# Discrete-Time Integrator Compare

Perform discrete-time integration or accumulation of signalexpand
>https://www.mathworks.com/help/simulink/slref/discretetimeintegrator.html

$$
\text{gain}\\
K = 1
$$

## Forward Euler Method
$$
\begin{align}
x(n+1) &=& x(n) + K*T*u(n) \tag{1}\\
y(n) &=& x(n) \tag{2}
\end{align}
$$

find the $y(n)$

---
$$
\begin{align}
y(n+1) &=& x(n+1) && \text{from $(2)$}\\
y(n+1) &=& x(n) + K*T*u(n+1) && \text{from $(1)$}\\
y(n+1) &=& y(n) + K*T*u(n+1) && \text{substitute $x(n)$ from $(2)$}\\
\hline
y(n) &=& y(n-1) + K*T*u(n) && \text{make $u(n)$ is now}\\
\hline
\end{align}
$$

tag_angle_integral = tag_angle_integral + tag_angle * update_rate_;

## Trapezoidal Method

$$
\begin{align}
x(n) &=& y(n-1) + K*T/2*u(n-1) \tag{3}\\
y(n) &=& x(n) + K*T/2*u(n) \tag{4}
\end{align}
$$

find $y(n)$

---

$$
\begin{align}
\hline
y(n) &=& y(n-1) + K*T/2*u(n-1) + K*T/2*u(n) && \text{substitute $(4)$ into $x(n)$}\\
\hline
\end{align}
$$

tag_angle_integral =    tag_angle_integral + 
                        last_tag_angle * 0.5 * update_rate_ + 
                        tag_angle * 0.5 * update_rate_;

反正微分都會多浪費一個記憶體了，就用**Trapezoidal Method** 吧