Speaker
Description
Today, mathematical modeling is dominated by increasingly high-dimensional and complex dynamical systems. One special type of structure is the bilinear state equation, which either naturally appears in various applications or results from the Carleman bilinearization of the underlying nonlinear dynamics. Recently, dynamical systems with quadratic outputs have also gained significant attention as they appear, e.g., in the modeling of the variance of a quantity of interest of a stochastic model. As a combination, we study the so-called bilinear quadratic output (BQO) systems described as
$\dot{x}(t)=Ax(t)+Nx(t)u(t) + Bu(t), y(t)=Cx(t)+x(t)^TMx(t)$
with state $x(t)\in\mathbb{R}^n$, scalar input $u(t)$ and output $y(t)$ and matrices $A, B, C, N$, and $M$ of suitable dimensions (SISO case).
BQO systems generalize bilinear systems ($M=0$) and linear quadratic output systems ($N=0$). For large dimensional systems, the task of structure-preserving model order reduction arises. Thus, we seek a BQO system of reduced dimension that approximates the original input-output behavior well. Whereas a balancing approach has been already discussed in [2], here we focus on an $\mathcal{H}_2$ optimal approach, similar to $\mathcal{H}_2$ optimal reduction for bilinear systems; see, e.g., [3,1]. We will establish the framework for the $\mathcal{H}_2$ model order reduction of BQO systems, develop iterative algorithms for finding a locally optimal reduced system, and test these algorithms on several numerical examples.
References:
[1] P. Benner and T. Breiten. Interpolation-based $\mathcal{H}_2$-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl., 33(3):859-885, 2012.
[2] H. Faßbender, S. Gugercin, and T. Peters. Bilinear Quadratic Output Systems and Balanced Truncation, July 2025. arXiv:2507.03684 [math].
[3] L. Zhang and J. Lam. On $\mathcal{H}_2$ model reduction of bilinear systems. Automatica, 38(2):205-216, Feb. 2002.