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.     

Input output linearization approach to state observer design fornonlinear 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     

Predictor-based continuous-discrete nonlinear observer design subject to aperiodic sampled and delayed measurement

In this paper, a predictor-based continuous-discrete nonlinear observer is proposed for a class of nonaffine Lipschitz nonlinear systems with aperiodic sampled delayed output measurements and external disturbance. Firstly, this study introduces a class of delayed sampling hybrid nonlinear systems. By employing Lyapunov techniques and trajectory-based stability theory, sufficient conditions are established for ensuring input-to-state stability of the delayed sampling hybrid nonlinear system with respect to external disturbances. Then the predictor-based continuous-discrete nonlinear observer is designed which consists of a continuous-time observer, a compensating injector and a predictor. The continuous-time observer is utilized to obtain continuous and delay-free state estimation. The compensating injector is designed to compensate for output errors that occur between sampling instants. Furthermore, the predictor is employed to obtain delay-free output error information, which is then utilized by the continuous-time observer for feedback correction. The proposed observer's exponential input-to-state property against the disturbance is proved via the proposed hybrid system stability theory. The effectiveness of the proposed observers has been demonstrated through numerical simulations and performance comparisons with the zero-order holder-based observer and the output predictor-based observer designs in order to highlight the effectiveness and advantages of the proposed observer.     

Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach

In this paper the disturbance attenuation and rejection problem is investigated for a class of MIMO nonlinear systems in the disturbance‐observer‐based control (DOBC) framework. The unknown external disturbances are supposed to be generated by an exogenous system, where some classic assumptions on disturbances can be removed. Two kinds of nonlinear dynamics in the plants are considered, respectively, which correspond to the known and unknown functions. Design schemes are presented for both the full‐order and reduced‐order disturbance observers via LMI‐based algorithms. For the plants with known nonlinearity, it is shown that the full‐order observer can be constructed by augmenting the estimation of disturbances into the full‐state estimation, and the reduced‐order ones can be designed by using of the separation principle. For the uncertain nonlinearity, the problem can be reduced to a robust observer design problem. By integrating the disturbance observers with conventional control laws, the disturbances can be rejected and the desired dynamic performances can be guaranteed. If the disturbance also has perturbations, it is shown that the proposed approaches are infeasible and further research is required in the future. Finally, simulations for a flight control system is given to demonstrate the effectiveness of the results. Copyright © 2005 John Wiley & Sons, Ltd.     

A disturbance observer design for backlash compensation

Electromechanical systems usually have an innate backlash in their gear trains which is normally nonlinear and limits the performance of the system. In this paper, a disturbance observer is designed in order to estimate the properties of the backlash. Generally, such disturbance observers are designed with the inverse model of the nominal plant but since the backlash is an internal factor of the plant, its effect cannot be separated from the system output. In the proposed model, the plant is reconfigured to separate the effects of the backlash from the output, and a new type of disturbance observer is proposed to detect and compensate for the backlash in the remodelled plant. The suggested disturbance observer is applied to the knee joint of a humanoid robot. Using the proposed disturbance observer, it is shown that the backlash can be compensated for, and that the tracking error decreases in the knee pitch trajectory. Also, an error analysis of the input trajectory is carried out to verify the performance of the proposed disturbance observer and the validity of the proposed control scheme is discussed.     

Robust adaptive observer for nonlinear systems with unmodeled dynamics

Unmodeled dynamics exist in almost all applications of observers due to the impossibility of using exact and detailed models. It is highly desired that the observers can dominate the effects of unmodeled dynamics independently to prevent the state estimations from diverging and to get the precise estimations. Based on adaptive nonlinear damping, this paper presents a robust adaptive observer for multiple-input multiple-output nonlinear systems with unknown parameters, uncertain nonlinearities, disturbances and unmodeled dynamics. The observer only has one adaptive parameter no matter how high the order of the system is and how many unknown parameters there are. With the proposed observer, neither estimating the unknown parameters or solving linear matrix inequalities is needed. It is shown that the state estimation error is uniformly bounded and can be made arbitrarily small.     

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.     

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     

State estimation for nonlinear systems under model uncertainties: a class of sliding-mode observers

This paper deals with the classical problem of state estimation, considering partially unknown, nonlinear systems with noise measurements. Estimation of both, state variables and unstructured uncertain term, are performed simultaneously. In order to transform the measured disturbance into system disturbance, an alternative system representation is proposed, which lead a more advantageous observer structure. The observer proposed contains a proportional-type contribution and a sliding term for the measurement of error, which provides robustness against noisy measurements and model uncertainties. Convergence analysis of the estimation methodology proposed is performed, analysing the equation of the dynamics of the estimation error; it is shown that the observer exhibits asymptotic convergence. Estimation of monomer concentration, average molecular weight, polydispersity and filtering of temperature in a batch stirred polymerization reactor illustrates the good performance of the observer proposed.     

Observer forms for perspective systems

The estimation of three-dimensional position information from two-dimensional images in computer vision systems can be formulated as a state estimation problem for a nonlinear perspective dynamic system. The multi-output state estimation problem has been treated by several authors using methods for nonlinear observer design. This paper shows that a perspective system can be transformed to two observer forms, and provides constructive methods for arriving at the transformations. These observer forms lead to straightforward observer designs. First, it is shown that, using an output transformation, the system admits an observer form which leads to an observer with linear error dynamics. A second observer design is based on a time-scaled block triangular form. Both designs assume a commonly used observability condition. The designs are demonstrated in simulation.     

含不匹配扰动非严格反馈系统的输出反馈补偿控制

针对一类非严格反馈非线性系统,系统中包含不确定函数和未知外部扰动,提出一种带不匹配扰动补偿的输出反馈模糊控制器.采用模糊逻辑系统逼近未知的非线性函数,同时构造模糊状态观测器观测系统未知状态.考虑观测器和控制器会受到外部扰动和模糊逼近误差构成的不匹配总扰动信号影响,采用改进的扰动观测器对不匹配扰动进行估计和补偿,使扰动观...     

Fault detection for linear distributed-parameter systems using finite-dimensional functional observers

In this article, finite-dimensional residual generators are directly designed for Riesz-spectral systems with bounded input and output operators to detect faults. This is achieved by using finite-dimensional observers, that can estimate linear functionals of the state without spillover. These observers allow for a decoupling of the unknown disturbances from the estimation error dynamics under mild assumptions. Then, a finite-dimensional residual generator is obtained by approximately decoupling the state from the residual, that is generated by the observer states and the outputs. It is shown that the resulting approximation error can be made small by increasing the observer order. Then, fault detection with the finite-dimensional residual generator can be assured by introducing a time-varying threshold. A faulty Euler–Bernoulli beam with structural damping illustrates the proposed finite-dimensional fault detection approach.     

Robust observer design for uncertain one‐sided Lipschitz systems with disturbances

In this paper, we study the robust observer design problem for a class of uncertain one‐sided Lipschitz systems with disturbances. Not only the system matrices but also the nonlinear functions are assumed to be uncertain. The nominal models of nonlinearities are assumed to satisfy both the one‐sided Lipschitz condition and the quadratically inner‐bounded condition. By utilizing a novel approach, our observer designs are robust against unknown nonlinear uncertainties and system and measurement noises. The new approach also relaxes some conservativeness in related existing results, ie, less conservative observer design conditions are obtained. Furthermore, the problem of designing reduced‐order observers is considered in case the output measurement is not subject to uncertainty and disturbance. Two examples are provided to show the efficiency and advantages of our results over existing works.     

Study on Four Disturbance Observers for FO-LTI Systems

This paper addresses the problem of designing disturbance observer for fractional order linear time invariant (FO-LTI) systems, where the disturbance includes time series expansion disturbance and sinusoidal disturbance. On one hand, the reduced order extended state observer (ROESO) and reduced order cascade extended state observer (ROCESO) are proposed for the case that the system state can be measured directly. On the other hand, the extended state observer (ESO) and the cascade extended state observer (CESO) are presented for another case when the system state cannot be measured directly. It is shown that combination of ROCESO and CESO can achieve a highly effective observation result. In addition, the way how to tune observer parameters to ensure the stability of the observers and reduce the observation error is presented in this paper. Finally, numerical simulations are given to illustrate the effectiveness of the proposed methods.      

Linear extended state observer based sliding mode disturbance decoupling control for nonlinear multivariable systems with uncertainty

In this paper, the sliding mode dynamic disturbance decoupling tracking control method based on the linear extended state observer (LESO) is proposed for a class of square multivariable nonlinear uncertain system. The model plant contains the known linear dynamics, the unknown nonlinear dynamics and the internal and external disturbances, and the various input-output pairs are interacted. The system states are not available for measurement. An improved LESO is developed. The leading feature which is different from the typical ESO lies in that its extended state does not contain the known linear dynamics. The improved LESO can guarantee the error variables to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters. So this ball radius can be arbitrarily as small as desired by tuning design parameters. And we give a simple method by which the gain parameters of LESO can be computed easily. This estimation to the total disturbance of the original system is introduced into the sliding mode control design to complete disturbance rejection and decoupling. Rigorous stability analysis shows that the system output can track the desired signal closely. Finally, a class of mass–spring–damper system is taken to make the numerical simulation analysis to illustrate the effectiveness of the proposed control method.     

A modified sliding-mode observer design with application to diffusion equation

In many physical systems, the system's full state cannot be measured. An observer is designed to reconstruct the state from measurements. Disturbances often contribute to the dynamics of the system, and the designed observer must account for them. In this paper, a modified sliding-mode observer (SMO), a robust observer, is proposed that combines the efficiency of a nonlinear observer with the robustness of an SMO. The estimation error is proven to converge to zero under natural assumptions. This improved observer is compared with an extended Kalman filter and an unscented Kalman filter, as well as a standard SMO for three different versions of heat equation: a linear, a quasi-linear, and a nonlinear heat equation. The comparisons are done with and without an external disturbance. The simulations show improved performance of the modified SMO over other observers.     

Cooperative output regulation of heterogeneous multi-agent systems based on passivity

This paper investigates the problem of cooperative output regulation of heterogeneous linear multi-agent systems. A passive framework is presented for the stabilisation analysis of cooperative output regulation, which can overcome the difficulty caused by the fact that the global dynamics of heterogeneous multi-agent systems depends on the global communication structure. An adaptive distributed observer is proposed to estimate the state of the exosystem, and the proposed distributed observer is independent of any global information of the communication graph. Based on passivity design and adaptive distributed observer, both a distributed state feedback and a distributed output feedback protocol are designed for output synchronisation of heterogeneous multi-agent systems. The gain matrices of the distributed protocols and observers are obtained by a Riccati equation design approach. Furthermore, sufficient local conditions for solving the problem of cooperative output regulation of heterogeneous multi-agent systems are presented. Finally, numerical simulation results are given to illustrate the effectiveness of the proposed distributed control schemes.     

Nonlinear observer design for a general class of nonlinear systems with real parametric uncertainty

This paper is a geometric study of the local observer design for a general class of nonlinear systems with real parametric uncertainty. Explicitly, we study the observer design problem for a general class of nonlinear systems with real parametric uncertainty and with an input generator (exosystem). In this paper, we show that for the classical case, when the state equilibrium does not change with the parametric uncertainty, 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 parametric uncertainty, there typically exist local exponential observers even when the plant output is purely a function of the state. We also present a characterization and construction procedure for local exponential observers for the general class of nonlinear systems with real parametric uncertainty under some stability assumptions. We also show that for the general class of nonlinear systems considered, 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. Finally, we generalize our results to a general class of nonlinear systems with input generator, and with exogenous disturbance.     

On fault-tolerant observers

The problem of designing asymptotic observers for multioutput systems with one unreliable output measurement is addressed. It is shown that observers designed on the basis of observability indexes are not fault tolerant, and an observer for linear systems which tolerates failures in one a priori known output is given. The results are extended to nonlinear systems with no inputs: it is pointed out that the proposed design procedure may enlarge the class of nonlinear systems for which a nonlinear observer can be obtained     

Adaptive output feedback control of uncertain nonlinear systems using single-hidden-layer neural networks

We consider adaptive output feedback control of uncertain nonlinear systems, in which both the dynamics and the dimension of the regulated system may be unknown. However, the relative degree of the regulated output is assumed to be known. Given a smooth reference trajectory, the problem is to design a controller that forces the system measurement to track it with bounded errors. The classical approach requires a state observer. Finding a good observer for an uncertain nonlinear system is not an obvious task. We argue that it is sufficient to build an observer for the output tracking error. Ultimate boundedness of the error signals is shown through Lyapunov's direct method. The theoretical results are illustrated in the design of a controller for a fourth-order nonlinear system of relative degree two and a high-bandwidth attitude command system for a model R-50 helicopter.