Closed Loop

Note

Things with feedback. The system uses information from the computed output in its decision making process.

Stability

Transfer functions from external signals to internal signals (,closed-loop transfer functions):

• $T_{ry}$ $T_{re}$, $T_{ru}$, $T_{dy}$, $T_{de}$, $T_{du}$
• The reason why we do this is because we’re interested in the impact of the external environment to our control system internally

To get these transfer functions…
$\text{CT}=\left[\begin{array}{c}U\\ E\\ Y\end{array}\right]=\left[\begin{array}{cc}\frac{C}{1+PC}& \frac{1}{1+PC}\\ \frac{1}{1+PC}& \frac{-P}{1+PC}\\ \frac{PC}{1+PC}& \frac{P}{1+PC}\end{array}\right]\left[\begin{array}{c}r\\ d\end{array}\right]$

$\text{DT}=\left[\begin{array}{c}U\\ E\\ Y\end{array}\right]=\left[\begin{array}{cc}\frac{D}{1+GD}& \frac{1}{1+GD}\\ \frac{1}{1+GD}& \frac{-G}{1+GD}\\ \frac{GD}{1+GD}& \frac{G}{1+GD}\end{array}\right]\left[\begin{array}{c}r\\ d\end{array}\right]$

These transfer functions can be derived from the form $y=Pu=P\left(d+\left(e\right)\right)=P\left(d+c\left(r-y\right)\right)$ (expand and make TFs in front of multipliers d, r, etc…)

Summary

A feedback system is called well-posed if all closed loop transfer functions from external signals to internal signals are real, rational, and proper.

Note that this by itself does not imply closed-loop stability. This needs to be combined with the roots of $\Delta$ being in $\mathbb{D}$ to imply closed-loop stability.

Summary

The closed system is closed-loop stable or internally stable if all closed-loop transfer functions from external signals to internal signals (the $T$’s above) are BIBO stable.

This is equivalent to saying that for any bounded r, d, then u, e, y are bounded. (This is the definition of bounded since for any bounded input signals, all the external signals are bounded)

You can do things like:
$y=T_{ry}r+T_{dy}d$ to find out what happens to y by passing the inputs through their transfer functions. The output will be bounded. The same idea applies here: $e=r-y$, if we know r and y are bounded, then the error is also bounded. We don’t need to check this.

Hint

You don’t need to check all 6 to prove closed loop stability, you can do so with 4 TF’s (re, de, ru, du)

You can have an unstable plant and through the correct controller choice, create a stable system.

Concepts

• Proving Closed Loop Stability
• Note that this is different for Sampled Data Digital Control Systems since they are non-LTI, see Closed Loop Stability for SD Systems

Equations

• $\%OS=\frac{\bar{y}-y_{ss}}{y_{ss}-y_o}$
• $e_{ss}=\frac{r_{ss}-y_{ss}}{r_{ss}}$

If $r[k]=1[k]$ then $r_{ss}=1$

• $k_p$ is our rise time here where $\bar{y}$ is the bar or upper limit of $y\left[k\right]$
• $k_s$ is the settling time