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.     

Efficient uncertainty quantification with the polynomial chaos method for stiff systems

The polynomial chaos (PC) method has been widely adopted as a computationally feasible approach for uncertainty quantification (UQ). Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a non-linear system of equations at every time step. Using the Galerkin approach the size of the system state increases from n to S × n, where S is the number of PC basis functions. Solving such systems with full linear algebra causes the computational cost to increase from O(n³) to O(S³n³). The S³-fold increase can make the computation prohibitive. This paper explores computationally efficient UQ techniques for stiff systems using the PC Galerkin, collocation, and collocation least-squares (LS) formulations. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost. The numerical results show a run time reduction with no negative impact on accuracy. In the stochastic collocation formulation, we propose a least-squares approach based on collocation at a low-discrepancy set of points. Numerical experiments illustrate that the collocation least-squares approach for UQ has similar accuracy with the Galerkin approach, is more efficient, and does not require any modification of the original code.     

Robust nonlinear MPC based on Volterra series and polynomial chaos expansions

A robust nonlinear model predictive controller (NMPC) based on a Volterra series is proposed. Polynomial chaos expansions (PCE) are used to represent the uncertainty in the Volterra series coefficients and this uncertainty is then propagated onto the output predictions. The key advantage of the PCE is that it provides an analytical expression to compute the L²-norm of the output prediction error resulting in computational savings, compared to previously proposed techniques, which are essential for real time implementation. Terminal and input constraints based on Structured Singular Value based-norms are used to ensure convergence to a set-point and compliance with constraints in manipulated variables. The algorithm is applied to a multivariable pH neutralization system. A comparative study shows superior closed loop performance and computational efficiency of the proposed technique as compared to previously proposed algorithms.     

混沌优化与模糊控制在混沌控制中的应用

为了解决延迟反馈控制中参数难以选取的问题,将混沌优化和模糊控制的方法应用到混沌系统的延迟反馈控制中,采用混沌优化方法确定出延迟反馈控制的延迟时间,利用模糊控制技术确定出控制刚度等参数,从而得到延迟反馈控制方法的参数.同时,利用本文算法,可以一次确定出混沌系统的多个不稳定周期轨道的周期.仿真结果验证了本文算法的有效性.     

Robust dynamic optimization in heterogeneous multiscale catalytic flow reactors using polynomial chaos expansion

This paper explores the application of optimal design and operational strategies under uncertainty to a transient multiscale catalytic flow reactor system. The catalytic reactor is modeled using a spatially-dependent multiscale model that comprises lattice-based kinetic Monte Carlo (kMC) models coupled with continuum partial differential equations (PDEs) to account for the fine-scale and the macroscale reactor behaviour, respectively. This work compares two uncertainty propagation techniques, power series expansion (PSE) and polynomial chaos expansion (PCE), to assess their performance in multiscale process systems. The analysis reveals that PCE provides accurate results at minimal computational cost for the multiscale catalytic reactor model under the conditions considered in this work. PCE is subsequently used to perform robust dynamic optimization studies on the catalytic reactor system under uncertainty. The first study determines the optimal temperature trajectories that maximize the reactor’s performance under uncertainty. The second study aims to identify the optimal design and operating policies that allow the reactor, under uncertainty in the multiscale model parameters, to meet targeted performance specifications within a level of confidence. Both studies illustrate the benefits of performing dynamic optimization studies to improve performance for multiscale process systems under uncertainty.     

Gradient dynamic optimization with Legendre chaos

The polynomial chaos approach for stochastic simulation is applied to trajectory optimization, by conceptually replacing random variables with free variables. Using the gradient method, we generate with low computational cost an accurate parametrization of optimal trajectories.     

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.     

基于混沌多项式的指令鲁棒优化及在飞行控制中的应用

本文提出一种新的方法对随机系统进行运动预测和控制指令设计,该方法可以充分利用已知信息设计控制指令以提高闭环随机系统的鲁棒性.首先采用混沌多项式对随机信息进行数学表述,并利用Galerkin投影法将随机变量的混沌多项式引入常微分方程中.然后,将随机变量的均值和方差考虑至优化问题的成本函数中,并利用伪谱法对控制指令进行鲁棒优化.最后,将该方法应用于飞行器的动力学预测以及控制指令设计.仿真结果表明,该方法能够预测飞行器飞行过程中不确定性的演化,其精度与蒙特卡罗方法相当,并且计算效率更高.此外,获得的控制指令对存在不确定参数或初始条件的随机系统具有强鲁棒性.     

Robust stability of a special class of polynomial matrices with control applications

The robust stability problem of uncertain continuous-time systems described by higher-order dynamic equations is considered in this paper. Previous results on robust stability of Metzlerian matrices are extended to matrix polynomials, with the coefficient matrices having exactly the same Metzlerian structure. After defining the structured uncertainty for this class of polynomial matrices, we provide an explicit expression for the real stability radius and derive simplified formulae for several special cases. We also report on alternative approaches for investigating robust Hurwitz stability and strong stability of polynomial matrices. Several illustrative examples throughout the paper support the theoretical development. Moreover, an application example is included to demonstrate uncertainty modeling and robust stability analysis used in control design.     

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.     

Integration of fault diagnosis and control based on a trade-off between fault detectability and closed loop performance

This paper presents a novel methodology for simultaneous optimal tuning of a fault detection and diagnosis (FDD) algorithm and a feedback controller for a chemical plant in the presence of stochastic parametric faults. The key idea is to propagate the effect of time invariant stochastic uncertainties onto the measured variables by using a Generalized Polynomial Chaos (gPC) expansion and the nonlinear first principles’ model of the process. A bi-level optimization is proposed for achieving a trade-off between the fault detectability and the closed loop process variability. The goal of the outer level optimization is to seek a trade-off between the efficiency of detecting a fault and the closed loop performance, while the inner level optimization is designed to optimally calibrate the FDD algorithm. The proposed method is illustrated by a continuous stirred tank reactor (CSTR) system with a fault consisting of stochastic and intermittent variations in the inlet concentration. Beyond achieving improved trade-offs between fault detectability and control, it is shown that the computational cost of the gPC model based method is lower than the Monte Carlo type sampling based approaches, thus demonstrating the potential of the gPC method for dealing with large problems and real-time applications.     

对混沌自适应控制的控制强度的讨论之二

文中对文献[1]提出的混沌自适应控制方法中给出的控制强度取值范围的计算方法作出了重要修改,指出了上文忽略的与控制矩阵的复数本征值λ±对应的控制强度,对离散系统只要满足｜λ±｜＜1亦可控制混沌,且更有效。其次指出对连续性系统仅考虑文献[1]给出的条件是不够的,控制强度还要满足λ±值的实部小于零,即微分方程的李雅普诺夫渐近稳定条件才能有效控制混沌。     

Polynomial chaos for simulating random volatilities

In financial mathematics, the fair price of options can be achieved by solutions of parabolic differential equations. The volatility usually enters the model as a constant parameter. However, since this constant has to be estimated with respect to the underlying market, it makes sense to replace the volatility by an according random variable. Consequently, a differential equation with stochastic input occurs, whose solution determines the fair price in the refined model. Corresponding expected values and variances can be computed approximately via a Monte Carlo method. Alternatively, the generalised polynomial chaos yields an efficient approach for calculating the required data. Based on a parabolic equation modelling the fair price of Asian options, the technique is developed and corresponding numerical simulations are presented.     

基于多项式和混沌序列的无线传感器网络密钥管理方案*

针对现有密钥管理方案占据存储空间大和安全性差的缺点,提出了一种基于多项式和混沌序列的混合密钥管理方案。该方案使用多项式建立节点间的共享密钥减少网络的存储开销,利用混沌序列对初值的敏感性特点解决多项式安全门限的问题,从而防止密钥的泄露。结果表明,本方案具有很高的安全性能,不仅节约了网络的通信资源,而且降低了节点的存储开销。     

永磁同步电动机的混沌特性及其反混沌控制

永磁同步电动机在一定工作情况下将呈现出混沌运动. 而在其非混沌工作区, 通过在永磁同步电动机系统中增加一个线性控制项, 则将使原本稳定的永磁同步电动机系统呈现出混沌行为, 即实现了系统的混沌化或反混沌控制. 为此本文全面地研究了永磁同步电动机混沌及其反混沌控制特性, 这为永磁同步电动机混沌运动的抑制或利用打下了重要的研究基础.     

通过线性增量控制抑制混沌

本文基于三维Lorenz两翼系统、三维四翼系统以及四维四翼系统, 利用线性增量控制实现了减少吸引子涡 卷数以及抑制混沌的目的. 此控制是在多涡卷混沌系统中耦合一个线性系统, 通过不断增强两个系统之间的耦合 强度, 移动并且逐渐消除多涡卷混沌系统的部分平衡点, 从而达到控制目标. 利用平衡点坐标变化曲线、分岔图、Ly -apunov指数谱、以及相轨图等表征方法分析并刻画了被控制的多涡卷系统动力学行为变化的过程. 为了验证线性 增量控制对于实现吸引子退化以及抑制混沌的有效性和电路可实现性, 以三维四翼系统的控制为例设计了电路, Multism仿真结果和硬件实验结果均与在MATLAB中得到的数值仿真结果一致.     

A polynomial fuzzy neural network for modeling and control

This paper introduces a novel neurofuzzy system based on polynomial fuzzy neural network (PFNN) architecture. A PFNN consists of a set of if-then rules with appropriate membership functions (MFs) whose parameters are optimized via a hybrid genetic algorithm. A polynomial neural network is employed in the defuzzification scheme to improve output performance and to select appropriate rules. A performance criterion for model selection is defined to overcome the overfitting problem in the modeling procedure. For a performance assessment of the PFNN inference system, two well-known problems are employed for a comparison with other methods. The results of these comparisons show that the PFNN inference system out-performs the other methods and exhibits robustness characteristics. This work was presented in part at the Fourth International Symposium on Artificial Life and Robotics, Oita, Japan, January 19–22, 1999     

Sparse polynomial division using a heap

In 1974, Johnson showed how to multiply and divide sparse polynomials using a binary heap. This paper introduces a new algorithm that uses a heap to divide with the same complexity as multiplication. It is a fraction-free method that also reduces the number of integer operations for divisions of polynomials with integer coefficients over the rationals. Heap-based algorithms use very little memory and do not generate garbage. They can run in the CPU cache and achieve high performance. We compare our C implementation of sparse polynomial multiplication and division with integer coefficients to the routines of the Magma, Maple, Pari, Singular and Trip computer algebra systems.     

Reliability based optimization in aeroelastic stability problems using polynomial chaos based metamodels

In this work, reliability based design optimization (RBDO) of two aeroelastic stability problems is addressed: (i) divergence, which arises in static aeroelasticity, and (ii) flutter, which arises in dynamic aeroelasticity. A set of design variables is considered as random variables, and the mean mass is minimized for a given set of constraints — including the probability of failure by divergence or flutter. The optimization process requires repeated evaluation of reliability, which is a major contributor to the total computational cost. To reduce this cost, a polynomial chaos expansion (PCE)-based metamodel is created over a grid in the parameter space. These precomputed PCEs are then interpolated for reliability calculation at intermediate points in the parameter space, as demanded by the optimization algorithm. Two new modifications are made to this method in this work. First, the Gauss quadrature rule is used — instead of statistical simulation — to estimate the chaos coefficients for higher computational speed. Second, to increase this computational gain further, a non-uniform grid is chosen instead of a uniform one, based on relative importance of the design parameters. This relative importance is found from a global sensitivity analysis. This new modified method is applied on a rectangular unswept cantilever wing model. For both optimization problems, it is observed that the proposed method yields accurate results with a considerable computational cost reduction, when compared to simulation based methods. The effect of grid spacing is also explored to achieve the best computational efficiency.     

分形图形与混沌图案的应用

对分形与混沌的应用进行了研究,给出了分形曲线--(1～5次)封闭Koch曲线的生成结果,并讨论了利用牛顿法解方程及其混沌情况生成图案的方法.以封闭Koch曲线和牛顿法解方程及其混沌情况图案为研究对象,把分形几何图形和牛顿法解方程之混沌情况图案相结合,实现了对封闭Koch曲线区域的填充;为分形和混沌理论在艺术领域的应用,做出了初步的尝试,研究表明分形与混沌可以被运用于艺术上.