I have a decaying signal wherby I want to calculate the final steady state C as early as possible.
$$y = Ae^{-\alpha t} + Be^{-\beta t} + C$$
I am able to obtain measurements of $y$ and therefore form simultaneous equations.
Let $u = e^{-\alpha}$ and let $v = e^{-\beta}$
Obtain 5 evenly spaced samples of $y$ at times $t = 0, 1, 2, 3, 4$
$$y_1 = Au^0 + Bv^0 + C = A + B + C$$ $$y_2 = Au^1 + Bv^1 + C = Au + Bv + C$$ $$y_3 = Au^2 + Bv^2 + C$$ $$y_4 = Au^3 + Bv^3 + C$$ $$y_5 = Au^4 + Bv^4 + C$$
I know $A, B, \alpha$ and $\beta$ have multiple solutions that satisfy the equation, but I do not care about their values. I just need an expression for $c$ in terms of $y_1, y_2, y_3, y_4$ and $y_5$.
Trouble is, I haven't done much mathamatics for 20 years, and I'm getting into a terrible muddle trying to solve the 4th and 5th equations.
Any assistance would greatly appreciated.
I have solved the simpler version of the equation using this method, and it did indeed work.
$$y = Ae^{-\alpha t} + C$$
This of course only generated 3 simultaneous equations, and had no multiple roots, so was trivial to calculate.
This seems to be hard to solve by hand. Here is the solution from Mathematica:
$$C = \frac{y_3^3-2 y_2 y_4 y_3-y_1 y_5 y_3+y_1 y_4^2+y_2^2 y_5}{y_2^2-2 y_3 y_2-2 y_4 y_2+2 y_5 y_2+3 y_3^2+y_4^2-y_1 y_3+2 y_1 y_4-2 y_3 y_4-y_1 y_5-y_3 y_5}. $$