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Constructing nonlinear data-driven models from pitching wing experiments using multisine excitation signals
A known challenge when building nonlinear models from data is to limit the size of the model in terms of the number of parameters. Especially for complex nonlinear systems, which require a substantial number of state variables, the classical formulation of the nonlinear part (e.g. through a basis expansion) tends to lead to a rapid increase in the model size. In this work, we propose two strategies to counter this effect: 1) The introduction of a novel nonlinear-state selection algorithm. The method relies on the non-parametric nonlinear distortion analysis of the Best Linear Approximation framework to identify the state variables which are the most impacted by nonlinearities. Pre-selecting only the most appropriate states when constructing the nonlinear terms results in a considerable reduction of the model size. 2) The use of so-called 'decoupled' functions directly in the model estimation procedure. While it is known that function decoupling can reduce the model size in a secondary step, we show how a decoupled formulation can be imposed to advantage from the start. The results of this approach are benchmarked with the state-of-the-art a posteriori decoupling technique. Our strategies are demonstrated on real-life data of a multiple-input, multiple-output (MIMO) ground vibration test of an F-16 aircraft, a prime complex and nonlinear dynamic system. 1 INTRODUCTION Engineers and scientists want mathematical models of the observed system for understanding, design and control. Modeling nonlinear systems is essential because many systems are inherently nonlinear. The challenge lies in the fact that there are several differently behaving nonlinear structures and therefore modeling is very involved. As it becomes increasingly important to cope with nonlinear analysis and modeling, various approaches have been proposed; for a detailed overview we refer to [1] [2] and [3]. In this work, we propose a data-driven nonlinear modeling procedure where we build upon a number of well-known, matured, system identification techniques, and add two novel tools in order to overcome some of the drawbacks of the classical approach. In doing so, we provide a complete modeling strategy which allows retrieving compact nonlinear state-space models from data. The procedure combines both nonparametric and parametric nonlinear modeling techniques and is particularly useful when dealing with complex nonlinear systems, such as dynamic structures with many resonances. An important domain of application is found in the modeling of multiple-input, multiple-output (MIMO) real-life vibro-acoustic measurements. We illustrate the methodologies on a ground vibration test of an F-16 aircraft. The recommended nonlinear modeling procedure is listed below and illustrated in Figure 1. ▪ In the experiment design step, systems are excited by broadband (multisine) signals at multiple excitation levels. The recommended multisine (also known as pseudo-random noise) excitation signal consists of a series of periodic multisines that are mutually independent over the experiments. The main advantage of the recommended signals is that there is no problem with spectral leakage or transients. They deliver excellent linear models while providing useful information about the level and type of nonlinearities. ▪ In the second step, the measured signals are (nonparametrically) analyzed by applying the (multisine-driven) Best Linear Approximation (BLA) framework of MIMO systems as a generalization of the conceptual work [4]. Even though the technique works best with the recommended multisines, (with some loss of accuracy) any (orthogonal) signal can be applied. This (multisine-driven) BLA analysis differs from the classical H 1 Frequency Response Function (FRF) estimation process [5]. The key idea is to make use of the statistical features of the excitation signal. The outcome of the BLA analysis results in a series of nonparametric FRFs together with noise and nonlinear distortion estimates.
Estimating a nonlinear model from experimental measurements of a vibrating structure remains a challenge, despite huge progress in recent years. A major issue is that the dynamical behaviour of a nonlinear structure strongly depends on the magnitude of the displacement response. Thus, the validity of an identified model is generally limited to a certain range of motion. Also, outside this range, the stability of the solutions predicted by the model are not guaranteed. This raises the question as to how a nonlinear model derived using data from relatively low amplitude excitation can be used to predict the dynamical behaviour for higher amplitude excitation. This paper focuses on this problem, investigating the extrapolation capabilities of data-driven nonlinear state-space models based on a subspace approach. The experimental vibrating structure consists of a cantilever beam in which magnets are used to generate strong geometric nonlinearity. The beam is driven by an electrodynamic shaker using several levels of broadband random noise. Acceleration data from the beam tip are used to derive nonlinear state-space models for the structure. It is shown that model predictions errors generally tend to increase when extrapolating towards higher excitation levels. Furthermore, the validity of the estimated nonlinear models become poor for very strong nonlinear behaviour. Linearised models are also estimated to have a complete view of the performance of each candidate model for each level of excitation.
This work discusses the experimental challenges and processing of unsteady experiments for a pitching wing in the low-speed wind tunnel of the Vrije Universiteit Brussel. The setup used for unsteady experiments consisted of two independent devices: (a) a position control device to steer the pitch angle of the wing, and (b) a pressure measurement device to measure the aerodynamic loads. The position control setup can pitch the wing for a range of frequencies, amplitude, and offset levels. In this work, a NACA-0018 wing profile was used with an aspect ratio of 1.8. The position control and the pressure measurement setups operate independently of each other, necessitating advanced signal processing techniques to synchronize the pitch angle and the lift force. Furthermore, there is a (not well-documented) issue with the (sampling) clock frequency of the pressure measurement setup, which was resolved using a fully automated spectral analysis technique. The wing was pitched using a simple harmonic sine excitation signal at eight different offset levels (between 6° and 21°) for a fixed amplitude variation (std) of 6°. At each offset level, the wing was pitched at five different frequencies between 0.1 Hz and 2 Hz (that correspond to reduced frequencies k ranging from 0.006 to 0.125). All the experiments were conducted at a fixed chord-based Reynolds number of 2.85 × 105. The choice of operating parameters invokes the linear and nonlinear behavior of the wing. The linear unsteady measurements agreed with the analytical results. The unsteady pressure measurements at higher offset levels revealed the nonlinear aerodynamic phenomenon of dynamic stall. This confirms that a nonlinear and dynamic model is required to capture the salient characteristics of the lift force on a pitching wing.
This paper discusses the appropriate way of implementing the PID controller in the software. This work can be seen as a tutorial that teaches the important concepts of the PID controller. The PID controller equation presented in most undergraduate textbooks is in the continuous Laplace domain. To implement the PID controller in the software the continuous Laplace domain equation must be transformed into the difference equation. Once transformed into a difference equation, this equation can be implemented to develop a digital PID controller to control any closed-loop system. This concept is discussed in this paper with the help of software that is developed to control a ball balancing beam.
Scientists and engineers want accurate mathematical models of physical systems for understanding, design, and control. To obtain accurate models, persistently exciting rich signals are needed. The MUMI Matlab toolbox creates multisine signals to assess the underlying systems in a time efficient, user-friendly way. In order to avoid any spectral leakage, to reach full nonparametric characterization of the noise, and to be able to detect nonlinearities and time-variations, multisine signals can be used. By the use of the toolbox, inexperienced users can easily create state-of-the-art excitation signals to be used to perturbate almost any physical systems.
Nonlinear state-space modelling is a very powerful black-box modelling approach. However powerful, the resulting models tend to be complex, described by a large number of parameters. In many cases interpretability is preferred over complexity, making too complex models unfit or undesired. In this work, the complexity of such models is reduced by retrieving a more structured, parsimonious model from the data, without exploiting physical knowledge. Essential to the method is a translation of all multivariate nonlinear functions, typically found in nonlinear state-space models, into sets of univariate nonlinear functions. The latter is computed from a tensor decomposition. It is shown that typically an excess of degrees of freedom are used in the description of the nonlinear system whereas reduced representations can be found. The method yields highly structured state-space models where the nonlinearity is contained in as little as a single univariate function, with limited loss of performance. Results are illustrated on simulations and experiments for: the forced Duffing oscillator, the forced Van der Pol oscillator, a Bouc-Wen hysteretic system, and a Li-Ion battery model.
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.
This paper introduces a user-friendly estimation framework for industrial measurements of vibro-acoustic systems with multiple inputs. Many mechanical structures are inherently nonlinear and there is no unique solution for modeling nonlinear systems. This is especially true when multiple-input, multiple-output (MIMO) systems are considered. This paper addresses the questions related to the user-friendly semi-automatic processing of MIMO measurements with respect to the design of experiment and the analysis of the measured data. When the proposed framework is used, with a minimal user interaction, it is easily possible a) to decide, if the underlying system is linear or not, b) to decide if the linear framework is still accurate (safe) enough to be used, and c) to tell the inexperienced (non-expert) user how much can be gained using an advanced nonlinear framework. The proposed nonparametric industrial framework is illustrated on various real-life measurements.
This paper introduces a nonparametric, nonlinear system identification toolbox called SAMI (simplified analysis for multiple input systems) developed for industrial measurements of vibro-acoustic systems with multiple inputs. It addresses the questions related to the user-friendly (semi-)automatic processing of multiple-input, multiple-output measurements with respect to the design of an experiment and the analysis of the measured data. When the proposed toolbox is used, with minimal user interaction, it is easily possible (a) to decide whether the underlying system is linear or not, (b) to decide whether the linear framework is still adequate to be used, and (c) to tell an inexperienced user how much can be gained using an advanced nonlinear framework. The toolbox is illustrated on openly accessible F-16 ground vibration testing measurements.
Many mechanical structures are nonlinear and there is no unique solution for modeling nonlinear systems. When a single-input, single-output system is excited by special signals, it is easily possible to decide whether the linear framework is still accurate enough to be used, or a nonlinear framework must be used. However, for multiple-input, multiple-output (MIMO) systems, the design of experiment is not a trivial question since the input and output channels are not mutually independent. This paper shows the first results of an ongoing research project and it addresses the questions related to the user-friendly processing of MIMO measurements with respect to the design of experiment and the analysis of the measured data. When the proposed framework is used, it is easily possible a) to decide, if the underlying system is linear or not, b) to decide if the linear framework is still accurate (safe) enough to be used, and c) to tell the unexperienced user how much it can be gained using an advanced nonlinear framework. The proposed nonparametric industrial framework is illustrated on a ground vibration testing of an electrical airplane.
This paper presents an efficient nonparametric time domain nonlinear system identification method. It is shown how truncated Volterra series models can be efficiently estimated without the need of long, transient-free measurements. The method is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. To avoid the excessive memory needs in case of long measurements or large number of estimated parameters, a practical gradient-based estimation method is also provided, leading to the same numerical results as the proposed Volterra estimation method. Moreover, the transient effects in the simulated output are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Combining the proposed methodologies, the nonparametric Volterra models of the cascaded water tanks benchmark are presented in this paper. The results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series with and without transient removal are compared and studied. It is clear that the obtained models capture the system dynamics when tested on a validation dataset, and their performance is comparable with the white-box (physical) models.
Wind-tunnel experiments were conducted on two symmetric airfoils with different thickness ratios as a test of the ability of Goman–Khrabrov-type models to predict the lift coefficient history during pitching maneuvers. The primary difference between the two airfoils was that the thin airfoil had no hysteresis in its lift curve during quasi-steady maneuvers, but static hysteresis was observed with the thick airfoil over a range of 16 < α < 22 deg . Both airfoils exhibited dynamic hysteresis in the lift coefficient when the wing was pitching. The existence of static hysteresis had a strong effect on the lift response during periodic pitching and must be accounted for in the model. A modified version of the Goman–Khrabrov model was introduced that captured the static hysteresis behavior and the large-scale features of the dynamic hysteresis for the thick airfoil, even when the thick airfoil was in a deep-stall condition. Some deviations between model and experiment were observed at the highest angles of attack, which were attributed to the convection of the dynamic stall vortex and its backwash effect as it leaves the airfoil.
The ability to discover physical laws and governing equations from data is
one of humankind's greatest intellectual achievements. A quantitative
understanding of dynamic constraints and balances in nature has facilitated
rapid development of knowledge and enabled advanced technological achievements,
including aircraft, combustion engines, satellites, and electrical power. In
this work, we combine sparsity-promoting techniques and machine learning with
nonlinear dynamical systems to discover governing physical equations from
measurement data. The only assumption about the structure of the model is that
there are only a few important terms that govern the dynamics, so that the
equations are sparse in the space of possible functions; this assumption holds
for many physical systems. In particular, we use sparse regression to determine
the fewest terms in the dynamic governing equations required to accurately
represent the data. The resulting models are parsimonious, balancing model
complexity with descriptive ability while avoiding overfitting. We demonstrate
the algorithm on a wide range of problems, from simple canonical systems,
including linear and nonlinear oscillators and the chaotic Lorenz system, to
the fluid vortex shedding behind an obstacle. The fluid example illustrates the
ability of this method to discover the underlying dynamics of a system that
took experts in the community nearly 30 years to resolve. We also show that
this method generalizes to parameterized, time-varying, or externally forced
systems.
Roboticists are increasingly dealing with challenging complex problems in system identification for model-based control, and this book lays a foundation of knowledge for the reader to absorb, which can help address the said challenges. That being said, the reader should prepare for a challenge when reading this book. Though it is an excellent text, it is not for the casual reader. A background in Fourier series and Laplace transforms helps as well,though Ljung does a good job with the early chapters, providing an overview of the concepts that will be used throughout the book. This book covers parametric and nonparametric methods, parameter estimation methods in the prediction error framework, frequency domain data and interpretations, various ways to compute estimates, recursive estimation techniques, model validation,and case studies. Each chapter ends with an extensive bibliography for further reading and, often, has an appendix for proofs and derivations. The author presents a thorough and complete picture of system identification as a methodology, set of tools,and practical approach to generating models from data. The author's work has been influential for decades, and anyone interested in buildingmodels from data would be well served to put the effort into mastering thismaterial.
Dynamic stall on an oscillating airfoil was investigated by a combination of surface pressure measurements and time-resolved particle image velocimetry. Following up on previous work on the onset of dynamic stall (Mulleners and Raffel in Exp Fluids 52(3):779–793, 2012), we combined time-resolved imaging with an extensive coherent structure analysis to study various aspects of stall development. The formation of the primary dynamic stall vortex was identified as the growth of a recirculation region and the ensuing instability of the associated shear layer. The stall development can be subdivided into two stages of primary and secondary instability with the latter being the effective vortex formation stage. The characteristic time scales associated with the primary instability stage revealed an overall decrease in dynamic stall delay with increasing effective unsteadiness of the pitching airfoil. The vortex formation stage was found to be largely unaffected by variations of the airfoil’s dynamics.
The dynamic process of flow separation on a stationary airfoil in a uniform flow was investigated experimentally by means of time-resolved measurements of the velocity field and the airfoil's surface pressure distribution. The unsteady movement of the separation point, the growth of the separated region, and surface pressure fluctuations were examined in order to obtain information about the chronology of stall development. The early inception of stall is characterised with the abrupt upstream motion of the separation point. During subsequent stall development, large scale shear layer structures are formed and grow while travelling downstream. This leads to a gradual increase of the separated flow region reaching his maximum size at the end of stall development which is marked by a significant lift drop.
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.
Nonlinear system identification is an extremely broad topic, since every system that is not linear is nonlinear. That makes it impossible to give a full overview of all aspects of the fi eld. For this reason, the selection of topics and the organization of the discussion are strongly colored by the personal journey of the authors in this nonlinear universe.
Accurate unsteady aerodynamic models are essential to estimate the forces on rapidly pitching wings and to develop model-based controllers. As system identification is arguably the most successful framework for model predictive control in general, in this paper we investigate whether system identification can be used to build data-driven models of pitching wings. The forces acting on the pitching wing can be considered a nonlinear dynamic function of the pitching angle and therefore require a nonlinear dynamic model. In this work, a nonlinear data-driven model is developed for a pitching wing. The proposed model structure is a polynomial nonlinear state-space model (PNLSS), which is an extension of the classical linear state-space model with nonlinear functions. The PNLSS model is trained on experimental data of a pitching wing. The experiments are performed using a dedicated wind tunnel setup. The pitch angle is considered as the input to the model, while the lift coefficient is considered as the output. Three models are trained on swept-sine signals at three offset angles with a fixed pitch amplitude and a range of reduced frequencies. The three training datasets are selected to cover the linear and nonlinear operating regimes of the pitching wing. The PNLSS models are validated on single-sine experimental data at the respective pitch offset angles. The PNLSS models are able to capture the nonlinear aerodynamic forces more accurately than a linear and semi-empirical models, especially at higher offset angles.
This paper introduces a user-friendly estimation toolbox for (industrial) measurements of (vibro-acoustic) systems with multiple inputs. The vibration testing methods are very important because they help to improve the product quality and to avoid safety and comfort issues. The time-consuming testing procedures are nowadays fully substituted by techniques that evaluates the frequency response functions (FRFs).
The toolbox provides a user-friendly nonparametric estimation method that uses a special combined Local Polynomial/Rational approach. The toolbox can be used even in case of multiple inputs with only one measured period – that would be not possible with the classical FRF estimator frameworks.
This is a demonstration of the PNLSS Toolbox 1.0. The toolbox is designed to identify polynomial nonlinear state-space models from data. Nonlinear state-space models can describe a wide range of nonlinear systems. An illustration is provided on experimental data of an electrical system mimicking the forced Duffing oscillator, and on numerical data of a nonlinear fluid dynamics problem.
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.
Aerodynamic modeling plays an important role in multiphysics and design problems, in addition to experiment and numerical simulation, due to its low-dimensional representation of unsteady aerodynamics. However, in the traditional study of aerodynamics, developing aerodynamic and flow models relies on classical theoretical (potential flow) and empirical investigation, which limits the accuracy and extensibility. Recently, with significant progress in high-fidelity computational fluid dynamic simulation and advanced experimental techniques, very large and diverse fluid data becomes available. This rapid growth of data leads to the development of data-driven aerodynamic and flow modeling. Through advanced mathematical methods from control theory, data science and machine learning, a lot of data-driven aerodynamic models have been proposed. These models are not only more accurate than theoretical models, but also require very low computational cost compared with numerical simulation. At the same time, they help to gain physical insights on flow mechanism, and have shown great potential in engineering applications like flow control, aeroelasticity and optimization. In this review paper, we introduce three typical data-driven methods, including system identification, feature extraction and data fusion. In particular, main approaches to improve the performance of data-driven models in accuracy, stability and generalization capability are reported. The efficacy of data-driven methods in modeling unsteady aerodynamics is described by several benchmark cases in fluid mechanics and aeroelasticity. Finally, future development and potential applications in related areas are concluded.
Pitching airfoil measurements are known to exhibit significant scatter near stall angles of attack that can make meaningful correlation with modeling or simulation difficult. Application of data-driven clustering algorithms to dynamic stall experiments, conducted at two different research facilities, revealed the presence of furcation within the data scatter. Such furcation render the statistical mean and standard deviation as inadequate to represent the observed cycle-to-cycle variations. After ruling out facility effects, an alternative approach to conventional statistical analysis is developed through the use of cluster averages, associated variances, and group probability. Several existing clustering techniques are tested; however, their shortcomings led to the development of two new data-driven algorithms that use proper orthogonal decomposition to cluster data based on flow phenomena that contribute the most energy to flow variations. Several test cases are used to show the physical mechanisms leading to cycle-to-cycle variations, such as differences in separation location, boundary layer reattachment, occurrence of leading-edge/trailing-edge stall, and presence of a dynamic stall vortex (or vortices). In all cases, these physical processes and their effects are obscured by conventional phase-averaging. Further analyses on the effects of the Mach number, reduced frequency, mean angle, and amplitude of oscillation reveal trends in the probability of a given flow behavior. An initial step towards using these results for advanced semiempirical models using Markov analysis is discussed.
In this paper, we consider the unsteady aerodynamics of a two-dimensional airfoil as a dynamical system whose input is the angle of attack (or airfoil motion) and output is the lift force. Based on this view, we discuss the evolution of lift and circulation from a purely dynamical perspective through step response, frequency response, transfer function, etc. In particular, we point to the relation between the high-frequency gain of the transfer function and the physics of the development of lift and circulation. Based on this view, we show that the circulatory lift dynamics is different from the circulation dynamics. That is, we show that the circulatory lift is not lift due to circulation. In fact, we show that the circulatory–non-circulatory classification is arbitrary. By comparing the steady and unsteady thin airfoil theory, we show that the circulatory lift possesses some acceleration (added-mass) effects. Finally, we perform simulations of Navier–Stokes equations to show that a non-circulatory maneuver in the absence of a free stream induces viscous circulation over the airfoil.
The dynamic stall development on a pitching airfoil at Re = 10 6 was investigated by time-resolved surface pressure and velocity field measurements. Two stages were identified in the dynamic stall development based on the shear layer evolution. In the first stage, the flow detaches from the trailing edge and the separation point moves gradually upstream. The second stage is characterized by the roll up of the shear layer into a large scale dynamic stall vortex. The two-stage dynamic stall development was independently confirmed by global velocity field and local surface pressure measurements around the leading edge. The leading edge pressure signals were combined into a single leading edge suction parameter. We developed an improved model of the leading edge suction parameter based on thin airfoil theory that links the evolution of the leading edge suction and the shear layer growth during stall development. The shear layer development leads to a change in the effective camber and the effective angle of attack. By taking into account this twofold influence, the model accurately predicts the value and timing of the maximum leading edge suction on a pitching airfoil. The evolution of the experimentally obtained leading edge suction was further analyzed for various sinusoidal motions revealing an increase in the critical value of the leading edge suction parameter with increasing pitch unsteadiness. The characteristic dynamic stall delay decreases with increasing unsteadiness, and the dynamic stall onset is best assessed by critical values of the circulation and the shear layer height which are motion independent. Published under license by AIP Publishing. https://doi.org/10.1063/1.5121312 ., s
Turbulence is one of the main characteristics of urban flows, but its effect on the performance of small VAWTs (often used in urban wind installations) has not been thoroughly researched. This experimental study focuses in testing a H-Darrieus VAWT prototype in different turbulent conditions inside the wind tunnel, in order to study the influence of turbulence intensity, integral length scales and Reynolds number. Passive grids are used to increase the wind tunnel free stream turbulence up to Iu = 15%, with integral length scales of Lux = 0.18 m and Reynolds numbers of ReD ≈ 300,000. The measurements are repeated in two wind tunnels of different size, which strengthens the results and helps quantify the effect of blockage on the turbine ratings. The results show power coefficient increases up to 20% from smooth (Iu = 0.5%) to turbulent (Iu = 15%) flows, an effect that is enhanced at low λ but that fades as ReD > 400,000. The study of the crossed influence of ReD, Iu and Lux offers valuable data in the process of optimizing the operation of small VAWTs inside urban environments.
An investigation of dynamic stall on a pitching NACA 0012 aspect ratio 4 wing is performed by means of high-fidelity wall-resolved large-eddy simulations. The flow parameters are freestream Mach number M∞=0.1 and chord Reynolds number Rec=2×105. Three cycles of a sinusoidal pitching motion are considered with reduced frequency k=πfc/U∞=π/16 and minimum and maximum angles of attack of 4 and 22 deg, respectively. Details of the unsteady flow structure are elucidated to provide a model of three-dimensional dynamic stall at higher Reynolds number than previously studied using large-eddy simulation. The initial process involves the upstream movement of transition, the formation of a short laminar separation bubble (LSB), and the emergence of a dynamic stall vortex (DSV) following LSB bursting. The DSV is initially fairly regular in the spanwise direction; however, as it propagates downstream, it is distorted into a Λ-type shape with legs pinned at the wing front corners. A marked kink appears in the DSV core outboard and near the wing surface. This induces a circulatory motion that self-intensifies, leading to the formation of an arch-type vortex. The legs of the arch vortex move downstream and inboard, leaving a distinct imprint on the instantaneous surface pressure. Eventually, through a reconnection process, the arch vortex transforms into a ringlike structure that is shed into the wake. The effect of a simulated endwall is also investigated because this arrangement is used in experiments. It is found that, during the early stages of the DSV, the endwall has no significant impact on the flow structure and loading; however, following the formation of the arch vortex, the presence of the wall has a pronounced effect.
Nonlinear system identification is a vast research field, today attracting a great deal of attention in the structural dynamics community. Ten years ago, an MSSP paper reviewing the progress achieved until then [1] concluded that the identification of simple continuous structures with localised nonlinearities was within reach. The past decade witnessed a shift in emphasis, accommodating the growing industrial need for a first generation of tools capable of addressing complex nonlinearities in larger-scale structures. The objective of the present paper is to survey the key developments which arose in the field since 2006, and to illustrate state-of-the-art techniques using a real-world satellite structure. Finally, a broader perspective to nonlinear system identification is provided by discussing the central role played by experimental models in the design cycle of engineering structures.
Block-oriented models are popular in nonlinear modeling because of their advantages to be quite simple to understand and easy to use. Many different identification approaches were developed over the years to estimate the parameters of a wide range of block-oriented models. One class of these approaches uses linear approximations to initialize the identification algorithm. The best linear approximation framework and the -approximation framework, or equivalent frameworks, allow the user to extract important information about the system, guide the user in selecting good candidate model structures and orders, and they prove to be a good starting point for nonlinear system identification algorithms. This paper gives an overview of the different block-oriented models that can be modeled using linear approximations, and of the identification algorithms that have been developed in the past. A non-exhaustive overview of the most important other block-oriented system identification approaches is also provided throughout this paper.
This article addresses the following problems: 1) First, a nonlinearity analysis is made looking for the presence of nonlinearities in an early phase of the identification process. The level and the nature of the nonlinearities should be retrieved without a significant increase in the amount of measured data. 2) Next it is studied if it is safe to use a linear system identification approach, even if the presence of nonlinear distortions is detected. The properties of the linear system identification approach under these conditions are studied, and the reliability of the uncertainty bounds is checked. 3) Eventually, tools are provided to check how much can be gained if a nonlinear model were identified instead of a linear model. Addressing these three questions forms the outline of this article. The possibilities and pitfalls of using a linear identification framework in the presence of nonlinear distortions will be discussed and illustrated on lab-scale and industrial examples. In this article, the focus is on nonparametric and parametric black box identification methods, however the results might also be useful for physical modeling methods. Knowing the actual nonlinear distortion level can help to choose the required level of detail that is needed in the physical model. This will strongly influence the modeling effort. Also, in this case, significant time can be saved if it is known from experiments that the system behaves almost linearly. The converse is also true. If the experiments show that some (sub-)systems are highly nonlinear, it helps to focus the physical modeling effort on these critical elements.
This paper examines the dynamic stalling of a finite wing of aspect ratio 3.0 when subject to constant pitch motions up to and beyond stall. In particular, unsteady surface pressure data were obtained at 192 locations on the wing surface and these were then analysed to provide information on the nature and phasing of dynamic stall events in both the chordwise and spanwise directions. It was also possible to obtain sectional force and moment coefficients by integration of the pressures measured on specific chordal arrays. This provided valuable insight into the load distribution on the wing throughout the range of motion. On this basis, it was established that the wing loading distribution was consistent with conventional understanding of steady wing loading up to the incidence at which the dynamic stall vortex was initiated. Beyond this point, the formation and subsequent convection of the vortex structure was found to be strongly three-dimensional but, nevertheless, exhibited many of the features of two-dimensional dynamic stall.
A system of differential equations relating the aerodynamics forces and the variables defining the velocity of the airfoil section is employed to simulate the time delay effects of the flow. The model involves a number of identifications of test results of a 2-D airfoil in static and in small-amplitude harmonic oscillation or random vibration configurations. Tests at high-amplitude motions then permit a verification of the validity of the model.
A semi-empirical model is formulated to represent the unsteady lift, drag, and pitching moment characteristics of an airfoil undergoing dynamic stall. The model is presented in a form which is consistent with an indicial formulation for the unsteady aerodynamics under attached flow conditions. The onset of vortex shedding during dynamic stall is represented using a criterion for leading edge or shock induced separation based on the attainment of a critical leading edge pressure. The induced vortex lift is represented empirically along with the associated pitching moment which is obtained by allowing the center of pressure to move in a time dependent manner during dynamic stall. Significant nonlinearities in the airfoil behavior associated with trailing edge separation are represented using a Kirchhoff flow model in which the separation point is related to the airfoil behavior. These effects are represented in such a way as to allow progressive transition between the dynamic stall and the static stall characteristics. It is shown how the above features may be implemented as an algorithm suitable for inclusion within rotorcraft airloads or aeroelasticity analyses. Validation of the model is presented with force and moment data from two-dimensional unsteady tests on the NACA 0012, HH-02, and SC-1095 airfoils.
To prevent unstable behavior or saturation, a frequency response function (FRF) measurement is often performed under closed-loop conditions (e.g., open-loop gain measurements of an operational amplifier). The difficulty of such FRF measurements is that the nonlinear (NL) distortions also perturb the input via the feedback loop. The latter introduces a bias in the estimate of the best linear approximation (BLA) and jeopardizes the interpretation of the output NL distortions. In this paper, we solve these problems via a generalized definition of the BLA that is valid for NL systems operating in feedback. The classical definition for open-loop systems follows as a special case.
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.
Mathematical modeling of unsteady aerodynamic forces and moments plays an important role in aircraft dynamics investigation and stability analysis at high angles of attack. In this article the state-space representation of aerodynamic forces and moments for unsteady aircraft motion is proposed. Consideration of separated flow about an airfoil and flow with vortex breakdown about a slender delta wing gives the base for mathematical modeling using internal variables describing the flow state. Coordinates of separation points or vortex breakdown can be taken, e.g., as internal state-space variables. These variables are governed by some differential equations. Within the framework of the proposed mathematical model it is possible to achieve good agreement with different experimental data obtained in water and wind tunnels. These high angle-of-attack experimental results demonstrate considerable dependence of aerodynamic loads on motion time history.
A probabilistic network is a graphical model that encodes probabilistic relationships between variables of interest. Such a model records qualitative influences between variables in addition to the numerical parameters of the probability distribution. ...
This survey paper contains a review of the past and recent developments in system identification of nonlinear dynamical structures. The objective is to present some of the popular approaches that have been proposed in the technical literature, to illustrate them using numerical and experimental applications, to highlight their assets and limitations and to identify future directions in this research area. The fundamental differences between linear and nonlinear oscillations are also detailed in a tutorial.
This paper presents a new method which non-parametrically estimates a two dimensional impulse response function hLTV(t, τ) of slowly time-varying systems. A generalized B-spline technique is used for double smoothing: once over the different excitation times τ (which refers to the system memory) and once over the actual excitation time t (referring to the system behavior). If the change of the parameters of the observed system is sufficiently slow, with respect to the system dynamics, we will be able to 1) reduce the disturbing noise by additional smoothing 2) reduce the number of model parameters that need to be stored.
Electrochemical impedance spectroscopy (EIS) is a powerful technique to study electrochemical processes and to perform screening tasks. Recently an integrated measuring and modeling methodology for EIS based on a multisine excitation signal was developed. A key issue in this methodology is the data analysis, allowing us to rapidly quantify the reliability of the measured data. In this paper, a comparison is made between classical single-sine and the proposed multisine measurements on the same system. The fitting of the impedance data obtained by single-or multisine excitation and using different weighting factors is also discussed. In addition to the advantages reported in earlier work, it is concluded that, of all investigated frequencies, the odd random phase multisine excitation yields the highest quality data in the shortest measurement time.
Recently [1] a method has been developed to suppress nonparametrically the noise (and system) transients (leakage errors) in frequency response function and noise (co-)variance estimates of single-input, single-output systems excited by periodic signals. This paper extends the results of [1] to multiple-input, multiple-output systems where all inputs and outputs are disturbed by noise (i.e. an errors-in-variables framework). Two methods are presented: the first starts from multiple experiments with uncorrelated sets of inputs, and makes no assumption about the frequency response matrix (FRM); while the second only requires one single experiment, but assumes that the FRM can locally be approximated by a polynomial. Both methods estimate simultaneously the FRM, the noise level, and the level of the nonlinear distortions. For lightly damped systems, the proposed methods either significantly reduce the experiment duration or, for a given measurement time, significantly increase the frequency resolution of the FRM estimate. If the noise (and/or system) transients are the dominant error sources, then the proposed methods also significantly reduce the covariance matrix of the FRM estimates. The use of the nonparametric noise covariance estimates for parametric transfer function modelling is also discussed in detail.
A model predictive control technique based on a step response model is developed using state estimation techniques. The standard step response model is extended so that integrating systems can be treated within the same framework. Based on the modified step response model, it is shown how the state estimation techniques from stochastic optimal control can be used to construct the optimal prediction vector without introducing significant additional numerical complexity. In the case of integrated or double integrated white noise disturbances filtered through first-order dynamics and white measurement noise, the optimal filter gain is parametrized explicitly in terms of a single parameter between 0 and 1, thus removing the requirement for solving a Riccati equation and equipping the control system with useful on-line tuning parameters. Parallels are drawn to the existing MPC techniques such as Dynamic Matrix Control (DMC), Internal Model Control (IMC) and Generalized Predictive Control (GPC).
In [3] the measurement and modelling of linear systems in the presence of non-linear distortions has been studied for a special class of periodic excitation signals. This paper extends the theory of [3] to more general classes of (periodic) excitation signals. Also, enhanced properties of the best linear approximation and the stochastic non-linear distortions are obtained. The theory is illustrated on a simulation and a real measurement example.