Моделирование сложных информационных систем: Методические указания по изучению спецкурса

FBGBKL?JKL...

Author: Сирота А.А.

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FBGBKL?JKL

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15 lhd h[kem`_gguo aZy\hd beb aZy\hd ihdb^Zxsbo kbkl_fm g_h[kem`_g gufbDhgdj_lbaZpbyh[h[s_gghcko_fuKFHfh`_l[ulvijh\_^_gZim l_fdeZkkbnbdZpbb Ij_`^_\k_]h\k_KFH^_eylgZ^\Z[hevrbodeZkkZ:jZahfdgmlu_\ dhlhjuo \oh^ghc ihlhd g_ aZ\bkbl hl ijhp_kkZ h[kem`b\Zgby b aZfdgm lu_\dhlhjuokms_kl\m_lh[jZlgZyk\yavb\oh^ghcihlhdaZy\hdaZ\bkbl hlijhp_kkZh[kem`b\ZgbyIhfbfhwlh]h:ihgZebqbx\hafh`ghklbh`b ^ZgbyaZy\hd\hq_j_^b\u^_eyxlKFH[_ah`b^ZgbyKFHkhldZaZfbZ lZd`_kf_rZggh]hlbiZ;ihqbkemiZjZee_evgh\dexq_gguowe_f_glZjguo ijb[hjh\ h[kem`b\Zgby \u^_eyxl h^ghdZgZevgu_ b fgh]hdZgZevgu_ KFH;ihqbkemwlZih\nZa h[kem`b\Zgbyijbihke_^h\Zl_evghf\dex q_gbb ijb[hjh\ h[kem`b\Zgby hij_^_eyxl h^ghnZagu_ b fgh]hnZagu_ KFHbebk_lbfZkkh\h]hh[kem`b\Zgby Ex[Zy KFH ij_^gZagZq_gZ ^ey h[kem`b\Zgby \oh^gh]h ihlhdZ k aZ ^Zgghc wnn_dlb\ghklvx < dZq_kl\_ hkgh\guo ihdZaZl_e_c wnn_dlb\gh klb ^ey KFH k hldZaZfb bkihevamxl Z[khexlgmx ijhimkdgmx kihkh[ ghklvQ (kj_^g__qbkehh[kem`_gguoaZy\hdaZ\j_fyL bhlghkbl_evgmx ijhimkdgmx kihkh[ghklv beb \_jhylghklv h[kem`b\Zgby T = 4 λ ]^_ λ kj_^g__qbkehihklmib\rboaZy\hdaZ\j_fyL>eyKFHkh`b^Zgb_f^h ihegbl_evgh \ dZq_kl\_ ihdZaZl_e_c bkihevamxl \j_f_ggu_ oZjZdl_jb klbdbij_[u\ZgbyaZy\hd\hq_j_^bb\kbkl_f_\p_ehf 3.2. Hkh[_gghklbihkljh_gbybfblZpbhgguofh^_e_cKFH =_g_jZpbyihlhdh\kh[ulbc Kmsghklv f_lh^Z bfblZpbhggh]h fh^_ebjh\Zgby ijbf_gbl_evgh d KFHkhklhbl: 1)\ihkljh_gbbZe]hjblfh\nhjfbjh\Zgbyj_ZebaZpbcih lhdh\kh[ulbcaZy\hd lh_klvZe]hjblfh\©bklhqgbdh\ª; \ihkljh_gbb Ze]hjblfh\ ijhp_kkZ h[kem`b\Zgby hl^_evgufb we_f_glZfb dZgZeZfb gZdhibl_eyfb b KFH \ p_ehf; fgh]hdjZlghf \hkijhba\_^_gbb ijh p_kkZ h[kem`b\Zgby j_ZebaZpbc \oh^guo ihlhdh\ ijb nbdkbjh\Zgguo \gmlj_ggboiZjZf_ljZoKFHkp_evxklZlbklbq_kdhchp_gdbihdZaZl_e_c wnn_dlb\ghklb < jZfdZo bfblZpbhgghc fh^_eb KFH ijbgylh \u^_eylv ljb lbiZ we_f_glh\ fh^_eb: Zdlb\gu_ iZkkb\gu_ b Zdlb\gh-iZkkb\gu_ Baf_g_gb_ khklhygbc Zdlb\guo we_f_glh\ h[mkeh\e_gh lhevdh bo \gml j_ggbfb k\hckl\Zfb lh]^Z dZd iZkkb\gu_ we_f_glu baf_gyxl k\hb kh klhygby lhevdh ih^ \ha^_ckl\b_f \g_rgbo nZdlhjh\ D Zdlb\ghiZkkb\guf we_f_glZf hlghkylky lZdb_ dhlhju_ \ h^ghf ba \hafh`guo khklhygbcy\eyxlkyZdlb\gufbZ\^jm]boiZkkb\gufb Ih\_^_gb_kbkl_fijbBF\hkijhba\h^blkygZhij_^_e_gghfhlj_ad_ \j_f_gb[0,T]
16 ^mxsb_ hkgh\gu_ kihkh[u aZ^Zgby fh^_evgh]h \j_f_gb: kihkh[ © ∆W », ijbdhlhjhf\fh^_ebjmxs_fZe]hjblf_ijbjZs_gb_\j_f_gbhkms_kl\ eyxlknbdkbjh\Zggufbgl_j\Zehf ∆W ; kihkh[© ∆] ªijbdhlhjhfijbjZ s_gb_ fh^_evgh]h \j_f_gb hkms_kl\eyxl lhevdh \ fhf_gl gZklmie_gby hq_j_^gh]h kh[ulby k\yaZggh]h k baf_g_gb_f khklhygby ex[h]h ba we_ f_glh\kbkl_fub__\g_rg_ckj_^u;kihkh[© ∆W × ∆] ªj_Zebamxsbcdhf [bgZpbxwlbokihkh[h\\jZaebqguodhfihg_glZoBF g_aZ\bkbfu gZau\Z_lky ihlhdhf k h]jZgb q_gguf ihke_^_ckl\b_f Ihlhdhf [_a ihke_^_ckl\by [he__ kbevgh_ mkeh\b_ gZau\Z_lky ihlhd \ dhlhjhf \_jhylghklv 3 W τ gZklmie_gby k kh[ulbcgZbgl_j\Ze_ [W W + τ] g_aZ\bkblhlihy\e_gbykh[ulbcgZ^jm]bo g_i_j_k_dZxsboky k ^Zgguf bgl_j\ZeZo Ihlhd kh[ulbc gZau\Z_lky hj ^bgZjguf _keb \_jhylghklv ihy\e_gby [he__ h^gh]h kh[ulby \ fZehc hdj_klghklbfhf_glZtij_g_[j_`bfhfZeZihkjZ\g_gbxk\_jhylghklvx \uiZ^_gbyh^gh]hkh[ulby 3 (W ∆W ) >> 3 > W ∆W = Θ ∆W WlhhagZqZ_lqlh kh[ulby \ hkgh\ghf ijhbkoh^yl ih hl^_evghklb [_a h[jZah\Zgby k]mkl dh\ Ihlhd kh[ulbc gZau\Z_lkyklZpbhgZjguf\h\j_f_gb_keb\_jhyl ghklvihy\e_gbylh]hbebbgh]hqbkeZkh[ulbcgZnbdkbjh\Zgghfbgl_j \Ze_g_aZ\bkblhl\j_f_gb 3 (W τ) = 3 (τ) N = >ey hj^bgZjguo ihlhdh\ \\h^blky ihgylb_ bgl_gkb\ghklb ihlhdZ λ(W ) = OLP 3 (W ∆W ) ∆W . >ey klZpbhgZjguo ihlhdh\ λ W = λ = FRQVW Z _keb Q

Q

Q

Q

Q

Q

L

Q

L

L

N

N

N

∆→ W

N

wlh _s_ b ihlhd k h]jZgbq_gguf ihke_^_ckl\b_f lh ^ey fgh]hf_jghc iehlghklbbgl_j\Zeh\f_`^mkh[ulbyfb\uihegy_lky I (τ τ ) = I (τ )×× I (τ ) , Q

I (τ



) = λ − 

 τ τ I G ; ( ) ∫ 

τ

Q

Q

I (τ) = = I (τ) = I (τ) ;

Q

Q

∞

λ = ∫ τI (τ )Gτ .

17 H[sbcih^oh^dfh^_ebjh\Zgbxhj^bgZjguoklZpbhgZjguoihlhdh\ k h]jZgbq_gguf ihke_^_ckl\b_f ihlhdb IZevfZ aZdexqZ_lky \ ke_^mx s_f KqblZ_lky ba\_klguf aZdhg jZkij_^_e_gby bgl_j\ZeZ f_`^m kh[u lbyfb I τ Lh]^ZkgZqZeZfh^_ebjm_lkyfhf_glgZklmie_gbyi_j\h]hkh [ulby\khhl\_lkl\bbkkhhlghr_gb_f τ =ϕ

W = W +τ ,

−

ϕ (τ

([ ) ,

)=

τ

∫ I (X)GX ,

ZaZl_f\k_oihke_^mxsbo\khhl\_lkl\bbkkhhlghr_gb_f W =W L

L

+ τ τ = ϕ

−

L

τ

−

L

([ ) ϕ(τ ) = ∫ I (X)GX L = Q , L

L

L

]^_ [ L = Q ihke_^h\Zl_evghklvjZ\ghf_jghjZkij_^_e_gguokemqZcguo qbk_egZbgl_j\Ze_[0,1]. Ijbf_j Ihlhd ImZkkhgZ beb ijhkl_crbc hj^bgZjguc [_a ih ke_^_ckl\byklZpbhgZjguc ihlhdk I (τ) = λH − λτ Lh]^Z L

 τ  I (τ) = λ  − ∫ λH − λ GX = λH − λτ = I (τ ) ,   X

lh _klv \ ^Zgghf kemqZ_ i_j\uc bgl_j\Ze jZkij_^_e_g lZd `_ dZd b \k_ hklZevgu_Hlkx^Z^eyihke_^h\Zl_evghklbfhf_glh\kh[ulbc: + τ τ = − OQ [ L = Q . λ

τ

ϕ (τ) = ϕ(τ ) = ∫ I (X)GX = − H − λτ = [ , W = W

L

L

−

L

L

L

Ijbf_jIhlhdWjeZg]ZIhlhdWjeZg]Zm-]hihjy^dZh[jZam_lky ba ijhkl_cr_]h \ j_amevlZl_ _]h ©ijhj_`b\Zgbyª dh]^Z \u[jZku\Zxlky \k_lhqdbaZbkdexq_gb_fm-hc>eylZdh]hihlhdZ

Imklvm=2, lh]^Z

(λP) I (τ ) = (P − )

P

τH − Pλτ =

∗ λ∗  − λ∗τ + λ∗ τ H − λ τ I (τ ) =  H 

( )

λ∗ λ∗ τ

P

−

H − λ τ λ =

(P − )

 ϕ (τ 

λ∗ . P

 λ∗ τ  − λ∗τ H ) = −  +  

b^eygZoh`^_gby τ lj_[m_lkyj_rZlvmjZ\g_gb_

H−

]

(] + ) = [ ] = λ∗ τ

.

>eyi>1 bgl_j\Zef_`^mkh[ulbyfb\ihlhd_nhjfbjm_lkykmq_lhflh]h qlh τ = τ + τ _klvkmffZ^\moh^bgZdh\hjZkij_^_e_gguoihihdZaZl_ev ghfmaZdhgm\_ebqbgkiZjZf_ljhf λ∗ dhlhjucihkmlbhij_^_ey_lbgl_g kb\ghklvbkoh^gh]h©ijhj_`b\Z_fh]hªihlhdZImZkkhgZLh]^Z L

L

L

W =W L

L

−

+ τ τ = − L

L

OQ [ [ L = Q . λ∗

L

L

18 GZoh`^_gb_Ze]hjblfZ^eyfh^_ebjh\Zgbyfhf_glZi_j\h]hkh[ulby ihlhdZ m-]h ihjy^dZ k\yaZgh k j_r_gb_f _s_ [he__ keh`guo ljZgkp_g ^_glguo mjZ\g_gbc Ihwlhfm _keb j_ZebaZpby \h \j_f_gb ^hklZlhqgh ^ebggZylhfh`ghihevah\Zlvkyke_^mxsbfmgb\_jkZevgufZe]hjblfhf: W =W L

L

−

+ τ τ = − L

L

P OQ ∏ [ L = Q . λ = P

M L

M

Ijbgpbiu fh^_ebjh\Zgby g_klZpbhgZjguo ihlhdh\ fh]ml [ulv ijh beexkljbjh\ZgugZke_^mxs_fijbf_j_ Ijbf_j G_klZpbhgZjguc ihlhd ImZkkhgZ bgl_gkb\ghklv dhlh jh]h λ = λ W aZ\bkbl hl \j_f_gb Z iehlghklv jZkij_^_e_gby bgl_j\ZeZ f_`^mkh[ulbyfbijhbkoh^ysbfb\fhf_glu W ∗ b W ∗ + W bf__l\b^ I (W ∗ W ) = Λ (W ∗ W )H

−Λ

W

∗+

Λ(W W ) = ∫ λ(X)GX , W

W

W

W

∗

]^_ Λ (W W ) = λ(W ∗ + W ) Hlkx^Z^eyfhf_glZihy\e_gbyi_j\h]hkh[ulby

τ

W = τ ∫ I (X)GX = Λ( τ

τ

) = ∫ λ(X)GX = − OQ [ I (X) = Λ ( X)H − Λ X .

>Zevg_crZyf_lh^bdZgZoh`^_gbyfhf_glh\kh[ulbcZgZeh]bqgZ W =W L

L

−

+ τ Λ(W L

L

−

τ ) =

W L

−

L

W

L

+τ

∫

L

λ(X)GX = − OQ [ L = Q .

−

>ey

fh^_ebjh\Zgby g_hj^bgZjguo ihlhdh\ \ Q-ko_fZo OLP 3 W ∆W ≠ djhf_aZ^ZgbyaZdhgh\jZkij_^_e_gbyfhf_glh\ihy\e_ ∆→ W

P

gby W L = Q g_h[oh^bfhhij_^_eylvjZkij_^_e_gb_qbkeZkh[ulbc\jZk kfZljb\Z_fuc fhf_gl \j_f_gb :gZeh]bqgh ^ey g_h^ghjh^guo ihlhdh\ lj_[m_lkyaZ^Zgb_aZdhgZjZkij_^_e_gbyijbagZdh\kh[ulbc I L = Q Ihkljh_gb_fh^_ebjmxs_]hZe]hjblfZKFH\p_ehf Djhf_ aZ^Zgby ihlhdh\ kh[ulbc ijb nhjfZebaZpbb ijhp_kkZ nmgd pbhgbjh\Zgby kbkl_fu k ihfhsvx Q-ko_f b ihkljh_gbb __ fh^_ebjmx s_]hZe]hjblfZg_h[oh^bfhaZ^ZlvkljmdlmjmlZdhckbkl_fuHgZij_^\Z jbl_evgh hij_^_ey_lky kh[kl\_ggufb \gmlj_ggbfb iZjZf_ljZfb KFH: dhebq_kl\hf nZa / n ; dhebq_kl\hf dZgZeh\ h[kem`b\Zgby dZ`^hc nZau L

L

/ D M = / n ;dhebq_kl\hfgZdhibl_e_cdZ`^hcnZau / M = /n ;_fdhklvx M

+M

dZ`^h]h gZdhibl_ey / L = / M = / n Z lZd`_ k\yayfb f_`^m BG b D lh _klv hi_jZlhjhf khijy`_gbyR we_f_glh\kbkl_fuDjhf_lh]haZ ^Zxlky^bkpbiebguh`b^ZgbcaZy\hd\Hb\u[hjZbogZh[kem`b\Zgb_\ D>eyjZajZ[hldbfh^_ebjmxs_]hZe]hjblfZp_e_khh[jZaghbkihevah\Zlv ki_pbZevgu_]jZnbq_kdb_bah[jZ`_gbykfjbk kljmdlmjuKFH +LM

+M

H31

D11

D31

19 H11

H21 µ11

b1 λ1

l11

µ21

l21

D12

µ31

l31 H32

D21

D32

µ12

µ32

l32

Jbk A^_kvh[hagZq_gby λ ∗ hij_^_eyxlbgl_gkb\ghklb\oh^guoihlhdh\Z µ ∗∗ bgl_gkb\ghklb ihlhdh\ h[kem`b\Zgby Rljboh\ufb ebgbyfb ihdZaZgu \hafh`gu_[ehdbjh\db
{

L

L

P

L

P

}

L P

( ) bgl_gkb\ghklvihlhdZhl B .

e_gbyhq_j_^ghcaZy\dbZ λ W L

L

L

P

L

2. We_f_glu lbiZ H, y\eyxsb_ky iZkkb\gufb DZ`^uc i-uc gZdhib l_ev k-hc nZau + fh`_l gZoh^blvky \ l_dms_f khklhygbb ] = {O ∈ O } I_j_oh^ ba h^gh]h khklhygby \ ^jm]h_ ijhbkoh^bl LN

L

L

+N

N

+LN

f]gh\_gghijbihklmie_gbbmoh^_ aZy\dbbkhijh\h`^Z_lkym\_ebq_gb _fmf_gvr_gb_f O gZ_^bgbpmaZbkdexq_gb_f]jZgbqguokhklhygbc 3. We_f_glulbiZKy\eyxsb_kyZdlb\gh-iZkkb\gufbHgbfh]mlgZ oh^blvky \ khklhygbb ©aZgylª ]^_ y\eyxlky Zdlb\gufb b \ khklhygbb ©k\h[h^_gª]^_y\eyxlkyiZkkb\gufbL_dms__khklhygb_i-]hdZgZeZj-hc nZau . hij_^_ey_lky ljhcdhc \_ebqbg ] D = α W µ W ]^_ L N

L M

LM

L M

L M U

L M

LM

U

}

- \j_fy h[kem`b\Zgby hq_j_^ghc aZ y\dbZ µ W -bgl_gkb\ghklvihlhdZh[kem`b\ZgbyI_j_oh^bah^gh]h khklhygby\^jm]h_ijhbkoh^blf]gh\_gghijbihklmie_gbbaZy\dbgZh[ kem`b\Zgb_bebhdhgqZgbbh[kem`b\Zgby Djhf_ lh]h \ BF p_e_khh[jZagh \\_klb ^\Z lbiZ \kihfh]Zl_evguo we_f_glh\: hq_j_^v aZy\hd b hq_j_^v k\h[h^guo dZgZeh\ Hq_j_^v aZy\hd HA gZ\oh^_dZ`^hcnZauh[kem`b\Zgbyy\ey_lkyiZkkb\gufwe_f_glhf, b __ khklhygb_ hij_^_ey_lky \_ebqbghc h[s_c hq_j_^b α

L M

= ©aZgylª ©k\h[h^_gª W

{

L M U

L M

LM

U

 ](ha ) = O (ha ) O (ha ) = ∑ O ( ) O ( ) ∈ O =  /

N

N

N

+N

L

N

L

L

N

(

+LN

  Khklhygby baf_gyxlky f]gh\_ggh 

)

ijbihklmie_gbbgh\hcaZy\db O (ha ) = O (ha ) + bebijbhk\h[h`^_gbbdZgZeZ

(

)

nZau O (ha ) = O (ha ) − N

N

N

N

Hq_j_^v k\h[h^guo dZgZeh\ HD dZ`^hc j-hc nZau

y\ey_lky iZkkb\guf we_f_glhf b oZjZdl_jbam_lky qbkehf b \hafh`gh

20 lbihf k\h[h^guodZgZeh\>eyHD__khklhygb_hij_^_ey_lky\_ebqbghc

D   ](hd) = O (hd) O (hd) = ∑ α ( ) O (hd) ∈ / D  Khklhygb_ baf_gy_lky f]gh\_ggh =   /

M

M

M

L M

M

M

M

L

(

)

ijbihklmie_gbbgh\hcaZy\db O (hd) = O (hd) − bebijbhk\h[h`^_gbbex[h]h

(

)

dZgZeZ O (hd) = O (hd) + . M

M

M

M

LZdbf h[jZahf ijb bfblZpbb ijhp_kkZ nmgdpbhgbjh\Zgby Q-ko_fu gZW
{

{

L

Q

}

}

+

{

{

: D = ](d ) L / D M /n : ha = ](ha ) N L

M

M

N

}

L

+N

+N

{

}

}

/ n : hd = ](hd) M / n M

dhlhju_bkihevamxlkyijbj_ZebaZpbbex[h]hlbiZfh^_ebjmxs_]hZe]h jblfZ H[sZy [ehd-ko_fZ lZdh]h Ze]hjblfZ ijb j_ZebaZpbb f_lh^Z∆zaZ^Z gbyfh^_evgh]h\j_f_gbbf__l\b^ihdZaZggucgZjbk<[ehdZoko_fu: lbiZ O (ha) = O (ha) + aZihfbgZ_lkyfhf_glihklmie_gbyaZy\db\h^bgbagZdh ibl_e_c + b __ ijbhjbl_l \ fZkkb\_ hq_j_^b aZy\hd nZau; lbiZ M

M

LM

O (ha) = O (ha) − \u[bjZ_lky aZy\dZ ba gZdhibl_ey + nbdkbjm_lky \j_fy \u M

M

LM

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W = PLQPLQ W

L M U

PLQ W L P

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-

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+

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O ha = O ha + O hd = O hd − O

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M = O hd +

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nhjfbjh\Zgb_ gh\hcaZy\db W = W +τ

hq_j_^v aZy\

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+

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+

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+

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O ha+ = O ha+ + O hd

/

+

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(

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[

J

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]

+

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(

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+

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N

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ν(W ) = ν = FRQVW ] ν (W ) = ]

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ν

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(

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 ] 

ν

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( ) (

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ν

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ν

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( ) [

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D

D

khklhygb_za(t+0)=j\u[bjZ_lkykemqZcghk\_jhylghklvx 3

1

N

LM

≥ ∑ 3 =

N

LM

=

M

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ν ′ = ν + ] (W ν + ) = ] W ν ] (W ν + ) = K , ](W ν + ) = (ν + ] W ν K) . Ihke_ kdZqdZ ijb W ν < W ≤ W ν+ ^\b`_gb_kbkl_fuijhbkoh^bl\khhl\_lkl 7

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i=i+1

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Hij_^_e_gb_ kibkdZ \oh^guo b mijZ\eyxsbo \ha^_ckl\bc :-kbkl_fu hlg_k_gguo d fhf_glm tk GZqZeh pbdeZ ih qbkem Z]j_]Zlh\ :-kbkl_fu

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32

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Моделирование сложных информационных систем: Методические указания по изучению спецкурса

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