From df2d8f8f29b54d7e83d639cec9ca04a2b2d742f7 Mon Sep 17 00:00:00 2001 From: Christian Risi <75698846+CnF-Gris@users.noreply.github.com> Date: Tue, 14 Jan 2025 19:14:22 +0100 Subject: [PATCH 1/3] V0.8.0 --- docs/Chapters/6-2-SENSITIVITY.md | 17 + docs/Chapters/CONGESTION-AVOIDANCE.md | 192 ++ docs/Chapters/EXTRAS.md | 50 + docs/Chapters/KALLMAN-FILTER.md | 222 ++ docs/Chapters/VIDEO-STREAMING.md | 172 ++ docs/Formularies/CONTROL-FORMULARY.md | 13 +- .../packet-system.excalidraw.json | 1462 +++++++++ .../control-of-a-plant-in-z.excalidraw.json | 1238 ++++---- ...ock-diagram-state-feedback.excalidraw.json | 2356 +++++++-------- .../Video-Streaming/buffer.excalidraw.json | 2676 +++++++++++++++++ .../feedback-linearization.excalidraw.json | 1660 ++++++++++ docs/Images/Video-Streaming/buffer.png | Bin 0 -> 295988 bytes 12 files changed, 8260 insertions(+), 1798 deletions(-) create mode 100644 docs/Chapters/6-2-SENSITIVITY.md create mode 100644 docs/Chapters/EXTRAS.md create mode 100644 docs/Images/.excalidraw/Congestion-Avoidance/packet-system.excalidraw.json create mode 100644 docs/Images/.excalidraw/Video-Streaming/buffer.excalidraw.json create mode 100644 docs/Images/.excalidraw/Video-Streaming/feedback-linearization.excalidraw.json create mode 100644 docs/Images/Video-Streaming/buffer.png diff --git a/docs/Chapters/6-2-SENSITIVITY.md b/docs/Chapters/6-2-SENSITIVITY.md new file mode 100644 index 0000000..10444e2 --- /dev/null +++ b/docs/Chapters/6-2-SENSITIVITY.md @@ -0,0 +1,17 @@ +# Sensitivity +Whenever we introduce our reference input, +we should check the associated [Sensitivity Function](./../Formularies/CONTROL-FORMULARY.md/#sensitivity-function). + + + +As we can see, a reference put +just after the output og $G_p(s)$ +yealds a more robust system, overall. + +# Augmented state +Each time we need to make the system +faster, we need an `integrator`. +In order to being able to place it, we just +need to augment the state. By doing that +we introduce some ***helping*** +`state variables` and the rest is as it is. \ No newline at end of file diff --git a/docs/Chapters/CONGESTION-AVOIDANCE.md b/docs/Chapters/CONGESTION-AVOIDANCE.md index e69de29..7addeed 100644 --- a/docs/Chapters/CONGESTION-AVOIDANCE.md +++ b/docs/Chapters/CONGESTION-AVOIDANCE.md @@ -0,0 +1,192 @@ +# Congestion Avoidance +Internet can be represented as a huge number +of links and buffers, where each buffer is +technically a `router`. + +Each time we send a packet, the `time` it +gets to go from `Point S` to `Point D` is +called `One Way Delay` or `Forward Delay` +$T_{fw}$. + +From `Point D` to `Point S` is +called `Backward Delay` $T_{bw}$ + +The `Round Trip Time` (`RTT`) is: +$RTT = T_{fw} + T_{bw}$ + +## Phil Karn Algorithm +To get an RTT estimation, we sum all the +delays introduced to physically transfer +packets $T_{pi}$ and the queue time of those +$T_{qi}$ on each node $N_{i}$. +$$ +RTT = \sum_i +\underbrace{T_{pi}}_{ + \text{propagation delay} +} + +\sum_i \underbrace{T_{qi}}_{ + \text{queueing delay} +} +$$ + +While we can't control any $T_{pi}$ as it is +a physical constraint, we can control $T_{qi}$ + +## Internet Delay Model + + + +### Variables +- $Bw_Delay = e^{-sT_{bw}}$ : Backward Delay +- $Fw_Delay = e^{-sT_{fw}}$ : Forward Delay +- $q(t)$ : receiving data $bits$ +- $d(t)$ : disturbance or draining of the buffer $\frac{bits}{s}$ +- $AdW$ : Advertised Window (AKA free buffer space) +- $G_c(s)$ : Controller that checks how much data can be sent to the receiver + +### Control +In order to control this, we need to model a Smith Predictor, +working without delay first and then equaling the two systems +later. + +## TCP Models +TCP was modeled in order to be a `Greedy Protocol`, thus using as much as bandwidth +it could grab. In order to have congestions (when a flow can't be transmitted) +TCP had to be controlled. The difficulty in controlling such system lies in the +inability to know many variables such as intermediate buffers. + +Tee way we control TCP has several `flavours` like: +- TCP Tahoe +- TCP Reno +- TCP New Reno +- TCP Santa Cruz +- TCP New Vegas +- TCP Westwood+ + +### TCP Tahoe +This `flavour` has 2 phases to control the single TCP flow. +It introduces concepts used in many other `flavours`: +- $cwndw$ : Congestion Window | If you send data more than this window, you'll likely + end up having a congestion and lose packets +- $t_k$ : Time when you received $k$ RTTs +- $sstresh$: Slow Start Treshold | after this value, we need to avoid having a + congestion and so we increase the $cwndw$ linearly + +#### Slow Start (SS) +This is the fastest phase as it is exponential. + +Each time all packets sent are successfully received, the $cwndw$ is exponentially increased: +$$ +cwndw = cwndw(k - 1) * 2 = cwndw(0) * 2^k +$$ + +Or in other words, on each successful `ACK` : +$$ +cwndw_{n} = cwndw_{n-1} + 1 +$$ + +#### Congestion Avoidance +We are in this phase once $cwndw \geq sshtresh$. This phase `probes` +(check the max capacity) the available bandwidth. + +Here on each successful `ACK`: +$$ +cwndw_{n} = cwndw_{n-1} + \frac{1}{cwndw_{n-1}} +$$ + +Once we receive a `3DUPACK` or an `RTO`, we get that the bandwidth is somewhere +full and we are losing packets. + +#### Congestion Phase +Eithe rwe receive a `3DUPACK` or an `RTO`: +- $sstresh = cwndw / 2$ +- $cwndw = 2$ + +### TCP Reno / New Reno[^tcp-new-reno-freebsd] +They differ from TCP Tahoe only on the Congestion Phase + +#### `3DUPACK` Congestion +- $sstresh = cwndw / 2$ +- $cwndw = sshtresh$ + +#### `RTO` Congestion +- $sstresh = cwndw / 2$ +- $cwndw = 2$ + +#### TCP New Reno Throughput Formula (Tuenis Ott) +$$ +\frac{W}{2}\frac{W}{2} + \frac{1}{2}\frac{W}{2}\frac{W}{2} = +\frac{3}{8}W^2 +$$ + +This formula is equal to how much $window$ is acked before a loss. +Therefore, +$ +\frac{3}{8}W^2 = \frac{1}{p} \leftrightarrow W = \frac{2\sqrt{2}}{\sqrt{3p}} +$ + + +### TCP Westwood +This algorithm tries to estimate the available bandwidth without blindly `probing`: +$$ +bw_k = \frac{ + acked_{k} - acked_{k-1} +}{ + t_{k} - t_{k-1} +} = \frac{data_k}{time\_interval} +$$ + +The thing is that if you try to sample in such a short period, you'll +have spikes that can't be filtered because of the high frequency +components. + +So, we should sample at $f_c \geq 2f_{max}$ as per Nyquist/Shannon +Theorem: + +$$ +\begin{cases} +bw_k = \frac{ + acked_{k} - acked_{k-1} +}{ + t_{k} - t_{k-1} +} = \frac{data_k}{time\_interval} \\ +\hat{bw}_k = \alpha bw_k + (1 - \alpha) \hat{bw}_{k-1} \rightarrow +\text{this is also called moving average} +\end{cases} \\ +cwndw = \hat{bw}_k RTT_{min} \rightarrow +\text{we are assuming $T_q = 0$} +$$ + +#### `3DUPACK` Congestion +- $sstresh = \hat{bw}_k RTT_{min}$ +- $cwndw = sshtresh$ + +#### `RTO` Congestion +- $sstresh = \hat{bw}_k RTT_{min}$ +- $cwndw = 2$ + +### TCP Cubic +This method has many differences with all preceding ones[^tcp-cubic-linux]. +Here this algorithm is `real-time` dependent and not `RTT` one. + +This algorithm is slow at start, after $t=K$ it becomes exponential and +`probes` the bandwidth + +#### Parameters +- $\beta$ : Multiplicative decrease factor | Set to $0.7$, $0.75$, $0.8$ +- $W_{max}$ : Windows size `right-before` its last reduction +- $T$ : Time elapsed since last reduction +- $C$ : Scaling constant | Set to $0.4$ +- $cwndw$: Congestion Window at current time + +# Congestion +- $cwndw = C(t-k)^3+ W_{max}$ +- $K = \sqrt[3]{\frac{(1 -\beta) W_{max}}{C}}$ + +## See also +- Statistical Multiplexing +- Congestion Avoidance: Van Jacobson ACM CCR 2004 + + +[^tcp-new-reno-freebsd]: TCP New Reno is used as the default algorithm on FreeBSD +[^tcp-cubic-linux]: TCP Cubic is used as the default in Linux \ No newline at end of file diff --git a/docs/Chapters/EXTRAS.md b/docs/Chapters/EXTRAS.md new file mode 100644 index 0000000..8d20a10 --- /dev/null +++ b/docs/Chapters/EXTRAS.md @@ -0,0 +1,50 @@ +# Extras + +## WebRTC + +Read GCC + + +## PageRank +Whenever we need to search for a result on Google, this one +of the most famous algorithm used. + +First we take some pages and all the links available +on that page linking to the other gathered pages: +```bash +p1, p2, p3, p4, ..., pn + +# Links found on the page +p1: +- a1 -> p2 +- a3 -> p4 +... +- al -> p8 + +p2: +... +``` + +Then we make an oriented graph where all the `points` are pages +and links are `arrows` pointing to the ***landing page*** + +### Variables +- $B_p$ : Set of pages that backlinks to $p$ +- $|a|$ : Cardinality of $a$ | Number of outgoing links from + $a$ + +### Formula +$$ +rank_k(p) = \sum_{a \in B_p}\frac{ + rank_{k-1}(a) +}{ +|a| +} \in [0, \dots, 1] +$$ + +Here the state is composed of all ranks for our gathered pages. +If $| rank_k(p) - rank_{k-1}(p)| < \epsilon$ we will stop +iterating. + +At time $k = 0$ all pages have the same importance that is +$rank_0(p) = \frac{1}{n}$ \ No newline at end of file diff --git a/docs/Chapters/KALLMAN-FILTER.md b/docs/Chapters/KALLMAN-FILTER.md index e69de29..c43a1ac 100644 --- a/docs/Chapters/KALLMAN-FILTER.md +++ b/docs/Chapters/KALLMAN-FILTER.md @@ -0,0 +1,222 @@ +# Kalman Filter +Until now, we dealed with ***ideal*** conditions, such as the ***complete +absence of noise***. + +Let's now put ourselves in in this position: +$$ +\begin{cases} +\dot{x}(t) = Ax(t) + Bu(t) + Gw(t) \\ + y(t) = C{x}(t) + Du(t) + v(t) +\end{cases} +$$ + +- $w(t)$ : Process Noise +- $v(t)$ : Measurement Noise + +## Analyzing the noise[^unimore] +Usually the Expectation of Gaussian Nose is $0$, but if that's not the case, +we just need to shift coordinates. + +> [!NOTE] +> $w_k$ and $v_k$ are $w(t)$ and $v(t)$ with $t = k$ +> + +> [!TIP] +> if $E[X] = 0$ and $E[Y] = 0$, since $Cov(X,Y) = E[(X - E[X])(Y - E[Y])]$, +> then $Cov(X,Y) = E[XY]$ + +- $E[w_k] = E[v_k] = 0 \rightarrow$ If this is not true, we can shift +coordinates +- $E[w_k w_q^T] = \begin{cases}R_w\; \text{if} \;q = k \\0\; \text{if} \;q \neq k\end{cases}$ +- $E[v_k v_q^T] = \begin{cases}R_v\; \text{if} \;q = k \\0\; \text{if} \;q \neq k\end{cases}$ +- $E[w_k v_q^T] = 0 \;\forall k,q$ + +In particular, $v_k$ and $w_k$ are incorrelated between +themselves, their states in time and the initial state +$x(0)$ + +Since our state is affected by noise, we make an estimation of it, and it will be affected by an error: +$$ +\hat{x}_k = x_k + e_k \leftrightarrow e_k = \hat{x}_k - x_k +$$ + +The error is measured as $e_k^T e_k = \sum_i e_i^2 \in \R$ + +Now we define some extra errors. The `a priori` happens +before measuring the output, the `a posteriori` happens +just right after: +$$ +\begin{align*} +e_k^+ &= \hat{x}_k^+ - x_k = e_k\rightarrow +\text{error a posteriori} \\ + + +e_k^- &= \hat{x}_k^- - x_k = \epsilon_k \rightarrow +\text{error a priori} \\ + + +\hat{x}_k^+ &= \hat{x}_k^- - L_k(C\hat{x}_k^- - y_k) +\rightarrow \text{Observed value} \\ + +\end{align*} + +$$ + +Now, let's expand the formula: + +$$ +\begin{align*} +e_k^+ &= e_k = \hat{x}_k^+ - x_k = \\ + + +&= \hat{x}_k^- - L_k(C\hat{x}_k^- - y_k) - x_k = \\ + + +&= \hat{x}_k^- - x_k - L_k(C\hat{x}_k^- - y_k) = \\ + + +&= \epsilon_k - L_k(C\hat{x}_k^- - y_k) = \\ + + +&= \epsilon_k - L_k(C\hat{x}_k^- - Cx_k -v_k) = \\ + + +&= \epsilon_k - L_k(C(\epsilon_k) -v_k) = \\ + + +&= \epsilon_k - L_k C \epsilon_k -Lv_k = \\ + + +&= \epsilon_k - L_k C \epsilon_k -Lv_k = \\ + + +&= (I - L_k C\textcolor{red}{_k} )\epsilon_k -L +\textcolor{red}{_k}v_k +\end{align*} \\ + +\text{Now let's compute } Cov(e_k^+)\\ + +\begin{align*} +Cov(e_k^+) &= E[e_k e_k^T] = \\ + + +&= E \{[(I - L_k C_k )\epsilon_k -L_kv_k] [(I - L_k C_k ) +\epsilon_k -L_kv_k ]^T\} = \\ + + +&= E \{[(I - L_k C_k )\epsilon_k -L_kv_k] +[\epsilon_k^T (I - L_k C_k )^T -v_k^T L_k^T ]\} = \\ + + +&= E [ + (I - L_k C_k )\epsilon_k \epsilon_k^T (I - L_k C_k )^T - + (I - L_k C_k )\epsilon_k v_k^T L_k^T - + L_k v_k \epsilon_k^T (I - L_k C_k )^T + + L_k v_k v_k^T L_k^T +]\rightarrow \\ + + +&\text{Noises and errors are not correlated } \\ + + +&E[ + (I - L_k C_k )\epsilon_k \epsilon_k^T (I - L_k C_k )^T +] = (I - L_kC_k) P_k^- (I - L_kC_k)^T\\ +&-E[ +(I - L_k C_k )\epsilon_k v_k^T L_k^T +] = 0 \rightarrow +\text{ +Remember that only $\epsilon_k$ and $v_k$ are random variables +}\\ +&-E[ + L_k v_k \epsilon_k^T (I - L_k C_k )^T +] = 0 \rightarrow +\text{ +Remember that only $\epsilon_k$ and $v_k$ are random variables +}\\ +&E[ + L_k v_k v_k^T L_k^T +] = L_k R_v L_k^T \rightarrow \\ + +P_k^+ &= (I - L_kC_k) P_k^- (I - L_kC_k)^T + L_k R_v L_k^T +\end{align*} +$$ + +We can see that our cost is +$E[e_k^T e_k] = \sum_i e_i^2 = trace(P_k^+)$. +To minimize this we derive over $L_k$ and equal to $0$, as this will be our +`minimum point` (assuming `Hessian` defined ***semi-positive***) + +$$ +\begin{align*} +\frac{d\,tr(P_k^+)}{dL_k} &= -2(I - L_kC_k)P_k^- C_k + 2L_kR_v += 0 \rightarrow\\ + +&\rightarrow (-I +L_kC_k)P_k^- C_k^T + L_kR_v =0 \rightarrow\\ + + +&\rightarrow -P_k^- C_k^T +L_kC_k P_k^- C_k^T + L_kR_v =0 \rightarrow\\ + + +&\rightarrow L_kC_k P_k^- C_k^T + L_kR_v = + P_k^- C_k^T\rightarrow\\ + + +&\rightarrow L_k (C_k P_k^- C_k^T + R_v) = P_k^- C_k^T\rightarrow\\ + + +&\rightarrow L_k = P_k^- C_k^T +(\underbrace{C_k P_k^- C_k^T + R_v}_{S_k})^{-1} + +\end{align*} +$$ + +Substitute for this minimum value: +> [!WARNING] +> You just need to expand this equationto get to the final solution +$$ +\begin{align*} +P_k^+ &= (I - L_kC_k) P_k^- (I - L_kC_k)^T + L_k R_v L_k^T = \\ + + +&= (I - C_kP_k^- C_k^T S_k^{-1}) P_k^- (I - P_k^- C_k^T S_k^{-1}C_k)^T + P_k^- +C_k^T S_k^{-1} +R_v (P_k^- C_k^T S_k^{-1})^T = \\ + + +&= (P_k^- - C_kP_k^- C_k^T S_k^{-1}P_k^-) (I - P_k^- C_k^T S_k^{-1}C_k)^T + P_k^- +C_k^T S_k^{-1} +R_v (P_k^- C_k^T S_k^{-1})^T= \\ + + +&= \dots = (I - L_kC_K)P_k^- \rightarrow \\ + + +\rightarrow L_kC_kP_k^- &= P_k^- - P_k^+ \rightarrow \\ + + +\rightarrow +L_kC_kP_k^-C_k^T &= (P_k^- - P_k^+)C_k^T =- L_kR_v + P_k^-C_k^T +\rightarrow \\ + +\rightarrow P_k^+C_k^T &= L_kR_v \rightarrow \\ +\rightarrow L_k &= P_k^+C_k^TR_v^{-1}\\ + +&\dots \\ + +P_K^- &= AP_{k-1}^+A^T + GR_wG^T + +\end{align*} +$$ + +### The actual Algorithm +0. $P_0^- = P_0$ + +Loop: +1. $P_k^+ = P_k^- - P_k^-C^T_k(C_kP^-_kC_k^T + R_v)^{-1}C_kP_k^-$ +2. $\hat{x}_k^+ = \hat{x}_k^- + P_k^+C^T_kR_v^{-1}(y_k - C_k\hat{x}_k^-)$ +3. $\hat{x}_{k+1}^- = A\hat{x}_k^+ + Bu_k$ +4. $P_{k+1}^- = AP_k^+A^T + GR_wG^T$ + + + +[^unimore]: [Unimore | 14 January 2025](http://www.dii.unimore.it/~lbiagiotti/MaterialeTSC1516/TSC-03-KalmanFilter.pdf) \ No newline at end of file diff --git a/docs/Chapters/VIDEO-STREAMING.md b/docs/Chapters/VIDEO-STREAMING.md index e69de29..20c87ae 100644 --- a/docs/Chapters/VIDEO-STREAMING.md +++ b/docs/Chapters/VIDEO-STREAMING.md @@ -0,0 +1,172 @@ +# Video Streaming +Nowadays video streaming works by downloading +$n$ chunks of a video $v$ at different qualities $l_{i}$. + +In order to model a device reproducing a video, we can consider the device +having a `buffer` that has a dimension $b_{max}$. Whenever we play the video +we are removing content from the `buffer`, making space for new chunks. + +When the `buffer` is full, we don't donwload new chunks. When it's empty the +reproduction will stop. + +Our objective is to keep the `buffer size` in this range: +$$ +0 < b < b_{max} +$$ + +## Downard Spiraling +The reason why we should never get our buffer full is as we will stop +using bandwidth. This means that if there's another `greedy flow` going +on, it will perceive an increase in bandwidth, getting more and more of it. + +This means that our video streaming will, on the contrary, perceive a lower +bandwidth and the quality will drop. + +## Existing Protocols to Control Buffer Size + +### DASH (Dynamic Adaptive Streaming over Http) + +### HLS (Http Live Streaming) +Made by Apple, have a length of chunks equal to $10s$ + + +## Controlling Video Streaming +First of all, we must say that Video Streaming should be controlled +with a controller ***built-on-top*** of the TCP controller, as the TCP +controller itself was ***built-on-top*** the IP controller, etc... + +This is a generic scheme: +![video streaming model](./../Images/Video-Streaming/buffer.png) + +Our Player buffer must be measured in **seconds!!!** +$$ +\begin{align*} +r(t) &= \frac{ + \Delta D +}{ + \Delta t +} = \frac { + \text{data\_quality} +}{ + \text{time\_interval} +} bps\\ + +\Delta t_f &= \text{frame\_time} \\ +l(t) &= \frac{ + \Delta D +}{ + \Delta t_f +}bps = \text{coded\_movie} \\ +t &= \text{real\_time} \\ +t_f &= \text{move\_time} \\ +\frac{ + \Delta t_f +} { + \Delta t +} &= \left\{ 1, < 1, > 1\right\} \rightarrow +\text{When we 1.5x or 0.5x on video} \\ +\dot{t}_f &= \frac{dt_f}{dt} + + +\end{align*} +$$ + +From here we have: +$$ +\begin{align*} +r(t) &= \frac{ + \Delta D +}{ + \Delta t +} = \frac{ + \Delta D +}{ + \Delta t +} +\frac{ + \Delta t_f +}{ + \Delta t_f +} \\ + + +&= \frac{ + \Delta D +}{ + \Delta t_f +} +\frac{ + \Delta t_f +}{ + \Delta t +} \\ + + +&= l(t)\dot{t}_f \\ +\dot{t}_f &= \frac{r(t)}{l(t)} +\end{align*} +$$ + +Now, let's suppose we are reproducing the video: +$$ +\dot{t}_f = \frac{r(t)}{l(t)} -o(t) \rightarrow \text{$o(t) = 1$ 1s to see 1s} +$$ + +> [!TIP] +> $o(t)$ is just how many seconds we are watching per seconds. It basically +> depends on the multiplier of speed we put on the video: +> ``` +> 0.5x -> 0.5s of video seen per second +> 2x -> 2s of video seen per second +> ... +> ``` + +### Feedback Linearization +Whenever we have a `non-linear-system` we can substitute the `input` for +a function dependent on the `state` +$$ +\begin{align*} +\dot{x} &= f(x,u) \\ +\dot{x} &= Ax + Bu(x) +\end{align*} +$$ + +Coming back to our goal of controlling video streaming. We want to have a +set point for our system $t^s_f$. As usual, we get the error and, in this +case, we want to have a 0 position error over the input, so we add an +integrator: +$$ +t_I = \int^t_{- \infty} t^s_f - t_f(\tau) d\tau +$$ + +Because we have integrated, we are augmenting our state space + +$$ +\begin{align*} +x &= \begin{bmatrix} +t_f = \frac{r(t)}{l(t)} - o(t)\\ +t_I +\end{bmatrix} && \text{as we can see, $t_f$ is non linear} \\ +\end{align*} +$$ + +Because $t_f$ is non linear, we must find a way to linearize it[^feedback-linearization]: +$$ +\frac{r(t)}{l(t)} - o(t) = k_pt_f + k_It_I \rightarrow \\ +l(t) = \frac{ + r(t) +}{ + o(t) + k_pt_f + k_It_I +} +$$ + +Now, simply, this algorithm will take the video quality that is nearer $l(t)$ + + + + +## See also +- Automatica 1999 c3lab +- Control Engineering Practice c3lab + +[^feedback-linearization]: [Feedback Linearization | 13 January 2025](https://aleksandarhaber.com/introduction-to-feedback-linearization/#google_vignette) \ No newline at end of file diff --git a/docs/Formularies/CONTROL-FORMULARY.md b/docs/Formularies/CONTROL-FORMULARY.md index 0e9e70e..d6ed2e6 100644 --- a/docs/Formularies/CONTROL-FORMULARY.md +++ b/docs/Formularies/CONTROL-FORMULARY.md @@ -32,4 +32,15 @@ $X = X_no \bigoplus X_{no}^{\perp} = X_no \bigoplus X_{o}$ > [!TIP] > Since $X_{o} = X_no^{\perp}$ we can find a set of perpendicular -> vectors by finding $Ker(X_{no}^{T})$ \ No newline at end of file +> vectors by finding $Ker(X_{no}^{T})$ + +## Sensitivity Function +This function tells us how much `disturbances` in our system +affects our $G(s)$ (here $T$):\ +$ +S = \frac{dG}{dT} \frac{G}{T} +$ + +Once found $S$ we do a `Bode Plot` of it to +see how much we differ from our original +`system` \ No newline at end of file diff --git a/docs/Images/.excalidraw/Congestion-Avoidance/packet-system.excalidraw.json b/docs/Images/.excalidraw/Congestion-Avoidance/packet-system.excalidraw.json new file mode 100644 index 0000000..e6058c8 --- /dev/null +++ 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zo4ZO=H9sr$ua69Y6t|<|Ug9trKVNxYA@irAgDvvbKjbnt#3R}r7UGkW^n6e}Y>)Y#`N_V^yWj2Am zgJ}W+GLm-TQgW)?f-tP40Tq(}Bx`~9Hl6rg2t=48`yJszbVaU^a3q1VGR5jv7E<+V9EiV`A@?-+KrBz?KrS_ZmwOwa0JGQF( zI%~ddMl#jlhN-JIG62|qkvo$lYvzs$~SvMBQ}0KR?6Qvi?PCvh0W|QqmLaN6bDy=z(MK2W z9v7AR8we?MsYN7=A*LVmmcEef8M`4{Q2(-7^j#ilRGX_W1uIudq8dXC+Z1IYh~rEG z6C@S+%|6XPzpKj$07X6FV*8JM*?z#Y)ulb6?pO2e^AG+n_#CoB?53)1j?=w)qe`Zm z$NKCO4FzqtzRotEj6wLNcWz;sSG3QJxP) Date: Wed, 15 Jan 2025 12:25:08 +0100 Subject: [PATCH 2/3] V0.8.2 --- README.md | 1 + ...KALLMAN-FILTER.md => 10-KALLMAN-FILTER.md} | 0 ...=> 11-CRYPTOGRAPHY-WITH-STATE-OBSERVER.md} | 0 ...IZATION.md => 12-KALLMAN-DECOMPOSITION.md} | 0 ...AVOIDANCE.md => 7-CONGESTION-AVOIDANCE.md} | 0 ...MITH-PREDICTOR.md => 8-SMITH-PREDICTOR.md} | 0 ...IDEO-STREAMING.md => 9-VIDEO-STREAMING.md} | 0 docs/README.md | 23 +++++++++++++++++++ 8 files changed, 24 insertions(+) rename docs/Chapters/{KALLMAN-FILTER.md => 10-KALLMAN-FILTER.md} (100%) rename docs/Chapters/{CRYPTOGRAPHY-WITH-STATE-OBSERVER.md => 11-CRYPTOGRAPHY-WITH-STATE-OBSERVER.md} (100%) rename docs/Chapters/{FEEDBACK-LINEARIZATION.md => 12-KALLMAN-DECOMPOSITION.md} (100%) rename docs/Chapters/{CONGESTION-AVOIDANCE.md => 7-CONGESTION-AVOIDANCE.md} (100%) rename docs/Chapters/{SMITH-PREDICTOR.md => 8-SMITH-PREDICTOR.md} (100%) rename docs/Chapters/{VIDEO-STREAMING.md => 9-VIDEO-STREAMING.md} (100%) diff --git a/README.md b/README.md index 9d321c1..c6df6f7 100644 --- a/README.md +++ b/README.md @@ -1,2 +1,3 @@ # Control-Network-Systems +All the info [here](./docs/README.md) \ No newline at end of file diff --git a/docs/Chapters/KALLMAN-FILTER.md b/docs/Chapters/10-KALLMAN-FILTER.md similarity index 100% rename from docs/Chapters/KALLMAN-FILTER.md rename to docs/Chapters/10-KALLMAN-FILTER.md diff --git a/docs/Chapters/CRYPTOGRAPHY-WITH-STATE-OBSERVER.md b/docs/Chapters/11-CRYPTOGRAPHY-WITH-STATE-OBSERVER.md similarity index 100% rename from docs/Chapters/CRYPTOGRAPHY-WITH-STATE-OBSERVER.md rename to docs/Chapters/11-CRYPTOGRAPHY-WITH-STATE-OBSERVER.md diff --git a/docs/Chapters/FEEDBACK-LINEARIZATION.md b/docs/Chapters/12-KALLMAN-DECOMPOSITION.md similarity index 100% rename from docs/Chapters/FEEDBACK-LINEARIZATION.md rename to docs/Chapters/12-KALLMAN-DECOMPOSITION.md diff --git a/docs/Chapters/CONGESTION-AVOIDANCE.md b/docs/Chapters/7-CONGESTION-AVOIDANCE.md similarity index 100% rename from docs/Chapters/CONGESTION-AVOIDANCE.md rename to docs/Chapters/7-CONGESTION-AVOIDANCE.md diff --git a/docs/Chapters/SMITH-PREDICTOR.md b/docs/Chapters/8-SMITH-PREDICTOR.md similarity index 100% rename from docs/Chapters/SMITH-PREDICTOR.md rename to docs/Chapters/8-SMITH-PREDICTOR.md diff --git a/docs/Chapters/VIDEO-STREAMING.md b/docs/Chapters/9-VIDEO-STREAMING.md similarity index 100% rename from docs/Chapters/VIDEO-STREAMING.md rename to docs/Chapters/9-VIDEO-STREAMING.md diff --git a/docs/README.md b/docs/README.md index e69de29..9600ae4 100644 --- a/docs/README.md +++ b/docs/README.md @@ -0,0 +1,23 @@ +# Control Network Systems +This dcoument has been made by following notes +made by Prof. Saverio Mascolo. + +## Chapters +1. [Modern Control](./Chapters/1-MODERN-CONTROL.md) +2. [Relation to Classical Control](./Chapters/2-RELATION-TO-CLASSICAL-CONTROL.md) +3. [Canonical Forms](./Chapters/3-CANONICAL-FORMS.md) +4. [Reachability and observability](./Chapters/4-REACHABILITY-AND-OBSERVABILITY.md) +5. [State Feedback](./Chapters/5-STATE-FEEDBACK.md) +6. [Sensitivity](./Chapters/6-2-SENSITIVITY.md) +7. [Congestion Avoidance](./Chapters/7-CONGESTION-AVOIDANCE.md) +8. [Smith Predictor](./Chapters/8-SMITH-PREDICTOR.md) +9. [Video Streaming](./Chapters/9-VIDEO-STREAMING.md) +10. [Kallman Filter](./Chapters/10-KALLMAN-FILTER.md) +11. [Crypthography](./Chapters/11-CRYPTOGRAPHY-WITH-STATE-OBSERVER.md) +12. [Kallman Decomposition](./Chapters/12-KALLMAN-DECOMPOSITION.md) +13. [Extras](./Chapters/EXTRAS.md) + +## Formularies +- [Geometry](./Formularies/GEOMETRY-FORMULARY.md) +- [Physics](./Formularies/PHYSICS-FORMULARY.md) +- [Control Theory](./Formularies/CONTROL-FORMULARY.md) \ No newline at end of file From 344641b0a40c8cf9703ae59e79d8711b37526945 Mon Sep 17 00:00:00 2001 From: Christian Risi <75698846+CnF-Gris@users.noreply.github.com> Date: Wed, 15 Jan 2025 14:49:52 +0100 Subject: [PATCH 3/3] V0.8.9 --- docs/Chapters/12-KALLMAN-DECOMPOSITION.md | 48 +++++ docs/Chapters/Examples/EXAMPLE-12.md | 230 ++++++++++++++++++++++ 2 files changed, 278 insertions(+) create mode 100644 docs/Chapters/Examples/EXAMPLE-12.md diff --git a/docs/Chapters/12-KALLMAN-DECOMPOSITION.md b/docs/Chapters/12-KALLMAN-DECOMPOSITION.md index e69de29..f2d54a7 100644 --- a/docs/Chapters/12-KALLMAN-DECOMPOSITION.md +++ b/docs/Chapters/12-KALLMAN-DECOMPOSITION.md @@ -0,0 +1,48 @@ +# Kallman Decomposition + +## Some background +- $X_r = R(K_c)$ : Reachable space is the range of the +`controllable matrix` +- $X_{no} = Ker(K_o)$ : Not observable Space is the kernel +of the `observability matrix` +- $X = X_r \bigoplus X_{nr}$ : Each possible state is sum of +bases of `reachable` and `not-reachable states` + +## Full Decomposition +> [!TIP] +> Follow [Example 12](./Examples/EXAMPLE-12.md/#kallman-full-decomposition) To understand this part +- $X_1 = X_r \cap X_{nr}$ : `Reachable` but `Not-Observable` + space +- $X_2 =$ Complement of $X_1$ to cover $X_r$ : Both `Reachable` + and `Observable` +- $X_3 =$ Complement of $X_1$ to cover $X_{no}$ : Both + `Not-Reachable` and `Not-Observable` +- $X_4 =$ Complement of all the others to cover $X$ : Bot + `Not-Reachable` and `Observable` + +From here we have these blocks: +$$ +\begin{align*} + \hat{A} &= \begin{bmatrix} + \hat{A}_{11} & \hat{A}_{12} & \hat{A}_{13} & \hat{A}_{14} \\ + 0 & \hat{A}_{22} & 0 & \hat{A}_{24} \\ + 0 & 0 & \hat{A}_{33} & \hat{A}_{34} \\ + 0 & 0 & 0 & \hat{A}_{44} + \end{bmatrix}\\ + + \hat{B} &= \begin{bmatrix} + \hat{B}_{1} \\ \hat{B}_{2} \\ 0 \\ 0 + \end{bmatrix}\\ + + \hat{C} &= \begin{bmatrix} + 0 & \hat{C}_{2} & 0 & \hat{C}_{4} + \end{bmatrix} +\end{align*} +$$ + +Now, the eigenvalues of $\hat{A} = \cup_i^4 \hat{A}_{ii}$ and: +- $eig(\hat{A}_{11})$: `Reachable` and `Not-Observable` +- $eig(\hat{A}_{22})$: `Reachable` and `Not-Observable` +- $eig(\hat{A}_{33})$: `Not-Reachable` and `Not-Observable` +- $eig(\hat{A}_{44})$: `Not-Reachable` and `Observable` + diff --git a/docs/Chapters/Examples/EXAMPLE-12.md b/docs/Chapters/Examples/EXAMPLE-12.md new file mode 100644 index 0000000..de738c7 --- /dev/null +++ b/docs/Chapters/Examples/EXAMPLE-12.md @@ -0,0 +1,230 @@ +# Example 12 +## Kallman Full Decomposition + +$$ +\begin{align*} + A &= \begin{bmatrix} + -4 & -3 & 0 & -2 \\ + 6 & 5 & 0 & 2 \\ + 4 & 1 & 1 & -6 \\ + -1 & -1 & 0 & -3 \\ + \end{bmatrix}\\ + + B &= \begin{bmatrix} + -1 \\ + 1 \\ + 2 \\ + 0 \\ + \end{bmatrix}\\ + + C &= \begin{bmatrix} + -3 & -2 & 0 & 1 \\ + \end{bmatrix}\\ + + D &= \begin{bmatrix} + 0 + \end{bmatrix}\\ + + K_r &= \begin{bmatrix} + -1 & 1 & -1 & 1 \\ + 1 & -1 & 1 & -1 \\ + 2 & -1 & 2 & -1\\ + 0 & 0 & 0 & 0 + \end{bmatrix}\\ + + K_{no} &= \begin{bmatrix} + +-3 &-2 &0 &1\\ +-1 &-2 &0 &-1\\ +-7 &-6 &0 &1\\ +-9 &-10& 0 &-1\\ + \end{bmatrix}\\ + +\end{align*}\\ +\text{To find all the bases, for $P_{K_r}$ you should find the}\\ +\text{independent columns and find the system}\\ + +\text{For $P_{K_{no}}$ you should find the system by looking at}\\ +\text{rows, solve it and then find some bases}\\ +\begin{align*} + + P_{K_r} &= \begin{bmatrix} + 0&1\\ + 0&-1\\ + 1&0\\ + 0&0 + \end{bmatrix}\\ + + X_r &= \begin{cases} + x_1 =-x_2 \\ + x_4 = 0 + \end{cases}\\ + + X_{no} &= \begin{cases} + -3x_1 - 2x_2 + x_4 = 0 \\ + -x_1 -2x_2 -x_4 = 0 + \end{cases} \rightarrow \\ + + &\rightarrow \begin{cases} + x_1 = x_4 \\ + x_1 = -x_2 + \end{cases} \\ + + P_{K_{no}} &= \begin{bmatrix} + 1 & 0\\ + -1 & 0\\ + 0 & 1 \\ + 1 & 0 + \end{bmatrix} + + +\end{align*} \\ +\text{Now, let's get all $X_r$ and $X_{no}$ elements}\\ + +\begin{align*} + X_r &= \left\{ + \alpha \begin{bmatrix} + 0\\ + 0\\ + 1\\ + 0 + \end{bmatrix} + + + \beta \begin{bmatrix} + 1\\ + -1\\ + 0\\ + 0 + \end{bmatrix} + \right\}\\ + + X_{no} &= \left\{ + \alpha \begin{bmatrix} + 1 \\ + -1\\ + 0 \\ + 1 + \end{bmatrix} + + + \beta \begin{bmatrix} + 0\\ + 0\\ + 1 \\ + 0 + \end{bmatrix} + \right\} +\end{align*}\\ +$$ +Now, let's get $X_1$, $X_2$, $X_3$, $X_4$ + +$$ +\begin{align*} +X_1 &= X_r \cap X_{no} = \begin{cases} + x_1 =-x_2 \\ + x_4 = 0 \\ + x_1 = x_4 + \end{cases}\\ + +X_1 &= \begin{bmatrix} + 0 \\ 0 \\ x_3 \\ 0 +\end{bmatrix}\\ + +0 &= X_1^T X_1^\perp \rightarrow \\ + +&\rightarrow \begin{bmatrix} + 0 & 0 & x_3 & 0 +\end{bmatrix} +\begin{bmatrix} + x_a \\ x_b \\ x_c \\ x_d +\end{bmatrix} = 0 \rightarrow \\ +&\rightarrow\begin{cases} + x_c = 0 +\end{cases} \\ + +X_1^\perp &= \begin{bmatrix} + x_a \\ x_b \\ 0 \\ x_d +\end{bmatrix}\\ + +X_2 &= X_r \cap X_1^\perp = \begin{cases} + x_1 = -x_2 \\ + x_3 = 0 \\ + x_4 = 0 \\ +\end{cases}\\ + +X_2 &= \begin{bmatrix} + x_1 \\ -x_1 \\ 0 \\ 0 +\end{bmatrix}\\ + +X_3 &= X_{no} \cap X_1^\perp = \begin{cases} + x_1 = -x_2 \\ + x_1 = x_4 \\ + x_3 = 0 \\ +\end{cases}\\ + +X_3 &= \begin{bmatrix} + x_1 \\ -x_1 \\ 0 \\ x_1 +\end{bmatrix}\\ + +0 &= X_2^T X_2^\perp = +\begin{bmatrix} + 1 & -1 & 0 & 0 +\end{bmatrix} +\begin{bmatrix} + x_a \\ x_b \\ x_c \\ x_d +\end{bmatrix}\rightarrow \\ +&\rightarrow\begin{cases} + x_a - x_b = 0 +\end{cases} \\ +\\ + +0 &= X_3^T X_3^\perp = +\begin{bmatrix} + 1 & -1 & 0 & 1 +\end{bmatrix} +\begin{bmatrix} + x_a \\ x_b \\ x_c \\ x_d +\end{bmatrix}\rightarrow \\ +&\rightarrow\begin{cases} + x_a - x_b + x_d = 0 +\end{cases} \\ +\\ + +X_4 &= X_1^\perp \cap X_2^\perp \cap X_3^\perp = \\ + &= \begin{cases} + x_3 = 0 \\ + x_1 = x_2 \\ + x_4 = 0\\ + \end{cases} = \\ + &= \begin{bmatrix} + x_1 \\ x_2 \\ 0 \\ 0 + \end{bmatrix} + + +\end{align*} + + +$$ + +Now we find the change state matrix: +$$ +\begin{align*} + Q^{-1} &= \begin{bmatrix} + X_1 & X_2 & X_3 & X_4 + \end{bmatrix} = \\ + &= \begin{bmatrix} + 0 & 1 & 1 & 1\\ + 0 & -1 & -1 & 1\\ + 1 & 0&0&0\\ + 0&0&1&0 + \end{bmatrix} +\end{align*} +$$ + +Compute: +$$ +\begin{align*} + \hat{A} &= QAQ^{-1}\\ + \hat{B} &= QB \\ + \hat{C} &= CQ^{-1} +\end{align*} +$$