├── .gitignore
├── 0. LP Intro - Blending.ipynb
├── 1. LP Problem - DMC.ipynb
├── 1.2 LP Problem - DMC - 3x3 and 4x4 (Generalized)-Copy1.ipynb
├── 1.2 LP Problem - DMC - 3x3 and 4x4 (Generalized).ipynb
├── 1.2 LP Problem - DMC - 3x3 and 4x4 (Generalized).py
├── 2. DMC Overview.ipynb
├── DMC.ipynb
├── DMC.py
├── README.md
├── WhiskasModel.lp
├── assets
└── img
│ ├── debut.png
│ └── ranade.png
├── main.png
├── requirements.txt
└── visualize_lp.ipynb
/.gitignore:
--------------------------------------------------------------------------------
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--------------------------------------------------------------------------------
/1.2 LP Problem - DMC - 3x3 and 4x4 (Generalized)-Copy1.ipynb:
--------------------------------------------------------------------------------
1 | {
2 | "cells": [
3 | {
4 | "cell_type": "markdown",
5 | "id": "460e40f9",
6 | "metadata": {},
7 | "source": [
8 | "# Linear Programming: DMC Formulation (3x3)\n",
9 | "- Author: Siang Lim, Shams Elnawawi\n",
10 | "- Last Updated: June 7th 2022\n",
11 | "- Created March 2nd 2022\n",
12 | "\n",
13 | "## References\n",
14 | "- Morshedi, A. M., Cutler, C. R., & Skrovanek, T. A. (1985). Optimal solution of dynamic matrix control with linear programing techniques (LDMC). In 1985 American Control Conference (pp. 199-208). IEEE.\n",
15 | "- Sorensen, R. C., & Cutler, C. R. (1998). LP integrates economics into dynamic matrix control. Hydrocarbon Processing, 77(9), 57-65.\n",
16 | "- Ranade, S. M., & Torres, E. (2009). From dynamic mysterious control to dynamic manageable control. Hydrocarbon Processing, 88(3), 77-81.\n",
17 | "- Godoy, J. L., Ferramosca, A., & González, A. H. (2017). Economic performance assessment and monitoring in LP-DMC type controller applications. Journal of Process Control, 57, 26-37."
18 | ]
19 | },
20 | {
21 | "cell_type": "markdown",
22 | "id": "c4f71c0d",
23 | "metadata": {},
24 | "source": [
25 | "## Before starting\n",
26 | "\n",
27 | "If this is your first time using this notebook, you may need to install a couple of packages to be able to run the code. If you end up with a ```ModuleNotFoundError```, go to the below cell (with the ```pip install``` lines) and uncomment both lines, then run the cell. \n",
28 | "\n",
29 | "You will have to restart the kernel after running installation commands, under the \"Kernel\" tab in the toolbar."
30 | ]
31 | },
32 | {
33 | "cell_type": "code",
34 | "execution_count": 1,
35 | "id": "6ecd4b52",
36 | "metadata": {},
37 | "outputs": [],
38 | "source": [
39 | "# pip install pulp\n",
40 | "# pip install matplotlib-label-lines"
41 | ]
42 | },
43 | {
44 | "cell_type": "code",
45 | "execution_count": 2,
46 | "id": "7e804c41",
47 | "metadata": {},
48 | "outputs": [],
49 | "source": [
50 | "# Just importing libraries and tweaking the plot settings\n",
51 | "import numpy as np\n",
52 | "import matplotlib.pyplot as plt\n",
53 | "%matplotlib widget\n",
54 | "\n",
55 | "from matplotlib.ticker import StrMethodFormatter\n",
56 | "import matplotlib.gridspec as gridspec\n",
57 | "from labellines import labelLine, labelLines\n",
58 | "\n",
59 | "from ipywidgets.widgets.interaction import interact\n",
60 | "import ipywidgets.widgets as widgets\n",
61 | "from ipywidgets import Layout\n",
62 | "\n",
63 | "# Import PuLP modeler functions\n",
64 | "from pulp import *\n",
65 | "\n",
66 | "fsize = 8\n",
67 | "tsize = 12\n",
68 | "tdir = 'in'\n",
69 | "major = 5.0\n",
70 | "minor = 3.0\n",
71 | "lwidth = 0.8\n",
72 | "lhandle = 2.0\n",
73 | "plt.style.use('default')\n",
74 | "plt.rcParams['font.size'] = fsize\n",
75 | "plt.rcParams['legend.fontsize'] = tsize\n",
76 | "plt.rcParams['xtick.direction'] = tdir\n",
77 | "plt.rcParams['ytick.direction'] = tdir\n",
78 | "plt.rcParams['xtick.major.size'] = major\n",
79 | "plt.rcParams['xtick.minor.size'] = minor\n",
80 | "plt.rcParams['ytick.major.size'] = 5.0\n",
81 | "plt.rcParams['ytick.minor.size'] = 3.0\n",
82 | "plt.rcParams['axes.linewidth'] = lwidth\n",
83 | "plt.rcParams['legend.handlelength'] = lhandle"
84 | ]
85 | },
86 | {
87 | "cell_type": "markdown",
88 | "id": "368d8fa3",
89 | "metadata": {},
90 | "source": [
91 | "## MV-CV equations\n",
92 | "\n",
93 | "$$\n",
94 | "G =\n",
95 | " \\begin{bmatrix}\n",
96 | " -0.200 & -0.072 & 0.0774 \\\\\n",
97 | " 0.125 & -0.954 & 0.0063 \\\\\n",
98 | " 0.025 & 0.101 & −0.0143\n",
99 | " \\end{bmatrix}\n",
100 | "$$\n",
101 | "\n",
102 | "Using the gain matrix, the CV relationship can be written in terms of its MVs, starting with:\n",
103 | "\n",
104 | "$$\n",
105 | "\\Delta \\text{CV}_{1} = G_{11} \\Delta \\text{MV}_{1} + G_{12} \\Delta \\text{MV}_{2} + G_{13} \\Delta \\text{MV}_{3} \\\\ \n",
106 | "\\Delta \\text{CV}_{2} = G_{21} \\Delta \\text{MV}_{1} + G_{22} \\Delta \\text{MV}_{2} + G_{23} \\Delta \\text{MV}_{3} \\\\\n",
107 | "\\Delta \\text{CV}_{3} = G_{31} \\Delta \\text{MV}_{1} + G_{32} \\Delta \\text{MV}_{2} + G_{33} \\Delta \\text{MV}_{3}\n",
108 | "$$\n",
109 | "\n",
110 | "We can impose upper and lower limits on the MVs:\n",
111 | "\n",
112 | "$$\n",
113 | "\\text{MV}_{1, \\text{Lo}} \\leq \\text{MV}_{1} \\leq \\text{MV}_{1, \\text{Hi}}\\\\ \n",
114 | "\\text{MV}_{2, \\text{Lo}} \\leq \\text{MV}_{2} \\leq \\text{MV}_{2, \\text{Hi}}\\\\\n",
115 | "\\text{MV}_{3, \\text{Lo}} \\leq \\text{MV}_{3} \\leq \\text{MV}_{3, \\text{Hi}}\\\\\n",
116 | "$$\n",
117 | "\n",
118 | "As well as the CVs:\n",
119 | "\n",
120 | "$$\n",
121 | "\\text{CV}_{1, \\text{Lo}} \\leq \\text{CV}_{1} \\leq \\text{CV}_{1, \\text{Hi}}\\\\ \n",
122 | "\\text{CV}_{2, \\text{Lo}} \\leq \\text{CV}_{2} \\leq \\text{CV}_{2, \\text{Hi}}\\\\\n",
123 | "\\text{CV}_{3, \\text{Lo}} \\leq \\text{CV}_{3} \\leq \\text{CV}_{3, \\text{Hi}}\\\\\n",
124 | "$$\n",
125 | "\n",
126 | "Since the CVs are related to the MVs by the gain matrix, we can substitute the equations to get CV limits in terms of MV movements:\n",
127 | "\n",
128 | "$$\n",
129 | "G_{11} \\Delta \\text{MV}_{1} + G_{12} \\Delta \\text{MV}_{2} + G_{13} \\Delta \\text{MV}_{3} \\leq \\Delta \\text{CV}_{1, \\text{Hi}}\\\\\n",
130 | "G_{11} \\Delta \\text{MV}_{1} + G_{12} \\Delta \\text{MV}_{2} + G_{13} \\Delta \\text{MV}_{3} \\geq \\Delta \\text{CV}_{1, \\text{Lo}}\\\\\n",
131 | "G_{21} \\Delta \\text{MV}_{1} + G_{22} \\Delta \\text{MV}_{2} + G_{23} \\Delta \\text{MV}_{3} \\leq \\Delta \\text{CV}_{2, \\text{Hi}}\\\\\n",
132 | "G_{21} \\Delta \\text{MV}_{1} + G_{22} \\Delta \\text{MV}_{2} + G_{23} \\Delta \\text{MV}_{3} \\geq \\Delta \\text{CV}_{2, \\text{Lo}}\\\\\n",
133 | "G_{31} \\Delta \\text{MV}_{1} + G_{32} \\Delta \\text{MV}_{2} + G_{33} \\Delta \\text{MV}_{3} \\leq \\Delta \\text{CV}_{3, \\text{Hi}}\\\\\n",
134 | "G_{31} \\Delta \\text{MV}_{1} + G_{22} \\Delta \\text{MV}_{2} + G_{33} \\Delta \\text{MV}_{3} \\geq \\Delta \\text{CV}_{3, \\text{Lo}}\\\\\n",
135 | "$$\n",
136 | "\n",
137 | "Let's see what this looks like:"
138 | ]
139 | },
140 | {
141 | "cell_type": "markdown",
142 | "id": "90f372d2",
143 | "metadata": {},
144 | "source": [
145 | "# Gains and CV Limits"
146 | ]
147 | },
148 | {
149 | "cell_type": "code",
150 | "execution_count": 3,
151 | "id": "6873757d",
152 | "metadata": {},
153 | "outputs": [],
154 | "source": [
155 | "G11 = -0.200\n",
156 | "G12 = -0.072\n",
157 | "G13 = 0.0774\n",
158 | "G14 = 0.0574\n",
159 | "\n",
160 | "G21 = 0.125\n",
161 | "G22 = -0.954\n",
162 | "G23 = 0.0063\n",
163 | "G24 = 0.0374\n",
164 | "\n",
165 | "G31 = 0.025\n",
166 | "G32 = 0.101\n",
167 | "G33 = -0.0143\n",
168 | "G34 = -0.1143\n",
169 | "\n",
170 | "G41 = 0.120\n",
171 | "G42 = 0.150\n",
172 | "G43 = -0.1143\n",
173 | "G44 = -0.0943\n",
174 | "\n",
175 | "CV1Lo = -6\n",
176 | "CV1Hi = 6\n",
177 | "\n",
178 | "CV2Lo = -10\n",
179 | "CV2Hi = 10.5\n",
180 | "\n",
181 | "CV3Lo = -3\n",
182 | "CV3Hi = 3.5\n",
183 | "\n",
184 | "CV4Lo = -4.5\n",
185 | "CV4Hi = 4\n",
186 | "\n",
187 | "cost_MV1 = -1\n",
188 | "cost_MV2 = -1\n",
189 | "cost_MV3 = -1\n",
190 | "cost_MV4 = 1\n",
191 | "\n",
192 | "limits = 10\n",
193 | "plot_limits = 20\n",
194 | "\n",
195 | "MV1Lo = -limits\n",
196 | "MV1Hi = limits\n",
197 | "MV2Lo = -limits\n",
198 | "MV2Hi = limits\n",
199 | "MV3Lo = -limits\n",
200 | "MV3Hi = 15\n",
201 | "MV4Lo = -limits\n",
202 | "MV4Hi = -limits"
203 | ]
204 | },
205 | {
206 | "cell_type": "code",
207 | "execution_count": 4,
208 | "id": "562bb6e4",
209 | "metadata": {},
210 | "outputs": [],
211 | "source": [
212 | "G = [(G11, G12, G13, G14), \n",
213 | " (G21, G22, G23, G24), \n",
214 | " (G31, G32, G33, G34), \n",
215 | " (G41, G42, G43, G44)]\n",
216 | "\n",
217 | "CV_values = [(CV1Lo, CV1Hi), \n",
218 | " (CV2Lo, CV2Hi), \n",
219 | " (CV3Lo, CV3Hi)] \n",
220 | "# (CV4Lo, CV4Hi)]\n",
221 | "CV_init_vals= [(-2.5,2.0), \n",
222 | " (-4,4.5), \n",
223 | " (-0.9,0.2)] \n",
224 | "# (-4,4.5)]\n",
225 | "MV_costs = [cost_MV1, \n",
226 | " cost_MV2, \n",
227 | " cost_MV3]\n",
228 | "# cost_MV4]\n",
229 | "MV_values = [(MV1Lo, MV1Hi), \n",
230 | " (MV2Lo, MV2Hi), \n",
231 | " (MV3Lo, MV3Hi)] \n",
232 | "# (MV4Lo, MV4Hi)]\n",
233 | "\n",
234 | "# (x,y,c) triplets of MVs, c for constant, i.e. \n",
235 | "# (MV1, MV2, MV3) for the first plot, \n",
236 | "# (MV1, MV3, MV2) for the second, \n",
237 | "# (MV2, MV3, MV1) for the third\n",
238 | "# plot_MV_indices = [(0,1), (0,2), (1,2)]\n",
239 | "\n",
240 | "nCVs = len(CV_values)\n",
241 | "nMVs = len(MV_values)"
242 | ]
243 | },
244 | {
245 | "cell_type": "code",
246 | "execution_count": 5,
247 | "id": "a7fdf889-850d-40db-93d0-a93bd835d92e",
248 | "metadata": {},
249 | "outputs": [
250 | {
251 | "name": "stdout",
252 | "output_type": "stream",
253 | "text": [
254 | "{'CV1 Limits': (-2.5, 2.0), 'CV2 Limits': (-4.0, 4.5), 'CV3 Limits': (-0.9, 0.2), 'MV1 Cost': -1.0, 'MV2 Cost': -1.0, 'MV3 Cost': -1.0, 'MV1 Limits': (-10.0, 10.0), 'MV2 Limits': (-10.0, 10.0), 'MV3 Limits': (-10.0, 15.0)}\n"
255 | ]
256 | }
257 | ],
258 | "source": [
259 | "CV_widgets = []\n",
260 | "MV_widgets = []\n",
261 | "cost_widgets = []\n",
262 | "stepsize = 0.2\n",
263 | "\n",
264 | "# Make CV sliders\n",
265 | "for i in range(nCVs):\n",
266 | " widget = widgets.FloatRangeSlider(\n",
267 | " value=CV_init_vals[i], min=CV_values[i][0], max=CV_values[i][1], step=stepsize,\n",
268 | " description=f'CV{i+1} Limits', continuous_update=False)\n",
269 | " CV_widgets.append(widget)\n",
270 | "\n",
271 | "# Make MV sliders\n",
272 | "for i in range(nMVs):\n",
273 | " widget = widgets.FloatRangeSlider(\n",
274 | " value=MV_values[i], min=MV_values[i][0], max=MV_values[i][1], step=stepsize,\n",
275 | " description=f'MV{i+1} Limits', continuous_update=False)\n",
276 | " MV_widgets.append(widget)\n",
277 | " \n",
278 | "# Make cost sliders\n",
279 | "for i in range(nMVs):\n",
280 | " widget = widgets.FloatSlider(\n",
281 | " value=MV_costs[i], min=-3, max=3, step=stepsize/2, # finer step for costs\n",
282 | " description=f'MV{i+1} Cost', continuous_update=True)\n",
283 | " cost_widgets.append(widget)\n",
284 | " \n",
285 | "sliders = CV_widgets + cost_widgets + MV_widgets\n",
286 | "values = [slider.value for slider in sliders]\n",
287 | "values_dict = {}\n",
288 | "for slider in sliders:\n",
289 | " values_dict[slider.description] = slider.value\n",
290 | "print(values_dict)"
291 | ]
292 | },
293 | {
294 | "cell_type": "code",
295 | "execution_count": 6,
296 | "id": "cb507e02-fab8-41d7-a338-89f60cbe41f2",
297 | "metadata": {},
298 | "outputs": [],
299 | "source": [
300 | "# display ui\n",
301 | "ncols = 3\n",
302 | "\n",
303 | "nrows = nCVs // ncols + 1\n",
304 | "CV_row = []\n",
305 | "for i in range(nrows):\n",
306 | " CV_row.append(widgets.VBox(CV_widgets[(ncols)*i:(ncols)*i+ncols]))\n",
307 | " \n",
308 | "nrows = nMVs // ncols + 1\n",
309 | "MV_row = []\n",
310 | "cost_row = []\n",
311 | "for i in range(nrows):\n",
312 | " MV_row.append(widgets.VBox(MV_widgets[(ncols)*i:(ncols)*i+ncols]))\n",
313 | " cost_row.append(widgets.VBox(cost_widgets[(ncols)*i:(ncols)*i+ncols]))\n",
314 | "\n",
315 | "ui = widgets.VBox([widgets.VBox(CV_row),\n",
316 | " widgets.VBox(MV_row),\n",
317 | " widgets.VBox(cost_row)])"
318 | ]
319 | },
320 | {
321 | "cell_type": "code",
322 | "execution_count": 7,
323 | "id": "337d52ee",
324 | "metadata": {},
325 | "outputs": [],
326 | "source": [
327 | "# Solve the LP\n",
328 | "def run_lp(values_dict):\n",
329 | " prob = LpProblem(\"DMC_problem\",LpMinimize)\n",
330 | " \n",
331 | " # How many MVs\n",
332 | " MVs = []\n",
333 | " for i in range(nMVs):\n",
334 | " MVs.append(LpVariable(f\"MV{i+1}\",-limits))\n",
335 | " \n",
336 | " # the objective function\n",
337 | " obj = 0\n",
338 | " for indx, MV in enumerate(MVs):\n",
339 | " obj += values_dict[f\"MV{indx+1} Cost\"]*MV\n",
340 | " \n",
341 | " prob += obj, \"Cost function of MVs\"\n",
342 | " \n",
343 | " # constraint formulation in terms of MV1 and MV2\n",
344 | " CV_contraint_lo = []\n",
345 | " CV_contraint_hi = []\n",
346 | " \n",
347 | " for i in range(nCVs):\n",
348 | " c = 0\n",
349 | " for indx, MV in enumerate(MVs):\n",
350 | " c += G[i][indx]*MV\n",
351 | " prob += c <= values_dict[f'CV{i+1} Limits'][1], f'CV{i+1} High Limit'\n",
352 | " prob += c >= values_dict[f'CV{i+1} Limits'][0], f'CV{i+1} Low Limit'\n",
353 | " \n",
354 | "# prob += G[i][0]*MV1 + G[i][1]*MV2 <= values_dict[f'CV{i+1} Limits'][1], f'CV{i+1} High Limit'\n",
355 | "# prob += G[i][0]*MV1 + G[i][1]*MV2 >= values_dict[f'CV{i+1} Limits'][0], f'CV{i+1} Low Limit'\n",
356 | " \n",
357 | " for indx, MV in enumerate(MVs):\n",
358 | " prob += MV <= values_dict[f'MV{indx+1} Limits'][1], f'MV{indx+1} High Limit'\n",
359 | " prob += MV >= values_dict[f'MV{indx+1} Limits'][0], f'MV{indx+1} Low Limit' \n",
360 | " \n",
361 | " if (prob.solve(PULP_CBC_CMD(msg=0)) == 1):\n",
362 | "# print([v.varValue for v in prob.variables()])\n",
363 | "# print([v for v in prob.variables()])\n",
364 | " return [v.varValue for v in prob.variables()], value(prob.objective)\n",
365 | " else:\n",
366 | " print(\"NOT SOLVED - Infeasibility!\")\n",
367 | " return np.zeros(nMVs), 0"
368 | ]
369 | },
370 | {
371 | "cell_type": "code",
372 | "execution_count": 8,
373 | "id": "f0e50a42",
374 | "metadata": {},
375 | "outputs": [],
376 | "source": [
377 | "d = np.linspace(-plot_limits, plot_limits, 1000)\n",
378 | "x,y = np.meshgrid(d,d)\n",
379 | "\n",
380 | "# Recalculate shaded regions\n",
381 | "constraints = []\n",
382 | "for i in range(nCVs):\n",
383 | " c_hi = G[i][0]*x + G[i][1]*y <= values_dict[f'CV{i+1} Limits'][1]\n",
384 | " c_lo = G[i][0]*x + G[i][1]*y >= values_dict[f'CV{i+1} Limits'][0]\n",
385 | " constraints.append(c_lo)\n",
386 | " constraints.append(c_hi)\n",
387 | "\n",
388 | "# the MVs\n",
389 | "for indx, var in enumerate([x,y]):\n",
390 | " c_hi = var <= values_dict[f'MV{indx+1} Limits'][1]\n",
391 | " c_lo = var >= values_dict[f'MV{indx+1} Limits'][0]\n",
392 | " constraints.append(c_lo)\n",
393 | " constraints.append(c_hi)"
394 | ]
395 | },
396 | {
397 | "cell_type": "code",
398 | "execution_count": 9,
399 | "id": "3b8daec8",
400 | "metadata": {},
401 | "outputs": [],
402 | "source": [
403 | "def handle_slider_change(change):\n",
404 | " ## grab slider vals\n",
405 | " values_dict = {}\n",
406 | " for slider in sliders:\n",
407 | " values_dict[slider.description] = slider.value\n",
408 | " \n",
409 | " # Find the LP soln\n",
410 | " soln, V = run_lp(values_dict)\n",
411 | " \n",
412 | " for indx, key in enumerate(axs_dict):\n",
413 | " ax = axs_dict[key]\n",
414 | " \n",
415 | " # remove previous line labels (TODO could be more efficient, modify labelLine library to return handles)\n",
416 | " # for txt in ax.texts:\n",
417 | " # txt.remove()\n",
418 | " \n",
419 | " # figure out which MVs we are plotting, and which ones are constant\n",
420 | " x_MV = key[1] # column is x-axis\n",
421 | " y_MV = key[0]+1 # row is y-axis\n",
422 | " c_MV = [a for a in range(nMVs) if a != x_MV and a != y_MV] \n",
423 | "\n",
424 | " # Recalculate shaded regions \n",
425 | " constraints = []\n",
426 | " \n",
427 | " # the obj func\n",
428 | " y_obj = (V - sum([values_dict[f'MV{cmv+1} Cost']*soln[cmv] for cmv in c_MV]) - values_dict[f'MV{x_MV+1} Cost']*d)/values_dict[f'MV{y_MV+1} Cost']\n",
429 | " \n",
430 | " # Update CV constraint lines\n",
431 | " for i in range(nCVs):\n",
432 | " cv_lim_lo = values_dict[f'CV{i+1} Limits'][0]\n",
433 | " cv_lim_hi = values_dict[f'CV{i+1} Limits'][1]\n",
434 | "\n",
435 | " # for CV constraint lines\n",
436 | " y_lo = (cv_lim_lo - G[i][x_MV]*d - sum([G[i][cmv]*soln[cmv] for cmv in c_MV]))/G[i][y_MV]\n",
437 | " y_hi = (cv_lim_hi - G[i][x_MV]*d - sum([G[i][cmv]*soln[cmv] for cmv in c_MV]))/G[i][y_MV]\n",
438 | " lines_handler_dict[key]['CV_lines_lo'][i].set_data(d, y_lo)\n",
439 | " lines_handler_dict[key]['CV_lines_hi'][i].set_data(d, y_hi)\n",
440 | "\n",
441 | " # for shading\n",
442 | " c_hi = G[i][x_MV]*x + G[i][y_MV]*y + sum([G[i][cmv]*soln[cmv] for cmv in c_MV]) <= cv_lim_hi\n",
443 | " c_lo = G[i][x_MV]*x + G[i][y_MV]*y + sum([G[i][cmv]*soln[cmv] for cmv in c_MV]) >= cv_lim_lo\n",
444 | " constraints.append(c_lo)\n",
445 | " constraints.append(c_hi)\n",
446 | " \n",
447 | " # search for valid xvals where the yvals are within plots limits\n",
448 | " # x_valid_lo = d[(y_lo < plot_limits) & (y_lo > -plot_limits)]\n",
449 | " # x_valid_hi = d[(y_hi < plot_limits) & (y_hi > -plot_limits)]\n",
450 | " # offset_lo = max(0,min(10, len(x_valid_lo)-3))\n",
451 | " # offset_hi = max(0,min(10, len(x_valid_hi)-3))\n",
452 | " # if len(x_valid_lo) > 0:\n",
453 | " # txt_cv_lo = labelLine(line_lo, x_valid_lo[offset_lo], fontsize=5, zorder=2.5)\n",
454 | " # if len(x_valid_hi) > 0:\n",
455 | " # txt_cv_hi = labelLine(line_hi, x_valid_hi[-offset_hi], fontsize=5, zorder=2.5) \n",
456 | "\n",
457 | " # Update MV constraint lines\n",
458 | " lines_handler_dict[key]['v_lo'].set_data([values_dict[f'MV{x_MV+1} Limits'][0],values_dict[f'MV{x_MV+1} Limits'][0]], [-limits, limits])\n",
459 | " lines_handler_dict[key]['v_hi'].set_data([values_dict[f'MV{x_MV+1} Limits'][1],values_dict[f'MV{x_MV+1} Limits'][1]], [-limits, limits])\n",
460 | " lines_handler_dict[key]['h_lo'].set_data([-limits, limits], [values_dict[f'MV{y_MV+1} Limits'][0],values_dict[f'MV{y_MV+1} Limits'][0]])\n",
461 | " lines_handler_dict[key]['h_hi'].set_data([-limits, limits], [values_dict[f'MV{y_MV+1} Limits'][1],values_dict[f'MV{y_MV+1} Limits'][1]]) \n",
462 | "\n",
463 | " # the 4 MV limits for this ax\n",
464 | " constraints.append(x >= values_dict[f'MV{x_MV+1} Limits'][0])\n",
465 | " constraints.append(x <= values_dict[f'MV{x_MV+1} Limits'][1])\n",
466 | " constraints.append(y >= values_dict[f'MV{y_MV+1} Limits'][0])\n",
467 | " constraints.append(y <= values_dict[f'MV{y_MV+1} Limits'][1]) \n",
468 | "\n",
469 | " # Shade the right regions\n",
470 | " lines_handler_dict[key]['im'].set_data((np.logical_and.reduce(constraints)).astype(float))\n",
471 | " \n",
472 | " # the quiver/vector field\n",
473 | " z_obj = (values_dict[f'MV{x_MV+1} Cost'] * xv) + (values_dict[f'MV{y_MV+1} Cost'] * yv) + sum([values_dict[f'MV{cmv+1} Cost']*soln[cmv] for cmv in c_MV])\n",
474 | " lines_handler_dict[key]['quiver'].set_UVC(-values_dict[f'MV{x_MV+1} Cost'],-values_dict[f'MV{y_MV+1} Cost'], z_obj)\n",
475 | "\n",
476 | " # the soln\n",
477 | " lines_handler_dict[key]['soln_marker'].set_data(soln[x_MV], soln[y_MV]);\n",
478 | " lines_handler_dict[key]['soln_text'].set_position((soln[x_MV], soln[y_MV]));\n",
479 | " lines_handler_dict[key]['soln_text'].set_text(\"({:.1f}, {:.1f})\".format(soln[x_MV], soln[y_MV]))\n",
480 | " lines_handler_dict[key]['soln_func'].set_data(d, y_obj);\n",
481 | " \n",
482 | " fig.canvas.draw()"
483 | ]
484 | },
485 | {
486 | "cell_type": "code",
487 | "execution_count": 10,
488 | "id": "8236d3a0-005d-440d-9482-15ec484d788a",
489 | "metadata": {},
490 | "outputs": [
491 | {
492 | "data": {
493 | "application/vnd.jupyter.widget-view+json": {
494 | "model_id": "9fb33089b9d34fc9b14df25515cf3d64",
495 | "version_major": 2,
496 | "version_minor": 0
497 | },
498 | "text/plain": [
499 | "HBox(children=(Output(), VBox(children=(HTMLMath(value='Use the controls to interact with this linear progr…"
500 | ]
501 | },
502 | "metadata": {},
503 | "output_type": "display_data"
504 | }
505 | ],
506 | "source": [
507 | "# Initialize plots\n",
508 | "d = np.linspace(-plot_limits, plot_limits, 100)\n",
509 | "x,y = np.meshgrid(d,d)\n",
510 | "\n",
511 | "gs = gridspec.GridSpec(nMVs-1, nMVs-1)\n",
512 | "gs.update(wspace=0.05, hspace=0.05)\n",
513 | "gs_indices = [(row, col) for row in range(nMVs-2,-1,-1) for col in range(row+1)]\n",
514 | "axs_dict = {}\n",
515 | "\n",
516 | "# mesh for vector field\n",
517 | "dvec = np.linspace(-plot_limits + (0.1*plot_limits), plot_limits - (0.1*plot_limits), 12)\n",
518 | "xv, yv = np.meshgrid(dvec, dvec)\n",
519 | "\n",
520 | "# init soln\n",
521 | "soln, V = run_lp(values_dict)\n",
522 | "\n",
523 | "# initialize line handlers\n",
524 | "CV_lines_lo = []\n",
525 | "CV_lines_hi = []\n",
526 | "MV_lines_lo = []\n",
527 | "MV_lines_hi = []\n",
528 | "lines_handler_dict = {}\n",
529 | "\n",
530 | "colors = ['r', 'b', 'y', 'g'] # TODO generalize this using a cmap instead of defining manual colors??!\n",
531 | "\n",
532 | "# plot as widget\n",
533 | "output = widgets.Output()\n",
534 | "with output:\n",
535 | " fig = plt.figure(figsize=(6,6), dpi=100, facecolor='white')\n",
536 | " plt.show()\n",
537 | " \n",
538 | "for r,c in gs_indices:\n",
539 | " # build the shared axes correctly and handle the tick labels\n",
540 | " if (r == nMVs-2 and c == 0): # no shared axis for the ax, bottom left corner\n",
541 | " ax = plt.subplot(gs[r,c])\n",
542 | " elif(c != 0): # all non-first columns share a y-axis with stuff to the left of it\n",
543 | " ax = plt.subplot(gs[r,c], sharey=axs_dict[(r,0)])\n",
544 | " plt.setp(ax.get_yticklabels(), visible=False)\n",
545 | " elif(r != nMVs-2): # all non-first rows share a x-axis with stuff below it\n",
546 | " ax = plt.subplot(gs[r,c], sharex=axs_dict[(nMVs-2,c)])\n",
547 | " plt.setp(ax.get_xticklabels(), visible=False)\n",
548 | "\n",
549 | " # add the labels\n",
550 | " if(r == nMVs-2):\n",
551 | " ax.set_xlabel(f'$\\Delta MV_{c+1}$')\n",
552 | " if(c == 0):\n",
553 | " ax.set_ylabel(f'$\\Delta MV_{r+2}$')\n",
554 | "\n",
555 | " axs_dict[(r,c)] = ax\n",
556 | "\n",
557 | "for indx, key in enumerate(axs_dict):\n",
558 | " ax = axs_dict[key]\n",
559 | " lines_handler_dict[key] = {}\n",
560 | "\n",
561 | " # figure out which MVs we are plotting, and which ones are constant\n",
562 | " x_MV = key[1] # column is x-axis\n",
563 | " y_MV = key[0]+1 # row is y-axis\n",
564 | " c_MV = [a for a in range(nMVs) if a != x_MV and a != y_MV]\n",
565 | "\n",
566 | " # plot the MV limits\n",
567 | " v_lo = ax.axvline(x=values_dict[f'MV{x_MV+1} Limits'][0], color='gray', lw=1, label=f'MV{x_MV+1} Lo')\n",
568 | " v_hi = ax.axvline(x=values_dict[f'MV{x_MV+1} Limits'][1], color='gray', lw=1, label=f'MV{x_MV+1} Hi')\n",
569 | " h_lo = ax.axhline(y=values_dict[f'MV{y_MV+1} Limits'][0], color='gray', lw=1, label=f'MV{y_MV+1} Lo')\n",
570 | " h_hi = ax.axhline(y=values_dict[f'MV{y_MV+1} Limits'][1], color='gray', lw=1, label=f'MV{y_MV+1} Hi')\n",
571 | "\n",
572 | " # labelLines([v_lo, v_hi], fontsize=4)\n",
573 | " # labelLine(h_hi, fontsize=4)\n",
574 | " \n",
575 | " # store the line handlers per ax in a dict\n",
576 | " lines_handler_dict[key]['v_lo'] = v_lo\n",
577 | " lines_handler_dict[key]['v_hi'] = v_hi\n",
578 | " lines_handler_dict[key]['h_lo'] = h_lo\n",
579 | " lines_handler_dict[key]['h_hi'] = h_hi\n",
580 | "\n",
581 | " # Recalculate shaded regions \n",
582 | " constraints = []\n",
583 | "\n",
584 | " # calculate CV lines\n",
585 | " lines_handler_dict[key]['CV_lines_lo'] = []\n",
586 | " lines_handler_dict[key]['CV_lines_hi'] = [] \n",
587 | " for i in range(nCVs):\n",
588 | " cv_lim_lo = values_dict[f'CV{i+1} Limits'][0]\n",
589 | " cv_lim_hi = values_dict[f'CV{i+1} Limits'][1]\n",
590 | "\n",
591 | " # for CV constraint lines\n",
592 | " y_lo = (cv_lim_lo - G[i][x_MV]*d - sum([G[i][cmv]*soln[cmv] for cmv in c_MV]))/G[i][y_MV]\n",
593 | " y_hi = (cv_lim_hi - G[i][x_MV]*d - sum([G[i][cmv]*soln[cmv] for cmv in c_MV]))/G[i][y_MV]\n",
594 | " line_lo, = ax.plot(d, y_lo, f'--{colors[i]}', label=f'$CV_{i+1}$ Lo');\n",
595 | " line_hi, = ax.plot(d, y_hi, f'-{colors[i]}', label=f'$CV_{i+1}$ Hi');\n",
596 | " lines_handler_dict[key]['CV_lines_lo'].append(line_lo)\n",
597 | " lines_handler_dict[key]['CV_lines_hi'].append(line_hi)\n",
598 | " \n",
599 | " # # search for valid xvals where the yvals are within plots limits\n",
600 | " # x_valid_lo = d[(y_lo < plot_limits) & (y_lo > -plot_limits)]\n",
601 | " # x_valid_hi = d[(y_hi < plot_limits) & (y_hi > -plot_limits)]\n",
602 | " # offset_lo = max(0,min(10, len(x_valid_lo)-3))\n",
603 | " # offset_hi = max(0,min(10, len(x_valid_hi)-3))\n",
604 | " # if len(x_valid_lo) > 0:\n",
605 | " # labelLine(line_lo, x_valid_lo[offset_lo], fontsize=5, zorder=2.5)\n",
606 | " # if len(x_valid_hi) > 0:\n",
607 | " # labelLine(line_hi, x_valid_hi[-offset_hi], fontsize=5, zorder=2.5)\n",
608 | " \n",
609 | " # for shading\n",
610 | " c_hi = G[i][x_MV]*x + G[i][y_MV]*y + sum([G[i][cmv]*soln[cmv] for cmv in c_MV]) <= cv_lim_hi\n",
611 | " c_lo = G[i][x_MV]*x + G[i][y_MV]*y + sum([G[i][cmv]*soln[cmv] for cmv in c_MV]) >= cv_lim_lo\n",
612 | " constraints.append(c_lo)\n",
613 | " constraints.append(c_hi)\n",
614 | " \n",
615 | "\n",
616 | " # the 4 MV limits for this ax\n",
617 | " constraints.append(x >= values_dict[f'MV{x_MV+1} Limits'][0])\n",
618 | " constraints.append(x <= values_dict[f'MV{x_MV+1} Limits'][1])\n",
619 | " constraints.append(y >= values_dict[f'MV{y_MV+1} Limits'][0])\n",
620 | " constraints.append(y <= values_dict[f'MV{y_MV+1} Limits'][1]) \n",
621 | "\n",
622 | " # the obj function \n",
623 | " y_obj = (V - sum([values_dict[f'MV{cmv+1} Cost']*soln[cmv] for cmv in c_MV]) - values_dict[f'MV{x_MV+1} Cost']*d)/values_dict[f'MV{y_MV+1} Cost']\n",
624 | " soln_func, = ax.plot(d, y_obj, '--k')\n",
625 | " soln_marker, = ax.plot(soln[x_MV], soln[y_MV], 'ok', zorder=10)\n",
626 | " soln_text = ax.text(soln[x_MV], soln[y_MV], '({:.2f},{:.2f})'.format(soln[x_MV], soln[y_MV]))\n",
627 | "\n",
628 | " lines_handler_dict[key]['soln_func'] = soln_func\n",
629 | " lines_handler_dict[key]['soln_marker'] = soln_marker\n",
630 | " lines_handler_dict[key]['soln_text'] = soln_text\n",
631 | " \n",
632 | " z_obj = (values_dict[f'MV{x_MV+1} Cost'] * xv) + (values_dict[f'MV{y_MV+1} Cost'] * yv) + sum([values_dict[f'MV{cmv+1} Cost']*soln[cmv] for cmv in c_MV])\n",
633 | " lines_handler_dict[key]['quiver'] = ax.quiver(xv, yv,-values_dict[f'MV{x_MV+1} Cost'],-values_dict[f'MV{y_MV+1} Cost'], z_obj, cmap='gray', headwidth=4, width=0.003, scale=40, alpha=0.5)\n",
634 | "\n",
635 | " ax.plot(0,0,'kx');\n",
636 | " ax.set_aspect('equal')\n",
637 | " im = ax.imshow(np.logical_and.reduce(constraints).astype(int), extent=(x.min(),x.max(),y.min(),y.max()), origin=\"lower\", cmap=\"Blues\", alpha=0.1)\n",
638 | " lines_handler_dict[key]['im'] = im\n",
639 | " \n",
640 | "####################################################\n",
641 | "# END OF INIT FUNC\n",
642 | "####################################################\n",
643 | "\n",
644 | "# register slides\n",
645 | "for widget in sliders:\n",
646 | " widget.observe(handle_slider_change, names='value')\n",
647 | "\n",
648 | "widgets.HBox([output, \n",
649 | " widgets.VBox([widgets.HTMLMath(\n",
650 | " value=r\"
Use the controls to interact with this linear program
\"), \n",
651 | " ui])])\n"
652 | ]
653 | },
654 | {
655 | "cell_type": "code",
656 | "execution_count": null,
657 | "id": "c752d9b1-788f-4c3d-b3ae-4f98877f6d83",
658 | "metadata": {},
659 | "outputs": [],
660 | "source": []
661 | }
662 | ],
663 | "metadata": {
664 | "kernelspec": {
665 | "display_name": "Python 3 (ipykernel)",
666 | "language": "python",
667 | "name": "python3"
668 | },
669 | "language_info": {
670 | "codemirror_mode": {
671 | "name": "ipython",
672 | "version": 3
673 | },
674 | "file_extension": ".py",
675 | "mimetype": "text/x-python",
676 | "name": "python",
677 | "nbconvert_exporter": "python",
678 | "pygments_lexer": "ipython3",
679 | "version": "3.8.8"
680 | }
681 | },
682 | "nbformat": 4,
683 | "nbformat_minor": 5
684 | }
685 |
--------------------------------------------------------------------------------
/1.2 LP Problem - DMC - 3x3 and 4x4 (Generalized).py:
--------------------------------------------------------------------------------
1 | # -*- coding: utf-8 -*-
2 | # ---
3 | # jupyter:
4 | # jupytext:
5 | # text_representation:
6 | # extension: .py
7 | # format_name: light
8 | # format_version: '1.5'
9 | # jupytext_version: 1.4.0
10 | # kernelspec:
11 | # display_name: Python 3 (ipykernel)
12 | # language: python
13 | # name: python3
14 | # ---
15 |
16 | # # Linear Programming: DMC Formulation (3x3)
17 | # - Author: Siang Lim, Shams Elnawawi
18 | # - Last Updated: June 7th 2022
19 | # - Created March 2nd 2022
20 | #
21 | # ## References
22 | # - Morshedi, A. M., Cutler, C. R., & Skrovanek, T. A. (1985). Optimal solution of dynamic matrix control with linear programing techniques (LDMC). In 1985 American Control Conference (pp. 199-208). IEEE.
23 | # - Sorensen, R. C., & Cutler, C. R. (1998). LP integrates economics into dynamic matrix control. Hydrocarbon Processing, 77(9), 57-65.
24 | # - Ranade, S. M., & Torres, E. (2009). From dynamic mysterious control to dynamic manageable control. Hydrocarbon Processing, 88(3), 77-81.
25 | # - Godoy, J. L., Ferramosca, A., & González, A. H. (2017). Economic performance assessment and monitoring in LP-DMC type controller applications. Journal of Process Control, 57, 26-37.
26 |
27 | # +
28 | # # !pip install pulp
29 |
30 | # +
31 | # Just importing libraries and tweaking the plot settings
32 | import numpy as np
33 | import matplotlib.pyplot as plt
34 | # %matplotlib widget
35 |
36 | from matplotlib.ticker import StrMethodFormatter
37 | import matplotlib.gridspec as gridspec
38 | from labellines import labelLine, labelLines
39 |
40 | from ipywidgets.widgets.interaction import interact
41 | import ipywidgets.widgets as widgets
42 | from ipywidgets import Layout
43 |
44 | # Import PuLP modeler functions
45 | from pulp import *
46 |
47 | fsize = 8
48 | tsize = 12
49 | tdir = 'in'
50 | major = 5.0
51 | minor = 3.0
52 | lwidth = 0.8
53 | lhandle = 2.0
54 | plt.style.use('default')
55 | plt.rcParams['font.size'] = fsize
56 | plt.rcParams['legend.fontsize'] = tsize
57 | plt.rcParams['xtick.direction'] = tdir
58 | plt.rcParams['ytick.direction'] = tdir
59 | plt.rcParams['xtick.major.size'] = major
60 | plt.rcParams['xtick.minor.size'] = minor
61 | plt.rcParams['ytick.major.size'] = 5.0
62 | plt.rcParams['ytick.minor.size'] = 3.0
63 | plt.rcParams['axes.linewidth'] = lwidth
64 | plt.rcParams['legend.handlelength'] = lhandle
65 | # -
66 |
67 | # ## MV-CV equations
68 | #
69 | # $$
70 | # G =
71 | # \begin{bmatrix}
72 | # -0.200 & -0.072 & 0.0774 \\
73 | # 0.125 & -0.954 & 0.0063 \\
74 | # 0.025 & 0.101 & −0.0143
75 | # \end{bmatrix}
76 | # $$
77 | #
78 | # Using the gain matrix, the CV relationship can be written in terms of its MVs, starting with:
79 | #
80 | # $$
81 | # \Delta \text{CV}_{1} = G_{11} \Delta \text{MV}_{1} + G_{12} \Delta \text{MV}_{2} + G_{13} \Delta \text{MV}_{3} \\
82 | # \Delta \text{CV}_{2} = G_{21} \Delta \text{MV}_{1} + G_{22} \Delta \text{MV}_{2} + G_{23} \Delta \text{MV}_{3} \\
83 | # \Delta \text{CV}_{3} = G_{31} \Delta \text{MV}_{1} + G_{32} \Delta \text{MV}_{2} + G_{33} \Delta \text{MV}_{3}
84 | # $$
85 | #
86 | # We can impose upper and lower limits on the MVs:
87 | #
88 | # $$
89 | # \text{MV}_{1, \text{Lo}} \leq \text{MV}_{1} \leq \text{MV}_{1, \text{Hi}}\\
90 | # \text{MV}_{2, \text{Lo}} \leq \text{MV}_{2} \leq \text{MV}_{2, \text{Hi}}\\
91 | # \text{MV}_{3, \text{Lo}} \leq \text{MV}_{3} \leq \text{MV}_{3, \text{Hi}}\\
92 | # $$
93 | #
94 | # As well as the CVs:
95 | #
96 | # $$
97 | # \text{CV}_{1, \text{Lo}} \leq \text{CV}_{1} \leq \text{CV}_{1, \text{Hi}}\\
98 | # \text{CV}_{2, \text{Lo}} \leq \text{CV}_{2} \leq \text{CV}_{2, \text{Hi}}\\
99 | # \text{CV}_{3, \text{Lo}} \leq \text{CV}_{3} \leq \text{CV}_{3, \text{Hi}}\\
100 | # $$
101 | #
102 | # Since the CVs are related to the MVs by the gain matrix, we can substitute the equations to get CV limits in terms of MV movements:
103 | #
104 | # $$
105 | # G_{11} \Delta \text{MV}_{1} + G_{12} \Delta \text{MV}_{2} + G_{13} \Delta \text{MV}_{3} \leq \Delta \text{CV}_{1, \text{Hi}}\\
106 | # G_{11} \Delta \text{MV}_{1} + G_{12} \Delta \text{MV}_{2} + G_{13} \Delta \text{MV}_{3} \geq \Delta \text{CV}_{1, \text{Lo}}\\
107 | # G_{21} \Delta \text{MV}_{1} + G_{22} \Delta \text{MV}_{2} + G_{23} \Delta \text{MV}_{3} \leq \Delta \text{CV}_{2, \text{Hi}}\\
108 | # G_{21} \Delta \text{MV}_{1} + G_{22} \Delta \text{MV}_{2} + G_{23} \Delta \text{MV}_{3} \geq \Delta \text{CV}_{2, \text{Lo}}\\
109 | # G_{31} \Delta \text{MV}_{1} + G_{32} \Delta \text{MV}_{2} + G_{33} \Delta \text{MV}_{3} \leq \Delta \text{CV}_{3, \text{Hi}}\\
110 | # G_{31} \Delta \text{MV}_{1} + G_{22} \Delta \text{MV}_{2} + G_{33} \Delta \text{MV}_{3} \geq \Delta \text{CV}_{3, \text{Lo}}\\
111 | # $$
112 | #
113 | # Let's see what this looks like:
114 |
115 | # # Gains and CV Limits
116 |
117 | # +
118 | G11 = -0.200
119 | G12 = -0.072
120 | G13 = 0.0774
121 | G14 = 0.0574
122 |
123 | G21 = 0.125
124 | G22 = -0.954
125 | G23 = 0.0063
126 | G24 = 0.0374
127 |
128 | G31 = 0.025
129 | G32 = 0.101
130 | G33 = -0.0143
131 | G34 = -0.1143
132 |
133 | G41 = 0.120
134 | G42 = 0.150
135 | G43 = -0.1143
136 | G44 = -0.0943
137 |
138 | CV1Lo = -6
139 | CV1Hi = 6
140 |
141 | CV2Lo = -10
142 | CV2Hi = 10.5
143 |
144 | CV3Lo = -3
145 | CV3Hi = 3.5
146 |
147 | CV4Lo = -4.5
148 | CV4Hi = 4
149 |
150 | cost_MV1 = -1
151 | cost_MV2 = -1
152 | cost_MV3 = -1
153 | cost_MV4 = 1
154 |
155 | limits = 10
156 | plot_limits = 20
157 |
158 | MV1Lo = -limits
159 | MV1Hi = limits
160 | MV2Lo = -limits
161 | MV2Hi = limits
162 | MV3Lo = -limits
163 | MV3Hi = 15
164 | MV4Lo = -limits
165 | MV4Hi = -limits
166 |
167 | # +
168 | G = [(G11, G12, G13, G14),
169 | (G21, G22, G23, G24),
170 | (G31, G32, G33, G34),
171 | (G41, G42, G43, G44)]
172 |
173 | CV_values = [(CV1Lo, CV1Hi),
174 | (CV2Lo, CV2Hi),
175 | (CV3Lo, CV3Hi)]
176 | # (CV4Lo, CV4Hi)]
177 | CV_init_vals= [(-2.5,2.0),
178 | (-4,4.5),
179 | (-0.9,0.2)]
180 | # (-4,4.5)]
181 | MV_costs = [cost_MV1,
182 | cost_MV2,
183 | cost_MV3]
184 | # cost_MV4]
185 | MV_values = [(MV1Lo, MV1Hi),
186 | (MV2Lo, MV2Hi),
187 | (MV3Lo, MV3Hi)]
188 | # (MV4Lo, MV4Hi)]
189 |
190 | # (x,y,c) triplets of MVs, c for constant, i.e.
191 | # (MV1, MV2, MV3) for the first plot,
192 | # (MV1, MV3, MV2) for the second,
193 | # (MV2, MV3, MV1) for the third
194 | # plot_MV_indices = [(0,1), (0,2), (1,2)]
195 |
196 | nCVs = len(CV_values)
197 | nMVs = len(MV_values)
198 |
199 | # +
200 | CV_widgets = []
201 | MV_widgets = []
202 | cost_widgets = []
203 | stepsize = 0.2
204 |
205 | # Make CV sliders
206 | for i in range(nCVs):
207 | widget = widgets.FloatRangeSlider(
208 | value=CV_init_vals[i], min=CV_values[i][0], max=CV_values[i][1], step=stepsize,
209 | description=f'CV{i+1} Limits', continuous_update=False)
210 | CV_widgets.append(widget)
211 |
212 | # Make MV sliders
213 | for i in range(nMVs):
214 | widget = widgets.FloatRangeSlider(
215 | value=MV_values[i], min=MV_values[i][0], max=MV_values[i][1], step=stepsize,
216 | description=f'MV{i+1} Limits', continuous_update=False)
217 | MV_widgets.append(widget)
218 |
219 | # Make cost sliders
220 | for i in range(nMVs):
221 | widget = widgets.FloatSlider(
222 | value=MV_costs[i], min=-3, max=3, step=stepsize/2, # finer step for costs
223 | description=f'MV{i+1} Cost', continuous_update=True)
224 | cost_widgets.append(widget)
225 |
226 | sliders = CV_widgets + cost_widgets + MV_widgets
227 | values = [slider.value for slider in sliders]
228 | values_dict = {}
229 | for slider in sliders:
230 | values_dict[slider.description] = slider.value
231 | print(values_dict)
232 |
233 | # +
234 | # display ui
235 | ncols = 3
236 |
237 | nrows = nCVs // ncols + 1
238 | CV_row = []
239 | for i in range(nrows):
240 | CV_row.append(widgets.VBox(CV_widgets[(ncols)*i:(ncols)*i+ncols]))
241 |
242 | nrows = nMVs // ncols + 1
243 | MV_row = []
244 | cost_row = []
245 | for i in range(nrows):
246 | MV_row.append(widgets.VBox(MV_widgets[(ncols)*i:(ncols)*i+ncols]))
247 | cost_row.append(widgets.VBox(cost_widgets[(ncols)*i:(ncols)*i+ncols]))
248 |
249 | ui = widgets.VBox([widgets.VBox(CV_row),
250 | widgets.VBox(MV_row),
251 | widgets.VBox(cost_row)])
252 |
253 |
254 | # +
255 | # Solve the LP
256 | def run_lp(values_dict):
257 | prob = LpProblem("DMC_problem",LpMinimize)
258 |
259 | # How many MVs
260 | MVs = []
261 | for i in range(nMVs):
262 | MVs.append(LpVariable(f"MV{i+1}",-limits))
263 |
264 | # the objective function
265 | obj = 0
266 | for indx, MV in enumerate(MVs):
267 | obj += values_dict[f"MV{indx+1} Cost"]*MV
268 |
269 | prob += obj, "Cost function of MVs"
270 |
271 | # constraint formulation in terms of MV1 and MV2
272 | CV_contraint_lo = []
273 | CV_contraint_hi = []
274 |
275 | for i in range(nCVs):
276 | c = 0
277 | for indx, MV in enumerate(MVs):
278 | c += G[i][indx]*MV
279 | prob += c <= values_dict[f'CV{i+1} Limits'][1], f'CV{i+1} High Limit'
280 | prob += c >= values_dict[f'CV{i+1} Limits'][0], f'CV{i+1} Low Limit'
281 |
282 | # prob += G[i][0]*MV1 + G[i][1]*MV2 <= values_dict[f'CV{i+1} Limits'][1], f'CV{i+1} High Limit'
283 | # prob += G[i][0]*MV1 + G[i][1]*MV2 >= values_dict[f'CV{i+1} Limits'][0], f'CV{i+1} Low Limit'
284 |
285 | for indx, MV in enumerate(MVs):
286 | prob += MV <= values_dict[f'MV{indx+1} Limits'][1], f'MV{indx+1} High Limit'
287 | prob += MV >= values_dict[f'MV{indx+1} Limits'][0], f'MV{indx+1} Low Limit'
288 |
289 | if (prob.solve(PULP_CBC_CMD(msg=0)) == 1):
290 | # print([v.varValue for v in prob.variables()])
291 | # print([v for v in prob.variables()])
292 | return [v.varValue for v in prob.variables()], value(prob.objective)
293 | else:
294 | print("NOT SOLVED - Infeasibility!")
295 | return np.zeros(nMVs), 0
296 |
297 |
298 | # +
299 | d = np.linspace(-plot_limits, plot_limits, 1000)
300 | x,y = np.meshgrid(d,d)
301 |
302 | # Recalculate shaded regions
303 | constraints = []
304 | for i in range(nCVs):
305 | c_hi = G[i][0]*x + G[i][1]*y <= values_dict[f'CV{i+1} Limits'][1]
306 | c_lo = G[i][0]*x + G[i][1]*y >= values_dict[f'CV{i+1} Limits'][0]
307 | constraints.append(c_lo)
308 | constraints.append(c_hi)
309 |
310 | # the MVs
311 | for indx, var in enumerate([x,y]):
312 | c_hi = var <= values_dict[f'MV{indx+1} Limits'][1]
313 | c_lo = var >= values_dict[f'MV{indx+1} Limits'][0]
314 | constraints.append(c_lo)
315 | constraints.append(c_hi)
316 |
317 |
318 | # -
319 |
320 | def handle_slider_change(change):
321 | ## grab slider vals
322 | values_dict = {}
323 | for slider in sliders:
324 | values_dict[slider.description] = slider.value
325 |
326 | # Find the LP soln
327 | soln, V = run_lp(values_dict)
328 |
329 | for indx, key in enumerate(axs_dict):
330 | ax = axs_dict[key]
331 |
332 | # remove previous line labels (TODO could be more efficient, modify labelLine library to return handles)
333 | # for txt in ax.texts:
334 | # txt.remove()
335 |
336 | # figure out which MVs we are plotting, and which ones are constant
337 | x_MV = key[1] # column is x-axis
338 | y_MV = key[0]+1 # row is y-axis
339 | c_MV = [a for a in range(nMVs) if a != x_MV and a != y_MV]
340 |
341 | # Recalculate shaded regions
342 | constraints = []
343 |
344 | # the obj func
345 | y_obj = (V - sum([values_dict[f'MV{cmv+1} Cost']*soln[cmv] for cmv in c_MV]) - values_dict[f'MV{x_MV+1} Cost']*d)/values_dict[f'MV{y_MV+1} Cost']
346 |
347 | # Update CV constraint lines
348 | for i in range(nCVs):
349 | cv_lim_lo = values_dict[f'CV{i+1} Limits'][0]
350 | cv_lim_hi = values_dict[f'CV{i+1} Limits'][1]
351 |
352 | # for CV constraint lines
353 | y_lo = (cv_lim_lo - G[i][x_MV]*d - sum([G[i][cmv]*soln[cmv] for cmv in c_MV]))/G[i][y_MV]
354 | y_hi = (cv_lim_hi - G[i][x_MV]*d - sum([G[i][cmv]*soln[cmv] for cmv in c_MV]))/G[i][y_MV]
355 | lines_handler_dict[key]['CV_lines_lo'][i].set_data(d, y_lo)
356 | lines_handler_dict[key]['CV_lines_hi'][i].set_data(d, y_hi)
357 |
358 | # for shading
359 | c_hi = G[i][x_MV]*x + G[i][y_MV]*y + sum([G[i][cmv]*soln[cmv] for cmv in c_MV]) <= cv_lim_hi
360 | c_lo = G[i][x_MV]*x + G[i][y_MV]*y + sum([G[i][cmv]*soln[cmv] for cmv in c_MV]) >= cv_lim_lo
361 | constraints.append(c_lo)
362 | constraints.append(c_hi)
363 |
364 | # search for valid xvals where the yvals are within plots limits
365 | # x_valid_lo = d[(y_lo < plot_limits) & (y_lo > -plot_limits)]
366 | # x_valid_hi = d[(y_hi < plot_limits) & (y_hi > -plot_limits)]
367 | # offset_lo = max(0,min(10, len(x_valid_lo)-3))
368 | # offset_hi = max(0,min(10, len(x_valid_hi)-3))
369 | # if len(x_valid_lo) > 0:
370 | # txt_cv_lo = labelLine(line_lo, x_valid_lo[offset_lo], fontsize=5, zorder=2.5)
371 | # if len(x_valid_hi) > 0:
372 | # txt_cv_hi = labelLine(line_hi, x_valid_hi[-offset_hi], fontsize=5, zorder=2.5)
373 |
374 | # Update MV constraint lines
375 | lines_handler_dict[key]['v_lo'].set_data([values_dict[f'MV{x_MV+1} Limits'][0],values_dict[f'MV{x_MV+1} Limits'][0]], [-limits, limits])
376 | lines_handler_dict[key]['v_hi'].set_data([values_dict[f'MV{x_MV+1} Limits'][1],values_dict[f'MV{x_MV+1} Limits'][1]], [-limits, limits])
377 | lines_handler_dict[key]['h_lo'].set_data([-limits, limits], [values_dict[f'MV{y_MV+1} Limits'][0],values_dict[f'MV{y_MV+1} Limits'][0]])
378 | lines_handler_dict[key]['h_hi'].set_data([-limits, limits], [values_dict[f'MV{y_MV+1} Limits'][1],values_dict[f'MV{y_MV+1} Limits'][1]])
379 |
380 | # the 4 MV limits for this ax
381 | constraints.append(x >= values_dict[f'MV{x_MV+1} Limits'][0])
382 | constraints.append(x <= values_dict[f'MV{x_MV+1} Limits'][1])
383 | constraints.append(y >= values_dict[f'MV{y_MV+1} Limits'][0])
384 | constraints.append(y <= values_dict[f'MV{y_MV+1} Limits'][1])
385 |
386 | # Shade the right regions
387 | lines_handler_dict[key]['im'].set_data((np.logical_and.reduce(constraints)).astype(float))
388 |
389 | # the quiver/vector field
390 | z_obj = (values_dict[f'MV{x_MV+1} Cost'] * xv) + (values_dict[f'MV{y_MV+1} Cost'] * yv) + sum([values_dict[f'MV{cmv+1} Cost']*soln[cmv] for cmv in c_MV])
391 | lines_handler_dict[key]['quiver'].set_UVC(-values_dict[f'MV{x_MV+1} Cost'],-values_dict[f'MV{y_MV+1} Cost'], z_obj)
392 |
393 | # the soln
394 | lines_handler_dict[key]['soln_marker'].set_data(soln[x_MV], soln[y_MV]);
395 | lines_handler_dict[key]['soln_text'].set_position((soln[x_MV], soln[y_MV]));
396 | lines_handler_dict[key]['soln_text'].set_text("({:.1f}, {:.1f})".format(soln[x_MV], soln[y_MV]))
397 | lines_handler_dict[key]['soln_func'].set_data(d, y_obj);
398 |
399 | fig.canvas.draw()
400 |
401 |
402 | # +
403 | # Initialize plots
404 | d = np.linspace(-plot_limits, plot_limits, 100)
405 | x,y = np.meshgrid(d,d)
406 |
407 | gs = gridspec.GridSpec(nMVs-1, nMVs-1)
408 | gs.update(wspace=0.05, hspace=0.05)
409 | gs_indices = [(row, col) for row in range(nMVs-2,-1,-1) for col in range(row+1)]
410 | axs_dict = {}
411 |
412 | # mesh for vector field
413 | dvec = np.linspace(-plot_limits + (0.1*plot_limits), plot_limits - (0.1*plot_limits), 12)
414 | xv, yv = np.meshgrid(dvec, dvec)
415 |
416 | # init soln
417 | soln, V = run_lp(values_dict)
418 |
419 | # initialize line handlers
420 | CV_lines_lo = []
421 | CV_lines_hi = []
422 | MV_lines_lo = []
423 | MV_lines_hi = []
424 | lines_handler_dict = {}
425 |
426 | colors = ['r', 'b', 'y', 'g'] # TODO generalize this using a cmap instead of defining manual colors??!
427 |
428 | # plot as widget
429 | output = widgets.Output()
430 | with output:
431 | fig = plt.figure(figsize=(6,6), dpi=100, facecolor='white')
432 | plt.show()
433 |
434 | for r,c in gs_indices:
435 | # build the shared axes correctly and handle the tick labels
436 | if (r == nMVs-2 and c == 0): # no shared axis for the ax, bottom left corner
437 | ax = plt.subplot(gs[r,c])
438 | elif(c != 0): # all non-first columns share a y-axis with stuff to the left of it
439 | ax = plt.subplot(gs[r,c], sharey=axs_dict[(r,0)])
440 | plt.setp(ax.get_yticklabels(), visible=False)
441 | elif(r != nMVs-2): # all non-first rows share a x-axis with stuff below it
442 | ax = plt.subplot(gs[r,c], sharex=axs_dict[(nMVs-2,c)])
443 | plt.setp(ax.get_xticklabels(), visible=False)
444 |
445 | # add the labels
446 | if(r == nMVs-2):
447 | ax.set_xlabel(f'$\Delta MV_{c+1}$')
448 | if(c == 0):
449 | ax.set_ylabel(f'$\Delta MV_{r+2}$')
450 |
451 | axs_dict[(r,c)] = ax
452 |
453 | for indx, key in enumerate(axs_dict):
454 | ax = axs_dict[key]
455 | lines_handler_dict[key] = {}
456 |
457 | # figure out which MVs we are plotting, and which ones are constant
458 | x_MV = key[1] # column is x-axis
459 | y_MV = key[0]+1 # row is y-axis
460 | c_MV = [a for a in range(nMVs) if a != x_MV and a != y_MV]
461 |
462 | # plot the MV limits
463 | v_lo = ax.axvline(x=values_dict[f'MV{x_MV+1} Limits'][0], color='gray', lw=1, label=f'MV{x_MV+1} Lo')
464 | v_hi = ax.axvline(x=values_dict[f'MV{x_MV+1} Limits'][1], color='gray', lw=1, label=f'MV{x_MV+1} Hi')
465 | h_lo = ax.axhline(y=values_dict[f'MV{y_MV+1} Limits'][0], color='gray', lw=1, label=f'MV{y_MV+1} Lo')
466 | h_hi = ax.axhline(y=values_dict[f'MV{y_MV+1} Limits'][1], color='gray', lw=1, label=f'MV{y_MV+1} Hi')
467 |
468 | # labelLines([v_lo, v_hi], fontsize=4)
469 | # labelLine(h_hi, fontsize=4)
470 |
471 | # store the line handlers per ax in a dict
472 | lines_handler_dict[key]['v_lo'] = v_lo
473 | lines_handler_dict[key]['v_hi'] = v_hi
474 | lines_handler_dict[key]['h_lo'] = h_lo
475 | lines_handler_dict[key]['h_hi'] = h_hi
476 |
477 | # Recalculate shaded regions
478 | constraints = []
479 |
480 | # calculate CV lines
481 | lines_handler_dict[key]['CV_lines_lo'] = []
482 | lines_handler_dict[key]['CV_lines_hi'] = []
483 | for i in range(nCVs):
484 | cv_lim_lo = values_dict[f'CV{i+1} Limits'][0]
485 | cv_lim_hi = values_dict[f'CV{i+1} Limits'][1]
486 |
487 | # for CV constraint lines
488 | y_lo = (cv_lim_lo - G[i][x_MV]*d - sum([G[i][cmv]*soln[cmv] for cmv in c_MV]))/G[i][y_MV]
489 | y_hi = (cv_lim_hi - G[i][x_MV]*d - sum([G[i][cmv]*soln[cmv] for cmv in c_MV]))/G[i][y_MV]
490 | line_lo, = ax.plot(d, y_lo, f'--{colors[i]}', label=f'$CV_{i+1}$ Lo');
491 | line_hi, = ax.plot(d, y_hi, f'-{colors[i]}', label=f'$CV_{i+1}$ Hi');
492 | lines_handler_dict[key]['CV_lines_lo'].append(line_lo)
493 | lines_handler_dict[key]['CV_lines_hi'].append(line_hi)
494 |
495 | # # search for valid xvals where the yvals are within plots limits
496 | # x_valid_lo = d[(y_lo < plot_limits) & (y_lo > -plot_limits)]
497 | # x_valid_hi = d[(y_hi < plot_limits) & (y_hi > -plot_limits)]
498 | # offset_lo = max(0,min(10, len(x_valid_lo)-3))
499 | # offset_hi = max(0,min(10, len(x_valid_hi)-3))
500 | # if len(x_valid_lo) > 0:
501 | # labelLine(line_lo, x_valid_lo[offset_lo], fontsize=5, zorder=2.5)
502 | # if len(x_valid_hi) > 0:
503 | # labelLine(line_hi, x_valid_hi[-offset_hi], fontsize=5, zorder=2.5)
504 |
505 | # for shading
506 | c_hi = G[i][x_MV]*x + G[i][y_MV]*y + sum([G[i][cmv]*soln[cmv] for cmv in c_MV]) <= cv_lim_hi
507 | c_lo = G[i][x_MV]*x + G[i][y_MV]*y + sum([G[i][cmv]*soln[cmv] for cmv in c_MV]) >= cv_lim_lo
508 | constraints.append(c_lo)
509 | constraints.append(c_hi)
510 |
511 |
512 | # the 4 MV limits for this ax
513 | constraints.append(x >= values_dict[f'MV{x_MV+1} Limits'][0])
514 | constraints.append(x <= values_dict[f'MV{x_MV+1} Limits'][1])
515 | constraints.append(y >= values_dict[f'MV{y_MV+1} Limits'][0])
516 | constraints.append(y <= values_dict[f'MV{y_MV+1} Limits'][1])
517 |
518 | # the obj function
519 | y_obj = (V - sum([values_dict[f'MV{cmv+1} Cost']*soln[cmv] for cmv in c_MV]) - values_dict[f'MV{x_MV+1} Cost']*d)/values_dict[f'MV{y_MV+1} Cost']
520 | soln_func, = ax.plot(d, y_obj, '--k')
521 | soln_marker, = ax.plot(soln[x_MV], soln[y_MV], 'ok', zorder=10)
522 | soln_text = ax.text(soln[x_MV], soln[y_MV], '({:.2f},{:.2f})'.format(soln[x_MV], soln[y_MV]))
523 |
524 | lines_handler_dict[key]['soln_func'] = soln_func
525 | lines_handler_dict[key]['soln_marker'] = soln_marker
526 | lines_handler_dict[key]['soln_text'] = soln_text
527 |
528 | z_obj = (values_dict[f'MV{x_MV+1} Cost'] * xv) + (values_dict[f'MV{y_MV+1} Cost'] * yv) + sum([values_dict[f'MV{cmv+1} Cost']*soln[cmv] for cmv in c_MV])
529 | lines_handler_dict[key]['quiver'] = ax.quiver(xv, yv,-values_dict[f'MV{x_MV+1} Cost'],-values_dict[f'MV{y_MV+1} Cost'], z_obj, cmap='gray', headwidth=4, width=0.003, scale=40, alpha=0.5)
530 |
531 | ax.plot(0,0,'kx');
532 | ax.set_aspect('equal')
533 | im = ax.imshow(np.logical_and.reduce(constraints).astype(int), extent=(x.min(),x.max(),y.min(),y.max()), origin="lower", cmap="Blues", alpha=0.1)
534 | lines_handler_dict[key]['im'] = im
535 |
536 | ####################################################
537 | # END OF INIT FUNC
538 | ####################################################
539 |
540 | # register slides
541 | for widget in sliders:
542 | widget.observe(handle_slider_change, names='value')
543 |
544 | widgets.HBox([output,
545 | widgets.VBox([widgets.HTMLMath(
546 | value=r"Use the controls to interact with this linear program
"),
547 | ui])])
548 |
549 | # -
550 |
551 |
552 |
--------------------------------------------------------------------------------
/DMC.ipynb:
--------------------------------------------------------------------------------
1 | {
2 | "cells": [
3 | {
4 | "cell_type": "markdown",
5 | "metadata": {},
6 | "source": [
7 | "# Implementing Dynamic Matrix Control in Python\n",
8 | "\n",
9 | "#### Author: Siang Lim, February 2020\n",
10 | "\n",
11 | "** work in progress **\n",
12 | "\n",
13 | "- Contents of this notebook are derived from DMC literature:\n",
14 | " - Hokanson, D. A., & Gerstle, J. G. (1992). **Dynamic Matrix Control Multivariable Controllers.** Practical Distillation Control, 248–271.\n",
15 | " - Sorensen, R. C., & Cutler, C. R. (1998). **LP integrates economics into dynamic matrix control.** Hydrocarbon processing (International ed.), 77(9), 57-65.\n",
16 | " - Morshedi, A. M., Cutler, C. R., & Skrovanek, T. A. (1985, June). **Optimal solution of dynamic matrix control with linear programing techniques (LDMC).** In 1985 American Control Conference (pp. 199-208). IEEE."
17 | ]
18 | },
19 | {
20 | "cell_type": "code",
21 | "execution_count": 1,
22 | "metadata": {
23 | "collapsed": true
24 | },
25 | "outputs": [],
26 | "source": [
27 | "import numpy as np\n",
28 | "import matplotlib.pyplot as plt"
29 | ]
30 | },
31 | {
32 | "cell_type": "markdown",
33 | "metadata": {},
34 | "source": [
35 | "# Definitions\n",
36 | "\n",
37 | "#### Independent Variables\n",
38 | "\n",
39 | "- MV (Manipulated Variables)\n",
40 | " - Variables that we can move directly, valves, feed rate etc.\n",
41 | "\n",
42 | "- FF (Feedforward Variables)\n",
43 | " - Disturbances that we can't control directly - ambient temperature.\n",
44 | "\n",
45 | "#### Dependent Variables\n",
46 | "- Variables that change due to a change in the independent variables, e.g. temperatures, pressures\n"
47 | ]
48 | },
49 | {
50 | "cell_type": "markdown",
51 | "metadata": {},
52 | "source": [
53 | "## Step Response Convolution Model"
54 | ]
55 | },
56 | {
57 | "cell_type": "markdown",
58 | "metadata": {},
59 | "source": [
60 | "The sequence $a_j$, where $j = 0,1,2,3 \\dots$ is the step response convolution model, which is the basis for DMC.\n",
61 | "\n",
62 | "Remembering that everything so far is in deviation variables, at is no more than a series of numbers based on\n",
63 | "a set length (time to steady state) and a set sampling interval.\n",
64 | "\n",
65 | "There is no set form of the model, other than that steady state is reached by the last interval. In fact, the last\n",
66 | "value $a_t$ is the process steady-state gain. "
67 | ]
68 | },
69 | {
70 | "cell_type": "code",
71 | "execution_count": 2,
72 | "metadata": {
73 | "collapsed": true
74 | },
75 | "outputs": [],
76 | "source": [
77 | "a = -np.array([0.5,1.0,1.5,1.8,1.9,1.95,1.98,1.99,1.999,2])"
78 | ]
79 | },
80 | {
81 | "cell_type": "code",
82 | "execution_count": 3,
83 | "metadata": {},
84 | "outputs": [
85 | {
86 | "data": {
87 | "text/plain": [
88 | "Text(0,0.5,'Response')"
89 | ]
90 | },
91 | "execution_count": 3,
92 | "metadata": {},
93 | "output_type": "execute_result"
94 | },
95 | {
96 | "data": {
97 | "image/png": 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98 | "text/plain": [
99 | ""
100 | ]
101 | },
102 | "metadata": {},
103 | "output_type": "display_data"
104 | }
105 | ],
106 | "source": [
107 | "%matplotlib inline\n",
108 | "plt.plot(a, '-or')\n",
109 | "plt.xlabel(\"Time (s)\")\n",
110 | "plt.ylabel(\"Response\")"
111 | ]
112 | },
113 | {
114 | "cell_type": "markdown",
115 | "metadata": {},
116 | "source": [
117 | "## Inputs"
118 | ]
119 | },
120 | {
121 | "cell_type": "markdown",
122 | "metadata": {},
123 | "source": [
124 | "For a lined out process with a single step in $u$ at time 0:\n",
125 | "\n",
126 | "$$ \\Delta u_0 = u_0 - u_{-1} $$\n",
127 | "\n",
128 | "then\n",
129 | "\n",
130 | "$$ y_1 - y_0 = a_1 \\Delta u_0 $$\n",
131 | "$$ y_2 - y_0 = a_2 \\Delta u_0 $$\n",
132 | "$$ y_3 - y_0 = a_3 \\Delta u_0 $$\n",
133 | "$$ y_4 - y_0 = a_4 \\Delta u_0 $$\n",
134 | "$$ \\vdots $$\n",
135 | "$$ y_t - y_0 = a_t \\Delta u_0 $$\n",
136 | "\n",
137 | "This is essentially the definition of the step response convolution model described before. What is interesting about this is that we can calculate what y will be at each interval for successive moves in u.\n",
138 | "\n",
139 | "For example purposes only, let us examine a series of three moves \"into the future\": one now, one at the next time interval, and one two time intervals into the future: \n",
140 | "\n",
141 | "$$ \\Delta u_0 = u_0 - u_{-1} $$\n",
142 | "$$ \\Delta u_1 = u_1 - u_{0} $$\n",
143 | "$$ \\Delta u_2 = u_2 - u_{1} $$"
144 | ]
145 | },
146 | {
147 | "cell_type": "code",
148 | "execution_count": 4,
149 | "metadata": {},
150 | "outputs": [
151 | {
152 | "name": "stdout",
153 | "output_type": "stream",
154 | "text": [
155 | "[[ 0]\n",
156 | " [ 1]\n",
157 | " [ 0]\n",
158 | " [ 0]\n",
159 | " [-1]]\n"
160 | ]
161 | }
162 | ],
163 | "source": [
164 | "U = np.atleast_2d(np.array([0,1,0,0,-1])).T\n",
165 | "print(U)"
166 | ]
167 | },
168 | {
169 | "cell_type": "code",
170 | "execution_count": 5,
171 | "metadata": {},
172 | "outputs": [
173 | {
174 | "data": {
175 | "text/plain": [
176 | "[]"
177 | ]
178 | },
179 | "execution_count": 5,
180 | "metadata": {},
181 | "output_type": "execute_result"
182 | },
183 | {
184 | "data": {
185 | "image/png": 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186 | "text/plain": [
187 | ""
188 | ]
189 | },
190 | "metadata": {},
191 | "output_type": "display_data"
192 | }
193 | ],
194 | "source": [
195 | "I_obs = np.pad(np.cumsum(U),(0,8),'constant')\n",
196 | "plt.step(I_obs, '-bx', where=\"post\")"
197 | ]
198 | },
199 | {
200 | "cell_type": "markdown",
201 | "metadata": {},
202 | "source": [
203 | "then\n",
204 | "\n",
205 | "$$ \n",
206 | "\\begin{align}\n",
207 | "y_1 - y_0 & = a_1 \\Delta u_0 \\\\\n",
208 | "y_2 - y_0 & = a_2 \\Delta u_0 + a_1 \\Delta u_1 \\\\\n",
209 | "y_3 - y_0 & = a_3 \\Delta u_0 + a_2 \\Delta u_1 + + a_1 \\Delta u_2 \\\\\n",
210 | "y_4 - y_0 & = a_4 \\Delta u_0 + a_3 \\Delta u_1 + + a_2 \\Delta u_2 \\\\\n",
211 | "\\vdots & \\\\\n",
212 | "y_t - y_0 & = a_t \\Delta u_0 + a_{t-1} \\Delta u_1 + + a_{t-2} \\Delta u_2 \\\\\n",
213 | "\\end{align}\n",
214 | "$$\n"
215 | ]
216 | },
217 | {
218 | "cell_type": "code",
219 | "execution_count": 6,
220 | "metadata": {},
221 | "outputs": [
222 | {
223 | "name": "stdout",
224 | "output_type": "stream",
225 | "text": [
226 | "[[-0.5 0. 0. 0. 0. ]\n",
227 | " [-1. -0.5 0. 0. 0. ]\n",
228 | " [-1.5 -1. -0.5 0. 0. ]\n",
229 | " [-1.8 -1.5 -1. -0.5 0. ]\n",
230 | " [-1.9 -1.8 -1.5 -1. -0.5 ]\n",
231 | " [-1.95 -1.9 -1.8 -1.5 -1. ]\n",
232 | " [-1.98 -1.95 -1.9 -1.8 -1.5 ]\n",
233 | " [-1.99 -1.98 -1.95 -1.9 -1.8 ]\n",
234 | " [-1.999 -1.99 -1.98 -1.95 -1.9 ]\n",
235 | " [-2. -1.999 -1.99 -1.98 -1.95 ]]\n"
236 | ]
237 | }
238 | ],
239 | "source": [
240 | "# Build dynamic matrix\n",
241 | "t = a.shape[0]\n",
242 | "u = U.shape[0]\n",
243 | "\n",
244 | "A = np.tile(a, (u,1)).T\n",
245 | "row_indices = np.atleast_2d(np.arange(t)).T - np.arange(u)\n",
246 | "col_indices = np.arange(u)\n",
247 | "A = np.tril(A[row_indices,col_indices])\n",
248 | "print(A)"
249 | ]
250 | },
251 | {
252 | "cell_type": "code",
253 | "execution_count": 7,
254 | "metadata": {
255 | "collapsed": true
256 | },
257 | "outputs": [],
258 | "source": [
259 | "Y = np.multiply(A,U.T)\n",
260 | "Y_obs = np.matmul(A,U)\n",
261 | "U_obs = np.pad(np.cumsum(U),(0,a.shape[0]-U.shape[0]),'constant')"
262 | ]
263 | },
264 | {
265 | "cell_type": "code",
266 | "execution_count": 8,
267 | "metadata": {},
268 | "outputs": [
269 | {
270 | "data": {
271 | "text/plain": [
272 | "array([[-0. , 0. , 0. , 0. , -0. ],\n",
273 | " [-0. , -0.5 , 0. , 0. , -0. ],\n",
274 | " [-0. , -1. , -0. , 0. , -0. ],\n",
275 | " [-0. , -1.5 , -0. , -0. , -0. ],\n",
276 | " [-0. , -1.8 , -0. , -0. , 0.5 ],\n",
277 | " [-0. , -1.9 , -0. , -0. , 1. ],\n",
278 | " [-0. , -1.95 , -0. , -0. , 1.5 ],\n",
279 | " [-0. , -1.98 , -0. , -0. , 1.8 ],\n",
280 | " [-0. , -1.99 , -0. , -0. , 1.9 ],\n",
281 | " [-0. , -1.999, -0. , -0. , 1.95 ]])"
282 | ]
283 | },
284 | "execution_count": 8,
285 | "metadata": {},
286 | "output_type": "execute_result"
287 | }
288 | ],
289 | "source": [
290 | "Y"
291 | ]
292 | },
293 | {
294 | "cell_type": "code",
295 | "execution_count": 9,
296 | "metadata": {},
297 | "outputs": [
298 | {
299 | "data": {
300 | "text/plain": [
301 | "(array([1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9]),\n",
302 | " array([1, 1, 1, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4]))"
303 | ]
304 | },
305 | "execution_count": 9,
306 | "metadata": {},
307 | "output_type": "execute_result"
308 | }
309 | ],
310 | "source": [
311 | "np.nonzero(Y)"
312 | ]
313 | },
314 | {
315 | "cell_type": "code",
316 | "execution_count": 10,
317 | "metadata": {},
318 | "outputs": [
319 | {
320 | "data": {
321 | "text/plain": [
322 | "array([1, 4])"
323 | ]
324 | },
325 | "execution_count": 10,
326 | "metadata": {},
327 | "output_type": "execute_result"
328 | }
329 | ],
330 | "source": [
331 | "np.nonzero(np.any(Y != 0, axis=0))[0]"
332 | ]
333 | },
334 | {
335 | "cell_type": "code",
336 | "execution_count": 11,
337 | "metadata": {},
338 | "outputs": [
339 | {
340 | "data": {
341 | "image/png": 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fkmXJZvaGmX0euo/QPH/V7wSfx/+E/q8sNbOXzCyxuuqp8eFsZtHAI8AwIBO4zswyI1tV\nRBUCdznnugHnA7fV8c8D4EfAqkgXERB/Bl5zznUFzqaOfi5m1hq4HchyznUHovGT9tQlTwBDj1l2\nD/CWc64T8FboeV3xBMd/Hm8A3Z1zPfHzS9xbXcXU+HAG+gI5zrl1zrl84DlgdIRrihjn3Fbn3OLQ\n4/34P76tI1tV5JhZG2AE8K9I1xJpZpYADAAeBXDO5Tvn9kS2qoiKAeqbWQzQANgS4XqqlXPuXWDX\nMYtHA0+GHj8JjKnWoiKoos/DOfe6c64w9HQB0Ka66qkN4dwa2FTmeS51OIzKMrMMoDfwUWQriaiH\ngbuB4kgXEgDtge3A46Fu/n+ZWcNIFxUJzrnNwJ+AjcBWYK9z7vXIVhUIzZ1zW8F/0QeaRbieIPkW\n8O/qerPaEM5WwbI6fwq6mTUCXgDucM7ti3Q9kWBmI4FtzrlFka4lIGKAPsDfnHO9gYPUrW7Lo0LH\nUkfj56lvBTQ0sxsiW5UElZndjz9kOLW63rM2hHMukFbmeRvqWPfUscwsFh/MU51zL0a6ngi6EBhl\nZl/gD3dcYmZPR7akiMoFcp1zJT0pM/BhXRcNBtY757Y75wqAF4F+Ea4pCL4ys5YAofttEa4n4szs\nJmAkMM5V47XHtSGcFwKdzKydmcXhT+qYFeGaIsbMDH9McZVzblKk64kk59y9zrk2zrkM/L+Lt51z\ndbZ15Jz7EthkZl1Ciy4FVkawpEjaCJxvZg1C/2cupY6eHHeMWcBNocc3AS9HsJaIM7OhwE+BUc65\nvOp87xofzqGD9ROBufj/XNOccysiW1VEXQiMx7cSPwndhke6KAmMHwJTzWwp0Av4fYTriYhQ78EM\nYDGwDP+3sE6NjGVmzwIfAl3MLNfMvg38AbjMzD4HLgs9rxNO8HlMBhoDb4T+lv692urRCGEiIiLB\nUuNbziIiIrWNwllERCRgFM4iIiIBo3AWEREJGIWziIhIwCicRUREAkbhLCIiEjAKZxERkYBROIuI\niASMwllERCRgFM4iIiIBo3AWEREJGIWziIhIwCicRUREAiYmUm+ckpLiMjIyIvX2IiIi1WrRokU7\nnHOpldk2LOFsZmnAU0ALoBiY4pz788lek5GRQXZ2djjeXkREJPDMbENltw1Xy7kQuMs5t9jMGgOL\nzOwN59zKMO1fRESkzgjLMWfn3Fbn3OLQ4/3AKqB1OPYtIiJSbd5/GNa/W37Z+nf98moU9hPCzCwD\n6A18FO59i4iIVKnWfWD6zaUBvf5d/7x1n2otI6wnhJlZI+AF4A7n3L4K1k8AJgCkp6eH861FRCRo\n3n/Yh1q7AaXL1r8LmxdD/zsiU9PhvZCfBwV5UHAICg9DXENo1s2vP7ANelwDz3wTsr4Nnz4DY58o\n/zNUg7CFs5nF4oN5qnPuxYq2cc5NAaYAZGVluXC9t4iIBFBJK7Qk3EpaoWOfKN2muBiKjkBsff98\nby7k7fTBWRKgUbHQeYhfv3Q67FobWnfY3zduAZf8zK+feRt8tTz0+tA+Wp8D46b59f8YCLvXl6+z\n8zC4/jn/eO59cOAr//jD/4MBd1d7MEP4ztY24FFglXNuUjj2KSIiNUB+HhzaBU3a+Oe52bBlCeTt\n8iHbvDs8fRVceAdkPwoZF8GL3/OhWXjY3xqmwk9y/Ovn3A2rXy3/HoltofNS//iTp2HdOxBdzwd6\nbANo0b1027gG0Kh5aF3oltKldP3An0LhIf+62PoQU9+He4nvvOlb9rN/DFnf8jW3u6jaA9qcO/MG\nrJn1B94DluEvpQK4zzk350SvycrKcrqUSkSkkqqrizhvl29Z5u32oZu30y+78EdQrxEsehI+/mfp\nusLD/nX3bfXB+Np9sOARv6xeE2iQBIVHYP9W3wpt0toHeElwxtSH+CZwwQ/8azZ+BHk7StfF1od6\njaFpB7++8IhvSUdV0RhaZVv3x7b2zzCgzWyRcy6rMtuGpeXsnHsfsHDsS0REKlCZLuISxcVweA8c\n2l0arm3OhYZNIXcRLH7SLz+0u7SFe9Mr0KwrLJsO/777mB0a9LrOh3NcQ99KbtkTGiRD/WR/b6EI\nuOgu/2WhfhJEx5bWOeBu3wod+wScc/OJf870807+OcTUq9zn9XVtXlw+iNsN8M83L67W1nNYWs5f\nh1rOIiKnaf27MO1GaNMXvngPOl8OsQ19uPa/A9LPh5y3YOrV4IrLv/aGF6HjpfDZq/DKHdCgaShc\nk/zji+6EpAzYtR62f1Yaug2a+pZtVPTXq7eKWqE1UbW3nEVEJMycg72bYNPHsOkj6DQEOl0GmaNh\n0RN+m8/mhEK2KRw54Jclt/et1/rJZQI4GVI6+fVdR/jbiSS387dwCEgrtCZSOIuIBEnBYXjpez6U\n92/xy2Ib+lZtTD1Y9Qqc9z1/1vLYJ6D9wPKvT25XeuZypFV0LLzdAAVzJSicRUQi4eBOyA21ijd9\n7M9IvvJvEBvvL+XJuBDSzoO0vtDsLNg4v3yXcNeRdbqLuLZTOIuIVLXiYtiXC4mhwZeeGwefzfaP\no2Kh5dmlZyMDfOu14/ehLuI6ReEsIhJuRw7A5kWlx4tzP4biIvjpBoiOgY6DoU0WpJ0PrXqVDsBx\nMuoirlMUziIiZ8I52LPRB3GXof6a3Pn/B//5g1+f2g3OutJ3UbsiIAayboloyRJ8CmcRkdO1byus\neLH0ePH+rX75+JegwyXQY6y/rrhNFtRPjGytUiMpnEVETubgjtLu6Q6X+LOjD3zpx2BObOuHo0zr\n61vGzTL9a1I6+pvI16RwFpG651RDYRYcgtl3+kDetdavj4qFhik+nJv3gLtWlx+TWSSMFM4iUveU\nDIU55u8QEwdLp/lbp8v8+ph4+HIppHaFc27yreKWvfxlTuBP6lIwSxVSOItI3VNylvMzY0uXJWZA\nix7+sRnc+kFEShMBqKJpPUREAmbPRvjgz/6SJvDHh1ue7R/3ux3u+BQG3Re5+kTKUDiLSO2Vn+e7\nq58cBQ/3hDd+AVs+8evSz4e9uX62pE+m+mPOIgGhbm0RqZ22rYJHh8CRff6s6ovv9dMeJqYfPztS\nu4s0FKYEisJZRGqHfVth6XMQ28BPDJHSGc6+FrqNgrYXQlSZjkINhSkBp/mcRaTmKjwCq+fAkqmw\n9i0/h3HmGLjmyUhXJnIczecsInXDq3fCkqchoTX0/zH0Gld+AgmRGkrhLCI1w8EdsPR5+OQZ+MY/\noXkm9P2eH7e6/SCIio50hSJho3AWkeAqKoTPX/dnU695DYoLoVVvf5IXQMueka1PpIoonEUkeI7s\n97M7FeTBjG9BvUZw3vd9t3XzzEhXJ1LlFM4iEgyHdsPyF/zJXa4IvvcuxCfAt+f6AUOiYyNdoUi1\nUTiLSGTlLoIFj8Cq2VB0BJqdBb3H+ZG8oqJLR/ESqUMUziJS/XauhQbJUD8Jtq2EtW/7CSZ6jfNh\nbBbpCkUiSuEsItXjyH5YMdOf3LXxQ7j8AbjgB9DzGn+LqRfpCkUCI2zhbGaPASOBbc657uHar4jU\ncMVFMOt2WPGiP8GraScY/Ct/CRQolEUqEM6JL54AhoZxfyJSE7z/8PGTRiydDjO+7R9HRcPhPdBj\nLHz7DZi40A8YktCy+msVqSHC1nJ2zr1rZhnh2p+I1BCt+/hJI8b8Aw7tgg8nw5dLwaJhxENQPxGu\nnRrpKkVqFB1zFpEz024AXHAbPDMWcGBRcPb1cPE9PphF5LRV63zOZjbBzLLNLHv79u3V+dYiEk77\ntvru7LVv++c9rvHXIgP0vwuu/BsktY1cfSI1XLWGs3NuinMuyzmXlZqaWp1vLSJnqvCIP9t66lj4\n30x485ewdp5ft3s9HPgSBtwNix47/hi0iJwWdWuLSOU8MRJyP4bGreDCO/w1ySkdfRBPv7l0fuR2\nF5V/LiKnLZyXUj0LXAykmFku8Evn3KPh2r+IVKODO2DpNPhsNtzwIsTGQ/87ILoedDhmBqjNi8sH\ncbsB/vnmxQpnka/JnHMReeOsrCyXnZ0dkfcWkQoUFULOG35+5DVzobjAzwB19WOQ3D7S1YnUeGa2\nyDmXVZlt1a0tUtcVFfhJJbZ+Cs9eCw1T4bzvaQYokQhSOIvURYf2wPIZ8Mkz0KIHXPFnf73yDS/6\nrmjNACUSUQpnkbpk/Xuw6PHyM0C17OXXmUHHSyNbn4gACmeR2m/XekjK8OG7cibkvAV9bvTTMrbs\npRmgRAJI4SxSGx3ZDytfhiVTYeN8+NZcSD8fBt0PQ37nz74WkcBSOIvUJgd3wus/88FccBCadoRL\nf1l6tnWD5MjWJyKVonAWqen2bIK9m6BtP6jXGDZ8AD2ugl43QFpfdVuL1EAKZ5GaKD/PDxCy5Gk/\nQldSBty+BGLi4PZPIKpaR+YVkTBTOIvUNAsfhTd/BUf2QWK6n/3p7OtKW8gKZpEaT+EsEiTvP+yv\nNy477OWKmfDJ0zD8IT/TU5M06DLcn23dtr/CWKQWUjiLBEnrPn7SiG/8E/IPwAd/gc2hYW7XvgVZ\n34LOQ/xNRGothbNIkLQb4IP56asABxYF3a/2XdcpnSJdnYhUE/WHiUTawZ2w4G8w63b/vOOl0P5i\n/7j/nXD1owpmkTpGLWeRSCiZAeqTqbD6tdIZoPLzfDf2l0thwN2Q/Si0H6ipF0XqGIWzSHVyzp9V\nveQpmP1jaJASmgHqemh+lr8savrNpfMjt7uo/HMRqRMUziJV7dAeWP6CbyX3uRHOuRkyx0Cj5tBp\nSPkZoDYvLh/E7Qb455sXK5xF6hCFs0hVWfu2H9v6s9lQeBiaZUK9BL+uQTJ0HXH8a/rfcfyydgMU\nzCJ1jMJZJJwO7oSGTf3jt38LO9dC7/G+27pVbw2lKSKVonAWOVNHDvipGJdMhS1L4K7PoH4iXPUo\nNG6pGaBE5LQpnEW+rp1r4b2H/AheBQchuQMM/Enp+uR2katNRGo0hbPI6dizCQqPQEpHcMWwchZ0\n/wb0vgHSzlO3tYiEhcJZ5FQKDsGq2X5863X/gcxRcM1TfmCQn+So21pEwk7hLHIy8x7wo3cd2QtN\n0mHgT6HXdaXrFcwiUgUUziJl7f8Slr8Ifb/rrz+OjoUuQ6HXOMi4SDNAiUi1UDiLFObDmn/7s61z\n3gRXBC3PhowLYcB/Rbo6EamDwhbOZjYU+DMQDfzLOfeHcO1bpMrsWgf/vBQO7fKXPV14u28la6IJ\nEYmgsPTRmVk08AgwDMgErjOzzHDs+2QWPPVzlv/l15CR4bsbMzJY/pdfs+Cpn1f1W3997z/sx08u\na/27fnmQ1cS6K6p51Wx45lp/HBkgMQPOuhLGzYAfr4DBv1Iwi0jEhesAWl8gxzm3zjmXDzwHjA7T\nvk+o0Z4YWu+YwvLmceAcy5vH0XrHFBrtCXBvfes+fiKDktAomeigdZ9IVnVqNbHukprXzoM1c+Gx\nYfD8ON+FnfOm3yYqCkZOgk6XQVR0RMsVESlhzrkz34nZ1cBQ59x3Qs/HA+c55yae6DVZWVkuOzv7\nzN44I8MH8uUH2X2kIRnx2ygsMOKKHDRtCg2awsSP/bYv3Qqfv17+9YlpMOEd//j58bBhfvn1qV3h\nllf94//3Ddj6afn1rc+BcdP840cvh5055de3GwBjH/eP/94f9m31j4vy4ch+aNEd9m3xExu88iM4\nvK/863tcDcMe9I//1BmKi8qvP+cmuPQX/lKf/+1+/OdzwQ/gorvg4A545Lzj1w+828+ItHsD/POS\n49df9mt//e62VfDEyNK6Y+L9WNEDfgKX3A+52fDMN49//Zi/QufLYd07MOPbx6+/5knI6A+fvVo6\nl3FZN8zwQ14unQ6v3XP8+m+95lu52Y/7oTKP9b13YddaX1tBHmDQbZT/uVtU8HmJiABMnQr33w8b\nN0J6OvzudzBu3Bnv1swWOeeyKrNtuJqYFY28cFzqm9kEYAJAenr6mb/rxo103+BY0PUczm/3OUuL\nM/g0qiMph/bQIb8+bTt3o17JtmnnQmz98q9vkFz6uG0/aJhafn1Cy9LH7QZAUkb59WVHgOpwiZ/y\nr6zUrqWPOw3xsxOV2LwItn7i5+xtNwC6DPchW1ar3qWPu43yg16U1aKHv7doyKygoyK1m7+PqVfx\n+qYd/X1co4rXl/y89RJK15fU3bKX/5kA6idV/PrGoc+vUfOK1zds5u8TWlW8vn6Sv09Mr3h9vcb+\nPrl9xetj6/vPtusIWDYd+t87oB8KAAAdSklEQVQJg39x/HYiUnWqKOjCorgYDh+GI0f8/eHDMGMG\n/OIX/jHAhg0wYYJ/XI11h6vlfAHwK+fc5aHn9wI45x440WvC3XJevaElndtu5aWV5zEtcTCrU9oy\ncvUHTG6wAW65BTdoEBYdkG7Lki7hrG9D9qM1Z67emlh3TaxZpLaYOtUHW15e6bIGDWDKFB90RUWl\noXiyW9nwDOetoKDyP0vbtvDFF2f0cZxOyzlc4RwDrAEuBTYDC4HrnXMrTvSacITz8r/8mtY7prB5\nbkO6f/w5y/t2ovXlB9ncdALF3YYQ+8rLdHv6H6yzBowf9wDfaHyIq8cOoG3vbmf0vmekJCxKQuLY\n50FVE+uuiTWL1ATOwf79sGsX7N59/H3J46lTywdzCTOIjobCwjOrwwzi40/vVq/eydffcov/+Sp6\nr+Li45efVrnVHM6hNx0OPIy/lOox59zvTrZ9OMJ5wVM/p9GeGLpPevxol8nyO2/hQGIh59/4336j\nw4dZ+ewr/GHhdt5rnIazKPru28TYzk244qbhxCc1OaMaTtv7D/sTlcqGw/p3YfPiiufyDYqaWHdN\nrFnkZMLdRXzoUMXhemzIHnu/Z49v9Z5IXBwkJ8OXX554m/vuO/NgjY0N/3j2GRm+K/tYNbHl/HWE\npVv7NG39bB0vPvs2M3ZEs7l+Igsf/z5NRo9gx/U30fSSizCN/iQiQXWiLuK//x2GDv16IXvkyInf\nLyoKEhN9yCYl+VvJ41Pd16/vQ7MKg67KnKor/gwonE/BFRez/o33aT/9KXj+ecZc+St2JzTl6uQC\nrrr2ElpldohIXSIiFdq5EzIzYdu2039to0YnDtGTBWxCwpkPV1uFQVelAnC2dp0M57LcgQO89Nhs\npq/azYdN0jFXTP/9m/herxT6j7/CfwMUEakue/fC4sWwcCFkZ/vb+vUnf81f/nLi8I2NrZ66TyTI\nZ2tXM4Xz17Tpk8+YMf1dZuyL57b3nuH6Lxaw97rxrB99LWcPuUDd3iISXgcPwpIlpSG8cCGsWVO6\nPiMDzj0XsrLgoYcqbjkHuYtYyonEdc61Qlqvrvy4V1d+VFhE0Tut4cknmLV4Ez9vsodOrzzO2ObG\nmHGX0axDWqRLFZGa5vBhWLq0NISzs2HlytIzgFu39iE8frwP5HPOgZSU0te3bl1xF/HvTnrurdRQ\najmfwr6vdvLq068xPecAi5u0Ibq4iEEHNvHIJS2pN2qkPytRRKSsggJYsaJ81/SyZaXX1aamlraI\nS24tW558n6Au4hpO3dpVJGfBUmbMnM/GL77ir8//ClJSmHbjTzhr5MWcNahvpMsTkUgoKoLPPivf\nNf3JJ6VnQicmlgZwSSCnpYX/EiAJPIVzVSsshDfe4NATT3Fuq29woF4DMvdtYWybOMaMH0JSWiW+\nAYtIzVNcDGvXlm8RL17sjx2DPzO6T5/yreIOHRTEAiicq9Xu3C+Z9fRcpm/MZ3lCK2KLCnho+weM\nGnsxXH45xOiwvkggnaqL2Dl/jW7ZFvGiRf5savCDYPTuXb5V3LmzH/lKpAIK5whZ9Z9sZsxeyE0v\nTiZ93Ure63MJ7w8ey9gxF9DxgrMjXZ6IlKjo+tv69eHWW/1JViWBvGOHXxcbCz17lu+azsyM/GVK\nUqMonCMtPx/mzOGRWUuYlHIORVHR9NqXy9h2Ddne7Wz6bl5Jvwd+evQb+/x7H2Rp1yy+P1CDn4hU\nixONXAW+5XvWWeVP1urZ0w8dKXIGdClVpMXFwZgx3DZmDNesy2Xm1DeYvi+G+3cm0frVhTwVG89k\nl0A/55jvEpi4spjJZIPCWaRq5ebC9OknDmYz2LfPt55FIkgt52riiotZ9uZH7Lzzburt38tto+8h\ntqiQffEN+cO//8KYvA0aSECkKnz1lZ+j9/nn4b33/LLY2IqnC9SAHlKFTqflrCGvqolFRdFzyAUM\nWvkB/TYu48oV89jWuCmHY+pxx6i7+Wa/7zPjz8+St2tvpEsVqfl27PDjN196KbRqBRMn+vGpf/1r\nf9nT448f3zrWgB4SIOrWrm7p6cx3Ccw8axC3f/AsT/UZweVrPuSj9B7819YE0s8fTN/+PcgbfzP1\nB/bXkKEilbVnD8yc6VvIb7zhrz/u1MlPTfjNb0L37qXbduni7zWghwSUwrmazb/3QX+MeeYD9Nu4\njPM3LmXimHuZ3C2Keq0b0mfH2fD88/xhUz3efWEdVycX8o1rB2mmLJGK7N8Pr7wCzz0Hc+f6kzHb\ntoW77oJrr4VevU58jfG4cQpjCSyFczVb2jWLyWTT7+V9YEY/28fkzKjSs7WvuBj+/GfOf/wV1qzc\nzZ/y03noyZX03/8643qmMPSmkZopS+q2vDx49VXfQn71VT9mdevWcNttvoXct68G/ZAaTyeEBdzG\n0ExZL+yN56K12fzhw6dw113Hiitv4KzBmilL6ogjR+C113wgz5rlR+Rq3hyuvtoH8oUXnvncwyJV\nTJdS1SLpvbpyZ6+u3FFYxMG3WkHTXXzy7w+4sslIOr2smbKkFisogDff9F3WM2f6S5yaNvVd0d/8\nJgwcqNG4pNZSy7kG2r9tJ6/8v9eYvvYASxL8TFkXH8zltxe1pOU3RmimLKm5CgvhnXd8C/nFF2HX\nLmjSBK680h9DvuQSjcolNZZGCKtDchYsZfrM+by1J5rZj04kPjGBD8b/kKRRw8i8+NxIlydyasXF\n8P77PpBnzIBt2/wEEqNH+xbykCEanUtqBYVzHeQKCrA33oDHH2dowsV8lprBWfu2MDYtjtHjLyep\nTYtIlyhSyjlYsMAH8vTpsGWLP9Fx5EgfyMOH68RHqXUUznXc7twvefn/zWX6pnxWJLQirrCAO/Yu\n5QdXZvlWiGbKkkhwzk+v+Pzz/rZxo28RDxvmA3nkSN9iFqmlFM5y1Mp3FjJjdjYXvv0Cly55iy0d\nMnnyqh8y9sp+dDy/Z6TLk9rOOVi2zIfxtGmQk+O/HA4Z4gN59Gh/TFmkDlA4y/Hy8+HVV5n1wrv8\nuNUgiqKi6b0vl7HtGzJy/DASmjeNdIVSkx07N/Jtt/nrkZ9/Hlat8pc5XXqpD+Qrr4Tk5EhXLFLt\nFM5yUtvWbuLlqW8w/UvHmoQWNMrP46Ptr9Lw5vEwaJCuF5XTU9HcyCUGDvSBfNVV0KxZ9dcmEiDV\nGs5mNhb4FdAN6Oucq1TiKpwjzxUXs/T1D1n67/cZ/9QfYM8e7rrmZ7TumMbV1wwk/ewukS5Rgqyo\nCD76yB8z3rfv+PWtW/spGkUEqP5w7gYUA/8A/kvhXEMdPkzBSzP5zns7ebdxOs6iOG/fJsZ2SWT4\njcN5aul2en6WTb8Hfnq063L+vQ+WDjsqdcOBA/D6634869mz/exPJ2LmL5MSEaCaRwhzzq0KvemZ\n7koiKT6e2Ouu5cnrYMvKtbz43DxmRMXzX1sbs3fkd+jZtB63ZQznEZdAP+eY7xL8BB5kg8K5dtu0\nyQfxrFnw9tv+/IWkJH+50xVXwE9+4rc5Vnp69dcqUkvomho5TqvMDkz8TQduKy5m4Sv/odP2ZJIe\nn8I1A6K54Zu/pe+m5XzWrB1/nfmAn8Dje9+MdMkSTiWXPM2a5VvIS5b45R07+nmRR43yY1mXXJJX\nWHj8MWfNjSxyRioVzmb2JlDRKBb3O+deruybmdkEYAJAur5VB55FRdF39CAYPQgen0K/jcuY3W0A\nC9qeDc7xt/PHsmPZm1xx6BCmASNqtkOHfKv4lVf8bcsWf2Jgv37wxz/6FnKXLhXP9lQy7aLmRhYJ\nm0qFs3NucDjezDk3BZgC/phzOPYp1SQ9ndiiAg7FxnPjoleY1vMyVqW24+/nXcWoVq3guuvYdM2N\ntBnQVzNl1RRffeWnXJw1C954w7d8GzWCoUN9GA8fDikplduX5kYWCSt1a0ulzL/3QX+MeeYD9Nu4\njKFr5jNxzL3c1jwfhg9n/9TnuKz+paRPe5yxLaMYM+4yUtu3iXTZUpZzsGJFaXf1Rx/5ZenpcMst\nvrt64ECNYy0SAGfcxDGzK80sF7gAeNXM5p55WRI0S7tmMTkzin62D8zoZ/uYnBnFpmFXwtSpxOR8\nzi9aHqIhhfzuYDPO//tivnPbX1k1daY/gUgiIz/fT7v4ox9B+/bQo4fvfi4qgt/8Bj75BL74AiZP\n1gQTIgGiQUgk7HI+/JTpMz/kpUONefrZ++hMHjnjJ1Aw+kq6DazUVQRyJnbtgn//27eQX3vNX4Mc\nHw+XXea7q0eOhJYtI12lSJ2jEcIkEIryC4h+43V4/HF+nN+OlzIvpvu+LYxNj2P0+KEktm4e6RJr\nj88/L+2ufv993zJu0cIH8RVXwODB/gxqEYkYhbMEzq5NX/Ly03OZUWamrOsO5vDrKzI1U9bXUVgI\nH37ow3jWLFi92i/v2dOH8ahRkJWloVhFAuR0wln/c6VaJKe14JZ7b+LVv36XOUObMY4tNFuzHEaM\noCgjgz/f81fWfrws0mUGw9SpkJHhgzUjwz8H2L8fZsyAG2/0reIBA+Dhh/0JXf/3f/7Y8aefwm9/\nC337KphFajC1nCVy8vNh9myWPTebMRljKIqKps/eXMZ2bMTIG4fROLUOzlxU0SQSsbH+GuPVq6Gg\nwM/oNGKEbyFffjkkJESuXhGpNHVrS42zbe0mZk59g+lfOT5v3IL4giPM3PkWXcd/o+7MlLV7N2Rm\nwpdfHr8uJgbuuMN3V19wgQ4DiNRACmepsVxxMZ++/iFz5i7ip0/8kug9e/jXkFvYl3U+Y68ZSFpN\nnykrPx/WrvWt4DVr/H3JTZNIiNRqCmepHQ4dgpkzufPtzbyU3AVnUZy/byNXd01m+PhhNEgKaHeu\nc7B16/EBvGYNrF/vz6Qu0by577IuuT34IGzffvw+27b1x5RFpMZSOEuts3nlWl56bh4zdsXwRaNU\nrlr1Hx5K2Qm33ILr1y8yQ4YeOOAvYSrb+l2zxt/27y/drn596NzZ38oGcefO0KRJ+X1WdMy5QQOY\nMkXDY4rUcApnqbVcaKashNkv0/XZf7G6fgq3XvNLrkop4qrrLqVF13bhfcOiItiwoXz4ljzevLl0\nOzN/1vSx4dulC7Rpc3rHzKdO1SQSIrWQwlnqhgMH+OTpl/n90v18nJBGVHER/Q/kMrZncy6/YRiP\nLdpKz8+y6ffAT48G3fx7H2Rp1yy+f+wc1Dt3VnwcOCen/PCjiYnHh2+XLn46Rc3MJSIncTrhrFM+\npeZq1Ihe3x/HNGDDklXMmP4eL0Q35u41MKh9B3peeDm3pV/OIy6Bfs4x3yUwcUURk1dPhQ/jyreG\nd+4s3W9MDHTo4EN3xIjyYZyaWvG0iSIiYaSWs9QqRYVFrJ39Fp2nPwnPPMPF3/0HmxJbcPaWNaxJ\nTeePcx5m+JoP/cYtWlTcCm7XTpcqiUjYqVtbBCiOiua5npcx+YJr2NKkdBzvmxe9wq9mPIhLSGD1\nV/vp1Kwx0VFqDYtI1VK3tggQlZ5Gxu4tHI6N5wcfPs/TvUdwxcp3GXhgIzRpwubdeQx9+D0a1Yuh\nd3oifdKTOKdtEn3aJtGonv5riEjk6C+Q1Frz732QiSuLmTzzAfptXEb/Lz5h4ph7GZE5AIAm9WN5\n+Ju9WLRhN4s27Ob/3v6cYgd/vrYXo3u1ZtOuPD5ev4tz2ibRtmkDTMeaRaSaKJyl1lraNYvJZNPv\n5X1gRj/bx+TMKJZ2zaIf0Dg+ljG9WzOmd2sADhwp5NNNe8hs6Qc3eWf1Nn7+8goAUhrFHW1ZX9s3\nnSb1YyP1Y4lIHaBjziInUFzs+HzbARZt2E32hl0s3rCbjbvy+PSXQ2gcH8v07E2s+Wq/7wpPT6JZ\nQnykSxaRANMxZ5EwiIoyurRoTJcWjbn+vHQA9uTl0zjet5pXf7mfpxZs4J/vrQcgLbk+/Tum8sA3\nekSsZhGpHRTOIqchsUHc0cc/G5nJ3UO7snzLXhaHjlvvP1xwdP03//EhMdHGOelJnJORTK+0RHWH\ni0ilKJxFzkBcTBR90n239ncuKl3unCOzVQIfrdvF5Hk5FDs/dskt/drxiysyAdi4M4+05PrlTjT7\n+3/W0rNNE/p1SDm6bP7aHSzN3Xv8qGYiUmspnEWqgJnxyyvOAkpPNFu8YTedWzQG4Kt9hxnwP/No\n2jCOPm39iWbntE2ia4vGTHxmCZOv702/DinMX7vj6HMRqTt0QphIBOw9VMCcZVuPXsa1fsdBACZd\nczYtmsTzg6cX07NNExZt3M0PL+nEhR1SSGwQS4sm8cRGR2AGLhE5YxohTKSG2XngCIs37qFXWiKp\njetxy+MLmbd623HbzZp4IT3bJPLyJ5v52ztrSWwQS1KDOJIaxpHUIJbv9G9PUsM4cnfnsW3/EZJD\n6xLiY3SdtkiE6WxtkRqmaaN6XJbphxidv3YHn27aw3cvase07Fx+PLgTrZMasPtgPm2TGwKQUD+W\ntOQG7MnL5/NtB9h9MJ89hwq46YIMAGYsyuXhNz8/uv/oKCOpQSxv3XUxTerH8tKSXBas3UViQx/u\nyQ3iSGwQy+BuzYmKMg7lFxEbbcRUopWu4+Qi4adwFgmQo8eYx/ljzoO6Njt6zPmyzLSj2w3q0oxB\nXZqVe21xsTs6YdZVfdpwdptEdufls+tgPnvyCtidl390WNINO/OYt3obe/IKyC8qBvzJbav/eygA\nP5u5nBcW59KkfixJDWJJahhH68T6TL6+DwBzV3zJzgP5JDWIJSbKuPXpxTx4VQ+Gdm9ZI46T6wuF\nBN0Zd2ub2f8AVwD5wFrgFufcnlO9Tt3aIser7tBwzpGXX8Sug/nsP1xIZis/Otpbq75iae5edufl\nszuvgN0H84mLieKxm88F4IZ/fcT7OTvK7Sva4LZBHXn6o420bBJP7u5DxMdGUT82mvjYaHq2acIf\nrz4bgN/PWcWOA0eoHxt9dH3HZo2OjtY2d8WXFBc74kPr6sdF07RhHGnJDQDYf7iAejHRxEbb1+qu\nL/sF4tgT78p+9kGiLxQ1X7UeczazIcDbzrlCM3sQwDn301O9TuEsUnMdLihiT15BqFXuA/zVpVuY\ns/xLbr+kI0kN49iwM49D+UUcKvC3dikNuW94NwBufvxjPv/qAIcLijgcWj+wcyqP39IXgPN+/yZf\n7TtS7j1H9GzJI6GWe49fzWX/4UKioywU7lF8o0+bo/u/dsqHxMVEEx8TRf04/wVgYOdUhvVoSUFR\nMY9/sJ4tew4zPXsT/Tul8EHOTn51RSZXZ6WRX1jMp7l7iI2OIibKiIuJIjY6iqaN4kiIj6Wo2JGX\nX0hstF9eXTOa6QtF9ajKmqv1mLNz7vUyTxcAV5/pPkUk2OJjo2nRJJoWTfyQpfPX7mDB+l3cfolv\nOU++vje3XNjuhK9/IhTCJZxzFBWXNhRmfL8fB/MLOZRfxOGCYg4XFNG0UekAMHde1pmDRwo5XFDM\noVDAd2nuL1Mr2c++QwVsKxP+LZrEM6xHS/KOFPH7OZ8d3dfcFV8BsHnPYQB2HjzC2L9/eFzNPxvR\nje9c1J71Ow4weNK7R5dHGcRGR/HbMd0Zm5XGyi37+O5T2aFQNx/y0VH8ZEgX+ndKYeWWfUx6Y83R\ndbHRUcTFGDf3a0eXFo3J2baflz/ZcnRdbLT/gnD5WS2YfH1vbn16MRd0aMr7n+/guxe1Y//hQt5Y\n+RXntU8mIT6WzXsOkbPtAFEGUWZY6L5XWiLxsdF8te8wW/YcIsqs3PrOzRsREx3FzgNH2J1XcPT1\nJdu0TqxPVJSx/3ABhwqKjq6LMn/pYMkAO/mFxRQ7R5QZZ7VKYOLU0BeIjjXj0sCebZoE4nLGcB9z\n/hbwfJj3KSIBdmwL7vwOTU+7RWdmxESXtkBLuq9P5GTBHx1lPDfhghOuT6gfw4pfX857n2/npy8s\nY9TZLZn16VY6Nfcn2yU1iGPqd84jv6iYgsJiCoocBUXFnBXq8k9uWI/7h3ejoLiYgkK/rqComE6h\nLwcN4qI5r30yBUWOwtC6/CJHbOjnO1xYxJY9h46+rmT/V5zdCmhMzraDTJ6Xw7Gdmme1SqBfhxTO\nzUjiteVfAvC/ZU76+/ePLiKhZSxvrvyKX85acdzP/d7dg0hLbsALi3P542urj1u/+OeXkdwwjsc+\nWM8j89Yet/6z/x5KfFQ0D72+hifmf1FuXUyUkfP74QDc99IyZizKLbd+3L8+4oehL26dmzfipsc+\nxkLBHmVGq8T6vHnnQABum7qY+Wt3hL4U+C8GHVIbHv2d3vr0IpZv2Vv6xQHIbJVw9HyIW59exBc7\n88p9OemdlsivR3c/uv/t+49gxtEvJudmJPPjyzoD8OzHm2ib3ICbHvuY6/um88rSrRHpnahUOJvZ\nm0CLClbd75x7ObTN/UAhMPUk+5kATABIT08/7WJFJHiW5u4t98erX4cUJl/fm6W5ewPZ3WpmfJq7\nh/teWs7fbuhDvw4pDOvRkonPLCGxQRz9OqRwYccT153cMI7vDmh/wvUZKQ2ZdE2vE67vk57EnB9d\ndML1Q7u3YP0DIygqdqFg918SGsfHMn/tDhZt2M2489KZvXQr9w7rSo82TXAOMpr6LxfDerSge+sm\nOOcodlDsHMXOkdq4HgAjerSkW8sEv764ZD00rBcNwMierejcvDEu9NqS+5Lr60f0bEmHZo2gzP7L\nGt6jBe1TG/rXFfttPlq/k7+8ncPtl3SkQ7NG9E5P8q8Lvb5kvHqA89onk9wwDod/rXOO1Malk8p0\na5lAfGx0uZ+v5GcHaJ4QT0GRA0rXN4ovjbqYaCM6yvzPXQxF+C9JJXYdPEJhsSOxfixPfriB2y/p\nGJF/x2G5ztnMbgK+D1zqnMurzGt0zFlEIqUmHguticecobTuG85LP3rII8j1QtXVXN0nhA0FJgED\nnXPbK/s6hbOISOXpC0X1qMqaqzucc4B6wM7QogXOue+f6nUKZxGR2q0mfqEIytnaGr5TRESkGpxO\nOGsEfRERkYBROIuIiARMxLq1zWw7sCGMu0wBdpxyKwkHfdbVQ59z9dDnXD30OUNb51xqZTaMWDiH\nm5llV7YvX86MPuvqoc+5euhzrh76nE+PurVFREQCRuEsIiISMLUpnKdEuoA6RJ919dDnXD30OVcP\nfc6nodYccxYREaktalPLWUREpFaoFeFsZkPNbLWZ5ZjZPZGupzYyszQzm2dmq8xshZn9KNI11WZm\nFm1mS8xsdqRrqa3MLNHMZpjZZ6F/1yeeZ1LOiJn9OPR3Y7mZPWtm8ad+Vd1W48PZzKKBR4BhQCZw\nnZllRraqWqkQuMs51w04H7hNn3OV+hGwKtJF1HJ/Bl5zznUFzkafd5Uws9bA7UCWc647EA1cG9mq\ngq/GhzPQF8hxzq1zzuUDzwGjI1xTreOc2+qcWxx6vB//h6x1ZKuqncysDTAC+Feka6mtzCwBGAA8\nCuCcy3fO7YlsVbVaDFDfzGKABsCWCNcTeLUhnFsDm8o8z0WhUaXMLAPoDXwU2UpqrYeBu4HiU20o\nX1t7YDvweOjwwb/MrGGki6qNnHObgT8BG4GtwF7n3OuRrSr4akM4WwXLdAp6FTGzRsALwB3OuX2R\nrqe2MbORwDbn3KJI11LLxQB9gL8553oDBwGdr1IFzCwJ35vZDmgFNDSzGyJbVfDVhnDOBdLKPG+D\nukyqhJnF4oN5qnPuxUjXU0tdCIwysy/wh2guMbOnI1tSrZQL5DrnSnp/ZuDDWsJvMLDeObfdOVcA\nvAj0i3BNgVcbwnkh0MnM2plZHP5Eg1kRrqnWMTPDH59b5ZybFOl6aivn3L3OuTbOuQz8v+W3nXNq\nZYSZc+5LYJOZdQktuhRYGcGSarONwPlm1iD0d+RSdPLdKcVEuoAz5ZwrNLOJwFz8WYCPOedWRLis\n2uhCYDywzMw+CS27zzk3J4I1iZyJHwJTQ1/q1wG3RLieWsk595GZzQAW46/6WIJGCzsljRAmIiIS\nMLWhW1tERKRWUTiLiIgEjMJZREQkYBTOIiIiAaNwFhERCRiFs4iISMAonEVERAJG4SwiIhIw/x92\nNkUOBhyXKgAAAABJRU5ErkJggg==\n",
342 | "text/plain": [
343 | ""
344 | ]
345 | },
346 | "metadata": {},
347 | "output_type": "display_data"
348 | }
349 | ],
350 | "source": [
351 | "plt.figure(figsize=(8,6));\n",
352 | "plt.subplot(2, 1, 1)\n",
353 | "plt.step(I_obs, '-bx', where=\"post\")\n",
354 | "plt.subplot(2, 1, 2)\n",
355 | "plt.plot(Y_obs,'-or')\n",
356 | "\n",
357 | "# Plot non-zero observations\n",
358 | "for i in np.nonzero(np.any(Y != 0, axis=0))[0]:\n",
359 | " plt.plot(Y[:,i],'--x')"
360 | ]
361 | },
362 | {
363 | "cell_type": "markdown",
364 | "metadata": {},
365 | "source": [
366 | "Unfortunately, the process is rarely at steady state when control is initiated. If we somehow knew the past 10 moves in $u$, we could get a series of 10 future predictions of $y$ formulated by extending the preceding logic\n",
367 | "and going back in time. \n",
368 | "\n",
369 | "Let us call this the vector $y_p$"
370 | ]
371 | },
372 | {
373 | "cell_type": "code",
374 | "execution_count": 12,
375 | "metadata": {
376 | "collapsed": true
377 | },
378 | "outputs": [],
379 | "source": [
380 | "U_hist = np.array([1,0,0,0,-1,0,0,0,0,0])"
381 | ]
382 | },
383 | {
384 | "cell_type": "code",
385 | "execution_count": 13,
386 | "metadata": {},
387 | "outputs": [
388 | {
389 | "name": "stdout",
390 | "output_type": "stream",
391 | "text": [
392 | "[[-0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 ]\n",
393 | " [-1. -1. -1. -1. -1. -1. -1. -1. -1. -1. ]\n",
394 | " [-1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 ]\n",
395 | " [-1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 ]\n",
396 | " [-1.9 -1.9 -1.9 -1.9 -1.9 -1.9 -1.9 -1.9 -1.9 -1.9 ]\n",
397 | " [-1.95 -1.95 -1.95 -1.95 -1.95 -1.95 -1.95 -1.95 -1.95 -1.95 ]\n",
398 | " [-1.98 -1.98 -1.98 -1.98 -1.98 -1.98 -1.98 -1.98 -1.98 -1.98 ]\n",
399 | " [-1.99 -1.99 -1.99 -1.99 -1.99 -1.99 -1.99 -1.99 -1.99 -1.99 ]\n",
400 | " [-1.999 -1.999 -1.999 -1.999 -1.999 -1.999 -1.999 -1.999 -1.999 -1.999]\n",
401 | " [-2. -2. -2. -2. -2. -2. -2. -2. -2. -2. ]]\n"
402 | ]
403 | }
404 | ],
405 | "source": [
406 | "A_hist = np.tile(a, (U_hist.shape[0], 1)).T\n",
407 | "print(A_hist)"
408 | ]
409 | },
410 | {
411 | "cell_type": "code",
412 | "execution_count": 14,
413 | "metadata": {
414 | "collapsed": true
415 | },
416 | "outputs": [],
417 | "source": [
418 | "row_shift_indices = np.atleast_2d(np.arange(a.shape[0])).T + np.flip(np.arange(A_hist.shape[0]))\n",
419 | "row_shift_indices = np.triu(row_shift_indices)\n",
420 | "col_indices = np.arange(A_hist.shape[0])"
421 | ]
422 | },
423 | {
424 | "cell_type": "code",
425 | "execution_count": 15,
426 | "metadata": {},
427 | "outputs": [
428 | {
429 | "name": "stdout",
430 | "output_type": "stream",
431 | "text": [
432 | "[[-2. -1.999 -1.99 -1.98 -1.95 -1.9 -1.8 -1.5 -1. -0.5 ]\n",
433 | " [-2. -2. -1.999 -1.99 -1.98 -1.95 -1.9 -1.8 -1.5 -1. ]\n",
434 | " [-2. -2. -2. -1.999 -1.99 -1.98 -1.95 -1.9 -1.8 -1.5 ]\n",
435 | " [-2. -2. -2. -2. -1.999 -1.99 -1.98 -1.95 -1.9 -1.8 ]\n",
436 | " [-2. -2. -2. -2. -2. -1.999 -1.99 -1.98 -1.95 -1.9 ]\n",
437 | " [-2. -2. -2. -2. -2. -2. -1.999 -1.99 -1.98 -1.95 ]\n",
438 | " [-2. -2. -2. -2. -2. -2. -2. -1.999 -1.99 -1.98 ]\n",
439 | " [-2. -2. -2. -2. -2. -2. -2. -2. -1.999 -1.99 ]\n",
440 | " [-2. -2. -2. -2. -2. -2. -2. -2. -2. -1.999]\n",
441 | " [-2. -2. -2. -2. -2. -2. -2. -2. -2. -2. ]]\n"
442 | ]
443 | }
444 | ],
445 | "source": [
446 | "M = A_hist[row_shift_indices,col_indices]\n",
447 | "M[np.tril_indices(M.shape[0])] = a[-1]\n",
448 | "print(M)"
449 | ]
450 | },
451 | {
452 | "cell_type": "code",
453 | "execution_count": 16,
454 | "metadata": {
455 | "collapsed": true
456 | },
457 | "outputs": [],
458 | "source": [
459 | "y_initial = 0"
460 | ]
461 | },
462 | {
463 | "cell_type": "code",
464 | "execution_count": 17,
465 | "metadata": {},
466 | "outputs": [
467 | {
468 | "name": "stdout",
469 | "output_type": "stream",
470 | "text": [
471 | "[-0.05 -0.02 -0.01 -0.001 0. 0. 0. 0. 0. 0. ]\n"
472 | ]
473 | }
474 | ],
475 | "source": [
476 | "Y_pred = np.matmul(M,U_hist) + y_initial\n",
477 | "print(Y_pred)"
478 | ]
479 | },
480 | {
481 | "cell_type": "markdown",
482 | "metadata": {},
483 | "source": [
484 | "The preceding convolution model will be wrong due to unmodelled load disturbances, nonlinearity in the models, and\n",
485 | "modelling errors. One way to correct for these is by calculating a prediction error $b$ at the current time and then adjusting the predictions. In equation form:\n",
486 | "\n",
487 | "$$ b = y_0^{actual} - y_0^{pred} $$"
488 | ]
489 | },
490 | {
491 | "cell_type": "code",
492 | "execution_count": 18,
493 | "metadata": {
494 | "collapsed": true
495 | },
496 | "outputs": [],
497 | "source": [
498 | "y0_actual = 0"
499 | ]
500 | },
501 | {
502 | "cell_type": "markdown",
503 | "metadata": {},
504 | "source": [
505 | "Then the vector y^{pred} is corrected by $b$ for predictions 1 to 10 as \n",
506 | "\n",
507 | "$$ y_i^{pc} = y_i^{pred} + b $$\n",
508 | "\n",
509 | "Where $y^{pc}$ is the vector of corrected predictions."
510 | ]
511 | },
512 | {
513 | "cell_type": "code",
514 | "execution_count": 19,
515 | "metadata": {
516 | "collapsed": true
517 | },
518 | "outputs": [],
519 | "source": [
520 | "b = y0_actual - Y_pred[0]"
521 | ]
522 | },
523 | {
524 | "cell_type": "markdown",
525 | "metadata": {},
526 | "source": [
527 | "The equation becomes\n",
528 | "\n",
529 | "\n",
530 | "$$ \n",
531 | "\\begin{align}\n",
532 | "y_1 - y_0 & = a_1 \\Delta u_0 + y_1^{pc} \\\\\n",
533 | "y_2 - y_0 & = a_2 \\Delta u_0 + a_1 \\Delta u_1 + y_2^{pc} \\\\\n",
534 | "y_3 - y_0 & = a_3 \\Delta u_0 + a_2 \\Delta u_1 + + a_1 \\Delta u_2 + y_3^{pc} \\\\\n",
535 | "y_4 - y_0 & = a_4 \\Delta u_0 + a_3 \\Delta u_1 + + a_2 \\Delta u_2 + y_4^{pc} \\\\\n",
536 | "\\vdots & \\\\\n",
537 | "y_t - y_0 & = a_t \\Delta u_0 + a_{t-1} \\Delta u_1 + + a_{t-2} \\Delta u_2 + y_t^{pc} \\\\\n",
538 | "\\end{align}\n",
539 | "$$\n",
540 | "\n",
541 | "In vector form:\n",
542 | "\n",
543 | "$$ y = y^{pc} + A\\Delta u $$"
544 | ]
545 | },
546 | {
547 | "cell_type": "markdown",
548 | "metadata": {},
549 | "source": [
550 | "# Control Law\n",
551 | "\n",
552 | "From the preceding text, we have a set of\n",
553 | "future predictions of $y$ based on past moves\n",
554 | "and a set of future values of $y$ with a set of\n",
555 | "three moves. \n",
556 | "\n",
557 | "Let us now assume that we\n",
558 | "have a desired target for $y$ called $y^{SP}$. (We,\n",
559 | "of course, need this for any control action!)\n",
560 | "\n",
561 | "One criterion we can use is to minimize the\n",
562 | "error squared over the time between now\n",
563 | "and the future (that is, to use the least squares criterion). In vector equation form:\n",
564 | "\n",
565 | "$$ \\min (y^{sp} - y)^2 $$\n",
566 | "\n",
567 | "Substituting the previous equation:\n",
568 | "\n",
569 | "$$ \\min (y^{sp} - y^{pc} + A\\Delta u)^2 $$"
570 | ]
571 | },
572 | {
573 | "cell_type": "code",
574 | "execution_count": null,
575 | "metadata": {
576 | "collapsed": true
577 | },
578 | "outputs": [],
579 | "source": []
580 | },
581 | {
582 | "cell_type": "code",
583 | "execution_count": null,
584 | "metadata": {
585 | "collapsed": true
586 | },
587 | "outputs": [],
588 | "source": []
589 | }
590 | ],
591 | "metadata": {
592 | "kernelspec": {
593 | "display_name": "Python 3",
594 | "language": "python",
595 | "name": "python3"
596 | },
597 | "language_info": {
598 | "codemirror_mode": {
599 | "name": "ipython",
600 | "version": 3
601 | },
602 | "file_extension": ".py",
603 | "mimetype": "text/x-python",
604 | "name": "python",
605 | "nbconvert_exporter": "python",
606 | "pygments_lexer": "ipython3",
607 | "version": "3.6.7"
608 | }
609 | },
610 | "nbformat": 4,
611 | "nbformat_minor": 2
612 | }
613 |
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/DMC.py:
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1 | # -*- coding: utf-8 -*-
2 | # ---
3 | # jupyter:
4 | # jupytext:
5 | # text_representation:
6 | # extension: .py
7 | # format_name: light
8 | # format_version: '1.5'
9 | # jupytext_version: 1.4.0
10 | # kernelspec:
11 | # display_name: Python 3
12 | # language: python
13 | # name: python3
14 | # ---
15 |
16 | # # Implementing Dynamic Matrix Control in Python
17 | #
18 | # #### Author: Siang Lim, February 2020
19 | #
20 | # ** work in progress **
21 | #
22 | # - Contents of this notebook are derived from DMC literature:
23 | # - Hokanson, D. A., & Gerstle, J. G. (1992). **Dynamic Matrix Control Multivariable Controllers.** Practical Distillation Control, 248–271.
24 | # - Sorensen, R. C., & Cutler, C. R. (1998). **LP integrates economics into dynamic matrix control.** Hydrocarbon processing (International ed.), 77(9), 57-65.
25 | # - Morshedi, A. M., Cutler, C. R., & Skrovanek, T. A. (1985, June). **Optimal solution of dynamic matrix control with linear programing techniques (LDMC).** In 1985 American Control Conference (pp. 199-208). IEEE.
26 |
27 | import numpy as np
28 | import matplotlib.pyplot as plt
29 |
30 | # # Definitions
31 | #
32 | # #### Independent Variables
33 | #
34 | # - MV (Manipulated Variables)
35 | # - Variables that we can move directly, valves, feed rate etc.
36 | #
37 | # - FF (Feedforward Variables)
38 | # - Disturbances that we can't control directly - ambient temperature.
39 | #
40 | # #### Dependent Variables
41 | # - Variables that change due to a change in the independent variables, e.g. temperatures, pressures
42 | #
43 |
44 | # ## Step Response Convolution Model
45 |
46 | # The sequence $a_j$, where $j = 0,1,2,3 \dots$ is the step response convolution model, which is the basis for DMC.
47 | #
48 | # Remembering that everything so far is in deviation variables, at is no more than a series of numbers based on
49 | # a set length (time to steady state) and a set sampling interval.
50 | #
51 | # There is no set form of the model, other than that steady state is reached by the last interval. In fact, the last
52 | # value $a_t$ is the process steady-state gain.
53 |
54 | a = -np.array([0.5,1.0,1.5,1.8,1.9,1.95,1.98,1.99,1.999,2])
55 |
56 | # %matplotlib inline
57 | plt.plot(a, '-or')
58 | plt.xlabel("Time (s)")
59 | plt.ylabel("Response")
60 |
61 | # ## Inputs
62 |
63 | # For a lined out process with a single step in $u$ at time 0:
64 | #
65 | # $$ \Delta u_0 = u_0 - u_{-1} $$
66 | #
67 | # then
68 | #
69 | # $$ y_1 - y_0 = a_1 \Delta u_0 $$
70 | # $$ y_2 - y_0 = a_2 \Delta u_0 $$
71 | # $$ y_3 - y_0 = a_3 \Delta u_0 $$
72 | # $$ y_4 - y_0 = a_4 \Delta u_0 $$
73 | # $$ \vdots $$
74 | # $$ y_t - y_0 = a_t \Delta u_0 $$
75 | #
76 | # This is essentially the definition of the step response convolution model described before. What is interesting about this is that we can calculate what y will be at each interval for successive moves in u.
77 | #
78 | # For example purposes only, let us examine a series of three moves "into the future": one now, one at the next time interval, and one two time intervals into the future:
79 | #
80 | # $$ \Delta u_0 = u_0 - u_{-1} $$
81 | # $$ \Delta u_1 = u_1 - u_{0} $$
82 | # $$ \Delta u_2 = u_2 - u_{1} $$
83 |
84 | U = np.atleast_2d(np.array([0,1,0,0,-1])).T
85 | print(U)
86 |
87 | I_obs = np.pad(np.cumsum(U),(0,8),'constant')
88 | plt.step(I_obs, '-bx', where="post")
89 |
90 | # then
91 | #
92 | # $$
93 | # \begin{align}
94 | # y_1 - y_0 & = a_1 \Delta u_0 \\
95 | # y_2 - y_0 & = a_2 \Delta u_0 + a_1 \Delta u_1 \\
96 | # y_3 - y_0 & = a_3 \Delta u_0 + a_2 \Delta u_1 + + a_1 \Delta u_2 \\
97 | # y_4 - y_0 & = a_4 \Delta u_0 + a_3 \Delta u_1 + + a_2 \Delta u_2 \\
98 | # \vdots & \\
99 | # y_t - y_0 & = a_t \Delta u_0 + a_{t-1} \Delta u_1 + + a_{t-2} \Delta u_2 \\
100 | # \end{align}
101 | # $$
102 | #
103 |
104 | # +
105 | # Build dynamic matrix
106 | t = a.shape[0]
107 | u = U.shape[0]
108 |
109 | A = np.tile(a, (u,1)).T
110 | row_indices = np.atleast_2d(np.arange(t)).T - np.arange(u)
111 | col_indices = np.arange(u)
112 | A = np.tril(A[row_indices,col_indices])
113 | print(A)
114 | # -
115 |
116 | Y = np.multiply(A,U.T)
117 | Y_obs = np.matmul(A,U)
118 | U_obs = np.pad(np.cumsum(U),(0,a.shape[0]-U.shape[0]),'constant')
119 |
120 | Y
121 |
122 | np.nonzero(Y)
123 |
124 | np.nonzero(np.any(Y != 0, axis=0))[0]
125 |
126 | # +
127 | plt.figure(figsize=(8,6));
128 | plt.subplot(2, 1, 1)
129 | plt.step(I_obs, '-bx', where="post")
130 | plt.subplot(2, 1, 2)
131 | plt.plot(Y_obs,'-or')
132 |
133 | # Plot non-zero observations
134 | for i in np.nonzero(np.any(Y != 0, axis=0))[0]:
135 | plt.plot(Y[:,i],'--x')
136 | # -
137 |
138 | # Unfortunately, the process is rarely at steady state when control is initiated. If we somehow knew the past 10 moves in $u$, we could get a series of 10 future predictions of $y$ formulated by extending the preceding logic
139 | # and going back in time.
140 | #
141 | # Let us call this the vector $y_p$
142 |
143 | U_hist = np.array([1,0,0,0,-1,0,0,0,0,0])
144 |
145 | A_hist = np.tile(a, (U_hist.shape[0], 1)).T
146 | print(A_hist)
147 |
148 | row_shift_indices = np.atleast_2d(np.arange(a.shape[0])).T + np.flip(np.arange(A_hist.shape[0]))
149 | row_shift_indices = np.triu(row_shift_indices)
150 | col_indices = np.arange(A_hist.shape[0])
151 |
152 | M = A_hist[row_shift_indices,col_indices]
153 | M[np.tril_indices(M.shape[0])] = a[-1]
154 | print(M)
155 |
156 | y_initial = 0
157 |
158 | Y_pred = np.matmul(M,U_hist) + y_initial
159 | print(Y_pred)
160 |
161 | # The preceding convolution model will be wrong due to unmodelled load disturbances, nonlinearity in the models, and
162 | # modelling errors. One way to correct for these is by calculating a prediction error $b$ at the current time and then adjusting the predictions. In equation form:
163 | #
164 | # $$ b = y_0^{actual} - y_0^{pred} $$
165 |
166 | y0_actual = 0
167 |
168 | # Then the vector y^{pred} is corrected by $b$ for predictions 1 to 10 as
169 | #
170 | # $$ y_i^{pc} = y_i^{pred} + b $$
171 | #
172 | # Where $y^{pc}$ is the vector of corrected predictions.
173 |
174 | b = y0_actual - Y_pred[0]
175 |
176 | # The equation becomes
177 | #
178 | #
179 | # $$
180 | # \begin{align}
181 | # y_1 - y_0 & = a_1 \Delta u_0 + y_1^{pc} \\
182 | # y_2 - y_0 & = a_2 \Delta u_0 + a_1 \Delta u_1 + y_2^{pc} \\
183 | # y_3 - y_0 & = a_3 \Delta u_0 + a_2 \Delta u_1 + + a_1 \Delta u_2 + y_3^{pc} \\
184 | # y_4 - y_0 & = a_4 \Delta u_0 + a_3 \Delta u_1 + + a_2 \Delta u_2 + y_4^{pc} \\
185 | # \vdots & \\
186 | # y_t - y_0 & = a_t \Delta u_0 + a_{t-1} \Delta u_1 + + a_{t-2} \Delta u_2 + y_t^{pc} \\
187 | # \end{align}
188 | # $$
189 | #
190 | # In vector form:
191 | #
192 | # $$ y = y^{pc} + A\Delta u $$
193 |
194 | # # Control Law
195 | #
196 | # From the preceding text, we have a set of
197 | # future predictions of $y$ based on past moves
198 | # and a set of future values of $y$ with a set of
199 | # three moves.
200 | #
201 | # Let us now assume that we
202 | # have a desired target for $y$ called $y^{SP}$. (We,
203 | # of course, need this for any control action!)
204 | #
205 | # One criterion we can use is to minimize the
206 | # error squared over the time between now
207 | # and the future (that is, to use the least squares criterion). In vector equation form:
208 | #
209 | # $$ \min (y^{sp} - y)^2 $$
210 | #
211 | # Substituting the previous equation:
212 | #
213 | # $$ \min (y^{sp} - y^{pc} + A\Delta u)^2 $$
214 |
215 |
216 |
217 |
218 |
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/README.md:
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1 | # DMC-Python
2 |
3 | A series of educational notebook tutorials for implementing Dynamic Matrix Control (DMC) from scratch in Python.
4 |
5 | 
6 |
7 | ## Usage
8 |
9 | `!git clone https://github.com/csianglim/DMC-Python.git`
10 |
11 | `!pip install -r 'DMC-Python/requirements.txt'`
12 |
13 | Note to developers: `requirements.txt` is generated automatically using `ipyreqsnb`.
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/WhiskasModel.lp:
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1 | \* The_Whiskas_Problem *\
2 | Minimize
3 | Total_cost_of_ingredients_per_100g_bag_of_Whiskas: 0.008 BeefPercent
4 | + 0.013 ChickenPercent
5 | Subject To
6 | FatRequirement: 0.1 BeefPercent + 0.08 ChickenPercent >= 6
7 | FibreRequirement: 0.005 BeefPercent + 0.001 ChickenPercent <= 2
8 | PercentagesSum: BeefPercent + ChickenPercent = 100
9 | ProteinRequirement: 0.2 BeefPercent + 0.1 ChickenPercent >= 8
10 | SaltRequirement: 0.005 BeefPercent + 0.002 ChickenPercent <= 0.4
11 | End
12 |
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/assets/img/debut.png:
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https://raw.githubusercontent.com/csianglim/DMC-Python/c68b4c0cc069ccda81587023ac940b40d013dc38/assets/img/debut.png
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/assets/img/ranade.png:
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https://raw.githubusercontent.com/csianglim/DMC-Python/c68b4c0cc069ccda81587023ac940b40d013dc38/assets/img/ranade.png
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/main.png:
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https://raw.githubusercontent.com/csianglim/DMC-Python/c68b4c0cc069ccda81587023ac940b40d013dc38/main.png
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/requirements.txt:
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1 | ipywidgets==7.7.0
2 | matplotlib==3.5.2
3 | numpy==1.21.6
4 | pandas==1.3.5
5 | pulp==2.6.0
6 | ipympl==0.9.1
7 |
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/visualize_lp.ipynb:
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1 | {
2 | "cells": [
3 | {
4 | "cell_type": "code",
5 | "execution_count": 16,
6 | "id": "04ae301a",
7 | "metadata": {},
8 | "outputs": [],
9 | "source": [
10 | "import numpy as np\n",
11 | "limits = 10\n",
12 | "d = np.linspace(-limits, limits, 200)\n",
13 | "x,y = np.meshgrid(d,d)"
14 | ]
15 | },
16 | {
17 | "cell_type": "code",
18 | "execution_count": 17,
19 | "id": "ae3ed735",
20 | "metadata": {},
21 | "outputs": [],
22 | "source": [
23 | "G11 = -0.200\n",
24 | "G12 = -0.072\n",
25 | "G21 = 0.125\n",
26 | "G22 = -0.954\n",
27 | "\n",
28 | "CV1Lo = -1\n",
29 | "CV1Hi = 1\n",
30 | "\n",
31 | "CV2Lo = -2\n",
32 | "CV2Hi = 2.5\n"
33 | ]
34 | },
35 | {
36 | "cell_type": "code",
37 | "execution_count": 18,
38 | "id": "5f8c6d07",
39 | "metadata": {},
40 | "outputs": [],
41 | "source": [
42 | "c1 = G11*x+G12*y <= CV1Hi\n",
43 | "y_c1 = (CV1Hi - G11*d)/G12"
44 | ]
45 | },
46 | {
47 | "cell_type": "code",
48 | "execution_count": 27,
49 | "id": "7eec6df2",
50 | "metadata": {},
51 | "outputs": [
52 | {
53 | "data": {
54 | "text/plain": [
55 | ""
56 | ]
57 | },
58 | "execution_count": 27,
59 | "metadata": {},
60 | "output_type": "execute_result"
61 | },
62 | {
63 | "data": {
64 | "image/png": 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\n",
65 | "text/plain": [
66 | ""
67 | ]
68 | },
69 | "metadata": {
70 | "needs_background": "light"
71 | },
72 | "output_type": "display_data"
73 | }
74 | ],
75 | "source": [
76 | "import matplotlib.pyplot as plt\n",
77 | "plt.figure(figsize=(5,5))\n",
78 | "plt.imshow(c1.astype(int), extent=(x.min(),x.max(),y.min(),y.max()), origin=\"lower\", cmap=\"Blues\", alpha=0.1)"
79 | ]
80 | },
81 | {
82 | "cell_type": "code",
83 | "execution_count": 29,
84 | "id": "962a58c0",
85 | "metadata": {},
86 | "outputs": [
87 | {
88 | "data": {
89 | "text/plain": [
90 | "(-10.0, 10.0)"
91 | ]
92 | },
93 | "execution_count": 29,
94 | "metadata": {},
95 | "output_type": "execute_result"
96 | },
97 | {
98 | "data": {
99 | "image/png": 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\n",
100 | "text/plain": [
101 | ""
102 | ]
103 | },
104 | "metadata": {
105 | "needs_background": "light"
106 | },
107 | "output_type": "display_data"
108 | }
109 | ],
110 | "source": [
111 | "plt.figure(figsize=(5,5))\n",
112 | "plt.plot(d, y_c1)\n",
113 | "plt.xlim([-limits, limits])\n",
114 | "plt.ylim([-limits, limits])"
115 | ]
116 | },
117 | {
118 | "cell_type": "code",
119 | "execution_count": 30,
120 | "id": "5a8cc977",
121 | "metadata": {},
122 | "outputs": [
123 | {
124 | "data": {
125 | "text/plain": [
126 | "array([ 13.88888889, 13.60971524, 13.3305416 , 13.05136795,\n",
127 | " 12.7721943 , 12.49302066, 12.21384701, 11.93467337,\n",
128 | " 11.65549972, 11.37632607, 11.09715243, 10.81797878,\n",
129 | " 10.53880514, 10.25963149, 9.98045784, 9.7012842 ,\n",
130 | " 9.42211055, 9.14293691, 8.86376326, 8.58458961,\n",
131 | " 8.30541597, 8.02624232, 7.74706868, 7.46789503,\n",
132 | " 7.18872138, 6.90954774, 6.63037409, 6.35120045,\n",
133 | " 6.0720268 , 5.79285315, 5.51367951, 5.23450586,\n",
134 | " 4.95533222, 4.67615857, 4.39698492, 4.11781128,\n",
135 | " 3.83863763, 3.55946399, 3.28029034, 3.00111669,\n",
136 | " 2.72194305, 2.4427694 , 2.16359576, 1.88442211,\n",
137 | " 1.60524846, 1.32607482, 1.04690117, 0.76772753,\n",
138 | " 0.48855388, 0.20938023, -0.06979341, -0.34896706,\n",
139 | " -0.6281407 , -0.90731435, -1.186488 , -1.46566164,\n",
140 | " -1.74483529, -2.02400893, -2.30318258, -2.58235623,\n",
141 | " -2.86152987, -3.14070352, -3.41987716, -3.69905081,\n",
142 | " -3.97822446, -4.2573981 , -4.53657175, -4.81574539,\n",
143 | " -5.09491904, -5.37409269, -5.65326633, -5.93243998,\n",
144 | " -6.21161362, -6.49078727, -6.76996092, -7.04913456,\n",
145 | " -7.32830821, -7.60748185, -7.8866555 , -8.16582915,\n",
146 | " -8.44500279, -8.72417644, -9.00335008, -9.28252373,\n",
147 | " -9.56169738, -9.84087102, -10.12004467, -10.39921831,\n",
148 | " -10.67839196, -10.95756561, -11.23673925, -11.5159129 ,\n",
149 | " -11.79508654, -12.07426019, -12.35343384, -12.63260748,\n",
150 | " -12.91178113, -13.19095477, -13.47012842, -13.74930207,\n",
151 | " -14.02847571, -14.30764936, -14.586823 , -14.86599665,\n",
152 | " -15.1451703 , -15.42434394, -15.70351759, -15.98269123,\n",
153 | " -16.26186488, -16.54103853, -16.82021217, -17.09938582,\n",
154 | " -17.37855946, -17.65773311, -17.93690676, -18.2160804 ,\n",
155 | " -18.49525405, -18.77442769, -19.05360134, -19.33277499,\n",
156 | " -19.61194863, -19.89112228, -20.17029592, -20.44946957,\n",
157 | " -20.72864322, -21.00781686, -21.28699051, -21.56616415,\n",
158 | " -21.8453378 , -22.12451145, -22.40368509, -22.68285874,\n",
159 | " -22.96203238, -23.24120603, -23.52037968, -23.79955332,\n",
160 | " -24.07872697, -24.35790061, -24.63707426, -24.91624791,\n",
161 | " -25.19542155, -25.4745952 , -25.75376884, -26.03294249,\n",
162 | " -26.31211614, -26.59128978, -26.87046343, -27.14963707,\n",
163 | " -27.42881072, -27.70798437, -27.98715801, -28.26633166,\n",
164 | " -28.5455053 , -28.82467895, -29.1038526 , -29.38302624,\n",
165 | " -29.66219989, -29.94137353, -30.22054718, -30.49972083,\n",
166 | " -30.77889447, -31.05806812, -31.33724176, -31.61641541,\n",
167 | " -31.89558906, -32.1747627 , -32.45393635, -32.73310999,\n",
168 | " -33.01228364, -33.29145729, -33.57063093, -33.84980458,\n",
169 | " -34.12897822, -34.40815187, -34.68732552, -34.96649916,\n",
170 | " -35.24567281, -35.52484645, -35.8040201 , -36.08319375,\n",
171 | " -36.36236739, -36.64154104, -36.92071468, -37.19988833,\n",
172 | " -37.47906198, -37.75823562, -38.03740927, -38.31658291,\n",
173 | " -38.59575656, -38.87493021, -39.15410385, -39.4332775 ,\n",
174 | " -39.71245114, -39.99162479, -40.27079844, -40.54997208,\n",
175 | " -40.82914573, -41.10831937, -41.38749302, -41.66666667])"
176 | ]
177 | },
178 | "execution_count": 30,
179 | "metadata": {},
180 | "output_type": "execute_result"
181 | }
182 | ],
183 | "source": [
184 | "y_c1"
185 | ]
186 | },
187 | {
188 | "cell_type": "code",
189 | "execution_count": null,
190 | "id": "be279192",
191 | "metadata": {},
192 | "outputs": [],
193 | "source": [
194 | "def visualize_lp(contraints, limits=10):\n",
195 | " # standard figure parameters (constraints)\n",
196 | " limits = 10\n",
197 | " d = np.linspace(-limits, limits, 100)\n",
198 | " x,y = np.meshgrid(d,d)\n",
199 | "\n",
200 | " # constraint formulation in terms of x and ya\n",
201 | " c1 = G11*x+G12*y <= CV1Hi\n",
202 | " c2 = G11*x+G12*y >= CV1Lo\n",
203 | " c3 = G21*x+G22*y <= CV2Hi\n",
204 | " c4 = G21*x+G22*y >= CV2Lo\n",
205 | "\n",
206 | " # equation of a line, y = mx + c\n",
207 | " y_c1 = (CV1Hi - G11*d)/G12\n",
208 | " y_c2 = (CV1Lo - G11*d)/G12\n",
209 | " y_c3 = (CV2Hi - G21*d)/G22\n",
210 | " y_c4 = (CV2Lo - G21*d)/G22 "
211 | ]
212 | }
213 | ],
214 | "metadata": {
215 | "kernelspec": {
216 | "display_name": "Python 3 (ipykernel)",
217 | "language": "python",
218 | "name": "python3"
219 | },
220 | "language_info": {
221 | "codemirror_mode": {
222 | "name": "ipython",
223 | "version": 3
224 | },
225 | "file_extension": ".py",
226 | "mimetype": "text/x-python",
227 | "name": "python",
228 | "nbconvert_exporter": "python",
229 | "pygments_lexer": "ipython3",
230 | "version": "3.8.8"
231 | }
232 | },
233 | "nbformat": 4,
234 | "nbformat_minor": 5
235 | }
236 |
--------------------------------------------------------------------------------