochastic projections derived from wiener expansions involving haar wavelet and b spline bases while wiener hermite expansions of the cyclic equations are employed intrusively and non intrusively application of the b spline stochastic projection is extended to the treatment of aerodynamic nonlinearities as modeled through the discrete euler equations the methodologies are compared in terms of computational cost convergence properties ease of implementation and potential for application to complex aeroelastic systems

functional coefficient autoregressive models in this article we propose a new class of models for nonlinear time series analysis investigate properties of the proposed model and suggest a modeling procedure for building such a model the proposed modeling procedure makes use of ideas from both parametric and nonparametric statistics a consistency result is given to support the procedure for illustration we apply the proposed model and procedure to several data sets and show that the resulting models substantially improve postsample multi step ahead forecasts over other models

non intrusive generalized polynomial chaos for the robust stability analysis of uncertain nonlinear dynamic friction systems this paper is devoted to the stability analysis of uncertain nonlinear dynamic dry friction systems the stability property of dry friction systems is known to be very sensitive to the variations of friction laws moreover the friction coefficient admits dispersions due to the manufacturing processes therefore it becomes necessary to take this uncertainty into account in the stability analysis of dry friction systems to ensure robust predictions of stable and instable behaviours the generalized polynomial chaos formalism is proposed to deal with this challenging problem treated in most cases with the prohibitive monte carlo based techniques two equivalent methods presented here combine the non intrusive generalized polynomial chaos with the indirect lyapunov method both methods are shown to be efficient in the estimation of the stability and instability regions with high accuracy and high confidence levels and at lower cost compared with the classic monte carlo based method

adaptive surrogate modeling by anova and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation the polynomial dimensional decomposition pdd is employed in this work for the global sensitivity analysis and uncertainty quantification uq of stochastic systems subject to a moderate to large number of input random variables due to the intimate connection between the pdd and the analysis of variance anova approaches pdd is able to provide a simpler and more direct evaluation of the sobol sensitivity indices when compared to the polynomial chaos expansion pc unfortunately the number of pdd terms grows exponentially with respect to the size of the input random vector which makes the computational cost of standard methods unaffordable for real engineering applications in order to address the problem of the curse of dimensionality this work proposes essentially variance based adaptive strategies aiming to build a cheap meta model i e surrogate model by employing the sparse pdd approach with its coefficients computed by regression three levels of adaptivity are carried out in this paper 1 the truncated dimensionality for anova component functions 2 the active dimension technique especially for second and higher order parameter interactions and 3 the stepwise regression approach designed to retain only the most influential polynomials in the pdd expansion during this adaptive procedure featuring stepwise regressions the surrogate model representation keeps containing few terms so that the cost to resolve repeatedly the linear systems of the least squares regression problem is negligible the size of the finally obtained sparse pdd representation is much smaller than the one of the full expansion since only significant terms are eventually retained consequently a much smaller number of calls to the deterministic model is required to compute the final pdd coefficients

spectral element method in time for rapidly actuated systems in this paper the spectral element se method is applied in time to find the entire time periodic or transient solution of time dependent differential equations the time periodic solution is computed by enforcing periodicity of the element set of particular interest are periodic forcing functions possessing high frequency content to maintain the spectral accuracy for such forcing functions an h refinement scheme is employed near the semi discontinuity without increasing the number of degrees of freedom time discretization by spectral elements is applied initially to a standard form of a set of linear first order differential equations subject to harmonic excitation and an excitation admitting rapid variation other case studies include the application of the se approach to parabolic and hyperbolic partial differential equations the first order form of these equations is obtained through semi discretization using conventional finite element spectral element and finite difference schemes element clustering h refinement is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation the convergence in time of the method is demonstrated in some cases machine precision is obtained with 25 degrees of freedom per cycle finally the method is applied to a weakly nonlinear problem with time periodic solution to demonstrate its future applicability to the analysis of limit cycle oscillations in aeroelastic systems

an adaptive least squares global sensitivity method and application to a plasma coupled combustion prediction with parametric correlation abstract we introduce an efficient non intrusive surrogate based methodology for global sensitivity analysis and uncertainty quantification modified covariance based sensitivity indices mcov si are defined for outputs that reflect correlated effects the overall approach is applied to simulations of a complex plasma coupled combustion system with disparate uncertain parameters in sub models for chemical kinetics and a laser induced breakdown ignition seed the surrogate is based on an analysis of variance anova expansion such as widely used in statistics with orthogonal polynomials representing the anova subspaces and a polynomial dimensional decomposition pdd representing its multi dimensional components the coefficients of the pdd expansion are obtained using a least squares regression which both avoids the direct computation of high dimensional integrals and affords an attractive flexibility in choosing sampling points this facilitates importance sampling using a bayesian calibrated posterior distribution which is fast and thus particularly advantageous in common practical cases such as our large scale demonstration for which the asymptotic convergence properties of polynomial expansions cannot be realized due to computation expense effort instead is focused on efficient finite resolution sampling standard covariance based sensitivity indices cov si are employed to account for correlation of the uncertain parameters magnitude of cov si is unfortunately unbounded which can produce extremely large indices that limit their utility alternatively mcov si are then proposed in order to bound this magnitude 0 1 the polynomial expansion is coupled with an adaptive anova strategy to provide an accurate surrogate as the union of several low dimensional spaces avoiding the typical computational cost of a high dimensional expansion it is also adaptively simplified according to the relative contribution of the different polynomials to the total variance the approach is demonstrated for a laser induced turbulent combustion simulation model which includes parameters with correlated effects

stochastic nonlinear aeroelastic analysis of a supersonic lifting surface using an adaptive spectral method an adaptive stochastic spectral projection method is deployed for the uncertainty quantification in limit cycle oscillations of an elastically mounted two dimensional lifting surface in a supersonic flow field variabilities in the structural parameters are propagated in the aeroelastic system which accounts for nonlinear restoring force and moment by means of hardening cubic springs the physical nonlinearities promote sharp and sudden flutter onset for small change of the reduced velocity in a stochastic context this behavior translates to steep solution gradients developing in the parametric space a remedy is to expand the stochastic response of the airfoil on a piecewise generalized polynomial chaos basis accurate approximation andaffordable computational costs are obtained using sensitivity based adaptivity for various types of supersonic stochastic responses depending on the selected values of the mach number on the bifurcation map sensitivity analysis via sobol indices shows how the probability density function of the peak pitch amplitude responds to combined uncertainties e g the elastic axis location torsional stiffness and flap angle we believe that this work demonstrates the capability and flexibility of the approach for more reliable predictions of realistic aeroelastic systems subject to a moderate number of uncertainties

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

hopf bifurcation analysis of an aeroelastic model using stochastic normal form abstract we investigate the effects of parameter uncertainties on the dynamical response of an aeroelastic model representing an oscillating airfoil in pitch and plunge with linear aerodynamics and cubic structural nonlinearities an approach based on the stochastic normal form is proposed to determine the effects due to the variations in the flow speed and the structural stiffness terms on the stability of the aeroelastic system near the hopf bifurcation point this approach allows us to study analytically the bifurcation scenario and to predict the amplitude and frequency of the limit cycle oscillation lco the results show that the amplitude of lco corresponding to the supercritical hopf bifurcation increases with the intensity of the noise perturbing the pitch and plunge cubic terms but there is almost no effect on the lco frequency uncertainties in the flow speed produce a shift in the bifurcation point and unstable subcritical behavior may occur for values of parameters for which the corresponding deterministic model is stable the stochastic normal form confirms and extends previously known numerical results regarding the effect of parameter variations and offers an effective way to perform sensitivity analysis of the system s response

stochastic collocation method for secondary bifurcation of a nonlinear aeroelastic system abstract a stochastic collocation method is proposed to investigate the secondary bifurcation of a two dimensional aeroelastic system with structural nonlinearity represented by cubic restoring forces and uncertainties expressed by random parameters in the cubic stiffness coefficient and in the initial pitch angle the accuracy of the stochastic collocation method is improved by incorporating higher order schemes such as piecewise cubic interpolation and piecewise cubic spline interpolation instead of a piecewise linear interpolation formula for an aeroelastic problem with the uncertainty expressed by a time dependent combination of five random variables an efficient collocation method is developed using a sparse grid approach with a dimension adaptive strategy numerical simulations are carried out to demonstrate the effectiveness of the proposed method for long term computation and discontinuous problems

enforcing positivity in intrusive pc uq methods for reactive ode systems we explore the relation between the development of a non negligible probability of negative states and the instability of numerical integration of the intrusive galerkin ordinary differential equation system describing uncertain chemical ignition to prevent this instability without resorting to either multi element local polynomial chaos pc methods or increasing the order of the pc representation in time we propose a procedure aimed at modifying the amplitude of the pc modes to bring the probability of negative state values below a user defined threshold this modification can be effectively described as a filtering procedure of the spectral pc coefficients which is applied on the fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold we demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor this is carried out by comparing different observables and error measures as obtained by non intrusive monte carlo and gauss quadrature integration and the filtered intrusive procedure the filtering procedure has been shown to effectively stabilize divergent intrusive solutions and also to improve the accuracy of stable intrusive solutions which are close to the stability limits

uncertainty analysis near bifurcation of an aeroelastic system variations in structural and aerodynamic nonlinearities on the dynamic behavior of an aeroelastic system are investigated the aeroelastic system consists of a rigid airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads we follow two approaches to determine the effects of variations in the linear and nonlinear plunge and pitch stiffness coefficients of this aeroelastic system on its stability near the bifurcation the first approach is based on implementation of intrusive polynomial chaos expansion pce on the governing equations yielding a set of nonlinear coupled ordinary differential equations that are numerically solved the results show that this approach is capable of determining sensitivity of the flutter speed to variations in the linear pitch stiffness coefficient on the other hand it fails to predict changes in the type of the instability associated with randomness in the cubic stiffness coefficient in the second approach the normal form is used to investigate the flutter hopf bifurcation boundary that occurs as the freestream velocity is increased and to analytically predict the amplitude and frequency of the ensuing lco the results show that this mathematical approach provides detailed aspects of the effects of the different system nonlinearities on its dynamic behavior furthermore this approach could be effectively used to perform sensitivity analysis of the system s response to variations in its parameters

effect of stiffness uncertainties on the flutter of a cantilever wing this paper deals with the investigation of the influence of spanwise distribution of bending and torsion stiffness uncertainties on the flutter behavior of an aeroelastic wing using a stochastic finite element approach the analysis adopted a numerical algorithm to simulate unsteady nonlinear incompressible flow based on the unsteady vortex lattice method interacting with linear aeroelastic structure in the absence of uncertainties the airflow and wing structure are treated as elements of a single dynamic system parameter uncertainties are represented by a truncated karhunen love expansion both perturbation technique and monte carlo simulation are used to establish the boundary of stiffness uncertainty level at which the wing exhibits flutter in the form of limit cycle oscillations and above which the wing experiences dynamic instability the analysis also includes the limitation of perturbation solution for a relatively large level of stiffness uncertainty

spectral and multiresolution wiener expansions of oscillatory stochastic processes abstract wiener chaos expansions are being evaluated for the representation of stochastic variability in the response of nonlinear aeroelastic systems which often exhibit limit cycles preliminary studies with a simple nonlinear aeroelastic computational model have shown that the standard non intrusive wiener hermite expansion fails to maintain time accuracy as the simulation evolves wiener hermite expansions faithfully reproduce the short term characteristics of the process but consistently lose energy after several mean periods of oscillation this energy loss remains even for very high order expansions to uncover the cause of this energy loss and to explore potential remedies the more elementary problem of a sinusoid with random frequency is used herein to simulate the periodic response of an uncertain system as time progresses coefficients of the higher order terms in both the wiener hermite and wiener legendre expansions successively gain and lose dominance over the lower order coefficients in a manner that causes any fixed order expansion in terms of global basis functions to fail over a simulation time of sufficient duration this characteristic behavior is attributed to the continually increasing frequency of the process in the random dimension the recently developed wiener haar expansion is found to almost entirely eliminate the loss of energy at large times both for the sinusoidal process and for the response of a two degree of freedom nonlinear system which is examined as a prelude to the stochastic simulation of aeroelastic limit cycles it is also found that mallat s pyramid algorithm is more efficient and accurate for evaluating wiener haar expansion coefficients than monte carlo simulation or numerical quadrature

derper an algorithm for the continuation of periodic solutions in ordinary differential equations abstract an algorithm for the continuation of periodic solutions of ode was derived and tested the algorithm is based on the shooting method and a standard continuation algorithm variational variables are used to compute the jacobi matrix the starting point on the periodic orbit is adaptively changed in the algorithm by fixing one component of the solution stability of periodic solutions along the continuous branch of solutions is determined by computing characteristic multipliers without difficulty the algorithm crosses limit points and at bifurcation points double period bifurcation points and points of tori bifurcation it proceeds along the original branch of solutions the usefulness of the algorithm is demonstrated in two examples one of them describes two stirred tank reactors with mutual mass exchange the other one is the lorenz model of turbulence