Parameter Identification Of Nonlinear Systems Research Articles

Comparison of Constrained Unscented and Cubature Kalman Filters for Nonlinear System Parameter Identification

Comparison of Constrained Unscented and Cubature Kalman Filters for Nonlinear System Parameter Identification

Design and Development of Structural Parameter Identification Software for Nonlinear Vibration Isolators

This paper developed an object-oriented software to facilitate parameter calibration of the quasi-zero stiffness (QZS) vibration isolator for engineering applications. Considering its advantages of convenient programming and high noise resistance, we employ the harmonic balance method to construct the identification core algorithm. The developed software consists of data pre-processing, parameter identification, and post-processing modules. The data pre-processing module imports and classifies all the required data. Then, the parameter identification module conducts preliminary nonlinearity analysis so that nonlinear structural parameter identification can be successfully carried out. After parameter identification, the data post-processing module displays and stores the identified results. The performance of the software is tested by a numerical example of a two-degree-of-freedom QZS vibration isolator, proving the software can give accurate identification results within a few user-friendly graphical interfaces. Based on this software, we hope that the application and promotion of nonlinear system parameter identification become more convenient.

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.

Parameter identification of nonlinear structural systems through frequency response sensitivity analysis

Nonlinearity is ubiquitously encountered in structural systems, and it may have a great and complicated influence on the dynamic behaviours, including bifurcation, internal resonance, load history dependence, etc. Identifying the nonlinear system parameters is essential for analysis and design of the structure. To this end, a new approach is developed in this paper for nonlinear system parameter identification from frequency response sensitivity analysis. At first, the harmonic balance equation is established to govern the frequency response of the nonlinear system, upon which the frequency response and sensitivity analysis can be conducted. A remarkable feature is that the harmonic balance equation is algebraic so that the sensitivity analysis, pertaining to a linearized equation, is rather simple and straightforward. Then, parameter identification is modelled as a nonlinear least-squares problem, and the sensitivity approach is adopted in conjunction with the trust-region constraint for convergent solution. Numerical examples are conducted to demonstrate the feasibility and performance of the proposed approach.

Parameter identification of nonlinear systems with time-delay from time-domain data

The present paper develops an enhanced response sensitivity approach for parameter identification of nonlinear time-delay systems from the time-domain data. Accurate and rapid identification of the time-delay system’s parameters is the key to achieve effective control and synchronization, especially for the delay parameters. The work of this paper is mainly threefold. Firstly, taking a general nonlinear time-delay system as the investigated object, the characteristics of the nonlinear time-delay system are introduced, and the response sensitivity analysis with respect to the system parameters is derived. Then, the problem of parameter identification is modeled as a nonlinear least-squares optimization function. To this end, a novel iteration-based approach is developed to solve such problem, and the ill-posed situation is tackled by Tikhonov regularization. Besides, the trust-region constraint is implemented to enhance the convergence of the algorithm. Finally, the feasibility of the proposed approach is verified by two numerical examples and a real electronic circuit experiment. The results proved that the proposed approach can identify parameters of nonlinear time-delay systems accurately, robustly and effectively.

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.

Parameter identification of nonlinear system via a dynamic frequency approach and its energy harvester application

A dynamic frequency-based parameter identification approach is applied for the nonlinear system with periodic responses. Starting from the energy equation, the presented method uses a dynamic frequency to precisely obtain the analytical limit cycle expression of nonlinear system and utilizes it as the mathematic foundation for parameter identification. Distinguished from the time-domain approaches, the strategy of using limit cycle to describe the system response is unaffected by the influence of phase change. The analytical expression is fitted with the value sets from phase coordinates measured in periodic oscillation of the nonlinear systems, and the unknown parameters are identified with the interior-reflective Newton method. Then the performance of this identification methodology is verified by an oscillator with nonlinear stiffness and damping. Besides, numerical simulations under noisy environment also verify the efficiency and robustness of the identification procedure. Finally, we apply this parameter identification method to the modeling of a large-amplitude energy harvester, to improve the accuracy of mechanical modeling. Not surprisingly, good agreement is achieved between the experimental data and identified parameters. It also verifies that the proposed approach is less time-consuming and more accuracy in identification procedure.

Parameter Identification of Nonlinear Systems with Time-delay Based on the Multi-innovation Stochastic Gradient Algorithm

This paper considers the parameter identification problem of block-oriented Hammerstein nonlinear systems with time-delay. Firstly, we adopt the data filtering technique to transform the identification model so that all the parameters will be separated in the resulting identification model which has no redundant parameters. Secondly, a multi-innovation stochastic gradient algorithm is used to estimate the system parameters. The proposed method has high computational efficiency and good accuracy. Simulation results are presented to demonstrate the effectiveness of the proposed algorithm.

Parameter identification of nonlinear system using an improved Lozi map based chaotic optimization algorithm (ILCOA)

In this paper, an efficient stochastic optimization algorithm is presented for parameter identification of nonlinear systems. Due to its robust performance, short running time and desirable potency to find local minimums the Lozi map-based chaotic optimization algorithm is an appropriate choice to estimate unknown parameters of nonlinear dynamic systems. To enhance the identification efficacy and in order to escape local minimum, a modified version of this algorithm with higher stability and better performance is rendered in this paper. An Improved Lozi map-based chaotic optimization algorithm (ILCOA) is employed to identify three nonlinear systems and the performance of the proposed algorithm is compared with other optimization algorithms. The simulation results of identification endorse the effectiveness of the proposed method.

Inverse problem of the Holling-Tanner model and its solution

In this paper we undertake to consider the inverse problem of parameter identification of nonlinear system of ordinary differential equations for a specific case of complete information about solution of the Holling-Tanner model for finite number of points for the finite time interval. In this model the equations are nonlinearly dependent on the unknown parameters. By means of the proposed transformation the obtained equations become linearly dependent on new parameters functionally dependent on the original ones. This simplification is achieved by the fact that the new set of parameters becomes dependent and the corresponding constraint between the parameters is nonlinear. If the conventional approach based on introduction of the Lagrange multiplier is used this circumstance will result in a nonlinear system of equations. A novel algorithm of the problem solution is proposed in which only one nonlinear equation instead of the system of six nonlinear equations has to be solved. Differentiation and integration methods of the problem solution are implemented and it is shown that the integration method produces more accurate results and uses less number of points on the given time interval.

EM‐based adaptive divided difference filter for nonlinear system with multiplicative parameter

SummaryThis paper is concerned with application of expectation maximization (EM) algorithm for deriving an adaptive version of divided difference filter for joint state estimation and multiplicative parameter identification of nonlinear system with the colored measurement noise. Owing to the fact that there exist a mutual coupling and interaction of state and parameter on each other, it requires a joint or simultaneous estimation of both state and parameter by a mutual iteration, and justly, EM iterates Expectation (E‐)step and Maximization (M‐)step to meet such requirement. Firstly, E‐step involves state filtering and smoothing issues under knowing the previous parameter identification results, which is well solved by resorting to the Gaussian approximation with a trade‐off between accuracy and complexity. Further, such Gaussian approximation estimators are applied for evaluating the condition expectation of complete‐data likelihood function, nonlinearly characterized by the multiplicative parameter needed to be optimized. Secondly, M‐step deals with the maximization of the condition expectation by directly making its derivative as zero to obtain the current general parameter identification equation as the nonlinear integral. Thirdly, by iteratively operating E‐step and M‐step, an adaptive divided difference filter is proposed for joint state estimation and parameter identification by using the second‐order Stirling interpolation to compute the associated nonlinear integral. Finally, the robust performance of the EM‐based adaptive version of divided difference filter to the unknown or time‐varying multiplicative parameter, as compared with the standard augmentation method, is demonstrated by a maneuvering target tracking example. Copyright © 2016 John Wiley & Sons, Ltd.

Parametric Identification of Nonlinear Vibration Systems Via Polynomial Chirplet Transform

The response of a nonlinear oscillator is characterized by its instantaneous amplitude (IA) and instantaneous frequency (IF) features, which can be significantly affected by the physical properties of the system. Accordingly, the system properties could be inferred from the IA and IF of its response if both instantaneous features can be identified accurately. To fulfill such an idea, a nonlinear system parameter identification method is proposed in this paper with the aid of polynomial chirplet transform (PCT), which has been proved a powerful tool for processing nonstationary signals. First, the PCT is used to extract the instantaneous characteristics, i.e., IA and IF, from nonlinear system responses. Second, instantaneous modal parameters estimation was adopted to extract backbone and damping curves, which characterize the inherent nonlinearities of the system. Third, the physical property parameters of the system were estimated through fitting the identified average nonlinear characteristic curves. Finally, the proposed nonlinear identification method is experimentally validated through comparing with two Hilbert transform (HT) based methods.

Hilbert Transform and Its Engineering Applications

The Hilbert transform is useful in calculating instantaneous attributes of a time series, especially the envelope amplitude and instantaneous frequency. The instantaneous envelope is the amplitude of the complex Hilbert transform; the instantaneous frequency is the time rate of change of the instantaneous phase angle. These properties can be applied to identify dynamic characteristics of a linear as well as a nonlinear system. However, the conventional discrete Hilbert transform, which is based on fast Fourier transform and inverse Fourier transform, has shown the lack of accuracy for time-derivative calculations. In this paper, we first introduce the Hilbert transform and its applications to the nonlinear system parameter identification. Then we address the practical issues in applying the Hilbert transform to engineering applications. To increase the accuracy of the envelope detection, we propose a Hilbert transform technique based on local-maxima interpolation. Analyses and simulations are carried out to demonstrate the advantages of the proposed technique. Finally, we employ the proposed local-maxima-interpolation technique in identifying the nonlinear dynamic characteristics of industrial examples.

Force State Maps Using Reproducing Kernel Particle Method and Kriging Based Functional Representations

The problem of identification of nonlinear system parameters from measured time histories of response under known excitations is considered. Solutions to this problem are obtained by using the force state mapping technique with two alternative functional representation schemes. These schemes are based on the application of reproducing kernel particle method (RKPM) and kriging techniques to fit the force state map. The RKPM has the capability to reproduce exactly polynomials of specified order at any point in a given domain. The kriging based methods represent the function under study as a random field and the parameters describing this field are optimally determined based on available observations. The present study investigates the performance of RKPM and kriging based fits to the force state maps for a variety of nonlinear dynamical systems. The study also examines the application of force state maps in (a) determining the fixed points limit cycles of the system and their stability, (b) determining the properties of the linear system which would result if nonlinearity were to be absent, and (c) dealing with nonlinearities that are continuous but not differentiable and nonlinearities that are discontinuous at a set of points within the domain of interest. Illustrative examples on single and multi-degrees of freedom nonlinear systems are presented to demonstrate the scope of the proposed procedures.

Parameter identification for nonlinear systems: Guaranteed confidence regions through LSCR

In this paper we consider the problem of constructing confidence regions for the parameters of nonlinear dynamical systems. The proposed method uses higher order statistics and extends the LSCR (leave-out sign-dominant correlation regions) algorithm for linear systems introduced in Campi and Weyer [2005, Guaranteed non-asymptotic confidence regions in system identification. Automatica 41(10), 1751–1764. Extended version available at 〈 http://www.ing.unibs.it/ ∼ campi 〉 ]. The confidence regions contain the true parameter value with a guaranteed probability for any finite number of data points. Moreover, the confidence regions shrink around the true parameter value as the number of data points increases. The usefulness of the proposed approach is illustrated on some simple examples.

Some Results on Parameter Identification of Nonlinear Systems

A priori global identifiability deals with the (theoretical) uniqueness of solutions of model parameters from noise free input–output data. This issue is particularly critical when dealing with physiological systems where a different numerical estimate can distinguish a pathological state from a normal state. Recently, differential algebra tools have been applied to study identifiability of dynamic systems described by polynomial or rational equations. These methods are based on elimination theory for algebraic differential systems, the main tool being the computation of the so-called characteristic set of a differential ideal associated to the polynomials defining the dynamic system. This characteristic set can be found by symbolic computation and provides the so-called exhaustive summary of the model. The reparametrization of the input–output relation of the system by the exhaustive summary plays a major role not only in a priori identifiability but also in parameter estimation of nonlinear models. It is in fact a linear reparametrization and can be used to derive explicit one-shot least-squares estimates of the parameters. This allows to avoid the usual bottleneck of nonlinear parameter optimization which has to be performed by iterative optimization routines which are often unreliable, e.g. they give no guarantee of converging to a true minimum, and hence require time-consuming random search in the parameter space. Our algorithm has been tested in one- and two-dimensional Michaelis–Menten model where the choice of initial values of the parameters is critical since nonlinear least-squares minimization problem shows many local minima. Our method structurally has only one minimum and does not require initial values for the unknown parameters. However, when noise in the data is present, the optimization problem is highly ill-conditioned and the solution of the problem in the general case is still under study.

Physical Parameter Identification of Non-Linear System via Pool Genetic Algorithm

In this paper, we propose a new method of estimating the physical parameters for a non-linear system with known structure by using a genetic algorithm (GA). Since a simple GA is empirically known in many cases to cause initial convergence or fall into a local minimum, we introduce a new algorithm named Pool GA and verify its effectiveness for the identification.

Modeling and parameter identification of systems with multisegment piecewise-linear characteristics

This paper deals with the modeling and parameter identification of nonlinear systems having multi-segment piecewise-linear characteristics. The decomposition of the corresponding mapping provides a new form of multi-segment nonlinearity representation, leading to an output equation where all the parameters to be estimated are separated. Hence, an iterative method with internal variable estimation can be applied for parameter identification using input/output data records. The only required a-priori knowledge of the nonlinear characteristic represents the limits for the domain partition. The proposed model of given static nonlinearity is also incorporated into the Hammerstein model. Examples of parameter identification for static and dynamic systems with multi-segment piecewise-linear characteristics are presented.

A wavelet-based approach for model and parameter identification of non-linear systems

A procedure is presented for identifying the mechanical parameters of zero-memory non-linear discrete structural systems. The procedure allows both the parameter estimation of a priori known dynamical models as well as the identification of classes of suitable non-linear models based on input–output data. The method relies on a wavelet-based discretization of the non-linear governing differential equation of motion. Orthogonal Daubechies scaling functions are used in the analysis. The scaling functions localization properties permit the tracking of fast variations of the state of the dynamical system which may be associated with unmodeled dynamics of measurement noise. The method is based on the knowledge of measured state variables and excitations and applies to single and multi-degree-of-freedom systems under either free or forced vibrations.

Parametric Identification of Systems With Dry Friction and Nonlinear Stiffness Using a Time Series Model

A new type of time series model, the AVD model which accommodates the Acceleration, Velocity and Displacement simultaneously, has been used to estimate the characteristics of nonlinear single and multi-degree of freedom systems with dry Coulomb friction, viscous damping and nonlinear stiffness. Numerical results via simulation are compared with those from a Fourier transform method, which suggests that the AVD model is a powerful technique for nonlinear system parameter identification.