Modeling physical uncertainties in dynamic stall induced fluid–structure interaction of turbine blades using arbitrary polynomial chaos

A nonlinear dynamic problem of stall induced flutter oscillation subject to physical uncertainties is analyzed using arbitrary polynomial chaos. A single-degree-of-freedom stall flutter model with torsional oscillation is considered subject to nonlinear aerodynamic loads in the dynamic stall regime and nonlinear structural stiffness. The analysis of the deterministic aeroelastic response demonstrated that the problem is sensitive to variations in structural natural frequency and structural nonlinearity. The effect of uncertainties in these parameters is studied. Arbitrary polynomial chaos is employed in which appropriate expansion polynomials are constructed based on the statistical moments of the uncertain input. The arbitrary polynomial chaos results are compared with Monte Carlo simulations.     

Modeling multibody systems with uncertainties. Part II: Numerical applications

This study applies generalized polynomial chaos theory to model complex nonlinear multibody dynamic systems operating in the presence of parametric and external uncertainty. Theoretical and computational aspects of this methodology are discussed in the companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects”.In this paper we illustrate the methodology on selected test cases. The combined effects of parametric and forcing uncertainties are studied for a quarter car model. The uncertainty distributions in the system response in both time and frequency domains are validated against Monte-Carlo simulations. Results indicate that polynomial chaos is more efficient than Monte Carlo and more accurate than statistical linearization. The results of the direct collocation approach are similar to the ones obtained with the Galerkin approach. A stochastic terrain model is constructed using a truncated Karhunen-Loeve expansion. The application of polynomial chaos to differential-algebraic systems is illustrated using the constrained pendulum problem. Limitations of the polynomial chaos approach are studied on two different test problems, one with multiple attractor points, and the second with a chaotic evolution and a nonlinear attractor set.The overall conclusion is that, despite its limitations, generalized polynomial chaos is a powerful approach for the simulation of multibody dynamic systems with uncertainties.     

Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects

This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”. The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions. The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties.     

Component mode synthesis and polynomial chaos expansions for stochastic frequency functions of large linear FE models

This paper presents a methodological approach for the numerical investigation of frequency transfer functions for large FE systems with linear and nonlinear stochastic parameters. The component mode synthesis methods are used to reduce the size of the model and are extended to stochastic structural vibrations. The statistical first two moments of frequency transfer functions are obtained by an adaptive polynomial chaos expansion. Free and fixed interface methods with and without reduction of interface dof are used. The coupling with the first and second order polynomial chaos expansion is elaborated for beams and assembled plates with linear and nonlinear stochastic parameters.     

Application of global sensitivity analysis to a tire model with correlated inputs

When a vehicle equipped with tire is manoeuvred on the ground, the tires are submitted to a number of forces – longitudinal force when driving or braking torque is applied to the wheel and/or lateral force when the wheel is steered to turn at a corner. Pacejka model describes these forces that represent the reaction of the road onto the tire. This nonlinear model depends on correlated parameters such as the friction coefficient, the vertical load, and the cornering stiffness, which have to be identified from some measurements. The sensitivity of Pacejka model to these correlated parameters are studied using an approach based on polynomial chaos. It consists in decorrelating the parameters using the Nataf transformation and then, in expanding the model output onto polynomial chaos. The sensitivity indices are then obtained straightforwardly from the algebraic expression of the coefficients of the polynomial expansion.     

Iterative learning of model reference adaptive controller for uncertain nonlinear systems with only output measurement

In this paper, a model reference adaptive control strategy is used to design an iterative learning controller for a class of repeatable nonlinear systems with uncertain parameters, high relative degree, initial output resetting error, input disturbance and output noise. The class of nonlinear systems should satisfy some differential geometric conditions such that the plant can be transformed via a state transformation into an output feedback canonical form. A suitable error model is derived based on signals filtered from plant input and output. The learning controller compensates for the unknown parameters, uncertainties and nonlinearity via projection type adaptation laws which update control parameters along the iteration domain. It is shown that the internal signals remain bounded for all iterations. The output tracking error will converge to a profile which can be tuned by design parameters and the learning speed is improved if the learning gain is large.     

Parameter estimation using polynomial chaos and maximum likelihood

When modelling biological processes, there are always errors, uncertainties and variations present. In this paper, we consider the coefficients in the mathematical model to be random variables, whose distribution and moments are unknown a priori, and need to be determined by comparison with experimental data. A stochastic spectral representation of the parameters and the solution stochastic process is used, based on polynomial chaoses. The polynomial chaos representation generates a system of equations of the same type as the original model. The inverse problem of finding the parameters is reduced to establishing the best-fit values of the random variables that represent them, and this is done using maximum likelihood estimation. In particular, in modelling biofilm growth, there are variations, measurement errors and uncertainties in the processes. The biofilm growth model is given by a parabolic differential equation, so the polynomial chaos formulation generates a system of partial differential equations. Examples are presented.     

Distributional uncertainty analysis using power series and polynomial chaos expansions

This paper provides an overview of computationally efficient approaches for quantifying the influence of parameter uncertainties on the states and outputs of nonlinear dynamical systems with finite-time control trajectories, focusing primarily on computing probability distributions. The advantages and disadvantages of various uncertainty analysis approaches, which use approximate representations of the full nonlinear model using power series or polynomial chaos expansions, are discussed in terms of computational cost and accuracy in computing the shape and tails of the state and output distributions. Application of the uncertainty analysis methods to a simulation study is used to provide advice as to which uncertainty analysis methods to select for a particular application. In particular, the results indicate that first-order series analysis can be accurate enough for the design of real-time robust feedback controllers for batch processes, although it is cautioned that the accuracy of such analysis should be confirmed a posteriori using a more accurate uncertainty analysis method. The polynomial chaos expansion is well suited to robust design and control when the objectives are strongly dependent on the shape or tails of the distributions of product quality or economic objectives.     

Recursive maximum likelihood parameter estimation for state space systems using polynomial chaos theory

This paper combines polynomial chaos theory with maximum likelihood estimation for a novel approach to recursive parameter estimation in state-space systems. A simulation study compares the proposed approach with the extended Kalman filter to estimate the value of an unknown damping coefficient of a nonlinear Van der Pol oscillator. The results of the simulation study suggest that the proposed polynomial chaos estimator gives comparable results to the filtering method but may be less sensitive to user-defined tuning parameters. Because this recursive estimator is applicable to linear and nonlinear dynamic systems, the authors portend that this novel formulation will be useful for a broad range of estimation problems.     

基于双增益方法的不确定非线性系统的输出反馈调节问题

研究一类具有多种不确定性的非线性系统的全局输出反馈调节问题.所研究系统的一个显著特点是非线性项被未知增长率和多项式形式的输出函数的乘积界定,难点是在输出受不确定参数摄动的情况下如何抑制非线性项.提出一种改进的双增益方法来设计输出反馈控制器,可以确保闭环系统所有信号全局一致有界并且原系统状态收敛到零.最后,采用质量弹簧机械系统的输出反馈镇定问题来说明控制策略的有效性.     

Multiple model-based event-triggered adaptive control of a class of discrete-time nonlinear systems

In this study, the problem of event-triggered-based adaptive control (ETAC) for a class of discrete-time nonlinear systems with unknown parameters and nonlinear uncertainties is considered. Both neural network (NN) based and linear identifiers are used to approximate the unknown system dynamics. The feedback output signals are transmitted, and the parameters and the NN weights of the identifiers are tuned in an aperiodic manner at the event sample instants. A switching mechanism is provided to evaluate the approximate performance of each identifier and decide which estimated output is utilised for the event-triggered controller design, during any two events. The linear identifier with an auxiliary output and an improved adaptive law is introduced so that the nonlinear uncertainties are no longer assumed to be Lipschitz. The number of transmission times are significantly reduced by incorporating multiple model schemes into ETAC. The boundedness of both the parameters of identifiers and the system outputs is demonstrated though the Lyapunov approach. Simulation results demonstrate the effectiveness of the proposed method.     

Robust Control System Design for an Uncertain Nonlinear System Using Minimax LQG Design Method

A systematic approach to design a nonlinear controller using minimax linear quadratic Gaussian regulator (LQG) control is proposed for a class of multi‐input multi‐output nonlinear uncertain systems. In this approach, a robust feedback linearization method and a notion of uncertain diffeomorphism are used to obtain an uncertain linearized model for the corresponding uncertain nonlinear system. A robust minimax LQG controller is then proposed for reference command tracking and stabilization of the nonlinear system in the presence of uncertain parameters. The uncertainties are assumed to satisfy a certain integral quadratic constraint condition. In this method, conventional feedback linearization is used to cancel nominal nonlinear terms and the uncertain nonlinear terms are linearized in a robust way. To demonstrate the effectiveness of the proposed approach, a minimax LQG‐based robust controller is designed for a nonlinear uncertain model of an air‐breathing hypersonic flight vehicle (AHFV) with flexibility and input coupling. Here, the problem of constructing a guaranteed cost controller which minimizes a guaranteed cost bound has been considered and the tracking of velocity and altitude is achieved under inertial and aerodynamic uncertainties.     

Robust LQR design for systems with probabilistic uncertainty

In this paper, we consider the design of robust quadratic regulators for linear systems with probabilistic uncertainty in system parameters. The synthesis algorithms are presented in a convex optimization framework, which optimize with respect to an integral cost. The optimization problem is formulated as a lower‐bound maximization problem and developed in the polynomial chaos framework. Two approaches are considered here. In the first approach, an exact optimization problem is formulated in the infinite‐dimensional space, which is solved approximately using polynomial‐chaos expansions. In the second approach, an approximate problem is formulated using a reduced‐order model and solved exactly. The robustness of the controllers from these two approaches are compared using a realistic flight control problem based on an F16 aircraft model. Linear and nonlinear simulations reveal that the first approach results in a more robust controller.     

基于自适应神经网络的不确定非线性系统的模糊跟踪控制

提出了一种基于模糊模型和自适应神经网络的跟踪控制方法.在系统具有未知不确定非线性特性的情况下,首先利用T＿S模糊模型对系统的已知特性进行近似建模,对基于模糊模型的模糊H∞跟踪控制律进行输出跟踪控制.并在此基础上,进一步采用RBF神经网络完全自适应控制,通过在线自适应调整RBF神经网络的权重、函数中心和宽度,从而有效地消除系统的未知不确定性和模糊建模误差的影响,保证了非线性闭环系统的稳定性和系统的H∞跟踪性能,而不要求系统的不确定项和模糊建模误差满足任何匹配条件或约束.最后,将所提出的方法应用到一非线性混沌系统,仿真结果表明了所提出的方案不仅能够有效地稳定该混沌系统,而且能使系统输出跟踪期望输出.     

基于扰动补偿的无微分模型参考自适应控制系统设计

针对一类含有参数不确定性和未知非线性扰动的系统,本文提出一种基于扰动补偿的无微分模型参考自适应控制方法,实现系统输出对参考模型输出信号的高精度跟踪.首先,利用被控对象模型信息设计扰动估计器,对系统非线性扰动进行在线估计;其次,基于非线性扰动估计值设计参考模型和无微分参数更新律,构建无微分模型参考自适应控制器,建立基于扰动补偿和状态反馈的自适应控制律,以消除参数不确定性和非线性扰动对系统输出的影响,保证系统输出对参考模型输出的准确跟踪;然后,给出闭环系统误差信号收敛条件和控制器参数整定方法;最后,通过数值仿真验证所提方法的有效性和优越性.     

Global adaptive output regulation of a class of nonlinear systems with nonlinear exosystems

This paper deals with global output regulation with nonlinear exosystems for a class of uncertain nonlinear output feedback systems. The circle criterion is exploited for the internal model design to accommodate the nonlinearities in the exosystems, and the explicit conditions are given for the exosystems such that the proposed internal model design can be applied. The uncertainties of the output feedback systems are in the form of unknown constant parameters, and adaptive control techniques are used to ensure the global stability of the proposed control design for output regulation.     

MEMS stochastic model order reduction method based on polynomial chaos expansion

Modeling and simulation of MEMS devices is a very complex task which involve the electrical, mechanical, fluidic and thermal domains, and there are still some uncertainties need to be accounted because of uncertain material and/or geometric parameters factors. According to these problems, we put forward to stochastic model order reduction method under random input conditions to facilitate fast time and frequency domain analyses, the method firstly process model order reduction by Structure Preserving Reduced-order Interconnect Macro Modeling method, then makes use of polynomial chaos expansions in terms of the random input and output variables for the matrices of a finite element model of the system; at last we give the expected values and standard deviations computing method to MEMS stochastic model. The simulation results verify the method is effective in large scale MEMS design process.     

Fundamental limitations of discrete-time adaptive nonlinear control

A particular polynomial is introduced which can be used to determine under what conditions a typical class of discrete-time nonlinear systems with uncertainties in both parameters and noises is not stabilizable by feedback, thus demonstrating the fundamental limitations of discrete-time adaptive nonlinear control. As a consequence, it is shown that for nonlinear systems with unknown parameters and noises, the systems may indeed be nonstabilizable, in general, whenever the usual linear growth condition is relaxed and the number of unknown parameters is large, even though the corresponding noise-free systems are globally stabilizable     

Adaptive Iterative Learning Control for a Class of Nonlinear Time-varying Systems with Unknown Delays and Input Dead-zone

This paper presents an adaptive iterative learning control (AILC) scheme for a class of nonlinear systems with unknown time-varying delays and unknown input dead-zone. A novel nonlinear form of dead-zone nonlinearity is presented. The assumption of identical initial condition for iterative learning control (ILC) is removed by introducing boundary layer function. The uncertainties with time-varying delays are compensated for by using appropriate Lyapunov-Krasovskii functional and Young0s inequality. Radial basis function neural networks are used to model the time-varying uncertainties. The hyperbolic tangent function is employed to avoid the problem of singularity. According to the property of hyperbolic tangent function, the system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapunov-like composite energy function (CEF) in two cases, while keeping all the closedloop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.      

Evolution of Probability Distribution in Time for Solutions of Hyperbolic Equations

We investigate the evolution of the probability distribution function in time for some wave and Maxwell equations in random media for which the parameters, e.g. permeability, permittivity, fluctuate randomly in space; more precisely, two different media interface randomly in space. We numerically compute the probability distribution and density for output solutions. The underlying numerical and statistical techniques are the so-called polynomial chaos Galerkin projection, which has been extensively used for simulating partial differential equations with uncertainties, and the Monte Carlo simulations.