Parameter identification for nonlinear systems: Guaranteed confidence regions through LSCR

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.

Non-Asymptotic Confidence Regions for Errors-In-Variables Systems

Asymptotic properties of SPS confidence regions

This paper describes a novel offline approach to estimate the parameters of a nonlinear system using the optimal control theory. In the conventional methods, only the output is considered in the identification process and the problem is stated as a nonlinear optimization problem. Nevertheless, the proposed method here is based on the least square error of both output and input. To do this, the problem is first stated as a tracking problem with some unknown parameters. The necessary and sufficiency conditions of optimality conditions are derived using the fundamental theorem of variational calculus. The problem is deployed as an effort optimal control problem which finally leads to a two point boundary value problem. To solve the resultant equations, a secondary optimal control problem is deployed to be solved using the homotopy algorithm. So the parameters are gradually converged to the real values. The proposed algorithm is analyzed for a one-link manipulator with the full measurements. Thereafter, the proposed method is applied on a real one-link manipulator and nonlinear model of permanent magnet dc motor. Application of the method for the identification of a few test beds at the AUT proves the validity and efficiency of the proposed approach.

Control and State Space Adjustments for Nonlinear Systems

This paper considers the parameter identification of nonlinear systems using Hammerstein model and in the presence of correlated output noise. Existing identification methods are all iterative. The proposed method, called MSLS, is a noniterative four-stage least square solution procedure. Therefore, it is computationally simpler. The estimates so obtained are statistically consistent. Two examples are included to demonstrate the utility of this method.

In this paper we will investigate some non-asymptotic properties of the modified least squares estimates for the non-linear function f(λ*) by observations that nonlinearly depend on the parameter λ*. Non-asymptotic confidence regions with fixed sizes for the modified least squares estimate are used. The obtained confidence region is valid for a finite number of data points when the distributions of the observations are unknown. Asymptotically the suggested estimates represent usual estimates of the least squares. The paper presents the results of practical applications of the proposed method in C-OTDR monitoring systems.

Solution of multipoint boundary value problems via block-pulse functions

The paper presents a method for the identification of nonlinear system parameters by using an improved Genetic Algorithm and Grey Function. The paper firstly outlines several commonly used nonlinear identification methods such as RLS, RIV and COR and also their drawbacks. Then, a method based on the Genetic Algorithm and Grey Function is proposed and given in detail in the paper. Finally, a simulation experiment to TV set production data of an electronic factory was carried out. The simulations show that the method can gain good results and is also simple and effective.

The use of neural networks in system identification is an emerging field. Neural networks have become popular in recent years as a means to identify linear and non-linear systems whose characteristics are unknown. The success of sigmoidal networks in parameter identification has been limited. However, harmonic activation-based neural networks, recent arrivals in the field of neural networks, have shown excellent promise in linear and non-linear system parameter identification. They have been shown to have excellent generalization capability, computational parallelism, absence of local minima, and good convergence properties. They can be used in the time and frequency domain. This paper presents the application of a special class of such networks, namely Fourier Series neural networks (FSNN) to vehicle system identification. In this paper, the applications of the FSNNs are limited to the frequency domain. Two examples are presented. The results of the identification are based on simulation data. The first example demonstrates the transfer function identification of a two-degree-of freedom lateral dynamics model of an automobile. The second example involves transfer function identification for a quarter car model. The network set-up for such identification is described. The results of the network identification are compared with theory. The results indicate excellent prediction properties of such networks.

In this paper we consider the problem of estimating confidence regions for the parameters of ARMA models. Based on subsampling techniques and building on earlier exact finite sample results due to Hartigan, we compute the exact probability that the true parameters belong to certain regions in the parameter space. By intersecting these regions, a confidence region containing the true parameters with guaranteed probability is then obtained. All results hold true for a finite number of data points and no asymptotic theory is used. The usefulness of the approach is illustrated in a simulation example.

This work presents a new strategy in design optimization of nonlinear vibration absorbers with continuous and discontinuous motions. A cluster-based harmonic balance aided optimization technique using force balance or energy balance as the basis is generalized and adapted for nonlinear systems. It is found that optimal design parameters form a cluster in the parameter space and points from the parameter space inside the cluster satisfies design considerations. One of the main disadvantages of using existing optimization methods in nonlinear systems is that the parameter regimes, which provide periodic solutions, are not known beforehand, so one has to first do bifurcation studies to arrive at periodic regimes and optimization has to be conducted in the range. Proposed method combines these two steps as it converges to periodic clusters alone. Since the method admits only periodic solutions, occurrence of conditions such as chaos and quasi periodicity can be eliminated from the dynamics of the system. The proposed method can also be used to find the optimal parameters of both linear and nonlinear dynamical systems.

Finite-sample system identification algorithms can be used to build guaranteed confidence regions for unknown model parameters under mild statistical assumptions. It has been shown that in many circumstances these rigorously built regions are comparable in size and shape to those that could be built by resorting to the asymptotic theory. The latter sets are, however, not guaranteed for finite samples and can sometimes lead to misleading results. The general principles behind finite-sample methods make them virtually applicable to a large variety of even nonlinear systems. While these principles are simple enough, a rigorous treatment of the attendant technical issues makes the corresponding theory complex and not easy to access. This is believed to be one of the reasons why these methods have not yet received widespread acceptance by the identification community and this letter is meant to provide an easy access point to finite-sample system identification by presenting the fundamental ideas underlying these methods in a simplified manner. We then review three (classes of) methods that have been proposed so far—1) Leave-out Sign-dominant Correlation Regions (LSCR); 2) Sign-Perturbed Sums (SPS); 3) Perturbed Dataset Methods (PDMs). By identifying some difficulties inherent in these methods, we also propose in this letter a new sign-perturbation method based on correlation which overcome some of these difficulties.

The on-line or adaptive identification of parameters in abstract linear and nonlinear infinite-dimensional dynamical systems is considered. An estimator in the form of an infinite-dimensional linear evolution system having the state and parameter estimates as its states is defined. Convergence of the state estimator is established via a Lyapunov estimate. The finite-dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions. Convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited. A finite-dimensional approximation theory is developed, and convergence results are established. Numerical results for examples involving the estimation of both constant and functional parameters in one-dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one-dimensional wave equation with Kelvin--Voigt viscoelastic damping are presented.

The particular concern of this paper is the construction of a confidence region with pointwise asymptotically correct size for the true value of a parameter of interest based on the generalized Anderson-Rubin (GAR) statistic when the moment variance matrix is singular. The large sample behaviour of the GAR statistic is analysed using a Laurent series expansion around the points of moment variance singularity. Under a condition termed first order moment singularity the GAR statistic is shown to possess a limiting chi-square distribution on parameter sequences converging to the true parameter value. Violation, however, of this condition renders the GAR statistic unbounded asymptotically. The paper details an appropriate discretisation of the parameter space to implement a feasible GAR-based confidence region that contains the true parameter value with pointwise asymptotically correct size. Simulation evidence is provided that demonstrates the efficacy of the GAR-based approach to moment-based inference described in this paper.