All info from [[doyleFeedbackControlTheory2009]]
# Chapter 1 - Introduction
![[Pasted image 20241012132644.png]]
Notable signals:
- r - reference or command input
- e - tracking error
- u - control signal, controller output
- d - plant distrurbance
- y - plant output
- n - sensor
# Chapter 2 - Norms
>[!note] Signal Norms
>1-Norm:
> $$||u||_1 = \int_{-\infty}^{\infty} |u(t)|dt$$
> 2-Norm:
> $$||u||_2 = \left(\int_{-\infty}^{\infty} u(t)^2 dt \right)^{1/2}$$
> $\infty$-Norm
> $$||u||_\infty = \sup_t |u(t)|$$
> Power Signals (Not really a norm):
> $$pow(u) = \left( \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T u(t)^2 dt \right)^{1/2}$$
> If the limit exists, u is called a *power signal*
> > [!caution] Some Implications
> >1. $||u||_2 < \infty \rightarrow pow(u) = 0$
> >2. u is a power signal and $||u||_\infty < \infty \rightarrow pow(u) \leq ||u||_\infty$
> >3. There's a third one in the book about the one norm. I'm ignoring it.

>[!nnote] System Norms
>$\hat G$ means the transfer function $G$ in the frequency domain.
>2-Norm:
>$$||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} $$
>$\infty$-norm
>$$||\hat G||_\infty = \sup_{\omega} |\hat G(j\omega)|$$
>>[!hint] Parseval's Theorem
>> If $\hat G$ is stable, then
>> $$ ||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} = \left( \int_{\infty}^\infty |G(t)|^2 dt \right)^{1/2}$$

![[Pasted image 20241012135404.png]]
# Chapter 3 - Basic Feedback Loop
![[Pasted image 20241014145054.png]]
P, C, and F are system transfer functions. For a system to be **well-posed**, they cannot all be strictly proper. P is almost always strictly proper while the others aren't. 

Something interesting - If $\frac{1}{1+PCF}$ is proper, then this suggests that the system output goes to zero when $j\omega\rightarrow \infty$. This isn't true in reality, because real systems will behave in a not linear way at high frequencies.

>[!tip] Nine System Transfer Functions
>$$\left(\matrix{x_1 \\ x_2 \\ x_3}\right) = \frac{1}{1+PCF} \left[\matrix{1 & -PF & -F \\ C & 1 & -CF \\ PC & P & 1}\right] \left(\matrix{r \\ d \\ n}\right) $$
>**Notable Properties**:
>- All 9 transfer functions are strictly proper if 1+PCF is not strictly proper.
>- If all 9 transfer functions are stable, then the system is **internally stable**

>[!note] Internal Stability
>A couple of theorems:
>>[!hint] Theorem 1
>>The feedback system is internally stable iff there are no closed-loop poles in $Res \geq 0$.
>
>>[!hint] Theorem 2
>>The feedback system is internally stable iff the following two conditions hold:
>>a) The transfer function 1+PCF has no zeros in $Res \geq 0$.
>>b) There is no pole-zero cancellation in $Res \geq 0$ when the product PCF is formed.
>
>>[!warning] Nyquist Criterion 
>>The feedback system is internally stable iff the Nyquist plot does not pass through the point -1 and encircles it exactly n times counterclockwise.
>

The sensitivity function is defined as:
$$ S = \frac{1}{1+L}$$ where L is the loop gain. The sensitivity function **is the transfer function from the reference input r to the tracking error e**.
The number of zeros at the origin of S has a lot to do with asymptotic tracking. For perfect step tracking, this means one zero at the origin. For a ramp, this means two zeros at the origin.

There is another function to understand. The complementary sensitivity function is defined as:
$$ T = 1-S = \frac{L}{1+L}$$
This function **is the transfer function from the reference input r to the output y**.

Generally speaking:
1. Performance specs that involve e result in weights on S
2. Performance specs that involve u result in weights on S

$$ \left[\matrix{e \\u}\right] = -\left[\matrix{PS & S \\ T & CS}\right] \left[\matrix{d \\ n}\right] $$
# Chapter 4 - Uncertainty and Robustness
>[!important] Multiplicative Disk Perturbation
>$$\tilde{P} = (1+\Delta W_2)P$$
> - P is a nominal plant transfer function
> - $\Delta$ is a variable stable transfer function s.t. $||\Delta||_\infty <1$
> - P and $\tilde P$ have the same unstable poles.
> If $||\Delta||_\infty <1$, $W_2$ should be chosen s.t.:
> $$ \left| \frac{\tilde P (j\omega)}{P(j\omega)} - 1 \right| \leq | W_2(j\omega) | \text{ , } \forall \omega$$

$|W_2(j\omega)|$ is the uncertainty profile. This inequality describes a disk in teh complex plane: at each frequency the point P~/P lies in the disk with center 1, radius |W_2|. 

W_2 is basically a transfer function that will always be greater in magnitude than that P~/P -1

>[!note] Robustness
>A controller $C$ is robust to set of plants $\mathcal{P}$ with respect to a characteristic if this characteristic holds for every plant in $\mathcal{P}$. 
>> [!important] Robust Stability
>> A system is robustly stable if it is internally stable for every plant in the set $\mathcal{P}$:
>> $$|| \Delta W_2 T ||_\infty < 1$$
>> ![[Pasted image 20241015172652.png]]
>> Nominal performance is achieved simultaneously when
>> $$ || \text{max} (|W_1S|, |W_2 T|)||_\infty < 1$$>
>
>>[!important] Robust Performance
>>$$ |||W_1 S | + |W_2 T| ||_\infty < 1 $$
>>![[Pasted image 20241015172708.png]]

Something really helpful to think about came to mind as a result of watching a Steve Brunton video[^1]. Think about the way that loop gain works:
$$ y = \frac{L}{1+L} r $$
If at a certain frequency $\omega$, L approaches -1, big problems happen. What this means is that the denominator in the above equation gets really small, which means the gain from r to y actually gets really big. If it IS -1, immediate undefined blow up. 

This is where robustness comes from. The distance between L and -1 for all frequencies is what robustness is. Less distance, less room for plant perturbation that could make you unstable. More distance, safer response. This gets integrated when you start thinking about $W_2$ and $\Delta$. These two things are how you account for the uncertainty and look at how that gets you closer to -1 or not. 

[^1]: [[stevebruntonControlBootcampSensitivity2017]]