## Feedback Control
![[Pasted image 20240930211415.png|350]]
is equivalent to
$Y_{1}(s)=\frac{G_{1}(s)}{1+G_{1}(s)G_{2}(s)}R_{1}(s).$
### Second Order Response
| Metric | Formula | Description |
| ----------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------- |
| **Damping Coefficient** | $\begin{align}\zeta &= \cos(\theta) \\ &\approx \sqrt{ \frac{\ln(OS)^{2}}{\pi^{2}+\ln(OS)^{2}} } \\ &\approx \frac{\tau}{\omega_{n}}\end{align}{}$ | Percent critical damping |
| **Damped Frequency** | ${} \omega_{d}=\sin(\theta_{s})\omega_{n} {}$ | Frequency of damped system |
| **Rise Time** | $t_{r} \approx \frac{1.8}{w_{n}}$ | Time between 10% and 90% of setpoint |
| **Overshoot** | ${} \begin{align}\%OS &\approx 100 e^{-\pi\zeta/\sqrt{ 1-\zeta^{2} }} \\&= 100 e^{-\pi/\tan\theta}\end{align} {}
lt;br> | Max value divided by setpoint |
| **Peak Time** | $t_{p} = \frac{\pi}{\omega_{d}}$ | Time to reach max overshoot point |
| **Settling Time** | $t_{s}= \frac{\ln(0.02)}{\zeta\omega_{n}} \approx \frac{3.91}{\sigma}$ | Time for transients to decay to 2% of setpoint |
![[Pasted image 20240927135513.png|500]]
### PID Control
| | PD | PI | PID |
| ----------------- | ----------------------------------- | ---------------------------------------------- | -------------------------------------------------------------------- |
| **$C(s)$** | $K(s+z)$ | ${} \frac{K(s+z)}{s} {}$ | $\frac{K(s+z)^{2}}{s}$ |
| **$C(s)$** | $K_{p}+K_{d}s$ | $K_{p}+ \frac{K_{i}}{s}$ | $K_{p}+\frac{K_{i} }{s} +K_{d}s$ |
| **$e_{ss}$** Step | ${} \frac{1}{1+G(0)} {}$ | 0 | 0 |
| **$e_{ss}$** Ramp | $\infty$ | C | 0 |
| | Steady state error | overshoot or instability | |
| | Weak to noise <br>(derivative term) | | |
| Root Locus Design | Deficiency angle | Add $\frac{1}{s}$ to plant, then use PD method | Add integrator to plant, then solve double PD with defficiency angle |
### Lead-Lag Control
General controller design with frequency response targets.