Feedback Controls

AKA Closed Loop

Summary

Here, we use the output in addition to our reference signal to control our system
The reference minus the sensor (the output) is the error

The large advantage to using this is:

• Robust to uncertainty in the plant
• Reduces effects of disturbances
• Stabilizes unstable systems and improves stability margins
• Improves transient performance by reducing overshoot and settling time
• Reduces or eliminates steady state error

Feedback Control Systems

• Proportional Controls
• Closed Loop

Performance of Feedback System

Performance specifications provide quantitative requirements for control systems:

• Overshoot
• Settling time (read steady state)
• Rise time (reach max in transient)
• Steady state error (steady state needs to be correct)
• Bandwidth
• Gain Margin and Phase Margin (how far are we from losing stability)
• Limits on internal signals (r, u, e)
  • Saturation (physical) limits
  • Safety (engineered) limits

Definition of $g\left[k\right]$
Let $G[z]$ be real, rational, proper

• Then $g\left[k\right]=z^{-1}\left(G\left[z\right]\right)$
• We can take a fractional decomp of $G\left[z\right]$
• Then we can distribute the $z^{-1}$ (using linearity) to the $G\left[\infty \right]$ and each partial fraction
• Here, The $G\left[z\right]$ can be taken out of the $z^{-1}$ because its a constant, so that term becomes $G\left[z\right]z^{-1}\left(1\right)$ where the $z^{-1}\left(0\right)$ becomes $\partial \left[k\right]$ which is 1 or 0
• And the sum over the fractional decomposition terms can be written as $\left(\genfrac{}{}{0ex}{}{}{\begin{array}{c}k-1\\ j-1\end{array}{p}_i^{k-j}}\right)$ but still as the double sum
• Therefore, $g\left[k\right]=G\left[\infty \right]\partial \left[k\right]+{\sum }_{i=1}^n{\sum }_{j=1}^{n_i}\left(\begin{array}{c}k-1\\ j-1\end{array}\right)p_i^{k-j}$
• Where $\left(\begin{array}{c}k-1\\ j-1\end{array}\right)p_i^{k-j}=\frac{\left(k-1\right)!}{\left(j-1\right)!\left(k-j\right)!}$

Concepts

• Final Value Theorem