diff --git a/jupyter_notebooks/02_BangBang.ipynb b/jupyter_notebooks/02_BangBang.ipynb
index c6cdb3bf84d68fbc126aed4be8a867aa96f88def..05189e109ac23879924702afd4069b6df9211930 100644
--- a/jupyter_notebooks/02_BangBang.ipynb
+++ b/jupyter_notebooks/02_BangBang.ipynb
@@ -19,7 +19,8 @@
"from tuto_control_lib.plot import plot_u_y\n",
"\n",
"import matplotlib.pyplot as plt\n",
- "import numpy as np"
+ "import numpy as np\n",
+ "from statistics import mean"
]
},
{
@@ -29,13 +30,14 @@
"tags": []
},
"source": [
- "One way to do this would be to have two bounds for the system output:\n",
+ "One way to do regulate the output of a system would be to have two bounds for the system sensor:\n",
"\n",
"- one upper bound\n",
"- one lower bound\n",
"\n",
"When the system output is greater than the upper bound, we decrease the input.\n",
- "And when the system output is lower than the lower bound, we increse the input."
+ "And when the system output is lower than the lower bound, we increase the input.\n",
+ "Else, we keep the previous input."
]
},
{
@@ -45,7 +47,14 @@
"tags": []
},
"source": [
- "Say that we want to regulate our system around the value 6.\n",
+ "Say that we want to regulate our system around the value 1.\n",
+ "\n",
+ "We now have to chose the values of the bounds.\n",
+ "\n",
+ "The issue is that there is no protocol to find the values of the bounds and the incremental part.\n",
+ "\n",
+ "So, we have to proceed by try-and-error.\n",
+ "\n",
"\n",
"We can say take as lower bound 5 and as upper bound 7."
]
@@ -62,17 +71,18 @@
"system, u, y_values, u_values, max_iter = IntroSystem(), 0, [], [], 100\n",
"\n",
"reference_value = 1\n",
- "upper_bound = 0.75\n",
- "lower_bound = 1.25\n",
+ "upper_bound = 0.5\n",
+ "lower_bound = 1.5\n",
+ "increment = 0.5\n",
"\n",
"for i in range(max_iter):\n",
" y = system.sense()\n",
" y_values.append(y)\n",
" \n",
" if y < lower_bound:\n",
- " u += 0.25\n",
+ " u += increment\n",
" elif y > upper_bound:\n",
- " u -= 0.25\n",
+ " u -= increment\n",
" else:\n",
" pass\n",
" system.apply(u)\n",
@@ -88,7 +98,31 @@
"tags": []
},
"source": [
- "As we can see, the system converges, but not to the reference value..."
+ "As we can see, the system is somewhat under control, but oscillate a lot."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ecdc7121-d253-477d-92dc-4485abd97d29",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "mean_error = mean(map(lambda x: abs(reference_value - x), y_values))\n",
+ "max_overshoot = (max(y_values) - reference_value) / reference_value\n",
+ "\n",
+ "print(f\"Mean Error: {mean_error}\")\n",
+ "print(f\"Max. Overshoot: {max_overshoot}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2b58c216-0ef8-4d17-8b35-514974b47570",
+ "metadata": {},
+ "source": [
+ "<div class=\"alert alert-info\">\n",
+ "Try changing the values of the bounds and the increment to see the behaviour of the system.\n",
+ "</div>"
]
},
{
diff --git a/jupyter_notebooks/03_PController.ipynb b/jupyter_notebooks/03_PController.ipynb
index 8c48125743dfcb1399716ac5dd7c56549266d6c0..485503f4d994ccdb6185c92ef9ad27f018017fe3 100644
--- a/jupyter_notebooks/03_PController.ipynb
+++ b/jupyter_notebooks/03_PController.ipynb
@@ -25,12 +25,32 @@
},
{
"cell_type": "markdown",
- "id": "125b3cfd-1f55-4202-8f50-97d0cafb5b87",
+ "id": "028756f1-641f-4c33-9879-cbb53ee0f256",
"metadata": {},
"source": [
+ "We have seen that a Bang-Bang solution manages to roughly get the system in the desired state.\n",
+ "\n",
+ "However, the lack of protocol and guarentees of this solution limits its adoption on production systems.\n",
+ "\n",
+ "In this section, we introduce the most basic controller from Control Theory: the *Proportional Controller*.\n",
+ "\n",
"The idea of the proportional controller is to have a response proportional to the control error.\n",
"\n",
- "The control error is the distance between the desired behaviour and the current state of the system."
+ "The control error is the distance between the desired behaviour and the current state of the system.\n",
+ "\n",
+ "The equation of a P Controller is the following:\n",
+ "\n",
+ "$$\n",
+ "u(k) = K_p \\times e(k) = K_p \\times \\left(y_{ref} - y(k)\\right)\n",
+ "$$\n",
+ "\n",
+ "where:\n",
+ "\n",
+ "- $K_p$ is the propotional gain of the controller\n",
+ "- $y_{ref}$ is the reference value for our system (i.e. the desired value of the system output)\n",
+ "- $y(k)$ is the system output at iteration $k$\n",
+ "- $e(k)$ is the control error at iteration $k$\n",
+ "- $u(k)$ is the input at iteration $k$"
]
},
{
@@ -40,13 +60,10 @@
"metadata": {},
"outputs": [],
"source": [
- "max_iter = 100\n",
- "system = IntroSystem()\n",
+ "system, u_values, y_values, u, max_iter = IntroSystem(), [], [], 0, 100\n",
+ "\n",
"reference_value = 1\n",
"kp = 3.3\n",
- "y_values = []\n",
- "u_values = []\n",
- "u = 0\n",
"\n",
"for i in range(max_iter):\n",
" y = system.sense()\n",
@@ -61,6 +78,15 @@
"plot_u_y(u_values, y_values, reference_value)"
]
},
+ {
+ "cell_type": "markdown",
+ "id": "63203ee3-be90-4e9a-92d4-57a08d042631",
+ "metadata": {},
+ "source": [
+ "As we can see, the system converges, but to a values different than the reference value.\n",
+ "The controller introduces oscillations before converging."
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,
@@ -68,7 +94,17 @@
"metadata": {},
"outputs": [],
"source": [
- "y_values[-1]"
+ "print(f\"Steady state value: {y_values[-1]}\\nReference value: {reference_value}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f664e07a-5ea6-4d0f-b826-beb0caeb6c90",
+ "metadata": {},
+ "source": [
+ "<div class=\"alert alert-info\">\n",
+ "Try changing the values of the proportional gain $K_p$ and the reference value.\n",
+ "</div>"
]
},
{
@@ -79,6 +115,39 @@
"# Design of a Proportional Controller"
]
},
+ {
+ "cell_type": "markdown",
+ "id": "49936477-c739-476b-a351-c59ec23f9d68",
+ "metadata": {},
+ "source": [
+ "To design a Proportional Controller with guarentees, we must have a model of our system.\n",
+ "\n",
+ "A model, in the sense of Control Theory, is a relation between the inputs and the outputs.\n",
+ "\n",
+ "The general form of a model is the following:\n",
+ "\n",
+ "$$\n",
+ "y(k + 1) = \\sum_{i = 0}^k a_i y(k - i) + \\sum_{i = 0}^k b_i u(k - i)\n",
+ "$$\n",
+ "\n",
+ "where:\n",
+ "\n",
+ "- $y(k + 1)$ is the next value of the output\n",
+ "- $y(k-i)$ and $u(k-i)$ are previous values of the output and the input\n",
+ "- $a_i$ and $b_i$ are the coefficients of the model ($\\forall i, (a_i, b_i) \\in (\\mathbb{R}, \\mathbb{R})$)\n",
+ "\n",
+ "Usually, and to simplify this introduction, we consider *first order models*.\n",
+ "\n",
+ "This means that the model only considers the last values of $y$ and $u$ to get the next value of $y$.\n",
+ "\n",
+ "$$\n",
+ "y(k + 1) = a y(k) + b u(k)\n",
+ "$$\n",
+ "\n",
+ "In this section, we will suppose that we have a first order model which we know the coefficients.\n",
+ "In a [future section](./05_Identification.ipynb), we will look at how to find these coefficients."
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,
@@ -86,7 +155,9 @@
"metadata": {},
"outputs": [],
"source": [
+ "# Our system\n",
"system = IntroSystem()\n",
+ "# The coefficients\n",
"a = 0.8\n",
"b = 0.5"
]
@@ -139,31 +210,25 @@
},
{
"cell_type": "markdown",
- "id": "ff825fc1-825d-4d0f-9172-cc3d25edb83b",
+ "id": "35f5b723-9eb7-4314-8fe1-7989b612e1fe",
"metadata": {},
"source": [
"Proportional Controllers are inheritly imprecise.\n",
- "But we can tune their precision based on the reference value.\n",
+ "But we can tune their precision ($e_{ss}$) based on the reference value ($r_{ss}$).\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"e_{ss} &= r_{ss} ( 1 - F_R(1)) \\\\\n",
- " &= r_{ss} \\left(1 - \\frac{b K_p}{1 - (a - b K_p)}\\right)\n",
+ " &= r_{ss} \\left(1 - \\frac{b K_p}{1 - (a - b K_p)}\\right) < e_{ss}^*\n",
"\\end{aligned}\n",
- "$$"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "f86f5dcd-564c-4f73-8156-cc0a5b2c685c",
- "metadata": {},
- "source": [
- "Say we want the steady state error to be less that $e_{target}$.\n",
+ "$$\n",
+ "\n",
+ "Say we want the steady state error to be less that $e_{ss}^*$.\n",
"\n",
"Then,\n",
"\n",
"$$\n",
- "K_p > \\frac{\\left(1 - \\frac{e_t}{r_{ss}}\\right)\\left(1 - a\\right)}{b\\frac{e_t}{r_{ss}}}\n",
+ "K_p > \\frac{\\left(1 - \\frac{e_{ss}^*}{r_{ss}}\\right)\\left(1 - a\\right)}{b\\frac{e_{ss}^*}{r_{ss}}}\n",
"$$\n",
"\n",
"In our case:"
@@ -177,9 +242,9 @@
"outputs": [],
"source": [
"r_ss = 1\n",
- "e_t = 0.15\n",
+ "e_star = 0.15\n",
"\n",
- "precision_lower_bound = (1 - e_t/r_ss) * (1 - a)/(b * (e_t/r_ss))\n",
+ "precision_lower_bound = (1 - e_star/r_ss) * (1 - a)/(b * (e_star/r_ss))\n",
"\n",
"print(f\"K_p > {precision_lower_bound}\")"
]
@@ -197,6 +262,8 @@
"id": "320e2695-9a0b-4a8d-909b-4385dbebf6af",
"metadata": {},
"source": [
+ "The settling time, or the time to reach the steady state value is defined as follows:\n",
+ "\n",
"$$\n",
"k_s \\simeq \\frac{-4}{\\log | a - b K_p| }\n",
"$$\n",
@@ -219,9 +286,9 @@
"metadata": {},
"outputs": [],
"source": [
- "k_s = 10\n",
- "settling_time_lower_bound = (a - exp(-4/k_s)) / b\n",
- "settling_time_upper_bound = (a + exp(-4/k_s)) / b\n",
+ "ks_star = 10\n",
+ "settling_time_lower_bound = (a - exp(-4/ks_star)) / b\n",
+ "settling_time_upper_bound = (a + exp(-4/ks_star)) / b\n",
"\n",
"print(f\"{settling_time_lower_bound} < K_p < {settling_time_upper_bound}\")"
]
@@ -239,6 +306,9 @@
"id": "a04de440-6087-4f45-a302-02c0f3bfdd20",
"metadata": {},
"source": [
+ "The maximum overshoot is the maximum error above the reference value.\n",
+ "It is defined as:\n",
+ "\n",
"$$\n",
"M_p = | a - b K_p|\n",
"$$\n",
@@ -249,6 +319,8 @@
"\\frac{a - M_p^*}{b} < K_p < \\frac{a + M_p^*}{b}\n",
"$$\n",
"\n",
+ "But we really are only interested in the upper bound.\n",
+ "\n",
"In our case:"
]
},
@@ -259,8 +331,8 @@
"metadata": {},
"outputs": [],
"source": [
- "mp = 0.1\n",
- "max_overshoot_upper_bound = (a + mp) / b\n",
+ "mp_star = 0.1\n",
+ "max_overshoot_upper_bound = (a + mp_star) / b\n",
"print(f\"K_p < {max_overshoot_upper_bound}\")"
]
},
@@ -294,9 +366,9 @@
"source": [
"As we can see, there is no value of $K_p$ that satisfies all the properties.\n",
"\n",
- "You can play with the different values defining the desired behavior to find a $K_p$ value that satisfies them.\n",
- "\n",
- "The key point is that implementing a Proportional controller requires some **trade-off**!\n",
+ "<div class=\"alert alert-danger\" role=\"alert\">\n",
+ " The key point is that implementing a Proportional controller requires some <b>trade-off</b>!\n",
+ "</div>\n",
"\n",
"In the example above, the value $K_p = 2.5$ seems to statisfy most of the properties."
]
@@ -308,10 +380,9 @@
"metadata": {},
"outputs": [],
"source": [
- "max_iter = 20\n",
"reference_value = 1\n",
"kp = 2.5\n",
- "y_values, u_values, u, system = [], [], 0, IntroSystem()\n",
+ "y_values, u_values, u, system, max_iter = [], [], 0, IntroSystem(), 20\n",
"\n",
"for i in range(max_iter):\n",
" y = system.sense()\n",
@@ -342,12 +413,12 @@
"outputs": [],
"source": [
"e_ss = reference_value - y_values[-1]\n",
- "max_overshoot = (max(y_values) - reference_value) / reference_value\n",
+ "max_overshoot = (max(y_values) - y_values[-1]) / y_values[-1]\n",
"settling_time = len([x for x in y_values if abs(x - y_values[-1]) > 0.05])\n",
"\n",
- "print(f\"Precision: {e_ss} -> desired: < {e_t}\")\n",
- "print(f\"Settling Time: {settling_time} -> desired: < {k_s}\")\n",
- "print(f\"Max. Overshoot: {max_overshoot} -> desired: < {mp}\")"
+ "print(f\"Precision: {e_ss} -> desired: < {e_star}\")\n",
+ "print(f\"Settling Time: {settling_time} -> desired: < {ks_star}\")\n",
+ "print(f\"Max. Overshoot: {max_overshoot} -> desired: < {mp_star}\")"
]
},
{
@@ -358,6 +429,16 @@
"As expected, the closed loop system overshoots too much, but the other properties are respected."
]
},
+ {
+ "cell_type": "markdown",
+ "id": "01c36250-4482-47d9-a1ab-6451f7470ca7",
+ "metadata": {},
+ "source": [
+ "<div class=\"alert alert-info\" role=\"alert\">\n",
+ " Try to change the requirements on the closed-loop properties to find different values of $K_p$ and plot the system.\n",
+ "</div>"
+ ]
+ },
{
"cell_type": "markdown",
"id": "09a14ea2-b496-4d8a-b5d7-155a5b4d60c9",
diff --git a/jupyter_notebooks/04_PIController.ipynb b/jupyter_notebooks/04_PIController.ipynb
index 9ab06292c42a62d184de7fba3080ac43627db886..6c82620c52c7692a37288fe5992ae01d83056c0b 100644
--- a/jupyter_notebooks/04_PIController.ipynb
+++ b/jupyter_notebooks/04_PIController.ipynb
@@ -30,8 +30,10 @@
"metadata": {},
"source": [
"As we have seen before, a Proportional controller is inheritly imprecise.\n",
+ "\n",
"One way to improve the precision of the closed loop system is to add an integral term to the controller.\n",
- "The integral term aims at cancel the steady state error.\n",
+ "\n",
+ "The integral term aims at canceling the steady state error.\n",
"\n",
"The form of the controller (in discrete time) is the following:\n",
"\n",
@@ -55,7 +57,7 @@
"ki = 1.5\n",
"y_values, u_values, u, system, integral = [], [], 0, IntroSystem(), 0\n",
"\n",
- "for i in range(max_iter):\n",
+ "for _ in range(max_iter):\n",
" y = system.sense()\n",
" y_values.append(y)\n",
" \n",
@@ -78,6 +80,16 @@
"However, there are some oscillations and overshooting..."
]
},
+ {
+ "cell_type": "markdown",
+ "id": "cb414899-941e-4652-93c1-df6ab939c6f8",
+ "metadata": {},
+ "source": [
+ "<div class=\"alert alert-info\" role=\"alert\">\n",
+ " Try to change the values of $K_p$ and $K_i$ to observe the change of behavior.\n",
+ "</div>"
+ ]
+ },
{
"cell_type": "markdown",
"id": "dc185bc1-7b31-41df-8dd3-b260aca6097b",
@@ -92,7 +104,14 @@
"metadata": {},
"source": [
"As for the P Controller, we have to chose the desired closed loop behavior.\n",
- "In the case of a PI Controller, we have the precision by the integral term.\n",
+ "\n",
+ "In the case of a PI Controller, we have the precision by the integral term, and the precision as for the P Controller.\n",
+ "\n",
+ "There are several methods to find gains for a PI Controller.\n",
+ "In the following we use the *pole placement method*.\n",
+ "The idea is to chose the poles of the closed-loop system to fit the desired behavior.\n",
+ "\n",
+ "Without too much details to avoid being too \"mathy\", we give the equations leading to the gains.\n",
"\n",
"Given the desired values for $k_s$ (settling time) and $M_p$ (max. overshoot), we get:\n",
"\n",
@@ -118,11 +137,13 @@
"metadata": {},
"outputs": [],
"source": [
+ "# The coefficients of our system\n",
"a = 0.8\n",
"b = 0.5\n",
"\n",
- "k_s = 10\n",
- "mp = 0.05\n",
+ "# Our desired properties\n",
+ "ks = 10\n",
+ "mp = 0.01\n",
"\n",
"r = exp(-4/k_s)\n",
"theta = pi * log(r) / log(mp)\n",
@@ -142,15 +163,15 @@
"source": [
"max_iter = 50\n",
"reference_value = 1\n",
- "y_values, u_values, u, system, integral = [], [], 0, IntroSystem(), 0\n",
+ "y_values, u_values, u, system, integral_error = [], [], 0, IntroSystem(), 0\n",
"\n",
"for i in range(max_iter):\n",
" y = system.sense()\n",
" y_values.append(y)\n",
" \n",
" error = reference_value - y\n",
- " integral += error\n",
- " u = kp * error + ki * integral\n",
+ " integral_error += error\n",
+ " u = kp * error + ki * integral_error\n",
" \n",
" system.apply(u)\n",
" u_values.append(u)\n",
@@ -166,14 +187,24 @@
"outputs": [],
"source": [
"e_ss = reference_value - y_values[-1]\n",
- "max_overshoot = (max(y_values) - reference_value) / reference_value\n",
+ "max_overshoot = (max(y_values) - y_values[-1]) / y_values[-1]\n",
"settling_time = len([x for x in y_values if abs(x - y_values[-1]) > 0.05])\n",
"\n",
"print(f\"Precision: {e_ss}\")\n",
- "print(f\"Settling Time: {settling_time} -> desired: < {k_s}\")\n",
+ "print(f\"Settling Time: {settling_time} -> desired: < {ks}\")\n",
"print(f\"Max. Overshoot: {max_overshoot} -> desired: < {mp}\")"
]
},
+ {
+ "cell_type": "markdown",
+ "id": "41159696-e75e-4ae5-8b69-44799bf482d9",
+ "metadata": {},
+ "source": [
+ "<div class=\"alert alert-info\" role=\"alert\">\n",
+ " Try to change the requirements on the closed-loop properties to find different values of $K_p$ and $K_i$ and plot the system.\n",
+ "</div>"
+ ]
+ },
{
"cell_type": "markdown",
"id": "e8e31f05-7d99-4793-801e-51c292c7c8ec",
diff --git a/jupyter_notebooks/05_Identification.ipynb b/jupyter_notebooks/05_Identification.ipynb
index 803378de86de49fb4dead7b2018b5b788dee4e2b..92e3fde4a56252eba005a6329263bbf047202c87 100644
--- a/jupyter_notebooks/05_Identification.ipynb
+++ b/jupyter_notebooks/05_Identification.ipynb
@@ -30,15 +30,15 @@
"metadata": {},
"source": [
"For moment, we supposed the model of the system known (i.e. the coeficients $a$ and $b$).\n",
- "But in practice, we do not know.\n",
+ "But in practice, we do not know them.\n",
"\n",
- "In this Section, we will perform what is called the *identification* to get the model of the system.\n",
+ "In this Section, we will perform what is called the *Identification* to get the coefficient of the system model.\n",
"\n",
"The idea of the identification phase is simple: \"Get the relation between the input and output\".\n",
"\n",
"To do this, the most basic way is to perform a serie of step inputs and observe the output.\n",
"\n",
- "In this Section, we will use the `UnknownSystem`."
+ "In this Section, we will use the `UnknownSystem` and try to find its coefficients."
]
},
{
@@ -48,20 +48,18 @@
"metadata": {},
"outputs": [],
"source": [
- "max_iter = 200\n",
"system = UnknownSystem()\n",
- "y_values, u_values, u, system, integral = [], [], 0, UnknownSystem(), 0\n",
- "#y_values, u_values, u, system, integral = [], [], 0, System(0.314, 0.628), 0\n",
+ "y_values_ident, u_values_ident, u, system, max_iter = [], [], 0, UnknownSystem(), 200\n",
"for i in range(max_iter):\n",
" y = system.sense()\n",
- " y_values.append(y)\n",
+ " y_values_ident.append(y)\n",
" \n",
" u = (i + 20) // 20\n",
" \n",
" system.apply(u)\n",
- " u_values.append(u)\n",
+ " u_values_ident.append(u)\n",
"\n",
- "plot_u_y(u_values, y_values)"
+ "plot_u_y(u_values_ident, y_values_ident)"
]
},
{
@@ -69,9 +67,7 @@
"id": "161a6257-f786-4437-b264-678feadd1863",
"metadata": {},
"source": [
- "Let us first look at the steady state.\n",
- "\n",
- "We are looking for an expression of the following form:\n",
+ "We are looking for an expression for a first order system of the following form:\n",
"\n",
"$$\n",
"y(k+1) = a y(k) + b u(k)\n",
@@ -97,12 +93,13 @@
"source": [
"previous_u = 1\n",
"gain_ss = None\n",
- "for (u, y) in zip(u_values, y_values):\n",
+ "for (u, y) in zip(u_values_ident, y_values_ident):\n",
" if u != previous_u:\n",
" print(f\"u: {previous_u} -> {gain_ss}\")\n",
" previous_u = u\n",
" else:\n",
- " gain_ss = y / u"
+ " gain_ss = y / u\n",
+ "print(f\"u: {previous_u} -> {gain_ss}\")"
]
},
{
@@ -118,7 +115,7 @@
"id": "3e9f8daf-b9c4-476b-b6f1-c5ea97b74574",
"metadata": {},
"source": [
- "We can try to perform a Least Mean Square to get an estimation of the model:"
+ "We will perform a Least Mean Square to get an estimation of the model:"
]
},
{
@@ -128,8 +125,8 @@
"metadata": {},
"outputs": [],
"source": [
- "u_values_identification = u_values\n",
- "y_values_identification = y_values\n",
+ "u_values_identification = u_values_ident\n",
+ "y_values_identification = y_values_ident\n",
"\n",
"s1 = sum(y * y for y in y_values_identification)\n",
"s2 = sum(u * y for (u, y) in zip(u_values_identification, y_values_identification))\n",
@@ -186,7 +183,7 @@
"metadata": {},
"outputs": [],
"source": [
- "ks = 20\n",
+ "ks = 10\n",
"mp = 0.05\n",
"\n",
"r = exp(-4/ks)\n",
@@ -205,17 +202,16 @@
"metadata": {},
"outputs": [],
"source": [
- "max_iter = 100\n",
"reference_value = 1\n",
- "y_values, u_values, u, system, integral = [], [], 0, UnknownSystem(), 0\n",
+ "y_values, u_values, u, system, integral_error, max_iter = [], [], 0, UnknownSystem(), 0, 50\n",
"\n",
"for i in range(max_iter):\n",
" y = system.sense()\n",
" y_values.append(y)\n",
" \n",
" error = reference_value - y\n",
- " integral += error\n",
- " u = kp * error + ki * integral\n",
+ " integral_error += error\n",
+ " u = kp * error + ki * integral_error\n",
" \n",
" system.apply(u)\n",
" u_values.append(u)\n",
@@ -223,6 +219,22 @@
"plot_u_y(u_values, y_values, reference_value)"
]
},
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "2759322c-239d-4376-a70a-e9fc6346f0e8",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "e_ss = reference_value - y_values[-1]\n",
+ "max_overshoot = (max(y_values) - y_values[-1]) / y_values[-1]\n",
+ "settling_time = len([x for x in y_values if abs(x - y_values[-1]) > 0.05])\n",
+ "\n",
+ "print(f\"Precision: {e_ss}\")\n",
+ "print(f\"Settling Time: {settling_time} -> desired: < {ks}\")\n",
+ "print(f\"Max. Overshoot: {max_overshoot} -> desired: < {mp}\")"
+ ]
+ },
{
"cell_type": "markdown",
"id": "d916d270-f6c7-4f0c-9f19-869a9c1aac84",
diff --git a/jupyter_notebooks/06_RealSystem.ipynb b/jupyter_notebooks/06_RealSystem.ipynb
index 8deeddd1e1fd9224d399e05b28c542c0ce97932c..4400929a65964484d67b86dec5f99f76f1cb5655 100644
--- a/jupyter_notebooks/06_RealSystem.ipynb
+++ b/jupyter_notebooks/06_RealSystem.ipynb
@@ -10,7 +10,7 @@
},
{
"cell_type": "markdown",
- "id": "ed9286f1-96d4-4713-8c63-d9c735c8ed63",
+ "id": "e38ada7d-aae2-4ff0-bd34-20d4c1bed366",
"metadata": {},
"source": [
"We provide a semi-real system.\n",
@@ -20,19 +20,41 @@
"We want to compute an estimation of $\\pi$.\n",
"One way to do this is to use Monte-Carlo simulations.\n",
"\n",
- "The idea of Monte-Carlo simulations is to execute *a lot* of small **independant** simulations and compute the final result based on the results of the simulations."
+ "The idea of Monte-Carlo simulations is to execute **a lot of small and independant simulations** and compute the final result based on the results of the simulations.\n",
+ "\n",
+ "In our case, each simulation we draw a random number $x$ in $[-1, 1]$, and then compute (in a very inefficient way) $\\sqrt{1 - x^2}$.\n",
+ "\n",
+ "The final result is the sum of each simulation, which is an approximation of $\\int_{-1}^1 \\sqrt{1-x^2}dx = \\frac{\\pi}{2}$\n",
+ "\n",
+ "Our sensor is the `loadavg` of the machine. The `loadavg` is a metric representing the CPU utilization.\n",
+ "\n",
+ "Our actuator is the number of threads excuting simulations in parallel (between 0 and 8).\n",
+ "\n",
+ "> **Our control objective is to control the `loadavg` metric around a given value by adapting the number of threads executing simulations.**\n",
+ "\n",
+ "You can run the system with the following `docker` command:\n",
+ "\n",
+ "```sh\n",
+ "docker run --privileged -it -v $(pwd):/data registry.gitlab.inria.fr/control-for-computing/tutorial/system:v0.0 tuto-ctrl main.lua 1000000\n",
+ "```\n",
+ "\n",
+ "The `main.lua` file is the file containing the controller code, and the last parameter is the total number of iterations to do."
]
},
{
"cell_type": "markdown",
- "id": "6d42e211-c4d6-4426-a811-2727a47d5db8",
+ "id": "8a1ea320-905c-4784-94ca-71c622046559",
"metadata": {},
"source": [
- "You can run the system with the following `docker` command:\n",
- "\n",
- "```sh\n",
- "docker run --privileged -it -v $(pwd):/data guilloteauq/tuto-ctrl-docker:july2022 tuto-ctrl main.lua 1000000\n",
- "```"
+ "<div class=\"alert alert-info\" role=\"alert\">\n",
+ " Your tasks:\n",
+ " <ol>\n",
+ " <li>Play with the system: Try different constant inputs and see the output</li>\n",
+ " <li>Assuming the underlying model is a first order model: Perform the identification</li>\n",
+ " <li>Design a PI Controller for this systems</li>\n",
+ " <li>Introduce pertubations: At some point in your experiment, run `yes` in another terminal to act as a disturbance, and see how your controller reacts.\n",
+ " </ol> \n",
+ "</div>"
]
},
{
@@ -40,12 +62,12 @@
"id": "629824b3-eeb5-40c2-a55b-922def8fdfbd",
"metadata": {},
"source": [
- "The `main.lua` file is the file containing the controller code.\n",
- "\n",
- "Why `lua`?\n",
+ "## Why `lua`?\n",
"\n",
"Because it is a simple, small language that integrate with `C` easily!\n",
- "Instead of giving you the source code and asking you to write `C` code to implement the controller, we can just write the controller in `lua` and pass the `lua` file as an argument of the `C` binary to be loaded."
+ "Instead of giving you the source code and asking you to write `C` code to implement the controller, we can just write the controller in `lua` and pass the `lua` file as an argument of the `C` binary to be loaded.\n",
+ "\n",
+ "You can find a `lua` cheat sheet [here](https://devhints.io/lua)."
]
},
{