Observer Design For Nonlinear Systems Research Articles

New results and examples on observer designof nonlinear systems around equilibria

In this paper, we present some new results and examples on nonlinear observer designfor continuous-time and discrete-time nonlinear systems defined in the neighborhood of equilibria. We also show that detectability is a necessary condition for the existence of nonlinear observers for nonlinear systems around equilibria. Using this necessary condition, we establish that, for the classical bifurcation case when the state equilibrium point does not change with the bifurcation parameter, and when the plant output function is purely a function of the state vector, there does not exist even a local asymptotic observer for the plant. Finally, we illustrate our results with classical examples from bifurcation theory.

Observer design for nonlinear system

A design method is presented that extends the least mean squared (LMS) algorithm of time–varying parameters which incorporate an internal model. The aim is to track the varying parameters of linear regression models in situations where regressors have not only slowly changing parameters but also rapidly time–varying ones. This algorithm is given from the standpoint that a change in a varying parameter can be approximated as a polynomial function of time with a finite degree. Under this assumption, the proposed algorithm includes an internal model of the polynomial function in a conventional LMS algorithm so as to compensate the influence of the varying parameters. The design method is based on a loop transformation for the dynamics and small–gain theorem. Furthermore, a sufficient condition for assuring stability of the proposed algorithm is given by the output strictly passive and small–gain theorem. Compared with conventional LMS adaptive algorithms, the tracking performance of varying parameters improves in numerical simulations.

Adaptive observer design for nonlinear systems using generalized nonlinear observer canonical form

In this paper, we present an adaptive observer for nonlinear systems that include unknown constant parameters and are not necessarily observable. Sufficient conditions are given for a nonlinear system to be transformed by state-space change of coordinates into an adaptive observer canonical form. Once a nonlinear system is transformed into the proposed adaptive observer canonical form, an adaptive observer can be designed under the assumption that a certain system is strictly positive real. An illustrative example is included to show the effectiveness of the proposed method.

Erratum Observer design for nonlinear systems using linear approximations

The publisher regrets that in the above article published in IMA Journal of Mathematical Control and Information, Volume 20 Number 3, September 2003, pp. 359–370, Fig. 2 was printed three times and Fig. 3 and Fig. 4 were omitted. Figures 3 and 4 are now reproduced correctly, on the following pages.

Observer design for nonlinear systems using linear approximations

There exist several approaches to the design of observers for nonlinear systems, including the separation of the nonlinear system into a linear part and a nonlinear perturbation of the system with a bounded condition [2], the use of Lie derivatives and the inversion of the Jacobian of a coordinate transformation to obtain the gain of the nonlinear observer [10] and the use of a Lyapunov equation to design the observer for a nonlinear system represented in a special canonical form [8]. The design of observers for linear systems is better understood (see [11], [14]) since in nonlinear theory there is a necessity to use more complex mathematics. Therefore, there exist interest in the development of simpler and general methods to solve the problem of nonlinear state reconstruction. This paper deals with the design of observers for nonlinear systems by using a recent technique in which the nonlinear dynamical system is represented as the limit of a sequence of linear time-varying approximations that converge to the solution of the nonlinear system under a local Lipschitz condition.

A relation between the output regulation and the observer design for nonlinear systems

Output regulation and observer design are two important problems for nonlinear systems, and there is a vast literature addressing each problem separately in the control literature. Isidori and Byrnes [1] have solved the output regulation problem for nonlinear systems with a Poisson stable exosystem, and Sundarapandian [2] has solved the exponential observer design problem for Lyapunov stable nonlinear systems. In this paper, we demonstrate that for a special class of Lyapunov stable nonlinear systems, namely neutrally stable systems, the exponential observer design problem can be solved by converting it into an output regulation problem and then solving the new problem using the output regulation techniques of Isidori and Byrnes [1]. Finally, we present the corresponding results for the discrete-time case.

Exponential observer design for nonlinear systems with real parametric uncertainty

In this paper, we present a characterization and construction procedure of local exponential observers for nonlinear systems with real parametric uncertainty under some stability assumptions. We also show that for the classical case when the state equilibrium does not change with the disturbance, and when the plant output is purely a function of the state, there is no local asymptotic observer for the plant. Next, we show that in sharp contrast to this case, for the general case of problems where we allow the state equilibrium to change with the disturbance, there typically exist local exponential observers even when the plant output is purely a function of the state. Finally, we generalize our results for nonlinear systems with real parametric uncertainty to derive corresponding observer design results for nonlinear systems with exogenous disturbance.

Global observer design for nonlinear systems

This paper is a geometric study of the global observer design for nonlinear systems. Using the theory of foliations, we derive necessary and sufficient conditions for global exponential observers for nonlinear systems under some assumptions. Our proof for these necessary and sufficient conditions for global exponential observers if via defining two equivalence relations known as horizontal and vertical equivalence relations, and constructing two foliations known as horizontal and vertical foliations from these equivalence relations. Finally, as a corollary of our global theorem, we derive necessary and sufficient conditions for local exponential observers of critically Lyapunov stable nonlinear systems.

Local observer design for nonlinear systems

This paper is a geometric study of the local observer design for nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable nonlinear systems. As an application of our local observer design, we consider a class of nonlinear systems with an input generator ( exosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs.

Observer design for non-linear systems that are not uniformly observable

The observer design problem is studied for a class of single-output non-linear systems that are not necessarily uniformly observable. In the first part of this paper, the non-linear observer canonical form is generalized to the partial non-linear observer canonical form which is quite similar to the decomposition of a linear system into its observable and unobservable states. A necessary and sufficient condition is given for a non-linear system to be transformable into this canonical form by the change of coordinates. Based on this canonical form, a sufficient condition is given for the existence of observers. In the second part, a high gain observer for a class of non-linear systems that are not uniformly observable will be developed in order to relax the system structure of the partial non-linear observer canonical form. To this end, a new canonical form is proposed that generalizes the canonical form of uniform observability and a sufficient condition is provided based on the proposed canonical form.

FURTHER DEVELOPMENTS ON ADAPTIVE OBSERVERS FOR NONLINEAR SYSTEMS WITH APPLICATION IN FAULT DETECTION

From previous studies of (Besançon, 1999; Besançon, 2000) on adaptive observer design for nonlinear systems, and the contribution of (Zhang, 2001) on exponential adaptive observers for linear time-varying systems, the purpose here is to propose some new observer design for a class of nonlinear systems. This observer can in turn be made adaptive, and in particular can help in detecting changes in some parameters.

Decomposition State Observer Design for Nonlinear Systems

This paper is introduced the decomposition principle to state observer design for nonlinear systems. The block-observed form of nonlinear systems is offered. On the basis of this form state observer synthesis is partitioned on sequentially separately solved elementary subproblems of smaller dimension. The hierarchy of control actions of state observer in the class of systems with high gains and discontinuous controls (the systems with separated motions on temps) is obtained. The outcomes on quantitative assessment from below of state observers gains with preservation of synthesis decomposition are resulted. It allows to decide an estimation problem of state vector with given accuracy by finite gains.

Input output linearization approach to state observer design for nonlinear 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.

Sliding Controller-Sliding Observer Design for Non-linear Systems

We study the combination of sliding observers and sliding controllers for affine non-linear systems. First, for systems in controllable canonical form we derive conditions for convergence of the observer and the combined sliding controller-sliding observer scheme. The main result is that under certain conditions uniform ultimately boundedness of the tracking error can be achieved. The analysis is extended to affine feedback linearisable systems. Affine systems with a relative degree lower than the system order are also studied, showing the necessity of some ‘matching’ conditions on the uncertainties to allow a theoretically perfect performance of the sliding observer. If those restrictive matching conditions are not imposed, the observer-controller scheme is still adequate if some bounds on the uncertainties are fulfilled, provided the sliding variable depends on the ‘certain’ part of the state. The application to robot manipulators is developed in some detail. The performance of the observer-controller scheme is tested by simulations on a two degree of freedom rigid robot and a one-link flexible robot, showing the effects of finite switching frequency and measurelnent noise.

Observer design for nonlinear systems with discrete-time measurements

This paper focuses on the development of asymptotic observers for nonlinear discrete-time systems. It is argued that instead of trying to imitate the linear observer theory, the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations. In particular, it is shown that the discrete Newton method, properly interpreted, yields an asymptotic observer for a large class of discrete-time systems, while the continuous Newton method may be employed to obtain a global observer. Furthermore, it is analyzed how the use of Broyden's method in the observer structure affects the observer's performance and its computational complexity. An example illustrates some aspects of the proposed methods; moreover, it serves to show that these methods apply equally well to discrete-time systems and to continuous-time systems with sampled outputs. >

Riccati不等式アプローチによる非線形システムのオブザーバ設計法

Riccati不等式アプローチによる非線形システムのオブザーバ設計法

Asymptotic observers for non-linear systems†

Abstract Observer design for non-linear systems is discussed. Some recent approaches are based on state and output change of coordinates to transform a non-linear system into a particular observer form, from which an asymptotic observer can be designed ensuring the asymptotic stability of the error dynamics in the new coordinates. In this paper, the stability properties of the error dynamics are studied in the original coordinates. With some examples, it is shown how the asymptotic stability in the new coordinates does not imply, in general, the asymptotic stability in the original ones. Some general results are stated and proved to guarantee the asymptotic stability of the error dynamics in the original coordinates.

Observer design for nonlinear systems

This paper deals with the observer design problem of a wide class of nonlinear systems subjected to bounded nonlinearities. A sufficient Liapunovlike condition is provided and the proposed dynamic observer is a direct extension of the one in linear case.

Robust observer design for a fluid pipeline

Observer design for non-linear systems is difficult work, especially for systems of high order. In this paper, we first present a state-space representation of a fluid pipeline. Then we turn the non-linear observer design task into optimization problems in order to bridge the gap between non-linear observer theory and its applications. In applying this method to the discrete high-order model of a pipeline, analytic solutions can easily be obtained. A structural condition for observers to be robust to pipeline friction and diameter variations is also proved through steady-state analysis. Simulation studies and experiments on a water pipeline show that: (a) the observers designed converge rapidly and reliably; (b) the computational expenditure is very small in comparison with the Kalman filter; and (c) friction and diameter variations may have little influence on the estimation accuracy.

Comments on ‘Comparative study of non-linear state-observation techniques’

Recent modifications, developed for the canonical form observer design of non-linear systems, lead to a modified result for the comparative study of Walcott el al. (1987).