Data-driven modeling of linear dynamical systems with quadratic output in the AAA framework

The AAA framework for modeling linear dynamical systems with quadratic output

We consider linear dynamical systems with quadratic output. We first define the two transfer functions, a single-variable and a multivariate one, that fully describe the dynamics of these special nonlinear systems. Then, using the samples of these two transfer functions, we extend the AAA algorithm to model linear systems with quadratic output from data.

The p-AAA algorithm for data driven modeling of parametric dynamical systems

The AAA algorithm has become a popular tool for data-driven rational approximation of single variable functions, such as transfer functions of a linear dynamical system. In the setting of parametric dynamical systems appearing in many prominent applications, the underlying (transfer) function to be modeled is a multivariate function. With this in mind, we develop the AAA framework for approximating multivariate functions where the approximant is constructed in the multivariate barycentric form. The method is data-driven, in the sense that it does not require access to full state-space model an…

通过多元有理插值的线性二次输出系统的$\mathcal{H}_2$-最优模型降阶

$\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation

本文针对线性二次输出系统，推导了$\mathcal{H}_2$最优模型降阶的插值最优性条件，提出了基于Petrov-Galerkin投影的迭代有理Krylov算法（LQO-IRKA），并验证了其在大规模问题中的有效性。

双线性二次输出系统与平衡截断

Bilinear Quadratic Output Systems and Balanced Truncation

本文针对具有二次输出的双线性系统，通过建立原始-对偶形式、定义并求解Gramian矩阵及其截断版本，提出了一种计算高效的平衡截断模型降阶框架，并用数值算例验证了其有效性。

Balanced truncation for model order reduction of linear dynamical systems with quadratic outputs

We investigate model order reduction (MOR) for linear dynamical systems, where a quadratic output is defined as a quantity of interest. The system can be transformed into a linear dynamical system with many linear outputs. MOR is feasible by the method of balanced truncation, but suffers from the large number of outputs in approximate methods. To ameliorate this shortcoming we derive an equivalent quadratic-bilinear system with a single linear output and analyze the properties of this system. We examine MOR for this system via the technique of balanced truncation, which requires a stabilizatio…

Sensitivity analysis of random linear dynamical systems using quadratic outputs

In uncertainty quantification, a stochastic modelling is often applied, where parameters are substituted by random variables. We investigate linear dynamical systems of ordinary differential equations with a quantity of interest as output. Our objective is to analyse the sensitivity of the output with respect to the random variables. A variance-based approach generates partial variances and sensitivity indices. We expand the random output using the generalised polynomial chaos. The stochastic Galerkin method yields a larger system of ordinary differential equations. The partial variances repre…