Observer Design Procedure Research Articles

Symmetries of differential delay systems with applications to observer design

AbstractIn this paper we construct a general observer theory for differential delay systems based on different types of symmetries (exact symmetry or asymptotic symmetry), ending up with a certain number of semi‐global and global observers, with bounded or unbounded system's solutions. We introduce the notions of symmetry for a differential delay system, being inspired by well‐known definitions of symmetry for an ordinary or partial differential system, and variational symmetry for the associated variational differential delay system. We illustrate observer design procedures in details, by proving that the existence of a (variational asymptotic) symmetry with system's detectability in the first approximation have a central role in the design of a state observer. The symmetry is a one‐parameter group of transformations which maps the system into itself (exact symmetry) or into a different system (asymptotic symmetry), approximating the original one with better and better accuracy as the parameter of the symmetry is larger. The types of symmetries we consider here show an important contractive action on the state and input spaces for which the system's solutions are mapped into arbitrarily small neighbourhoods of the origin in which the transformed system can be well‐approximated by its linearization. The parameter of the symmetry may be constant (semiglobal observers) or updated on‐line by a state norm estimator (global observers).

A Position-Sensorless Control of Doubly Fed Induction Machine by Stator-Equation-Based Adaptive Reduced-Order Observer

A position-sensorless control is developed for doubly fed induction machines (DFIM). In this article, an adaptive reduced-order observer is used to estimate both rotor speed and position. The proposed reduced-order observer is based on the stator equation, and four issues are explored. First, the stator equation is used to realize the adaptive reduced-order observer, by which the structure is simple without flux calculation. Second, the stability of adaptive reduced-order observer is analyzed by the linearization method, which paves the way for design procedure of observer's feedback gain. Third, the tracking performance of the rotor-position estimation is investigated through the acceleration/deceleration of rotor speed; an explicit guideline is provided to design the adaptation integral gain to satisfy the specified tracking response. Fourth, sensitivity to machine's parameter variations is studied; the robust stability property is enhanced by the observer's feedback gain; and the robust sensitivity to inductance variation is strengthened by the adjustment of mutual inductance with stator voltage accordingly. Experiment is conducted with a 4-kW DFIM drive system to verify the validation of the proposed theoretical concepts.

On high-gain observer design for nonlinear systems with delayed output measurements

In this paper, we propose a high-gain observer design for nonlinear systems with time-varying delayed output measurements. Based on a recent high-gain like observer design method, called HG/LMI observer, a larger bound of the time-delay is allowed compared to that obtained by using the standard high-gain methodology. Such a HG/LMI observer adopts a lower value of the tuning parameter, which results in the reduction of the value of the observer gain, and an increase in the maximum bound of the delay required to ensure exponential convergence. Indeed, an explicit relation between the maximum bound of the delay and the observer tuning parameter is derived by using a Lyapunov–Krasovskii functional jointly with the Halanay inequality. Such a relation shows clearly the superiority of HG/LMI observer design methodology. An application to nonlinear systems with sampled measurements is provided. Furthermore, the proposed methodology is extended to systems with nonlinear outputs. This extension provides more general synthesis conditions and encompasses the linear case as a particular situation. Finally, two numerical examples are proposed to illustrate the performance of the proposed observer design procedure, and comparison to standard approaches is also provided.

Hysteresis Compensation and Nonlinear Observer Design for State-of-Charge Estimation Using a Nonlinear Double-Capacitor Li-Ion Battery Model

This article focuses on state-of-charge (SoC) estimation for a lithium-ion battery modeled using a recently developed nonlinear double-capacitor representation that has been shown to be highly accurate. The measurement equation of the model has two nonlinear functions, one of them being significant hysteresis in voltage as a function of the SoC. The hysteresis term is modeled using a physically intuitive modified Preisach representation consisting of a series of hysterons which get switched <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> or <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> to produce the hysteresis phenomenon. The proposed model for the hysteresis term is not differentiable but is Lipschitz bounded. A custom nonlinear observer is designed for this class of nonlinear systems using Lyapunov analysis to guarantee asymptotic stability. The observer design procedure consists of satisfying a linear matrix inequality on a set of vertices of a convex function. The developed observer is evaluated using experimental battery data. Experimental data is also used to determine the weights of the hysterons in the data-based modified Preisach model. The nonlinear observer performs well, yielding accurate SoC estimates in the presence of hysteresis. The nonlinear observer is also compared with the extended Kalman filter and the unscented Kalman filter, and is seen to provide superior performance for the same application due to its guaranteed global convergence properties. A broad technical contribution of this article is the development of an observer design method to handle the presence of hysteresis in nonlinear systems.

A Contribution in the Observer Design for a Class of Bilinear Singular Delayed Systems

This paper proposes a functional observer for a bilinear singular systems with constant time delay present in both state and known input vectors. The considered system has transformed to an updated form in order to estimate both a state vector and an unknown inputs functional vector independently from the considered time delay. The necessary conditions for the existence of the functional observer are formulated. The proposed observer design procedure is based on the Lyapunov–Krasovskii theory. Numerical example is given to prove the new approach.

Performance analysis of typical linear augmented observers for a class of MIMO systems with nonlinear uncertainty

To estimate the states and the uncertainties for multi-input–multi-output (MIMO) systems is significant in control engineering. The paper investigates the estimating performance of a widely-used linear augmented observer (LAO) by quantitatively analyzing the biased term of estimation error, which generalizes the previous studies. Then the necessary and sufficient condition for unbiased estimation of LAO is thoroughly discussed. More importantly, by investigating the observability of MIMO uncertain systems, the paper proves that the estimation of LAO is unbiased if and only if the states and the uncertainties of MIMO uncertain systems are observable. The presented theoretical analysis can help practitioners evaluate the feasibility of estimating the specific uncertainties via LAO in practical systems. Finally, even if the estimation of LAO is biased, the design procedure of extended state observer is presented, which offers the unbiased estimation of total disturbance and the derivatives of output.

Three-channel Synergetic State Observer for Data Transmission System with Chaotic Dynamics

The paper presents the first developed procedure of state variables three-channel observer design for a nonlinear dynamic system. The state observer allows to estimate the values of parameters that are not available for direct measurement. Here, the observer is used to build a data secure communication system with a chaotic carrier bearing signal. As a chaotic carrier generator, we use the model of the novel chaotic attractor. The data transmission system is implemented by nonlinear mixing of useful signals into parameters of the chaotic generator model on the transmitter side and by reconstructing the useful signals on the receiver side with the designed state observer. The introduction of the state observer into chaotic generator data transmission system provides the system implementation without the use of chaotic synchronization. We provide step-by-step general description for a synergetic state observer design as well as example of the chaotic carrier bearing signal observer design procedure along with closed-loop system computer simulation results.

Attack Detection and Isolation for Distributed Load Shedding Algorithm in Microgrid Systems

Load shedding is usually adopted as an emergency management to cope with large frequency deviation and supply–demand imbalance in a microgrid (MG). As generators and loads are usually highly distributed in the MG, a distributed load shedding strategy is considered in this article. In the proposed strategy, each agent first locally discovers the system's global knowledge. Efficient load shedding decisions are then made with the acquired information. Inspired by the fact that the communication channels are very vulnerable to malicious attacks, we further address the security issue of the considered strategy. It is assumed that an attacker intends to disrupt the information discovery procedure and further deteriorates the system operation by injecting malicious signals. By considering the injected signal as an external input with no prior knowledge, we first establish necessary and sufficient conditions for the misbehaviors to be observed. The design procedure of an unknown input observer is then presented, based on which a detect and isolate mechanism is further developed to distributively detect and isolate the misbehaving agent and mitigate the induced negative effects. The simulation and experimental results finally validate the effectiveness of our schemes.

High-Order Polynomial Observer Design for Robust Adaptive Synchronization of Uncertain Fractional-Order Chaotic Systems

Our focus in this paper is to present a new procedure of designing a high-order robust observer for chaos synchronization of a general class of uncertain nonlinear system with fractional-order derivative, using an adaptive strategy together with some parameter adjusting mechanisms. Some less stringent conditions for the exponential and asymptotic stability of adaptive robust control systems with fractional order are derived. A criterion for robust stability of an error system is obtained using the master–slave synchronization concept together with the Lyapunov stability theory associated with some algebraic manipulations. The high polynomial observer which can guarantee the robust stability of the closed loop system also rejects the effect of perturbations on the system dynamics within a prescribed level. The findings of this research are illustrated using computer simulations for the control problem of fractional Genesio–Tesi system. The proposed approach offers a systematic design procedure of a robust polynomial observer for the chaos synchronization of a large class of nonlinear systems.

A Data Secured Communication System Design Procedure with a Chaotic Carrier and Synergetic Observer

We have considered the problem of secure communication by means of chaotic carrier communication channels. Data protection is provided if the signal in the communication channel looks like a stochastic or noisy one, i.e., by the steganography property, and also by using advanced and secured approaches of mixing a useful signal into a chaotic carrier. The problem is to find effective and efficient techniques for useful signal reconstruction at the receiver side. We firstly use a synergetic nonlinear mathematical observer to recover the two useful signals transmitted simultaneously over a single communication channel. Compared to a known observer, the synergetic one operates with initial nonlinear models, i.e., it may be applied directly to chaotic systems. In that system structure, we consider the useful signals as unobservable variables and estimate them by the given model of a chaotic generator. A chaotic data transmission system with two channels is demonstrated; the presented mathematical procedure explains the steps of synergetic two-channel observer design. Computer simulation results prove the viability of our approach. The proposed data transmission scheme and the observer design procedure provide effective transmission and reconstruction of informational signals. Reconstruction error is up to approximately 25% but has a very short pulses shape and could be eliminated with further analog or digital filtering algorithms. As an example application, simulation of image transmission and recovery is demonstrated. The study’s results may be used as a basis for further research into secure data transmission system design.

Design of state‐dependent impulsive observer for non‐linear time‐delay systems

In this study, a new state-dependent impulsive observer (SDIO) is proposed for a class of non-linear time-delay systems. The proposed observer is based on the extended pseudo-linearisation technique that parameterises the non-linear time-delay system to a pseudo-linear structure with time delay and state-dependent coefficients. Applying this technique, the presented observer is utilised for non-linear systems with multiple, time-varying and distributed delays. Furthermore, the extended pseudo-linearisation technique simplifies the procedure of impulsive observer design for non-linear time-delay systems. The proposed SDIO is capable of continuously estimating system states using discrete samples of the system output that are available at discrete impulse times. The stability and convergence of the proposed observer are proven via a theorem utilising time-varying and delay-independent Lyapunov function and the comparison system theory of impulsive systems. It is guaranteed that the estimation error asymptotically converges to zero under well-defined and less-conservative sufficient conditions that are presented in terms of feasible linear matrix inequalities. In addition, the stability theorem specifies an upper bound on the time intervals between consecutive impulses. Results are simulated on Congo Ebola disease model which is an epidemic non-linear time-delay system. Simulation results confirm the effectiveness and performance of the proposed SDIO.

Real-time performance of DAE and ODE based estimators evaluated on a diesel engine

Computation and sampling time requirements for real-time implementation of observers is studied. A common procedure for state estimation and observer design is to have a system model in continuous time that is converted to sampled time with Euler forward method and then the observer is designed and implemented in sampled time in the real time system. When considering state estimation in real time control systems for production there are often limited computational resources. This becomes especially apparent when designing observers for stiff systems since the discretized implementation requires small step lengths to ensure stability. One way to reduce the computational burden, is to reduce the model stiffness by approximating the fast dynamics with instantaneous relations, transforming an ordinary differential equations (ODE) model into a differential algebraic equation (DAE) model. Performance and sampling frequency limitations for extended Kalman filter (EKF)’s based on both the original ODE model and the reduced DAE model are here analyzed and compared for an industrial system. Furthermore, the effect of using backward Euler instead of forward Euler when discretizing the continuous time model is also analyzed. The ideas are evaluated using measurement data from a diesel engine. The engine is equipped with throttle, exhaust gas recirculation (EGR), and variable geometry turbines (VGT) and the stiff model dynamics arise as a consequence of the throttle between two control volumes in the air intake system. The process of simplifying and modifying the stiff ODE model to a DAE model is also discussed. The analysis of the computational effort shows that even though the ODE, for each time-update, is less computationally demanding than the resulting DAE, an EKF based on the DAE model achieves better estimation performance than one based on the ODE with less computational effort. The main gain with the DAE based EKF is that it allows increased step lengths without degrading the estimation performance compared to the ODE based EKF.

A Simple Spectral Observer

The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. A spectral observer can be catalogued as an online algorithm for time-frequency analysis because is a method that can compute on the fly the Fourier Transform (FT) of a signal, without having the entire signal available from the start. In this regard, this paper presents a novel spectral observer with an adjustable constant gain for reconstructing a given signal by means of the recursive identification of the coefficients of a Fourier series. The reconstruction or estimation of a signal in the context of this work means to find the coefficients of a linear combination of sines a cosines that fits a signal such that it can be reproduced. The design procedure of the spectral observer is presented along with the following applications: (1) the reconstruction of a simple periodical signal, (2) the approximation of both a square and a triangular signal, (3) the edge detection in signals by using the Fourier coefficients, (4) the fitting of the historical Bitcoin market data from 1 December 2014 to 8 January 2018 and (5) the estimation of a input force acting upon a Duffing oscillator. To round out this paper, we present a detailed discussion about the results of the applications as well as a comparative analysis of the proposed spectral observer vis-à-vis the Short Time Fourier Transform (STFT), which is a well-known method for time-frequency analysis.

An efficient observer design method for singular discrete-time systems with time-delay and nonlinearity: LMI approach

In this paper, the observer design method for linear and nonlinear singular discrete-time systems with constant time-delays is proposed. By constructing appropriate Lyapunov–Krasovskii functional and using linear matrix inequality (LMI) technique, the asymptotic convergence criterion is developed in terms of LMIs, which can be solved numerically using MATLAB® LMI® toolbox. The sufficient condition for the existence of a full-order observer is obtained and the states are estimated using Schur complement and S-procedure lemma very well. Moreover, an extension procedure for observer design of singular linear system with time-varying delay is presented. Simulation results are included to prove the efficiency of the suggested approach.

Unknown Input Observer Design for Interval Type-2 T-S Fuzzy Systems With Immeasurable Premise Variables.

This paper deals with the problem of robust unknown input fault detection observers (UIFDOs) design for interval type-2 Takagi-Sugeno (T-S) fuzzy systems with immeasurable premise variables. It has been shown that choosing the system states, which may be immeasurable, as premise variables, will able us to model a larger class of nonlinear systems. Accordingly, the premise variables of underlying system are considered to be immeasurable. However, the design procedure of a stable observer for such systems are more challenging. Furthermore, the system is supposed be affected by time-varying delays and unknown inputs. The UIFDO is exploited so as to generate a residual signal with the most possible sensitivity to fault and the least sensitivity to exogenous signals. In this paper, this issue is investigated thoroughly, in this respect the design procedure consists of two sections: 1) measurable and 2) immeasurable premise variables. Sufficient design conditions are provided in terms of linear matrix inequalities for both cases. The effectiveness of the proposed UIFDO in detection of two different kinds of faults is illustrated during the simulation of a numerical example. Moreover, a fair comparison has been drawn between the proposed UIFDO and an existent reference to indicate that interval type-2 T-S fuzzy model is more superior than type-1. Finally, the faulty behavior of a one-link manipulator is investigated to express the applicability of the proposed method.

Fractional neural observer design for a class of nonlinear fractional chaotic systems

In this paper, a novel observer structure for nonlinear fractional-order systems is presented to estimate the states of fractional-order nonlinear chaotic system with unknown dynamical model. A new fractional error back-propagation learning algorithm is derived to adapt weights of the artificial neural network, by taking advantage of the Lyapunov stability strategy of fractional-order systems which is called Miattag–Leffler stability. The main contribution is the extension of neural observer for fractional dynamics in a way that satisfies Miattag–Leffler conditions. Observer design procedure guarantees the convergence of observer error to the neighborhood of zero. Furthermore, the robustness of the proposed estimator against uncertainties and external disturbances are the main benefits of the proposed method. Simulation results show the effectiveness and capabilities of the proposed observer.

A neural network approach to simultaneous state and actuator fault estimation under unknown input decoupling

The paper deals with the problem of a neural network-based robust state and actuator fault estimator design for non-linear discrete-time systems. It starts from a review of recent developments in the area of robust estimators and observers for non-linear discrete-time systems and proposes less restrictive procedure for designing a neural network-based H∞ observer. The proposed approach guaranties a predefined disturbance attenuation level and convergence of the observer, as well as unknown input decoupling and state and actuator fault estimation. The main advantage of the design procedure is its simplicity. The paper presents an observer design procedure that is reduced to solving a set of linear matrix inequalities. The final part of the paper presents an illustrative example concerning an application of the proposed approach to the multi-tank system benchmark.

A new methodology for an adaptive state observer design for a class of nonlinear systems with unknown parameters in unmeasured state dynamics

An adaptive state observer is an adaptive observer that does not require the persistent excitation condition to estimate the state. The usual structural requirement for designing this kind of observers is that the unknown parameters explicitly appear in the measured state dynamics. This paper deals with the problem of adaptive state observer synthesis for a class of nonlinear systems with unknown parameters in unmeasured state dynamics. The novelty of the proposed approach is that it requires neither a canonical form nor the approximation of some of the output’s time derivatives. Firstly, we establish a new matrix equality that characterizes the structure of almost all systems found in the very small literature dealing with this problem. Then, this equality is exploited in the construction of the adaptation law. This simplifies the design procedure and makes it very similar to the conventional adaptive state observer design procedure. The problem of finding the observer gains is expressed as a linear matrix inequalities optimization problem. Two examples are given to demonstrate the validity of the proposed scheme.

Extended state observer–based nonlinear cascade proportional–integral–derivative control of the nano quadrotor

The nano quadrotor is a nonlinear multi-input and multi-output system with strong coupling, which causes difficulties in control law design. In order to achieve a favorable performance, an extended state observer–based nonlinear cascade proportional–integral–derivative controller is proposed in this article. First, the nano quadrotor platform is built, and the dynamic model is established. Second, a novel and practical measuring method is given to obtain model parameters. Then, based on the active disturbance rejection control method, the design procedure of the extended state observer–based nonlinear cascade proportional–integral–derivative controller is presented. In the developed controller, a tracking differentiator is involved to extract the signals of gyroscope, and extended state observer is used to estimate the disturbance. To obtain a better performance of tracking differentiator and extended state observer, a systematic parameter-tuning method is studied. Finally, simulation results are given to demonstrate the efficiency of the proposed controller.

Functional observer design for retarded system with interval time-varying delays

ABSTRACTFunctional observers are the key solution to numerous practical estimation problems, wherein full-order observers cannot be applied. This paper studies the novel problem of minimum-order functional observer design for time-delay systems with interval time-varying state delays. Moreover, unlike the majority of the existing papers on this topic, which consider either small or unknown delay rates, the delay derivative is assumed to possess an upper bound not limited to be less than one. A new augmented Lyapunov–Krasovskii functional with triple integral terms is employed to derive less conservative delay-dependent sufficient conditions for the stability of the closed-loop observer dynamics, which are expressed in terms of linear matrix inequalities. In addition, contemporary techniques such as Wirtinger-based single- and double-integral inequalities, delay splitting scheme, reciprocally convex approach and convex combination technique, along with the descriptor transformation, are adopted in constructing the new stability criterion that is exploited for obtaining the observer parameters. A genetic-algorithm-based searching schema is proposed to optimally tune a number of weighting parameters in the observer design procedure. Numerical examples and simulation results are demonstrated to confirm the effectiveness and the superiority of the proposed observer design algorithm.