Adaptive observers for a class of uniformly observable systems with nonlinear parametrization and sampled outputs

In this paper, we propose an adaptive observer for a class of uniformly observable nonlinear systems with nonlinear parametrization and sampled outputs. A high gain adaptive observer is first designed under the assumption that the output is continuously measured and its exponential convergence is investigated, thanks to a well defined persistent excitation condition. Then, we address the case where the output is available only at (non uniformly spaced) sampling instants. To this end, the continuous-time output observer is redesigned leading to an impulsive observer with a corrective term involving instantaneous state impulses corresponding to the measured samples and their estimates. Moreover, it is shown that the proposed impulsive observer can be put under the form of a hybrid system composed of a continuous-time observer coupled with an inter-sample output predictor. Two design features are worth to be emphasized. Firstly, the observer calibration is achieved through the tuning of a scalar design parameter. Secondly, the exponential convergence to zero of the observation and parameter estimation errors is established under a well defined condition on the maximum value of the sampling partition diameter. More specifically, the observer design is firstly carried out in the case of linear parametrization before being extended to the nonlinear one. The theoretical results are corroborated through simulation results involving a typical bioreactor.     

Adaptive NN output‐feedback decentralized stabilization for a class of large‐scale stochastic nonlinear strict‐feedback systems

In this paper, the decentralized adaptive neural network (NN) output‐feedback stabilization problem is investigated for a class of large‐scale stochastic nonlinear strict‐feedback systems, which interact through their outputs. The nonlinear interconnections are assumed to be bounded by some unknown nonlinear functions of the system outputs. In each subsystem, only a NN is employed to compensate for all unknown upper bounding functions, which depend on its own output. Therefore, the controller design for each subsystem only need its own information and is more simplified than the existing results. It is shown that, based on the backstepping method and the technique of nonlinear observer design, the whole closed‐loop system can be proved to be stable in probability by constructing an overall state‐quartic and parameter‐quadratic Lyapunov function. The simulation results demonstrate the effectiveness of the proposed control scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

严格反馈互联系统分散式backstepping自适应迭代学习控制

基于Lyapunov分析方法,针对具有严格反馈形式的非线性互联系统,本文设计了一种分散式backstepping自适应迭代学习控制器.子系统之间的互联项为所有子系统输出项线性有界,为每个子系统设计的控制器仅采用该子系统的信息,不需要子系统之间相互传递信息.在控制器中,引入在时间轴和迭代轴上同时更新的自适应参数,以补偿子系统之间的互联项影响.通过采用本文给出的控制器,可使得每个子系统的输出跟踪相应的参考模型输出,仿真结果验证了本文算法的有效性.     

High order sliding mode observer for linear systems with unbounded unknown inputs

A global observer is designed for strongly detectable systems with unbounded unknown inputs. The design of the observer is based on three steps. First, the system is extended taking the unknown inputs (and possibly some of their derivatives) as a new state; then, using a global high-order sliding mode differentiator, a new output of the system is generated in order to fulfil, what we will call, the Hautus condition, which finally allows decomposing the system, in new coordinates, into two subsystems; the first one being unaffected directly by the unknown inputs, and the state vector of the second subsystem is obtained directly from the original system output. Such decomposition permits designing of a Luenberger observer for the first subsystem, which satisfies the Hautus condition, i.e. all the outputs have relative degree one w.r.t. the unknown inputs. This procedure enables one to estimate the state and the unknown inputs using the least number of differentiations possible. Simulations are given in order to show the effectiveness of the proposed observer.     

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.     

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.     

基于高增益鲁棒滑模观测器的故障检测和隔离

针对一类同时具有执行器和传感器故障的不确定线性系统,讨论了基于观测器的故障检测和隔离方法.首先,通过引入增维向量,使得在构造的增维系统中,故障向量包含了原系统的执行器故障和传感器故障.通过构造辅助输出使增维系统的观测器匹配条件得以满足,同时设计高增益滑模观测器对辅助输出进行估计.然后,对增维系统构造鲁棒滑模观测器并用作故障检测观测器,通过滑模控制项来抑制干扰,使观测器具有鲁棒性.在此基础上,结合多观测器故障隔离思想,提出了可以同时对执行器故障和传感器故障进行检测和隔离的方法. 最后,通过对一个五阶飞行器模型进行仿真,证明了所提方法的有效性.     

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     

Decentralized Observer Design for a Class of Nonlinear Uncertain Large Scale Systems with Lumped Perturbations

Many valuable properties of the state feedback method can not be applied to some class of control systems while some of the system states cannot be measured directly. An attractive alternative approach is to make good use of a state observer. In this paper, a new decentralized sliding mode observer (DSMO) is proposed for a class of nonlinear uncertain large‐scale systems (LSS) with lumped perturbations based on the sliding mode control (SMC) theory. Our main result presented here is that we introduce a new switching term to the traditional LSS observer design for a class of large‐scale system to generate a new decentralized sliding mode observer. The generalized matrix inverse concept is adopted to avoid using the un‐measurable state and the global reaching condition of the sliding mode for each error subsystem is guaranteed. The stability of each equivalent error subsystem is verified based on the strictly positive real concept. It also shows that the investigated uncertain large‐scale systems still possesses the property of insensitivity to the lumped perturbations as does the traditional linear system. Moreover, the state transformation approach is no longer needed as there is no longer concern about the problems of finding a suitable transformation or indirect estimated states, since the proposed DSMO is not based on the transformed system model. Finally, a numerical example with a series of computer simulations is given to demonstrate the validity of the proposed decentralized sliding mode observer.     

一类非线性网络控制系统的故障检测

针对远程被控对象为非线性模型，且具有输出诱导时延的一类网络控制系统（NCS），假设可能发生执行器故障，对系统进行故障检测：利用T—S模糊模型将对象线性化，建立了模糊观测器，并给出了观测器系统渐近稳定的条件。进一步地，又考虑到满足该观测器稳定条件的矩阵难以寻找到，因此利用模糊模型中的主导子系统思想，重新设计了观测器，并基于Lyapunov稳定性定理，推导出了整个观测器系统全局渐近稳定的充分条件。最后，通过一个仿真例子验证了方法的有效性。     

A filtered high gain observer for a class of non uniformly observable systems – Application to a phytoplanktonic growth model

The design of a high gain observer with filtering capabilities for a class of non uniformly observable systems is proposed. The filtering feature enables to perform a smooth estimation in the presence of noisy output measurements. Of fundamental interest, the Filtered High Gain Observer (FHGO) design is first carried out by assuming that the output measurements are available in a continuous manner before being extended to the case of sampled outputs leading thereby to a continuous-discrete time FHGO which inherits the main properties of the original FHGO. The performance of the FHGO and its main properties are highlighted and compared to those of a Standard High Gain Observer (SHGO) through a bioprocess dealing with the growth of a phytoplanktonic population.     

Stabilizing a class of nonlinear parameter-varying systems using interval observer based on a piecewise framework

This article addresses an interval observer-based control for stabilizing a class of nonlinear parameter-varying systems with noisy output by designing a switching surface. An input-dependent interval observer is firstly developed to estimate the lower and upper bounds of the states. Next, a switching-based controller is designed to stabilize the interval observer which implies the stability of the main parameter-varying system. The developed stabilizing switching surfaces are designed based on the outputs of the main system and the bounds of the states of the observer. By choosing an appropriate piecewise Lyapunov function, the closed-loop stability analysis of the interval observer system leads to a set of linear matrix inequalities including stability and Metzler constraints, simultaneously. The effectiveness of the proposed method is verified using the simulation results.     

Global tracking sliding mode control for a class of nonlinear systems via variable gain observer

A novel output‐feedback sliding mode control strategy is proposed for a class of single‐input single‐output (SISO) uncertain time‐varying nonlinear systems for which a norm state estimator can be implemented. Such a class encompasses minimum‐phase systems with nonlinearities affinely norm bounded by unmeasured states with growth rate depending nonlinearly on the measured system output and on the internal states related with the zero‐dynamics. The sliding surface is generated by using the state of a high gain observer (HGO) whereas a peaking free control amplitude is obtained via a norm observer. In contrast to the existing semi‐global sliding mode control solutions available in the literature for the class of plants considered here, the proposed scheme is free of peaking and achieves global tracking with respect to a small residual set. The key idea is to design a time‐varying HGO gain implementable from measurable signals. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Unknown inputs observer for a class of nonlinear uncertain systems: An LMI approach

This paper deals with the simultaneous estimation of states and unknown inputs for a class of Lipschitz nonlinear systems using only the measured outputs. The system is assumed to have bounded uncertainties that appear on both the state and output matrices. The observer design problem is formulated as a set of linear constraints which can be easily solved using linear matrix inequalities (LMI) technique. An application based on manipulator arm actuated by a direct current (DC) motor is presented to evaluate the performance of the proposed observer. The observer is applied to estimate both state and faults.     

Continuous–discrete-time adaptive observers for nonlinear systems with sampled output measurements

This paper considers the continuous–discrete-time adaptive observer (CDAO) design for a class of nonlinear systems with unknown constant parameters and sampled output measurements. The proposed observer is actually an impulsive system, since the observer state flows according to a set of differential equations and with instantaneous state jumps corresponding to measured samples and their estimates, and an inter-sample output predictor is used to predict the output during sampling intervals. By assuming appropriate persistent excitation conditions and following a technical lemma, an upper bound of the sampling intervals is derived, with which the convergence of the observer state and unknown parameters can be ensured. Finally, the proposed observer is used in examples of chaotic oscillators and single-link flexible-joint robot manipulator to show the effectiveness.     

Prescribed performance adaptive neural output control for a class of switched nonstrict‐feedback nonlinear time‐delay systems: State‐dependent switching law approach

This paper investigates the tracking problem for a class of uncertain switched nonlinear delayed systems with nonstrict‐feedback form. To address this problem, by introducing a new common Lyapunov function (CLF), an adaptive neural network dynamic surface control is proposed. The state‐dependent switching rule is designed to orchestrate which subsystem is active at each time instance. In order to compensate unknown delay terms, an appropriate Lyapunov‐Krasovskii functional is considered in the constructing of the CLF. In addition, a novel switched neural network–based observer is constructed to estimate system states through the output signal. To maintain the tracking error performance within a predefined bound, a prescribed performance bound approach is employed. It is proved that by the proposed output‐feedback control, all the signals of the closed‐loop system are bounded under the switching law. Moreover, the transient and steady‐state tracking performance is guaranteed by the prescribed performance bound. Finally, the effectiveness of the proposed method is illustrated by two numerical and practical examples.     

High gain observer for a class of non-triangular systems

This paper presents a high gain observer for a class of MIMO nonlinear systems involving some uncertainties. The latter is particularly composed of cascade subsystems where each subsystem is associated with a subset of the output variables, and assumes a triangular dependence on its own state variables and may depend on the state variables of all other subsystems. The main contribution consists in extending the available results to allow more interconnections between the subsystems. Of fundamental interest, it is shown that the underlying observation error exponentially converges to zero in the absence of uncertainties. Moreover, the observation error can be made as small as desired by properly specifying the observer design parameter in the case where uncertainties are considered.     

Reduced-order dynamic observer error linearisation: a natural extension of observer error linearisation

This paper introduces the concept of reduced-order dynamic observer error linearisation (RDOEL) for multi-output systems, which is the problem of transforming a nonlinear system into a nonlinear observer canonical form with the aid of auxiliary dynamics. The proposed RDOEL framework is not only a modified version of dynamic observer error linearisation (DOEL) but also a natural extension of observer error linearisation (OEL). We provide three necessary conditions for the RDOEL problem, and then derive a necessary and sufficient condition described in terms of Lie algebras of vector fields. Furthermore, from the result, we also give a geometric necessary and sufficient condition for the OEL problem, which has not yet been completely established in the case where a diffeomorphism on the output of general form is considered.     

Cascade high-gain observers for a class of nonlinear systems with large delayed measurements

This paper proposes a cascade high-gain observer for a class of triangular nonlinear systems with large and diverse time delays at states and output equation. Compared with some existing results in the literature, the main contribution is to consider the simultaneous and diverse delays in both states and output and further, to propose a cascade observer in dealing with arbitrarily large delayed measurements. By choosing a suitable Lyapunov-Krasovskii functional, the sufficient condition is presented that guarantees the exponential convergence of observation error to the origin. Simulation results on a numerical example and a practical case study involving inverted pendulum are finally given to illustrate the effectiveness of the proposed approach.