I am studying control theory, and I am focusing on dynamical systems. While studying, I have met the topic of bilinear systems and of the bilinear realization.
What I cannot understand is the really beginning, so I do not understand what a bilinear system is.
In my notes I have that it is something expressed as: $$ f(x,u)=f_1(x,u)+f_2(x,u) $$
where it is said that $f_1$ is linear on the set $X\times U$ where $X$ is the set of state variables and $U$ is the set of inputs, and $f_2$ is bilinear on the set $X\times U$, and $h$ is linear on $X$, so I have a system of the type: $$ \begin{cases} f(x,u)=Ax+Bu+\sum_{i=1}^{p}N_ixu_i\\ h(x)=Cx \end{cases}. $$ (sorry for the bad indentation, I don't understand how to fix it)
Now, I honestly don't understand how to interpret this system. Infact, there is a summation and a matrix $N$ which I don't understand what it is, and moreover I don't have an intuition of what a bilinear system is.
About the bilinear realization, I mentioned it since I am studying bilinear systems in the context of dynamical system, and so in the next chapter i have to study it is present a description on how to pass from a state representation(explicit representaion) to a differential equation(implicit representation).
Can somebody please help me?