root/applications/dual/texts/simple_system_LIDP.lyx @ 964

Revision 754, 6.4 kB (checked in by smidl, 15 years ago)

fixes + LQ for dual control

Line 
1#LyX 1.6.4 created this file. For more info see http://www.lyx.org/
2\lyxformat 345
3\begin_document
4\begin_header
5\textclass scrartcl
6\options DIV=12
7\use_default_options true
8\language english
9\inputencoding auto
10\font_roman ae
11\font_sans default
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14\font_sc false
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17\font_tt_scale 100
18
19\graphics default
20\paperfontsize default
21\spacing single
22\use_hyperref false
23\papersize default
24\use_geometry false
25\use_amsmath 1
26\use_esint 1
27\cite_engine basic
28\use_bibtopic false
29\paperorientation portrait
30\secnumdepth 3
31\tocdepth 3
32\paragraph_separation indent
33\defskip medskip
34\quotes_language english
35\papercolumns 1
36\papersides 1
37\paperpagestyle default
38\tracking_changes false
39\output_changes false
40\author ""
41\author ""
42\end_header
43
44\begin_body
45
46\begin_layout Title
47System pro simulaci
48\end_layout
49
50\begin_layout Section*
51Puvodni zadani:
52\end_layout
53
54\begin_layout Standard
55Vychazime ze zadani
56\begin_inset CommandInset citation
57LatexCommand cite
58key "ThoClu:05"
59
60\end_inset
61
62:
63\end_layout
64
65\begin_layout Standard
66\begin_inset Formula \begin{eqnarray}
67SYSTEM:\,\, y_{t+1} & = & y_{t}+bu_{t}+\sigma e_{t},\,\,\, e_{t}\sim\mathcal{N}(0,1),\label{eq:sys}\\
68ZTRATA:\,\,\,\,\, L_{t} & = & (y_{t+1}-r_{t+1})^{2}\label{eq:los}\end{eqnarray}
69
70\end_inset
71
72Reseni schematicky:
73\end_layout
74
75\begin_layout Standard
76\begin_inset Formula \begin{equation}
77V_{t}=\min_{u_{t}}\mathsf{E}_{e_{t},b}\left\{ L_{t}+V_{t+1}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} \label{eq:dp}\end{equation}
78
79\end_inset
80
81kde stredni hodnota se pocita pres neurcitost v
82\begin_inset Formula $e_{t}$
83\end_inset
84
85 a pres neurcitost v
86\begin_inset Formula $b$
87\end_inset
88
89.
90\end_layout
91
92\begin_layout Standard
93Pro linearni a Gausovsky system (
94\begin_inset CommandInset ref
95LatexCommand ref
96reference "eq:sys"
97
98\end_inset
99
100) je k dispozici konjugovana hustota ve forme Normalniho rozlozeni pravdepodobno
101sti
102\begin_inset Formula $f(b_{t})=\mathcal{N}(\hat{b}_{t},P_{t}),$
103\end_inset
104
105 jejiz parametry se vyvijeji rekurzivne, rovnice (28) v
106\begin_inset CommandInset citation
107LatexCommand cite
108key "ThoClu:05"
109
110\end_inset
111
112.
113 Tim padem je mozne vycislit ocekavanou hodnotu pres
114\begin_inset Formula $b$
115\end_inset
116
117 v (
118\begin_inset CommandInset ref
119LatexCommand ref
120reference "eq:dp"
121
122\end_inset
123
124) analyticky:
125\begin_inset Formula \begin{eqnarray*}
126V_{t} & = & \min_{u_{t}}\mathsf{E}_{e_{t},b}\left\{ (y_{t}+bu_{t}+\sigma e_{t}-r_{t+1})^{2}+V_{t+1}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} \\
127 & = & \min_{u_{t}}\mathsf{E}_{e_{t}}\left\{ (y_{t}+\hat{b}u_{t}+\sigma e_{t}-r_{t+1})^{2}+P_{t}u_{t}^{2}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} +\\
128 &  & +\mathsf{E}_{e_{t},b}\left\{ V_{t+1}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} \end{eqnarray*}
129
130\end_inset
131
132Muzeme provest preznaceni
133\begin_inset Formula \[
134V_{t+1}(H_{t})=\mathsf{E}_{e_{t},b}\left\{ V_{t+1}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} \]
135
136\end_inset
137
138kde
139\begin_inset Formula $H_{t}=[y_{t},\hat{b}_{t},P_{t}]$
140\end_inset
141
142.
143 Vysledna uloha je ekvivalentni tomu, kdyby zadani bylo:
144\end_layout
145
146\begin_layout Standard
147\begin_inset Formula \begin{eqnarray}
148SYSTEM:\,\, H_{t+1} & = & \left[\begin{array}{c}
149y_{t+1}\\
150\hat{b}_{t+1}\\
151P_{t+1}\end{array}\right]=\left[\begin{array}{c}
152y_{t}+\hat{b}_{t}u_{t}\\
153(28)\\
154(28)\end{array}\right]+\left[\begin{array}{c}
155\sigma e_{t}\\
1560\\
1570\end{array}\right]\label{eq:sys2}\\
158ZTRATA:\,\,\,\,\, L_{t} & = & (y_{t+1}-r_{t+1})^{2}+P_{t}u_{t}^{2}.\label{eq:los2}\end{eqnarray}
159
160\end_inset
161
162
163\end_layout
164
165\begin_layout Standard
166(28) je opet rovnice (28) z
167\begin_inset CommandInset citation
168LatexCommand cite
169key "ThoClu:05"
170
171\end_inset
172
173.
174 Takto upravenou ulohu muzeme resit pomoci algoritmu
175\begin_inset CommandInset citation
176LatexCommand cite
177key "TodTas:09"
178
179\end_inset
180
181.
182\end_layout
183
184\begin_layout Subsection*
185LQ rizeni
186\end_layout
187
188\begin_layout Standard
189Algoritmus LQ rizeni je aplikovatelny v pripade, ze
190\begin_inset Formula $b$
191\end_inset
192
193 v (
194\begin_inset CommandInset ref
195LatexCommand ref
196reference "eq:sys"
197
198\end_inset
199
200) je zname.
201 V pripade, ze
202\begin_inset Formula $b$
203\end_inset
204
205 nezmame je mozne optimalni rizeni aproximovat tzv.
206 receding horizon strategii.
207 Tato strategie spociva v nahrazeni
208\begin_inset Formula $b\equiv\hat{b}_{t}$
209\end_inset
210
211, spocteni optimalniho zasahu, provedeni
212\begin_inset Formula $u_{t}$
213\end_inset
214
215 , oprava
216\begin_inset Formula $b_{t}$
217\end_inset
218
219 a opetovne prepocteni strategie.
220\end_layout
221
222\begin_layout Standard
223Tomuto postupu se rika certainty equivalence.
224 Nevyhodou tohoto pristupu je, ze chyba rizeni pro chybny odhad
225\begin_inset Formula $\hat{b}$
226\end_inset
227
228 je znacna.
229\end_layout
230
231\begin_layout Standard
232Druhou moznosti aproximace je pouziti systemu (
233\begin_inset CommandInset ref
234LatexCommand ref
235reference "eq:sys2"
236
237\end_inset
238
239) s nahradou
240\begin_inset Formula $\hat{b}_{t+1}=\hat{b}_{t}$
241\end_inset
242
243,
244\begin_inset Formula $P_{t+1}=P_{t}$
245\end_inset
246
247.
248 Vysledek je velmi podobny jako u CE strategie, avsak do ztratove funkce
249 pribyl penalizacni clen
250\begin_inset Formula $P_{t}u_{t}^{2}$
251\end_inset
252
253, ktery penalizuje velke hodnoty
254\begin_inset Formula $u_{t}$
255\end_inset
256
257.
258 Pro velke hodnoty
259\begin_inset Formula $P_{t}$
260\end_inset
261
262 tak vznika preference pro male hodnoty
263\begin_inset Formula $u_{t}$
264\end_inset
265
266.
267 Vysledne strategii rizeni se proto rika cautious, tedy opatrna.
268 Nevyhodou teto strategie je prilisna
269\begin_inset Quotes eld
270\end_inset
271
272opatrnost
273\begin_inset Quotes erd
274\end_inset
275
276, ktera vychazi z predpokladu konstantnosti
277\begin_inset Formula $P_{t}$
278\end_inset
279
280, tedy velke penalizace
281\begin_inset Formula $u_{t}$
282\end_inset
283
284 na celem horizontu.
285 Kvuli aproximaci neni ve strategii zohlednen vliv
286\begin_inset Formula $u_{t}$
287\end_inset
288
289 na
290\begin_inset Formula $P_{t}$
291\end_inset
292
293, a tim i fakt, ze vhodne zvolene
294\begin_inset Formula $u_{t}$
295\end_inset
296
297 muze hodnoty
298\begin_inset Formula $P_{t}$
299\end_inset
300
301 snizit.
302\end_layout
303
304\begin_layout Standard
305Tento efekt se da kompenzovat tim, ze predpokladame, ze
306\begin_inset Formula $P_{t}$
307\end_inset
308
309 bude s casem klesat, napr:
310\begin_inset Formula \[
311P_{t+1}=\frac{1}{2}P_{t}.\]
312
313\end_inset
314
315pripadne az do krajnosti:
316\begin_inset Formula \[
317P_{t+1}=0.\]
318
319\end_inset
320
321
322\end_layout
323
324\begin_layout Standard
325\begin_inset CommandInset bibtex
326LatexCommand bibtex
327bibfiles "vs-world,world_classics,mk,world"
328options "plain"
329
330\end_inset
331
332
333\end_layout
334
335\end_body
336\end_document
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