A second order ODE is sufficient for modelling of many periodic signals

Which is the minimum order an autonomous non-linear ordinary differential equation (ODE) needs to have to be able to model a periodic signal? This question is motivated by recent research on periodic signal analysis, where non-linear ODEs are used as models. The results presented here show that a second order ODE is sufficient for a large class of periodic signals. More precisely, conditions on a periodic signal are established that imply the existence of an ODE that has the periodic signal as a solution. A criterion that characterizes the above class of periodic signals by means of the overtone contents of the signals is also presented. The reason why higher order ODEs are sometimes needed is illustrated with geometric arguments. Extensions of the theoretical analysis to cases with orders higher than two are developed using this insight.     

Mean value theorem for boundary value problems with given upper and lower solutions

An approach based on successive application of the mean value theorem or, equivalently, a successive linear interpolation that excludes extrapolation, is described for two-point boundary value problem (BVP) associated with nonlinear ordinary differential equations (ODEs). The approach is applied to solve numerically a two-point singular BVP associated with a second-order nonlinear ODE which is a mathematical model in membrane response of a spherical cap that arises in nonlinear mechanics. The upper and lower bounds on solution for the foregoing second-order ODE are assumed known analytically. Other possible methods such as the successive bisection for the BVP associated with second-order nonlinear ODE and a multivariable Taylor series for the second or higher-order nonlinear ODEs are also discussed to solve two-point BVP. The scope/limitation of the later methods and other possible higher-order methods in the present context are stressed.     

Least-squares periodic signal modeling using orbits of nonlinear ODEs and fully automated spectral analysis

Periodic signals can be modeled by means of second-order nonlinear ordinary differential equations (ODEs). The right-hand side function of the ODE is parameterized in terms of known basis functions. The least-squares algorithm developed for estimating the coefficients of these basis functions gives biased estimates, especially at low signal-to-noise ratios. This is due to noise contributions to the periodic signal and its derivatives evaluated using finite difference approximations. In this paper a fully automated spectral analysis (ASA) technique is used to eliminate these noise contributions. A simulation study shows that using the ASA technique significantly improves the performance of the least-squares estimator.     

Unsupervised kernel least mean square algorithm for solving ordinary differential equations

In this paper a novel method is introduced based on the use of an unsupervised version of kernel least mean square (KLMS) algorithm for solving ordinary differential equations (ODEs). The algorithm is unsupervised because here no desired signal needs to be determined by user and the output of the model is generated by iterating the algorithm progressively. However, there are several new approaches in literature to solve ODEs but the new approach has more advantages such as simple implementation, fast convergence and also little error. Furthermore, it is also a KLMS with obvious characteristics. In this paper the ability of KLMS is used to estimate the answer of ODE. First a trial solution of ODE is written as a sum of two parts, the first part satisfies the initial condition and the second part is trained using the KLMS algorithm so as the trial solution solves the ODE. The accuracy of the method is illustrated by solving several problems. Also the sensitivity of the convergence is analyzed by changing the step size parameters and kernel functions. Finally, the proposed method is compared with neuro-fuzzy [21] approach.     

A homotopy-based moving horizon estimation

Moving horizon estimation (MHE) solves a constrained dynamic optimisation problem. Including nonlinear dynamics into an optimal estimation problem generally comes at the cost of tackling a non-convex optimisation problem. Here, a particular model formulation is proposed in order to convexify a class of nonlinear MHE problems. It delivers a linear time-varying (LTV) model that is globally equivalent to the nonlinear dynamics in a noise-free environment, hence the optimisation problem becomes convex. On the other hand, in the presence of unknown disturbances, the accuracy of the LTV model degrades and this results in a less accurate solution. For this purpose, some assumptions are imposed and a homotopy-based approach is proposed in order to transform the problem from convex to non-convex, where the sequential implementation of this technique starts with solving the convexified MHE problem. Two simulation studies validate the efficiency and optimality of the proposed approach with unknown disturbances.     

L1 adaptive control for general partial differential equation (PDE) systems

This paper addresses the L₁ adaptive control problem for general Partial Differential Equation (PDE) systems. Since direct computation and analysis on PDE systems are difficult and time-consuming, it is preferred to transform the PDE systems into Ordinary Differential Equation (ODE) systems. In this paper, a polynomial interpolation approximation method is utilized to formulate the infinite dimensional PDE as a high-order ODE first. To further reduce its dimension, an eigenvalue-based technique is employed to derive a system of low-order ODEs, which is incorporated with unmodeled dynamics described as bounded-input, bounded-output (BIBO) stable. To establish the equivalence with original PDE, the reduced-order ODE system is augmented with nonlinear time-varying uncertainties. On the basis of the reduced-order ODE system, a dynamic state predictor consisting of a linear system plus adaptive estimated parameters is developed. An adaptive law will update uncertainty estimates such that the estimation error between predicted state and real state is driven to zero at each time-step. And a control law is designed for uncertainty handling and good tracking delivery. Simulation results demonstrate the effectiveness of the proposed modeling and control framework.     

Robust adaptive neural observer design for a class of nonlinear parabolic PDE systems

A robust adaptive neural observer design is proposed for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities and bounded disturbances. The modal decomposition technique is initially applied to the PDE system to formulate it as an infinite-dimensional singular perturbation model of ordinary differential equations (ODEs). By singular perturbations, an approximate nonlinear ODE system that captures the dominant (slow) dynamics of the PDE system is thus derived. A neural modal observer is subsequently constructed on the basis of the slow system for its state estimation. A linear matrix inequality (LMI) approach to the design of robust adaptive neural modal observers is developed such that the state estimation error of the slow system is uniformly ultimately bounded (UUB) with an ultimate bound. Furthermore, using the existing LMI optimization technique, a suboptimal robust adaptive neural modal observer can be obtained in the sense of minimizing an upper bound of the peak gains in the ultimate bound. In addition, using two-time-scale property of the singularly perturbed model, it is shown that the resulting state estimation error of the actual PDE system is UUB. Finally, the proposed method is applied to the estimation of temperature profile for a catalytic rod.     

Optimal determination of rate coefficients in multiple-reaction systems

An efficient methodology is developed for parameter estimation and is applied by fitting 6 unknown rate coefficients. The estimation procedure is generally applicable to any system, although development has currently been limited to first-order systems of ordinary differential equations (ODE), such as those describing multiple chemical reactions. The objective is to find parameter values so as to minimize the sum of squared error (SSE), where each error term is the difference between the calculated system solution at a point and a selected data value. Since the calculated solution is generally quite nonlinear, an iterative solution is required. At each iteration, parameter values are supplied, the system is solved, and the SSE is determined. In addition, efficient algorithms require the SSE gradient (with respect to the vector of unknown parameters) in order to provide updated parameter estimates. Using conventional techniques, determination of this gradient involves solution of an ODE system for each parameter to be estimated. ff more than a few parameters are involved, the cost could be prohibitive. However, a procedure using adjoint operators is developed in which the SSE gradient can be calculated by solving only one additional ODE system, regardless of the number of parameters being optimized. Combined with a quasi-Newton updating system, an efficient methodology results. This methodology has been applied to a set of six chemical reactions describing the aqueous speciation (hydrolysis) of iodine.     

Parameter identification of periodical signals: Application to measurement and analysis of ocean wave forces

This article presents an approach based on state observers to identify the parameters of an unknown periodic force exerted on a mechanical system. This approach comprises two stages and can be executed in real time by using only displacement measurements. The first stage goal is the estimation of the coefficients of a Fourier series that approximates the periodic force. From the estimated coefficients, the phase and the amplitude of the signal can be simultaneously computed; and from the estimated force, in a second stage, the frequencies of the signal can be estimated. To perform the tasks at each stage, two state observers were designed. To show the applicability of the proposed approach, the reconstruction of a wave force affecting a marine structure as well as the computation of the amplitude and phase of its spectral components was taken as case of study. The performance of the state observer was examined by means of simulations and off-line tests carried out with experimental data. Such data were obtained by executing laboratory tests and measuring waves in the Caribbean sea.     

归一化频率自适应梳状滤波器的稳定性分析

采用多个归一化频率估计器并联形成梳状滤波器, 以跟踪和检测平稳概周期信号各正弦成分的未知频率和未知幅值. 滤波器包括相互耦合的状态估计和频率估计两个非线性微分方程. 运用慢积分流形实现两个微分方程之间的解耦, 获得关于多个频率估计值的概周期非线性动力系统, 再应用平均方法导出估计频率的非线性自治方程. 分析了自治系统的三种局部稳定性: 孤立平衡点的指数稳定性, 中心流形存在性与半稳定性以及结构扰动下的有界性. 说明幅值估计与信号跟随的收敛性和有界性. 给出滤波器参数对频率跟踪和幅值估计的暂态和稳态性能的影响. 算法实现了在给定频率区间而不是给定数值条件下的正弦分量及其幅值的准确跟随, 并且响应速度不受正弦分量幅值大小的影响. 通过仿真验证了算法的有效性.     

Design of fault diagnosis filters and fault-tolerant control for aclass of nonlinear systems

This paper presents a set of algorithms for fault diagnosis and fault tolerant control strategy for affine nonlinear systems subjected to an unknown time-varying fault vector. First, the design of fault diagnosis filter is performed using nonlinear observer techniques, where the system is decoupled through a nonlinear transformation and an observer is used to generate the required residual signal. By introducing an extra input to the observer, a direct estimation of the time-varying fault is obtained when the residual is controlled, by this extra input, to zero. The stability analysis of this observer is proved and some relevant sufficient conditions are obtained. Using the estimated fault vector, a fault tolerant controller is established which guarantees the stability of the closed loop system. The proposed algorithm is applied to a combined pH and consistency control system of a pilot paper machine, where simulations are performed to show the effectiveness of the proposed approach     

一类非线性连续系统参数估计方法

本文提出一类非线性连续系统的参数估计方法.该方法巧妙利用不同幅值的伪随机输入信号,从原系统中离析出线性子系统.基于线性子系统的参数估计值和集合理论,提出一个零极点划分的寻优准则.在此准则下,实现对系统参数的估计.仿真结果表明,该方法的计算速度较快且估值精度较高.     

State and parameter estimation for nonlinear biological phenomena modeled by S-systems

Biological pathways can be modeled as a nonlinear system described by a set of nonlinear ordinary differential equations (ODEs). A central challenge in computational modeling of biological systems is the determination of the model parameters. In such cases, estimating these variables or parameters from other easily obtained measurements can be extremely useful. For example, time-series dynamic genomic data can be used to develop models representing dynamic genetic regulatory networks, which can be used to design intervention strategies to cure major diseases and to better understand the behavior of biological systems. Unfortunately, biological measurements are usually highly affected by errors that hide the important characteristics in the data. Therefore, these noisy measurements need to be filtered to enhance their usefulness in practice. This paper addresses the problem of state and parameter estimation of biological phenomena modeled by S-systems using Bayesian approaches, where the nonlinear observed system is assumed to progress according to a probabilistic state space model. The performances of various conventional and state-of-the-art state estimation techniques are compared. These techniques include the extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and the developed improved particle filter (IPF). Specifically, two comparative studies are performed. In the first comparative study, the state variables (the enzyme CadA, the transport protein CadB, the regulatory protein CadC and lysine Lys for a model of the Cad System in E. coli (CSEC)) are estimated from noisy measurements of these variables, and the various estimation techniques are compared by computing the estimation root mean square error (RMSE) with respect to the noise-free data. In the second comparative study, the state variables as well as the model parameters are simultaneously estimated. In this case, in addition to comparing the performances of the various state estimation techniques, the effect of the number of estimated model parameters on the accuracy and convergence of these techniques is also assessed. The results of both comparative studies show that the UKF provides a higher accuracy than the EKF due to the limited ability of EKF to accurately estimate the mean and covariance matrix of the estimated states through lineralization of the nonlinear process model. The results also show that the IPF provides a significant improvement over PF because, unlike the PF which depends on the choice of sampling distribution used to estimate the posterior distribution, the IPF yields an optimum choice of the sampling distribution, which also accounts for the observed data. The results of the second comparative study show that, for all techniques, estimating more model parameters affects the estimation accuracy as well as the convergence of the estimated states and parameters. However, the IPF can still provide both convergence as well as accuracy related advantages over other estimation methods.     

重复学习控制及其在一类非线性系统中的应用

为跟踪或抑制仅周期已知的未知周期参考或扰动信号,提出一种新的重复学习控制方法,利用系统的稳态误差并通过迭代学习构造前馈补偿,实现了误差的渐近收敛,将所提出方法应用于一类常见的扰动信号和系统输出具有未知非线性关系的非线性系统,假设其满足连续里普希斯条件,利用重复学习控制器,系统的稳态误差可以减小到极低的程度,该方法控制精度高,实现简单,与传统的基于时延内模的重复控制方法相比,具有对非重复性干扰不敏感的优点,仿真结果验证了该方法的有效性。     

带有色厚尾量测噪声的鲁棒高斯近似滤波器和平滑器

为了解决带有色厚尾量测噪声的非线性状态估计问题,本文提出了新的鲁棒高斯近似（Gaussian approximate,GA）滤波器和平滑器.首先,基于状态扩展方法将量测差分后带一步延迟状态和白色厚尾量测噪声的非线性状态估计问题,转化成带厚尾量测噪声的标准非线性状态估计问题.其次,针对量测差分后模型中的噪声尺度矩阵和自由度（Degrees of freedom,DOF）参数未知问题,设计了新的高斯近似滤波器和平滑器,通过建立未知参数和待估计状态的共轭先验分布,并利用变分贝叶斯方法同时估计未知的状态、尺度矩阵、自由度参数.最后,利用目标跟踪仿真验证了本文提出的带有色厚尾量测噪声的鲁棒高斯近似滤波器和平滑器的有效性以及与现有方法相比的优越性.     

Bayesian sparse solutions to linear inverse problems with non-stationary noise with Student-t priors

Bayesian approach has become a commonly used method for inverse problems arising in signal and image processing. One of the main advantages of the Bayesian approach is the possibility to propose unsupervised methods where the likelihood and prior model parameters can be estimated jointly with the main unknowns. In this paper, we propose to consider linear inverse problems in which the noise may be non-stationary and where we are looking for a sparse solution. To consider both of these requirements, we propose to use Student-t prior model both for the noise of the forward model and the unknown signal or image. The main interest of the Student-t prior model is its Infinite Gaussian Scale Mixture (IGSM) property. Using the resulted hierarchical prior models we obtain a joint posterior probability distribution of the unknowns of interest (input signal or image) and their associated hidden variables. To be able to propose practical methods, we use either a Joint Maximum A Posteriori (JMAP) estimator or an appropriate Variational Bayesian Approximation (VBA) technique to compute the Posterior Mean (PM) values. The proposed method is applied in many inverse problems such as deconvolution, image restoration and computed tomography. In this paper, we show only some results in signal deconvolution and in periodic components estimation of some biological signals related to circadian clock dynamics for cancer studies.     

Periodic signal modeling based on Lie/spl acute/nard's equation

The problem of modeling periodic signals is considered. The approach taken here is motivated by the well known theoretical results on the existence of periodic orbits for Lie/spl acute/nard systems and previous results on modeling periodic signals by means of second-order nonlinear ordinary differential equations. The approach makes use of the appropriate conditions imposed on the polynomials of the Lie/spl acute/nard's system to guarantee the existence of a unique and stable limit cycle. These conditions reduce the number of parameters required to generate accurate models.     

具有输入死区的非线性系统的鲁棒重复控制

针对一类输入含死区非线性特性的周期时变系统, 在周期时变参数不可参数化的情形下设计鲁棒重复控制器. 采用微分自适应律估计未知死区参数, 剩余的有界项通过鲁棒方法予以消除, 为避免出现颤振现象, 采用饱和函数替代符号函数. 在系统输出跟踪周期轨迹的情形下, 将非参数化不确定项转化为含周期时变参数的形式, 以达到利用周期学习律进行估计的目的. 理论分析与仿真结果表明, 采用部分饱和或全饱和学习算法均能实现输出误差有界收敛, 并保证闭环系统所有信号有界.     

Lie- and chronologico-algebraic tools for studying stability of time-varying systems

We will study stability and asymptotic stability for time-varying systems described by ODEs of the form , where f(t,x) is 1-periodic with respect to t and >0 is a small parameter. Since the discovery of stabilizing effect of vibration in the reverse pendulum example, there have been a lot of study regarding stability of such systems and design of fast-oscillating stabilizing feedback laws. In this paper we suggest an approach which is kind of high-order averaging procedure based on Lie algebraic formalism and the formalism of chronological calculus. This latter is a method of asymptotic analysis for flows generated by time-variant ODE. We apply the approach to study stability issues for linear and nonlinear systems. In particular, we derive conditions of stability for the second- and third-order linear differential equations with periodic fast-oscillating coefficients, we study output-feedback stabilization of bilinear systems and consider high-order averaging procedure for nonlinear systems under homogeneity assumptions. At the end we study the problem of stabilization of nonholonomic (control-linear) systems by means of time-varying feedbacks.     

Nonlinear robust fault reconstruction and estimation using a sliding mode observer

This paper considers fault detection and estimation issues for a class of nonlinear systems with uncertainty, using an equivalent output error injection approach. A particular design of sliding mode observer is presented for which the parameters can be obtained using LMI techniques. A fault estimation approach is presented to estimate the fault and the estimation error is dependent on the bounds on the uncertainty. For a special class of uncertainty, a fault reconstruction scheme is presented where the reconstructed signal can approximate the fault signal to any accuracy. The proposed fault estimation/reconstruction signals are only based on the available plant input/ouput information and can be calculated on-line. Finally, a simulation study on a robotic arm system is presented to show the effectiveness of the scheme.