Nonlinear Subspace Identification Method Research Articles

A subspace parameter identification method for nonlinear structures under oversampling conditions

In engineering practice, sampling frequencies much higher than the Nyquist frequency are often used to obtain accurate measurements of the dynamic characteristics of nonlinear systems. However, it may lead to oversampling when using the nonlinear subspace identification method (NSIM). This study proposes a subspace parameter identification method to address the identification of nonlinear structures under oversampling conditions. The proposed method first employs the prediction error method (PEM) to estimate the linear part of the state space model. In noisy environments, the PEM can achieve more accurate results than the NSIM by minimizing an error criterion function. The initial values fed into the PEM are obtained by the stochastic subspace identification method. The nonlinear part of the state space model is estimated by solving a linear regression equation. The proposed method effectively reduces the impact of high sampling frequencies on identification accuracy by combining data with different sampling frequencies to construct the linear regression equation. Among them, the low sampling frequency data is obtained by downsampling the high sampling frequency measurement data. The proposed method effectively reduces the effect of high sampling frequencies and environmental noise on identification accuracy by accurately estimating both the linear and nonlinear parts of the state space model. The identification accuracy of the proposed method is numerically verified by a chain-type structure with five degrees of freedom and a cantilever beam structure. Additionally, experimental verification is carried out on a two degrees of freedom floor structure with local nonlinearities. The results show that the proposed method achieves high identification accuracy in oversampling conditions and noisy environments.

Automatic Nonlinear Subspace Identification Using Clustering Judgment Based on Similarity Filtering

Accurately determining system order plays a vital role in system identification directly related to the accuracy of identification results, especially for nonlinear system identification. Due to the need for human subjective judgment, the traditional sequence determination method easily causes uncertainty in the results; and the phenomenon of the virtual mode or omission occurs. An automatic nonlinear subspace identification method is proposed to address the aforementioned problems. When the eigenvalue decomposition of the constructed Hankel matrix is performed, the calculation range of the modal order of the system is estimated. The similarity coefficient and distance function are introduced to cluster the identified modal results, the poles of the false modes are removed to obtain the cluster stabilization diagram, and the best order of the system is received. Then, the modal parameters and nonlinear coefficients are obtained. Simulation examples are carried out to verify the effectiveness and robustness of the proposed method. An experimental study is carried out on a multilayer building with nonlinear characteristics. Compared with the traditional stabilization graph, the accuracy of the automatic order determination proposed in this paper is proven.

Identification of Nonlinear Stiffness and Damping Parameters Using a Hybrid Approach

Robustly identifying nonlinear mechanical systems is generally a challenging task, and this is particularly true when the structure under test exhibits nonlinear behaviors related to both stiffness and damping. In this work, a hybrid nonlinear identification approach is proposed by combining the restoring force surface (RFS) method and the nonlinear subspace identification method. The multiparameter nonlinear identification strategy is based on a first characterization conducted using the RFS method, followed by a nonlinear state-space representation using subspace algorithms. Two common friction simulation examples and one complex multi-degree-of-freedom system are employed to verify the proposed method. The effect of the measurement noise on the parameter estimation results is investigated by corrupting the previously generated output, adding different levels of Gaussian zero-mean noise. Results show that the nonlinear coefficients associated with the stiffness and damping nonlinearities can be identified with a high level of confidence, and the proposed method works well under different noise-level contaminations.

Control oriented data driven linear parameter varying model for proton exchange membrane fuel cell systems

Proton exchange membrane fuel cell systems are widely used to drive vehicles. It is indispensable to establish accurate and simple control oriented model of the system. However, this control oriented model is difficult to build due to the characteristic of strongly nonlinear, coupled and time - varying in proton exchange membrane fuel cell systems. To solve this problem, nonlinear subspace identification method is proposed to discover linear parameter varying model for the proton exchange membrane fuel cell system. The modeling process can be divided into two stages. In the first stage, algebraic transformation method is proposed to transform the nonlinear model to linear parameter varying model. In the second stage, kernel method based subspace identification is presented to identify parameter matrices of the proton exchange membrane fuel cell system. The established model is demonstrated by a 50 kW on board proton exchange membrane fuel cell system under real driving condition. The results show that mean absolute percent error and root mean squared error are as low as 9.6941e−06 and 0.0957, respectively. Therefore, the control oriented model perfectly agrees with real data. Compared with the prediction error method and recursive neural network nonlinear auto - regressive moving average model with exogenous method, modeling accuracy and speed of the proposed method are the best. Moreover, linear control theory can be employed to design controller based on the linear parameter varying model, which greatly simplifies controller design for proton exchange membrane fuel cell systems.

A modified time domain subspace method for nonlinear identification based on nonlinear separation strategy

Nonlinear factors existing in engineering structures have drawn considerable attention, and nonlinear identification is a competent technique to understand the dynamic characteristics of nonlinear structures. Therefore, in this paper, a novel nonlinear separation subspace identification (NSSI) algorithm based on subspace algorithm and nonlinear separation strategy is proposed to conduct nonlinear parameter identification of nonlinear structures. For the proposed NSSI algorithm, the low-level excitation test is firstly conducted to obtain the transfer matrix in the linear response formula. Then, the obtained transfer matrix is used in the high-level excitation test to calculate the nonlinear response part by the proposed nonlinear separation strategy, and the subspace algorithm is utilized to identify the nonlinear parameter on the modified state-space model including only the nonlinear part. The proposed NSSI algorithm can reduce the coupling error caused by simultaneously processing both the large number part (corresponding to the linear part) and small number part (corresponding to the nonlinear part) in the traditional nonlinear subspace identification (NSI) algorithm. At last, two numerical experiments are given to validate the effectiveness of the developed novel nonlinear identification method. Furthermore, some influence factors are discussed to show the stability of the identification algorithm, and some comparisons between the proposed NSSI method and traditional NSI method are also conducted to demonstrate the advantages of the novel method.

A nonlinear subspace-prediction error method for identification of nonlinear vibrating structures

The industrial structural systems always contain various kinds of nonlinear factors. Recently, a number of new approaches have been proposed to identify those nonlinear structures. One of the promising methods is the nonlinear subspace identification method (NSIM). The NSIM is derived from the principals of the stochastic subspace identification method (SSIM) and the internal feedback formulation. First, the nonlinearities in the system are regarded as internal feedback forces to its underlying linear dynamic system. The linear and nonlinear components of the identified system can be decoupled. Second, the SSIM is employed to identify the nonlinear coefficients and the frequency response functions of the underlying linear system. A typical SSIM always consists of two steps. The first step makes a projection of certain subspaces generated from the data to identify the extended observability matrix. The second one is to estimate the system matrices from the identified observability matrix. Since the calculated process of the NSIM is non-iterative and this method poses no additional problems on the part of parameterization, the NSIM becomes a promising approach to identify nonlinear structural systems. However, the result generated by the NSIM has its deficiency. One of the drawbacks is that the identified results calculated by the NSIM are not the optimal solutions which reduce the identified accuracy. In this study, a new time-domain subspace method, namely the nonlinear subspace-prediction error method (NSPEM), is proposed to improve the identified accuracy of nonlinear systems. In the improved version of the NSIM, the prediction error method (PEM) is used to reestimate those estimated coefficient matrices of the state-space model after the application of NSIM. With the help of the PEM, the identified results obtained by the NSPEM can truly become the optimal solution in the least square sense. Two numerical examples with local nonlinearities are provided to illustrate the effectiveness and accuracy of the proposed algorithm, showing advantages with respect to the NSIM in a noise environment.

Data-Driven Nonlinear Subspace Modeling for Prediction and Control of Molten Iron Quality Indices in Blast Furnace Ironmaking

Blast furnace (BF) in ironmaking is a nonlinear dynamic process with complicated physical-chemical reactions, where multiphases and multifields interactions with long time delay phenomena take place during its operation. In BF operation, the molten iron temperature as well as the Si content ([Si]), the phosphorus content ([P]), and the sulfur content ([S]) is the most essential quality (MIQ) indices. The measurement, modeling, and control of these indices have always been important issues in metallurgic engineering and automation. This paper proposes a novel data-driven nonlinear state-space modeling method for the prediction and control of multivariate MIQ indices by integrating hybrid modeling and control techniques together. First, to improve modeling efficiency, a data-driven hybrid method that combines canonical correlation analysis and correlation analysis is established to identify the most influential controllable variables as the modeling inputs from multitudinous factors. Then, a Hammerstein model for the prediction of MIQ indices is established using the Least squares support vector machine-based nonlinear subspace identification method. Such a model is further simplified by using piecewise cubic Hermite interpolating polynomial method. Compared with the original Hammerstein model, it has been shown that this simplified model can not only significantly reduce the computational complexity, but can also exhibit a good reliability and accuracy for a stable prediction of MIQ indices. In order to verify the practicability of the developed model, it is applied to the design of a genetic algorithm based nonlinear predictive controller for the multivariate MIQ indices by directly taking the established model as a predictor. Industrial experiments show the advantages and effectiveness of the proposed approaches.

Theoretical and Experimental Identification of Cantilever Beam With Clearances Using Statistical and Subspace-Based Methods

Clearance turns up in a large number of engineering structures because of the errors during assembling, manufacturing, and wearing. The presence of clearance in engineering structures changes the normal dynamic response and will result in low precision and short lifetime. The clearance parameter identification of such nonlinear system is the prerequisite to control and eliminate the effect of clearance nonlinearity. In this paper, a derivative plot of probability density function (DPPDF) for displacement response is proposed to precisely identify the clearance value of continuous system, and the nonlinear subspace identification (NSI) method is modified to recognize the related contact stiffness based on the frequency response function (FRF) equations of continuous system. The DPPDF method is carried out by analyzing the distribution characteristic of displacement response, and the clearance value is derived through inspecting the probability density function (PDF) plot and the second derivative plot of the PDF. Based on the identified clearance, the clearance nonlinearity is regarded as external force, and the relationship between the dynamic responses and the external forces in frequency domain can be expressed as the form of FRF equations. Based on the FRF equations, the contact stiffness in continuous system is obtained with modified NSI method. This combined identification process is verified by a single-degree-of-freedom (SDOF) system and a cantilever beam system with clearances, and some influence factors of this identification process, including noise, transfer error, and force level, are discussed in detail. In the end, an experiment device with changeable clearance and contact stiffness was designed to conduct identification experiments, and the results show that the proposed methods perform effectively in identifying the clearance parameters.

Subspace-Based Identification of Nonlinear Structures

Conventional linear estimators give results contaminated in presence of nonlinearities and the extraction of underlying linear system properties is thus difficult. To overcome this problem, the implementation of a recently developed method, called Nonlinear Subspace Identification (NSI), is considered in this paper, by using the perspective of nonlinearities as unmeasured internal feedback forces. Although its formulation is very simple, particular care has to be taken to reduce the ill-conditioning of the problem, in order to find numerically stable solutions. To this purpose, the robustness and the high numerical performances of the subspace algorithms are successfully exploited, as shown by the implementation of the proposed method on simulated multi-degree-of-freedom systems with typical nonlinear characteristics as well as on an experimental case. These examples demonstrate that the application of subspace algorithms to nonlinear system identification gives better conditioning and computational efficiency with respect to the most recent nonlinear techniques. Moreover, the capability of the NSI method of simultaneously dealing with several nonlinear terms, with a light computational effort, may be also exploited in those situations where no a priori knowledge of the location and the type of nonlinearities is given, being this method well capable of detecting the contribution of the dominant nonlinearities.On the basis of the results discussed in this paper, and compared with those of other well-assessed nonlinear techniques, the proposed method appears having the chances to become a robust procedure to be widely exploited in many industrial fields, being its capability of separating linear and nonlinear contribution terms widely requested in mechanical and civil engineering field.

Wiener model identification and predictive control of a pH neutralisation process

Wiener model identification and predictive control of a pH neutralisation process is presented. Input-output data from a nonlinear, first principles simulation model of the pH neutralisation process are used for subspace-based identification of a black-box Wiener-type model. The proposed nonlinear subspace identification method has the advantage of delivering a Wiener model in a format which is suitable for its use in a standard linear-model-based predictive control scheme. The identified Wiener model is used as the internal model in a model predictive controller (MPC) which is used to control the nonlinear white-box simulation model. To account for the unmeasurable disturbance, a nonlinear observer is proposed. The performance of the Wiener model predictive control (WMPC) is compared with that of a linear MPC, and with a more traditional feedback control, namely a PID control. Simulation results show that the WMPC outperforms the linear MPC and the PID controllers.