Constructive algorithm for system immersion into non-linear observer form

There exists a class of non-linear systems which cannot be transformed into a non-linear observer form (an observable linear system up to output injection) via diffeomorphism, but can be immersed into a higher dimensional non-linear observer form. This class of systems can be characterized by a differential equation called characteristic equation. If the system is an n dimensional system and it is immersible into n?+?m dimensional observer form, the characteristic equation involves n?+?m?+?1 unknowns where n?+?m unknowns are for the state immersion and one for the output transformation. In general, one should solve these unknowns simultaneously which makes the characterization difficult. After investigating the algebraic structure of the characteristic equation, we present an algorithm to check the immersibility under a constant rank assumption. Using the algorithm, one can check the immersibility iteratively since only one unknown is involved at each step of the algorithm.     

Dynamic observer error linearization

In this paper, a new framework of observer error linearization problem is proposed. The main idea of our approach is twofold. The one is to introduce an auxiliary dynamics whose input is the system output, and the other is to transform the augmented system into an observable linear system with an injection term which contains the system output as well as the state of the auxiliary dynamics. It is a natural extension of the recently developed dynamic observer error linearization where the injection term contains only newly defined output. It is also shown that whenever an n dimensional system is immersible into an n+d dimensional linear system up to an output injection, then it can be also dynamically observer error linearizable in our sense with a d dimensional auxiliary dynamics. Moreover, we show that the converse is not true by providing a counterexample, which implies that our approach is applicable to a strictly wider class of systems than that of the system immersion method.     

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.     

State observer based dynamic fuzzy logic system for a class of SISO nonlinear systems

This paper presents an observer based dynamic fuzzy logic system (DFLS) scheme for a class of unknown single-input single-output (SISO) nonlinear dynamic systems with external disturbances. The proposed approach does not need the availability of the state variables. Within this scheme, the DFLS is employed to identify the unknown nonlinear dynamic system. The control law and parameter adaptation laws of the DFLS are derived based on Lyapunov synthesis approach. The control law is robustfied in H _∞ sense to attenuate external disturbance, model uncertainties, and fuzzy approximation errors. It is shown that under appropriate assumptions, it guarantees the boundedness of all the signals in the closed-loop system and the asymptotic convergence to zero of tracking errors. The proposed method is applied to an inverted pendulum system to verify the effectiveness of the proposed algorithms.     

Interval observer design for consistency checks of nonlinear continuous-time systems

This paper deals with fault detection for nonlinear continuous-time systems. A procedure based on interval analysis is proposed to build a guaranteed qLPV (quasi-Linear Parameter-Varying) approximation of the nonlinear model. The interval qLPV approximation makes it possible to derive two point observers which estimate respectively the lower and the upper bound of the state vector using cooperativity theory. A set guaranteed to contain the actual value of the residual is then designed. The modelling uncertainties and measurement errors are taken into account at the design stage. The proposed methodology is illustrated through numerical simulations.     

Modified observer backstepping controller for a dynamic positioning system

A procedure for attenuating the control law of a vessel dynamic positioning system, based on the observer backstepping methodology, is proposed. The motivation is the appearance of an undesirable on–off behavior on the signal sent to the actuators when their saturation is considered and the control law is dependent on estimated state variables. Two gain matrices associated with the error variables are introduced to achieve the desired attenuation. Stability is proven through Lyapunov stability analysis. Numerical simulations confirm the effectiveness of the proposed controller to render the control law compatible with the limitations of the actuators.     

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.     

A globally exponentially convergent immersion and invariance speed observer for mechanical systems with non-holonomic constraints

The problem of velocity estimation for general, n degrees-of-freedom, mechanical systems, is of great practical and theoretical interest. For unconstrained systems many partial solutions have been reported in the literature. However, even in this case, the basic question of whether it is possible to design a globally convergent speed observer remains open. In this paper, an affirmative answer to the question is given for general mechanical systems with knon-holonomic constraints, by proving the existence of a 3n−2k+1-dimensional globally exponentially convergent speed observer. An observer for unconstrained mechanical systems is obtained as a particular case of this general result. Instrumental for the construction of the speed observer is the use of the Immersion and Invariance technique, in which the observer design problem is recast as a problem of rendering attractive and invariant a manifold defined in the extended state-space of the plant and the observer.     

基于系统浸入和流形不变自适应方法的静止无功补偿器非线性鲁棒自适应控制方法

本文提出一种将系统浸入和流形不变(I&I)自适应控制方法与L2-增益抑制鲁棒控制方法相结合的静止无功补偿器(SVC)的非线性鲁棒自适应控制方法.所提方法首先通过参数估计误差和鲁棒控制律的设计,使得所构造的表示参数估计误差函数的流形不变且吸引,从而使参数估计误差在这一流形上收敛于零.然后,通过所设计的可调参数对参数估计误差的收敛性能进行控制,以此来保证参数估计器对不确定参数的自适应估计能力.最后,采用自适应逆推算法推导鲁棒控制律,并通过使不确定外部扰动满足从输入到输出的耗散性来保证系统对不确定扰动的鲁棒性.仿真结果表明,利用所提方法设计的SVC控制器和参数替换律在参数估计、发电机功角动态响应方面优于传统自适应逆推算法,从而提高了输电系统的稳定水平.     

Full-order observer design for a class of port-Hamiltonian systems

We consider a special class of port-Hamiltonian systems for which we propose a design methodology for constructing globally exponentially stable full-order observers using a passivity based approach. The essential idea is to make the augmented system consisting of the plant and the observer dynamics to become strictly passive with respect to an invariant manifold defined on the extended state-space, on which the state estimation error is zero. We first introduce the concept of passivity of a system with respect to a manifold by defining a new input and output on the extended state-space and then perform a partial state feedback passivation which leads to the construction of the observer. We then illustrate this observer design procedure on two physical examples, the magnetic levitation system and the inverted pendulum on the cart system.     

A constructive solution for stabilization via immersion and invariance: The cart and pendulum system

Immersion and Invariance (I&I) is the method to design asymptotically stabilizing control laws for nonlinear systems that was proposed in [Astolfi, A., & Ortega, R. (2003). Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems. IEEE Transactions on Automatic Control, 48, 590-606]. The key steps of I&I are (i) the definition of a target dynamics, whose order is strictly smaller than the order of the system to be controlled; (ii) the construction of an invariant manifold such that the restriction of the system dynamics to this manifold coincides with the target dynamics; (iii) the design of a control law that renders the manifold attractive and ensures that all signals are bounded. The second step requires the solution of a partial differential equation (PDE) that may be difficult to obtain. In this short note we use the classical cart and pendulum system to show that by interlacing the first and second steps, and invoking physical considerations, it is possible to obviate the solution of the PDE. To underscore the generality of the proposed variation of I&I, we show that it is also applicable to a class of n-dimensional systems that contain, as a particular case, the cart and pendulum system.     

The extended Luenberger observer for nonlinear systems

For nonlinear single-input single-output systems , the relationships for a state transformation into the nonlinear observer canonical form are developed. It is possible to dimension a nonlinear observer by an eigenvalue assignment without solving the nonlinear partial differential equations for the transformation, if the transformed nonlinearities are linearized about the reconstructed state. With reference to the extended Kalman filter algorithm, this nonlinear observer design is called the extended Luenberger observer.     

一类非线性系统有限时间流形吸引的浸入与不变控制

本文对一类非线性系统,提出了一种设计渐近稳定控制律的有效方法.其中,通过更新系统浸入与不变流形理论的应用方法,流形的吸引坐标可以在有限时间内收敛到平衡点.为了得到闭环系统的稳定性,增广系统的各个信号被证明是有界的.本文得出的一个重要成果是流形吸引有限时间的计算方法.此外,在施加了有限时间流形吸引控制器之后,流形对外部有界未知扰动具有不敏感性.最后利用车摆系统来论述所提出的控制方法的设计步骤,以及通过仿真验证控制器的性能.     

Relaxed observer design of discrete-time nonlinear systems via a novel ranking-based switching mechanism

In this paper, the problem of relaxed observer design of discrete-time nonlinear systems is studied by developing a novel ranking-based switching mechanism. To do this, the useful ranking information of the normalized fuzzy weighting functions is utilized in order to give a denser subdivision of the normalized fuzzy weighting function space and therefore essentially yields the proposed ranking-based switching mechanism. Based on the obtained switching mechanism, a family of switching observers can be developed for the purpose of guaranteeing the estimation error system to be asymptotically stable with less conservatism than the existing results available in the references. Finally, two numerical examples are presented to illustrate the advantages of the proposed method.     

多变量非线性系统的间接模糊输出反馈自适应控制

针对一类多输入多输出非线性不确定系统，提出一种基于观测器的模糊间接自适应控制方法，并基于李亚普诺夫函数方法，导出了输出反馈控制律以及参数的自适应律，证明了整个控制方案不但能保证闭环系统稳定，而且取得了良好的跟踪控制性能。     

Visually guided ranging from observations of points, lines and curves via an identifier based nonlinear observer

In this paper we consider the problem of estimating the range information of features on an affine plane in by observing its image with the aid of a CCD camera, wherein we assume that the camera is undergoing a known motion. The features considered are points, lines and planar curves located on planar surfaces of static objects. The dynamics of the moving projections of the features on the image plane have been described as a suitable differential equation on an appropriate feature space. This dynamics is used to estimate feature parameters from which the range information is readily available. In this paper the proposed identification has been carried out via a newly introduced identifier based observer. Performance of the observer has been studied via simulation.     

Direct transformation of nonlinear systems into state affine MISO form for observer design

This paper gives necessary and sufficient conditions to solve in a constructive way the transformation of a general multiple-input-single-output nonlinear system into a state affine system. This result is direct i.e., without solving the state elimination problem when computing the input-output differential equation. A Kalman-like observer can be designed for the obtained state affine system.     

An RBF neural network-based adaptive control for SISO linearisable nonlinear systems

An RBF neural network-based adaptive control is proposed for Single-Input and Single-Output (SISO) linearisable nonlinear systems in this paper. It is shown that a SISO nonlinear system is first linearised by using the differential geometric approach in the state space, and the linearised nonlinear system is then treated as a partially known system. The known dynamics are used to design a nominal feedback controller to stabilise the nominal system, and an adaptive RBF neural network-based compensator is then designed to compensate for the effects of uncertain dynamics. The main function of the RBF neural network in this work is to adaptively learn the upper bound of the system uncertainty, and the output of the neural network is then used to adaptively adjust the gain of the compensator so that the strong robustness with respect to unknown dynamics can be obtained, and the tracking error between the plant output and the desired reference signal can asymptotically converge to zero. A simulation example is performed in support of the proposed scheme.     

New algorithm for observer error linearization with a diffeomorphism on the outputs

In this paper, we give the necessary and sufficient conditions that guarantee the existence of a diffeomorphism which allows one to transform a multi-output nonlinear dynamical system into a normal observable canonical form. In particular, we propose an algorithm that permits one to derive such diffeomorphism. The main feature of the canonical form is that it is obtained by allowing a diffeomorphism on the outputs and it also allows one to design an observer with linear error dynamics. We first consider multi-output nonlinear dynamical system without inputs. We then extend the results obtained to multi-output nonlinear dynamical system with inputs.