On exponential observers for nonlinear systems

The aim of the paper is to identify the class of nonlinear systems that have exponential observers - a concept introduced by previous authors. It is shown that a necessary condition for the existence of an exponential observer for a nonlinear system is that the corresponding linearized system is detectable, and for local exponential stabilization problems, the condition is also sufficient.This paper gives also a theorem on the separation property for the exponential design problem, and it enables us to tell exactly to what extent the classical local linearization approach is applicable.     

Interval estimation for uncertain systems with time-varying delays

The estimation problem for uncertain time-delay systems is addressed. A design method of reduced-order interval observers is proposed. The observer estimates the set of admissible values (the interval) for the state at each instant of time. The cases of known fixed delays and uncertain time-varying delays are analysed. The proposed approach can be applied to linear delay systems and nonlinear time-delay systems in the output canonical form. It involves the properties of quasi-monotone/Metzler/cooperative systems. In this framework, it is shown that if under a suitable coordinate transformation the delay-free subsystem is cooperative, then the delayed estimation error dynamics inherits this property. The conditions to find the observer gains are formulated in the form of LMI. The framework efficiency is demonstrated on examples of nonlinear systems.     

An algorithm for the structural analysis of state space: synthesis of nonlinear observers

The problem addressed is the linearization of multi‐input multi‐output (MIMO) nonlinear systems by a generalized state coordinates transformation and generalized input–output injection, in order to design an observer. This observer will have linear error dynamics. The goal is to bring together two observers design approaches: a structural one and a numerical one. Necessary and sufficient conditions for the existence of a linearizing generalized state transformation are obtained by an algebraic way and without computing the input–output differential equations. The main result tests integrability conditions of differential one‐forms derived from the state space representation and is applicable to a large subclass of nonlinear systems. Copyright © 2001 John Wiley & Sons, Ltd.     

线性描述系统有限时间函数观测器设计及存在性讨论

本文针对线性描述系统研究了其有限时间函数观测器的设计方法和存在条件.首先,通过构造两个结构上完全相同的渐近收敛观测器得到了一个新的有限时间函数观测器,它可以在任意给定的时间内达到对目标函数的精确估计,且估计精度和收敛时间不依赖于原系统的初始条件.随后,对该函数观测器的存在条件进行了详细讨论,给出了函数观测器存在的充分必...     

Observers design for linear time-delay systems

In this contribution we propose a simple and useful approach to design observers for discrete-time systems with delays in the state and output variables. The main feature is that the necessary and sufficient conditions for the existence of such observer are derived. The stability analysis is performed by the Lyapunov approach, where the obtained conditions are expressed in terms of a modified Riccati equation. Numerical examples are provided to show efficiency of the proposed observer.     

Adaptive observers for single output nonlinear systems

An adaptive version of the nonlinear observer obtained by A.J. Krener et al. (1983) is presented. This version involves the cancellation of nonlinear terms by output injection. As an intermediate step, necessary and sufficient conditions are given for transforming a nonlinear system by state-space change of coordinates into the special adaptive observer form that was used by Y. Bastin et al. (1988) to design adaptive observers     

Active state estimation of nonlinear systems

In this paper, state observers for control systems with nonlinear outputs are studied. For such systems, the observability does not only depend on the initial conditions, but also on the exciting control used. Thus, for such systems, design of active control is an integral part of the design for state observers. Here some sufficient conditions are given for the convergence of an observer. It is also discussed, via a camera example, how to actively excite a system in order to improve the observability.     

Observer design for continuous-time dynamical systems

We review the main design techniques of state observer design for continuous-time dynamical systems, namely algorithms which reconstruct online the full information of a dynamical process on the basis of partially measured data. Starting from necessary conditions for the existence of such asymptotic observers, we classify the available methods depending on the detectability/observability assumptions they require. We show how each class of observer relies on transforming the system dynamics in a particular normal form which allows the design of an observer, and how each observability condition guarantees the invertibility of its associated transformation and the convergence of the observer. Finally, some implementation aspects and open problems are briefly discussed.     

Observers and output feedback stabilising controllers for nonlinear strict-feedback systems with sampled observation

The design of observers and output feedback stabilising controllers for continuous-time strict-feedback systems with sampled observation is considered. First two types of observers are designed. One is a discrete-time semiglobal and practical reduced-order observer for the exact model and the other is a continuous-time semiglobal and practical full-order observer for continuous-time strict feedback systems with sampled observation. Then by combining the designed continuous-time observers and continuous-time state feedback laws that are continuous, zero at the origin, and uniformly globally asymptotically stabilise continuous-time systems, output feedback semiglobally practically uniformly asymptotically stabilising controllers are constructed. Numerical examples are given to illustrate the proposed design of observers and output feedback controllers.     

Peaking free output‐feedback exact tracking of uncertain nonlinear systems via dwell‐time and norm observers

The control of uncertain nonlinear systems by high‐gain observer based output feedback is addressed. Two tracking sliding mode controllers are designed for a broad class of uncertain nonlinear systems with arbitrary relative degree and unmatched polynomial nonlinearities in the unmeasured states. The proposed strategies are based either on dwell‐time for control activation or on simple norm state observers to remove the peaking phenomenon related with high‐gain observers, depending on the nonlinearity growth conditions. In contrast with previous works, exact tracking is also achieved by means of a switching strategy based on locally exact differentiators. Global or semi‐global stability is proved by using Lyapunov theory and on small‐gain analysis. Simulations show that the proposed methodologies provide better and uniform transient behavior, larger regions of attraction, performance recovery with significantly smaller observer gains and good robustness properties with respect to exogenous disturbances and measurement noise. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability of a class of discrete-time nonlinear recursive observers

This work provides a framework for nominal and robust stability analysis for a class of discrete-time nonlinear recursive observers (DNRO). Given that the system has linear output mapping, local observability and Jacobian matrices satisfying certain conditions, the nominal and robust stability of the DNRO is defined by the property of estimation error dynamics and is analyzed using Lyapunov theory. Moreover, a simultaneous state and parameter estimation scheme is shown to be Input-to-State Stable (ISS), and adaptively reduce plant-model mismatch on-line. Three design strategies of the DNRO that satisfy the stability results are given as examples, including the widely used extended Kalman filter, extended Luenberger observer, and the fixed gain observer.     

A New Algorithm to Design Minimal Multi‐Functional Observers for Linear Systems

Designing minimum possible order (minimal) observers for multi‐input multi‐output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi‐functional observers for linear time invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings.     

Nonlinear observer design via passivation of error dynamics

We present a new design scheme of nonlinear state observers (global, full order, asymptotic observers) through passivation of the error dynamics. In order to consider passivity of the error dynamics for the observer problem, we place a conceptual input and output on the generalized error dynamics which also includes the plant, and the strictness of passivity is extended with respect to a set in which the estimation error becomes zero. Then, output feedback passivation for the error dynamics will lead to the construction of a state observer. It is also shown that a nonlinear observer is generally vulnerable to measurement disturbance, in the sense that even an arbitrarily small measurement disturbance can lead to a blowup of the error state. However, due to the passivity of the error dynamics, the proposed nonlinear injection gain can be easily modified for the observer to be robust to measurement disturbances.     

Robust fault detection in linear systems based on PI observers

A parametric approach for robust fault detection in linear systems with unknown disturbances is presented. The residual is generated using full-order proportional-integral (PI) observers. The approach is based on a result for PI observer design recently proposed. In terms of the design degrees of freedom provided by the parametric PI observer design and a group of introduced parameter vectors, a sufficient and necessary condition for PI observer design with disturbance decoupling is established. By properly constraining the design parameters according to this proposed condition, the effect of the disturbance to the residual signal is decoupled, and a simple algorithm is presented. The presented approach offers all the degrees of design freedom. A numerical example illustrates the effect of the proposed approach.     

Nonlinear Luenberger‐like observers for nonlinear MIMO systems

This paper deals with the design of observers for a class of continuous time nonlinear multi‐input multi‐output systems with nonlinear outputs. Geometric tools are used to transform the original system into an appropriate observer canonical form. Furthermore, a pole placement technique is used to obtain a desired transient response of resulting error dynamics. The observer design is presented for two cases. In both cases, it is shown that the observer gain can be obtained from the solution of a Riccati equation. An illustrative example of state estimation in induction motors is presented to explain the proposed observer design. The performance of the method is also verified by numerical simulations. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Semi‐global finite‐time observers for multi‐output nonlinear systems

Finite‐time stability is investigated for nonlinear systems, which satisfy uniqueness of solution. First, a new sufficient condition for local finite‐time stability is presented. Next, by using the high‐gain observers and carefully selecting the homogeneity powers and weights, the problem of semi‐global and finite‐time stable observers is studied for multi‐output nonlinear systems with uniform observability and a triangular structure. Then, a design procedure is worked out for such observers. Finally, two numerical examples further verify the validity of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Separation results for the stabilization of nonlinear systems using different high-gain observer designs

High-gain observers have been used in the design of output feedback controllers due to their ability to robustly estimate the unmeasured states while asymptotically attenuating disturbances. The available techniques for the design of high-gain observers can be classified into three groups: pole-placement algorithms, Riccati equation-based algorithms, and Lyapunov equation-based algorithms. In [1], we presented separation results for globally bounded stabilizing state feedback controllers when the high-gain observer is designed using pole-placement so as to create a closed-loop system with two-time-scale structure. In this paper, we show that the separation results of [1] hold for the other observer designs.     

Sampled-data chain-observer design for a class of delayed nonlinear systems

The problem of observer design is addressed for a class of triangular nonlinear systems with not-necessarily small delay and sampled output measurements. One more difficulty is that the system state matrix is dependent on the un-delayed output signal which is not accessible to measurement, making existing observers inapplicable. A new chain observer, composed of m elementary observers in series, is designed to compensate for output sampling and arbitrary large delays. The larger the time-delay the larger the number m. Each elementary observer includes an output predictor that is conceived to compensate for the effects of output sampling and a fractional delay. The predictors are defined by first-order ordinary differential equations (ODEs) much simpler than those of existing predictors which involve both output and state predictors. Using a small gain type analysis, sufficient conditions for the observer to be exponentially convergent are established in terms of the minimal number m of elementary observers and the maximum sampling interval.     

Sliding mode observers: a survey

Sliding mode observers have unique properties, in that the ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set of state estimates that are precisely commensurate with the actual output of the plant. It is also the case that analysis of the average value of the applied observer injection signal, the so-called equivalent injection signal, contains useful information about the mismatch between the model used to define the observer and the actual plant. These unique properties, coupled with the fact that the discontinuous injection signals which were perceived as problematic for many control applications have no disadvantages for software-based observer frameworks, have generated a ground swell of interest in sliding mode observer methods in recent years. This article presents an overview of both linear and non-linear sliding mode observer paradigms. The use of the equivalent injection signal in problems relating to fault detection and condition monitoring is demonstrated. A number of application specific results are also described. The literature in the area is presented and qualified in the context of continuing developments in the broad areas of the theory and application of sliding mode 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.