Identification of second-order kernels in aerodynamics

Volterra series is one of the powerful system identification methods for representing the nonlinear dynamic system behavior. The methods of step response and impulse response are commonly applied to a discrete aerodynamic Computational Fluid Dynamic (CFD) to identify the first- and second-order Volterra kernels. A critical problem, however, is the difficulty of identifying the second-order Volterra kernels correctly in CFD-based method. In this paper the second-order Volterra kernel function is expanded in terms of Chebyshev functions to reduce the size of the problem and the accuracy of the identification is also improved based on a third-order reduced model of Volterra series

Tensor Network alternating linear scheme for MIMO Volterra system identification

This article introduces two Tensor Network-based iterative algorithms for the identification of high-order discrete-time nonlinear multiple-input multiple-output (MIMO) Volterra systems. The system identification problem is rewritten in terms of a Volterra tensor, which is never explicitly constructed, thus avoiding the curse of dimensionality. It is shown how each iteration of the two identification algorithms involves solving a linear system of low computational complexity. The proposed algorithms are guaranteed to monotonically converge and numerical stability is ensured through the use of orthogonal matrix factorizations. The performance and accuracy of the two identification algorithms are illustrated by numerical experiments, where accurate degree-10 MIMO Volterra models are identified in about 1 second in Matlab on a standard desktop pc

A methodology for using nonlinear aerodynamics in aeroservoelastic analysis and design

A methodology is presented for using the Volterra-Wiener theory of nonlinear systems in aeroservoelastic (ASE) analyses and design. The theory is applied to the development of nonlinear aerodynamic response models that can be defined in state-space form and are, therefore, appropriate for use in modern control theory. The theory relies on the identification of nonlinear kernels that can be used to predict the response of a nonlinear system due to an arbitrary input. A numerical kernel identification technique, based on unit impulse responses, is presented and applied to a simple bilinear, single-input single-output (SISO) system. The linear kernel (unit impulse response) and the nonlinear second-order kernel of the system are numerically-identified and compared with the exact, analytically-defined and linear and second-order kernels. This kernel identification technique is then applied to the CAP-TSD (Computational Aeroelasticity Program-Transonic Small Disturbance) code for identification of the linear and second-order kernels of a NACA64A010 rectangular wing undergoing pitch at M = 0.5, M = 8.5 (transonic), and M = 0.93 (transonic). Results presented demonstrate the feasibility of this approach for use with nonlinear, unsteady aerodynamic responses

Compositional nonlinear audio signal processing with Volterra series

We develop a compositional theory of nonlinear audio signal processing based on a categorification of the Volterra series. We augment the classical definition of the Volterra series to be functorial with respect to a base category whose objects are temperate distributions and whose morphisms are certain linear transformations. This leads to formulae describing how the outcomes of nonlinear transformations are affected if their input signals are first linearly processed. We then consider how nonlinear audio systems change, and introduce as a model thereof the notion of morphism of Volterra series. We show how morphisms can be parameterized and used to generate indexed families of Volterra series, which are well-suited to model nonstationary or time-varying nonlinear phenomena. We describe how Volterra series and their morphisms organize into a functor category, Volt, whose objects are Volterra series and whose morphisms are natural transformations. We exhibit the operations of sum, product, and series composition of Volterra series as monoidal products on Volt and identify, for each in turn, its corresponding universal property. We show, in particular, that the series composition of Volterra series is associative. We then bridge between our framework and a subject at the heart of audio signal processing: time-frequency analysis. Specifically, we show that an equivalence between a certain class of second-order Volterra series and the bilinear time-frequency distributions (TFDs) can be extended to one between certain higher-order Volterra series and the so-called polynomial TFDs. We end with prospects for future work, including the incorporation of nonlinear system identification techniques and the extension of our theory to the settings of compositional graph and topological audio signal processing.Comment: Master's thesi

Regularized Nonparametric Volterra Kernel Estimation

In this paper, the regularization approach introduced recently for nonparametric estimation of linear systems is extended to the estimation of nonlinear systems modelled as Volterra series. The kernels of order higher than one, representing higher dimensional impulse responses in the series, are considered to be realizations of multidimensional Gaussian processes. Based on this, prior information about the structure of the Volterra kernel is introduced via an appropriate penalization term in the least squares cost function. It is shown that the proposed method is able to deliver accurate estimates of the Volterra kernels even in the case of a small amount of data points

A kernel method for non-linear systems identification – infinite degree volterra series estimation

Volterra series expansions are widely used in analyzing and solving the problems of non-linear dynamical systems. However, the problem that the number of terms to be determined increases exponentially with the order of the expansion restricts its practical application. In practice, Volterra series expansions are truncated severely so that they may not give accurate representations of the original system. To address this problem, kernel methods are shown to be deserving of exploration. In this report, we make use of an existing result from the theory of approximation in reproducing kernel Hilbert space (RKHS) that has not yet been exploited in the systems identification field. An exponential kernel method, based on an RKHS called a generalized Fock space, is introduced, to model non-linear dynamical systems and to specify the corresponding Volterra series expansion. In this way a non-linear dynamical system can be modelled using a finite memory length, infinite degree Volterra series expansion, thus reducing the source of approximation error solely to truncation in time. We can also, in principle, recover any coefficient in the Volterra series

Estimation of generalised frequency response functions

Volterra series theory has a wide application in the representation, analysis, design and control of nonlinear systems. A new method of estimating the Volterra kernels in the frequency domain is introduced based on a non-parametric algorithm. Unlike the traditional non-parametric methods using the DFT transformed input-output data, this new approach uses the time domain measurements directly to estimate the frequency domain response functions

Versatile surrogate models for IC buffers

In previous papers [1,2] the authors have investigated the use of Volterra series in the identification of IC buffer macro-models. While the approach benefited from some of the inherent qualities of Volterra series it preserved the two-state paradigm of earlier methods (see [3] and its references) and was thus limited in its versatility. In the current paper the authors tackle the challenge of going beyond an application or device-oriented approach and build versatile surrogate models that mimic the behavior of IC buffers over a wide frequency band and for a variety of loads thus achieving an unprecedented degree of generality. This requires the use of a more general system identification paradig