## 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.
## Root Locus Design
### Root Locus
The root locus is the complex-plane plot of all values $s$ for which the *characteristic equation* holds for some positive real value of $K$:
$
1 + K G(s)=0
$
The poles of $G_{cl}$ must satisfy the characteristic equation.
This equation can be broken down into
- the **magnitude condition**: $\lvert G(s) \rvert =\frac{1}{K}$
- the **angle condition:** $\angle G(s) = \sum ^{m}_{i=1} \angle(s-z_{i}) - \sum_{i=1}^{n} \angle(s-p_{i})$
The angle condition can be rewritten as
$
\sum \phi_{i}-\sum\theta_{i} = (2n + 1)\pi
$
Which can be used to derive rules for drawing the root locus but honestly let's just use `rlocus(sys)` in Matlab.
![[Pasted image 20240927005834.png|500]]
### Controller Design
1. Compute desired closed-loop pole to satisfy steady state error requirement
2. Compute deficiency angle for the desired pole
$
\angle \psi = -180° - \angle P(s)|_{s=d}
$
3. Determine $p$ and $z$ such that $\angle C(s)|_{c=d} = \phi$ using angle condition or bisection method
4. Determine gain $K$ from magnitude condition
$
K = \frac{1}{\left\lvert \frac{s+z}{s+p} \cdot P(s)|_{s=d} \right\rvert }
$
## Frequency Response
### Bode Plots
### Controller Design