Nonlinear Identification Problem Research Articles

On Local Unique Solvability for a Class of Nonlinear Identification Problems

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders.

Parametric identification of flat stochastic systems for effective connectivity characterization

Some of the mechanisms that generate neuronal signals are known at the cellular level and rest on a balance of excitatory and inhibitory interactions within and between populations of neurons. Neural mass models assume that a neuronal population can be approximated using very few state variables, generally limited to mean membrane currents, potentials, and firing rates. This article deals with nonlinear parametric identification problems in neurophysiologically based models simulating brain effective connectivity. We propose a novel approach which utilizes optimal control theory for partially flat stochastic differential systems. The optimization-based approach to effective connectivity characterization has been tested through simulation experiments and compared with the extended and unscented Kalman filters. A variety of case studies have been successfully used for connectivity parameter identification: constant functions, step functions, periodic functions and random functions.

Structural nonlinear damage identification based on the information distance of GNPAX/GARCH model and its experimental study

In the structural health monitoring (SHM) of civil engineering, most of the structural damage is nonlinear damage, such as breathing cracks and bolt looseness. Under the excitation of external loads, the time-domain response data of the structure produced by these nonlinear damages have nonlinear features. In order to solve the time-domain nonlinear damage identification problem of complex structures, this paper proposes a nonlinear damage identification method based on the information distance of GNPAX/GARCH (general expression of system identification for linear and nonlinear with polynomial approximation and exogenous inputs/generalized autoregressive conditional heteroskedasticity) model. First, an order determination method based on Bayesian optimization to select the order of the GNPAX/GARCH model was proposed, and the GNPAX/GARCH model was established for damage identification. Then, the redundant structural items of GNPAX/GARCH model were removed by the model optimization method based on the structural pruning algorithm. Finally, the information distance of the GNPAX/GARCH model conditional heteroscedasticity series between the baseline state and test state was derived, and the structural damage source locations were determined according to the information distance. A three-story frame structure experiment and a stand structure experiment were used to verify the effectiveness of the proposed method. The results show that the proposed method can effectively identify the nonlinear damages caused by the component breathing crack and joint bolt looseness, verifying its robustness to the nonlinear damage identification of the multi-story and multi-span complex structures.

Determination of the space-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

PurposeWhen modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer, i.e. thermal waves propagate at a finite velocity. To accommodate for this feature that is characteristic to heat transfer in biological bodies, the classical Fourier's law has to be modified resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the space-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from final time temperature measurements.Design/methodology/approachThe non-linear and ill-posed blood perfusion coefficient identification problem is reformulated as a non-linear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Both exact and numerically simulated noisy input data are inverted.FindingsThe reconstruction of the unknown blood perfusion coefficient for three benchmark numerical examples is illustrated and discussed to verify the proposed numerical procedure. Moreover, the proposed algorithm is tested on a physical example which consists of identifying the blood perfusion rate of a biological tissue subjected to an external source of laser irradiation. The numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueAlthough previous studies estimated the important thermo-physical blood perfusion coefficient, they neglected the wave-like nature of heat conduction present in biological tissues that are captured by the more accurate thermal-wave model of bio-heat transfer. The originalities of the present paper are to account for such a more accurate thermal-wave bio-heat model and to investigate the possibility of determining its space-dependent blood perfusion coefficient from temperature measurements at the final time.

Structural Nonlinear Damage Identification Method Based on the Kullback–Leibler Distance of Time Domain Model Residuals

Under external load excitation, damage such as breathing cracks and bolt loosening will cause structural time domain acceleration to have nonlinear features. To solve the problem of time domain nonlinear damage identification, a damage identification method based on the Kullback–Leibler (KL) distance of time domain model residuals is proposed in this paper. First, an autoregressive (AR) model order was selected using the autocorrelation function (ACF) and Akaike information criterion (AIC). Then, an AR model was obtained based on the structural acceleration response time series, and the AR model residual was extracted. Finally, the KL distance was used as a damage indicator to judge the structural damage source location. The effectiveness of the proposed method was verified by using a multi-story, multi-span stand model experiment and a simulated eight-story shear structure. The results show that the proposed structural nonlinear damage identification method can effectively distinguish the structural damage location of multi-degree-of-freedom shear structures and complex stand structures, and it is robust enough to detect environmental noise and small damage.

Nonlinear control object identification problems: Methods and approaches

In the study of nonlinear systems, the development of control descriptions in accordance with the model of the process depends on the tasks of solving the problems of identification of control objects. Control object modeling is limited to using the same methods for both linear and nonlinear types. Non-linear control is complicated due to the scope of application of object identification methods and the nature of their use algorithms. In the article, modern methodological approaches to the research of the identification system of nonlinear control objects are considered. The importance of building an interpolator based on the principle of obtaining its original description and transition to an image based on the acquired digital information about the object in the discretely determined system is explained. In the process of research, the authors pay attention to the possibility of creating a mathematical description in several variants of this method and distinguishing the optimal one. In this comparison, the linearized model of technological objects is based on maintaining the dominance feature.

Structural system identification and damage detection using adaptive hybrid Jaya and differential evolution algorithm with mutation pool strategy

Engineering applications of swarm intelligence optimization algorithms have been widely employed to identify structural systems and damages owing to their merits of simplicity, flexibility, robustness etc., while they often suffer from the defects of slow efficiency, premature convergence or even trapping into local optima in solving the inverse problem of nonlinear optimization-based parameter identification with partial noise-contaminated measurements. To deal with this issue, an adaptive hybrid Jaya and differential evolution (AHJDE) algorithm is proposed based on Jaya algorithm and differential evolution (DE) by effectively combining the advantages of both algorithms. In the proposed AHJDE, four improvements including adaptive mutation strategy, dynamic mutation and crossover operators, sampling-based resizing search space and linear resizing population size are integrated. The effectiveness of the proposed method is verified using a numerical example of 20-DOF linear system by comparing its performance with particle swarm optimization, modified artificial bee colony algorithm, clustering tree seeds algorithm, improved butterfly optimization algorithm etc. considering known mass case and unknown mass case. In addition, numerical studies on a nonlinear single degree-of-freedom system with classical Bouc-Wen hysteretic model and improved Bouc-Wen model are implemented to investigate the applicability in the field of nonlinear system identification. Finally, a series of experimental tests on a five-story steel frame structure are conducted in the laboratory to further validate the performance of the proposed approach in damage identification. Identification results demonstrate the proposed AHJDE can accurately and effectively identify the unknown system parameters and damages with limited sensors and noise-polluted responses.

Maximum likelihood-based extended Kalman filter for soft tissue modelling

Realistic modelling of human soft tissue is very important in medical applications. This paper proposes a novel method by dynamically incorporating soft tissue characterisation in the process of soft tissue modelling to increase the modelling fidelity. This method defines nonlinear tissue deformation with unknown mechanical properties as a problem of nonlinear filtering identification to dynamically identify mechanical properties and further estimate nonlinear deformation behaviour of soft tissue. It combines maximum likelihood theory, nonlinear filtering and nonlinear finite element method (NFEM) for modelling of nonlinear tissue deformation behaviour based on dynamic identification of homogeneous tissue properties. On the basis of hyperelasticity, a nonlinear state-space equation is established by discretizing tissue deformation through NFEM for dynamic filtering. A maximum likelihood algorithm is also established to dynamically identify tissue mechanical properties during the deformation process. Upon above, a maximum likelihood-based extended Kalman filter is further developed for dynamically estimating tissue nonlinear deformation based on dynamic identification of tissue mechanical properties. Simulation and experimental analyses reveal that the proposed method not only overcomes the NFEM limitation of expensive computations, but also absorbs the NFEM merit of high accuracy for modelling of homogeneous tissue deformation. Further, the proposed method also effectively identifies tissue mechanical properties during the deformation modelling process.

Inverse two-phase nonlinear Stefan and Cauchy-Stefan problems: A phase-wise approach

We develop a novel phase-wise sequential numerical approach based on the method of fundamental solutions (MFS) for inverse two-phase nonlinear Stefan and Cauchy-Stefan problems in one dimension (1D). By treating each phase independently, the inverse two-phase nonlinear Stefan problem splits into two single-phase inverse problems: an inverse nonlinear boundary identification problem and an inverse linear one-phase Stefan problem. Along with the reconstruction of boundary data, the simultaneous reconstruction of the boundary and initial data is also considered. Numerical investigations show the robustness and efficiency of the proposed method in reconstructing the data.

Structural nonlinear damage identification based on Bayesian optimization GNAR/GARCH model and its experimental study

Cracks and bolt looseness are common damages in a steel structure. Under the time-history loading, nonlinear characteristics of variable stiffness exist in cracks, and bolt looseness has complex nonlinear characteristics of stiffness and damping. To effectively solve the problem of structural nonlinear damage identification, an output-only damage identification method based on the GNAR/GARCH (general expression for linear and nonlinear autoregressive/general autoregressive conditional heteroskedasticity) model with Bayesian optimization is proposed. First, the stationarity test and continuous difference processing was carried out on the measured acceleration data, and an order determination method based on Bayesian optimization was proposed to automatically select the optimal model order. Then, a model structure optimization method based on Bayesian optimization was proposed to select the optimal model structure of a GNAR/GARCH model, and a GNAR/GARCH model was established based on the measured acceleration data for damage identification. Finally, a new damage indicator PIDI (probabilistic Itakura distance indicator) was proposed to judge the structural healthy status. The nonlinear damage experiment of three-story frame and relay tower was used to verify the effectiveness of the proposed method. In addition, in order to illustrate the efficiency of the proposed method for nonlinear damage identification, a comparative study was carried out using other two damage identification methods. The results show that the nonlinear damage identification method based on the GNAR/GARCH model with Bayesian optimization can easily identify the nonlinear damage of the frame and relay tower structure, and it can effectively identify the structure damage with single or multiple nonlinear damage sources.

An Efficient Numerical Scheme Based on Radial Basis Functions and a Hybrid Quasi-Newton Method for a Nonlinear Shape Optimization Problem

The purpose of this work is to construct a robust numerical scheme for a class of nonlinear free boundary identification problems. First, a shape optimization problem is constructed based on a least square functional. Schauder’s fixed point theorem is manipulated to show the existence solution for the state solution. The existence of an optimal solution of the optimization problem is proved. The proposed numerical scheme is based on the Radial Basis Functions method as a discretization approach, the minimization process is a hybrid Differential Evolution heuristic method and the quasi-Newton method. At the end we establish some numerical examples to show the validity of the theoretical results and robustness of the proposed scheme.

Parametric Neural Network-Based Model Free Adaptive Tracking Control Method and Its Application to AFS/DYC System.

This paper deals with adaptive nonlinear identification and trajectory tracking problem for model free nonlinear systems via parametric neural network (PNN). Firstly, a more effective PNN identifier is developed to obtain the unknown system dynamics, where a parameter error driven updating law is synthesized to ensure good identification performance in terms of accuracy and rapidity. Then, an adaptive tracking controller consisting of a feedback control term to compensate the identified nonlinearity and a sliding model control term to deal with the modeling error is established. The Lyapunov approach is synthesized to ensure the convergence characteristics of the overall closed-loop system composed of the PNN identifier and the adaptive tracking controller. Simulation results for an AFS/DYC system are presented to confirm the validity of the proposed approach.

A projective two-point gradient Kaczmarz iteration for nonlinear ill-posed problems

In this paper, we propose and analyze a fast Kaczmarz type method for solving multiple nonlinear ill-posed problems in Hilbert spaces. The method is the combination of the projective two-point gradient method and the Kaczmarz method. The key idea, in contrast to the standard two-point gradient method, is to use modified discrete backtracking search algorithm in each iteration in combination with metric projection of the step size to reduce the total number of performed steps and the computation time. Under reasonable conditions used in this work, we establish the strong convergence result of the method in the noise-free case. Moreover, we present the stability and regularity of the proposed method terminated by the discrepancy principle for the case of noisy data. Finally, some numerical experiments on a nonlinear parameter identification problem are presented, which exhibit that the effectiveness of reconstruction results and the acceleration effect of the method.

Parameter identification of Bouc–Wen type models using a modified experience-based learning algorithm

The experience-based learning (EBL) algorithm is a new global optimization algorithm that is free from any algorithm-specific control parameters and has been applied to solve structural damage identification problems. However, similar to other metaheuristic algorithms, the EBL algorithm has its disadvantages. To obtain better searching performance, a modified EBL algorithm is proposed for solving the nonlinear hysteretic parameter identification problem with a Bouc–Wen type model. A new updating equation is introduced to improve the global optimization ability of the algorithm. Numerical studies on a single-degree-of-freedom system with or without degradation and pinching are conducted to investigate the efficiency and robustness of the proposed algorithm. A laboratory test of four precast concrete infill walls is presented and their hysteretic parameters are also successfully identified. Both numerical and laboratory results are compared with those obtained from the original EBL algorithm, a cloud model based fruit fly optimization algorithm, the Jaya algorithm, a particle swarm optimization algorithm and a squirrel search algorithm, demonstrating the superiority of the proposed method for nonlinear hysteretic parameter identification.

Determination of the time-dependent thermal grooving coefficient

Changes in morphology of a polycrystalline material may occur through interface motion under the action of a driving force. An important special case that is considered in this paper is the thermal grooving that occurs when a grain boundary intersects the flat surface of a recently solidified metal slab giving rise to the formation of a thin symmetric groove. In case the transient surface diffusion is the main forming mechanism this yields a fourth-order time-dependent partial differential equation with unknown time-dependent surface diffusivity. In order to determine it, the profile of the free grooving surface at a fixed location is recorded in time. The grooving boundaries are supported by self-adjoint boundary conditions. We provide sufficient conditions on the input data for which the resulting coefficient identification problem is proved to be well-posed. Furthermore, we develop a predictor–corrector finite-difference spline method for obtaining an accurate and stable numerical solution to the nonlinear coefficient identification problem. Numerical results illustrate the performance of the inversion of both exact and noisy data.

An adaptive technique based blood glucose control in type-1 diabetes mellitus patients.

This study proposes Grasshopper Optimization Algorithm (GOA) based type 1 diabetes mellitus system utilizing the nonlinear Bergman minimal model with proportional integral derivative (PID) controller. GOA is the optimization algorithm, which is utilized for selecting the optimized tuning parameters of the PID controller also solves the nonlinear system parameter identification problem. The novelty of the proposed study is to stabilize the glucose level in blood for type 1 diabetic patients by infusion of insulin in reduced time with optimal quantity. Without any intervention to the normal activities of patients, the supply of insulin injection and glucose monitoring is performed automatically for type 1 diabetic patients using this controller. In between the measured variable and set point, the difference is calculated by the PID controller to evaluate an error values. In realistic patient oriented conditions, the control performance evaluation, control optimization, and advanced patient modelling should be highly concentrated during the research/analysis on blood glucose control. Evaluation is performed to analyze control performances and implementation is done on Simulink/MATLAB environment. The performance analysis of the type 1 diabetes mellitus system with GOA technique is also discussed and to improve the control performance, to optimize the controller parameters. The simulation results have proved the substantial improvement in the performance of proposed algorithm with the better results achieved than the other conventional controllers such as PSO-PID and EHO-PID.

Polynomial nonlinear state space identification of an aero-engine structure

Most nonlinear identification problems often require prior knowledge or an initial assumption of the mathematical law (model structure) and data processing to estimate the nonlinear parameters present in a system, i.e. they require the functional form or depend on a proposition that the measured data obey a certain nonlinear function. However, obtaining prior knowledge or performing nonlinear characterisation can be difficult or impossible for certain identification problems due to the individualistic nature of practical nonlinearities. For example, joints between substructures of large aerospace design frequently feature complex physics at local regions of the structure, making a physically motivated identification in terms of nonlinear stiffness and damping impossible. As a result, black-box models which use no prior knowledge can be regarded as an effective method. This paper explores the pragmatism of a black-box approach based on Polynomial Nonlinear State Space (PNLSS) models to identify the nonlinear dynamics observed in a large aerospace component. As a first step, the Best Linear Approximation (BLA), noise and nonlinear distortion levels are estimated over different amplitudes of excitation using the Local Polynomial Method (LPM). Next, a linear state space model is estimated on the non-parametric BLA using the frequency domain subspace identification method. Nonlinear model terms are then constructed in the form of multivariate polynomials in the state variables while the parameters are estimated through a nonlinear optimisation routine. Further analyses were also conducted to determine the most suitable monomial degree and type required for the nonlinear identification procedure. Practical application is carried out on an Aero-Engine casing assembly with multiple joints, while model estimation and validation is achieved using measured sine-sweep and broadband data obtained from the experimental campaign.

Fault Diagnosis of Planetary Roller Screw Mechanism Based on Bird Swarm Algorithm and Support Vector Machine

Intelligent fault diagnosis of rotating machinery has been widely developed in recent years due to the improvement of computing power, but how to identify the fault states of planetary roller screw mechanism is a difficult problem in practical industrial applications. A fault diagnosis method for planetary roller screw mechanism is proposed by combining with bird swarm algorithm (BSA) and support vector machine (SVM), which shows strong advantages in solving small sample, nonlinear and high-dimensional identification problems, and the bird swarm algorithm with high optimization accuracy and good robustness. In this paper, the vibration data of the planetary roller screw mechanism in two states with and without grease are collected, and features are extracted from the time domain, frequency domain and time- frequency domain, respectively. The predicted accuracy of SVM and BSA-SVM is compared, and the feasibility of the proposed method is verified.

Nonlinear identification of Bouc-Wen hysteretic parameters using improved experience-based learning algorithm

In this paper, an improved experience-based learning algorithm (EBL), termed as IEBL, is proposed to solve the nonlinear hysteretic parameter identification problem with Bouc-Wen model. A quasi-opposition-based learning mechanism and new updating equations are introduced to improve both the exploration and exploitation abilities of the algorithm. Numerical studies on a single-degree-of-freedom system without/with viscous damping are conducted to investigate the efficiency and robustness of the proposed algorithm. A laboratory test of seven lead-filled steel tube dampers is presented and their hysteretic parameters are also successfully identified with normalized mean square error values less than 2.97%. Both numerical and laboratory results confirm that, in comparison with EBL, CMFOA, SSA, and Jaya, the IEBL is superior in nonlinear hysteretic parameter identification in terms of convergence and accuracy even under measurement noise.

Parameter Estimation of the ALBA Autonomous Surface Craft

Arequipa region holds the largest extension of the Peruvian littoral at the Pacific sea, has also fresh water resources composed of rivers and lagoons from the coast to the Andes highland. The ALBA vehicle is a low-cost autonomous surface vessel with open source architecture that is being developed to support water monitoring tasks in the region. This article deals with the nonlinear identification problem for an autonomous surface craft and the maximum likelihood estimation approach is used to estimate its parameters. The parametric nonlinear model is considered with simulated and experimental data. The results shows good fitting values when two, three and a maximum four parameters are estimated.