子系统为ARMA和分段线性函数的Wiener系统的参数辨识

本文研究ARMA线性子系统串联分段线性函数的Wiener系统的递推辨识问题.利用相关分析法和Yule-Walker方程给出线性部分参数的递推辨识算法,而对非线性部分参数用递推的最小二乘(LS)算法给出估计,并证明了这些算法都以概率1收敛到真值.     

微生物发酵非线性系统辨识、控制及 并行优化研究

主要研究微生物发酵过程中不同工况下的非线性、非光滑且无法求得解析解的动力系统及其主要性质,建立了具有数百个不同动力系统为主要约束、有连续与离散两种辨识参量、依据实验数据与生物系统鲁棒性为性能指标的辨识模型,阐述了此类辨识模型与最优控制模型的建立方法、数值模拟方法及并行优化计算方法,并介绍了笔者的著作《非线性发酵动力系统——辨识、控制与并行优化》的基本内容。     

基于分段常值水平集的抛物型方程的参数识别算法

研究了线性抛物型方程不连续参数的识别算法.根据原有算法对于加噪观测数据计算不收敛的问题,本文基于分段常值水平集方法,根据水平集函数和优化过程的特点,修正原有Uzawa型算法中的带有总变差(TV)正则化的极小化模型和对常值向量的极小化模型,并且利用分裂Bregman迭代算法处理TV范数的优越性,构造一种新的参数识别算法格式.数值实验结果显示,新算法具有计算时间短、精度高、抗噪性强的优点.     

基于时间满意度的应急物资储备库选址问题

基于应急物资配送过程中时间因素的重要性,将时间满意度引人应急物资储备库选址问题中.针对时间满意度为线性分段函数,建立了以时间满意度最小的需求点的时间满意度尽量大以及系统总费用最小为目标的双目标混合整数规划模型,对目标函数的最小最大值问题进行转化,在此基础上构造新的优化模型,并设计了相应的启发式算法求解.最后通过算例说明算法的可行性和有效性.     

分段线性神经网络的逼近理论

随着分段线性函数的广泛应用,本文尝试研究浅层和深层的分段线性神经网络的逼近理论.作者将应用于三层感知机模型的万能逼近定理拓展到分段线性神经网络中,并给出与隐藏神经元个数相关的逼近误差估计.利用分段线性函数构造锯齿函数的显式方法,证明解析函数可以通过分段线性神经网络的深度堆叠以指数速率逼近,并辅以相应的数值实验.     

估计海冰厚度函数的一种辨识方法

提出一种采用海冰和海水温度观测数据来估计海冰厚度的辨识方法, 避免了因使用厚度数据所带来的种种局限性. 首先建立一个拟线性海冰-海水热力学系统, 得到了系统解的存在唯一性; 然后以该系统中描述海冰厚度函数的参数为辨识量, 以系统输出的温度和实际观测温度的偏差为目标泛函, 建立了以目标泛函为最小的参数辨识模型; 最后构造了以半隐式差分格式、遗传算法和Hooke-Jeeves算法相结合的数值算法, 得到了海冰厚度函数, 并对辨识量做了敏感性分析. 结果表明: 这种方法是有效可行的.     

微生物批式流加发酵多阶段最优控制的最优性条件

在微生物批式流加发酵生产1,3一丙二醇(1,3-PD)过程中,关键是如何控制甘油和碱的流加速度.本文将流加速度看成一个随时间变化的控制函数,提出一个带控制的多阶段动力系统描述批式发酵过程,并证明了系统的一些性质.以终端时刻1,3-PD的生产强度最大为性能指标,以上述动力系统和连续状态不等式为约束条件建立了最优控制模型,最后利用不可微优化理论得到了最优控制问题的最优性条件,并证明了最优性条件和最优性函数零点的等价性.     

分段低次插值多项式序列的一致收敛性

利用分段线性与三次Hermite插值基函数以及连续模概念,分别推导出分段线性与三次Hermite插值多项式序列一致收敛于被插函数.     

一类非光滑凸优化问题的邻近梯度算法

考虑求解目标函数为光滑损失函数与非光滑正则函数之和的凸优化问题的一种基于线搜索的邻近梯度算法及其收敛性分析,证明了在梯度局部Lipschitz连续条件下该算法是R-线性收敛的,并在非光滑部分为稀疏块LASSO正则函数情况下给出了误差界条件成立的证明,得到了线性收敛率.最后,数值实验结果验证了方法的有效性.     

解约束优化的分段线性有理NCP函数

本文给出新的NCP函数,这些函数是分段线性有理正则伪光滑的,且具有良好的性质.把这些NCP函数应用到解非线性优化问题的方法中.例如,把求解非线性约束优化问题的KKT点问题分别用QP-free方法,乘子法转化为解半光滑方程组或无约束优化问题.然后再考虑用非精确牛顿法或者拟牛顿法来解决该半光滑方程组或无约束优化问题.这个方法是可实现的,且具有全局收敛性.可以证明在一定假设条件下,该算法具有局部超线性收敛性.     

Modelling of dynamical systems with motion dependent discontinuities

The paper introduces two concepts for describing and solving dynamical systems with motion dependent discontinuities such as clearances, impacts, dry friction, or combination of these phenomena. The first approach assumes any dynamic system can be considered as continuous in a finite number of continuous subspaces, which together form so-called global hyperspace. Global solution is obtained by “gluing” local solutions obtained by solving the problem in the continuous subspaces. An efficient numerical algorithm is presented, and then used to solve dynamics of a piecewise oscillator, which has been also verified experimentally. The second approach considers that in reality the system parameters do not change in an abrupt manner. Therefore, a smooth contiunuous function is used to model a transition between the subspaces, in particular the sigmoid function is employed. This allows to control the degree of abruptness on the intersections of the continuous subspaces. An asymmetrical, piecewise linear oscillator has been examined to provide recommendations regarding validity of this approach.     

Piecewise linear approximation for the dynamical Φ_3~4 model

We construct a piecewise linear approximation for the dynamicalΦ3~4 model on T^3.The approximation is based on the theory of regularity structures developed by Hairer(2014).They proved that renormalization in a dynamicalΦ3~4 model is necessary for defining the nonlinear term.In contrast to Hairer(2014),we apply piecewise linear approximations to space-time white noise,and prove that the solutions of the approximating equations converge to the solution of the dynamicalΦ_3~4 model.In this case,the renormalization corresponds to multiplying the solution by a t-dependent function,and adding it to the approximating equation.     

Identification of Parameters of Nonlinear Dynamical Systems Simulated by Volterra Polynomials

We study the identification methods for the nonlinear dynamical systems described by Volterra series. One of the main problems in the dynamical system simulation is the problem of the choice of the parameters allowing the realization of a desired behavior of the system. If the structure of the model is identified in advance, then the solution to this problem closely resembles the identification problem of the system parameters. We also investigate the parameter identification of continuous and discrete nonlinear dynamical systems. The identification methods in the continuous case are based on application of the generalized Borel Theorem in combination with integral transformations. To investigate discrete systems, we use a discrete analog of the generalized Borel Theorem in conjunction with discrete transformations. Using model examples, we illustrate the application of the developed methods for simulation of systems with specified characteristics.     

Pathway identification using parallel optimization for a complex metabolic system in microbial continuous culture

The bio-dissimilation of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae) is a complex bioprocess due to the multiple inhibitions of substrate and products onto the cell growth. In consideration of the fact that both the inhibition mechanisms of 3-hydroxypropionaldehyde (3-HPA) onto the cell growth and the transport systems of glycerol and 1,3-PD across the cell membrane are still unclear, we consider 72 possible metabolic pathways, and establish a novel mathematical model which is represented by an eight-dimensional nonlinear dynamical system. The existence, uniqueness, continuous dependence of solutions to the system and the compactness of the solution set are explored. On the basis of biological robustness, we give a quantitative definition of robustness index of the intracellular substances. Taking the robustness index of the intracellular substances together with the relative error between the experimental data and the computational values of the extracellular substances as a performance index, a parameter identification model is proposed for the nonlinear dynamical system, in which 43848 continuous variables and 1152 discrete variables are involved. A parallel particle swarm optimization — pathways identification algorithm (PPSO-PIA) is constructed to find the optimal pathway and parameters under various experiments conditions. Numerical results show that the optimal pathway and the corresponding dynamical system can describe the continuous fermentation reasonably.     

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.     

Extracting interpretable fuzzy models for nonlinear systems using gradient-based continuous ant colony optimization

This paper exploits the ability of a novel ant colony optimization algorithm called gradient-based continuous ant colony optimization, an evolutionary methodology, to extract interpretable first-order fuzzy Sugeno models for nonlinear system identification. The proposed method considers all objectives of system identification task, namely accuracy, interpretability, compactness and validity conditions. First, an initial structure of model is obtained by means of subtractive clustering. Then, an iterative two-step algorithm is employed to produce a simplified fuzzy model in terms of number of fuzzy sets and rules. In the first step, the parameters of the model are adjusted by utilizing the gradient-based continuous ant colony optimization. In the second step, the similar membership functions of an obtained model merge. The results obtained on three case studies illustrate the applicability of the proposed method to extract accurate and interpretable fuzzy models for nonlinear system identification.     

An algorithm for the estimation of a regression function by continuous piecewise linear functions

The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L ₂ risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.     

Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol

A nonlinear dynamical system was established in our preceding work to describe the batch and continuous bioconversions of glycerol to 1,3-propanediol by Klebsiella pneumoniae. The purpose of this article is to analyze the sensitivity of kinetic parameters of the dynamical system and identify their values from experiment. A global sensitivity analysis approach is constructed by combining the local technique with the Monte Carlo method. With only those parameters of higher sensitivity as design variables, we propose a parameter identification model and solve it by a gradient-based simulated annealing algorithm. Numerical results show that our methods are feasible and efficient.     

Periodic orbits in two classes of piecewise smooth maps with positive nonlinear parts

In this paper we consider two classes of one dimensional piecewise smooth continuous maps that have been derived as normal forms for grazing bifurcations of piecewise smooth dynamical systems. These maps are linear on one side of the phase space and nonlinear on the other side. The case of nonlinear parts with negative coefficients has been studied previously and it is proved that period-adding scenarios are generic in this case. In contrast to this result, in our analytical and numerical results, the period-adding scenarios are not observed when the nonlinear parts have positive coefficients. Furthermore, our results suggest that the typical bifurcation scenario is period doubling cascade leading to chaos in this case, which is similar to that of the smooth logistic map.