PID Controller from input signals

Transfer function for angular speed:

\frac{A}{1 + \tau s}

The input signal that begins at time $t_{0}$ and its minimum and maximum values are given by $u_\text{min}, u_\text{max}$ .

The resulting output signal is initially at $y_{0}$ and eventually settles down for a steady state value of $y_\text{ss}$ .

The steady state gain $A$ is given by:

A = \frac{y_\text{ss} - y_0}{u_\text{max} - u_\text{min}} = \frac{\triangle y}{\triangle u}

Time constant $\tau$ is time required for output to increase from initial value to $0.632 \times \triangle y$

Let $t_1$ is time when change in output is $0.632 \times \triangle y$ :

\begin{aligned} y(t_{1}) &= 0.632 \times (y_\text{ss} -y_{0}) + y_{0} \\[8pt] \tau &= t_{1} - t_{0} \end{aligned}

find the transfer function

\begin{align*} \triangle y &= 3V \\[6pt] \triangle v &= 8.285 - 2.454 = 5.831 \text{rad/s} \\[6pt] A &= \frac{5.831}{3} = 1.9436666667 \text{rad/s} \\[12pt] \text{target velocity} &= 2.454 + 0.632 * 5.831 = 6.139192 \text{rad/s} \\[8pt] \tau \approx 0.029 \text{secs} \end{align*}

note: reach it at around 5.029 sec

proportional $P = 350 * \frac{\pi}{180}$