Feedback Controls

P Controls

Proportional Controls

• In a block diagram, this is summing the output to the input
• This is controlled using a $k_p$ or some proportional gain which controls our “flow”
• As you increase $k_p$ you can make the DC gain as close as possible to 1 (This is the servo performance or why we use servo motors)
• Mathematically, $P_{out}=k_{p}e\left(t\right)+p_0$
• Increasing $k_p$ decreases $\zeta$, making the transient response more oscillatory

PD Controls

Proportional Derivative Controls

• A PD controller adds a derivative term to the P controller, which reacts to the rate of change of the error
• This is like implementing a virtual spring and damper
• Mathematically, $u\left(t\right)=K_{p}e\left(t\right)+K_d\frac{d}{dt}e\left(t\right)$ where we still have our proportional gain, but now have a derivative gain connected to the rate of change of the error
  • In LaPlace… $F\left(s\right)=G_K\left(s\right)E\left(s\right)$
• We use this when we need faster responses and to reduce overshoot
• Designing a PD controller means finding $k_p$ and $k_d$ which we can do by putting our transfer function into standard 2nd Order form and when doing a comparison of coefficients