Model based learning of sigma points in unscented Kalman filtering

The unscented Kalman filter (UKF) is a widely used method in control and time series applications. The UKF suffers from arbitrary parameters necessary for sigma point placement, potentially causing it to perform poorly in nonlinear problems. We show how to treat sigma point placement in a UKF as a learning problem in a model based view. We demonstrate that learning to place the sigma points correctly from data can make sigma point collapse much less likely. Learning can result in a significant increase in predictive performance over default settings of the parameters in the UKF and other filters designed to avoid the problems of the UKF, such as the GP-ADF. At the same time, we maintain a lower computational complexity than the other methods. We call our method UKF-L. © 2011 Elsevier B.V.

Measurements of triggering and transient growth in a model lean-premixed gas turbine combustor

Instability triggering and transient growth of thermoacoustic oscillations were experimentally investigated in combination with linear/nonlinear flame transfer function (FTF) methodology in a model lean-premixed gas turbine combustor operated with CH 4 and air at atmospheric pressure. A fully premixed flame with 10kW thermal power and an equivalence ratio of 0.60 was chosen for detailed characterization of the nonlinear transient behaviors. Flame transfer functions were experimentally determined by simultaneous measurements of inlet velocity fluctuations and heat release rate oscillations using a constant temperature anemometer and OH */CH * chemiluminescence emissions, respectively. The phase-resolved variation of the local flame structure at a limit cycle was measured by planar laser-induced fluorescence of OH. Simultaneous measurements of inlet velocity, OH */CH * emission, and acoustic pressure were performed to investigate the temporal evolution of the system from a stable to a limit cycle operation. This measurement allows us to describe an unsteady instability triggering event in terms of several distinct stages: (i) initiation of a small perturbation, (ii) exponential amplification, (iii) saturation, (iv) nonlinear evolution of the perturbations towards a new unstable periodic state, (v) quasi-steady low-amplitude periodic oscillation, and (vi) fully-developed high-amplitude limit cycle oscillation. Phase-plane portraits of instantaneous inlet velocity and heat release rate clearly show the presence of two different attractors. Depending on its initial position in phase space at infinitesimally small amplitude, the system evolves towards either a high-amplitude oscillatory state or a low-amplitude oscillatory state. This transient phenomenon was analyzed using frequency- and amplitude-dependent damping mechanisms, and compared to subcritical and supercritical bifurcation theories. The results presented in this paper experimentally demonstrate the hypothesis proposed by Preetham et al. based on analytical and computational solutions of the nonlinear G-equation [J. Propul. Power 24 (2008) 1390-1402]. Good quantitative agreement was obtained between measurements and predictions in terms of the conditions for the onset of triggering and the amplitude of triggered combustion instabilities. © 2011 The Combustion Institute.

Tsunami wave runup on the clustered islands using boussinesq-type model

This paper describes the implementation of the Boussinesq-type model and extends its application to the tsunami wave runup on the clustered islands (multiple adjacent conical islands), in turn, an extensively validated two-dimensional Boussinesq-type model is employed to examine the interaction between a propagating solitary wave and multiple idealised conical islands, with particular emphasis on a combination effect of two adjustable parameters for spacing interval/diameter ratio between the adjacent conical islands, S/D, and the rotating angle of the structural configuration,θ on maximum soliton runup heights. An extensive parameter study concerning the combination effect of alteringθ and S/D on the maximum soliton runup with the multi-conical islands is subsequently carried out and the distributions of the maximum runup heights on each conical island are obtained and compared for the twin-island cases. The worst case study is performed for each case in respect of the enhancement in the maximum wave runup heights by the multi-conical islands. It is found that the nonlinear wave diffraction, reflection and refraction play a significant role in varying the maximum soliton runup heights on multiconical islands. The comparatively large maximum soliton runups are generally predicted for the merged and bottom mounted clusteredislands. Furthermore, the joints of the clustered-merged islands are demonstrated to suffer the most of the tsunami wave attack. The conical islands that position in the shadow regions behind the surrounding islands are found to withstand relatively less extreme wave impact. Although, these numerical investigations are considerable simplifications of the multi conical islands, they give a critical insight into certain important hydrodynamic characteristics of the interaction between an extreme wave event and a group of clustered conical islands, and thus providing a useful engineering guidance for extreme wave mitigation and coastal development. Copyright © 2012 by the International Society of Offshore and Polar Engineers (ISOPE).

Mixed nonlinear LES for DES suitable flows

It is shown that a new mixed nonlinear/eddy viscosity LES model reproduces profiles better than a number of competing nonlinear and mixed models for plane channel flow. The objective is an LES method that produces a fully resolved turbulent boundary layer and could be applied to a variety of aerospace problems that are currently studied with RANS, RANS-LES, or DES methods that lack a true turbulent boundary layer. There are two components to the new model. One an eddy viscosity based upon the advected subgrid scale energy and a relatively small coefficient. Second, filtered nonlinear terms based upon the Leray regularization. Coefficients for the eddy viscosity and nonlinear terms come from LES tests in decaying, isotropic turbulence. Using these coefficients, the velocity profile matches measurements data at Reτ ≈ 1000 exactly. Profiles of the components of kinetic energy have the same shape as in the experiment, but the magnitudes differ by about 25%. None of the competing LES gets the shape correct. This method does not require extra operations at the transition between the boundary layer and the interior flow.

Nonlinear optics in Silicon photonic crystal cavities

Silicon is known to be a very good material for the realization of high-Q, low-volume photonic cavities, but at the same it is usually considered as a poor material for nonlinear optical functionalities like second-harmonic generation, because its second-order nonlinear susceptibility vanishes in the dipole approximation. In this work we demonstrate that nonlinear optical effects in silicon nanocavities can be strongly enhanced and even become macroscopically observable. We employ photonic crystal nanocavities in silicon membranes that are optimized simultaneously for high quality factor and efficient coupling to an incoming beam in the far field. Using a low-power, continuous-wave laser at telecommunication wavelengths as a pump beam, we demonstrate simultaneous generation of second- and third harmonics in the visible region, which can be observed with a simple camera. The results are in good agreement with a theoretical model that treats third-harmonic generation as a bulk effect in the cavity region, and second-harmonic generation as a surface effect arising from the vertical hole sidewalls. Optical bistability is also observed in the silicon nanocavities and its physical mechanisms (optical, due to two-photon generation of free carriers, as well as thermal) are investigated. © 2011 IEEE.

Studying adsorbent dynamics on a quartz crystal resonator using its nonlinear electrical response

The quartz crystal resonator has been traditionally employed in studying surface-confined physisorbed films and particles by measuring dissipation and frequency shifts. However, theoretical interpretation of the experimental observations is often challenged due to limited understanding of physical interaction mechanisms at the interfaces involved. Here we model a physisorbed interaction between particles and gold electrode surface of a quartz crystal and demonstrate how the nonlinear modulation of the electric response of the crystal due to the nonlinear interaction forces may be used to study the dynamics of the particles. In particular, we show that the graphs of the deviation in the third Fourier harmonic response versus oscillation amplitude provide important information about the onset, progress and nature of sliding of the particles. The graphs also present a signature of the surface-particle interaction and could be used to estimate the interaction energy profile. Interestingly, the insights gained from the model help to explain some of the experimental observations with physisorbed streptavidin-coated polystyrene microbeads on quartz resonators. © 2012 Elsevier B.V. All rights reserved.

Nonlinear phenomena in thermoacoustics: A comparison between single-mode and multi-mode methods

Nonlinear analysis of thermoacoustic instability is essential for prediction of frequencies and amplitudes of limit cycles. In frequency domain analyses, a quasi-linear transfer function between acoustic velocity and heat release rate perturbations, called the flame describing function (FDF), is obtained from a flame model or experiments. The FDF is a function of the frequency and amplitude of velocity perturbations but only contains the heat release response at the forcing frequency. While the gain and phase of the FDF provide insight into the nonlinear dynamics of the system, the accuracy of its predictions remains to be verified for different types of nonlinearity. In time domain analyses, the governing equations of the fully coupled problem are solved to find the time evolution of the system. One method is to discretize the governing equations using a suitable basis, such as the natural acoustic modes of the system. The number of modes used in the discretization alters the accuracy of the solution. In our previous work we have shown that predictions using the FDF are almost exactly the same as those obtained from the time-domain using only one mode for the discretization. We call this the single-mode method. In this paper we compare results from the single-mode and multi-mode methods, applied to a thermoacoustic system of a premixed flame in a tube. For some cases, the results differ greatly in both amplitude as well as frequency content. This study shows that the contribution from higher and subharmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems. Hence multi-mode simulations are necessary, and the single-mode method or the FDF may be insufficient to capture some of the complex nonlinear behaviour in fhermoacoustics.

A flexural crack model for damage detection in reinforced concrete structures

The use of changes in vibration data for damage detection of reinforced concrete structures faces many challenges that obstruct its transition from a research topic to field applications. Among these is the lack of appropriate damage models that can be deployed in the damage detection methods. In this paper, a model of a simply supported reinforced concrete beam with multiple cracks is developed to examine its use for damage detection and structural health monitoring. The cracks are simulated by a model that accounts for crack formation, propagation and closure. The beam model is studied under different dynamic excitations, including sine sweep and single excitation frequency, for various damage levels. The changes in resonant frequency with increasing loads are examined along with the nonlinear vibration characteristics. The model demonstrates that the resonant frequency reduces by about 10% at the application of 30% of the ultimate load and then drops gradually by about 25% at 70% of the ultimate load. The model also illustrates some nonlinearity in the dynamic response of damaged beams. The appearance of super-harmonics shows that the nonlinearity is higher when the damage level is about 35% and then decreases with increasing damage. The restoring force-displacement relationship predicted the reduction in the overall stiffness of the damaged beam. The model quantitatively predicts the experimental vibration behaviour of damaged RC beams and also shows the damage dependency of nonlinear vibration behaviour. © 2011 Published under licence by IOP Publishing Ltd.

Seafloor-riser interaction model

Fatigue stresses associated with extreme storms, vessel movements, and vortex-induced vibrations are critical to the performance of steel catenary risers. The critical location for fatigue damage often occurs within the touchdown zone, where cyclic interaction of the riser with the seabed occurs. Developing a model for seabed stiffness requires characterization of a number of complex nonlinear processes including trench formation, nonlinear soil stiffness, soil suction, and breakaway of the riser from the seafloor. The analytical framework utilized in this research considers the riser-seafloor interaction problem in terms of a pipe resting on a bed of springs, the stiffness characteristics of which are described by nonlinear load-deflection (P-y) curves. The P-y model allows for first penetration and uplift, as well as repenetration and small range motions within the bounding loop defined by extreme loading. The backbone curve is constructed from knowledge of the soil strength, the rate of strength increase with depth, trench width, and two additional parameters, while three parameters are necessary for the cyclic response. © ASCE 2009.

Interaction model for steel compliant riser on soft seabed

The use of catenary steel-compliant-riser (SCR) systems has increased as hydrocarbon production has moved progressively farther offshore and into deeper waters. The issue of fatigue damage caused by cyclic interaction of a riser with the seabed has gained prominence with the widespread use of SCRs and with the lengthening of the spans. The problem involves a number of complex factors, including trench configuration, nonlinear soil stiffness, breakaway of the riser from the seafloor, and degradation of soil resistance during cyclic loading. This paper presents a soilinteraction model capable of modeling these complexities, using input parameters that can be obtained with reasonable expenditure. Model simulations for typical offshore soft-soil conditions indicate that the model is capable of realistic predictions of cyclic bending moments. The degradation of soil resistance has a major effect on cyclic bending moments, particularly when uplift motions at the riser touchdown point (TDP) are large. © 2008 Society of Petroleum Engineers.

Wave propagation due to sinusoidal excitation in nonlinear one-dimensional soil column

The objective of the author's on-going research is to explore the feasibility of determining reliable in situ curves of shear modulus as a function of strain using the dynamic test. The purpose of this paper is limited to investigating what material stiffness is measured from a dynamic test, focusing on the harmonic excitation test. A one-dimensional discrete model with nonlinear material properties is used for this purpose. When a sinusoidal load is applied, the cross-correlation of signals from different depths estimates a wave velocity close to the one calculated from the secant modulus in the stress-strain loops under steady-state conditions. The variables that contributed to changing the average slope of the stress-strain loop also influence the estimate of the wave velocity from cross-correlation. Copyright ASCE 2007.

Nonlinear drillstring dynamics analysis

This paper studies the dynamical response of a rotary drilling system with a drag bit, using a lumped parameter model that takes into consideration the axial and torsional vibration modes of the bit. These vibrations are coupled through a bit-rock interaction law. At the bit-rock interface, the cutting process introduces a state-dependent delay, while the frictional process is responsible for discontinuous right-hand sides in the equations governing the motion of the bit. This complex system is characterized by a fast axial dynamics compared to the slow torsional dynamics. A dimensionless formulation exhibits a large parameter in the axial equation, enabling a two-time-scales analysis that uses a combination of averaging methods and a singular perturbation approach. An approximate model of the decoupled axial dynamics permits us to derive a pseudoanalytical expression of the solution of the axial equation. Its averaged behavior influences the slow torsional dynamics by generating an apparent velocity weakening friction law that has been proposed empirically in earlier work. The analytical expression of the solution of the axial dynamics is used to derive an approximate analytical expression of the velocity weakening friction law related to the physical parameters of the system. This expression can be used to provide recommendations on the operating parameters and the drillstring or the bit design in order to reduce the amplitude of the torsional vibrations. Moreover, it is an appropriate candidate model to replace empirical friction laws encountered in torsional models used for control. © 2009 Society for Industrial and Applied Mathematics.

Slow peaking and low-gain designs for global stabilization of nonlinear systems

This paper presents an analysis of the slow-peaking phenomenon, a pitfall of low-gain designs that imposes basic limitations to large regions of attraction in nonlinear control systems. The phenomenon is best understood on a chain of integrators perturbed by a vector field up(x, u) that satisfies p(x, 0) = 0. Because small controls (or low-gain designs) are sufficient to stabilize the unperturbed chain of integrators, it may seem that smaller controls, which attenuate the perturbation up(x, u) in a large compact set, can be employed to achieve larger regions of attraction. This intuition is false, however, and peaking may cause a loss of global controllability unless severe growth restrictions are imposed on p(x, u). These growth restrictions are expressed as a higher order condition with respect to a particular weighted dilation related to the peaking exponents of the nominal system. When this higher order condition is satisfied, an explicit control law is derived that achieves global asymptotic stability of x = 0. This stabilization result is extended to more general cascade nonlinear systems in which the perturbation p(x, v) v, v = (ξ, u) T, contains the state ξ and the control u of a stabilizable subsystem ξ = a(ξ, u). As an illustration, a control law is derived that achieves global stabilization of the frictionless ball-and-beam model.

Delayed control of a Moore-Greitzer axial compressor model

Several feedback control laws have appeared in the literature concerning the stabilization of the nonlinear Moore-Greitzer axial compression model. Motivated by magnitude and rate limitations imposed by the physical implementation of the control law, Larsen et al. studied a dynamic implementation of the S-controller suggested by Sepulchre and Kokotović. They showed the potential benefit of implementing the S-controller through a first-order lag: while the location of the closed-loop equilibrium achieved with the static control law was sensitive to poorly known parameters, the dynamic implementation resulted in a small limit cycle at a very desirable location, insensitive to parameter variations. In this paper, we investigate the more general case when the control is applied with a time delay. This can be seen as an extension of the model with a first-order lag. The delay can either be a result of system constraints or be deliberately implemented to achieve better system behavior. The resulting closed-loop system is a set of parameter-dependent delay differential equations. Numerical bifurcation analysis is used to study this model and investigate whether the positive results obtained for the first-order model persist, even for larger values of the delay.