High gain observer based extended generic model control with application to a reactive distillation column

This article aims at synthesizing an estimator based hybrid control scheme that consists of a high gain nonlinear observer and the extended generic model controller (EGMC) that is developed by the application of differential geometry theory. The model-based EGMC control system demands the knowledge of some physical state variables of the process and therefore, the development of a suitable algorithm to perform the state estimation has captured the attention. Here, we design a high gain observer so that it estimates a limited number of states which are solely required for the controller simulation. As a consequence, there exists a significant structural discrepancy. Despite this large mismatch, the state observer performs satisfactorily in converging the estimation error for the case of an example ethylene glycol reactive distillation system. With the same reduced-order predictor model, a comparison is also made between the high gain observer and the extended Kalman filter (EKF). Finally, the high gain observer based EGMC control structure shows promising performance in regulating the ethylene glycol column.     

Transformation of nonlinear discrete-time system into the extended observer form

The paper addresses the problem of transforming discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of both the extended coordinate and output transformations, solving the problem, are formulated in terms of differential one-forms, associated with the input–output equation, corresponding to the state equations. An algorithm for transformation of state equations into the extended observer form is proposed and illustrated by an example. Moreover, the considered approach is compared with the method of dynamic observer error linearisation, which likewise is intended to enlarge the class of systems transformable into an observer form.     

Extended state observer for uncertain lower triangular nonlinear systems subject to stochastic disturbance

The extended state observer (ESO) is the most important part of an emerging control technology known as active disturbance rejection control to this day, aiming at estimating “total disturbance” from observable measured output. In this paper, we construct a nonlinear ESO for a class of uncertain lower triangular nonlinear systems with stochastic disturbance and show its convergence, where the total disturbance includes internal uncertain nonlinear part and external stochastic disturbance. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.     

线性/非线性切换扩张状态观测器

为了解决非线性扩张状态观测器(NLESO)对大幅度扰动估计能力有限的问题,本文提出一种线性/非线性切换扩张状态观测器.首先分析了非线性扩张状态观测器对大幅度扰动估计能力有限的原因,然后提出在NLESO的非线性区间引入一段线性扩张状态观测器(LESO)弥补NLESO的缺陷,其次从理论上证明了提出的线性/非线性切换扩张状态观测器的收敛性.最后,通过数值仿真验证了提出的线性/非线性切换扩张状态观测器的可行性.     

Extended observer form: simple existence conditions

The paper focuses on the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of certain partial derivatives, related to the input–output equation, corresponding to the state equations. Moreover, a certain algorithm for transforming the state equations into the observer form is proposed.     

Extended state observer for uncertain lower triangular nonlinear systems

The extended state observer (ESO) is a key part of the active disturbance rejection control approach, a new control strategy in dealing with large uncertainty. In this paper, a nonlinear ESO is designed for a kind of lower triangular nonlinear systems with large uncertainty. The uncertainty may come from unmodeled system dynamics and external disturbance. We first investigate a nonlinear ESO with high constant gain and present a practical convergence. Two types of ESO are constructed with explicit error estimations. Secondly, a time varying gain ESO is proposed for reducing peaking value near the initial time caused by constant high gain approach. The numerical simulations are presented to show visually the peaking value reduction. The mechanism of peaking value reduction by time varying gain approach is analyzed.     

Fal函数的改进及3种新型非线性扩张状态观测器

首先分析现有fal函数和传统非线性扩张状态观测器中存在的6类问题,提出一种固定时间收敛的fal函数,其收敛速度快于传统fal函数,且收敛时间上界与初始误差无关;然后,分别设计基于新型fal函数的非线性扩张状态观测器、指数收敛非线性扩张状态观测器和全新结构的固定时间收敛非线性扩张状态观测器;最后,通过仿真验证这3种新型的非线性扩张状态观测器的估计总误差相对于传统非线性扩张状态观测器分别减少15.4%、16.9%和93.8%,对应的被控状态总误差分别减少13.6%、30.4%和78.3%.     

Extended state observer for MIMO nonlinear systems with stochastic uncertainties

ABSTRACT

In this paper, both linear extended state observer (ESO) and nonlinear ESO with homogeneous weighted functions are proposed for a class of multi-input multi-output (MIMO) nonlinear systems composed of coupled subsystems with large stochastic uncertainties. The stochastic uncertainties in each subsystem including internal coupled unmodelled dynamics and external stochastic disturbance without known statistical characteristics are lumped together as the stochastic total disturbance (extended state) of each subsystem. The linear ESO and nonlinear ESO are designed separately for real-time estimation of not only the unmeasured state but also the stochastic total disturbance of each subsystem. The practical mean square convergence of these two classes of ESOs are developed. Some numerical simulations are presented to demonstrate the effectiveness of the ESOs with the advantages of smaller peaking values and more accurate estimation by the nonlinear ESO.     

单输入/单输出非线性系统的状态观测器设计*

本文提出了一类非线性系统的状态观测器设计方法，针对单输入／单输出非线性系统，首先通过非线性状态变换，把原非线性系统变换成简单的中间非线性模型；然后对该简单的中间非线性模型设计稳定的常增益状态观测器，从而经状态逆变换得到原非一系统的非线性状态探测器，最后通过一倒立摆非线性系统的仿真研究说明了该方法的有效性和可行性。     

Eigenspectrum-based extended Luenberger observers for a class of distributed parameter systems

State estimation is an important problem in distributed parameter system especially with nonlinear dynamics in industrial process. An extended Luenberger observer based on the eigen-spectrum of the system operator is developed in this paper to handle this problem. The distributed parameter system is projected into a finite-dimensional subspace where a low-order ordinary differential equation describing the dominant dynamics of the system is derived. A Luenberger observer extended with a nonlinear part is developed based on that dominant dynamics. A sufficient condition is given in this paper for the convergence of the estimated error. Finally, by applying the developed design method to the temperature estimation of a catalytic rod, the achieved simulation results show the effectiveness of the proposed observer.     

线性/非线性切换扩张状态观测器的收敛性和稳定性分析

线性/非线性切换扩张状态观测器(linear/nonlinear switching extended state observer,SESO)是为追求更高的控制精度和抗扰能力,在进行状态估计时根据状态误差大小在线性扩张状态观测器(linear extended state observer,LESO)和非线性扩张状态观测器(nonlinear extended state observer,NLESO)之间进行切换的一种扩张状态观测器.但由于该算法存在非线性结构,目前还没有从频域角度对该方法进行收敛性和控制系统稳定性分析的相关研究.针对此问题,以二阶系统为研究对象,首先对SESO的结构及参数选择进行了介绍,然后,分别采用等效增益法和描述函数法,从频域角度对了其状态估计误差的收敛性和基于SESO的自抗扰控制系统的稳定性进行了研究.结果表明,三阶SESO具有良好的收敛性,能实现对系统状态和"总扰动"的误差估计,并且选择适当的SESO参数,可避免极限环的产生.     

Extended nonlinear observer canonical form depending on system output and auxiliary state

This paper deals with the problem of transforming a single output nonlinear system with an auxiliary dynamics into an extended nonlinear observer canonical form (ENOCF). The proposed ENOCF depends on system output and auxiliary state, and admits a kind of high-gain observer. We provide two necessary conditions and an equivalent condition for the existence of such a transformation, and then apply the results to the Rössler system as a case study.     

On the convergence of an extended state observer for nonlinear systems with uncertainty

The extended state observer first proposed by Jingqing Han in [J.Q. Han, A class of extended state observers for uncertain systems, Control Decis. 10 (1) (1995) 85-88 (in Chinese)] is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering. Unfortunately, there is no rigorous proof of convergence to date. In this paper, we attempt to tackle this long unsolved extraordinary problem. The main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance, for which the effect of total disturbance error is eliminated by the high-gain.     

A direct algebraic approach to observer design under switching measurement equations

Based on the algebraic transformation of a switched linear measurement equation into a nonlinear, yet deterministic, equation, an asymptotic state observer is constructed for discrete-time linear systems whose observations are generated according to randomly switching measurement modes. The observer, which combines the algebraic transformation with a Newton observer applied to the resulting nonlinear measurement equation, is shown to be locally exponentially convergent under arbitrary mode sequences.     

一类不确定对象的扩张状态观测器

利用观测器形式的跟踪一微分器,对形如ｘ＾（ｎ）＝ｆ（ｘ,ｘ,．．．Ｘ＾（ｎ－１）,ｔ）＋Ｗ（ｔ）的不确定系统给出了“扩张状态观测器”。只要适当选取观测器中的非线性函数和相应参数,它能很好地跟踪一批不确定对象的扩张状态。     

广义非线性扩张状态观测器设计及性能分析

针对时变外扰,提出广义非线性扩张状态观测器设计方法.在分析传统扩张状态观测器的设计策略的基础上,通过对总扰动进行重构、引入广义扩张状态,设计反映扰动中已知分量的广义扩张状态观测器(扩张r+1阶).理论分析了观测器的收敛性,并得出了观测误差上界与扩张阶数的定量关系式.通过仿真对广义扩张状态观测器抑制外界正弦扰动的有效性进行检验,数值模拟结果表明,本文设计的观测器能够有效利用扰动中已知分量的信息,降低系统的不确定性,提高观测精度.     

Transformation of Nonlinear MIMO Discrete‐Time Systems into the Extended Observer Form

The paper addresses the problem of transforming discrete‐time multi‐input multi‐output nonlinear state equations into the extended observer form, which, besides the inputs and outputs, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of the extended coordinate transformation are formulated in terms of differential one‐forms, associated with the input‐output equations, corresponding to the state equations. The difference between the single‐input single‐output and multi‐input multi‐output cases is described. The applicability of the conditions is illustrated by an example.     

新型有限时间收敛滑模扩张状态观测器研究

针对一类具有量测噪声的非线性不确定系统,设计了基于新型滑模扩张状态观测器的Terminal滑模控制方案．首先对系统进行两次状态扩张,然后设计一种新型滑模扩张状态观测器,通过采用特殊的滑模面保证观测误差在有限时间内收敛到零．在此基础上,设计Terminal滑模控制器,使系统状态也能在有限时间内收敛到零．严格的理论证明和仿真结果均证明了所设计新型滑模观测器及闭环控制方案的有效性和快速性．     

An observation algorithm for nonlinear systems with unknown inputs

This paper deals with the design of unknown input observers for nonlinear systems. An algorithm which determines if the state and the unknown input of the system can be recovered in finite time is introduced. This algorithm leads to the transformation of the system into an extended block triangular observable form suitable for the design of finite time observers. The proposed method is useful to relax some restrictive conditions of the existing nonlinear unknown input observer design procedures.     

三阶扩张状态观测器的优化参数配置方法

研究三阶非线性扩张状态观测器扰动观测性能的优化问题。首先,采用系数冻结法固定与观测器状态相关的非线性系数,利用线性系统的极点配置方法进行频带拓展;然后,分析配置后的极点随非线性系数的变化规律,在此基础上,提出一种保证扰动观测带宽受系统状态变化影响最小的参数配置方法;最后,通过算例和仿真对比表明了所提出参数配置方法的有效性和优越性。