OBSERVER LINEARIZATION OF NONLINEAR SYSTEMS BY GENERALIZED TANSFORMATIONS

In this paper, we study the problem of observer linearization for single output dynamical systems in the presence of an output‐dependent time‐scaling transformation and a simultaneous output diffeomorphism. The approach, based on an exterior calculus approach, provides a constructive approach to the problem of equivalence of a locally observable nonlinear system to a linear observer form by means of an output dependent time‐scale transformation, an output diffeomorphism and a state‐space diffeomorphism. A generalization of existing results is obtained which allows the treatment of a larger class of locally observable nonlinear systems.     

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.     

State estimation for linear hybrid systems with periodic jumps and unknown inputs

In order to solve the state estimation problem for linear hybrid systems with periodic jumps and unknown inputs, some hybrid observers are proposed. The proposed observers admit a Luenberger‐like structure and the synthesis is given in terms of linear matrix inequalities (LMIs). Therefore, the proposed observer designs are completely constructive and provide some input‐to‐state stability properties with respect to unknown inputs. It is worth mentioning that the structure of the hybrid observers, as well as the structure of the LMIs, depends on some observability properties of the flow and jump dynamics, respectively. Then, in order to compensate the effect of the unknown inputs, a hybrid sliding‐mode observer is added to the Luenberger‐like observer structure, providing exponential convergence to zero of the state estimation error despite certain class of unknown inputs. The existence of the hybrid observers and the unknown input hybrid observer is guaranteed if and only if the hybrid system is observable and strongly observable, respectively. Some numerical examples illustrate the feasibility of the proposed estimation approach.     

匹配条件不满足时线性系统未知输入观测器设计

针对一类不满足观测器匹配条件的线性系统,讨论了未知输入观测器设计方法.首先,为了突破观测器匹配条件的限制,提出了一种与未知输入相对阶无关的辅助输出构造方法.然后,把未知输入看作系统状态的一部分,将原系统转化为一个不含未知输入的增维线性描述系统.针对这样的系统转化,对一系列等价前提条件进行了详细的讨论.之后,针对该增维线性描述系统,构造Luenberger观测器来估计原系统的状态和未知输入.同时,借助于高阶滑模微分器,来估计辅助输出中的未知信号.最后,对一个单连杆柔性机械手模型进行了数值仿真,仿真结果表明了方法的有效性.     

基于有限时间观测器的离散系统混沌同步

本文基于驱动–响应模型针对一类离散时间混沌系统提出了一种基于有限时间观测器的同步方法. 首先, 将混沌系统写成具有未知输入的线性系统形式. 随后, 给出了观测器匹配条件和强可观条件. 在观测器匹配条件的 假设下, 通过适当的状态变换, 给出了具有降维形式的有限时间观测器设计框架使得该观测器不再受到未知输入的 影响. 然后, 证明了强可观条件结合观测器匹配条件可以保证一个有限时间观测器的存在, 该观测器可以使得响应 系统达到对驱动系统的精确同步, 且达到同步所需要的时间可以任意设定, 不受观测器系统矩阵极点配置和初值条 件的影响. 最后, 给出了两个混沌系统的例子验证了所提方法的有效性.     

High gain observer based on a triangular structure

A high gain observer based on a triangular structure of nonlinear systems is proposed. An algorithm capable of calculating a gain of the observer is given. This observer synthesis is then extended to a class of multi‐output nonlinear systems which contains the model of binary distillation columns. Finally, we illustrate the performance of the estimator using numerical simulations of a methanol–ethanol distillation column. Copyright © 2002 John Wiley & Sons, Ltd.     

一类非线性系统的自适应观测器设计

在故障诊断应用中, 状态方程中的未知参数和输出方程中的未知参数分别表征执行机构故障和传感器故障, 所以研究状态方程和输出方程同时含有未知参数的自适应观测器有着实际的应用意义. 本文基于高增益观测器和自适应估计理论, 针对状态方程和输出方程同时含有未知参数的一类一致可观的非线性系统, 用构造性方法设计了一种联合估计状态和未知参数的自适应观测器. 该自适应观测器的参数估计采用时变增益矩阵, 结构形式及参数设置简单. 给出了使该自适应观测器满足全局指数收敛性的持续激励条件, 并在理论上简洁地证明了该自适应观测器的全局指数收敛性. 数值仿真结果表明该自适应观测器具有良好的快速收敛性、跟踪性等期望性能.     

基于辅助输出的线性系统状态和未知输入同时估计方法

在未知输入观测器匹配条件不满足的情况下, 针对一类线性时不变系统, 研究了同时估计系统 状态和未知输入的问题. 首先, 基于可测输出对未知输入的相关度的概念, 给出了辅助输出 的构造方法, 使得匹配条件得以满足. 为了处理辅助输出中的未知信息, 提出了一种高增益 观测器设计方法, 它不仅能估计辅助输出, 而且还能估计辅助输出的导数. 然后, 基于辅助 输出的估计值, 提出了一种降维观测器设计方法, 可以在不受未知输入影响的情况下估计系统 的状态; 接下来, 基于状态和辅助输出及其导数的估计值, 给出了未知输入估计. 最后, 对一个五 阶系统进行了数字仿真, 仿真结果表明所提出的方法是有效的.     

Estimation of states,faults and unknown disturbances in non-linear systems

In this paper, we extend existing theory on non-linear unknown input observer design to a wider class of non-linear systems where the systems are subject to unknown input and output disturbances and experience faults. The approach used is to decouple the faults and unknown disturbances from the rest of the system through a series of transformations on state and output equations. Once total disturbance decoupling is achieved, an appropriate observer for the disturbance free part of the non-linear system is proposed and designed. The designed observer is, subsequently, used for estimation of unknown inputs, outputs, and fault signals. Two examples are given for illustration.     

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.     

Output feedback quantized observer‐based synchronization of linear multi‐agent systems over jointly connected topologies

The synchronization problem of linear over‐actuated multi‐agent systems with unmeasurable states is studied in this paper, under both limited communication data rate and switching topology flows. A class of adaptive quantized observer‐based encoding–decoding schemes and a class of certainty equivalence principle‐based control protocols are proposed. By developing the graph‐based space decomposition technique and analyzing the closed‐loop quantized dynamic equations, it is shown that if the network topology flow is jointly connected, the communication channels are periodic active, and the agent dynamics is observable, and with the orthogonal system matrix, the proposed communication and control protocols can ensure the closed‐loop system to achieve synchronization exponentially fast with finite bits of information exchange per step. Copyright © 2015 John Wiley & Sons, Ltd.     

Constructive algorithm for dynamic observer error linearization via integrators: single output case

Dynamic observer error linearization which has been introduced recently is a new framework for observer design. Although this approach unifies several existing results on the problem and extends the class of systems that can be transformed into an observable linear system with an injection term of known signals, constructive algorithms to check the applicability are not available yet. In this paper, a constructive algorithm is proposed to solve the problem under some restrictions on the system structure and on the auxiliary dynamics introduced in the problem. The algorithm is constructive in the sense that the components of the transformation can be obtained step‐by‐step. Copyright © 2006 John Wiley & Sons, Ltd.     

Transformation of Nonlinear MIMO Discrete‐Time Systems into the Extended Observer Form

The paper addresses the problem of transforming discrete‐time multi‐input multi‐output nonlinear state equations into the extended observer form, which, besides the inputs and outputs, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of the extended coordinate transformation are formulated in terms of differential one‐forms, associated with the input‐output equations, corresponding to the state equations. The difference between the single‐input single‐output and multi‐input multi‐output cases is described. The applicability of the conditions is illustrated by an example.     

Nonidentifier‐based adaptive control for nonlinearly parameterized systems with measurement uncertainty

For a family of nonlinear systems with parametric uncertainty in both state and output equations, we prove that global adaptive regulation is still achievable by output feedback. The bounds of the time‐varying parameter at the system output are unknown, and the class of nonlinear systems is assumed to be dominated by a triangular system that satisfies a linear growth condition with a polynomial output‐dependent rate. The result presented in this article has incorporated and generalized recent advances on robust output feedback control of nonlinear systems with output uncertainty, all of them are required to satisfy a linear growth condition with a constant rate. A nonidentifier‐based universal controller is proposed with a high gain estimator, rather than observer, whose gain is updated in a dynamic fashion. It is shown that a single dynamic gain is sufficient for dealing with the unknown parameter at the system output and the system parametric uncertainty simultaneously.     

Recursive nonlinear observer design: beyond the uniform observability

We propose a recursive design scheme of a state observer for multiple-input-multiple-output, partly lower triangular nonlinear systems. The design begins from the subdynamics far from the output and propagates to the subdynamics close to the output, recalling the backstepping scheme for nonlinear control. The proposed class of systems is fairly general since it includes nonuniformly observable and/or detectable multioutput systems. Error convergence to zero is proved assuming boundedness of inputs a posteriori (i.e., after the design), which is preferable whereas most results in the literature assume the boundedness; a priori (i.e., before the design). A global observer is proposed with the global Lipschitz condition of the system, but without any restriction on the size of Lipschitz coefficient. The Lipschitz condition can be removed when a semiglobal observer is of interest.     

Further results on global stabilization of uncertain nonlinear systems by output feedback

For a family of uncertain nonlinear systems dominated by a triangular system that satisfies linear growth condition with an output dependent growth rate, we prove that global robust stabilization can be achieved by smooth output feedback. This conclusion has incorporated and generalized the recent output feedback stabilization results, for instance, the work (IEEE Trans. Automat. Control 2002; 47 :2068–2073) where the same conclusion was already shown to be true for planar systems, and the work (Proceedings of the 42nd IEEE, CDC, 2003; 1544–1549) where the growth rate is required to be a polynomial function of the system output. There are two key ingredients in the present contribution. One of them is the introduction of a rescaling transformation with a dynamic factor that is tuned on‐line through a Riccati‐like differential equation, which turns out to be extremely effective in dealing with the system uncertainty. The other one is the development of a recursive observer design algorithm making it possible to assign the robust observer gains in a step‐by‐step fashion. Both a smooth state feedback controller and a robust observer are explicitly constructed for the rescaled system using only the knowledge of the bounding system. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust control approach for input–output linearizable nonlinear systems using high‐gain disturbance observer

This paper addresses a robust control approach for a class of input–output linearizable nonlinear systems with uncertainties and modeling errors considered as unknown inputs. As known, the exact feedback linearization method can be applied to control input–output linearizable nonlinear systems, if all the states are available and modeling errors are negligible. The mentioned two prerequisites denote important problems in the field of classical nonlinear control. The solution approach developed in this contribution is using disturbance rejection by applying feedback of the uncertainties and modeling errors estimated by a specific high‐gain disturbance observer as unknown inputs. At the same time, the nonmeasured states can be calculated from the estimation of the transformed system states. The feasibility and conditions for the application of the approach on mechanical systems are discussed. A nonlinear multi‐input multi‐output mechanical system is taken as a simulation example to illustrate the application. The results show the robustness of the control design and plausible estimations of full‐rank disturbances.Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive output feedback regulation for a class of uncertain nonlinear systems

In this paper, the problem of global state regulation by output feedback is investigated for a class of uncertain nonlinear systems satisfying some relaxed upper‐triangular‐type condition. Using a linear dynamic gain observer with two dynamic gains and introducing two appropriate change of coordinates, we give a constructive design procedure for the linear‐like output feedback stabilizing controller. It is proved that the proposed controller globally regulates all the states of the uncertain system and maintains global boundedness of the closed‐loop system. An example is provided to demonstrate the effectiveness of the proposed design scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Reduced-order observer-based output feedback control of nonlinear time-delay systems with prescribed performance

This paper studies the problem of output feedback control for a class of nonlinear time-delay systems with prescribed performance. The system is in the form of triangular structure with unmodelled dynamics. First, we introduce a reduced-order observer to provide the estimate of the unmeasured states. Then, by setting a new condition with the performance function, we design the state transformation with prescribed performance control. By employing backstepping method, we construct the output feedback controller. It is proved that the resulting closed-loop system is asymptotically stable and both transient and steady-state performance of the output are preserved with the changing supply function idea. Finally, a simulation example is conducted to show the effectiveness of the main results.