Make Bode Diagram and look at -3 dB

Read Breakpoint Frequency, w_b $$s = a \pm jb$$

In transfer function G(s), substitute $$s=jw_b=\frac{j}{\tau}$$

Solve the Transfer Function

Convert to Polar Form $$z = a + jb , $$ $$z = r \angle \theta , \quad r = \sqrt{a^2 + b^2}$$ $$ \quad \theta = \tan^{-1}!\left(\frac{b}{a}\right)$$

$$r = k_dc$$

No transfer function for returning pass (unity)

Input=Output for returning pass.

ex: cruise control. r=desired speed, y=actual speed

• $$r-y = e$$

• \(e\) represents the tracking error, the difference between the desired output \(r\) and the actual output \(y\).

$$ u(t) = K_p e(t) + K_i \int e(t)\,dt + K_d \frac{de}{dt} $$

• The control signal \(u\) to the plant is equal to the proportional gain \(K_p\) times the magnitude of the error plus the integral gain \(K_i\) times the integral of the error plus the derivative gain \(K_d\) times the derivative of the error.

When you route error into the control system.

Example gravel depositing on weighted belt * Weight too heavy, system controls flow rate to reduce weight * Weight is sufficient, system does no work to correct error = neutral zone