Time scaling for observer design with linearizable error dynamics

In this paper, we consider the problem of observer design for dynamical systems with scalar output by linearization of the error dynamics via coordinate change, output injection, and time scaling. We present necessary and sufficient conditions which guarantee the existence of a coordinate change and output-dependent time scaling, such that in the new coordinates and with respect to the new time the system has linear error dynamics.     

Nonlinear observers with approximately linear error dynamics: themultivariable case

Exact error linearization uses nonlinear input-output injection to design observers with linear error dynamics in certain coordinates. This approach can only be applied nongenerically. We propose an observer for a wider class of multivariable systems which uniformly minimizes the nonlinear part of the system that cannot be canceled by nonlinear input-output injection. Our approach is numerical, constructive, and provides locally exponentially stable error dynamics. An example compares our design with a high-gain method     

Nonlinear discrete-time observer design with linearizable error dynamics

A necessary and sufficient condition for the existence of a discrete-time nonlinear observer with linearizable error dynamics is provided. The result can be applied to any real analytic nonlinear system whose linear part is observable. The necessary and sufficient condition is the solvability of a nonlinear functional equation. Furthermore, the well-known Siegel's theorem on the linearizability of a mapping is naturally reproduced in a corollary. The proposed observer design method is constructive and can be applied approximately to any sufficiently smooth, linearly observable system yielding a local observer with approximately linear error dynamics.     

An observer design for linear systems with unknown inputs

A method is presented for designing an observer for the state of a linear time-invariant system with unknown inputs. The structure algorithm developed by Silverman is applied to obtain the observer which estimates the maximum estimable subspace of the state. It is shown that the observer equation can be derived from the maximum uncontrollable subspace of the original system with the aid of a left inverse for a transposed linear system. This leads us to the necessary and sufficient condition for the existence of a state observer. An application to the insensitivity observer synthesis is also included.     

An observer design for linear time-delay systems

An observer design is proposed for linear systems with time delays. The key to the design is to find a generalized coordinate change such that, in the new coordinates, all the time-delay terms are injected by the system's output. The existence of such a coordinate change is guaranteed by a rank condition on the observability matrix. The novelty of the proposed design is clearly reflected in the multiple-output case, where a dimensional expansion in the coordinate change could become necessary, and hence is allowed     

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.     

Design of nonlinear observers with approximately linear error dynamics using multivariable Legendre polynomials

This paper presents a numerical approach to the design of nonlinear observers by approximate error linearization. By using a Galerkin approach on the basis of multivariable Legendre polynomials an approximate solution to the singular PDE of the observer design technique proposed by Kazantzis and Krener (see (Syst. Control Lett. 1998; 34 :241–247; SIAM J. Control Optim. 2002; 41 :932–953)) is determined. It is shown that the L₂‐norm of the remaining nonlinearity in the resulting error dynamics can be made small on a specified multivariable interval in the state space. Furthermore, a linear matrix equation is derived for determining the corresponding change of co‐ordinates and output injection such that the proposed design procedure can easily be implemented in a numerical software package. A simple example demonstrates the properties of the new numerical observer design. Copyright © 2006 John Wiley & Sons, Ltd.     

A direct method for the construction of nonlinear discrete-time observer with linearizable error dynamics

We provide a direct method for the design of nonlinear discrete-time observers by a construction of a change of variables. An explicit expression of the change of variables is given. Some simulations for chaotic systems, such as Lozi system and He/spl acute/non system, are provided to illustrate the proposed method.     

Nonlinear observer design by dynamic observer error linearization

In this note, we address the problem of transforming a nonlinear system into nonlinear observer canonical form in the extended state-space with the aid of dynamic system extension and introduction of virtual outputs. As an intermediate step for the general problem, we consider a restricted structure of dynamic extension, which is obtained, roughly speaking, by adding the chains of integrators to the outputs of original system. We propose sufficient conditions which can be verified using the system dynamics expressed in their original coordinates. An illustrative example is included that demonstrates the advantage of the proposed method over the conventional method.     

Full-order observer design for linear systems with unknown inputs

In this article, a full-order observer without unknown inputs reconstruction is suggested in order to achieve finite-time reconstruction of the state vector for a class of linear systems with unknown inputs. The observer is a simple one, its derivation being direct and easy. It will be shown that the problem of full-order observers for linear systems with unknown inputs can be reduced in this case to a standard one (the unknown input vector will not interfere in the observer equations). The effectiveness of the suggested design algorithm is illustrated by a numerical example (aircraft longitudinal motion), and, for the same aircraft dynamics, we make a comparison between our new observer and other already existing observers from the existence conditions and dynamic characteristics’ point of view; the superiority of the new designed observer is demonstrated.     

离散线性系统高阶积分观测器设计

针对离散线性系统提出一类高阶积分观测器。并且显示这类观测器满足极点配置分离原理,同时给出了这类观测器的存在条件．基于Sylvester矩阵方程的显式参数化通解提出了这类观测器的参数化设计方法．该方法不仅给出了观测器增益矩阵的参数表达式,而且还提供了观测器系统矩阵左特征向量的参数表达式．该设计方法给出了所有的设计自由度,为实现系统的其他性能提供了方便且强有力的工具．数值例子说明了设计过程,并表明了该方法的有效性．     

线性系统对偶Luenberger观测器设计

提出了线性系统对偶Luenberger观测器的参数化设计方法.基于一类广义Sylvester矩阵方程的显式参数化解,根据一些自由参数给出了对偶Luenberger观测器所有增益矩阵的参数化表达.该设计方法能提供所有的设计自由度,它们能进一步用来实现系统的其他性能指标.数值例子显示了该方法的设计过程.     

线性广义系统的区间观测器设计

本文研究了连续时间线性广义系统的区间观测器设计问题.首先根据正系统的稳定性判据提出了一种基于线性矩阵不等式的广义区间观测器直接设计法,然后通过引入更多的设计自由度进一步放宽了区间观测器的设计条件,扩大了设计方法的适用范围.所提出的设计方法无需坐标变换,是一种直接设计方法.最后,通过两个仿真算例验证了所提出方法的有效性.     

New scheme for discrete observer design with estimation error covariance assignment

This article presents an innovative method for solving an estimation error covariance assignment problem to design an observer for a stochastic linear system. In the proposed method, the covariance assignment problem is converted to the problem of finding an extra noise-like input to the observer. Using appropriate matrix manipulation, the Riccati equation of the estimation error covariance assignment problem, is converted to a new deterministic linear state-space model. Also, the extra noise-like input to the observer is modelled as an input to the new deterministic linear state-space model. Therefore, all the conventional and well-defined control strategies could be applied and there is no need to solve a complicated Riccati equation. Moreover, using the proposed method, a multi-objective estimation error covariance tracking problem would be easily converted to the problem of controlling a standard deterministic linear state-space system. Based on the integral control method, which is applied to the new state-space model, formulations for the proposed covariance feedback law are presented. The control law results in a stable closed-loop covariance system and assigns a pre-specified covariance matrix to the estimation errors.     

Observer with linear error dynamics for nonlinear multi-output systems

Based upon nonlinear state coordinate transformations and a condensed dual Brunovský form for multi-output systems, a method for designing nonlinear state observers with linear error dynamics is presented. Necessary and sufficient conditions are provided for identifying the class of nonlinear systems for which state observers with linear error dynamics can be obtained.     

Non-linear observer design by approximate error linearization

For a given non-linear system, an observer that provides exactly linear error dynamics can be computed by solving the so-called generalized characteristic equation (GCE). Unfortunately, the existence of a solution to the GCE is not a generic property. For unforced, scalar-output systems, we show how spline functions may be used to construct approximate solutions that minimize a norm of the non-linear terms obstructing linearization of the error dynamics. The resulting error dynamics are shown to be locally exponentially stable. A numerical example illustrates the design and performance of the observer.     

An output error method for reduced order controller design

An efficient computational technique is presented for the design of reduced order controllers for linear discrete-time systems. The technique is based on the minimization of the output error between the closed-loop system and a specified reference model.     

线性扩张状态观测器的观测误差讨论

给出一种量化表述线性扩张状态观测器(LESO)观测误差的近似方法.通过线性化“总扰”项,从时域内推导出LESO的观测误差解析式,包括动态响应部分和稳态静差部分.进而提出将静差解析式作为观测误差的量化表达式,并使用该方法讨论LESO取不同构建方式对观测精度产生的影响,分析建模不准确、输入量存在偏差情况下观测器的容错能力.仿真结果很好地验证了讨论结论的正确性,从而间接证明了将量化表达式作为一种描述LESO观测精度的近似方法的可行性.