Your formulas are discontinuous. You cannot use any of the ode* functions for them. The discontinuities occur exactly at the integer times and nowhere else.
If I recall correctly you asked about the same system with different phrasing before, talking about a system that is driven the first half of the day and not driven the second half. You were advised then on how to proceed, and you were advised then that ode* routines evaluate at intermediate times, not just the boundaries you give for tspan. For example ode45 will try to evaluate at roughly t=1.08 and your test is not prepared that; 1.08>round(1.08) yes but 1.0>round(1.0) false but 1.0 is part of the first half day and so should be forced.
Do not use an if in an ode routine. Instead have two different functions, one with forcing and one without, and alternate calling them for the half days.
