From d424d2e7aade355e02a1af8fd18067e309c22ef2 Mon Sep 17 00:00:00 2001 From: Christian Risi <75698846+CnF-Gris@users.noreply.github.com> Date: Sat, 4 Jan 2025 18:40:19 +0100 Subject: [PATCH] Added modern control and material --- docs/Chapters/MODERN-CONTROL.md | 98 ++ .../Modern-Control/state-space-discrete.png | Bin 0 -> 86368 bytes .../Modern-Control/state-space-time.png | Bin 0 -> 87124 bytes .../state-space-1.excalidraw.json | 1258 ++++++++++++++++ .../state-space-2.excalidraw.json | 1294 +++++++++++++++++ 5 files changed, 2650 insertions(+) create mode 100644 docs/Chapters/MODERN-CONTROL.md create mode 100644 docs/Images/Modern-Control/state-space-discrete.png create mode 100644 docs/Images/Modern-Control/state-space-time.png create mode 100644 docs/Images/excalidraw/Modern-Control/state-space-1.excalidraw.json create mode 100644 docs/Images/excalidraw/Modern-Control/state-space-2.excalidraw.json diff --git a/docs/Chapters/MODERN-CONTROL.md b/docs/Chapters/MODERN-CONTROL.md new file mode 100644 index 0000000..213bbf2 --- /dev/null +++ b/docs/Chapters/MODERN-CONTROL.md @@ -0,0 +1,98 @@ +# 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/Images/Modern-Control/state-space-discrete.png b/docs/Images/Modern-Control/state-space-discrete.png new file mode 100644 index 0000000000000000000000000000000000000000..6c335a944ae979c364cdb9ac43c5ea44215fa402 GIT binary patch literal 86368 zcmeFZcRbbq`#&D3q#-LJqa4X5tL$VQd#|jFV<%;Am9obP*@}ZQ<5&?#WrSlF8HZOW zo9s9azsD(h>GSU8uiu~FTen-UUgh~1*W-HJuj{&B*YyZdSCv0SLQ8V!(4kXM1sTml zhmLj}I&_$k_$c_B#8GG=_z%H7O?l}Jm-m-3#ULrqq8F){5NMS}WuG)ZW1FoC+dI+}#oTx3> zDn1Hjn#n%vg2wC@X^Cl5>KNSE@)!O=@lWe%*n*_OT<_`OYCEc(3jv1+iJAW2Z~spQ 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