Regularization aspects in continuous-time model identification

This paper presents an analysis of some regularization aspects in continuous-time model identification. The study particulary focuses on linear filter methods and shows that filtering the data before estimating their derivatives corresponds to a regularized signal derivative estimation by minimizing a compound criterion whose expression is given explicitly. A new structure based on a null phase filter corresponding to a true regularization filter is proposed and allows to discuss the filter phase effects on parameter estimation by comparing its performances with those of the Poisson filter-based methods. Based on this analysis, a formulation of continuous-time model identification as a joint system input-output signal and model parameter estimation is suggested. In this framework, two linear filter methods are interpreted and a compound criterion is proposed in which the regularization is ensured by a model fitting measure, resulting in a new regularization filter structure for signal estimation.     

Identification of a toroidal continuously variable transmission using continuous-time system identification methods

This paper deals with the identification of a toroidal continuously variable transmission (T-CVT). The model describing the T-CVT system is a model with exogenous time-varying parameters and, in order to facilitate the parameter estimation, the complete T-CVT model is split into two sub-models. The parameters of the first sub-model are estimated with the Poisson moment functionals (PMF) method around several operating points, while the parameter of the second sub-model, reduced to a pure integrator, is estimated with the linear integral filter (LIF) method. These two methods belong to the continuous-time system identification methods. The complete T-CVT model accuracy is finally verified with experiments carried out with a test-bed and an actual vehicle.     

Identification of time delays in linear stochastic systems

This paper presents a new approach to the explicit identification of an input time delay in continuous-time linear systems. The system model is converted to a discrete-time version, assuming that a digital computer is to be used for time delay estimation and control. A recursive identification algorithm based on parallel Kalman filtering and Bayes' estimation is developed. The sampling rate is adapted during the time delay estimation process using the most recent estimate of the time delay. This method assures that the estimate of the time delay approaches the true value with each successive iteration. The proposed method also has the advantage of a fast convergence rate because prior knowledge of the delay, if available, can be effectively utilized.     

Digital sliding mode controller design for multiple time-delay continuous-time transfer function matrices with a long input–output delay

This paper extends the dominant eigenvector-based sliding mode control (SMC) design methodology, which was originally developed for delay-free continuous-time processes with known parameters, to the case of multiple time-delay continuous-time processes with known/unknown parameters. In addition, this paper presents a new prediction-based Chebyshev quadrature digital redesign methodology for indirect design of the digital counterpart of the analog sliding mode controller (ASMC) for multiple time-delay continuous-time transfer function matrices with either a long input delay or a long output delay. An approximated discrete-time model and its corresponding continuous-time model are constructed for multiple time-delay continuous-time stable/unstable dynamical processes with known/unknown parameters, using first the conventional observer/Kalman filter identification (OKID) method. Then, an optimal ASMC is developed using the linear quadratic regulator (LQR) approach, in which the corresponding sliding surface is designed using the user-specified eigenvectors and the scalar sign function. For digital implementation of the proposed non-augmented low-dimensional ASMC, a digital counterpart is designed based on the existing prediction-based digital redesign method and the newly developed prediction-based Chebyshev quadrature digital redesign method. Finally, a non-augmented low dimensional digital observer with a long input or output dead time is constructed for the implementation of the digitally redesigned sliding mode controller, to improve the performances of multiple time-delay dynamical processes. The effectiveness of the proposed method has been verified by means of two illustrative examples.     

Optimal estimation for continuous-time Markovian jump linear systems with delayed measurements

This paper is to investigate the linear minimum mean square error estimation for continuous-time Markovian jump linear systems with delayed measurements. The key technique applied for treating the measurement delay is the reorganization innovation analysis, by which the state estimation with delayed measurements is transformed into a standard linear mean square filter of an associated delay-free system. The optimal filter is derived based on the innovation analysis method together with geometric arguments in Hilbert space. An analytical solution to the filter is obtained in terms of two Riccati differential equations, and hence is very simple in computation. Computer simulations are carried out to evaluate the performance of the proposed algorithms. The problem of tracking a maneuvering target is addressed.     

参数不确定系统的H∞估计问题的显式解和中心解

研究在连续时间情形下的具有部分参数不确定性的系统的H∞状态估计问题,它可 以被化简为带有一个自由可调参数对象的H∞状态估计,由此可得到滤波器的简洁通解显 式.并进一步研究了H∞估计的中心解,以及它与卡尔曼滤波器的关系.实例计算表明,对于 参数具有不确定性的系统,H∞滤波器的性能明显地优于卡尔曼滤波器.     

基于PSO的连续系统直接辨识及在带弹性负载电机的应用

针对连续方程误差模型辨识须引入数值滤波器而造成有偏估计,提出一种基于粒子群优化(PSO)算法和连续输出误差法的连续系统直接辨识方法.该方法用简化四阶龙格库塔法进行系统输出的数值逼近,并采用PSO进行数值优化以避免有偏估计.对所提出方法的估计性质进行分析,进而得出辨识问题的全局解在开环下参数估计的一致性.仿真案例表明,所提出方法对案例的辨识精度高于简化修正辅助变量法等几种连续系统辨识方法.将所提出的方法应用于带弹性负载的电机模型辨识,获得了良好的估计,从而表明了所提出方法在应用上的有效性.     

On-line estimation of time delay and continuous-time process parameters†

A simple technique is presented for on-line estimation of constant or slowly-varying continuous-time process parameters and time delay. The method is shown to allow considerable flexibility for application to systems of varying complexity. A major advantage of the algorithm lies in its ability to track time-delay variations over a practically unlimited range. The technique is based on approximation of time delay in the frequency domain by a rational transfer function, construction of the derivatives of process input and output using multiple filters, and estimation using a model which is non-linear in the desired parameters. In spite of this inherent non-linearity with respect to the sought parameters, in general the estimation schemes lead to the true, unique solution. The cases when this is not true are shown not to be of serious consequence.     

Identification of discrete-time output error model for industrial processes with time delay subject to load disturbance

In this paper, a bias-eliminated output error model identification method is proposed for industrial processes with time delay subject to unknown load disturbance with deterministic dynamics. By viewing the output response arising from such load disturbance as a dynamic parameter for estimation, a recursive least-squares identification algorithm is developed in the discrete-time domain to estimate the linear model parameters together with the load disturbance response, while the integer delay parameter is derived by using a one-dimensional searching approach to minimize the output fitting error. An auxiliary model is constructed to realize consistent estimation of the model parameters against stochastic noise. Moreover, dual adaptive forgetting factors are introduced with tuning guidelines to improve the convergence rates of estimating the model parameters and the load disturbance response, respectively. The convergence of model parameter estimation is analyzed with a rigorous proof. Illustrative examples for open- and closed-loop identification are shown to demonstrate the effectiveness and merit of the proposed identification method.     

Robust H ∞ filter design for time-delay systems with saturation

This paper investigates the H∞ filtering problem for a class of linear continuous-time systems with both time delay and saturation. Such systems have time delay in their state equations and saturation in their output equations, and their process and measurement noises have unknown statistical characteristics and bounded energies. Based on the Lyapunov-Krasovskii stability theorem and the linear matrix inequalities (LMIs) technique, a generalized dynamic filter architecture is proposed, and a filter design method is developed. The linear H∞ filter designed by the method can guarantee the H∞ performance. The parameters of the designed filter can be obtained by solving a kind of LMI. An illustrative example shows that the design method proposed in this paper is very effective.     

A PMF-based subspace method for continuous-time model identification. Application to a multivariable winding process

This paper presents a methodology for system identification of continuous-time state-space models from finite sampled input-output signals. The estimation problem of the consecutive time-derivatives and integrals of the input-output signals is considered. The appropriate frequency characteristcs of a linear filtering based on the Poisson moment functionals in regards to the derivative or integral estimation problem is shown. The proposed method combines therefore the Poisson moment functionals technique with subspace based state-space system identification methods. The developed algorithm is based on a generalized singular value decomposition to compensate the noise colouring caused by the linear prefiltering of the input-output data. Rules of thumb are presented to choose the design parameters and new regards to the selection of the Poisson filter cut-off frequency are introduced. Finally, the proposed method is applied to a multivariable winding processes. The experimental results emphasize the applicability of the developed methodology.     

Alternative model structure with simplistic noise model to identify linear time invariant systems subjected to non-stationary disturbances

Non-stationary disturbances are of common occurrence in chemical process industry. These cannot be modeled using constant parameterized models and hence pose a difficult problem in the identification of true process and disturbance dynamics. A simple system identification technique to identify the linear processes affected by non-stationary disturbances is proposed in this work. This uses a time varying bias term, a representative of the additive non-stationary external disturbance entering the process, in addition to the output predictions in an ARMAX or OE model framework. Decoupled loss function and covariance update with different forgetting factors for linear time invariant input–output dynamics part and time varying part (bias term) of the model ensures the unbiased estimation of true process dynamics along with disturbance dynamics. Practical issues such as time delay estimation, model order selection are discussed. Extensions for time varying processes and MIMO processes are also proposed. Validation is performed using various simulation studies.     

A frequency domain step response identification method for continuous-time processes with time delay

Step response test is widely practiced for model identification in process industry. A frequency domain step response identification method is proposed for obtaining a continuous-time process model with time delay. By introducing a damping factor to the step response for realization of Laplace transform, a frequency response estimation algorithm is first proposed, in which only single integral is needed for computation, compared to recently developed identification methods based on multiple integral in time domain. Based on the estimated frequency response, two model fitting algorithms are developed analytically for obtaining a time delay model of first-, second-, or higher order with repetitive poles. Another two algorithms based on fitting multiple frequency response points thus estimated are proposed for obtaining a time delay model of any order, the latter of which may also be used to improve fitting accuracy over a specified frequency range interested to control design. Meanwhile, practical strategies to consolidate identification robustness against measurement noise are given based on consistent estimation analysis, together with a guideline for model structure selection to realize optimal fitting for identification of a high order process. Illustrative examples from recent references are used to demonstrate the effectiveness and merits of the proposed identification algorithms.     

Robust identification of first-order plus dead-time model from step response

In this paper, a simple yet robust identification method for a linear monotonic process, derived from a step test, is proposed. New linear regression equations are derived, from which the parameters of a first-order plus dead-time model can be obtained directly. No iterations in calculation are needed. The proposed method outperforms the existing estimation methods that use step-test responses. The estimation error is smaller in both the time domain and the frequency domain. Furthermore, the method is robust in the presence of large amounts of measurement noise. The effectiveness of the identification method has been demonstrated through a number of simulation examples and a real-time test.     

连续时间模型的闭环子空间辨识

本文对直接使用采样数据进行连续系统的闭环子空间辨识问题进行了研究．将线性滤波方法与基于主 元分析的子空间辨识相结合,利用参考输入或者外部激励信号的高阶滤波变换的正交投影变量作为辅助变量,提出 了一种新的连续时间系统闭环子空间辨识算法．数值仿真表明了与其他算法相比,本文提出的算法具有很好的辨识 效果．     

Adaptive parameter estimation for a general dynamical system with unknown states

This paper is concerned with the design of a state filter for a time‐delay state‐space system with unknown parameters from noisy observation information. The key is to investigate new identification algorithms for interactive state and parameter estimation of the considered system. Firstly, an observability canonical state‐space model is derived from the original model by linear transformation for the purpose of simplifying the model structure. Secondly, a direct state filter is formulated by minimizing the state estimation error covariance matrix on the basis of the Kalman filtering principle. Thirdly, once the unknown states are estimated, a state filter–based recursive least squares algorithm is proposed for parameter estimation using the least squares principle. Then, a state filter–based hierarchical least squares algorithm is derived by decomposing the original system into several subsystems for improving the computational efficiency. Finally, the numerical examples illustrate the effectiveness and robustness of the proposed algorithms.     

含时延和丢失数据不确定系统的FIR滤波器

针对状态空间模型中存在服从伯努利分布的时延和随机观测丢失的情况,基于极大似然法则,分别设计有限脉冲响应(finite impulse response, FIR)滤波器的慢速率批处理形式和快速率迭代形式.首先,将时延和数据丢失情况下的模型表述为服从伯努利分布的概率线性函数;然后,通过极大似然处理从而得到所提出极大似然FIR算法;最后,将在相同条件下的极大似然FIR估计、改进型卡尔曼滤波以及无偏FIR估计3种滤波方法进行对比,从估计误差、均方根误差和不确定性影响等角度进行比较分析.实验部分通过3-DOF直升机模型仿真,可发现所提出极大似然FIR估计方法在处理时延和数据丢失问题时更加有效,鲁棒性更高.     

Identification of delayed system using instrumental variable method

System identification uses system inputs and outputs to raise mathematical models.Various techniques of system identification exist that offer a nominal model and an uncertainty bound.Many practical systems such as thermal processes & chemical processes have inbuilt time delay.If the time delay used in the system model for controller design does not concur with the actual process time delay,a closed-loop system may be unstable or demonstrate unacceptable transient response characteristics so here the time delay is assumed to be time-invariant. This paper proposes on-line identification of delayed complex/uncertain systems using instrumental variable(Ⅳ) method.Parametric uncertainty has been considered which may be represented by variations of certain system parameters over some possible range.This method allows consistent estimation when the system parameters are associated with the noise terms,as the IV methods(IVM’s)usually make no assumption on the noise correlation configuration.The faster convergence of the parameters including noise terms has been proved in this paper.Iterative prefiltering(IP)method has also been used for the identification of the delayed uncertain system and the graphical results given in this paper demonstrate that the convergence results are inferior to the instrumental variable method.     

Multipartitioning linear estimation algorithms: Continuous systems

The partitioned estimation algorithms of Lainiotis for the linear continuous-time state estimation problem have been generalized in this paper in two important ways. First, the initial condition of the estimation problem can, using the results of this paper, be partitioned into the sum of an arbitrary number of jointly Gaussian random variables; and second, these jointly Gaussian random variables may be statistically dependent. The form of the resulting algorithm consists of an imbedded Kalman filter with partial initial conditions and one correction term for each other partition or subdivision of the initial state vector. Emphasis in this paper is on ways in which this approach, called multipartitioning, can be used to provide added insight into the estimation problem. One significant application is in the parameter identification problem where identification algorithms can be formulated in which the inversion of the information matrix of the parameters is replaced by simple division by scalars. A second use of multipartitioning is to show the specific effects on the filtered state estimate of off-diagonal terms in the initial-state covariance matrix.