I have a question related with the problem shown in the title.
I got a state space model of my system through calculations and it turns out to have a form as
follows
$\dot x(t)=Ax(t)+Br(t)+c$ where A and B are constant matrixes and x and r are state variable
and input, respectivley. The final term is a constant vector with same size as $x$ and I believe it is a
disturbance term. As you can see, it strictly is not linear but I want to make it a linear
form. As I think, since $c$ can be regarded as an input(or disturbance) given to the system right at the
time reference denoted as $t = 0$, It may be represented as $w(t)=c*u(t)$ where
u(t) is a unit step function. In this way, it seems that I could get a state
space representation with multi-inputs
$\dot x(t)=Ax(t)+Br(t)+w(t)$
but I added a unit step function which was absent actually.
I want to know whether doing like this is theoretically correct (not practical view). And if it's
correct, can I get some reliable reference (sth like a paper) on this?
Thank you.
Best,
This term could be considered as a shift in the input: $$r(t)=r'(t)+a$$
$$ \dot x(t)=A x(t)+B(r'(t)+a)+c $$
then
$$\dot x(t)=A x(t)+B r'(t)+ \underbrace{B a+c}_{=0}$$
Set $a$ in the way that $$Ba=-c$$
In cases where B does not have enough degrees of freedom to do that then it will be questionable whether the input can compensate the effect to $c$ at all.