Nonlinear observer design for two-time-scale systems

A two-time-scale system involves both fast and slow dynamics. This article studies observer design for general nonlinear two-time-scale systems and presents two alternative nonlinear observer design approaches, one full-order and one reduced-order. The full-order observer is designed by following a scheme to systematically select design parameters, so that the fast and slow observer dynamics are assigned to estimate the corresponding system modes. The reduced-order observer is derived based on a lower dimensional model to reconstruct the slow states, along with the algebraic slow-motion invariant manifold function to reconstruct the fast states. Through an error analysis, it is shown that the reduced-order observer is capable of providing accurate estimation of the states for the detailed system with an exponentially decaying estimation error. In the last part of the article, the two proposed observers are designed for an anaerobic digestion process, as an illustrative example to evaluate their performance and convergence properties.     

一类不匹配不确定动态系统的鲁棒执行器故障检测与重构

针对一类不匹配不确定性动态系统,将不匹配不确定性的滑模控制方法与线性矩阵不等式（LMI）方法结合,设计一种新的鲁棒滑模观测器,提出了不匹配不确定动态系统滑模观测器稳定的充分必要条件以及LMI的存在定理,并证明了对系统不确定性以及外界干扰具有鲁棒性。无须对动态系统进行规范化处理,直接利用LMI方法求解鲁棒观测器增益矩阵,简化了滑模观测器设计过程。根据上述设计的鲁棒滑模观测器,应用等价输出误差介入原理和LMI方法,设计重构执行器故障的优化策略,提出在线获取故障信息的鲁棒执行器故障检测与重构方法,实现执行器故障的检测与重构。数字仿真验证了执行器故障重构方法的可靠性。     

Adaptive gain sliding mode observer for state of charge estimation based on combined battery equivalent circuit model

An adaptive gain sliding mode observer (AGSMO) for battery state of charge (SOC) estimation based on a combined battery equivalent circuit model (CBECM) is presented. The error convergence of the AGSMO for the SOC estimation is proved by Lyapunov stability theory. Comparing with conventional sliding mode observers for the SOC estimation, the AGSMO can minimise chattering levels and improve the accuracy by adaptively adjusting switching gains to compensate modelling errors. To design the AGSMO for the SOC estimation, the state equations of the CBECM are derived to capture dynamics of a battery. A lithium-polymer battery (LiPB) is used to conduct experiments for extracting parameters of the CBECM and verifying the effectiveness of the proposed AGSMO for the SOC estimation.     

Observer-based non-linear control of a fed-batch autoinductive fermentation process

Optimal regulatory control of an autoinductive recombinant culture in a fed-batch reactor is considered. End point optimization results in a three-stage process: biomass growth, inducer synthesis and product synthesis. It is shown that in the last stage the substrate concentration should be maintained constant. This is achieved using an input—output linearizing controller accompanied by a novel non-linear state observer for the estimation of unmeasured state variables on the basis of on-line off-gas carbon dioxide concentration measurements. Experimental runs of luminous recombinant E. coli strain in a laboratory fermenter demonstrate the rapid convergence of the observer estimates as well as the effectiveness and robustness of the overall control system.     

Iterative identification of output error model for industrial processes with time delay subject to colored noise

To dealwith colored noise and unexpected load disturbance in identification of industrial processes with time delay, a bias-eliminated iterative least-squares (ILS) identification method is proposed in this paper to estimate the output error model parameters and time delay simultaneously. An extended observation vector is constructed to establish an ILS identification algorithm. Moreover, a variable forgetting factor is introduced to enhance the convergence rate of parameter estimation. For consistent estimation, an instrumental variable method is given to deal with the colored noise. The convergence and upper bound error of parameter estimation are analyzed. Two illustrative examples are used to show the effectiveness and merits of the proposed method.     

未知输入扩展卡尔曼滤波的收敛性分析及其非线性故障诊断

Unknown input observer is one of the most famous strategies for robust fault diagnosis of linear systems, but studies on nonlinear cases are not sufficient. On the other hand, the extended Kalman filter （EKF） is wellknown in nonlinear estimation, and its convergence as an observer of nonlinear deterministic system has been derived recently. By combining the EKF and the unknown input Kalman filter, we propose a robust nonlinear estimator called unknown input EKF （UIEKF） and prove its convergence as a nonlinear robust observer under some mild conditions using linear matrix inequality （LMI）. Simulation of a three-tank system “DTS200”, a benchmark in process control, demonstrates the robustness and effectiveness of the UIEKF as an observer for nonlinear systems with uncertainty, and the fault diagnosis based on the UIEKF is found successful.     

A PROCEDURE FOR OBTAINING M-ESTIMATES IN REGRESSION MODELS WITH SERIALLY DEPENDENT ERRORS

Abstract. Applications where error terms in a regression model display both non-normal and serially dependent behavior are considered. For the estimation of the parameters, an iterative Cochrane-Orcutt type M-estimator is proposed. A proof of convergence of the iterative procedure is given. In a simulation experiment, where the least absolute error criterion is applied, the performance of the estimator is tested and the theoretical convergence properties illustrated. In particular, the existence of multiple stationary points in an iterative process is discussed.     

On-line estimation of VFA concentration in anaerobic digestion via methane outflow rate measurements

This paper deals with the design of a robust nonlinear observer as a software sensor to achieve the on-line estimation of the concentration of Volatile Fatty Acids (VFA) in a class of continuous anaerobic digesters (AD). Taking into account the limited availability of on-line sensors for AD process, in this contribution is assumed that only the methane outflow rate is available for on-line measurement. The estimation method is based on a modified version for a two-dimensional mathematical model of AD process. From the differential algebraic observability approach it is shown that the VFA concentration is detectable from the methane outflow rate measurements. The observer convergence is analyzed by using Lyapunov stability techniques. Numerical simulations illustrate the effectiveness of the proposed estimation method for a four-dimensional AD model with uncertainties associated with unmodeled dynamics and disturbances in the inflow composition.     

化学反应器的神经元自适应观测和非线性控制

提出了一种基于神经网络的自适应观测和非线性控制策略,证明了自适应观测器的收敛件和非线性控制系统的稳定性,将其用于连续搅拌釜式放热反应器的浓度控制。根据可在线测量的反应温度,在线估计不可在线测量的反应物浓度和辨识Arrhenius指前因子,并利用重构的状态信息设计出带约束的非线性控制策略。仿真结果表明,观测器/控制器的组合提供了满意的闭环特性,证实了本文方法的有效性。     

ESTIMATION AND LOCALIZATION OF MAXIMUM TEMPERATURE IN A TUBULAR CHEMICAL REACTOR BY LUENBERGER STATE OBSERVER

This study deals with an observer built for distributed parameter systems described by nonlinear representations. This observer was applied to a tubular chemical reactor in order to estimate the maximum temperature of the reactant mixture and its position in stationary and transient regimes. This estimation uses a Luenberger observer, while MacCormack's numerical method was used for resolution of the partial differential equations. Furthermore, to take into account the nonlinearity of the system, the gain of this estimator was proposed as a function of the position along the chemical reactor. This observer requires the values of both concentration and temperature at the inlet but only of temperature at the outlet. The convergence and robustness of this estimator were experimentally tested with initialization and modeling errors.     

Robust software sensor design for the state estimation in a sulfate-reducing bioreactor

The goal of this work is to present a class of state observer in order to infer unavailable concentrations in a sulfate-reducing bioreactor containing an anoxic culture of Desulfovibrio alaskensis 6SR from the measurements of the sulfate concentration. The methodology is applied to the corresponding sulfate-reducing model considering modeling uncertainties. The proposed observer design presents a proportional term plus a sigmoid function structure. The model’s observability was locally analyzed by the observability matrix test, concluding that the sulfate-reducing bioreactor is detectable. A sketch of proof of the observer’s convergence is depicted in order to show the asymptotic convergence characteristics. Numerical experiments provide an overview about the superior performance of the proposed observer methodology compared to a sliding-mode and high order sliding mode observers.     

Inference on Multivariate Heteroscedastic Time Varying Random Coefficient Models

In this article, we introduce the general setting of a multivariate time series autoregressive model with stochastic time‐varying coefficients and time‐varying conditional variance of the error process. This allows modelling VAR dynamics for non‐stationary time series and estimation of time‐varying parameter processes by the well‐known rolling regression estimation techniques. We establish consistency, convergence rates, and asymptotic normality for kernel estimators of the paths of coefficient processes and provide pointwise valid standard errors. The method is applied to a popular seven‐variable dataset to analyse evidence of time variation in empirical objects of interest for the DSGE (dynamic stochastic general equilibrium) literature.     

Parameter identification for state estimation—application to bioprocess software sensors

Together with some on-line measurements, a reliable process model is the key ingredient of a successful state observer design. In common practice, the model parameters are inferred from experimental data so as to minimize a model prediction error, e.g. so as to minimize an output least-squares criterion. In this procedure, no care is actually exercised to ensure that the unmeasured model states are sensitive to the measured states. In turn, if sensitivity is too low, the resulting state observer will probably generate poor estimates of the unmeasured states. To alleviate these problems, a new parameter identification procedure is proposed in this study, which is based on a cost function combining a conventional prediction error criterion with a state estimation sensitivity measure. Minimization of this combined cost function produces a model dedicated to state estimation purposes. A thorough analysis of the procedure is presented in the context of bioreactor modeling, including parameter identification, model validation and design of extended Kalman filters and full horizon observers.     

Saturated PI control of continuous bioreactors with Haldane kinetics

The problem of designing a saturated Output-Feedback (OF) controller for a continuous bioreactor with Haldane kinetics is addressed. The reactor must be operated at maximum biomass production rate by manipulating the feed rate on the basis of a (biomass or substrate) measurement. The consideration of the problem as an interlaced observer-control design leads to a saturated PI controller that recovers (up to observer convergence) the behavior of a detailed model-based robust nonlinear state-feedback (SF) globally stabilizing saturated controller. The saturated PI control scheme has: (i) closed-loop stability conditions in terms of control gains and limits, and (ii) a simple construction-tuning procedure. The proposed approach is illustrated and tested with a representative case example through numerical simulations.     

State estimation for high-dimensional chemical processes

The application of conventional observer designs for high-dimensional systems may not always be practical due to high computational requirements or the resulting observers being too sensitive to measurement noise. In order to address these issues, this paper presents two observer design techniques for state estimation of high-dimensional chemical processes. One technique is used for systems with inputs, whereas the other one is specifically geared towards systems that are not excited from the outside. Both of these observers are applicable to linear and with a modification to non-linear systems.The main idea behind the presented observer designs is that a reduced-order observer is implemented instead of a conventional state estimator. The motivation is that subspaces, which are close to being unobservable, cannot be correctly reconstructed in a realistic setting due to measurement noise and inaccuracies in the model. The presented approaches make use of this observation and only reconstruct the parts of the system where accurate state estimation is possible. The observer designs are illustrated on a 30-tray distillation column model. Additionally, it has been shown that the location of process measurements has a major effect on the performance of the presented reduced-order observers.     

EFFICIENT NON‐PARAMETRIC ESTIMATION OF THE SPECTRAL DENSITY IN THE PRESENCE OF MISSING OBSERVATIONS

The problem of non‐parametric spectral density estimation for discrete‐time series in the presence of missing observations has a long history. In particular, the first consistent estimators of the spectral density have been developed at about the same time as consistent estimators for non‐parametric regression. On the other hand, while for now, the theory of efficient (under the minimax mean integrated squared error criteria) and adaptive nonparametric regression estimation with missing data is well developed, no similar results have been proposed for the spectral density of a time series whose observations are missed according to an unknown stochastic process. This article develops the theory of efficient and adaptive estimation for a class of spectral densities that includes classical causal autoregressive moving‐average time series. The developed theory shows how a missing mechanism affects the estimation and what penalty it imposes on the risk convergence. In particular, given costs of a single observation in time series with and without missing data and a desired accuracy of estimation, the theory allows one to choose the cost‐effective time series. A numerical study confirms the asymptotic theory.     

Nonlinear adaptive control algorithm for a multicomponent batch distillation column

In this article, a nonlinear adaptive control strategy is proposed for a multicomponent batch distillation column. The hybrid control scheme consists of a generic model controller (GMC) and a nonlinear adaptive state estimator (ASE). In the first part of the study, an adaptive observer is designed aiming to estimate the partially known parameters based on the measured compositions in the presence of process/predictor mismatch. The open-loop dynamic behavior of the developed ASE estimator is investigated under initialization error, disturbance, and uncertain parameters. In the subsequent part, the adaptive GMC-ASE controller (GMC control structure in conjunction with ASE estimator) has been synthesized for the example distillation column. A simulation-based comparative study has been conducted between the derived nonlinear GMC-ASE control algorithm and a gain-scheduled proportional integral (GSPI) law in terms of constant composition control. The proposed adaptive control scheme is shown to be quite promising due to the exponential error convergence capability of the ASE estimator in addition to the high-quality performance of the GMC controller.     

Temperature regulation of a class of continuous chemical reactor based on a nonlinear Luenberger-like observer

In this paper is studied the temperature regulation problem of a continuous stirred tank reactor with two consecutive and oscillatory exothermic reactions in which the heat of the reactions is unknown. A strategy to estimate the reactions' heat from measurements of reactor temperature is proposed. This strategy is coupled with a linearizing-like feedback regulatory control. Practical convergence of this control scheme is proved when the uncertainty observer satisfies the high gain assumption. The performance of the uncertainty observer and the closed loop behavior is illustrated by means of numerical simulations. ©1997 SCI     

State observer scheme for joint kinetic parameter and state estimation

The development and acceptance of model-based monitoring tools in the bioprocess industry is made difficult by the usually large uncertainty associated with the process model. A natural approach to handle this issue is the design of adaptive state observers for the joint estimation of the process state and some of the uncertain model parameters. However, the state extension is often restricted to a few parameters only, for which observability conditions are satisfied with the available measurement information. In this study, this latter issue is circumvented by the combination of two observers: (a) a receding-horizon observer is designed for the joint estimation of the state and uncertain model parameters, and (b) an asymptotic observer, which provides state estimates independently of the kinetic model, is used to provide the missing additional information to the receding-observer, thus avoiding observability loss. This paper derives the properties of this combined state estimation scheme and demonstrates its performance with a realistic simulation case study of animal cell cultures.