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.     

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.     

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.     

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.     

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.     

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.     

An investigation of forced oscillation for signal stabilisation of two-dimensional nonlinear system

The phenomena of synchronisation, desynchronisation and forced oscillation has been studied using describing function theory for a two input and two output nonlinear system containing saturation-type nonlinearities and subjected to high-frequency deterministic signal (dither) for the purpose of limit cycle quenching. The analytical results have been compared with the results of digital simulation/MATLAB-SIMULINK for a typical example varying the nonlinear element.     

Further result on a dynamic recurrent neural-network-based adaptive observer for a class of nonlinear systems

In Kim et al. [(1997) A dynamic recurrent neural-network-based adaptive observer for a class of nonlinear systems. Automatica 33(8), 1539–1543], authors present an excellent neural network (NN) observer for a class of nonlinear systems. However, the output error equation in their paper is strictly positive real (SPR) which is restrictive assumption for nonlinear systems. In this note, by introducing a vector b₀ and Lyapunov equation, the observer design is obtained without requiring the SPR condition. Thus, our observer can be applied to a wider class of systems.     

An algorithm for nonlinear system parameter identification

An algorithm for system parameter identification will be presented in this paper. The method is applicable to linear and nonlinear systems with known structures. It is applied in this paper to systems in which both system and measurement noises can be neglected. The algorithm requires less time per iteration and less computer storage than the quasilinearization method. A shaft position control system with multiple nonlinearities will be used to illustrate the method.     

Observer-based H-infinity output feedback control with feedback gain and observer gain variations for Delta operator system

Considering that the controller feedback gain and the observer gain are of additive norm-bounded variations, a design method of observer-based H-infinity output feedback controller for uncertain Delta operator systems is proposed in this paper. A sufficient condition of such controllers is presented in linear matrix inequality （LMI） forms. A numerical example is then given to illustrate the effectiveness of this method, that is, the obtained controller guarantees the closed-loop system asymptotically stable and the expected H-infinity performance even if the controller feedback gain and the observer gain are varied.     

An adaptive high-gain observer for nonlinear systems

In this paper the authors provide a solution to the noise sensitivity of high-gain observers. The resulting nonlinear observer possesses simultaneously (1) extended Kalman filter’s good noise filtering properties, and (2) the reactivity of the high-gain extended Kalman filter with respect to large perturbations.The authors introduce innovation as the quantity that drives the gain adaptation. They prove a general convergence result, propose guidelines to practical implementation and show simulation results for an example.     

非线性系统模型参数估计的算法模型

针对非线性系统模型的多样性,提出了适用于多种非线性模型的基于粒子群优化算法的参数估计方法。计算结果表明,粒子群优化算法是非线性系统模型参数估计的有效工具。     

Hybrid dead-beat observers for a class of nonlinear systems

This paper studies both the full order and the reduced order dead-beat observer problem for a class of nonlinear systems, linear in the unmeasured states. A novel hybrid observer design strategy is proposed, with the help of the notion of strong observability in finite time. The proposed methodology is applied for the estimation of the frequency of a sinusoidal signal. The results show that accurate estimation can be provided even if the signal is corrupted by high frequency noise. A brief discussion of the robustness properties of the proposed observer with respect to measurement errors is also provided.