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1 Quick Reference

Standard Block Diagram

where

• $LaTeX: y$ is the system output,
• $LaTeX: r$ is the reference input or desired output,
• $LaTeX: r'$ is the filtered reference input,
• $LaTeX: e$ is the error between $LaTeX: y$ and $LaTeX: r'$,
• $LaTeX: P$ is the plant,
• $LaTeX: H$ is the feedback sensor,
• $LaTeX: W$ is the prefilter used shape the reference input, and
• $LaTeX: K$ is the controller.

$LaTeX: OL=PKH$

where

• $LaTeX: OL$ is the open loop transfer function

Note that $LaTeX: W$ is not part of the OL but is part of the tranfer fuction from $LaTeX: y$ to $LaTeX: r$. See below.

$LaTeX: CL=\frac{OL}{1+OL}=\frac{PKH}{1+PKH}$

where

• $LaTeX: CL$ is the closed loop transfer function

$LaTeX: DR=1-CL=\frac{1+OL}{1+OL}-\frac{OL}{1+OL}=\frac{1}{1+OL}=\frac{1}{1+PKH}$

where

• $LaTeX: DR$ is the disturbance rejection transfer function

Note that the $LaTeX: DR$ is from $LaTeX: y$ to $LaTeX: d_o$.

2 All Transfer Functions

Below is an equation relating every input to the output $LaTeX: y$

$LaTeX: y=\frac{PKH}{1+PKH}Wr+\frac{P}{1+PKH}d_i+\frac{1}{1+PKH}d_o+\frac{-PKH}{1+PKH}n$

The transfer function from r to y is

$LaTeX: T_{ry}=\frac{PKH}{1+PKH}W$

The transfer function from d_i to y is

$LaTeX: T_{d_iy}=\frac{P}{1+PKH}$

The transfer function from d_o to y is

$LaTeX: T_{d_oy}=\frac{1}{1+PKH}$

The transfer function from n to y is

$LaTeX: T_{ny}=\frac{-PKH}{1+PKH}$

For the system to be stable all transfer functions must be stable.