Bilinear realizability is equivalent to existence of a singular affine differential i/o equation
The paper discusses the following realization problem. Given an affine input-output equation of the form \[ \sum^{k}_{i=1}a_ i(u(t),u^ 1(t),...,u^{(k-1)}(t))y^{(i)}(t)=b(u(t),...,u^{(k-1)}(t)), \] under which conditions does there exist a state space system \[ \Sigma:\quad \dot x=f(x)+g(x)u,\quad y=h(x) \] realizing the same input-output pairs. It is shown that under the assumption that there does exist some state- space realization for the affine input-output equation, then this realization can immersed into a bilinear control system \[ \dot z=Az+(Bz)u,\quad y=Cz. \] Conversely, whenever the state space system \(\Sigma\) can be immersed into a bilinear system then the input-output pairs satisfy an affine input-output equation. An analogous result is stated and proved in terms of so-called generating series of a control system, an analog of the well-known Volterra kernels of a nonlinear system.
scientific article; zbMATH DE number 4074887
scientific article