Third-order Volterra Model Research Articles

A Data-Based Compact High-Order Volterra Model for Complex Blast Furnace System

In this article, a data-based compact Volterra model is constructed to carry out the silicon prediction task. The main motivations are to, on the one hand, make better use of the strong memory ability of the Volterra series, which is suitable for reflecting the large inertia of the blast furnace process, on the other hand, overcome the difficulty that high complexity of the original Volterra models may result in serious overfitting. Six kinds of models, including single-input and two-input linear, second-order and third-order compact Volterra models, are successively designed and verified through a real blast furnace case. The reasonable agreement between the predicted values and the observed values indicates the compact Volterra model, especially the single-input third-order Volterra model, are powerful and competitive for describing the complex blast furnace system. The experimental results can serve as a guide for the blast furnace operators to judge the in-furnace thermal state change in time and further provide an indication on how to control the blast furnace in advance.

Nonlinear identification of the total baroreflex arc: higher-order nonlinearity

The total baroreflex arc is the open-loop system relating carotid sinus pressure (CSP) to arterial pressure (AP). The nonlinear dynamics of this system were recently characterized. First, Gaussian white noise CSP stimulation was employed in open-loop conditions in normotensive and hypertensive rats with sectioned vagal and aortic depressor nerves. Nonparametric system identification was then applied to measured CSP and AP to establish a second-order nonlinear Uryson model. The aim in this study was to assess the importance of higher-order nonlinear dynamics via development and evaluation of a third-order nonlinear model of the total arc using the same experimental data. Third-order Volterra and Uryson models were developed by employing nonparametric and parametric identification methods. The R2 values between the AP predicted by the best third-order Volterra model and measured AP in response to Gaussian white noise CSP not utilized in developing the model were 0.69 ± 0.03 and 0.70 ± 0.03 for normotensive and hypertensive rats, respectively. The analogous R2 values for the best third-order Uryson model were 0.71 ± 0.03 and 0.73 ± 0.03. These R2 values were not statistically different from the corresponding values for the previously established second-order Uryson model, which were both 0.71 ± 0.03 (P > 0.1). Furthermore, none of the third-order models predicted well-known nonlinear behaviors including thresholding and saturation better than the second-order Uryson model. Additional experiments suggested that the unexplained AP variance was partly due to higher brain center activity. In conclusion, the second-order Uryson model sufficed to represent the sympathetically mediated total arc under the employed experimental conditions.

Constrained predictive control of a SISO nonlinear system based on third-order S-PARAFAC Volterra models

This paper is concerned with the identification and nonlinear predictive control approach for a nonlinear process based on a third-order reduced complexity, discrete-time Volterra model called the third-order S-PARAFAC Volterra model. The proposed model is given using the PARAFAC tensor decomposition that provides a parametric reduction compared with the conventional Volterra model. In addition, the symmetry property of the Volterra kernels allows us to further reduce the complexity of the model. These properties allow synthesizing a nonlinear model-based predictive control (NMBPC). Then we construct the general form of a new predictor and we propose an optimization algorithm formulated as a quadratic programming (QP) algorithm under linear and nonlinear constraints. The performance of the proposed third-order S-PARAFAC Volterra model and the developed NMBPC algorithm are illustrated on a numerical simulation and validated on a benchmark such as a continuous stirred-tank reactor system.

Time series prediction of nonlinear ship structural responses in irregular seaways using a third-order Volterra model

To predict the nonlinear structural responses of a ship traveling through irregular waves, a third-order Volterra model was applied based on the given irregular data. A nonlinear wave–body interaction system was identified using the nonlinear autoregressive with exogenous input (NARX) technique, which is one of the most commonly used nonlinear system identification schemes. The harmonic probing method was applied to extract the first-, second- and third-order frequency response functions of the system. To achieve this, a given set of time history data of both the irregular wave excitation and the corresponding midship vertical bending moment for a certain sea state was fed into the three-layer perceptron neural network. The network parameters are determined based on the supervised training. Next, the harmonic probing method was applied to the identified system to extract the frequency response function of each order. While applying the harmonic probing method, the nonlinear activation function (i.e., the hyperbolic tangent function) was expanded into a Taylor series for harmonic component matching. After the frequency response functions were obtained, the structural responses of the ship under an arbitrary random wave excitation were easily calculated with rapidity using a third-order Volterra series. Additionally, the methodology was validated through the in-depth analysis of a nonlinear oscillator model for a weak quadratic and cubic stiffness term, whose analytic solutions are known. It was confirmed that the current method effectively predicts the nonlinear structural response of a large container carrier under arbitrary random wave excitation.

Simulation of nonlinear bridge aerodynamics: A sparse third-order Volterra model

A sparse third-order Volterra model is utilized to simulate nonlinear bridge aerodynamics and attendant response under turbulent fluctuations. The Volterra model is pruned based on aerodynamic considerations, which significantly reduces computational effort needed for nonlinear analysis. The first-, second- and third-order Volterra kernels are identified using least-squares with the input–output pairs obtained from a numerical simulation and a wind-tunnel experiment. Both studies involving a simulation and an experiment show that the proposed sparse third-order Volterra model can adequately simulate nonlinear bridge aerodynamics with high fidelity. Finally, the robustness of the sparse Volterra model is verified by comparing it to an upgraded Volterra scheme.

Estimated electricity production for an anti-roll tank deployed offshore for use as a wave energy converter

Wave energy is the largest renewable energy resource; the total prospective annual wave energy resource offshore of Taiwan is approximately 16.9 TWh. To develop this energy potential, this study examines the anti-roll tank, which is used to reduce the excessive rolling of a ship, as a type of wave energy converter (WEC). From the perspective of energy interaction, the liquid in the tank absorbs part of the wave energy and converts it to a different type of kinetic energy during a ship's motion in the waves. Thus, for this study, a passive U-tube tank is chosen with a tank-carried buoy selected from a small craft. The liquid motion is coupled with the 5DOF equations of a ship's motion to calculate the frequency response functions (FRFs) of the tank's water height in regular waves, and then using the third-order Volterra model to construct the time histories of response at given seaway. The wave climate around Taiwan's coastal area is simulated using the Taiwan Coastal Ocean Modeling System (TaiCOMs) and the Longdong (25°05′43′′N, 121°55′21′′E), which is located in the northeastern coastal area of Taiwan, as a target testing area. Results show that the entire available electricity of 2009 was estimated to be approximately 7.2 × 104 kWh per deployed buoy.

Nonlinear Dynamic Modeling of Neuron Action Potential Threshold During Synaptically Driven Broadband Intracellular Activity

Activity-dependent variation of neuronal thresholds for action potential (AP) generation is one of the key determinants of spike-train temporal-pattern transformations from presynaptic to postsynaptic spike trains. In this study, we model the nonlinear dynamics of the threshold variation during synaptically driven broadband intracellular activity. First, membrane potentials of single CA1 pyramidal cells were recorded under physiologically plausible broadband stimulation conditions. Second, a method was developed to measure AP thresholds from the continuous recordings of membrane potentials. It involves measuring the turning points of APs by analyzing the third-order derivatives of the membrane potentials. Four stimulation paradigms with different temporal patterns were applied to validate this method by comparing the measured AP turning points and the actual AP thresholds estimated with varying stimulation intensities. Results show that the AP turning points provide consistent measurement of the AP thresholds, except for a constant offset. It indicates that 1) the variation of AP turning points represents the nonlinearities of threshold dynamics; and 2) an optimization of the constant offset is required to achieve accurate spike prediction. Third, a nonlinear dynamical third-order Volterra model was built to describe the relations between the threshold dynamics and the AP activities. Results show that the model can predict threshold accurately based on the preceding APs. Finally, the dynamic threshold model was integrated into a previously developed single neuron model and resulted in a 33% improvement in spike prediction.

Practical approach to investigate the statistics of nonlinear pressure on a high-speed ship by using the Volterra model

The present method takes the results of the verified theory for regular waves and bridges a high-order numerical model to construct the time histories of ship responses in irregular waves. The theory is based on the strip method, with the nonlinearities of the force components calculated over the hull's instantaneous wetted surface, and the numerical model adopts the nonlinear third-order Volterra model. The advantage of this method is systematically leading the contributions of the high-order harmonics excited in regular waves into the response predictions for irregular waves. This study focuses on the nonlinear behavior of the pressure on the hull surface of a high-speed vessel in irregular waves of sea state five. A series of experiments was carried out involving both regular and irregular head waves. The ISSC 1979 wave spectrum was followed, with a significant wave height ( H 1/3) of 3.2 m and an average zero up-crossing period ( T H 1 / 3 ) of 8.0 s. It was confirmed that even for the strong nonlinearities, such as the pressure responses near the waterline or near the keel on the bow zone, the present method is more realistic than that of the usual linear practice.

Experimental study on the nonlinear pressure acting on a high-speed vessel in regular waves

It is well known that the hydrodynamic responses of a high-speed vessel traveling in regular head waves of even moderate wave height can show significant nonlinear behavior, and so linear statistical techniques become insufficient for predicting the statistics of responses in irregular waves. On the other hand, it has been shown that an approximate third-order Volterra model is applicable to handling the statistics of some nonlinear seakeeping problems, such as motions and vertical hull girder loads. In the present study, the focus is on the nonlinear behavior of the pressure responses of the hull surface, especially on the pressures acting on alternately wet and dry areas near the waterline and on the bow zone with high deadrise angles that may be subject to slight impact and water pile-up effects. To clarify the validity of applying Volterra modeling to this problem, a series of experiments in regular and irregular head waves were carried out, and approximate third-order and fifth-order Volterra models with the proposed algorithm for finding frequency response functions (FRFs) were applied as a means of validation. In the present article, the first part of the validation was performed using experimental data in regular waves. It was confirmed that the third-order Volterra model has adequate accuracy to simulate deterministically the variation of pressure responses in regular waves of different wave steepness up to a wave amplitude to wavelength ratio of 0.01 even for the highly nonlinear pressures acting on the above-mentioned areas of the hull surface.

Tailored Sequence Design for Third-Order Volterra Model Identification

In this work, the problem of identifying third-order Volterra models from process input−output data is considered. As a metric of model fidelity, the prediction error variance expression is employed. Tailored input sequences are designed to estimate the linear, nonlinear diagonal, and subdiagonal kernels for the third-order Volterra model. Random binary signals (RBS) are used to estimate the second- and third-order off-diagonal kernels. These input sequences are plant-friendly, and they are designed to take advantage of the Volterra model structure. For comparison, Volterra kernels are identified using a cross-correlation approach. A mildly nonlinear system, which has been previously employed, serves as the simulation case study (Congalidis et al. AIChE J. 1989, 35, 891; and Parker et al. J. Process Control 2001, 11, 237). On the basis of both static and dynamic analyses, the tailored identification algorithm estimates kernels that capture the nonlinear system behavior. Furthermore, the estimates from this algorithm are superior to, and require less data than, estimates using the cross-correlation technique.

Non‐linear performance comparison for FD and PD SOI MOSFETs based on the integral function method and Volterra modelling

AbstractThe harmonic and intermodulation distortions of both fully‐depleted (FD) and partially‐depleted (PD) silicon‐on‐insulator (SOI) MOSFETs are studied. The analysis is based on the recently developed integral function method and the results are compared to a third‐order Volterra model of the MOSFET. This modelling helps us to understand the non‐linear mechanisms of the considered devices and to predict their frequency behaviour. The models are validated through large‐signal network analyser measurements. The devices performances are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

EFFICIENT INPUT SIGNAL DESIGN FOR THIRD-ORDER VOLTERRA MODEL IDENTIFICATION

The present work addresses the identification of third-order Volterra models from input-output process data. Building on previous studies regarding input sequence design for third-order Volterra models (Soni and Parker, 2004), a new reduced-length sequence is designed to identify third-order sub-diagonal kernels. This input sequence leads to a 30% reduction in the data required to accurately estimate the sub-diagonal kernel. Identification of second-and third-order off-diagonal kernels is carried out using a random binary sequence (RBS). These input sequences exploit the third-order Volterra model structure and use the prediction error variance expression as a measure of model fidelity. The utility of the proposed approach is demonstrated on an isothermal polymerization reactor case study. The reduced length sequence shows excellent performance and results in an 47% improvement in the sum-squared error (SSE) value of the third-order sub-diagonal kernel.

Nonlinear system analysis of renal autoregulation in normotensive and hypertensive rats.

We compared the dynamic characteristics in renal autoregulation of blood flow of normotensive Sprague-Dawley rats (SDR) and spontaneously hypertensive rats (SHR), using both linear and nonlinear systems analysis. Linear analysis yielded only limited information about the differences in dynamics between SDR and SHR. The predictive ability, as determined by normalized mean-square errors (NMSE), of a third-order Volterra model is better than for a linear model. This decrease in NMSE with a third-order model from that of a linear model is especially evident at frequencies below 0.2 Hz. Furthermore, NMSE are significantly higher in SHR than SDR, suggesting a more complex nonlinear system in SHR. The contribution of the third-order kernel in describing the dynamics of renal autoregulation in arterial blood pressure and blood flow was found to be important. Moreover, we have identified the presence of nonlinear interactions between the oscillatory components of the myogenic mechanism and tubuloglomerular feedback (TGF) at the level of whole kidney blood flow in SDR. An interaction between these two mechanisms had previously been revealed for SDR only at the single nephron level. However, nonlinear interactions between the myogenic and TGF mechanisms are not detected for SHR.

A sparse third-order orthogonal frequency-domain Volterra-like model

The purpose of this paper is twofold. The first objective is to develop a discrete frequency-domain orthogonal third-order Volterra model valid for both nonGaussian and Gaussian stationary random inputs in order to eliminate the presence of ‘interference’ terms. The second objective is to develop a sparse third-order frequency-domain Volterra model by identifying the most significant frequency-domain Volterra kernel coefficients using a given amount of raw experimental times series data. The concept of coherence function is extended to the orthogonal higher-order model in order to quantify the goodness of both the model and its constituent linear, quadratic, and cubic components and, then, is utilized as a criterion to select the most significant frequency-domain Volterra kernel coefficients to be included in the sparse Volterra model. Identification of the most significant frequency-domain Volterra coefficients is based upon a frequency-domain extension of the well-known orthogonal-search method. Finally, the practicality and feasibility of these approaches are demonstrated by utilizing them to model actual physical nonlinear systems given experimental input-output data from such systems.

On the selective counting of third-order products

A practical and efficient method is presented for the counting of third-order products generated by an arbitrary number of carriers with arbitrary power spectral shapes. In particular, three selective counting procedures (i.e. sorting by position, sorting by position and group, and sorting by position, group, and kind) are introduced in a systemic way, based upon a discrete third-order Volterra model. The advantage of the counting algorithm is that (1) the algorithm provides the exact counting of third-order products of all eleven possible groups; (2) the counting process requires only two multiplications for the selective counting of third-order products at each position, and the algorithm, which is thus efficient, can be easily implemented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>