Extended state observer for uncertain lower triangular nonlinear systems subject to stochastic disturbance

The extended state observer (ESO) is the most important part of an emerging control technology known as active disturbance rejection control to this day, aiming at estimating “total disturbance” from observable measured output. In this paper, we construct a nonlinear ESO for a class of uncertain lower triangular nonlinear systems with stochastic disturbance and show its convergence, where the total disturbance includes internal uncertain nonlinear part and external stochastic disturbance. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.     

On the convergence of an extended state observer for nonlinear systems with uncertainty

The extended state observer first proposed by Jingqing Han in [J.Q. Han, A class of extended state observers for uncertain systems, Control Decis. 10 (1) (1995) 85-88 (in Chinese)] is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering. Unfortunately, there is no rigorous proof of convergence to date. In this paper, we attempt to tackle this long unsolved extraordinary problem. The main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance, for which the effect of total disturbance error is eliminated by the high-gain.     

Sampled-data extended state observer for uncertain nonlinear systems

In this paper, we present a sampled-data nonlinear extended state observer (NLESO) design method for a class of nonlinear systems with uncertainties and discrete time output measurement. To accommodate the inter-sample dynamics, an inter-sample output predictor is employed in the structure of the NLESO to estimate the system output in the sampling intervals, where the prediction is used in the proposed observer instead of the system output. The exponential convergence of the sampled-data NLESO is also discussed and a sufficient condition is given by the Lyapunov method. A numerical example is provided to illustrate the performance of the proposed observer.     

Extended state observer for MIMO nonlinear systems with stochastic uncertainties

ABSTRACT

In this paper, both linear extended state observer (ESO) and nonlinear ESO with homogeneous weighted functions are proposed for a class of multi-input multi-output (MIMO) nonlinear systems composed of coupled subsystems with large stochastic uncertainties. The stochastic uncertainties in each subsystem including internal coupled unmodelled dynamics and external stochastic disturbance without known statistical characteristics are lumped together as the stochastic total disturbance (extended state) of each subsystem. The linear ESO and nonlinear ESO are designed separately for real-time estimation of not only the unmeasured state but also the stochastic total disturbance of each subsystem. The practical mean square convergence of these two classes of ESOs are developed. Some numerical simulations are presented to demonstrate the effectiveness of the ESOs with the advantages of smaller peaking values and more accurate estimation by the nonlinear ESO.     

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 robust adaptive observer for a class of singular nonlinear uncertain systems

This paper proposes a robust adaptive observer for a class of singular nonlinear non-autonomous uncertain systems with unstructured unknown system and derivative matrices, and unknown bounded nonlinearities. Unlike many existing observers, no strong assumption such as Lipschitz condition is imposed on the recommended system. An augmented system is constructed, and the unknown bounds are calculated online using adaptive bounding technique. Considering the continuous nonlinear gain removes the chattering which may appear in practical applications such as analysis of electrical circuits and estimation of interaction force in beating heart robotic-assisted surgery. Moreover, a simple yet precise structure is attained which is easy to implement in many systems with significant uncertainties. The existence conditions of the standard form observer are obtained in terms of linear matrix inequality and the constrained generalised Sylvester's equations, and global stability is ensured. Finally, simulation results are obtained to evaluate the performance of the proposed estimator and demonstrate the effectiveness of the developed scheme.     

Semi-global observer for multi-output nonlinear systems

We present an explicit form of nonlinear observers for a class of multi-input multi-output systems. Observer construction for multi-output nonlinear systems is not a trivial extension of single output case, especially when the global error convergence is of interest. In this paper, we consider a class of systems in which the subsystem for each output has a triangular dependence on the states of that subsystem itself, and the overall system has a block triangular form for each subsystem. Hence, the contribution is to extend the results existing in the literature in that interconnections between the subsystems are allowed. The construction is based on the saturation of some estimates, which is originated by Khalil and Esfandiari for the use of semi-global output feedback control.     

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.     

Adaptive fuzzy observer backstepping control for a class of uncertain nonlinear systems with unknown time-delay

In this paper, a new adaptive fuzzy backstepping control approach is developed for a class of nonlinear systems with unknown time-delay and unmeasured states. Using fuzzy logic systems to approximate the unknown nonlinear functions, a fuzzy state observer is designed for estimating the unmeasured states. On the basis of the state observer and applying the backstepping technique, an adaptive fuzzy observer control approach is developed. The main features of the proposed adaptive fuzzy control approach not only guarantees that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded, but also contain less adaptation parameters to be updated on-line. Finally, simulation results are provided to show the effectiveness of the proposed approach.     

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.     

Vehicle state estimation for anti-lock control with nonlinear observer

Vehicle state estimation during anti-lock braking is considered. A novel nonlinear observer based on a vehicle dynamics model and a simplified Pacejka tire model is introduced in order to provide estimates of longitudinal and lateral vehicle velocities and the tire-road friction coefficient for vehicle safety control systems, specifically anti-lock braking control. The approach differs from previous work on vehicle state estimation in two main respects. The first is the introduction of a switched nonlinear observer in order to deal with the fact that in some driving situations the information provided by the sensor is not sufficient to carry out state estimation (i.e., not all states are observable). This is shown through an observability analysis. The second contribution is the introduction of tire-road friction estimation depending on vehicle longitudinal motion. Stability properties of the observer are analyzed using a Lyapunov function based method. Practical applicability of the proposed nonlinear observer is shown by means of experimental results.     

基于扩张状态观测器的反馈线性化无人飞行器姿态控制

随着集成电子等技术逐渐成熟,无人飞行器得到了广泛的关注和应用;其中,由于飞行器在飞行阶段具有高响应速度、耦合动力学和非线性等特点,使其在飞行阶段的姿态控制成为该领域重点研究的方向之一;针对该研究方向,文章提出了一种基于反步法技术的反馈线性化无人飞行器姿态跟踪控制方案来解决其在受到环境干扰的条件下可以精准跟踪姿态角度的问题;在该方法中,设计了反步法技术拆分简化模型和反馈线性化减少调节参数,并利用扩张状态观测器(ESO)来对扰动进行估计和补偿,同时给出了ESO的收敛性和闭环系统的稳定性来证明该方法的可实施性;最后给出了仿真结果,验证了该方法在干扰的环境下仍可以精准控制无人飞行器姿态。     

Continuous adaptive observer for state affine sampled‐data systems

In this paper, a hybrid adaptive observer is designed for a class of nonlinear sampled‐data systems with constant unknown parameters. The proposed observer uses a predictor of the output between the sampling times. This predictor is re‐initialized at each sampling time. This observer is very simple to implement and converges exponentially under some sufficient conditions. An explicit relation between the bound of the maximum allowable sampling time (τ_MASP) and the parameters of the observer is also given. Copyright © 2012 John Wiley & Sons, Ltd.     

线性/非线性切换扩张状态观测器

为了解决非线性扩张状态观测器(NLESO)对大幅度扰动估计能力有限的问题,本文提出一种线性/非线性切换扩张状态观测器.首先分析了非线性扩张状态观测器对大幅度扰动估计能力有限的原因,然后提出在NLESO的非线性区间引入一段线性扩张状态观测器(LESO)弥补NLESO的缺陷,其次从理论上证明了提出的线性/非线性切换扩张状态观测器的收敛性.最后,通过数值仿真验证了提出的线性/非线性切换扩张状态观测器的可行性.     

Robust decentralised observer based guaranteed cost control for nonlinear uncertain interconnected systems. Application to multi-machine power systems

This article addresses the problem of a robust decentralised observer based guaranteed cost control for nonlinear interconnected systems with norm bounded time varying parameter uncertainties.

Sufficient conditions for the robust stabilisation and the guaranteed cost control performances of the closed-loop system are formulated in an optimisation problem in terms of linear matrix inequalities. Moreover, the designed problem is solved by a proposed iterative algorithm to compute the observer and the control gain matrices, to minimise the upper bound of a prescribed performance index and to maximise the nonlinearity domain, which leave the augmented system stable, despite the emerged uncertainties on the subsystems and on the interconnections between them. An extensive application example of a power system with three interconnected machines is discussed to demonstrate the reliability of the proposed control approach.     

Design of decentralised variable structure observer for mismatched nonlinear uncertain large-scale systems

Although the state feedback approach is quite popular in control engineering, it cannot be used while the system states cannot be measured. The state observer approach may be used to overcome such a shortcoming. Also, most control systems have become larger and more complicated; therefore, based on the variable structure control theory, a new decentralised variable structure observer (DVSO) for a class of nonlinear large-scale systems with mismatched uncertainties will be considered in this article. The switching surface function is determined such that the equivalent system will have the desired behaviour once the system reaches the switching surface. And then a new DVSO is designed such that the estimated states will approach the system states. Using the Lyapunov stability theory and using the generalised matrix inverse concept, the uncertain nonlinear error system trajectories can be driven onto the sliding manifold and then the existence of a sliding mode and the attractiveness to the sliding surface is ensured. With the proposed DVSO, the estimation errors asymptotically tend to zero if the matching condition is satisfied, and the effects of the mismatched parts can be uniformly ultimately bounded if the matching condition is not satisfied. Finally, a numerical example with a succession of computer simulations is given to demonstrate the effectiveness of the proposed approach.     

不确定随机非线性系统自适应观测器设计

考虑带非参数不确定项的随机非线性系统自适应观测器设计问题.不同于已有结果,系统的不确定项无需满足Lipschitz连续性条件,也不必要仅仅是系统输出的函数.通过设计一个带参数自适应律的非线性观测器来重构系统状态,该观测器结构简单目易于实现.应用Lyapunov稳定性理论和随机微分理论证明该观测器是最终有界的,并且它的界可以通过选取适当的参数进行调节.最后,数值仿真结果表明了该观测器的有效性.     

基于观测器非线性不确定系统的自适应模糊控制

针对一类单输入单输出不确定非线性系统，提出一种稳定的自适应模糊控制方法，该方法不需要系统状态可测的条件，而是通过设计模糊状态观测器来估计系统的状态，证明了所提出的控制方法不但能使闭环系统稳定，而且输出误差可取得H∞跟踪控制性能，仿真结果进一步验证了该控制算法的实用性和有效性。     

一类基于模糊逻辑系统的非线性不确定系统自适应观测器设计

利用模糊逻辑系统具有充分利用专家语言信息和逼近连续函数的性质, 分析和研究了一类具有未知不确定界的非线性系统的自适应状态观测器设计问题. 观测器和自适应律的构成直接利用了系统的数学结构信息和模糊逻辑系统对不确定性的输出信息, 在较弱的假设条件下, 这种观测器与被控系统状态间的误差及各参数估计误差一致终极有界. 最后的仿真实例说明了所采用的方法的有效性.     

Adaptive fuzzy controller design with observer for a class of uncertain nonlinear MIMO systems

In this paper, an adaptive fuzzy control approach is proposed to stabilize a class of uncertain nonlinear MIMO systems with the unmeasured states and the external disturbances. The fuzzy logic systems are used to approximate the unknown functions. Because it does not required to assume that the system states are measurable, it needs to design an observer to estimate the system unmeasured states. The considered MIMO systems are more general, i.e. they consist of N subsystems and each subsystem is in the non‐affine form. The stability of the closed‐loop system is verified by using Lyapunov analysis method. Two simulation examples are utilized to verify the effectiveness of the proposed approach. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society