Estimation Of Nonlinear Systems Research Articles

A sufficient condition to design unknown input observers for nonlinear systems with arbitrary relative degree

AbstractThis article addresses the problem of state and unknown inputs (UIs) estimation for nonlinear systems with arbitrary relative degree with respect to the UIs. For this purpose, a novel nonlinear unknown input observer (UIO) is proposed, which is able to decouple the UIs by using the derivatives of the output signal. The error dynamics is attained by an exact handling and a factorization of its gradient to obtain a local polytopic representation suitable for input‐affine nonlinear systems. For that representation, a novel design condition based on convex optimization and linear matrix inequalities is proposed to exponentially stabilize the estimation error and to guarantee the validity of the proposed nonlinear UIO. Numerical simulations indicate the effectiveness of the proposed approach for different classes of nonlinear systems, for which the UIs could be totally decoupled from the state estimation.

A multi-fidelity ensemble Kalman filter with hyperreduced reduced-order models

We present an efficient data assimilation framework for nonlinear dynamical systems that uses multi-fidelity statistical estimates based on a full-order model and a projection-based hyperreduced order model (ROM) that is trained on-the-fly. This framework is particularly applicable to conservation laws in aerodynamics that yield expensive forward models. The formulation comprises the following technical components: (i) an ensemble Kalman filter to tractably handle high-dimensional, strongly nonlinear dynamical models; (ii) multi-fidelity forecast models, where ROM-based coarse fidelities are constructed on-the-fly; and (iii) hyperreduction for the ROM, based on proper orthogonal decomposition and the empirical quadrature procedure, constructed using the ensemble of full order model trajectories. We show that the multi-fidelity statistical estimates based on efficient, on-the-fly construction of the ROM enables rapid and reliable state estimation for practical nonlinear dynamical systems. We demonstrate the effectiveness of our framework to estimate the state of a separated flow around an airfoil by (i) showing that the ROM-based multi-fidelity method is more accurate than the single-fidelity method for comparable computational cost and (ii) showing that multi-fidelity statistics enable the use of a less accurate surrogate when compared against ROM-only filters, at negligible cost.

Advanced-multi-step moving horizon estimation for large-scale nonlinear systems

Nonlinear Model Predictive Control (NMPC) is an optimization-based control strategy that directly incorporates nonlinear dynamic models and has desirable stability and robustness properties. State estimation is an essential counterpart to NMPC and Moving Horizon Estimation (MHE) is also an optimization-based strategy that directly incorporates the nonlinear dynamics and constraints. However, NMPC and MHE are challenged by the computational expense of solving NLPs at each time step. For NMPC, this is avoided by advanced-step and advanced-multi-step approaches, which solve the detailed optimization off-line (possibly over multiple sampling times) and perform sensitivity-based corrections to the optimal solution on-line, with over two orders of magnitude less computation. This work complements advanced-multi-step NMPC with an advanced-multi-step MHE approach. The development solves rigorous optimization problems in background along with detailed updates to the arrival cost. On-line corrections are enabled by fast sensitivity-based NLP. The amsMHE approach is demonstrated on two large-scale distillation case studies with hundreds of state variables, and shows that nonlinear state estimation for large-scale systems can be implemented with negligible on-line computation.

Normalized-Model Reference System for Parameter Estimation of Induction Motors

This manuscript proposes a short tuning march algorithm to estimate induction motors (IM) electrical and mechanical parameters. It has two main novel proposals. First, it starts by presenting a normalized-model reference adaptive system (N-MRAS) that extends a recently proposed normalized model reference adaptive controller for parameter estimation of higher-order nonlinear systems, adding filtering. Second, it proposes persistent exciting (PE) rules for the input amplitude. This N-MRAS normalizes the information vector and identification adaptive law gains for a more straightforward tuning method, avoiding trial and error. Later, two N-MRAS designs consider estimating IM electrical and mechanical parameters. Finally, the proposed algorithm considers starting with a V/f speed control strategy, applying a persistently exciting voltage and frequency, and applying the two designed N-MRAS. Test bench experiments validate the efficacy of the proposed algorithm for a 10 HP IM.

Bayesian Cramér-Rao Lower Bounds for Prediction and Smoothing of Nonlinear TASD Systems.

The performance evaluation of state estimators for nonlinear regular systems, in which the current measurement only depends on the current state directly, has been widely studied using the Bayesian Cramér-Rao lower bound (BCRLB). However, in practice, the measurements of many nonlinear systems are two-adjacent-states dependent (TASD) directly, i.e., the current measurement depends on the current state as well as the most recent previous state directly. In this paper, we first develop the recursive BCRLBs for the prediction and smoothing of nonlinear systems with TASD measurements. A comparison between the recursive BCRLBs for TASD systems and nonlinear regular systems is provided. Then, the recursive BCRLBs for the prediction and smoothing of two special types of TASD systems, in which the original measurement noises are autocorrelated or cross-correlated with the process noises at one time step apart, are presented, respectively. Illustrative examples in radar target tracking show the effectiveness of the proposed recursive BCRLBs for the prediction and smoothing of TASD systems.

Adaptive Masreliez–Martin fractional interpolatory cubature Kalman filter with recursive noise estimation

In this study, we propose a Masreliez–Martin fractional interpolatory cubature Kalman filter and an adaptive Masreliez–Martin fractional interpolatory cubature Kalman filter to design the estimators for fractional-order discrete-time nonlinear systems under non-Gaussian measurement noise and unknown process noise covariance. Ther Masreliez–Martin fractional interpolatory cubature Kalman filter is developed by extending the interpolatory cubature Kalman filter to the fractional nonlinear discrete system to increase the robustness of estimation under noisy environment by applying the Masreliez–Martin method, including both time update and measurement update. Meanwhile, the adaptive Masreliez–Martin fractional interpolatory cubature Kalman filter is put forward to enhance the state estimation adaptive to the fractional system with uncertain process noise through recursive estimation. Simulation results on the re-entry target tracking system have demonstrated the adaptiveness, robustness and effectiveness of the propose filters.

Reachable Set Estimation and Safety Verification of Nonlinear Systems via Iterative Sums of Squares Programming

Reachable Set Estimation and Safety Verification of Nonlinear Systems via Iterative Sums of Squares Programming

Image reconstruction algorithm of electromagnetic tomography based on fractional Kalman filter

Aiming at the problem of low quality of image reconstruction of electromagnetic tomography (EMT), in this paper, an image reconstruction algorithm of EMT based on fractional Kalman filter (FKF) is proposed. Firstly, the principle of EMT and the principle of state equation of FKF are expound respectively. FKF is often used in the state estimation of nonlinear systems. There is a nonlinear relationship between the object field distribution and the sensor signal in the EMT. Therefore, according to this feature, FKF is applied to the image reconstruction algorithm of EMT. The image reconstruction process of EMT is regarded as the state estimation process of FKF, the normalized measurement voltage is taken as the observation value, and the sensitivity matrix is taken as the measurement matrix. To establish the nonlinear state estimation equation of the FKF and a priori estimation error covariance equation in the EMT, the gray value of image obtained by LBP is used as the initial value of the state estimation, a prior estimation state vector and a priori estimation error covariance matrix are obtained by prediction update, the Kalman filter gain and the posterior estimation error covariance matrix are obtained by the correction feedback process. After repeated iterations, the final state vector, i.e. reconstructed image of EMT is obtained. Finally, simulation experiments are carried out for seven different flow patterns. The results show that the image error and correlation coefficients of the reconstructed image of this algorithm are better than traditional algorithms such as LBP, Landweber, Kalman filter, and have better anti-noise effect than Kalman filter. Therefore, the image reconstruction algorithm of FKF is a new method and means to study the image reconstruction of EMT.

A practical filter error method for aerodynamic parameter estimation of aircraft in turbulence

It is common for aircraft to encounter atmospheric turbulence in flight tests. Turbulence is usually modeled as stochastic process noise in the flight dynamics equations. In this paper, parameter estimation of nonlinear dynamic system with both process and measurement noise was studied, and a practical filter error method was proposed. The linearized Kalman filter of first-order approximation was used for state estimation, in which the filter gain, along with the system parameters and the initial states, constituted the parameter vector to be estimated. The unknown parameters and measurement noise covariance were estimated alternately by a relaxation iteration method, and the sensitivities of observations to unknown parameters were calculated by finite difference approximation. Some practical aspects of the method application were discussed. The proposed filter error method was validated by the flight simulation data of a research aircraft. Then, the method was applied to the flight tests of a subscale aircraft, and the aerodynamic stability and control derivatives were estimated. All the estimation results were compared with the results of the output error method to demonstrate the effectiveness of the approach. It is shown that the filter error method is superior to the output error method for flight tests in atmospheric turbulence.

Encoding–decoding strategy based resilient state estimation for bias-corrupted stochastic nonlinear systems

This paper is concerned with the resilient state estimation problem for a type of stochastic nonlinear systems, in which the possible dynamical bias is considered that is depicted by a dynamical equation. In pursuit of enhancing the robustness of the propagated data, a binary encoding strategy (BES) is exploited in the binary symmetric channel (BSC). While the random bit errors caused by the channel noise may take place during the propagation of the binary bit string via the memoryless BSC. To characterize the occurrence of the bit errors, a series of Bernoulli distributed random variables is adopted. More specifically, in order to deal with the possible gain fluctuation of the estimator in the execution process, a resilient state estimator is employed. This paper intends to put forward a novel resilient estimation scheme under the BES, which can assure that the estimation error dynamics is exponentially ultimately bounded in mean square. A sufficient criterion is first acquired for the existence of the expected resilient estimator and the estimator parameter is achieved by solving a convex optimization problem. Finally, an illustrative simulation example is provided to verify the validity of the theoretical results.

Sampled-data estimator for nonlinear systems with uncertainties and arbitrarily fast rate of convergence

We study a class of continuous-time nonlinear systems with discrete measurements, model uncertainty, and sensor noise. We provide an estimator of the state for which the observation error enjoys a variant of the exponential input-to-state stability property with respect to the model uncertainty and sensor noise. A valuable novel feature is that the overshoot term in this stability estimate only involves a recent history of uncertainty values. Also, the rate of exponential convergence can be made arbitrarily large by reducing the supremum of the sampling intervals. Our proof uses a recently developed trajectory based approach. We illustrate our work using a model for a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation.

Fuzzy-weighted differential evolution computing paradigm for fractional order nonlinear wiener systems

The parameter estimation of fractional order nonlinear Wiener system is complex and challenging task due to the presence of significant nonlinearity at the output block, unknown fractional order, unknown parameters of the linear/nonlinear blocks, and the unmeasurable intermediate variables as well as the states. In this study, a novel design of fuzzy-weighted differential evolution is presented for parameter estimation of fractional order nonlinear Wiener (FO-NW) systems which are designed as an extension of conventional Wiener type models subjected to be fractional order system by exploiting Grunwald-Letnikov fractional derivative. The parameter estimation problem of FO-NW systems is constructed by defining a merit/error function between the true and estimated response via knacks of approximation theory in mean square error sense. Fuzzy Weighted Differential evolution algorithm is employed as an optimization mechanism to estimate the parameters of FO-NW systems with various output nonlinearities of polynomial, sinusoidal and sigmoidal kernels for low/high noisy environments in the system dynamics. Comparative studies based on rigorous statistics endorse the accurate, effective, stable and robust performance of fuzzy-weighted differential evolution algorithm.

Efficient Implementation of Continuous-Discrete Extended Kalman Filters for State and Parameter Estimation of Nonlinear Dynamic Systems

The extended Kalman filter (EKF), one of the most popular state estimators for nonlinear uncertain dynamic systems, is considered here in its continuous-discrete (CD) form. The CD version enhances the EKF's accuracy but requires robust and computationally efficient numerical integration methods. The purpose of this article is to present several new efficient integration methods suited to the implementation of the CD-EKF and to compare them to classical implementations such as the Dormand–Prince 4/5 and Euler methods. They are first applied on two proposed nonlinear parameter estimation testbenches and to the stiff Curtiss and Hirschfelder model for state and parameter estimation. Then, they are applied to the estimation of the iron losses of a high-speed permanent-magnet synchronous motor. The results show that for highly nonlinear dynamic systems or for a low to medium sampling frequency, the implicit implementation of the EKF provides a better accuracy and convergence safety. For less nonlinear functions and a higher sampling frequency, classical explicit methods obviously give similar results in a lower computation time.

Modern Koopman Theory for Dynamical Systems

The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator-theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: (1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems; (2) the approach is formulated in terms of measurements, making it ideal for leveraging big data and machine learning techniques; and (3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.

Overall hyperbolic-singular-value-decomposition-based square-root solutions in Kalman filters with deterministically sampled mean and covariance for state estimation in continuous-discrete nonlinear stochastic systems

The paper aims at square-rooting continuous-discrete Kalman filters with deterministically sampled mean and covariance. These cover all methods devised within the quadrature, cubature and unscented Kalman filtering frameworks. The main problem addressed is a potential negativity of some weights used in calculations of the sampled expectations and covariances, which precludes from an orthogonal square-rooting procedure to be designed and applied. However, square-root solutions are often requested because of their exceptional robustness to round-off and other errors committed. These also preserve the symmetry and positivity of the covariances computed in automatic mode. Here, we apply the J-orthogonal square-rooting approach devised earlier to designing our overall square-root solutions in the realm of quadrature, cubature and unscented Kalman filtering techniques, which are adjusted easily to any state estimator in use by setting its weights and deterministically selected nodes. Our J-orthogonal square-rooting procedure is grounded on the hyperbolic singular value decomposition. It leads to two algorithms covering all Itô-Taylor-approximation-based Kalman filters with deterministically sampled mean and covariance. Performances of such square-root solutions are assessed and compared to their non-square-root counterparts in a simulated flight control scenario with ill-conditioned measurements.

Overall recursive least squares and overall stochastic gradient algorithms and their convergence for feedback nonlinear controlled autoregressive systems

AbstractThis article deals with the problems of the parameter estimation for feedback nonlinear controlled autoregressive systems (i.e., feedback nonlinear equation‐error systems). The bilinear‐in‐parameter identification model is formulated to describe the feedback nonlinear system. An overall recursive least squares algorithm is developed to handle the difficulty of the bilinear‐in‐parameter. For the purpose of avoiding the heavy computational burden, an overall stochastic gradient algorithm is deduced and the forgetting factor is introduced to improve the convergence rate. Furthermore, the convergence analysis of the proposed algorithms are established by means of the stochastic process theory. The effectiveness of the proposed algorithms are illustrated by the simulation example.

Neighborhood‐based distributed robust unknown input observer for fault estimation in nonlinear networked systems

In this paper, a robust distributed fault estimation is addressed for a class of nonlinear networked systems with actuator faults where influences from process disturbances are minimized. In real-time monitoring systems, a substantial challenge is to find out the size and the shape of the occurred faults, and consequently, it is significantly more difficult in multi-agent systems. To consider this concern, in this paper, a distributed fault estimation approach has been proposed for multi-agent systems such that each agent employs an augmented system owing to a designated communication graph for estimating the fault and states both in this agent and in its neighbors. Meanwhile, each agent is equipped with an unknown input observer (UIO) to decouple the partial disturbances of the process as much as possible and to reduce the disturbances that cannot be decoupled. In the form of linear matrix inequalities (LMIs), sufficient conditions are provided to guarantee stability and to obtain the parameter matrices of the introduced observer. Finally, two numerical examples are provided to verify the performance of the introduced fault estimation scheme.

Adaptive Evolutionary Computation for Nonlinear Hammerstein Control Autoregressive Systems with Key Term Separation Principle

The knacks of evolutionary and swarm computing paradigms have been exploited to solve complex engineering and applied science problems, including parameter estimation for nonlinear systems. The population-based computational heuristics applied for parameter identification of nonlinear systems estimate the redundant parameters due to an overparameterization problem. The aim of this study was to exploit the key term separation (KTS) principle-based identification model with adaptive evolutionary computing to overcome the overparameterization issue. The parameter estimation of Hammerstein control autoregressive (HC-AR) systems was conducted through integration of the KTS idea with the global optimization efficacy of genetic algorithms (GAs). The proposed approach effectively estimated the actual parameters of the HC-AR system for noiseless as well as noisy scenarios. The simulation results verified the accuracy, convergence, and robustness of the proposed scheme. While consistent accuracy and reliability of the designed approach was validated through statistical assessments on multiple independent trials.

Novel design of weighted differential evolution for parameter estimation of Hammerstein-Wiener systems

IntroductionKnacks of evolutionary computing paradigm-based heuristics has been exploited exhaustively for system modeling and parameter estimation of complex nonlinear systems due to their legacy of reliable convergence, accurate performance, simple conceptual design ease implementation ease and wider applicability. ObjectivesThe aim of the presented study is to investigate in evolutionary heuristics of weighted differential evolution (WDE) to estimate the parameters of Hammerstein-Wiener model (HWM) along with comparative evaluation from state-of-the-art counterparts. The objective function of the HWM for controlled autoregressive systems is efficaciously formulated by approximating error in mean square sense by computing difference between true and estimated parameters. MethodsThe adjustable parameters of HWM are estimated through heuristics of WDE and genetic algorithms (GAs) for different degrees of freedom and noise levels for exhaustive, comprehensive, and robust analysis on multiple autonomous trials. ResultsComparison through sufficient large number of graphical and numerical illustrations of outcomes for single and multiple execution of WDE and GAs through different performance measuring metrics of precision, convergence and complexity proves the worth and value of the designed WDE algorithm. Statistical assessment studies further prove the efficacy of the proposed scheme. ConclusionExtensive simulation based experimentations on measure of central tendency and variance authenticate the effectiveness of the designed methodology WDE as precise, efficient, stable, and robust computing platform for system identification of HWM for controlled autoregressive scenarios.

Dynamic state estimation in nonlinear stiff systems using implicit state space models

Dynamic state estimation in nonlinear stiff systems using implicit state space models