# What are the practical uses of a state transition matrix?

I am studying different dynamical models of relative motion of satellites. In my work, I've always just used a numerical integrator like RK78 to integrate the cartesian system of equations:

$$\dot{r}=v$$

$$\dot{v}=-\frac{\mu}{r^3}\bar{r}+a_{pert}$$

Where I can make the model as complex as I want by expanding the $$a_{pert}$$ term that stands for accelerations due to perturbing forces like nonspherical gravity, solar radiation pressure, atmospheric drag, etc.

However, in the field of relative motion, it is clear that countless researchers have spent an enormous amount of effort to create a myriad of state transition matrices to describe relative motion of satellite. These include the CW, Lawden, Gim-Alfriend, and Yan-Alfriend models, for example - the latter two being very complex.

My question is this: what are these used for? If you can easily integrate a nonlinear model numerically, what would I use these complicated STMs for? Are these derived purely so we can create a model that is computationally tractable for a flight computer so that they can be the backbone of a guidance algorithm? What reason would I want to use these STMs over the exact nonlinear model with perturbations if they are only approximations of that model?