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 - Plant Uncertainty