I have always used the following definition of controllability gramian of $(A,B)$ over the time window $[t_0,t_f]$ $$W_c[t_0,t_f] = \int_{t_0}^{t_f}e^{A(t_f - t)}BB'e^{A'(t_f - t)}$$ I have used ' as transpose, and $e^{A(t_f - t)}$ is matrix exponential
I just found out some books (Rugh Chapter 9) have the following definition $$W_c[t_0,t_f] = \int_{t_0}^{t_f}e^{A(t_0 - t)}BB'e^{A'(t_0 - t)}$$
I am unable to show that the two definition are exactly equal. With change of variable I end up with different limits. Any insights will be great.
These are defined as reachability and controllability Gramians respectively. In control theory reachability is defined as the ability to drive the system from $0$ initial state to any final state in a finite time, i.e. $x(t_0)=0$ and $x(t_f)=x_f$. Controllability is defined as the ability to drive the system from any initial state to $0$ final state in a finite time, i.e. $x(t_0)=x_0$ and $x(t_f)=0$. This is why $t_f-t$ occurs in the first matrix and $t_0-t$ occurs in the second matrix.
These concepts are equivalent in linear continuous time systems. To see that
$$ W_r[t_0,t_f] = e^{A (t_f-t_0)} W_c[t_0, t_f] e^{A' (t_f-t_0)} $$
where $W_r[t_0,t_f]$ is the reachability Gramian (the first matrix in the question). Since the matrix $e^{A (t_f-t_0)}$ is always invertible, $\text{rank} W_r = \text{rank} W_c$.
However, reachability and controllability are not equivalent in linear discrete time systems, because of the finite modes.