This model seems to be based on the assumption that for a discrete sequence x(k) governed by the difference equation
x(k+1) = A*x(k),x(0)=x0
that there is a continuous time solution between samples given by the equation
xdot(t) = A*x(t), t>=t(k), x(t(k)) = x(k)
The model posted above implements these two equations, but as you've noted it doesn't behave the way you expect. So the question is: why do you think these equations are correct for the process you want to model?
What would really be helpful is if you could explain the two light blue curves in the figure and how they relate to each other. I assume that the stair step curve is the solution to
x(k+1) = A*x(k) , x(0) = x0
But what is the other the light blue curve? Presumably it depends on the stair-step curve, but how? What is the mathematical relationship between them? Once that relationship is defined, it will be straightforward to model and simulate.
By the way, the posted model should be run with a much smaller step size to really see what's going on, maybe even use a variable step solver with a step size of 0.01
