- #1
Cyrus
- 3,238
- 17
I have a difference equation which is given as:
ΔP = e^P [1]
where we can re-write ΔP as: Δ P = P_2 - P_1, where the subscripts indicate two distinct discrete time indices.
What I would like to do: is to convert this into a continuous time expression and solve it, if possible.
In order to help give some insight, I will solve a similar type of problem where I know the solution.
ΔP = c_1 [2]
Note here, that in all cases we are running the recursive algorithm at a fixed data rate. Therefore, I can rewrite equation [1] as:
Δ P = P_2 - P_1 = c_2 ⋅ Δ t
where c_1 = c_2 ⋅ Δ t
This allows me to divide both sides by [equation] \Delta t [/equation]:
ΔP /Δt = c_2
And in the limit:
dP/dt = c_2
which then becomes:
P(t) - P(0) = c_2⋅(t - t_0)
And so the result is that this recursive equation [2] gives us a linear ramp if we were to implement it. What I am trying to do for equation [1] is figure out what this expression will look like.
ΔP = e^P [1]
where we can re-write ΔP as: Δ P = P_2 - P_1, where the subscripts indicate two distinct discrete time indices.
What I would like to do: is to convert this into a continuous time expression and solve it, if possible.
In order to help give some insight, I will solve a similar type of problem where I know the solution.
ΔP = c_1 [2]
Note here, that in all cases we are running the recursive algorithm at a fixed data rate. Therefore, I can rewrite equation [1] as:
Δ P = P_2 - P_1 = c_2 ⋅ Δ t
where c_1 = c_2 ⋅ Δ t
This allows me to divide both sides by [equation] \Delta t [/equation]:
ΔP /Δt = c_2
And in the limit:
dP/dt = c_2
which then becomes:
P(t) - P(0) = c_2⋅(t - t_0)
And so the result is that this recursive equation [2] gives us a linear ramp if we were to implement it. What I am trying to do for equation [1] is figure out what this expression will look like.