A novel linear algebra method for the determination of periodic steady states of nonlinear oscillators

Periodic steady-state (PSS) analysis of nonlinear oscillators has always been a challenging task in circuit simulation. We present a new way that uses numerical linear algebra to identify the PSS(s) of nonlinear circuits. The method works for both autonomous and excited systems. Using the harmonic balancing method, the solution of a nonlinear circuit can be represented by a system of multivariate polynomials. Then, a Macaulay matrix based root-finder is used to compute the Fourier series coefficients. The method avoids the difficult initial guess problem of existing numerical approaches. Numerical examples show the accuracy and feasibility over existing methods. © 2014 IEEE.postprin

An adaptive dynamical low-rank tensor approximation scheme for fast circuit simulation

Tensors, as higher order generalization of matrices, have received growing attention due to their readiness in representing multidimensional data intrinsic to numerous engineering problems. This paper develops an efficient and accurate dynamical update algorithm for the low-rank mode factors. By means of tangent space projection onto the low-rank tensor manifold, the repeated computation of a full tensor Tucker decomposition is replaced with a much simpler solution of nonlinear differential equations governing the tensor mode factors. A worked-out numerical example demonstrates the excellent efficiency and scalability of the proposed dynamical approximation scheme.postprin

Limit cycle identification in nonlinear polynomial systems

published_or_final_versio

Computing the state difference equations for discrete overdetermined linear mD systems

We derive an algorithm that computes the state difference equations for a given set of poles of linear discrete overdetermined autonomous mD systems. These difference equations allow the realization of the dynamical system by means of delay, multiplication and addition elements in simulation diagrams. In doing so we generalize the classical Cayley–Hamilton theorem to multivariate polynomial ideals and provide a system theoretic interpretation to the notion of polynomial ideals, leading monomials and Gröbner bases. Furthermore, we extend the problem to include poles at infinity and so arrive at a description of overdetermined descriptor systems. This results in a new state space description of autonomous mD descriptor systems. In addition, we discuss the separation of the state variables of singular mD systems into a regular and singular part. A sufficient condition under which these two state vector parts can be interpreted as a forward evolving regular part and a backward evolving singular part is given. The robustness and efficiency of the developed algorithms are demonstrated via numerical experiments.postprin

STORM: a nonlinear model order reduction method via symmetric tensor decomposition

Nonlinear model order reduction has always been a challenging but important task in various science and engineering fields. In this paper, a novel symmetric tensor-based orderreduction method (STORM) is presented for simulating largescale nonlinear systems. The multidimensional data structure of symmetric tensors, as the higher order generalization of symmetric matrices, is utilized for the effective capture of highorder nonlinearities and efficient generation of compact models. Compared to the recent tensor-based nonlinear model order reduction (TNMOR) algorithm [1], STORM shows advantages in two aspects. First, STORM avoids the assumption of the existence of a low-rank tensor approximation. Second, with the use of the symmetric tensor decomposition, STORM allows significantly faster computation and less storage complexity than TNMOR. Numerical experiments demonstrate the superior computational efficiency and accuracy of STORM against existing nonlinear model order reduction methods.postprin

Low-Rank Tensor Decompositions for Nonlinear System Identification: A Tutorial with Examples

Nonlinear parametric system identification is the estimation of nonlinear models of dynamical systems from measured data. Nonlinear models are parameterized, and it is exactly these parameters that must be estimated. Extending familiar linear models to their nonlinear counterparts quickly leads to practical problems. For example, the generalization of a multivariate linear function to a multivariate polynomial implies that the number of parameters grows exponentially with the total degree of the polynomial. This exponential explosion of model parameters is an instance of the so-called curse of dimensionality. Both the storage and computational complexities are limiting factors in the development of system identification methods for such models.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Team Kim Batselie

Enforcing symmetry in tensor network MIMO Volterra identification

The estimation of an exponential number of model parameters in a truncated Volterra model can be circumvented by using a low-rank tensor decomposition approach. This low-rank property of the tensor decomposition can be interpreted as the assumption that all Volterra parameters are structured. In this article, we investigate whether it is possible to explicitly enforce symmetry of the Volterra kernels to the low-rank tensor decomposition. We show that low-rank symmetric Volterra identification is an ill-conditioned problem as the low-rank property of the exact symmetric kernels cannot be upheld in the presence of measurement noise. Furthermore, an algorithm is derived to compute the symmetric Volterra kernels directly in tensor network form.Team Kim Batselie