Bilinear PMOR method

The model reduction method we present here is applicable to linear parameter-varying (LPV) systems of the form

$\dot{x}(t)=Ax(t)+\sum _{i=1}^d{p}_i(t)A_{i}x(t)+B_0{u}_0(t),y(t)=Cx(t),$

where $A,A_i\in {\mathbb{ℝ}}^{n\times n},B_0\in {\mathbb{ℝ}}^{n\times m}$ and $C\in {\mathbb{ℝ}}^{p\times n}.$

The main idea is that the structure of the above type of systems is quite similar to so-called bilinear control systems. Although belonging to the class of nonlinear control systems, the latter exhibit many features of linear time-invariant systems. In more detail, a bilinear control system is given as follows

$\dot{x}(t)=Ax(t)+\sum _{i=1}^m{N}_{i}x(t)u_i(t)+Bu(t),$

where $A,N\in {\mathbb{ℝ}}^{n\times n},B\in {\mathbb{ℝ}}^{n\times m}$ and $C\in {\mathbb{ℝ}}^{p\times n}.$