I want to write the following solely in terms of its initial value $O_1$ $$ O_t = - \frac{1}{T_w} \sum_{i=t-T_p}^{t-1} O_{i} - \frac{1}{T_i} \sum_{i=1}^{t-T_p-1} O_i + B_t $$ where
- $T_w , T_i$ are non zero constants
- $T_p$ is positive integer
- $B_t$ is some function of t
- Define $O_i = 0 , \; B_i = 0\quad \forall \; i < 1$
I'd expanded the first 3 terms in order to find pattern , we may compare the blue terms .
I found a pattern , like the coefficients have the form $a_{n+1} = a_n + a_n^2 $ but I don't have a close form for it . Furthermore , there seems to be other patterns as well .
Context : above is actually a model for supply chain , with $\frac{1}{T_w} ,\frac{1}{T_i} $ as proportional gain (they are 2 indepedent feedback controllers) , you don't see the error because I've already simplified the expression . I want to do this because I do have the analytic expressions for $O_1$ , so this is the first step to prepare for optimization .
Hint.
Considering first $T_p = 3$ from
$$ \left\{ \begin{array}{rcl} \frac{o_0+o_1+o_2}{T_w}+o_3 &=& b_3 \\ \frac{o_1+o_2+o_3}{T_w}+o_4&=&b_4 \\ \frac{o_1}{T_i}+\frac{o_2+o_3+o_4}{T_w}+o_5 &=& b_5\\ \frac{o_1+o_2}{T_i}+\frac{o_3+o_4+o_5}{T_w}+o_6 &=& b_6\\ \end{array} \right. $$
we can represent a matrix formula as
$$ M_1 O_{t+T_p} + M_2O_t=B_{t+T_p} $$
with
$$ \cases{ O_t = (o_{-1},o_0,o_1,o_2)^T\\ M_1 = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ \frac{1}{T_w} & 1 & 0 & 0 \\ \frac{1}{T_w} & \frac{1}{T_w} & 1 & 0 \\ \frac{1}{T_w} & \frac{1}{T_w} & \frac{1}{T_w} & 1 \\ \end{array} \right)\\ M_2 = \left( \begin{array}{cccc} 0 & \frac{1}{T_w} & \frac{1}{T_w} & \frac{1}{T_w} \\ 0 & 0 & \frac{1}{T_w} & \frac{1}{T_w} \\ 0 & 0 & \frac{1}{T_i} & \frac{1}{T_w} \\ 0 & 0 & \frac{1}{T_i} & \frac{1}{T_i} \\ \end{array} \right) } $$
and as $M_1$ is invertible we have
$$ O_{t+T_p} + M_1^{-1}M_2O_t=M_1^{-1}B_{t+T_p} $$
NOTE
$$ M_1^{-1} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ -\frac{1}{T_w} & 1 & 0 & 0 \\ \frac{1-T_w}{T_w^2} & -\frac{1}{T_w} & 1 & 0 \\ -\frac{(T_w-1)^2}{T_w^3} & \frac{1-T_w}{T_w^2} & -\frac{1}{T_w} & 1 \\ \end{array} \right) $$
now assuming $O_t = 0$ we have
$$ O_{t+T_p} = \left( \begin{array}{c} b_3 \\ b_4-\frac{b_3}{T_w} \\ b_5+\frac{b_3 (1-T_w)}{T_w^2}-\frac{b_4}{T_w} \\ b_6-\frac{b_3 (T_w-1)^2}{T_w^3}+\frac{b_4 (1-T_w)}{T_w^2}-\frac{b_5}{T_w} \\ \end{array} \right) $$