I timed the DARE unit tests, and the new solver is 0 to 100% faster in
all cases (that is, it's at least as fast as Drake's and up to 2x faster
in some cases).
The new solver is also much simpler, takes less time to compile, and
drops the libwpimath.so size from 325 MB to 301 MB.
I think most of the compilation time is coming from the eigenvalue
decompositions used to enforce argument preconditions.
There were three options for where to put this function:
1. A free function in LinearQuadraticRegulator.h. Returning a K matrix
means the user can't use the LinearQuadraticRegulator in a loop
anymore.
2. A default argument added to ctors in LinearQuadraticRegulator for a
time delay (default of 0). This has the smallest API footprint from
the user perspective, but it bloats the already substantial
constructor overload set of LinearQuadraticRegulator.
3. A member function in LinearQuadraticRegulator that modifies the
internal K. This would still have to take in a LinearSystem or (A, B)
pair because the ctor doesn't store it. Storing it internally feels
like paying for what we don't use most of the time.
I went with option 3.
I verified the tests's expected values in Python with
scipy.linalg.fractional_matrix_power().
Closes#2877.
