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 $C\in {\mathbb{ℝ}}^{p\times n}.$

As one can see, there seems to be a close connection between LPV and bilinear systems which may be advantageous. To be more precise, we can interpret LPV systems as special bilinear system by simply setting

$\tilde{A}=A,{\tilde{N}}_i=0,i=1,\dots ,m,{\tilde{N}}_i=A_i,i=m+1,\dots ,m+d,\tilde{B}=\left[\begin{array}{cc}{B}_0& 0\end{array}\right],\tilde{C}=C,\tilde{u}=\left[\begin{array}{c}{u}_0\\ p_1(t)\\ ⋮\\ p_d(t)\end{array}\right].$

It is important to note that in this way the parameter dependency has been hidden in the structure of a bilinear systems and if we use a bilinear model reduction method, we automatically end up with a structure-preserving model reduction method for LPV systems as well.