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The associated `transfer function` $G(s)$ is: + +$$ +G(s) = \frac{Y(s)}{U(s)} +$$ + +### Root Locus + + +### Bode Diagram + +### Nyquist Diagram + +## State Space Representation + +### State Matrices +A state space representation has 4 Matrices: $A, B, C, D$ with coefficients in +$\R$: +- $A$: State Matrix `[x_rows, x_columns]` +- $B$: Input Matrix `[x_rows, u_columns]` +- $C$: Output Matrix `[y_rows, x_columns]` +- $D$: Direct Coupling Matrix `[y_rows, u_columns]` + +$$ +\begin{cases} +\dot{x}(t) = Ax(t) + Bu(t) \;\;\;\; \text{Dynamic of the system}\\ + y(t) = C{x}(t) + Du(t) \;\;\;\; \text{Static of the outputs} +\end{cases} +$$ + +This can be represented with the following diagrams: + +#### Continuous Time: +![continuous state space diagram](..\Images\Modern-Control\state-space-time.png) + +--- +#### Discrete time: + +![discrete state space diagram](..\Images\Modern-Control\state-space-discrete.png) + +### State Vector +This is a state vector `[x_rows, 1]`: +$$ +x(t) = \begin{bmatrix} + x_1(t)\\ + \dots\\ + x_x(t) + \end{bmatrix} + +\text{or} \: + +x(k) = \begin{bmatrix} + x_1(k)\\ + \dots\\ + x_x(k) + \end{bmatrix} +$$ + +Basically, from this we can know each next step of the state vector, represented +as: + +$$ +x(k + 1) = f\left( + x(k), u(k) +\right) = Ax(k) + Bu(k) +$$ + +### Examples + +#### Cart attached to a spring and a damper, pulled by a force +![cart being pulled image](../Images/Modern-Control/cart-pulled.png) + +##### Formulas +- Spring: $\vec{F} = -k\vec{x}$ +- Fluid Damper: $\vec{F_D} = -b \vec{\dot{x}}$ +- Initial Force: $\vec{F_p}(t) = \vec{F_p}(t)$ +- Total Force: $m \vec{\ddot{x}}(t) = \vec{F_p}(t) -b \vec{\dot{x}} -k\vec{x}$ + +> [!TIP] +> +> A rule of thumb is to have as many variables in our state as the max number +> of derivatives we encounter. In this case `2` +> +> Solve the equation for the highest derivative order +> +> Then, put all variables equal to the previous one derivated: +> +> $$ +x(t) = \begin{bmatrix} + x_1(t)\\ + x_2(t) = \dot{x_1}(t)\\ + \dots\\ + x_n(t) = \dot{x}_{n-1}(t) + \end{bmatrix} +\; +\dot{x}(t) = \begin{bmatrix} + \dot{x_1}(t) = x_2(t)\\ + \dot{x_2}(t) = x_3(t)\\ + \dots\\ + \dot{x}_{n-1}(t) = \dot{x}_{n}(t)\\ + \dot{x}_{n}(t) = \text{our formula} + \end{bmatrix} + +> $$ +> + +Now in our state we may express `position` and `speed`, while in our +`next_state` we'll have `speed` and `acceleration`: +$$ +x(t) = \begin{bmatrix} + x_1(t)\\ + x_2(t) = \dot{x_1}(t) + \end{bmatrix} +\; +\dot{x}(t) = \begin{bmatrix} + \dot{x_1}(t) = x_2(t)\\ + \dot{x_2}(t) = \ddot{x_1}(t) + \end{bmatrix} + +$$ +Our new state is then: +$$ +\begin{cases} +\dot{x}_1(t) = x_2(t)\\ +\dot{x}_2(t) = \frac{1}{m} \left( \vec{F}(t) - b x_2(t) - kx_1(t) \right) +\end{cases} +$$ + +let's say we only want to check for the `position` and `speed` of the system, our +State Space will be: +$$ +A = \begin{bmatrix} +0 & 1 \\ +- \frac{k}{m} & - \frac{b}{m} \\ +\end{bmatrix} + +B = \begin{bmatrix} +0 \\ +\frac{1}{m} \\ +\end{bmatrix} + +C = \begin{bmatrix} +1 & 0 \\ +0 & 1 +\end{bmatrix} + +D = \begin{bmatrix} +0 +\end{bmatrix} +$$ + +let's say we only want to check for the `position` of the system, our +State Space will be: +$$ +A = \begin{bmatrix} +0 & 1 \\ +- \frac{k}{m} & - \frac{b}{m} \\ +\end{bmatrix} + +B = \begin{bmatrix} +0 \\ +\frac{1}{m} \\ +\end{bmatrix} + +C = \begin{bmatrix} +1 & 0 +\end{bmatrix} + +D = \begin{bmatrix} +0 +\end{bmatrix} +$$ + +> [!TIP] +> In order to being able to plot the $\vec{x}$ against the time, you need to +> multiply $\vec{\dot{x}}$ for the `time_step` and then add it to the state[^so-how-to-plot-ssr] +> + +### Horner Factorization +let's say you have a complete polynomial of order `n`, you can factorize +in this way: + +$$ +\begin{align*} +p(s) &= s^5 + 4s^4 + 5s^3 + 2s^2 + 10s + 1 =\\ + +&= s ( s^4 + 4s^3 + 5s^2 + 2s + 10) + 1 = \\ + +&= s ( s (s^3 + 4s^2 + 5s + 2) + 10) + 1 = \\ + +&= s ( s (s (s^2 + 4s + 5) + 2) + 10) + 1 = \\ + +&= s ( s (s ( s (s + 4) + 5) + 2) + 10) + 1 + +\end{align*} +$$ + +If you were to take each s with the corresponding number in the parenthesis, +you'll make this block: +![horner factorization to diagram block](../Images/Modern-Control/horner-factorization.png) + + + + + + + + + + +### Case Studies + +- PAGERANK +- Congestion Control +- Video Player Control +- Deep Learning + +[^so-how-to-plot-ssr]: [Stack Exchange | How to plot state space variables against time on unit step input? | 05 January 2025 ](https://electronics.stackexchange.com/questions/307227/how-to-plot-state-space-variables-against-time-on-unit-step-input) \ No newline at end of file diff --git a/docs/Chapters/2-RELATION-TO-CLASSICAL-CONTROL.md b/docs/Chapters/2-RELATION-TO-CLASSICAL-CONTROL.md new file mode 100644 index 0000000..781792f --- /dev/null +++ b/docs/Chapters/2-RELATION-TO-CLASSICAL-CONTROL.md @@ -0,0 +1,160 @@ +# Relation to Classical Control + +## A Brief Recap of Discrete Control +Let's say we want to control something ***Physical***, hence intrinsically +***time continuous***, we can model our control in the `z` domain and make our +$G_c(z)$. But how do we connect these systems: + +![scheme of how digital control interconnects with classical control](../Images/Relation-to-classical-control/digital-control.png) + +#### Contraints +- $T_s$: Sampling time +- $f_s \geq 2f_m$: Sampling Frequency must be at least 2 times the max frequency + of the system + + +#### Parts of the system +1. Take `reference` and `output` and compute the `error` +2. Pass this signal into an `antialiasing filter` to avoid ***aliases*** +3. Trasform the `Laplace Transform` in a `Z-Transform` by using the following + relation:\ + $z = e^{sT}$ +4. Control everything through a `control block` engineered through + `digital control` +5. Transform the `digital signal` to an `analogic signal` through the use of a + `holder` (in this case a `zero order holder`) +6. Pass the signal to our `analogic plant` (which is our physical system) +7. Take the `output` and pass it in `retroaction` + +### Zero Order Holder +It has the following formula: +$$ +ZoH = \frac{1}{s} \left( 1 - e^{sT}\right) +$$ + +#### Commands: +- `c2d(sysc, Ts [, method | opts] )`[^matlab-c2d]: Converts `LTI` systems into + `Discrete` ones + + +## Relation between $S(A, B, C, D)$ to $G(s)$ + +### From $S(A, B, C, D)$ to $G(s)$ +Be this our $S(A, B, C, D)$ system: +$$ +\begin{cases} +\dot{x}(t) = Ax(t) + Bu(t) \;\;\;\; \text{Dynamic of the system}\\ + y(t) = C{x}(t) + Du(t) \;\;\;\; \text{Static of the outputs} +\end{cases} +$$ + +now let's make from this a `Laplace Transform`: +$$ +\begin{align*} +& \begin{cases} + sX(s) - x(0)= AX(s) + BU(s) \\ + Y(s) = CX(s) + DU(s) + \end{cases} \longrightarrow && \text{Normal Laplace Transformation}\\ + + +& \longrightarrow +\begin{cases} + sX(s) = AX(s) + BU(s) \\ + Y(s) = CX(s) + DU(s) + \end{cases} \longrightarrow && \text{Usually $x(0)$ is 0}\\ + + +& \longrightarrow +\begin{cases} + X(s) \left(sI -A \right) =BU(s) \\ + Y(s) = CX(s) + DU(s) + \end{cases} \longrightarrow && \text{$sI$ is technically equal to $s$}\\ + + +& \longrightarrow +\begin{cases} + X(s) = \left(sI - A\right)^{-1}BU(s) \\ + Y(s) = CX(s) + DU(s) + \end{cases} \longrightarrow && \\ + + +& \longrightarrow +\begin{cases} + X(s) = \left(sI - A\right)^{-1}BU(s) \\ + Y(s) = C\left(sI - A\right)^{-1}BU(s) + DU(s) + \end{cases} \longrightarrow && \text{Substitute for $X(s)$}\\ + + +& \longrightarrow +\begin{cases} + X(s) = \left(sI - A\right)^{-1}BU(s) \\ + Y(s) = \left(C\left(sI - A\right)^{-1}B + D\right)U(s) + \end{cases} \longrightarrow && \text{Group for $U(s)$}\\ + + +& \longrightarrow +\begin{cases} + X(s) = \left(sI - A\right)^{-1}BU(s) \\ + \frac{Y(s)}{U(s)} = \left(C\left(sI - A\right)^{-1}B + D\right) + \end{cases} \longrightarrow && \text{Get $G(s)$ from definition}\\ + +\longrightarrow \;& G(s) = \left(C\left(sI - A\right)^{-1}B + D\right) && +\text{Formal definition of $G(s)$}\\ + + +\end{align*} +$$ + +#### Properties +- Since $G(s)$ can be ***technically*** a matrix, this may represent a +`MIMO System` +- The system is ***always*** `proper` (so it's denominator is of an order + higher of the numerator) +- If $D$ is $0$, then the system is `strictly proper` and ***realizable*** + +- While each $S(A_i, B_i, C_i, D_i)$ can be transformed into a ***single*** +$G(s)$, this isn't true viceversa. + +- Any particular $S(A_a, B_a, C_a, D_a)$ is called `realization` +- $det(sI - A)$ := Characteristic Polinome +- $det(sI - A) = 0$ := Characteristic Equation +- $eig(A)$ := Solutions of the Characteristic Equation and `poles` of the system +- If the system is `SISO` and this means that $C \in \R^{1,x}$, + $B \in \R^{x,1}$ and $D \in \R$, meaning + that: +$$ +\begin{align*} + G(s) &= \left(C\left(sI - A\right)^{-1}B + D\right) =\\ + &= \left(C \frac{Adj\left(sI - A\right)}{det\left(sI - A\right)}B + D\right) + = && \text{Decompose the inverse in its formula}\\ + &= \frac{n(s)}{det\left(sI - A\right)} \in \R +\end{align*} +$$ + +> [!NOTE] +> As you can see here, by decomposing the inverse matrix in its formula it's +> easy to see that the divisor is a `scalar`, a `number`. +> +> Moreover, because of how $B$ and $C$ are composed, the result of this Matrix +> multiplication is a `scalar` too, hence we can write this as a single formula. +> +> Another thing to notice, regardless if this is a `MIMO` or `SISO` system is +> that at the divisor we have all `eigenvalues` of A as `poles` by +> [definition](../Formularies/GEOMETRY-FORMULARY.md/#eigenvalues) +> + +### Transforming a State-Space into Another one +We basically need to use some non singular `Permutation Matrices`: +$$ +\begin{align*} +&A_1, B_1, C_1, D_1 \\ +&A_2 = PAP^{-1} \\ +&B_2 = PB \\ +&C_2 = CP^{-1} \\ +&D_2 = D_1 +\end{align*} +$$ + + + +[^matlab-c2d]: [Matlab Official Docs | c2d | 05 January 2025](https://it.mathworks.com/help/control/ref/dynamicsystem.c2d.html) \ No newline at end of file diff --git a/docs/Chapters/3-CANONICAL-FORMS.md b/docs/Chapters/3-CANONICAL-FORMS.md new file mode 100644 index 0000000..7a1e07b --- /dev/null +++ b/docs/Chapters/3-CANONICAL-FORMS.md @@ -0,0 +1,67 @@ +# Canonical Forms + +In order to see if we are in one of these canonical forms, just write the +equations from the block diagram, and find the associated $S(A, B, C, D)$. + +> [!TIP] +> In order to find a rough diagram, use +> [Horner Factorization](MODERN-CONTROL.md/#horner-factorization) to find +> $a_i$ values. Then put all the $b_i$ to the right integrator by shifting them +> as many left places, starting from the rightmost, for the number of +> associated $s$ + +## Control Canonical Form +It is in such forms when: +$$ +A = \begin{bmatrix} +- a_1 & -a_2 & -a_3 & \dots & -a_{n-1} &-a_n\\ +1 & 0 & 0 & \dots & 0 & 0\\ +0 & 1 & 0 & \dots & 0 & 0\\ +\dots & \dots & \dots & \dots & \dots \\ +0 & 0 & 0 & \dots & 1 & 0 +\end{bmatrix} + +B = \begin{bmatrix} +1 \\ 0 \\ \dots \\ \dots \\ 0 +\end{bmatrix} + +C = \begin{bmatrix} +b_1 & b_2 & \dots & b_n +\end{bmatrix} + +D = \begin{bmatrix} +0 +\end{bmatrix} +$$ + +## Modal Canonical Forms +> [!CAUTION] +> This form is the most difficult to find, as this varies drastically in cases +> of double roots +> +$$ +A = \begin{bmatrix} +- a_1 & 0 & 0 & \dots & 0\\ +0 & -a_2 & 0 & \dots & 0\\ +0 & 0 & -a_3 & \dots & 0\\ +\dots & \dots & \dots & \dots & \dots \\ +0 & 0 & 0 & 0 & -a_n +\end{bmatrix} + +B = \begin{bmatrix} +1 \\ 1 \\ \dots \\ \dots \\ 1 +\end{bmatrix} + +C = \begin{bmatrix} +b_1 & b_2 & \dots & b_n +\end{bmatrix} + +D = \begin{bmatrix} +0 +\end{bmatrix} +$$ + +## Observable Canonical Form + + +[^reference-input-pole-allocation]: [MIT | 06 January 2025 | pg. 2](https://ocw.mit.edu/courses/16-30-feedback-control-systems-fall-2010/c553561f63feaa6173e31994f45f0c60_MIT16_30F10_lec11.pdf) \ No newline at end of file diff --git a/docs/Chapters/4-REACHABILITY-AND-OBSERVABILITY.md b/docs/Chapters/4-REACHABILITY-AND-OBSERVABILITY.md new file mode 100644 index 0000000..dd21f29 --- /dev/null +++ b/docs/Chapters/4-REACHABILITY-AND-OBSERVABILITY.md @@ -0,0 +1,140 @@ +# Reachability and Observability + +## Reachability +While in the non linear world, we can solve a `system` +***numerically***, through an `iterative-approach`: +$$ +\begin{align*} +\dot{x}(t) &= f(x(t), u(t)) && t \in \R \\ +x(k+1) &= f(x(k), u(k)) && t \in \N +\end{align*} +$$ + +In the linear world, we can do this ***analitically***: +> [!TIP] +> We usually consider $x(0) = 0$ +$$ +\begin{align*} +x(1) &= Ax(0) + Bu(0) \\ +x(2) &= Ax(1) + Bu(1) &&= A^{2}x(0) + ABu(0) + Bu(1) \\ +x(3) &= Ax(2) + Bu(2) &&= A^{3}x(0) + A^{2}Bu(0) + ABu(1) + Bu(2) \\ +\dots \\ +x(k) &= Ax(k-1) + Bu(k-1) &&= +\underbrace{A^{k}x(0)}_\text{Free Dynamic} + +\underbrace{A^{k-1}Bu(0) + \dots + ABu(k-2) + Bu(k-1) } +_\text{Forced Dynamic} \\[40pts] + +x(k) &= \begin{bmatrix} +B & AB & \dots & A^{k-2}B & A^{k-1}B +\end{bmatrix} + +\begin{bmatrix} +u(k-1) \\ u(k-2) \\ \dots \\ u(1) \\ u(0) +\end{bmatrix} +\end{align*} +$$ + +Now, there's a relation between the determinant and the matrix containing +$A$ and $B$: + +$$ +\begin{align*} + +&p_c(s) = det(sI -A) = +s^n + a_{n-1} s^{n-1} + a_{n-2}s^{n - 2} + \dots + a_1s + a_0 = 0 +\\[10pt] + +&\text{Apply Caley-Hamilton theorem:} \\ +&p_c(A) = A^{n} + a_{n-1}A^{n_1} + \dots + a_1A + a_0 = 0\\[10pt] + +&\text{Remember $G(s)$ formula and multiply $p_c(A)$ for $B$:} \\ +&p_c(A)B = A^{n}B + a_{n-1}A^{n_1}B + \dots + a_1AB + a_0B = 0 \rightarrow \\ + +&p_c(A)B = a_{n-1}A^{n_1}B + \dots + a_1AB + a_0B = -A^{n}B +\end{align*} +$$ + +All of these makes us conclude something about the Kallman +Controllability Matrix: +$$ +K_c = \begin{bmatrix} +B & AB & \dots & A^{k-2}B & A^{k-1}B +\end{bmatrix} +$$ + +Moreover, $x(n) \in range(K_c)$ and $range(K_c)$ is said +`reachable space`.\ +In particular if $rank(K_c) = n \rightarrow range(K_c) = \R^{n}$ this +is `fully reachable` or `controllable` +> [!TIP] +> Some others use `non-singularity` instead of the $range()$ definition + + +> [!NOTE] +> On the Franklin Powell there's another definition to $K_c$ that +> comes from the fact that we needed to find a way to transform +> ***any*** `realization` into the +> [`Control Canonical Form`](./CANONICAL-FORMS.md/#control-canonical-form) +> + +## Observability +This is the capability of being able to deduce the `initial state` by just +observing the `output`. + +Let's focus on the $y(t)$ part: +$$ +y(t) = +\underbrace{Cx(t)}_\text{Free Output} + +\underbrace{Du(t)}_\text{Forced Output} +$$ + +Assume that $u(t) = 0$: +$$ +\begin{align*} +& y(0) = Cx(0) && x(0) = x(0) \\ +& y(1) = Cx(1) && x(1) = Ax(0) && +\text{Since $u(t) = 0 \rightarrow Bu(t) = 0$} \\ +& y(2) = Cx(2) && x(2) = A^2x(0) \\ +& \vdots && \vdots \\ +&y(n) = Cx(n) && x(n) = A^nx(0) \rightarrow \\ +\rightarrow &y(n) = CA^nx(0) +\end{align*} +$$ + +Now we have that: +$$ +\begin{align*} +\vec{y} = &\begin{bmatrix} +C \\ +CA \\ +\vdots \\ +CA^{n} +\end{bmatrix} x(0) \rightarrow \\ + +\rightarrow x(0) = &\begin{bmatrix} +C \\ +CA \\ +\vdots \\ +CA^{n} +\end{bmatrix}^{-1}\vec{y} \rightarrow \\ + +\rightarrow x(0) = & \frac{Adj(K_o)}{det(K_o)} \vec{y} +\end{align*} +$$ + +For the same reasons as before, we can use Caley-Hamilton here too, also, we can see that if $K_o$ is `singular`, there can't be an inverse. + +As before, $K_o$ is such a matrix that allows us to see if there exists a +[`Canonical Observable Form`](CANONICAL-FORMS.md/#observable-canonical-form) + +The `non-observable-space` is equal to: +$$ +X_{no} = Kern(K_o) : \left\{ K_ox = 0 | x \in X\right\} +$$ + +## Decomposition of these spaces +The space of possible points is $X$ and is equal to +$X = X_r \bigoplus X_{r}^{\perp} = X_r \bigoplus X_{nr}$ + +Analogously we can do the same with the observable spaces +$X = X_no \bigoplus X_{no}^{\perp} = X_no \bigoplus X_{o}$ \ No newline at end of file diff --git a/docs/Chapters/5-STATE-FEEDBACK.md b/docs/Chapters/5-STATE-FEEDBACK.md new file mode 100644 index 0000000..a04f83e --- /dev/null +++ b/docs/Chapters/5-STATE-FEEDBACK.md @@ -0,0 +1,104 @@ +# State Feedback + +## State Feedback +When you have your $G(s)$, just put poles where you want them to be and compute the right $k_i$. + +What we are doing is to put $u(t) = -Kx(t)$, feeding back the `state` to the +`system`. + +> [!WARNING] +> While $K$ matrix is controllable, the A matrix is equivalent to our plant, so +> it's impossible to change $a_i$ without changing the `system` + +> [!CAUTION] +> We are doing some oversimplifications here, but this can't be used as formal +> proof for whatever we are saying. To see more, look at +> [Ackerman's formula](https://en.wikipedia.org/wiki/Ackermann%27s_formula). +> +> However is it possible to put a fictional reference input to make everything +> go back to a similar proof for the +> characteristic equation[^reference-input-pole-allocation] + + +While it is possible to allocate poles in each form, the Canonical Control +one makes it very easy to accomplish this task: + +$$ +A = \begin{bmatrix} +- a_1 \textcolor{#ff8382}{-k_1}& -a_2 \textcolor{#ff8382}{-k_2} +& -a_3 \textcolor{#ff8382}{-k_3}& \dots +& -a_{n-1}\textcolor{#ff8382}{-k_{n-1}} + &-a_n \textcolor{#ff8382}{-k_n}\\ +1 & 0 & 0 & \dots & 0 & 0\\ +0 & 1 & 0 & \dots & 0 & 0\\ +\dots & \dots & \dots & \dots & \dots \\ +0 & 0 & 0 & \dots & 1 & 0 +\end{bmatrix} +$$ + +This changes our initial considerations, however. The new Characteristi Equation +becomes: +$$ +det\left(sI - (A - BK)\right) +$$ + +## State Observer +What happens if we have not enough sensors to get the state in +`realtime`? + +> [!NOTE] +> Measuring the state involves a sensor, which most of the times is +> either inconvenient to place because of space, or inconvenient +> economically speaking + +In this case we `observe` our output to `estimate` our state +($\hat{x}$). THis new state will be used in our $G_c$ block: + +$$ +\dot{\hat{x}} = A\hat{x} + Bu + L(\hat{y} - y) = A\hat{x} + Bu + +LC(\hat{x} - x) +$$ + +Since we are estimating our state, we need the estimator to be fast, +**at least 6 times fater** than our `plant`, and $u = -K\hat{x}$. + +Let's compute the error we introduce in our system: + +$$ +\begin{align*} +e &= x - \hat{x} \rightarrow \\ +\rightarrow \dot{e} &= \dot{x} - \dot{\hat{x}} \rightarrow\\ + +&\rightarrow Ax - BK\hat{x} - A\hat{x} + BK\hat{x} - LCe \rightarrow\\ +&\rightarrow -Ae - LCe \rightarrow\\ +&\rightarrow eig(-A-LC) +\end{align*} +$$ + +## Making a Controller +So, this is how we will put all of these blocks saw until now. +![block diagram](..\Images\State-Feedback\block-diagram.png) + +From this diagram we can deduce immediately that to us our input is $y$ +and our output is $u$. By treating the blue part as a `black box`, we can +say that: +$$ +\begin{cases} +\dot{\hat{x}} = A\hat{x} - BK\hat{x} + LC\hat{x} - Ly \\ +u = -K\hat{x} +\end{cases} +$$ + +So our $S(A_c, B_c, C_c, D_c)$ is: + +$$ +\begin{align*} +&A_c = A_p - B_pK + LC_p \\ +&B_c = -L \\ +&C_c = -K \\ +&D_c = 0 \\ +\end{align*} +$$ + +and so, you just need to substitute these matrices to the equation of the +$G_p(s)$ and you get you controller \ No newline at end of file diff --git a/docs/Chapters/CONGESTION-AVOIDANCE.md b/docs/Chapters/CONGESTION-AVOIDANCE.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/Chapters/CRYPTOGRAPHY-WITH-STATE-OBSERVER.md b/docs/Chapters/CRYPTOGRAPHY-WITH-STATE-OBSERVER.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/Chapters/Examples/EXAMPLE-3.md b/docs/Chapters/Examples/EXAMPLE-3.md new file mode 100644 index 0000000..204db1d --- /dev/null +++ b/docs/Chapters/Examples/EXAMPLE-3.md @@ -0,0 +1,132 @@ +# Example 3 + +## Double Mass Cart + +![double mass cart](./../../Images/Examples/Example-3/double-cart.png) + +### Formulas + +- Resulting forces for cart 1:\ +$ +m_1 \ddot{p}_1 = k_2(p_2 - p_1) + b_2( \dot{p}_2 - \dot{p}_1) - + k_1 p_1 - b_1 \dot{p}_1 +$ + +- Resulting forces for cart 2:\ +$ +m_2 \ddot{p}_2 = F - k_2(p_2 - p_1) - b_2( \dot{p}_2 - \dot{p}_1) +$ + +### Reasoning +We now have 2 different accelerations. The highest order of derivatives is 2 for +2 variables, hence we need 4 variables in the `state`: + +$$ +x = \begin{bmatrix} +x_1 = p_1\\ +x_2 = p_2\\ +x_3 = \dot{p}_1\\ +x_4 = \dot{p}_2 +\end{bmatrix} + +\dot{x} = \begin{bmatrix} +\dot{x}_1 = \dot{p}_1 = x_3 \\ +\dot{x}_2 = \dot{p}_2 = x_4\\ +\dot{x}_3 = \ddot{p}_1 = + \frac{1}{m_1} \left[ k_2(x_2 - x_1) + b_2( x_4 - x_3) - + k_1 x_1 - b_1 x_3 \right]\\ +\dot{x}_4 = \ddot{p}_2 = + \frac{1}{m_2} \left[ F - k_2(x_2 - x_1) - b_2( x_4 - x_3) \right]\\ +\end{bmatrix} +$$ + +Let's write our $S(A, B, C, D)$: +$$ +A = \begin{bmatrix} +0 & 0 & 1 & 0 \\ +0 & 0 & 0 & 1 \\ +% 3rd row +- \frac{k_2 - k_1}{m_1} & +\frac{k_2}{m_1} & +-\frac{b_2 + b_1}{m_1} & +\frac{b_2}{m_1} \\ +% 4th row +\frac{k_2}{m_12} & +- \frac{k_2}{m_2} & +\frac{b_2}{m_2} & +- \frac{b_2}{m_2} \\ +\end{bmatrix} + +B = \begin{bmatrix} +0 \\ +0 \\ 0 \\ 1 +\end{bmatrix} + +C = \begin{bmatrix} +1 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 +\end{bmatrix} + +D = \begin{bmatrix} +0 +\end{bmatrix} +$$ + + +## Suspended Mass + + +> [!NOTE] +> For those of you the followed CNS course, refer to professor +> PDF for this excercise, as it has some unclear initial conditions +> +> However, in the formulas section, I'll take straight up his own + +![suspended mass](./../../Images/Examples/Example-3/suspended-mass.png) + +### Formulas + +- Resulting forces for mass:\ +$ +m \ddot{p} = -k(p - r) -b(\dot{p} - \dot{r}) +$ + +### Reasoning + + +$$ +x = \begin{bmatrix} +x_1 = p \\ +x_2 = \dot{x}_1 +\end{bmatrix} + +\dot{x} = \begin{bmatrix} +\dot{x}_1 = x_2 \\ +\dot{x}_2 = \frac{1}{m} \left[-k(x_1 - r) -b(x_2 - \dot{r}) \right] +\end{bmatrix} +$$ + + +> [!WARNING] +> Info here are wrong + +Let's write our $S(A, B, C, D)$: +$$ +A = \begin{bmatrix} +0 & 1\\ +-\frac{k}{m} & - \frac{b}{m} +\end{bmatrix} + +B = \begin{bmatrix} +0 \\ +\frac{k + sb}{m} +\end{bmatrix} + +C = \begin{bmatrix} +1 & 0 +\end{bmatrix} + +D = \begin{bmatrix} +0 & 0 +\end{bmatrix} +$$ \ No newline at end of file diff --git a/docs/Chapters/FEEDBACK-LINEARIZATION.md b/docs/Chapters/FEEDBACK-LINEARIZATION.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/Chapters/KALLMAN-FILTER.md b/docs/Chapters/KALLMAN-FILTER.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/Chapters/MODERN-CONTROL.md b/docs/Chapters/MODERN-CONTROL.md deleted file mode 100644 index 213bbf2..0000000 --- a/docs/Chapters/MODERN-CONTROL.md +++ /dev/null @@ -1,98 +0,0 @@ -# Modern Control - -Normally speaking, we know much about classical control, in the form -of: - -$$ -\dot{x}(t) = ax(t) + bu(t) \longleftrightarrow sX(s) - x(0) = aX(S) + bU(s) -$$ - -With the left part being a derivative equation in continuous time, while the -right being its tranformation in the complex domain field. - -> [!NOTE] -> -> $$ -> \dot{x}(t) = ax(t) + bu(t) \longleftrightarrow x(k+1) = ax(k) + bu(k) -> $$ -> -> These are equivalent, but the latter one is in discrete time. -> - -## A brief recap over Classical Control - -Be $Y(s)$ our `output variable` in `classical control` and $U(s)$ our -`input variable`. The associated `transfer function` $G(s)$ is: - -$$ -G(s) = \frac{Y(s)}{U(s)} -$$ - -### Root Locus - - -### Bode Diagram - -### Nyquist Diagram - -## State Space Representation - -### State Matrices -A state space representation has 4 Matrices: $A, B, C, D$ with coefficients in -$\R$: -- $A$: State Matrix `[x_rows, x_columns]` -- $B$: Input Matrix `[x_rows, u_columns]` -- $C$: Output Matrix `[y_rows, x_columns]` -- $D$: Direct Coupling Matrix `[y_rows, u_columns]` - -$$ -\begin{cases} -\dot{x}(t) = Ax(t) + Bu(t) \;\;\;\; \text{Dynamic of the system}\\ - y(t) = C{x}(t) + Du(t) \;\;\;\; \text{Static of the outputs} -\end{cases} -$$ - -This can be represented with the following diagrams: - -Continuous Time: -![continuous state space diagram](..\Images\Modern-Control\state-space-time.png) - ---- -Discrete time: - -![discrete state space diagram](..\Images\Modern-Control\state-space-discrete.png) - -### State Vector -This is a state vector `[x_rows, 1]`: -$$ -x(t) = \begin{bmatrix} - x_1(t)\\ - \dots\\ - x_x(t) - \end{bmatrix} - -\text{or} \: - -x(k) = \begin{bmatrix} - x_1(k)\\ - \dots\\ - x_x(k) - \end{bmatrix} -$$ - -Basically, from this we can know each next step of the state vector, represented -as: - -$$ -x(k + 1) = f\left( - x(k), u(k) -\right) = Ax(k) + Bu(k) -$$ - - -### Case Studies - -- PAGERANK -- Congestion Control -- Video Player Control -- Deep Learning \ No newline at end of file diff --git a/docs/Chapters/SMITH-PREDICTOR.md b/docs/Chapters/SMITH-PREDICTOR.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/Chapters/VIDEO-STREAMING.md b/docs/Chapters/VIDEO-STREAMING.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/Formularies/CONTROL-FORMULARY.md b/docs/Formularies/CONTROL-FORMULARY.md new file mode 100644 index 0000000..0e9e70e --- /dev/null +++ b/docs/Formularies/CONTROL-FORMULARY.md @@ -0,0 +1,35 @@ +# Control Formulary +## Settling time +$ +T_s = \frac{\ln(a_{\%})}{\zeta \omega_{n}} +$ + +- $\zeta$ := Damping ratio +- $\omega_{n}$ := Natural frequency + +## Overshoot +$ +\mu_{p}^{\%} = 100 e^{ + \left( + \frac{- \zeta \pi}{\sqrt{1 - \zeta^{2}}} + \right) +} +$ + +## Reachable Space +$X_r = Span(K_c)$ + +$X = X_r \bigoplus X_{r}^{\perp} = X_r \bigoplus X_{nr}$ + +> [!TIP] +> Since $X_{nr} = X_r^{\perp}$ we can find a set of perpendicular +> vectors by finding $Ker(X_r^{T})$ + +## Non Observable Space +$X_no = Kern(K_o)$ + +$X = X_no \bigoplus X_{no}^{\perp} = X_no \bigoplus X_{o}$ + +> [!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 diff --git a/docs/Formularies/GEOMETRY-FORMULARY.md b/docs/Formularies/GEOMETRY-FORMULARY.md new file mode 100644 index 0000000..9593b14 --- /dev/null +++ b/docs/Formularies/GEOMETRY-FORMULARY.md @@ -0,0 +1,49 @@ +# Geometry Formulary + +## Inverse of a Matrix +$A^{-1} = \frac{1}{det(A)} Adj(A)$ + +## Adjugate Matrix +The adjugate of a matrix $A$ is the `transpose` of the `cofactor matrix`:\ +$Adj(A) = C^{T}$ + +### $(i-j)$-minor (AKA $M_{ij}$) +$M_{ij}$ := Determinant of the matrix $B$ got by removing the +***$i$-row*** and the ***$j$-column*** from matrix $A$ + +### Cofactors +$C$ is the matrix of `cofactors` of a matrix $A$ where all the elements $c_{ij}$ +are defined like this:\ +$ +c_{ij} = \left( -1\right)^{i + j}M_{ij} +$ + +## Eigenvalues +By starting from the definition of `eigenvectors`:\ +$A\vec{v} = \lambda\vec{v}$ + +As we can see, the vector $\vec{v}$ was unaffected by this matrix +multiplication appart from a scaling factor $\lambda$, called `eigenvalue` + +By rewriting this formula we get:\ +$ +\left(A - \lambda I\right)\vec{v} = 0 +$ + +This is solved for:\ +$\det(A- \lambda I) = 0$ + +> [!NOTE] +> If the determinant is 0, then $(A - \lambda I )$ is not invertible, so +> we can't solve the previous equation by using the trivial solution (which +> can't be taken into account since $\vec{v}$ is not $0$ by definition) + +## Caley-Hamilton +Each square matrix over a `commutative ring` satisfies its own +characteristic equation $det(\lambda I - A)$ + +> [!TIP] +> In other words, once found the characteristic equation, we can +> substitute the ***unknown*** variable $\lambda$ with the matrix itself +> (***known***), +> powered to the correspondent power \ No newline at end of file diff --git a/docs/Formularies/PHYSICS-FORMULARY.md b/docs/Formularies/PHYSICS-FORMULARY.md new file mode 100644 index 0000000..d134818 --- /dev/null +++ b/docs/Formularies/PHYSICS-FORMULARY.md @@ -0,0 +1,37 @@ +# Physics Formulary + +> [!TIP] +> +> You'll often see $\vec{v}$ and $\vec{\dot x}$, and $\vec{a}$ and +> $\vec{\ddot{x}}$. +> +> These are equal, but the latter forms, express better the relation +> between state space variables + +## Hooke's Law (AKA Spring Formula) +$\vec{F} = -k\vec{x}$ +- $k$: Spring Constant +- $\vec{x}$: vector of spring stretch from rest position + +## Fluid drag +$\vec{F_D} = \frac{1}{2}b\vec{v}^2C_{D}A$ +- $b$: density of fluid +- $v$: speed of object ***relative*** to the fluid +- $C_D$: drag coefficient +- $A$: cross section area + +### Stokes Drag +$\vec{F_D} = -6 \pi R\mu \vec{v}$ +- $\mu$: dynamic viscosity +- $R$: radius (in meters) of the sphere +- $\vec{v}$: flow velocity ***relative*** to the fluid + +### Simplified Fluid Drag (Simplified Stokes Equation) +$\vec{F_D} = -b \vec{v}$ +- $b$: simplified coefficient that has everything else +- $\vec{v}$: flow velocity ***relative*** to the fluid +## + +## Newton force + + diff --git a/docs/Images/.excalidraw/Examples/Example-3/double-carts.excalidraw.json b/docs/Images/.excalidraw/Examples/Example-3/double-carts.excalidraw.json new file mode 100644 index 0000000..735443a --- /dev/null +++ b/docs/Images/.excalidraw/Examples/Example-3/double-carts.excalidraw.json @@ -0,0 +1,2512 @@ +{ + "type": "excalidraw", + "version": 2, + "source": "https://marketplace.visualstudio.com/items?itemName=pomdtr.excalidraw-editor", + "elements": [ + { + 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