Parameter Estimation Of Nonlinear Systems Research Articles

Efficient delay and parameter estimation for nonlinear systems based on lifted IRK integrators with inexact Jacobians

Efficient delay and parameter estimation for nonlinear systems based on lifted IRK integrators with inexact Jacobians

Performance Comparison of Meta-Heuristics Applied to Optimal Signal Design for Parameter Identification.

This paper presents a comparative study that explores the performance of various meta-heuristics employed for Optimal Signal Design, specifically focusing on estimating parameters in nonlinear systems. The study introduces the Robust Sub-Optimal Excitation Signal Generation and Optimal Parameter Estimation (rSOESGOPE) methodology, which is originally derived from the well-known Particle Swarm Optimization (PSO) algorithm. Through a real-life case study involving an Autonomous Surface Vessel (ASV) equipped with three Degrees of Freedom (DoFs) and an aerial holonomic propulsion system, the effectiveness of different meta-heuristics is thoroughly evaluated. By conducting an in-depth analysis and comparison of the obtained results from the diverse meta-heuristics, this study offers valuable insights for selecting the most suitable optimization technique for parameter estimation in nonlinear systems. Researchers and experimental tests in the field can benefit from the comprehensive examination of these techniques, aiding them in making informed decisions about the optimal approach for optimizing parameter estimation in nonlinear systems.

Parameter estimation of nonlinear systems: dwarf mongoose optimization algorithm with key term separation principle

Parameter estimation of nonlinear systems: dwarf mongoose optimization algorithm with key term separation principle

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.

Dynamically learning the parameters of a chaotic system using partial observations

<p style='text-indent:20px;'>Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we supply a rigorous analytical proof that guarantees the convergence of this algorithm to the true parameter values when the system in question is the classic three-dimensional Lorenz system. Such a result appears to be the first of its kind for dynamical parameter estimation of nonlinear systems. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We moreover probe the limitations of the algorithm by identifying dynamical regimes under which certain parameters cannot be effectively inferred having only observed certain state variables. In such cases, modifications to the algorithm are proposed that ultimately result in recovery of the parameter. Lastly, computational evidence is provided that supports the efficacy of the algorithm well beyond the hypotheses specified by the theorem, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.</p>

Parameter estimation for nonlinear systems with multirate measurements and random delays

AbstractWe present an approach for parameter estimation with multirate measurements, with the slow measurements having variable time delays due to laboratory analysis, and also being functions of all the states during the sample collection. We formulate a particle filter‐based approach under the framework of the expectation maximization algorithm to develop the estimates. The effectiveness and applicability of the proposed method are demonstrated though a simulation example, a hybrid tank experiment and an industrial case study; in each case, the slow and fast measurements are for the same variable. We show that this approach results in improved parameter estimation when the information from the delayed measurements is fused with the fast measurement information.

Global sensitivity analysis for the design of nonlinear identification experiments

Bayesian inference techniques have been used extensively in recent years for parameter estimation in nonlinear systems. Despite the many advances made in the field, highly nonlinear systems can still be challenging to identify. Of key interest is the challenge in establishing the identifiability of the model with respect to various excitation signals and, in particular, doing so prior to the collection of experimental data. Global sensitivity analysis techniques provide a perspective on this problem that is well-suited to informing the design of identification experiments for use with Bayesian inference techniques. These methods quantify the relative importance of the parameters to the model response by decomposing the variance of the response into contributions from the respective parameters. The sensitivities obtained provide a valuable indication of the information available for parameter estimation in the response of a system to a particular excitation. In this study, nonlinear model parameters are identified based on experimental responses from a nonlinear energy sink device with the unscented Kalman filter. The experimental identification results are compared with those of a Sobol’ sensitivity analysis on the system model to demonstrate how global sensitivity analysis can be used as a method to preselect experimental excitations for use with Bayesian inference techniques.

A fast-converge, real-time auto-calibration algorithm for triaxial accelerometer

Online parameter estimation for nonlinear systems are challenging, especially when only limited computational powers are available. The auto-calibration of triaxial accelerometers is essentially a nonlinear parameter estimation problem. In this paper, a simple but efficient auto-calibration method for micro triaxial accelerometers is proposed. In particular, this method is an online calibration approach based on the six-parameter auto-calibration model, which can be implemented in a wearable health-monitoring device to compensate the estimation errors due to parameter drift. To achieve online calibration, the nonlinear model together with its cost function for auto-calibration is linearized, and an online recursive method is exploited to identify the unknown parameters and remove the bias caused by linearization. This online recursive method is developed by modifying the damped recursive least square estimation (MDRLS), where the estimated unknown parameter can converge stably in a short period. The MDRLS-based method can significantly reduce the calculation complexity comparing to the commonly used nonlinear optimization method. Additionally, the proposed method is validated by both simulation and experiment. The simulation results show the estimated parameters are very close to their true values. By comparing the output results before and after calibration in different experimental conditions, the effectiveness of the proposed method is further verified by real-time experiments.

A novel data-driven model based parameter estimation of nonlinear systems

In practice, it is usually difficult to estimate the characteristic parameters of a system due to the system nonlinearity and uncertainty. To address this problem, in this study, the characteristic parameters of a nonlinear system are identified by using a data driven method based on the best discretization of the Nonlinear Differential Equation (NDE) model of the system. The best discretization of a NDE model is firstly determined, and then the discretized model, known as the Nonlinear Auto Regressive with eXegenous input (NARX) model, is determined by using the Least Squares (LS) algorithm from the input and output data of system. A case study is discussed to validate the proposed system parameter identification method, where the characteristic parameters of a rotating blade-casing system are evaluated under a bandwidth rub impact in the horizontal direction with noise. The result shows that the identified model can be used to describe the characteristics of the underlying system accurately, which provides a reliable model for the dynamic analysis, control of rotating blade-casing system.

State and parameter estimation using extended Kitanidis Kalman filter

State estimation in presence of unknown parameter variations has received considerable attention in literature. The conventional approach is to model the unknown drifting parameters as random walk model and estimate them simultaneously with states by augmenting them with the states of the system. However, the estimates critically depend on the tuning of the random walk model. Kitanidis Kalman Filter (KKF) [1] is an unbiased minimum variance estimator for only the states in presence of unknown inputs for linear systems. KKF allows optimal estimates of states to be obtained in presence of unknown inputs by appropriately choosing the gain matrix during the state update step. In this work, we investigate the problem of state and parameter estimation for nonlinear systems using extended Kitanidis Kalman Filter (EKKF). EKKF enables state and parameter estimation separately. In particular, we extend the work of Ganesh et al. [2] by providing a maximum likelihood based moving window framework for parameter estimation based on the innovations generated by EKKF. Towards this end, we characterise the innovations generated by KKF and show that unlike Kalman filter where the innovations are white, the innovations generated in KKF are a coloured Gaussian stochastic process. We also present a recursive approach for computing the covariance kernel of the innovations. The proposed method is applied to a simulation and an experimental case study to demonstrate its utility. The results indicate that the proposed method can be a useful tool for state and parameter estimation of nonlinear systems.

An efficient sliding mode observer-based model predictive control: experimental implementation on a DC/DC power converter

In this paper, a discretization-based sliding mode observer (DBSMO) is proposed for the state and parameter estimation of nonlinear systems. In the DBSMO structure, an accurate discretized dynamics are derived for the state and parameter update of the sliding mode observer (SMO) instead of integration-based state update. In this way, faster converging observer dynamics are obtained. The stability properties of the conventional SMO remain the same for DBSMO. In the application presented here, first a real-time DC/DC power converter is designed with a computer interface. Then, to show the enhancement of convergence properties, conventional SMO and DBSMO-based model predictive controllers are designed and applied to control of the DC/DC power converter. Through simulation and experimental results, it is shown that the estimation performance of SMO is greatly improved so that the tracking performance is also increased, which are the main contributions of the paper.

On the Performance of Online Parameter Estimation Algorithms in Systems with Various Identifiability Properties

In recent years, Bayesian inference has been extensively used for parameter estimation in nonlinear systems; in particular it has proved to be very useful for damage detection purposes. The problem of parameter estimation is inherently correlated with the issue of identifiability, i.e., is one able to learn uniquely the parameters of the system from available measurements? The identifiability properties of the system will govern the complexity of the posterior probability density functions (pdfs), and thus the performance of learning algorithms. Off-line methods such as Markov Chain Monte Carlo methods are known to be able to estimate the true posterior pdf, but can be very slow to converge. In this paper we study the performance of on-line estimation algorithms on systems which exhibit challenging identifiability properties, i.e., systems for which all parameters cannot be uniquely identified from the available measurements, leading to complex, possibly multi-modal posterior pdfs. We show that on-line methods are capable of correctly estimating the posterior pdfs of the parameters, even in challenging cases. We also show that a good trade-off can be obtained between computational time and accuracy by correctly selecting the right algorithm for the problem at hand, thus enabling fast estimation and subsequent decision making.

A Parameter Estimation Method for Nonlinear Systems Based on Improved Boundary Chicken Swarm Optimization

Parameter estimation is an important problem in nonlinear system modeling and control. Through constructing an appropriate fitness function, parameter estimation of system could be converted to a multidimensional parameter optimization problem. As a novel swarm intelligence algorithm, chicken swarm optimization (CSO) has attracted much attention owing to its good global convergence and robustness. In this paper, a method based on improved boundary chicken swarm optimization (IBCSO) is proposed for parameter estimation of nonlinear systems, demonstrated and tested by Lorenz system and a coupling motor system. Furthermore, we have analyzed the influence of time series on the estimation accuracy. Computer simulation results show it is feasible and with desirable performance for parameter estimation of nonlinear systems.

Parameter estimation for nonlinear systems by using the data filtering and the multi-innovation identification theory

For Hammerstein output-error autoregressive systems, a decomposition based multi-innovation stochastic gradient (D-MISG) identification algorithm and a data filtering based multi-innovation stochastic gradient (F-MISG) identification algorithm are derived by means of the key-term separation principle and the multi-innovation identification theory. The D-MISG algorithm uses the decomposition technique to transform a Hammerstein system into two subsystems and requires less computational cost, and the F-MISG algorithm uses a linear filter to filter the input-output data and has a higher estimation accuracy for larger innovation lengths. The simulation results show that the proposed two algorithm can give satisfactory parameter estimates.

Nonlinear joint state and parameter estimation: Application to a wastewater treatment plant

A systematic approach to joint state and time-varying parameter estimation for nonlinear systems is proposed in this paper. Applying the sector nonlinearity transformation to both the system nonlinearities and the time-varying parameters, the original system is equivalently rewritten as a Takagi–Sugeno system with unmeasurable premise variables. A joint state and parameter observer whose parameters are designed by solving an LMI optimization problem is then proposed. The target application is a realistic model of an activated sludge wastewater treatment plant, being an uncertain nonlinear system affected by a time-varying parameter.

Parameter estimation through ignorance

Dynamical modeling lies at the heart of our understanding of physical systems. Its role in science is deeper than mere operational forecasting, in that it allows us to evaluate the adequacy of the mathematical structure of our models. Despite the importance of model parameters, there is no general method of parameter estimation outside linear systems. A relatively simple method of parameter estimation for nonlinear systems is introduced, based on variations in the accuracy of probability forecasts. It is illustrated on the logistic map, the Henon map, and the 12-dimensional Lorenz96 flow, and its ability to outperform linear least squares in these systems is explored at various noise levels and sampling rates. As expected, it is more effective when the forecast error distributions are non-Gaussian. The method selects parameter values by minimizing a proper, local skill score for continuous probability forecasts as a function of the parameter values. This approach is easier to implement in practice than alternative nonlinear methods based on the geometry of attractors or the ability of the model to shadow the observations. Direct measures of inadequacy in the model, the "implied ignorance," and the information deficit are introduced.

Comparison of Expectation-Maximization based parameter estimation using Particle Filter, Unscented and Extended Kalman Filtering techniques

The primary requirement of filtering algorithms such as Particle Filter (PF), Unscented Kalman Filter (UKF) and Extended Kalman Filter (EKF) is the availability of an accurate nonlinear state space model. In the absence of good parameter values, one has to estimate both the hidden states and the unknown parameters in a joint framework using measurements available from the process. This problem of joint state and parameter estimation for nonlinear systems can be solved recursively through the combination of a nonlinear smoother and a maximum likelihood parameter estimation scheme. Expectation Maximization (EM) is an efficient optimization algorithm which can provide the maximum likelihood estimate of the model parameters even in the presence of missing data. The algorithm can generate parameter estimates that maximize the likelihood of all the data including those with missing output measurements. This paper presents an approach which combines the EM algorithm with a suitable nonlinear smoother, such as PF, UKF or EKF based smoother. An application of this method to a simulated Continuous Fermentor process with unknown model parameters is presented. A comparative study of the results, when the different smoothing schemes were used in this approach, is presented. The results show that the UKS based technique was able to generate unbiased parameter estimates. The Particle Smoother based parameter estimates converged in the neighbouhood of their true values, but the technique was found to be computationally intensive compared to the UKS and EKS based techniques.

Trajectory Sensitivity Method and Master‐Slave Synchronization to Estimate Parameters of Nonlinear Systems

A combination of trajectory sensitivity method and master‐slave synchronization was proposed to parameter estimation of nonlinear systems. It was shown that master‐slave coupling increases the robustness of the trajectory sensitivity algorithm with respect to the initial guess of parameters. Since synchronization is not a guarantee that the estimation process converges to the correct parameters, a conditional test that guarantees that the new combined methodology estimates the true values of parameters was proposed. This conditional test was successfully applied to Lorenz′s and Chua′s systems, and the proposed parameter estimation algorithm has shown to be very robust with respect to parameter initial guesses and measurement noise for these examples.

State and parameter estimation in nonlinear systems as an optimal tracking problem

In verifying and validating models of nonlinear processes it is important to incorporate information from observations in an efficient manner. Using the idea of synchronization of nonlinear dynamical systems, we present a framework for connecting a data signal with a model in a way that minimizes the required coupling yet allows the estimation of unknown parameters in the model. The need to evaluate unknown parameters in models of nonlinear physical, biophysical, and engineering systems occurs throughout the development of phenomenological or reduced models of dynamics. Our approach builds on existing work that uses synchronization as a tool for parameter estimation. We address some of the critical issues in that work and provide a practical framework for finding an accurate solution. In particular, we show the equivalence of this problem to that of tracking within an optimal control framework. This equivalence allows the application of powerful numerical methods that provide robust practical tools for model development and validation.

A robust estimation of the exponent function in the Gompertz law

The estimation of the solution of a system of two differential equations introduced by Norton et al. [Predicting the course of Gompertzian growth, Nature 264 (1976) 542–544] that is equivalent to the famous Gompertz growth law is performed by means of the recent adaptive scheme of Besançon and collaborators [High gain observer based state and parameter estimation in nonlinear systems, paper 204, the sixth IFAC Symposium, Stuttgart Symposium on Nonlinear Control Systems (NOLCOS), 2004, available at 〈 http://www.nolcos2004.uni-stuttgart.de 〉 ]. Results of computer simulations illustrate the robustness of the approach.