Speaker
Description
We study so called bilinear-quadratic output (BQO) systems of the form $\dot{x}(t)=Ax(t)+\sum_{k=1}^mN_kx(t)u_k(t)+Bu(t)$ with initial condition $x(0)=0$ and output $y(t)=Cx(t)+\begin{bmatrix} x(t)^TM_1x(t)& \dots & x(t)^TM_px(t) \end{bmatrix}^T$, where $A\in\mathbb{R}^{n\times n}, B\in \mathbb{R}^{n\times m}, C\in\mathbb{R}^{p\times n},N_k\in \mathbb{R}^{n\times n}$ for $k=1,\dots,m$, $M_j\in\mathbb{R}^{n\times n}$ for $j=1,\dots,p$, $t\in [0,\infty)$. Here, $x(t)\in\mathbb{R}^n$ describes the state, $u(t)\in\mathbb{R}^m$ the input and $y(t)\in \mathbb{R}^p$ the output of the system. Moreover, we assume $M_j$ to be symmetric due to $x(t)^TM_jx(t)=x(t)^TM_j^Tx(t)=\frac{1}{2}x(t)^T(M_j+M_j^T)x(t)$.
We present algebraic Gramians in different variants for these BQO systems, compare them and their relations to Lyapunov equations arising from bilinear ($M_j=0$) or linear-quadratic output ($N_k=0$) systems. Next, we propose a balancing algorithm for truncation of certain states with the goal of structure-preserving model order reduction. This algorithm is tested in several numerical experiments.
