Estimation Of Nonlinear Systems Research Articles

Event-triggered state and fault simultaneous estimation for nonlinear systems with time delays

This paper addresses the problem of event-triggered robust state and fault simultaneous estimation for nonlinear time-delay systems subject to actuator and sensor unknown disturbances. Based on a fault decomposition technique and some basic mathematical transformations, we obtain an augmented system where the state vector consists of the variable of the original system and the fault. Then a novel event-triggered state observer for the augmented system is proposed to robustly estimate the variable of the original system and the fault. We next established a sufficient condition for the existence of such an observer. We translated it into a linear matrix inequality (LMI), which can be effectively solved using the MATLAB LMI Control Toolbox. Finally, an illustrative example is applied to test the proposed method.

Joint state and fault estimation for nonlinear systems with missing measurements and random component faults under Round-Robin Protocol

This work studies the fault estimation problem for a class of nonlinear systems subjected to missing measurements and random component faults under the Round-Robin protocol. Considering that the communication bandwidth is not infinite in the actual projects, the Round-Robin protocol is used for data scheduling, in which sensor nodes cyclically transmit data to the controller/filter. Meanwhile, component faults are also considered which usually caused by actuator faults. It should be emphasized that the introduction of RR protocol will make the system output equation more complex, the MMs and component faults will generate more unknown parameters, which will bring new challenges during the design of the filter. In respect of the problems above, a recursive joint state and fault estimation (JSFE) algorithm is proposed to solve the difficulties caused by missing measurements, component faults, and nonlinearity. The performance of the designed JSFE filter is analyzed by using the boundedness theorem and the reasonable parameter condition is derived for this filter. Finally, an engineering example is given to verify the effectiveness and feasibility of the proposed joint state and fault estimation scheme. This study provides a new theoretical basis and reference for the JSFE of nonlinear systems with mixed faults under RR protocol.

Approximate current state observability of discrete-time nonlinear systems under cyber-attacks

Widespread use of the Internet and other means of communication has increased the risk of cyber-attacks that can affect the warranted operation of a system. To prevent damages from attacks, an assessment of its security plays an important role in providing fast and reliable solutions. This paper focuses on state estimation of nonlinear systems under attacks. We consider a plant represented by a discrete-time nonlinear system and an attacker modeled as a finite state machine. We propose a novel notion of observability, called approximate current state observability under attacks, which corresponds to the possibility of identifying the current state of the plant after a given transient despite the malicious action of an attacker that can replace the plant output symbols in the communication network infrastructure. To provide conditions for this property to hold, we resort to the use of formal methods and in particular, of symbolic models. Symbolic models provide an abstract description of purely continuous systems, where a symbolic state corresponds to an aggregate of continuous states in the original system. The approach we propose is particularly useful when dealing with cyber-security of continuous processes in that it offers a framework to deal with the heterogeneous models of the plant and the attacker. An academic example showing the applicability of the results presented is included.

Design of linear unknown input observers for sensor fault estimation in nonlinear systems

A systematic method for the design of linear unknown input observers (UIO) for sensor fault estimation in nonlinear systems is developed. Necessary and sufficient conditions for the existence of linear UIOs for nonlinear systems are derived. As long as these conditions are satisfied, we obtain explicit design formulas for the observer. The method is applied to a non-isothermal chemical reactor (CSTR) subject to a concentration sensor fault.

On derivative-free extended Kalman filtering and its Matlab-oriented square-root implementations for state estimation in continuous-discrete nonlinear stochastic systems

Recent research in nonlinear filtering and signal processing has suggested an efficient derivative-free Extended Kalman filter (EKF) designed for discrete-time stochastic systems. Such approach, however, has failed to address the estimation problem for continuous-discrete models. In this paper, we develop a novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs). Additionally, we derive their Cholesky-based square-root MDEs and SPDEs and obtain several numerically stable derivative-free EKF methods. Finally, we propose the MATLAB-oriented implementations for all continuous-discrete derivative-free EKF algorithms derived. They are easy to implement because of the built-in fashion of the MATLAB numerical integrators utilized for solving either the MDEs or SPDEs in use, which are the ordinary differential equations (ODEs). More importantly, these are accurate derivative-free EKF implementations because any built-in MATLAB ODE solver includes the discretization error control that bounds the discretization error arisen and makes the implementation methods accurate. Besides, this is done in automatic way and no extra coding is required from users. The new filters are particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, i.e. when the calculation of Jacobian matrices are either problematical or questionable. The performance of the novel filtering methods is demonstrated on several numerical tests.

Robust filtering for spacecraft attitude estimation systems with multiplicative noises, unknown measurement disturbances and correlated noises

Unknown measurement disturbances and correlated noise caused by jitter or vibration phenomena during spacecraft operation and the complex external environment are important topics for spacecraft attitude estimation. This paper studies the problem of spacecraft attitude estimation for nonlinear systems with multiplicative noises, unknown measurement disturbances and correlated noises. In addition, a two-step prediction framework is used to complete the noise decoupling. This paper aims to design a robust filter that minimizes the upper bound of the prediction error covariance and estimation error covariance in the presence of multiplicative noises, correlated noises and unknown measurement disturbances. The prediction gain and filtering gain of the robust filter are designed to minimize the upper bound. Finally, the simulation results verify the effectiveness of the proposed filter.

Sliding mode cubic observer for state estimation of linear and Lipschitz nonlinear systems

This paper proposes a composite state observer structure that combines the sliding mode and cubic observers for state estimation of linear time-invariant and nonlinear Lipschitz systems. The proposed observer generalises the aforementioned ones and can be reduced by the proper assignment of parameters. Convergence criteria and performance advantages are given for the proposed observer structure. The paper also provides a simple structural parameter selection framework for the design of the sliding mode and sliding mode cubic observers. Simulation examples show the superiority of the proposed observer to the existing ones.

Data-driven modeling and parameter estimation of nonlinear systems

Nonlinear systems play a significant role in numerous scientific and engineering disciplines, and comprehending their behavior is crucial for the development of effective control and prediction strategies. This paper introduces a novel data-driven approach for accurately modeling and estimating parameters of nonlinear systems utilizing trust region optimization. The proposed method is applied to three well-known systems: the Van der Pol oscillator, the Damped oscillator, and the Lorenz system, which find broad applications in engineering, physics, and biology. The results demonstrate the efficacy of the approach in accurately identifying the parameters of these nonlinear systems, enabling a reliable characterization of their behavior. Particularly in chaotic systems like the Lorenz system, capturing the dynamics on the attractor proves to be crucial. Overall, this article presents a robust data-driven approach for parameter estimation in nonlinear dynamical systems, holding promising potential for real-world applications.Graphic

On the design of an unknown input observer to fault detection, isolation, and estimation for uncertain multi-delay nonlinear systems

In this article, an unknown input observer (UIO) is proposed for fault detection and isolation (FDI) for uncertain nonlinear time-delay systems with multiple discrete delays. The presented UIO can be able to isolate both actuator and sensor faults and estimate them to identify the shape, time of occurrence, amplitude, and other features of the faults. Furthermore, the proposed FDI method is robust to both input/state and output disturbances. At first, the system dynamics and faults are augmented and then UIO is driven. A delay-independent Lyapunov–Krasovskii functional is utilized to guarantee the stability of the proposed UIO-based FDI method. The terms of stability and UIO gains are obtained by solving some feasible linear matrix inequalities (LMI). The proposed method is simulated on two interconnected well-mixed non-isothermal continuous stirred tank reactors (CSTRs) with recycling where there are three parallel, irreversible elemental exothermic reactions. Two point-wise time delays in the output temperature and concentration in both CSTRs are considered. Furthermore, parametric uncertainties in the modeling are investigated. The simulation and numerical results show the superiority of the proposed UIO for fault detection, isolation, and estimation in the presence of multiple discrete time delays, model uncertainty, and input/state and output disturbances.

FE-aided Kalman Filter for nonlinear state estimation with unknown input

A Kalman Filter-Finite Element (KF-FE) framework for joint input-state estimation of nonlinear systems is proposed in the current study. The KF-FE framework has been developed aiming to estimate the nonlinear structural behavior induced by excessive loading such as earthquake-induced ground motions or strong waves, during which a limited number of responses can be measured. The proposed framework involves the use of a Bayesian recursive filter, namely the Kalman Filter, in conjunction with a nonlinear Finite Element (FE) model, which allows to update the state space model formulation at each time step. The proposed KF-FE framework builds its novelty for nonlinear systems on the basis of existing methods for online state estimation of linear systems. The application of such a framework can be suitable for nonlinear state estimation when only output measurements are available. A numerical testbed is adopted herein consisting of a 2D steel moment resisting frame structure subjected to earthquake ground motions. The KF-FE results from the nonlinear numerical analysis scheme indicate estimation performance of high accuracy for both global (e.g., nodal displacements and rotations) and local (moment-curvature relationships) demand parameters while the computational cost is kept low.

Deception attacks on event-triggered distributed consensus estimation for nonlinear systems

This paper considers the security problem of distributed consensus estimation for nonlinear systems subject to deception attacks with enough/limited energy. Depending on whether the communication resources of the attacker are limited, two different attack strategies are introduced under the distributed consensus estimation framework. Then an event-triggered distributed consensus extended Kalman filter is devised for nonlinear systems, where the upper bound for the estimation error covariance matrix is derived in terms of the variance-constrained method. The stability analysis is executed to establish sufficient conditions that guarantee the boundedness of the estimation error. Finally, the results of a numerical simulation are analyzed to show the estimation performances of the proposed distributed consensus algorithm.

Parameter Estimation of Linear and Nonlinear Systems Connected in Parallel

Presently, the problem of parameter estimation is addressed for a more general nonlinear model. In this respect, the proposed nonlinear system is composed of the parallel connection of linear and nonlinear blocks. The interconnected structure of linear and nonlinear blocks systems leads to a highly nonlinear problem involving several unknown parameters and inaccessible internal signals. In this parameter estimation method, the linear and nonlinear block parameters are estimated in one stage by using simple a sine signal or multi-cosine wave.

Sequential Fusion Filter for State Estimation of Nonlinear Multi-Sensor Systems with Cross-Correlated Noise and Packet Dropout Compensation

This paper is concerned with the problem of state estimation for nonlinear multi-sensor systems with cross-correlated noise and packet loss compensation. In this case, the cross-correlated noise is modeled by the synchronous correlation of the observation noise of each sensor, and the observation noise of each sensor is correlated with the process noise at the previous moment. Meanwhile, in the process of state estimation, since the measurement data may be transmitted in an unreliable network, data packet dropout will inevitably occur, leading to a reduction in estimation accuracy. To address this undesirable situation, this paper proposes a state estimation method for nonlinear multi-sensor systems with cross-correlated noise and packet dropout compensation based on a sequential fusion framework. Firstly, a prediction compensation mechanism and a strategy based on observation noise estimation are used to update the measurement data while avoiding the noise decorrelation step. Secondly, a design step for a sequential fusion state estimation filter is derived based on an innovation analysis method. Then, a numerical implementation of the sequential fusion state estimator is given based on the third-degree spherical-radial cubature rule. Finally, the univariate nonstationary growth model (UNGM) is combined with simulation to verify the effectiveness and feasibility of the proposed algorithm.

Estimation and stochastic control of nonlinear dynamic systems over the AWGN channel: Application in tele‐presence and tele‐operation of autonomous vehicles

AbstractThis paper is concerned with tele‐presence and tele‐operation of autonomous vehicles over the Additive White Gaussian Noise (AWGN) channel. This wireless communication channel is subject to transmission noise and transmission power constraint. An encoder, decoder and controller are proposed by implementing a novel linearization method for linearizing the nonlinear dynamic systems at operating points. For the linearized systems, the encoder, decoder and controller are implemented that were previously proposed for output tracking as well as stability of linear dynamic systems. Two novel linearization methods are proposed and their satisfactory performances are proved for output tracking as well as stability of nonlinear dynamic system. The first method is based on the fixed linearization rate and the second one is based on the variable (optimal) linearization rate. The decoder that is based on the second method is in fact the extended Kalman filter with the optimal linearization rate. The performances of these two methods are compared with each other and the other proposed methods by implementing them to the problems of the tele‐presence and tele‐operation of autonomous vehicles over the AWGN channel. Their satisfactory performances are illustrated. It is illustrated that the second method has higher computational complexity with slightly better performance and is applicable to a larger class of reference trajectories.

Forward modeling and inverse estimation for nonlinear filtering

AbstractThe highly accurate state estimation of nonlinear systems is needed in many different application fields. In many existing nonlinear filters, the uncompensated nonlinear approximation error and widely used numerical sampling methods both seriously affect the estimation accuracy and numerical stability. In this paper, based on the variational Bayesian framework, a novel iterative nonlinear filter consisting of forward modeling and inverse estimation is proposed. In the forward modeling stage, a linear Gaussian regression process with distributed variational parameters (VPs) is designed to fit the whole nonlinear measurement likelihood. The distributed VPs consider both first and second moments information, so as to compensate for approximation error from the whole likelihood perspective, instead of only the nonlinear function approximation, which can further improve the accuracy of Bayesian posterior update. In the linear Gaussian regression process, the original nonlinear measurement function (NMF) is just viewed as a prior and no longer directly participates in the modeling of the measurement likelihood. Hence, in the inverse estimation stage, when maximizing the evidence lower bound to update the posteriors of the state and VPs, we can obtain closed‐form solutions without numerical sampling methods to approximate the integral of NMF required in many existing methods, which can guarantee numerical stability. Moreover, to avoid the random setting of hyperparameters and to block the propagation of the accumulated error, we propose two rules, estimation and lower bound consistencies, to conduct the hyperparameters' initialization at the beginning of iteration at each sampling time and also derive an optimization method to update hyperparameters during iteration. Finally, the performance of our proposed method is demonstrated in the nonlinear orbital estimation problems.

A Fixed-Point State observer with Steffensen-Aitken accelerated convergence

In this work, we present an alternative algorithm for the state estimation of discrete-time nonlinear systems, which we have called Fixed-Point state observer with Steffensen-Aitken accelerated convergence. This algorithm decomposes the state estimation task into a set of consecutive fixed point iteration problems. In other words, it considers a system of nonlinear equations for each time instant and uses a fixed point iteration method to solve it, such that the solution given by the method is actually a state estimation of the discrete-time nonlinear system at the current time instant. To increase the convergence speed of the fixed point iteration method, we propose to incorporate the Δ2-Aitken method. Nonetheless, later we show that is possible to increase even more this speed by means of the Steffensen’s method. The main advantage of our algorithm is the lack of complex calculations, such as the Jacobian matrix and its inverse, which are necessary for similar algorithms such as the Newton observer. Therefore, our proposal has a low computational cost, is free of singularities, and is easy to implement. Furthermore, unlike conventional estimators like the Luenberger observer and the Sliding Mode observer, it does not require to calculate gains. To prove the effectiveness of the Fixed-Point state observer, we estimate the unknown states of a modified Chua chaotic attractor and compare the numerical results with those obtained by Luenberger, Sliding Mode, and Newton observers.

An improved long short‐term memory based on global optimization square root extended Kalman smoothing algorithm for collaborative state of charge and state of energy estimation of lithium‐ion batteries

SummaryState of charge and state of energy are essential performance indicators of the battery management system and the key to reflecting the remaining capacity of batteries. Aiming at the problems of low precision, long time, and strongly nonlinear system estimation of state of charge and state of energy of lithium‐ion batteries based on traditional algorithm under complex working conditions, this paper proposes a hybrid method consisting of the long short‐term memory neural network and square root extended Kalman smoothing. The long short‐term memory neural network can enhance the memory ability of the previous time data. The sliding window technology is introduced into the network to improve the correlation between the last time and the subsequent time estimation. Based on the traditional Kalman filtering algorithm, the square root and reverse smoothing algorithms are introduced to solve the risk of the negative covariance matrix and the problems of slow convergence and significant estimation deviation caused by a strongly nonlinear system. According to experiments, under the hybrid pulse power characterization working condition at 25°C, the maximum absolute errors of state of charge and state of energy are 1.779% and 1.487%, and the mean absolute errors are 0.352% and 0.894%, respectively. Under the Beijing bus dynamic stress test working condition at 25°C, the maximum absolute errors of state of charge and state of energy are 2.703% and 2.369%, and the mean absolute errors are 0.462% and 0.621%, respectively. The experimental results show that this algorithm can obtain reliable state of charge and state of energy under different complex working conditions with high accuracy, convergence, and robustness.

Multifidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos-Kriging

We propose novel methods for Conditional Value-at-Risk (CVaR) estimation for nonlinear systems under high-dimensional dependent random inputs. We develop a novel DD-GPCE-Kriging surrogate that merges dimensionally decomposed generalized polynomial chaos expansion and Kriging to accurately approximate nonlinear and nonsmooth random outputs. We use DD-GPCE-Kriging (1) for Monte Carlo simulation (MCS) and (2) within multifidelity importance sampling (MFIS). The MCS-based method samples from DD-GPCE-Kriging, which is efficient and accurate for high-dimensional dependent random inputs, yet introduces bias. Thus, we propose an MFIS-based method where DD-GPCE-Kriging determines the biasing density, from which we draw a few high-fidelity samples to provide an unbiased CVaR estimate. To accelerate the biasing density construction, we compute DD-GPCE-Kriging using a cheap-to-evaluate low-fidelity model. Numerical results for mathematical functions show that the MFIS-based method is more accurate than the MCS-based method when the output is nonsmooth. The scalability of the proposed methods and their applicability to complex engineering problems are demonstrated on a two-dimensional composite laminate with 28 (partly dependent) random inputs and a three-dimensional composite T-joint with 20 (partly dependent) random inputs. In the former, the proposed MFIS-based method achieves 104x speedup compared to standard MCS using the high-fidelity model, while accurately estimating CVaR with 1.15% error.

Novel Physics-Informed Artificial Neural Network Architectures for System and Input Identification of Structural Dynamics PDEs

Herein, two novel Physics Informed Neural Network (PINN) architectures are proposed for output-only system identification and input estimation of dynamic systems. Using merely sparse output-only measurements, the proposed PINNs architectures furnish a novel approach to input, state, and parameter estimation of linear and nonlinear systems with multiple degrees of freedom. These architectures are comprised of parallel and sequential PINNs that act upon a set of ordinary differential equations (ODEs) obtained from spatial discretization of the partial differential equation (PDE). The performance of this framework for dynamic system identification and input estimation was ascertained by extensive numerical experiments on linear and nonlinear systems. The advantage of the proposed approach, when compared with system identification, lies in its computational efficiency. When compared with traditional Artificial Neural Networks (ANNs), this approach requires substantially smaller training data and does not suffer from generalizability issues. In this regard, the states, inputs, and parameters of dynamic state-space equations of motion were estimated using simulated experiments with “noisy” data. The proposed framework for PINN showed excellent great generalizability for various types of applications. Furthermore, it was found that the proposed architectures significantly outperformed ANNs in generalizability and estimation accuracy.

Finite-time observer–based robust fault estimation for nonlinear systems

This article investigates the finite-time fault-estimation problem for a class of Lipschitz nonlinear systems with actuator faults and unknown input disturbances. Specifically, the original system is preliminarily decoupled into two reduced-order subsystems using a linear nonsingular transformation. A novel finite-time observer is proposed to estimate actuator faults in finite time. Subsequently, the proposed finite-time observer is extended to nonlinear systems with unknown input disturbances to investigate the robust fault-estimation problem. The actuator fault and unknown input disturbance are estimated simultaneously using two finite-time observations. Furthermore, a fault-estimation scheme is designed based on finite-time observers. The finite-time stability of the observers is then analysed using the Moreno–Lyapunov function method. The analytical proofs presented herein demonstrate that estimation errors can converge to a region of zero in finite time. Finally, the effectiveness of the developed finite-time observers and fault-estimation scheme was validated through simulations of a sample single-link flexible-joint robot.