Polynomial Nonlinear State Space Research Articles

Constructing nonlinear data-driven models from pitching wing experiments using multisine excitation signals

Accurate modelling of unsteady nonlinear aerodynamic loads is crucial for the effective design and control of aerodynamic systems. Data-driven modelling approaches have emerged as powerful tools for capturing intricate system dynamics without prior knowledge of the underlying physical principles. This paper focuses on the construction of nonlinear data-driven models for pitching wing systems using multisine excitation signals and their relative merits compared to swept sine signals. The polynomial nonlinear state-space (PNLSS) modelling framework is used, which has been proven in the system identification community to offer flexibility and richness in representing diverse nonlinear phenomena. Data are gathered with a dedicated experimental wind tunnel setup. A NACA 0018 wing is made to oscillate in pitch according to a swept sine and a multisine with a frequency up to 2 Hz. The experiments are repeated to cover the attached flow, light stall, and deep stall regimes. PNLSS models are trained on multisines and benchmarked against models trained on swept sines to assess their comparative performance. Then, all models are successfully cross-validated using single sine data at different angles of attack. We conclude that one single data-driven model can accurately represent the lift force across different linear (attached flow) and nonlinear (light and deep stall) operating regimes.

Parallel distributed identification for block structure nonlinear system

The problem of identifying one block structure nonlinear system, consisting of four linear blocks and one static nonlinear block, is considered here. Under the assumptions that static nonlinear is a polynomial form, the block structure nonlinear system can be transformed into one state space form. Then this nonlinear system identification is formulated into identifying the state space form for these four linear blocks. After transforming the formal state space form, we obtain a polynomial nonlinear state space equation. Then the parallel distributed algorithm is used to identify each system matrix, while using the relationship between the formal state space equation and its transformed state space equation. Finally the simulation example is used to prove the efficiency of our algorithm.

State-Space Model Representation to Characterize an Energy-Harvesting Circuit

This paper presents a method to model an energy harvesting circuit and a propagation channel as parts of a wireless power transfer (WPT) system. Previously , an original solution was proposed by the authors to maximize the collected DC power level, implying to model the different parts of the system by using state space representation and using convex optimization algorithms to calculate multi carrier signal weights. Firstly, a state-space model of the radio channel is obtained by the use of multivariable output-error state space (MOESP) algorithm. Then, by the vector fitting method a state-space representation is obtained in order to model the linear part of the energy harvesting circuit. At last, by the polynomial-nonlinear state-space (PNLSS) method, the non-linear part of the rectifier is also modeled.

Extrapolation Behavior Comparison of Nonlinear State Space Models

Abstract The extrapolation behavior and its consequences of two different state space models is studied. More specifically, the extrapolation behavior of the local model state space network (LMSSN) is compared to the polynomial nonlinear state space model (PNLSS). Both approaches represent favorable alternatives in the light of their results on recent benchmark studies. The models are compared with regards to their static extrapolation behavior and their tendency to erratic behavior if test amplitudes grow larger than training amplitudes. It is shown on a Hammerstein and Wiener artificial test process that the PNLSS shows severe shortcomings concerning stability when operated close to or outside the boundaries in which it was trained. The LMSSN instead shows more favorable behavior in extrapolation and does only exhibit minor erratic behavior. If the process is inherently polynomial in some way, the PNLSS outperforms other models in extrapolation, as shown on the Silverbox benchmark test.

Nonlinear System Identification with Dominating Output Noise - A Case Study on the Silverbox

Abstract In the overview paper on nonlinear system identification Schoukens and Ljung (2019), it was indicated that reliable expressions to calculate the variance of an estimated nonlinear model are lacking, especially if the disturbing noise is entering the nonlinear regressors. In this study, we provide a better view on the driving mechanisms of the variability of estimated nonlinear models that is due to noise on the output. To do so, we follow a double approach. Firstly, a basic insight on the impact of disturbing noise on the monomial ynis studied. Next, these insights are used in a case study on the forced Duffing oscilator data, also called the Silver box (Schoukens and Noel, 2016). The following models are studied: Nonlinear autoregressive exogenous models (NARX) using a 2-layer Neural Net (NARX-NN) and a polynomial (NARX-poly) expansion; and polynomial nonlinear state space models (PNLSS). This limited study indicates that an output error criterion (PNLSS, and NARX used in a simulation mode) does better than minimizing the equation error (NARX-NN and NARX-poly in prediction mode). When the signal-to-noise ratio (SNR) drops below 20 dB, the reduction in the error is more than a factor 10. This is a strong indication that, just as for linear identification, it is very important to properly deal with the noise properties in the cost function whenever the SNR of the output measurements drops such that noise becomes more important than structural model errors.

Parallel distributed estimation for polynomial nonlinear state space models

In this paper, nonlinear system is modeled by one polynomial nonlinear state space model. Two polynomial functions of input or state are added in the common state space model and these two polynomial functions correspond to the nonlinear factors. This polynomial nonlinear state space model can be represented the special nonlinear closed feedback system. To identify every system matrix in the polynomial nonlinear state space model, firstly each system matrix is vectorized as an unknown parameter vector and then two parallel distribution algorithms are applied to identify this unknown parameter vector in the unconstrained or constrained conditions respectively. When some state equation equalities are deemed as the constrained conditions, the new optimization variables are the state instants and the unknown parameter vector, consisted by all system matrices. These complex optimization variables are solved by parallel distribution algorithm and the whole process about parallel distribution algorithm are given explicitly. Generally, the main contributions of this papre consist in two folds: one is to apply that polynomial function to represent the nonlinear factor, existing in the nonlinear state space model, and the other is to propose parallel distribution algorithm to identify the unknown parameter vector, while analyzing its property. Finally the simulation example is used to prove the efficiency of this parallel distribution algorithm.

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.

A nonlinear model of vortex-induced forces on an oscillating cylinder in a fluid flow

A nonlinear model relating the imposed motion of a circular cylinder, submerged in a fluid flow, to the transverse force coefficient is presented. The nonlinear fluid system, featuring vortex shedding patterns, limit cycle oscillations and synchronisation, is studied both for swept sine and multisine excitation. A nonparametric nonlinear distortion analysis (FAST) is used to distinguish odd from even nonlinear behaviour. The information which is obtained from the nonlinear analysis is explicitly used in constructing a nonlinear model of the polynomial nonlinear state-space (PNLSS) type. The latter results in a reduction of the number of parameters and an increased accuracy compared to the generic modelling approach where typically no such information of the nonlinearity is used. The obtained model is able to accurately simulate time series of the transverse force coefficient over a wide range of the frequency–amplitude plane of imposed cylinder motion.

Nonlinear system identification of fractional Wiener models

The Wiener model is a case of block-oriented model, which is suitable for modeling a large class of nonlinear systems. It consists of the cascade of two parts, where a linear dynamic part is followed by a static nonlinear part. On the other hand, fractional order models have gained increasing interest over the last decades due to their better representation of some physical phenomena. In this paper, the fractional Wiener system identification is aimed. The polynomial nonlinear fractional state space equations are used to describe the nonlinear model. Thus, the extension of an output error method (Levenberg Marquard algorithm) is developed to estimate the fractional Wiener system. The efficiency of the identification method is evaluated and confirmed by simulation examples.

Dealing with Transients due to Multiple Experiments in Nonlinear System Identification

Although in many static estimation problems the data are collected from repeated experiments, the default underlying (assumption) setting in most of the system identification tasks is that the data are generated from a single experiment. The rationale for this is that more data can be obtained by increasing the measurement time instead of doing more experiments. As the measurement time tends to infinity, under suitable assumptions it is possible to estimate consistently the model parameters. In most of the real life cases, however, increasing the measurement time is either not possible or it does not cover the whole operating range of the system to be identified, hence data from multiple experiments need to be combined. Furthermore, most of the real life systems are nonlinear and a large variety of nonlinear systems can be described by a nonlinear state space model structure. In this paper, a methodology to deal with the transients arising due to concatenating data from multiple experiments during the identification of Polynomial nonlinear state space (PNLSS) models is described. The methodology is validated on data generated from a laboratory experimental set-up.

On identification of well‐conditioned nonlinear systems: Application to economic model predictive control of nonlinear processes

The focus of this work is on economic model predictive control (EMPC) that utilizes well‐conditioned polynomial nonlinear state‐space (PNLSS) models for processes with nonlinear dynamics. Specifically, the article initially addresses the development of a nonlinear system identification technique for a broad class of nonlinear processes which leads to the construction of PNLSS dynamic models which are well‐conditioned over a broad region of process operation in the sense that they can be correctly integrated in real‐time using explicit numerical integration methods via time steps that are significantly larger than the ones required by nonlinear state‐space models identified via existing techniques. Working within the framework of PNLSS models, additional constraints are imposed in the identification procedure to ensure well‐conditioning of the identified nonlinear dynamic models. This development is key because it enables the design of Lyapunov‐based EMPC (LEMPC) systems for nonlinear processes using the well‐conditioned nonlinear models that can be readily implemented in real‐time as the computational burden required to compute the control actions within the process sampling period is reduced. A stability analysis for this LEMPC design is provided that guarantees closed‐loop stability of a process under certain conditions when an LEMPC based on a nonlinear empirical model is used. Finally, a classical chemical reactor example demonstrates both the system identification and LEMPC design techniques, and the significant advantages in terms of computation time reduction in LEMPC calculations when using the nonlinear empirical model. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3353–3373, 2015

Identification of a Wiener–Hammerstein system using the polynomial nonlinear state space approach

In this paper, the Polynomial NonLinear State Space (PNLSS) approach is applied to model a nonlinear system with a Wiener–Hammerstein structure. To obtain good initial estimates, the best linear approximation of the system under test is first identified. Next, this linear model is extended to a polynomial nonlinear state space model to capture also the system's nonlinear behavior. The identification procedure is applied to measurement data.

Nonlinear system-identification of the filling phase of a wet-clutch system

The work presented illustrates how the choice of input perturbation signal and experimental design improves the derived model of a nonlinear system, in particular the dynamics of a wet-clutch system. The relationship between the applied input current signal and resulting output pressure in the filling phase of the clutch is established based on bandlimited periodic signals applied at different current operating points and signals approximating the desired filling current signal. A polynomial nonlinear state space model is estimated and validated over a range of measurements and yields better fits over a linear model, while the performance of either model depends on the perturbation signal used for model estimation.

Identification of nonlinear systems using Polynomial Nonlinear State Space models

In this paper, we propose a method to model nonlinear systems using polynomial nonlinear state space equations. Obtaining good initial estimates is a major problem in nonlinear modelling. It is solved here by identifying first the best linear approximation of the system under test. The proposed identification procedure is successfully applied to measurements of two physical systems.

Identification of Nonlinear Feedback Systems Using a Structured Polynomial Nonlinear State Space Model

In this paper, we propose an identification procedure for nonlinear feedback systems by introducing a new type of model structure. The idea is to combine the advantages of two different model structures: the Polynomial NonLinear State Space (PNLSS) model and the block-structured feedback model. By imposing structural information to the existing PNLSS model, we obtain a flexible model with a lower number of parameters compared with the unconstrained structure. The structured-PNLSS model consists of a linear block in the feedforward path and a PNLSS block in the feedback path. The presented identification procedure for the structured-PNLSS model is applied to input/output measurements of a nonlinear device.

Identification of a Wiener-Hammerstein System Using the Polynomial Nonlinear State Space Approach

In this paper, we apply the Polynomial NonLinear State Space (PNLSS) approach to model a nonlinear system with a Wiener-Hammerstein structure. To obtain good initial estimates, the best linear approximation of the system under test is first identified. Next, this linear model is extended to a polynomial nonlinear state space model to capture also the system's nonlinear behaviour. The identification procedure is applied to measurement data.