A relaxed condition for stability of nonlinear observer-based controllers

We study certainty-equivalence design of observer-based controllers, and present a new stability condition that exploits state-dependent convergence properties of observers. This result eliminates the conservatism of treating the observer error as an exogeneous disturbance. As an application, we show that the reduced-order variant of the class of observers in [Automatica 37 (2001) 9231] preserves global asymptotic stability in a certainty-equivalence implementation. In another application we study a nonminimum phase system, which cannot be stabilized with the existing output-feedback designs.     

On emulated nonlinear reduced-order observers for networked control systems

We consider a general class of nonlinear reduced-order observers and show that the global asymptotic convergence of the observation error in the absence of network-induced constraints is maintained for the emulated observer semiglobally and practically (with respect to the maximum allowable transmission interval) when system measurements are sent through a communication channel. Networks governed by a Lyapunov uniformly globally asymptotically stable protocol are investigated. Our results can be used to synthesize various observers for networked control systems for a range of network configurations, as we illustrate it by considering classes of immersion and invariance observers which include the circle-criterion observers.     

On the almost sure and mean-square exponential convergence of somestochastic observers

It is pointed out that linear observers used for estimating the state of the discrete-time stochastic-parameter systems are both almost surely and mean-square (MS) exponentially convergent under the same conditions guaranteeing mean-square convergence. In addition to the mean-square convergence properties of linear observers constructed for mean-square stable stochastic-parameter systems, they also possess an almost-sure exponential convergence property, and the rate of MS convergence is exponential. This rate depends on the parameters used in the design     

Certainty-Equivalence Feedback Design With Polynomial-Type Feedbacks Which Guarantee ISS

The purpose of this note is to establish a certainty-equivalence feedback design for inverse optimally controlled affine systems. In particular, it is shown that a class of polynomial-type state feedbacks in conjunction with a globally asymptotically convergent observer leads to a globally asymptotically stable closed-loop. A key step in the proposed certainty-equivalence feedback design procedure is the identification of a new class of polynomial-type inverse optimal feedbacks which guarantees input-to-state stability (ISS) with respect to measurement errors. As a consequence, the proposed certainty-equivalence feedback design has the important feature that the state feedback is allowed to contain polynomial nonlinearities of arbitrarily high degree in the unmeasured states. This feature is illustrated on an example     

Disturbance attenuation using proportional integral observers

In this paper, we first propose a proportional integral observer design for single-output uncertain linear systems which permit us to attenuate either measurement noise or modelling errors. We show that, when only sensor noise is present in the system, an integral observer alone suffices to achieve good convergence and filtering properties. On the other hand, when modelling errors and sensor noise are present, we show that, for some classes of linear systems, the proportional integral observer allows us to decouple completely the modelling uncertainties while keeping satisfactory convergence properties. A comparison of the classical proportional observer to the proposed observers are given via academic simulation examples. We next extend the design to the class of single-output uniformly observable non-linear systems. We show through a practical simulation example, dealing with a flexible joint robot, that the non-linear proportional integral has very satisfactory disturbance attenuating properties.     

Sliding mode observers for nonlinear models with unbounded noise and measurement uncertainties

This work extends the applicability of variable structure observers designed for nonlinear systems in two ways. First, it is proved that these observers using a boundary-layer scheme can be applied to system models described by Ito differential equations, resulting in almost sure and mean square exponential estimation error. Second, the use of variable structure observers is extended to nonlinear measurement models containing disturbance effects. Also, a novel approach for obtaining the required parameters in the observer design is provided. Finally, two examples are given to illustrate the application and favorable convergence properties of these generalizations.     

Closed loop observers bundle for uncertain biotechnological models

We propose nonlinear observers for a class of biotechnological processes. These observers are an extension of the open loop asymptotic observers (observers with unknown inputs) devoted to biotechnological systems for which some parts of the model are unknown. We take benefit of the additional outputs which are (nonlinear) functions of the state to design a closed loop observer. The global convergence of these nonlinear observers is proven. We use these observers to design interval based observers which predict guaranteed intervals in which the state is lying. We run simultaneously a broad set of interval observers and we select the best ones. The method is illustrated with a model describing the bioconversion of a substrate using micro-organisms in a bioreactor.     

Redesign of adaptive observers for improved parameter identification in nonlinear systems

We propose a method for redesigning adaptive observers for nonlinear systems. The redesign uses an adaptive law that is based on delayed observers. This increases the computational burden, but gives significantly better parameter identification and robustness properties. In particular, given that a special persistency of excitation condition is satisfied, we prove uniform global asymptotic stability and semi-global exponential stability of the origin of the state and parameter estimation error, and give explicit lower bounds on the convergence rate of both the state and parameter estimation error dynamics. For initial conditions with a known upper bound, we prove tunable exponential convergence rate. To illustrate the use of the proposed method, we apply it to estimate the unmeasured flow rate and the uncertain friction parameters in a model of a managed pressure drilling system. The simulation results clearly show the improved performance of the redesigned adaptive observer compared to a traditional design.     

矩阵分解的多变量模型参考自适应控制

研究了基于高频增益矩阵分解的多变量模型参考自适应控制 .为此首先基于矩阵分解构造出新的参数化方程 ,选取必然等价自适应控制律 .然后 ,引入规范化信号和类Lyapunov函数 ,获得了规范化自适应律 .并进一步给出了虚拟规范化信号的有用性质 .最后 ,严格地给出了闭环系统的稳定性和收敛性分析     

Observers for Lipschitz non-linear systems

In an interesting paper R. Rajamani and Y. M. Cho have proposed a systematic methodology to design observers. They have introduced a new problem: relation between distance to unobservability and convergence of 'Luenberger-like' observers. A result for the convergence of the observer has also been given. They have presented a quantity denoted by i, claimed as the distance to unobservability. We show that this number is not actually the distance from the system to the set of unobservable systems. Moreover, the result for the observer is incorrect. We provide a counterexample for the result of convergence. In this paper results for convergence are obtained, with additional strengthened hypothesis, to correct Rajamani and Cho's result. The results are used to design an observer for a, now classical, single-link flexible robot joint. The behaviour of the observer to noisy output is quite satisfactory.     

Gradient-Like Observers for Invariant Dynamics on a Lie Group

This paper proposes a design methodology for non-linear state observers for invariant kinematic systems posed on finite dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialized to systems on Lie groups. For invariant systems this leads to a general factorization theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach.      

Adaptive observers for nonlinearly parameterized class of nonlinear systems

In this paper, one proposes adaptive observers for a class of uniformly observable MIMO nonlinear systems with general nonlinear parameterizations. The state and the unknown parameters of the considered systems are supposed to lie in bounded domains which size can be arbitrarily large and the exponential convergence of the observers is shown to result under a well-defined persistent excitation condition. Moreover, the gain of the observers involves a design function that has to satisfy a simple condition which is given. Different expressions of such a function are proposed and it is shown that adaptive high gain like observers and adaptive sliding mode like observers can be derived by considering particular expressions of the design function. The theory is supported by simulation results related to the estimation of the biomass concentration and the Contois model parameters in a bioreactor.     

Design of output feedback controllers for sampled-data fuzzy systems

In this paper, the design problem of output feedback controllers for sampled-data fuzzy models is considered. We consider the case where the premise variable coincides with the state variable. We give observers and output feedback controllers for an approximation of the original system. We then apply the observers and output feedback controllers to the original system and give sufficient conditions for the asymptotic convergence and stability respectively. A design example is given to illustrate the theory.     

TCSC控制的一种新自适应Backstepping方法

针对带有TCSC（Thyristor Controlled Series Compensator）的单机无穷大电力系统,提出一种新的TCSC自适应非线性控制Backstepping方法．该方法不仅保留了系统的非线性特性和对未知参数的实时在线估计,而且突破了经典的确定性等价性原理来设计参数估计器和动态反馈控制器．仿真结果表明,与基于传统自适应Backstepping得到的控制设计相比较,这种新方法在系统响应和自适应速度方面具有更优越的性能,从而为工程应用提供了一种有效的选择．     

Fault reconstruction for Takagi–Sugeno fuzzy systems via learning observers

This paper addresses the problem of observer-based fault reconstruction for Takagi–Sugeno fuzzy systems. Two types of fuzzy learning observers are constructed to achieve simultaneous reconstruction of system states and actuator faults. Stability and convergence of the proposed observers are proved using Lyapunov stability theory, and necessary conditions for the existence of the observers are further discussed. The design of fuzzy learning observers can be formulated in terms of a series of linear matrix inequalities that can be conveniently solved using convex optimisation technique. A single-link flexible manipulator is employed to verify the effectiveness of the proposed fault-reconstructing approaches.     

Design of first- and second-order sliding mode observers for induction motors using a stator-flux model

We extend current research in the area of ‘sensorless’ control of induction motors by presenting two observers based on first- and second-order sliding mode control theories for the simultaneous estimation of flux and speed. We base the observers on the stator-flux model of the motor instead of the usual rotor-flux model mainly because of the uncertain rotor resistance that plays a significant role in the latter. By designing the observers as if they are sliding mode controllers, we lend the properties of parameter insensitive closed-loop dynamics and finite time convergence to the stator flux and speed estimation schemes. We also present simulation and experimental results to validate the operation of the observers.     

Global observability and detectability analysis of uncertain reaction systems and observer design

Observation issues are of fundamental importance for reaction systems due to the limited availability of on-line sensors and the uncertainties related to the mathematical model. The objective of this work is to propose a methodology to make a global analysis of observability and detectability of such systems, with a particular concern about the design of robust observers. For this the uncertainties of the reaction system are modelled as arbitrary unknown inputs. These results include and generalize the standard asymptotic observers, well-known in the biotechnology. Moreover, the observability properties required for the existence of such observers and for the assignability of their convergence dynamics are clarified.     

非线性系统大范围渐近稳定状态观测器设计

提出了一类化工过程中非线性系统大范围渐近稳定状态观测器的构造方法。通过配置极点选择时变状态反馈增益阵保证观测器的性能,使观测器能在大范围内稳定工作。其稳定性可由Lyaptmov理论得到证明,存在模型误差和噪声干扰时也有较好的鲁棒性。从工程应用的角度,提出了分区域配置矩阵的策略,大大减少了在线计算量。同时改进了算法,保证能在应用中正确离线计算。在CSPR仿真器上的应用结果证明了该方法的良好性能。     

Cooperative observers and regulators for discrete‐time multiagent systems

This paper investigates the design of distributed observers for agents with identical linear discrete‐time state‐space dynamics networked on a directed graph interaction topology. The digraph is assumed to have fixed topology and contain a spanning tree. Cooperative observer design guaranteeing convergence of the estimates of all agents to their actual states is proposed. The notion of convergence region for distributed observers on graphs is introduced. It is shown that the proposed cooperative observer design has a robustness property. Application of cooperative observers is made to the synchronization problem. A command trajectory generator and pinning control are employed for synchronizing all the agents to a desired trajectory. Complete knowledge about the agent's state is not assumed. A duality principle is shown for observers and state feedback for distributed discrete‐time systems on graph topologies. Three different observer/controller architectures are proposed for dynamic output feedback regulator design, and they are shown to guarantee convergence of the estimate to the true state and synchronization of all the agents' states to the command state trajectory. This provides design methods for cooperative regulators based on a separation principle. It is shown that the observer convergence region and feedback control synchronizing region for discrete‐time systems are inherently bounded, so that the conditions for observer convergence and state synchronization are stricter than the results for the continuous‐time counterparts. This is in part remedied by using weighting of different feedback coupling gains for every agent. Copyright © 2012 John Wiley & Sons, Ltd.     

Preserving order observers for nonlinear systems

Preserving Order Observers provide an estimation that is always above or below the true variable, and in the absence of uncertainties/perturbations, the estimation converges asymptotically to the true value of the variable. In this paper, we propose a novel methodology to design preserving order observers for a class of nonlinear systems in the nominal case or when perturbations/uncertainties are present. This objective is achieved by combining two important systemic properties: dissipativity and cooperativity. Dissipativity is used to guarantee the convergence of the estimation error dynamics, whereas cooperativity of the error dynamics assures the order‐preserving properties of the observer. The use of dissipativity for observer design offers a big flexibility in the class of nonlinearities that can be considered while keeping the design simple: it leads in many situations to the solution of a linear matrix inequality (LMI). Cooperativity of the observer leads to an LMI. When both properties are considered simultaneously, the design of the observer can be reduced, in most cases, to the solution of both a bilinear matrix inequality and an LMI. Because a couple of preserving order observers, one above and one below, provide an interval observer, the proposed methodology unifies several interval observers design methods. The design methodology has been validated experimentally in a three‐tanks system, and it has also been tested numerically and compared with an example from the literature.Copyright © 2013 John Wiley & Sons, Ltd.