Sliding-mode observers for nonlinear systems with unknown inputs and measurement noise

This paper deals with the design of observers for Lipschitz nonlinear systems with not only unknown inputs but also measurement noise when the observer matching condition is not satisfied. First, an augmented vector is introduced to construct an augmented system, and an auxiliary output vector is constructed such that the observer matching condition is satisfied and then a high-gain sliding mode observer is considered to get the exact estimates of both the auxiliary outputs and their derivatives in a finite time. Second, for nonlinear system with both unknown inputs and measurement noise, an adaptive robust sliding mode observer is developed to asymptotically estimate the system’s states, and then an unknown input and measurement noise reconstruction method is proposed. Finally, a numerical simulation example is given to illustrate the effectiveness of the proposed methods.     

Fault detection and isolation design for uncertain nonlinear systems based on full-order,reduced-order and high-order high-gain sliding-mode observers

This paper considers the observer-based fault detection and isolation design problems when the observer matching condition is not satisfied. Based on the relative degree concept, an auxiliary output vector that may satisfy the observer matching condition is constructed. Since the auxiliary output vector contains unknown information, we use a high-order high-gain sliding-mode observer to exactly estimate not only the auxiliary outputs, but also their derivatives in a finite time. Then, an adaptive robust full-order observer is developed to serve as an actuator fault detection observer. For the actuator fault reconstruction purpose, a reduced-order observer is proposed to estimate the system states even if there are some actuator faults and an actuator fault reconstruction method is provided to reach the fault isolation purpose. A numerical simulation example is used to illustrate the effectiveness of the proposed methods.     

Designing fault detection observers for linear systems with mismatched unknown inputs

Algebraic unknown input observers (UIOs) that have been previously reported in the literature can be constructed under the assumption that linear systems with unknown inputs satisfy the so-called observer matching condition. This condition restricts practical applications of UIOs for fault detection and isolation (FDI). We present an algebraic design for fault detection observers (FDOs) for the case in which the observer matching condition is not satisfied. To loosen the restriction imposed by the observer matching condition, the UIO design method combined with the unknown input modeling technique is proposed to design an FDO that decouples the effect of mismatched unknown inputs. To do this, first, unknown inputs that denote the faults of no interest and process disturbances are decomposed into algebraically rejectable unknown inputs and modeled unknown inputs such that the observer matching condition is satisfied. Under the assumption that mismatched unknown inputs are deterministic and can be expressed as the responses of fictitious autonomous dynamical systems, an augmented system is obtained by combining the original system model with the unknown input model. Finally, through the design technique of a UIO for the augmented system, a reduced-order FDO is constructed to estimate an augmented state vector that consists of both the original state variables and the augmentative state variables. The estimated state is then used to generate the residual, which should be designed to be insensitive to unknown inputs while being sensitive to the faults of interest. Two numerical examples are provided to show the usefulness and the feasibility of the presented approach.     

基于辅助输出的线性系统状态和未知输入同时估计方法

在未知输入观测器匹配条件不满足的情况下, 针对一类线性时不变系统, 研究了同时估计系统 状态和未知输入的问题. 首先, 基于可测输出对未知输入的相关度的概念, 给出了辅助输出 的构造方法, 使得匹配条件得以满足. 为了处理辅助输出中的未知信息, 提出了一种高增益 观测器设计方法, 它不仅能估计辅助输出, 而且还能估计辅助输出的导数. 然后, 基于辅助 输出的估计值, 提出了一种降维观测器设计方法, 可以在不受未知输入影响的情况下估计系统 的状态; 接下来, 基于状态和辅助输出及其导数的估计值, 给出了未知输入估计. 最后, 对一个五 阶系统进行了数字仿真, 仿真结果表明所提出的方法是有效的.     

Adaptive fuzzy controller for non-affine systems with zero dynamics

In this article, the direct adaptive fuzzy control problem is investigated for a class of general non-linear systems with zero dynamics. The direct adaptive fuzzy controller is developed based on a unified observer which is used to estimate the time derivatives of the output. The corrective term of the proposed observer involves a well-defined design function which is shown to be satisfied by the commonly used high-gain-based observers, namely for the usual high-gain observers and the sliding-mode observers together with their implementable versions. By using a general error function, and without resorting to the famous strictly positive real condition or the filtering of the observation error, a general proportional–integral (PI) law for updating the fuzzy parameters is proposed. Ultimately boundedness of the error signals is shown through Lyapunov's direct method. Theoretical results are illustrated through two simulation examples.     

同时含有未知输入和测量干扰系统全维和降维观测器设计

针对同时含有未知输入和测量干扰的不确定系统研究了全维和降维观测器设计问题. 首先, 利用待定系数法给出了全维观测器的结构和存在条件. 该条件完全由原系统的系统矩阵给出, 易于检验. 对于降维观测器, 为了消除测量干扰的影响, 提出了一种新的测量输出构造方法, 使得新构造的测量输出不再包含干扰信号. 此外, 证明了全维和降维观测器存在条件的内在统一性, 即全维观测器所需要满足的观测器匹配条件和强可检测条件在研究降维观测器所要讨论的新的系统中都可以得到保持. 因而, 在全维观测器存在条件下, 也可以设计一个相应的降维观测器. 最后, 给出了一个数值例子验证所提方法的有效性.     

基于自适应滑模观测器的不匹配非线性系统执行器故障重构

针对一类含有未知干扰的不匹配非线性Lipschitz系统,提出了基于自适应滑模观测器的执行器故障重构方法.首先引入辅助输出矩阵,使得辅助输出系统的观测器匹配条件得以满足,同时设计了高增益观测器实现对未知辅助输出的精确估计;然后针对辅助输出系统建立故障重构滑模观测器,设计了自适应律在线修正滑模控制器增益,考虑故障上界未知的前提下,提出了观测器状态估计误差稳定的存在定理,运用Schur补引理将观测器反馈增益矩阵设计方法转化为求解线性矩阵不等式约束优化问题,同时引入线性变换矩阵,在故障上界未知的前提下设计了滑模控制增益,使得输出估计误差收敛稳定,确保了滑模运动在有限时间内发生,在此基础上利用等效控制输出误差注入原理实现了执行器故障重构;最后通过仿真算例验证了本文方法的有效性.     

Disturbance Decoupled Observers for Systems With Unknown Inputs

This note deals with the design of reduced-order disturbance decoupled scalar functional observers for linear systems with unknown inputs. Based on a parametric approach, existence conditions are derived and a design procedure for finding reduced-order scalar functional observers is given. The derived existence conditions are relaxed and the procedure can find first-order disturbance decoupled scalar functional observers for some cases where the number of unknown inputs is more than the number of outputs. Also, the observer matching condition, which is the necessary requirement for the design of state observers for linear systems with unknown inputs, is not required. Numerical examples are given to illustrate the attractiveness of the proposed design method.      

Adaptive fuzzy decentralized control for nonlinear large-scale systems based on high-gain observer

An adaptive fuzzy decentralized backstepping output-feedback control approach is proposed for a class of nonlinear large-scale systems with completely unknown functions,the interconnections mismatched in control inputs,and without the measurements of the states.Fuzzy logic systems are employed to approximate the unknown nonlinear functions,and an adaptive high-gain observer is developed to estimate the unmeasured states.Using the designed high-gain observer,and combining the fuzzy adaptive control theory with backstepping approach,an adaptive fuzzy decentralized backstepping output-feedback control scheme is developed.It is proved that the proposed control approach can guarantee that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded(SUUB),and that the observer errors and the tracking errors converge to a small neighborhood of the origin by appropriate choice of the design parameters.Finally,a simulation example is provided to show the eectiveness of the proposed approach.     

Nonlinear sliding mode high-gain observers for fault estimation

A robust high gain observer for state and unknown inputs/faults estimations for a special class of nonlinear systems is developed in this article. Ensuring the observability of the faults/unknown inputs with respect to the outputs, the faults can be estimated from the sliding surface. Under a Lipschitz condition for the nonlinear part, the high gain observers are designed under some regularity assumptions. In the sliding mode, the convergence of the estimation error dynamics is proven similar to the analysis of high-gain observers.     

Multi-modelling as new estimation schema for high-gain observers

The presented paper proposes a novel method of observer design. A new two-layer observer structure is introduced. The first layer consists of multiple high-gain observers. The latter is built to connect the first layer observers into single one. As the new contribution, the new mapping is defined between an unknown state and measurable outputs allowing to explore new estimation schema. Hence, the proposed method enhances the estimation process for linear and nonlinear systems. Furthermore, it is shown that the introduced observation scheme improves the transients. Illustrative examples are calculated to show the properties of the new observation method.     

On observer design for nonlinear systems

The main weakness of all control methodologies is the dependency of feedbacks to full state measurements. In practical situations, measuring the states of a given system may fail because sometimes the measurements are impossible and sometimes, possible, but too expensive. Observer design for highly nonlinear dynamics is an important issue, particularly when the locally observable dynamics are not linearly observable. In such circumstances the ability to reduce the system to observable or observer form is key to observer design. This paper provides two observers for nonlinear systems given in Brunovski form. The first observer is a high-gain observer with a classical output injection form, while the second is a high-gain observer with a q-integral path. Finally, the discrete-time implementation of the high-gain observer is discussed in linear matrix inequality framework. A motivating example is shown to highlight the efficacy of the developed observers.     

Fuzzy adaptive high-gain-based observer backstepping control for SISO nonlinear systems

In this paper, a new fuzzy adaptive control approach is developed for a class of SISO strict-feedback nonlinear systems, in which the nonlinear functions are unknown and the states are not available for feedback. By fuzzy logic systems to approximate the unknown nonlinear functions, a fuzzy adaptive high-gain observer is designed to estimate the unmeasured states. Under the framework of the backstepping design, fuzzy adaptive output feedback control is constructed recursively. It is shown that the proposed fuzzy adaptive control approach guarantees the semi-global boundedness property for all the signals of the resulting closed-loop system. Simulation results are included to illustrate the effectiveness of the proposed techniques.     

Input output linearization approach to state observer design fornonlinear system

Presents a state observer for a class of nonlinear systems based on the input output linearization. While the previous result presented state observers for nonlinear systems of full relative degree, we proposed a procedure fur the design of nonlinear state observers which do not require the hypothesis of full relative degree. Assuming that there exists a global state observer for internal dynamics and that some functions are globally Lipschitz, we can design a globally convergent state observer. It is also shown that if the zero dynamics are locally exponentially stable, then there exists a local state observer. An example is given to illustrate the proposed design of nonlinear state observers     

Robust fault detection in linear systems based on full-order state observers

A parametric approach to robust fault detection in linear systems with unknown disturbances is presented. The residual is generated using full-order state observers （FSO）. Based on an analytical solution to a type of Sylvester matrix equations, the parameterization of the observer gain matrix is given. In terms of the design degrees of freedom provided by the parametric observer design and a group of introduced parameter vectors, a sufficient and necessary condition for fullorder state observer design with disturbance decoupling is then established. By properly constraining the design parameters according to this proposed condition, the effect of the disturbance on the residual signal is also decoupled, and a simple algorithm is developed. The presented approach offers all the degrees of design freedom. Finally, a numerical example illustrates the effect of the proposed approach.     

基于有限时间观测器的离散系统混沌同步

本文基于驱动–响应模型针对一类离散时间混沌系统提出了一种基于有限时间观测器的同步方法. 首先, 将混沌系统写成具有未知输入的线性系统形式. 随后, 给出了观测器匹配条件和强可观条件. 在观测器匹配条件的 假设下, 通过适当的状态变换, 给出了具有降维形式的有限时间观测器设计框架使得该观测器不再受到未知输入的 影响. 然后, 证明了强可观条件结合观测器匹配条件可以保证一个有限时间观测器的存在, 该观测器可以使得响应 系统达到对驱动系统的精确同步, 且达到同步所需要的时间可以任意设定, 不受观测器系统矩阵极点配置和初值条 件的影响. 最后, 给出了两个混沌系统的例子验证了所提方法的有效性.     

Robust fault detection in linear systems based on full-order state observers

A parametric approach to robust fault detection in linear systems with unknown disturbances is presented. The residual is generated using full-order state observers (FSO). Based on an analytical solution to a type of Sylvester matrix equations, the parameterization of the observer gain matrix is given. In terms of the design degrees of freedom provided by the parametric observer design and a group of introduced parameter vectors, a sufficient and necessary condition for fullorder state observer design with disturbance decoupling is then established. By properly constraining the design parameters according to this proposed condition, the effect of the disturbance on the residual signal is also decoupled, and a simple algorithm is developed. The presented approach offers all the degrees of design freedom. Finally, a numerical example illustrates the effect of the proposed approach.     

Robust structure and motion recovery for monocular vision systems with noisy measurements

This study proposes a novel complete-order nonlinear structure and motion observer for monocular vision systems subjected to significant measurement noise. In contrast with previous studies that assume noise-free measurements, and require prior knowledge of either the relative motion of the camera or scene geometry, the proposed scheme assumes a single component of linear velocity as known. Under a persistency of excitation condition, the observer then relies on filtered estimates of optical flow to yield exponentially convergent estimates of the unknown motion parameters and feature depth that converge to a uniform, ultimate bound in the presence of measurement noise. The unknown linear and angular velocities are assumed to be generated using an imperfectly known model that incorporates a bounded uncertainty, and optical flow estimation is accomplished using a robust differentiator that is based on the sliding-mode technique. Numerical results are used to validate and demonstrate superior observer performance compared to an alternative leading design in the presence of model uncertainty and measurement noise.     

Multiple sliding mode observers and unknown input estimations for Lipschitz nonlinear systems

Multiple sliding mode observers for state and unknown input estimations of a class of MIMO nonlinear systems are systematically developed in this paper. A new nonlinear transformation is formulated to divide the original system into two interconnected subsystems. The unknown inputs are assumed to be bounded and not necessarily Lipschitz, and do not require any matching condition. Under structural assumptions for the unknown input distribution matrix, the sliding mode terms of the nonlinear observer are designed to track their respective unknown inputs. Also, the unknown inputs can be reconstructed from the multiple sliding mode structurally. The conditions for asymptotic stability of estimation error dynamics are derived. Finally, simulation results are given to demonstrate the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.