MPC and methodo

79ff7712 · CERF Sophie · 88861a10 · 79ff7712 · 79ff7712
Commit 79ff7712 authored 9 months ago by CERF Sophie
--- a/content/07_MPC.ipynb
+++ b/content/07_MPC.ipynb
@@ -10,57 +10,10 @@
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-      "\u001b[33mWARNING: You are using pip version 21.2.3; however, version 24.2 is available.\n",
-      "You should consider upgrading via the '/home/scerf/Documents/enseignements/Tuto/jupyter/tuto.venv/bin/python3 -m pip install --upgrade pip' command.\u001b[0m\n",
-      "Note: you may need to restart the kernel to use updated packages.\n",
-      "Successful loading!\n"
-     ]
-    }
-   ],
+   "outputs": [],
   "source": [
    "%pip install ipywidgets==8.0.6\n",
    "from tuto_control_lib.systems import IntroSystem\n",
@@ -79,21 +32,41 @@
   "id": "8b9d0e81",
   "metadata": {},
   "source": [
+    "# An Optimal Controller\n",
+    "\n",
+    "Optimal control is an alternative to PID controllers. Rather than computing the action as a function of the error, it is the solution of an optimization problem. Optimal control consists in finding the **control actions that optimize a cost function**, often formulated as a performance measuren and relying on the system's model.\n",
+    "\n",
+    "For linear systems, their exist an analytical solution. For non-linear systems (e.g. if we have a constraint on the control action) the optimal solution can be approached numerically. **Model-Predictive Control (MPC)** is a form of optimal control that allows a **trade-off between optimality and computation cost**, by considering optimization on a finite horizon.\n",
+    "\n",
+    "# Model-Predictive Control\n",
+    "\n",
+    "## Principle\n",
+    "\n",
+    "A Model-Predictive Controller consists in the following steps:\n",
+    "1. **Predict** the future evolution of the output, given the model and hypothetical control actions, over a finite horizon,\n",
+    "2. **Optimize** a cost function by soundly selecting a series of actions,\n",
+    "3. **Apply the first action**, and \n",
+    "4. **Repeat** each time step.\n",
+    "\n",
+    "Applying only the first action and optimizing at each step has the benefits of a **feedback approach**: it can correct uncertainties in the predictions or the model, and compensate for some disturbances.\n",
+    "\n",
+    "## Cost function\n",
+    "\n",
+    "The usual formulation of the MPC cost function is design to ensure reference tracking (the output signal should converge to the reference value $y - y_{ref}$) with reasonable efforts (reasonable actions $|u|$):\n",
+    "\n",
    "$$\n",
-    "J = \\mu \\sum_{i=1}^{H} (y(k+i) - y_{ref})^2  + \\sum_{i=1}^{H} u^2(k+i) ]\n",
-    "$$\n",
-    "$$\n",
-    "u[1:H] = argmin (J)\n",
+    "J = \\mu \\sum_{i=1}^{H} (y(k+i) - y_{ref})^2  + \\sum_{i=1}^{H} u(k+i)^2 )\n",
    "$$\n",
-    "use model (a and b) to compute future values of y\n",
    "\n",
-    "apply $u[1]$\n",
-    "\n"
+    "A weighting factor $\\mu$ allows specifying the relative important of both objectives. \n",
+    "$H$ is the lenght of the prediction horizon.\n",
+    "\n",
+    "Let us define the above cost function, replacing all future value of the output by their prediction using the model (e.g. $a$ and $b$):"
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": null,
   "id": "55c4897b",
   "metadata": {},
   "outputs": [],
@@ -113,27 +86,26 @@
    "    return cost_value"
   ]
  },
+  {
+   "cell_type": "markdown",
+   "id": "f074aecb",
+   "metadata": {},
+   "source": [
+    "## Implementation \n",
+    "\n",
+    "At each time step, the MPC consists in:\n",
+    "1. Computing the action trajectory that minimizes the cost over the horizon $H$: \n",
+    "$$u(1:H) = argmin (J)$$\n",
+    "2. apply $u(1)$\n",
+    "\n"
+   ]
+  },
  {
   "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": null,
   "id": "24a4b359",
   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "7b110d7a7c844d2f9b9a39ddecce8310",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(IntSlider(value=2, description='H', max=5), IntSlider(value=10, description='weight', ma…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
+   "outputs": [],
   "source": [
    "@interact(H=(0, 5, 1), weight=(1,50,1))\n",
    "def draw_p(H=2, weight=10):\n",
@@ -153,19 +125,41 @@
  },
  {
   "cell_type": "markdown",
-   "id": "ea0f209d",
+   "id": "3053e74f",
   "metadata": {},
   "source": [
-    "[Back to menu](./00_Main.ipynb) or [Next chapter](./08_FeedForward.ipynb)"
+    "<div class=\"alert alert-info\" role=\"alert\">    \n",
+    " Try different values of the horizon and the weight factor. How does the controller perform compared to a PID ?\n",
+    "</div>"
   ]
  },
  {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "19502aed",
+   "cell_type": "markdown",
+   "id": "d421db9d",
   "metadata": {},
-   "outputs": [],
-   "source": []
+   "source": [
+    "<div class=\"alert alert-danger\" role=\"alert\">\n",
+    "    You need to scroll until the end to see the plots!\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6c4ab817",
+   "metadata": {},
+   "source": [
+    "### Bibliography\n",
+    "\n",
+    "Kouvaritakis, B., & Cannon, M. (2016). Model predictive control. Switzerland: Springer International Publishing, 38, 13-56."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ea0f209d",
+   "metadata": {},
+   "source": [
+    "[Back to menu](./00_Main.ipynb) or [Next chapter](./08_FeedForward.ipynb)"
+   ]
  }
 ],
 "metadata": {

 %% Cell type:markdown id:5ecd8455-ffcf-4230-adaf-6514c725b40b tags:

 # Model-Predictive Control

 %% Cell type:code id:cbfec597 tags:

 ``` python
 %pip install ipywidgets==8.0.6
 from tuto_control_lib.systems import IntroSystem
 from tuto_control_lib.plot import *

 import matplotlib.pyplot as plt
 import numpy as np
 from ipywidgets import interact
 from scipy.optimize import fmin

 print("Successful loading!")
 ```

-%% Output
+%% Cell type:markdown id:8b9d0e81 tags:

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-    [33mWARNING: You are using pip version 21.2.3; however, version 24.2 is available.
-    You should consider upgrading via the '/home/scerf/Documents/enseignements/Tuto/jupyter/tuto.venv/bin/python3 -m pip install --upgrade pip' command.[0m
-    Note: you may need to restart the kernel to use updated packages.
-    Successful loading!
+# An Optimal Controller

-%% Cell type:markdown id:8b9d0e81 tags:
+Optimal control is an alternative to PID controllers. Rather than computing the action as a function of the error, it is the solution of an optimization problem. Optimal control consists in finding the **control actions that optimize a cost function**, often formulated as a performance measuren and relying on the system's model.
+
+For linear systems, their exist an analytical solution. For non-linear systems (e.g. if we have a constraint on the control action) the optimal solution can be approached numerically. **Model-Predictive Control (MPC)** is a form of optimal control that allows a **trade-off between optimality and computation cost**, by considering optimization on a finite horizon.
+
+# Model-Predictive Control
+
+## Principle
+
+A Model-Predictive Controller consists in the following steps:
+1. **Predict** the future evolution of the output, given the model and hypothetical control actions, over a finite horizon,
+2. **Optimize** a cost function by soundly selecting a series of actions,
+3. **Apply the first action**, and
+4. **Repeat** each time step.
+
+Applying only the first action and optimizing at each step has the benefits of a **feedback approach**: it can correct uncertainties in the predictions or the model, and compensate for some disturbances.
+
+## Cost function
+
+The usual formulation of the MPC cost function is design to ensure reference tracking (the output signal should converge to the reference value $y - y_{ref}$) with reasonable efforts (reasonable actions $|u|$):

 $$
-J = \mu \sum_{i=1}^{H} (y(k+i) - y_{ref})^2  + \sum_{i=1}^{H} u^2(k+i) ]
-$$
-$$
-u[1:H] = argmin (J)
+J = \mu \sum_{i=1}^{H} (y(k+i) - y_{ref})^2  + \sum_{i=1}^{H} u(k+i)^2 )
 $$
-use model (a and b) to compute future values of y

-apply $u[1]$
+A weighting factor $\mu$ allows specifying the relative important of both objectives.
+$H$ is the lenght of the prediction horizon.

+Let us define the above cost function, replacing all future value of the output by their prediction using the model (e.g. $a$ and $b$):

 %% Cell type:code id:55c4897b tags:

 ``` python
 def cost_function(u, parameters):
    y_old, reference_value, a, b, weight = parameters
    reference_tracking = 0
    action_cost = 0
    H = np.size(u)
    for i in range(H):
        sum_actions = 0
        for j in range(i+1):
            sum_actions += a**(i-j)*u[j]
        reference_tracking += (a**(i+1)*y_old + b*sum_actions - reference_value)**2
        action_cost += u[i]**2
    cost_value = weight*reference_tracking + action_cost
    return cost_value
 ```

+%% Cell type:markdown id:f074aecb tags:
+
+## Implementation
+
+At each time step, the MPC consists in:
+1. Computing the action trajectory that minimizes the cost over the horizon $H$:
+$$u(1:H) = argmin (J)$$
+2. apply $u(1)$
+
+
 %% Cell type:code id:24a4b359 tags:

 ``` python
 @interact(H=(0, 5, 1), weight=(1,50,1))
 def draw_p(H=2, weight=10):
    system, u_values, y_values, a, b, max_iter = IntroSystem(), [], [], 0.8, 0.5, 100

    reference_value = 1
    for i in range(max_iter):
        y = system.sense()
        y_values.append(y)

        u = fmin(cost_function,np.ones((H,1)),args=(np.array([y, reference_value, a, b, weight]),))

        system.apply(u[0])
        u_values.append(u[0])
    plot_u_y(u_values, y_values, reference_value)
 ```

-%% Output
+%% Cell type:markdown id:3053e74f tags:

+<div class="alert alert-info" role="alert">
+ Try different values of the horizon and the weight factor. How does the controller perform compared to a PID ?
+</div>

-%% Cell type:markdown id:ea0f209d tags:
+%% Cell type:markdown id:d421db9d tags:

-[Back to menu](./00_Main.ipynb) or [Next chapter](./08_FeedForward.ipynb)
+<div class="alert alert-danger" role="alert">
+    You need to scroll until the end to see the plots!
+</div>

-%% Cell type:code id:19502aed tags:
+%% Cell type:markdown id:6c4ab817 tags:

-``` python
-```
+### Bibliography
+
+Kouvaritakis, B., & Cannon, M. (2016). Model predictive control. Switzerland: Springer International Publishing, 38, 13-56.
+
+%% Cell type:markdown id:ea0f209d tags:
+
+[Back to menu](./00_Main.ipynb) or [Next chapter](./08_FeedForward.ipynb)

--- a/content/09_Methodology.ipynb
+++ b/content/09_Methodology.ipynb
@@ -5,26 +5,48 @@
   "id": "c3b876b7-b17f-4264-a7b9-9e11dc34ac18",
   "metadata": {},
   "source": [
-    "# Methodology"
+    "# Complete Methodology "
   ]
  },
  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "id": "eae61ca7-8af2-46fc-8524-69c39237eb0f",
+   "cell_type": "markdown",
+   "id": "314baaad",
+   "metadata": {},
+   "source": [
+    "In the previous chapters, you learned to identify a model, design controllers, and evaluate their performances. To be able to apply those methods on a system, an important step of formulation of the control problem is needed, for instance to define the output and control signals, the reference values, etc."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7257aff7",
+   "metadata": {},
+   "source": [
+    "## Formulate your own control problem !"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b489b3e7",
   "metadata": {},
-   "outputs": [],
   "source": [
-    "# TODO"
+    "You are now invited to consider your prefered self-adaptive systems to formulate it as a feedback loop and design a controller. \n",
+    "\n",
+    "If you have no specific system under study, pair up and join other groups !\n",
+    "\n",
+    "Here are the step to follow:\n",
+    "1. **Connect** to the shared note [here](https://notes.inria.fr/MRc7TYzXSg2PLwnwlpXkAg#).\n",
+    "2. Make a **copy** of the slides for your team at the bottom of the note. The slides to copy are: System, Objective, Challenges, Sensor, Action, Objectives in terms of control, Analysis, Control design, Evaluation.\n",
+    "3. Take your time to answer the questions and **fill the slides** with your responses.\n",
+    "4. At the end of the session, we will take few mintues for a **restitution**. Please remove or comment out the questions from your slides, and only let your formulation of your control problem."
   ]
  },
  {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "e3c6d1c3-e165-477f-9a69-a04ba0356eca",
+   "cell_type": "markdown",
+   "id": "8e6b48ee",
   "metadata": {},
-   "outputs": [],
-   "source": []
+   "source": [
+    "[Back to menu](./00_Main.ipynb) or [Optional Exercice](./10_CiGri.ipynb)"
+   ]
  }
 ],
 "metadata": {
@@ -43,7 +65,7 @@
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
-   "version": "3.10.8"
+   "version": "3.9.7"
  }
 },
 "nbformat": 4,

 %% Cell type:markdown id:c3b876b7-b17f-4264-a7b9-9e11dc34ac18 tags:

-# Methodology
+# Complete Methodology

-%% Cell type:code id:eae61ca7-8af2-46fc-8524-69c39237eb0f tags:
+%% Cell type:markdown id:314baaad tags:

-``` python
-# TODO
-```
+In the previous chapters, you learned to identify a model, design controllers, and evaluate their performances. To be able to apply those methods on a system, an important step of formulation of the control problem is needed, for instance to define the output and control signals, the reference values, etc.

-%% Cell type:code id:e3c6d1c3-e165-477f-9a69-a04ba0356eca tags:
+%% Cell type:markdown id:7257aff7 tags:

-``` python
-```
+## Formulate your own control problem !
+
+%% Cell type:markdown id:b489b3e7 tags:
+
+You are now invited to consider your prefered self-adaptive systems to formulate it as a feedback loop and design a controller.
+
+If you have no specific system under study, pair up and join other groups !
+
+Here are the step to follow:
+1. **Connect** to the shared note [here](https://notes.inria.fr/MRc7TYzXSg2PLwnwlpXkAg#).
+2. Make a **copy** of the slides for your team at the bottom of the note. The slides to copy are: System, Objective, Challenges, Sensor, Action, Objectives in terms of control, Analysis, Control design, Evaluation.
+3. Take your time to answer the questions and **fill the slides** with your responses.
+4. At the end of the session, we will take few mintues for a **restitution**. Please remove or comment out the questions from your slides, and only let your formulation of your control problem.
+
+%% Cell type:markdown id:8e6b48ee tags:
+
+[Back to menu](./00_Main.ipynb) or [Optional Exercice](./10_CiGri.ipynb)