Nonlinear State-space Systems Research Articles

Workpiece temperature control in friction stir welding of Inconel 718 through integrated numerical analysis and process control

Friction stir welding (FSW) offers significant advantages over fusion welding, particularly for high-strength alloys like Inconel 718. However, achieving optimal surface quality in Inconel 718 FSW remains challenging due to its sensitivity to temperature fluctuations during welding. This study integrates finite element simulations, statistical analysis, and advanced control methodologies to enhance weld surface quality through adequate thermal management. High-fidelity simulations of the FSW process were conducted using a validated 3D transient COMSOL Multiphysics model, producing a comprehensive dataset correlating process parameters (rotational speed, axial force, and welding speed) with workpiece temperature. This dataset facilitated statistical analysis and parameter optimization through Analysis of variance (ANOVA) method, leading to a deeper understanding of process variables. A nonlinear state-space system model was subsequently developed using experimental data and the system identification toolbox in Matlab, incorporating domain-specific insights. This model was rigorously validated with an independent dataset to ensure predictive accuracy. Utilizing the validated model, tailored control strategies, including proportional-integral-derivative (PID) and model predictive control (MPC) in both single and multivariable configurations, were designed and evaluated. These control strategies excelled in maintaining welding temperatures within optimal ranges, demonstrating robustness in response times and disturbance handling. This precision in thermal management is poised to significantly refine the FSW process, enhancing both surface integrity and microstructural uniformity. The strategic implementation of these controls is anticipated to substantially improve the quality and consistency of welding outcomes.

Nonessentiality of Reservoir's Fading Memory for Universality of Reservoir Computing.

This article describes a novel sufficient condition concerning approximations with reservoir computing (RC). Recently, RC using a physical system as the reservoir has attracted attention. Because many physical systems are modeled as state-space systems, it is necessary to guarantee the approximations given by reservoirs represented as nonlinear state-space systems. There are two problems with existing approaches: a reservoir must have a property called fading memory and must be represented as a set of maps between input and output signals on the bi-infinite-time (BIT) interval. These two conditions are too strict for reservoirs represented as nonlinear state-space systems as they require the reservoir to have a unique equilibrium state for the zero input. This article proposes an approach that employs operators from right-infinite-time (RIT) inputs to RIT outputs. Furthermore, we develop a novel extension of the Stone-Weierstrass theorem to handle discontinuous functions. To apply the extended theorem, we define functionals corresponding to operators and introduce a metric on the domain of the functionals. The resulting sufficient condition does not require the reservoir to have fading memory or continuity with respect to inputs and time. Therefore, our result guarantees the approximations with very common reservoirs and provides a rationale for physical RC. We present an example of a physical reservoir without fading memory. With the example reservoir, the RC model successfully approximates NARMA10, a benchmark task for time series predictions.

Reservoir computing for macroeconomic forecasting with mixed-frequency data

Macroeconomic forecasting has recently started embracing techniques that can deal with large-scale datasets and series with unequal release periods. Mixed-data sampling (MIDAS) and dynamic factor models (DFMs) are the two main state-of-the-art approaches to modeling series with non-homogeneous frequencies. We introduce a new framework, called the multi-frequency echo state network (MFESN), based on a relatively novel machine learning paradigm called reservoir computing. Echo state networks (ESNs) are recurrent neural networks formulated as nonlinear state-space systems with random state coefficients where only the observation map is subject to estimation. MFESNs are considerably more efficient than DFMs and can incorporate many series, as opposed to MIDAS models, which are prone to the curse of dimensionality. All methods are compared in extensive multistep forecasting exercises targeting U.S. GDP growth. We find that our MFESN models achieve superior or comparable performance over MIDAS and DFMs at a much lower computational cost.

Identification of structured nonlinear state–space models for hysteretic systems using neural network hysteresis operators

Hysteretic system behavior is ubiquitous in science and engineering fields including measurement systems and applications. In this paper, we put forth a nonlinear state–space system identification method that combines the state–space equations to capture the system dynamics with a compact and exact artificial neural network (ANN) representation of the classical Prandtl–Ishlinskii (PI) hysteresis. These ANN representations called PI hysteresis operator neurons employ recurrent ANNs with classical activation functions, and thus can be trained with classical neural network learning algorithms. The structured nonlinear state–space model class proposed in this paper, for the first time, offers a flexible interconnection of PI hysteresis operators with a linear state–space model through a linear fractional representation. This results in a comprehensive and flexible model structure. The performance is validated both on numerical simulation and on measurement data.

Variational Identification of Linearly Parameterized Nonlinear State-Space Systems

A variational Bayesian (VB) approach to the identification of linearly parameterized nonlinear state-space models (LP-NSSMs) is developed in this article. Conjugate priors over the unknown parameters are introduced, and the posterior distributions are approximated under the VB framework. In order to estimate the hidden states of the LP-NSSMs with random parameters, the augmented LP-NSSMs are established such that nonlinear smoothing methods can be applied. An extension to comprehensively tackling data anomalies, such as outliers and missing observations, is also considered to further improve the robustness of the presented method. Finally, a numerical example and the practical benchmarks are adopted to validate the efficacy of the proposed algorithm.

Sequential Monte Carlo: A Unified Review

Sequential Monte Carlo methods—also known as particle filters—offer approximate solutions to filtering problems for nonlinear state-space systems. These filtering problems are notoriously difficult to solve in general due to a lack of closed-form expressions and challenging expectation integrals. The essential idea behind particle filters is to employ Monte Carlo integration techniques in order to ameliorate both of these challenges. This article presents an intuitive introduction to the main particle filter ideas and then unifies three commonly employed particle filtering algorithms. This unified approach relies on a nonstandard presentation of the particle filter, which has the advantage of highlighting precisely where the differences between these algorithms stem from. Some relevant extensions and successful application domains of the particle filter are also presented.

Infinite-dimensional representation of nD nonlinear state space systems

Infinite-dimensional representation of nD nonlinear state space systems

Identification of Nonlinear State-Space Systems With Skewed Measurement Noises

In this paper, we consider the identification problem for nonlinear state-space models with skewed measurement noises. The generalized hyperbolic skew Student’s t (GHSkewt) distribution is employed to describe the skewed noises and formulate the hierarchical model of the considered system. A unified framework for estimating unknown states and model parameters is presented based on expectation-maximization (EM) algorithm, in which the forward filtering backward simulation with rejection sampling (RS-FFBSi) is employed to efficiently estimate the smoothing densities of the hidden states, and optimization method is adopted to update model parameters. One numerical study and the electro-mechanical positioning system (EMPS) are employed to verify the effectiveness of the developed approach.

Identification of Nonlinear State-Space Systems via Sparse Bayesian and Stein Approximation Approach

This paper is concerned with the parameter estimation of non-linear discrete-time systems from noisy state measurements in the state-space form. A novel sparse Bayesian convex optimisation algorithm is proposed for the parameter estimation and prediction. The method fully considers the approximation method, parameter prior and posterior, and adds Bayesian sparse learning and optimization for explicit modeling. Different from the previous identification methods, the main identification challenge resides in two aspects: first, a new objective function is obtained by our improved Stein approximation method in the convex optimization problem, so as to capture more information of particle approximation and convergence; second, another objective function is developed with L1-regularization, which is sparse method based on recursive least squares estimation. Compared with the previous study, the new objective function contains more information and can easily mine more important information from the raw data. Three simulation examples are given to demonstrate the proposed algorithm’s effectiveness. Furthermore, the performances of these approaches are analyzed, including parameter estimation of root mean squared error (RMSE), parameter sparsity and prediction of state and output result.

Parameter identification for Hammerstein nonlinear system with polynomial and state space model

This study investigates a two-stage parameter identification algorithm for the Hammerstein nonlinear system based on special test signals. The studied Hammerstein nonlinear system has a static nonlinear subsystem represented by polynomial basis function and a dynamic linear subsystem described by canonical observable state space model, and special test signals composed of binary signals and random signals are applied to parameter identification separation of the nonlinear subsystem and linear subsystem. The detailed identification procedures consist of two main steps. Firstly, using the characteristics that binary signals do not excite the static nonlinear subsystem, the dynamic linear subsystem parameters are identified through recursive least squares algorithm based on input-output data of binary signals. Secondly, unmeasurable state variables of the identified system are replaced with estimated values, thus the nonlinear subsystem parameters are obtained using recursive least squares algorithm with the help of input-output data of random signals. The efficiency and accuracy of proposed identification scheme are confirmed on experiment results of a numerical simulation and a practical nonlinear process, and experimental simulation results show that the developed two-stage identification algorithm has excellent predictive performance for identifying the Hammerstein nonlinear state space systems.

Maximum Likelihood recursive state estimation using the Expectation Maximization algorithm

A Maximum Likelihood recursive state estimator is derived for non-linear state–space models. The estimator iteratively combines a particle filter to generate the predicted/filtered state densities and the Expectation Maximization algorithm to compute the maximum likelihood filtered state estimate. Algorithms for maximum likelihood state filtering, prediction and smoothing are derived. The convergence properties of these algorithms, which are inherited from the Expectation Maximization algorithm and the particle filter, are examined in two examples. For nonlinear state–space systems with linear measurements and additive Gaussian noises, it is shown that the filtering and prediction algorithms reduce to gradient-free optimization in a form of a fixed-point iteration. It is also shown that, with randomized reinitialization, which is feasible because of the simplicity of the algorithm, these methods are able to converge to the Maximum Likelihood Estimate (MLE) of multimodal, truncated and skewed densities, as well as those of disjoint support.

An Algebraic Approach to Identifiability

This paper addresses the problem of identifiability of nonlinear polynomial state-space systems. Such systems have already been studied via the input-output equations, a description that, in general, requires differential algebra. The authors use a different algebraic approach, which is based on distinguishability and observability. Employing techniques from algebraic geometry such as polynomial ideals and Gröbner bases, local as well as global results are derived. The methods are illustrated on some example systems.

Adaptive control systems with linear regressor and parameter identification algorithm applied to a magnetic levitation system

Conventional regulators or controllers are used regularly in control engineering, however, there are limitations to their use. In this document we develop the essential mathematical concepts and expressions in the study of adaptive control techniques. To do this, we will initially make a brief deduction of the equations that are identified in a linear regressor control system, and also for the case of the parameter identification algorithm. Next, we will apply this control technique in a non-linear state space system for a magnetic levitation system, allowing us to conclude, among other things, that this technique makes the control more robust.

Nonlinear state-space system identification with robust laplace model

This paper investigates a robust identification solution for the nonlinear state-space model in which the outputs are polluted by unknown outliers. The problem of outliers is frequently encountered in practical industries that can greatly challenge the modelling of industrial processes. In order to overcome the obstacles brought by the outliers, the heavy-tailed Laplace distribution is applied to describe the output measurement process. Specifically, the Laplace distribution can be decomposed as a scale mixture of Gaussian distributions, which makes it robust for the outliers. The unknown model parameters are estimated with the expectation–maximisation algorithm while the particle smoother is used to solve the latent state estimation problem. The usefulness and robustness of the proposed algorithm are verified through the numerical examples including the model of a common chemical process.

Identification of nonlinear state-space time-delay system

PurposeThis paper aims to investigate an identification strategy for the nonlinear state-space model (SSM) in the presence of an unknown output time-delay. The equations to estimate the unknown model parameters and output time-delay are derived simultaneously in the proposed strategy.Design/methodology/approachThe unknown integer-valued time-delay is processed as a latent variable which is uniformly distributed in a priori known range. The estimations of the unknown time-delay and model parameters are both realized using the Expectation-Maximization (EM) algorithm, which has a good performance in dealing with latent variable issues. Moreover, the particle filter (PF) with an unknown time-delay is introduced to calculated the Q-function of the EM algorithm.FindingsAlthough amounts of effective approaches for nonlinear SSM identification have been developed in the literature, the problem of time-delay is not considered in most of them. The time-delay is commonly existed in industrial scenario and it could cause extra difficulties for industrial process modeling. The problem of unknown output time-delay is considered in this paper, and the validity of the proposed approach is demonstrated through the numerical example and a two-link manipulator system.Originality/valueThe novel approach to identify the nonlinear SSM in the presence of an unknown output time-delay with EM algorithm is put forward in this work.

Particle filter for aircraft mass estimation and uncertainty modeling

This article investigates the estimation of aircraft mass and thrust settings of departing aircraft using a recursive Bayesian method called particle filtering. The method is based on a nonlinear state-space system derived from aircraft point-mass performance models. Using only aircraft surveillance data, flight states such as position, velocity, wind speed, and air temperature are collected and used for the estimations. With the regularized Sample Importance Re-sampling particle filter, we are able to estimate the aircraft mass within 30 seconds once an aircraft is airborne. Using this short flight segment allows the assumption of constant mass and thrust setting. The segment at the start of the climb also represents the time when maximum thrust setting is most likely to occur. This study emphasizes an important aspect of the estimation problem, the observation noise modeling. Four observation noise models are proposed, which are all based on the native navigation accuracy parameters that have been obtained automatically from the surveillance data. Simulations and experiments are conducted to test the theoretical model. The results show that the particle filter is able to quantify uncertainties, as well as determine the noise limit for an accurate estimation. The method of this study is tested with a data-set consisting of 50 Cessna Citation II flights where true masses were recorded.

Constrained particle filtering for movement identification in forearm prosthesis

We formulate the problem of movement identification for the forearm prosthesis using a nonlinear state-space system and the hypothesis of muscle synergies. The synergy activation coefficients contain task-specific information and can be used to identify limb movements. In the proposed framework, the measurements are EMG data and the system state consists of muscle synergy activation coefficients, which are physiologically constrained to be nonnegative on average. Particle filters are the state-of-the-art techniques for optimal state estimation in nonlinear and non-Gaussian systems. However, the very numerical nature of the particle filters, which constitutes their strength, becomes their major weakness in handling constraints on the state. In this paper, we solve the movement identification problem by introducing a constrained particle filter termed as mean density truncation (MiND). We show that MiND minimally perturbs the unconstrained distribution of the state while simultaneously satisfying the desired constraints on the unknown state. We recorded EMG data from forearm muscles of 12 participants for identification of hand and wrist movements. The proposed particle filtering with MiND provided an accurate stream of synergy activation coefficients (p < 0.001) which were used for movement identification with error rates significantly lower (p < 0.05) than currently used heuristics and Linear Discriminate Analysis.

Relative degree of interconnected SISO nonlinear control systems

The concept of relative degree plays an important role in nonlinear control theory. It provides, for example, a necessary and sufficient condition for the existence of a feedback linearizing control law for a single-input, single-output (SISO) input-affine nonlinear state space system. It also gives a sufficient condition under which a left inverse exists. In applications it is common for systems to be composed of smaller interconnected subsystems. It is known that various feedback structures preserve relative degree, but it is largely unknown how to compute the relative degree of interconnected systems using only their properties. So the goal of this paper is to determine the relative degree of two nonlinear control systems interconnected in a variety of different ways. A collection of illustrative examples is given.

Identification of Nonlinear State-Space Systems From Heterogeneous Datasets

This paper proposes a new method to identify nonlinear state-space systems from heterogeneous datasets. The method is described in the context of identifying biochemical/gene networks (i.e., identifying both reaction dynamics and kinetic parameters) from experimental data. Simultaneous integration of various datasets has the potential to yield better performance for system identification. Data collected experimentally typically vary depending on the specific experimental setup and conditions. Typically, heterogeneous data are obtained experimentally through 1) replicate measurements from the same biological system or 2) application of different experimental conditions such as changes/perturbations in biological inductions, temperature, gene knock-out, gene over-expression, etc. We formulate here the identification problem using a Bayesian learning framework that makes use of “sparse group” priors to allow inference of the sparsest model that can explain the whole set of observed heterogeneous data. To enable scale up to large number of features, the resulting nonconvex optimization problem is relaxed to a reweighted Group Lasso problem using a convex–concave procedure. As an illustrative example of the effectiveness of our method, we use it to identify a genetic oscillator (generalized eight species repressilator). Through this example we show that our algorithm outperforms Group Lasso when the number of experiments is increased, even when each single time-series dataset is short. We additionally assess the robustness of our algorithm against noise by varying the intensity of process noise and measurement noise.

An Approach to the Design of Nonlinear State-Space Control Systems

An Approach to the Design of Nonlinear State-Space Control Systems