Observer design for some classes of uniformly observable nonlinear hybrid systems

This paper deals with the synthesis of high gain observers for uniformly observable nonlinear hybrid systems. First, the single output-case is treated and then an extension is made to a particular class of multi-output hybrid systems. In effect, a high gain observer is proposed allowing the estimation of both the continuous states and the discrete location of the system. This is done by using techniques borrowed from the area of model-based fault detection whereby observers are employed for residual generation. Simulation results on a tank process are given in order to highlight the performances of the proposed observer.     

Sampled-data nonlinear observer design for chaos synchronization: A Lyapunov-based approach

This paper considers sampled-data based chaos synchronization using observers in the presence of measurement noise for a large class of chaotic systems. We study discretized model of chaotic systems which are perturbed by white noise and employ Lyapunov-like theorems to come up with a simple yet effective observer design. For the choice of observer gain, a suboptimal criterion is obtained in terms of LMI. We present semiglobal as well as global results. The proposed scheme can also be extended for discrete-time chaotic systems. Numerical simulations have been carried out to verify the effectiveness of theoretical results.     

A semi-global finite-time convergent observer for a class of nonlinear systems with bounded trajectories

This paper considers the problem of designing finite-time convergent observers for a class of lower-triangular nonlinear systems with bounded solution trajectories. Using the homogeneous domination approach, we construct an observer with homogeneous structure and saturation design, whose states will converge to the real states in a finite time by adjusting the observer gain. Several application examples of this finite-time convergent observer are discussed in this paper.     

Hybrid observer design for linear switched system via Differential Petri Nets

The aim of this paper is to propose a hybrid observer design for linear switched systems modelled either via Differential Petri Nets (DPN) or via Timed Differential Petri Nets (TDPN). The switched systems, herein, considered are characterized by switching laws that can depend on the continuous states or on both of a given dwell time and the continuous states. In addition, the structure of the proposed observers is based on a discrete observer and a continuous observer on interaction. The discrete observer reconstructs the discrete mode, by estimating both of the discrete marking and the firing vector. Once, the active mode is obtained, the continuous states are estimated. Finally, the outputs of the continuous observer are used to update the marking and the firing vector. At the end of the paper, several simulation results are presented to illustrate the proposed approach.     

Some Remarks on High Gain Observer Design

High gain observer design can be carried out for nonlinear systems, whose dynamics can be decomposed into a linear and a Lipschitz continuous nonlinear part. The observer uses a linear output injection, which is specified by a constant gain matrix. Due to this simple structure, high gain observers are frequently used in practical applications. The observer gain is often chosen via eigenvalue placement. However, the formal existence conditions are more complicated. In many cases, there is a finite bound on the maximum feasible Lipschitz constant of the nonlinear part for which the error dynamics can be stabilized. The existing results can be improved significantly, if the structure of the linear part is taken into account. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations

This paper aims to design full-order and reduced-order observers for one-sided Lipschitz nonlinear systems. The system under consideration is an extension of its known Lipschitz counterpart and possesses inherent advantages with respect to conservativeness. For such system, we first develop a novel Riccati equation approach to design a full-order observer, for which rigorous mathematical analysis is performed. Consequently, we show that the conditions under which a full-order observer exists also guarantee the existence of a reduced-order observer. A design method for the reduced-order observer that is dependent on the solution of the Riccati equation is then presented. The proposed conditions are easily and numerically tractable via standard numerical software. Furthermore, it is theoretically proven that the obtained conditions are less conservative than some existing ones in recent literature. The effectiveness of the proposed observers is illustrated via a simulative example.     

Robust state estimation and fault diagnosis for uncertain hybrid systems

Robust state estimation and fault diagnosis are challenging problems in the research into hybrid systems. In this paper a novel robust hybrid observer is proposed for a class of hybrid systems with unknown inputs and faults. Model uncertainties, disturbances and faults are represented as structured unknown inputs. The robust hybrid observer consists of a mode observer for mode identification and a continuous observer for continuous state estimation and mode transition detection. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to model uncertainties and disturbances can be guaranteed for the hybrid observer by disturbance decoupling. Furthermore, the detectability and mode identifiability conditions are rigorously analyzed. On the basis of the robust hybrid observer, a robust fault detection and isolation scheme is presented also in the paper. Simulations of a hybrid four-tank system show the proposed approach is effective.     

An observer for a class of disturbance driven nonlinear systems

An observer design for a class of nonlinear systems driven by disturbances or uncertainties is presented. The design is based on a high gain strategy, and the gain of the proposed observer is explicitly given.     

Robust state estimation and fault diagnosis for uncertain hybrid nonlinear systems

Robust state estimation and fault diagnosis are challenging problems in the research of hybrid systems. In this paper, a novel robust hybrid observer is proposed for a class of uncertain hybrid nonlinear systems with unknown mode transition functions, model uncertainties and unknown disturbances. The observer consists of a mode observer for discrete mode estimation and a continuous observer for continuous state estimation. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to unknown transition functions, model uncertainties and disturbances can be guaranteed by disturbance decoupling and selecting proper thresholds. The transition detectability and mode identifiability conditions are rigorously analyzed. Based on the robust hybrid observer, a robust fault diagnosis scheme is presented for faults modeled as discrete modes with unknown transition functions, and the analytical properties are investigated. Simulations of a hybrid three-tank system demonstrate that the proposed approach is effective.     

Constant-gain observer for a class of multi-output nonlinear systems

A new approach to design a constant-high gain exponential observer for triangular multi-output nonlinear systems is proposed. This observer is designed to systems satisfying some regularity assumptions.     

Exact differentiation and sliding mode observers for switched Lagrangian systems

The main topic of this paper is the problem of constructing observers for switched mechanical systems, which includes, as a specific case, the design of observers based on the high order sliding mode technique. The high order sliding mode is used to overcome the chattering phenomena occurring, which induce some irrelevant and undesirable phenomena for mechanical systems. The proposed approach, based on the Fliess canonical form, also allows observers to give an estimate of the discrete location of the system, which indicates the dynamic evolution. The convergence of the observers is proved and a stick–mass–friction system is used to illustrate the efficiency of the proposed hybrid observers.     

A new model-free sliding observer to synchronization problem

In this paper, a new observer is proposed for the synchronization problem, this new observer presents a simple structure that contains a sliding mode term which turns out to be robust against output noises as well as sustained disturbances, the slave system is a pure sliding-mode observer. As far as we know in the literature this class of observers have not been used in the synchronization problem. Comparisons with other two model-based observers, Thau observer and Bestle-Zeitz observer, are proposed. The performances of these observers are shown by using Lorenz system and Chua’s circuit.     

Local observer design for nonlinear systems

This paper is a geometric study of the local observer design for nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable nonlinear systems. As an application of our local observer design, we consider a class of nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, 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.     

Observer design for discrete-time nonlinear systems

This paper is a geometric study of the observer design for discrete-time nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable discrete-time nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable discrete-time nonlinear systems. As an application of our local observer design, we consider a class of discrete-time nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, 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.     

Design of observers for nonlinear systems with H ∞ performance analysis

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time‐delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H _∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.     

Analytical failure-detection using adpative,secondary observers

The problem of designing analytical failure-detection systems, using adaptive observers, is addressed in this paper. Failure-detection systems can be applied to linear multi-input, multi-output systems and are related to the examination of then-dimensional observer error vector which carries the necessary information on possible failures. This approach leads toward the design of highly sensitive failure detection systems, obtaining a unique fingerprint for every possible failure (abrupt or soft). In order to keep the observer's false-alarm rate under a certain specified value, it is necessary to have an acceptable matching between the observer model and the system parameters. It is shown here that properly designed adaptive observers are able to detect abrupt changes in the system (actuator, sensor failures, etc.) with adequate reliability. Conditions for convergence for the adaptive observer algorithm are obtained. Good tracking performance with small observer output errors, coupled with accurate and fast parameter identification in both deterministic and stochastic cases, is obtained.Dedicated to G. LeitmannThis research was supported by a National Research Council Associateship at NASA Ames Research Center. The author is indebted to both NRC and NASA Ames Research Center.     

Adjustable dimension descriptor observer based fault estimation for switched nonlinear systems with partially unknown nonlinear dynamics

This paper focuses on the fault estimation problem for switched systems with partially unknown nonlinear dynamics, actuator and sensor faults, simultaneously. The fault estimation observers are constructed, in which the observer dimension is not fixed and can be selected in a certain range. Both the disturbance decoupling and disturbance attenuation are considered, where the unknown nonlinear dynamics can be decoupled and the effect of modeling error and measurement disturbance is attenuated. Based on the average dwell time and the piecewise Lyapunov function, the observer parameter matrices can be calculated by solving LMIs and matrix equations. Finally, two examples are listed to verify the proposed fault estimation approach.     

Methods for constructing observers for linear dynamical systems under uncertainty

We consider various methods for constructing asymptotic observers for linear stationary MIMO systems under an unknown external perturbation. Square systems (in which the number of measured outputs coincides with that of unknown inputs) are considered separately. For such systems we propose a method for constructing observers that yields either an asymptotic observer (for systems of maximal relative order) or an arbitrarily accurate observer. Hyperoutput systems (in which the number of measured outputs exceeds that of unknown inputs) are considered separately as well. For such systems we propose a method for synthesizing asymptotic observers of the phase vector.     

Further Improvements in High Gain Observer Design for Systems with Structured Nonlinearities

Due to their simple implementation based on a constant gain matrix, high gain observers are very common in practical applications. We consider systems whose dynamics can be decomposed into a linear and a nonlinear part, where the nonlinear part meets some Lipschitz condition. In many cases there exists a finite bound on the maximum feasible Lipschitz constant for which the error dynamics can be stabilized. Necessary and in some sense sufficient conditions for this maximum Lipschitz constant are given in [1]. These results has been improved in [2,3] by taking the structure of the linear part into account. Having a system with one single nonlinearity, the results given in [2,3] are strict. If multiple nonlinearities occur, even this approach tends to be to conservative. In this case, one could additionally take the internal structure of the nonlinearities into account which leads to a larger set of systems for which convergence of the observer error can be guaranteed. Our new approach is based on an approximation of the structured singular value [4] which yields existence conditions in terms of linear matrix inequalities (LMIs). These LMIs may as well be used for the numerical computation of the observer gain. We demonstrate the advantage of our method on an example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient Observer Design for Multirate Sampled-Data Systems

This contribution deals with the observer design for linear multirate sampled systems. The proposed algorithm is fea-tured by a time-invariant modelling approach. It is shown that a multirate state observer can be derived by gradually building up the observer gain matrix by means of optimization based design methods in combination with a pole placement observer design. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)