Аналитические и имитационные модели: Учебное пособие

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<ul><li><p>.. H .. .. </p><p> - ( ) , ( 210200 </p><p> ( )). </p><p> 2007 </p></li><li><p> 2 </p><p> 518.5.001.57(075.8) .., .. , ... -</p><p> : . - : - , 2007. - 310 . </p><p>ISBN 978-5-8327-0268-1 c p c </p><p> , x p cc, . . . 8. . 30. .: 22 . </p><p> .., -. . , , () , . ., -. . , , . ISBN 978-5-8327-0268-1 </p><p> , 2007 .., .., </p><p> .., 2007 </p></li><li><p> 3 </p><p> H </p><p> H... 6 1. ...... 9 </p><p>1.1. .. 9 1.2. . 18 2. ... 24 2.1. . 24 2.2. . 29 2.3. ..... 33 2.4. .. 38 2.5. . 40 3. ... 46 3.1. .. 46 3.2. .. 53 3.3. . 55 4. .... 59 4.1. .... 59 4.2. . 60 4.3. .... 69 4.4. .. 78 4.5. .. 83 4.6. .. 93 5. . 106 5.1. .. 106 5.2. . 111 5.3. .. 120 </p></li><li><p> 4 </p><p> 6. ... 132 6.1. ..... 132 6.2. 136 6.3. ..... 138 </p><p>7. ..... 147 7.1. ....... 147 7.2. ...... 151 7.3. ...... 153 7.4. ........ 156 7.5. - --..... 158 7.6. ...... 162 7.7. . 184 7.8. ... 190 7.9. . 211 8. P PCC P CC.. 221 8.1. p p... 221 8.2. p cp px p cx cc... 226 </p><p>9. UML... 229 9.1. ... 229 9.2. ... 231 9.3. . 238 </p></li><li><p> 5 </p><p>9.4. . 241 9.5. 249 9.6. UML. 255 9.7. . 258 9.8. .. 261 9.9. 267 9.10. ... 278 </p><p>10. - .. 282 10.1. .. 282 10.2. 295 10.3. 299 10.4. . 303 308 </p></li><li><p> 6 </p><p> , , . , , . </p><p> , , . , - . . , .. , . [1, 2]. - , </p><p> , , , [2, 3]. </p></li><li><p> 7 </p><p> . - , , . , , , . </p><p> , , Matlab-Simulink-Stateflow, UML, Simulink, Stateflow, Model Vision Studium, Modelia, AnyLogi . </p><p> - , Unified Modeling Language (UML) [4]. </p><p> . </p><p>. , . : ; ; ; . </p></li><li><p> 8 </p><p> . , , . . , , . . , . , , . , . . cp px p . UML, - . [4]. </p></li><li><p> 9 </p><p> 1. </p><p> 1.1. </p><p> 1.1.1. . , , . . , , , . , . </p><p> , .. , . (modulus (.) - ) - - [1]. , -. . , </p></li><li><p> 10 </p><p> , . </p><p> , . . , , . </p><p> . , , (), . . </p><p> . . </p><p> . , . , . </p><p> . , , </p></li><li><p> 11 </p><p> . . , . </p><p> : - ; - </p><p>; - ; </p><p> ; - ; - </p><p>; - ; - ; - </p><p>; - ; - ; - ; - . </p><p> - , , , . , .. , - . </p><p> . </p></li><li><p> 12 </p><p> : , , . </p><p> , , . </p><p> . , . , , . </p><p> . , - . </p><p> , , , , . </p></li><li><p> 13 </p><p> . , . , . , . . </p><p> . , . H (). Z, . , . 1.1, , Y, F - , . </p><p>X Y </p><p>F </p><p>Z </p><p> . 1.1 </p><p> 1.1.2. . </p><p> [1,2,3]. </p></li><li><p> 14 </p><p> , () Y: </p><p>Y=W(), (1.1) W . W(.) </p><p> ( , , , ..). W , . (t) Y(t). (). </p><p> , .. , , , . , . : . - , </p><p> , . , . </p></li><li><p> 15 </p><p> - , , .. . </p><p> , . </p><p> , , - . : </p><p>- , ; </p><p>- , , ; </p><p>- , , ( ..). , </p><p> , , . , </p><p> W(t) , . </p></li><li><p> 16 </p><p> () . . . 1.2 . </p><p>t1t </p><p>t </p><p>t2 t3 t4 t5 t6 t7 </p><p>t1t </p><p>t </p><p>x(t) </p><p>y(t) </p><p>t2 t3 t4 t5 t6 t7 </p><p>x(t) </p><p>y(t) </p><p> . 1.2 </p><p> y(t) ti (ti) t. </p></li><li><p> 17 </p><p> ( ) , . </p><p> ( ) . , </p><p> W y(t)=W(,t). </p><p> , </p><p>Y=W(Z,t,t-1,,t-k), (1.2) t-i=(t-i) - t-i. . , </p><p> - Y=W(,Z,), ={1,2,,k} - . , , </p><p> (, ), (, ). </p><p> , . , , </p></li><li><p> 18 </p><p>.yC)x(WC)xC(Wm</p><p>1i</p><p>m</p><p>1iiiii</p><p>m</p><p>1iii ==</p><p>= == (1.3) </p><p> Y=W. (1.3) , . </p><p> , . : , ( ), , - . , </p><p> , . </p><p> - . . </p><p> 1.2. </p><p> , </p><p> tT zZ . </p><p> : - </p><p> ( ); - </p><p> ( ); </p></li><li><p> 19 </p><p>- ( ). </p><p> , , , , . </p><p> x ={1,2,,m}, ii,, ( m,1i = ), i - . =12m , x . =L(t), </p><p> t x , L(t). y Y - </p><p> . , tT, )t(y . y y1,y2,,yr, , yjYj, ( r,1j = ), Yj - , Y=Y1Y2Yr . Y=M(t). </p><p> ( ). </p><p> . z zkZk, ( n,1k = ), Zk - , Z Z=Z1Z2Zn. </p></li><li><p> 20 </p><p> () . , , . 1.3. </p><p>X Y V </p><p>+ </p><p>Z </p><p>. 1.3 </p><p> , r- ( ) ={1,2,,r}, . ( ) V={v1,v2,,vr}. </p><p>V=B(Y), (1.4) B(Y) - , Y - V. . . [4] - () yY , () (,y), yY, . </p></li><li><p> 21 </p><p> Y. : - , ; - Y, ; - QY, , , .. (,y), . , q, q=(,Y,Q), , Y , Q , QY. , 1Q, 2QY, 1Q , 2Q . . - ={1,2,,n}, Y={y1,y2,,ym} Q={(i,yj)}, , yY )n,1i( = , )m,1j( = . RQ, nm. i RQ, yjY . i yj rij=1, (i,yj)Q, rij=0, (i,yj)Q. (. . 1.4.), i , yjY , (i,yj)Q , </p></li><li><p> 22 </p><p> i yj. . yY , q-1=(Y,,Q-1), q ( q). </p><p>y1 y2 y3 </p><p>x1 x2 x3 x4 </p><p> ={1,2,3,4}, Y={y1,y2,y3}, Q={(1,y1), (1,y2), (2,y1), </p><p>(2,y2), (3,y2), (4,y3)}. . 1.4 </p><p> =12m, Y=Y1Y2Yr, Z=Z1Z2Zn , . </p><p>f=(12m,Z1Z2Zn,F). (1.5) f x ={1,2,,m}12m z ={z1,z2,,zn}Z1Z2Zn. F f. </p><p>f={(12m),(Z1Z2Zn), (Z1Z2Zn),F}, (1.6) </p><p>.. f ( x , z )[(12m)(Z1Z2Zn)] </p></li><li><p> 23 </p><p> z ={z1,z2,,zn}Z1Z2Zn. </p><p>f={[(12m),(Z1Z2Zn)], (Y1Y2Yr),F}. (1.7) </p><p> f ( x , z ) [(12m), (Z1Z2Zn)] y ={y1,y2,,yr}Y1Y2 Yr. F f. : </p><p>f={[(12m)(Z1Z2Zn)],[Z1Z2Zn], (Y1Y2Yr),F), (1.8) </p><p>.. f {( x , z ), z } {[(12m), (Z1Z2Zn)], [Z1Z2Zn]} y ={y1,y2,,yr}Y1Y2 Yr. : </p><p>)f1 1 m 1 1 n{(X , X ,..., X ) (Z ,Z , ...,Z (1.9) </p><p>)} ) f1 1 m 1 1 n 1 1 n{(X , X , ..., X ) (Z ,Z , ...,Z (Z ,Z , ...,Z . (1.10) : </p><p>)} ) f1 1 m 1 1 n 1 1 r{(X , X ,..., X ) (Z ,Z ,...,Z (Y ,Y ,...,Y (1.11) </p><p>)] 1 1 m 1 1 n{[(X , X ,..., X ) (Z ,Z ,...,Z )} ) f1 1 n 1 1 r(Z ,Z , ...,Z (Y ,Y ,...,Y . (1.12) </p></li><li><p> 24 </p><p> 2. </p><p>2.1. </p><p> 2.1.1. . . , , . </p><p> , [5]. , , . , MatLab, Omola, Dymola, Dymosim, Model Vision Studium, UML . , , , . 2.1. : </p><p>- ; - L ; - U(t) ; - IL(t) ; - U(t) . </p><p> , . </p></li><li><p> 25 </p><p>U(t) UC(t) </p><p>C IL(t) </p><p>L </p><p>. 2.1 </p><p> : </p><p>LC I</p><p>dtdU</p><p> = , CL UU</p><p>dtdIL += . </p><p> z1=U, U/L=(t) : </p><p>dtz</p><p>dtdz 21 = , )t(x</p><p>Lz</p><p>dtdz 12 += . (2.1) </p><p> U=0, (t)=0 (2.1) . (t) , t. (2.1), z1(t) z2(t). U(t) </p><p>, R - , , [6]: </p><p>-tC 1 1</p><p>1</p><p>U (t) = 1- e (cos t + sin t)</p><p>, (2.2) </p><p> R =2L</p><p>, 201 =</p><p>LC, 2 2 21 0 = . </p></li><li><p> 26 </p><p> , U(t) R, L , ( ) . </p><p>2.1.2. . , . . E(t) t. E(t+t)-E(t) t t0. , ( ): </p><p>).t(kEt</p><p>)t(E)tt(E=</p><p>+ </p><p>)t(kEt</p><p>)t(E)tt(Elim0t</p><p>=</p><p>+</p><p> ( ): </p><p>)t(kEdt</p><p>)t(dE= . (2.3) </p><p> (2.3), . t=0, E(t=0)=E0 </p><p>E(t)=E0ekt. (2.4) (2.4) . 2.2. </p></li><li><p> 27 </p><p>t</p><p>(t) </p><p>0</p><p> . 2.2 </p><p> t=0 E=E0, , </p><p>2E0=E0ekt, 2=ekT, T=(1/k)ln2. , , , (, ). </p><p>2.1.3. . , [2]. m1 - ; m2 - </p><p> , t; 1 - ; 2 - . 1 - 2 - . , . </p></li><li><p> 28 </p><p>L1=11, L2=22. m1 </p><p>t 22tm2, m2 t 11tm1, </p><p>m1=-22tm2, m2=-11tm1. (2.4) (2.4) - . (2.4) . t, </p><p>12 2 2</p><p>m = - p mt</p><p>, 2 1 1 1m = - p mt</p><p>. </p><p> t0, , </p><p>12 2</p><p>dm = -L mdt</p><p>, 2 1 1dm = -L mdt</p><p> (2.5) </p><p> (2.5) . 2.1.4. . , </p><p> , y, V V VY. m - ; u ; - 0; f(u) - . , . </p><p> , </p><p>xVdtdx</p><p>= , yVdtdy</p><p>= . </p><p> : </p><p>;sinUFdt</p><p>dVm y</p><p>y += += cosUFdt</p><p>dVm x</p><p>x . </p></li><li><p> 29 </p><p>)u(fdt</p><p>dm= . </p><p> , : </p><p>,cosUFdt</p><p>dV m ,Vdtdy ,V</p><p>dtdx</p><p>xx</p><p>yx +=== </p><p>)u(fdtdm ,sinUF</p><p>dtdV</p><p> m yy =+=</p><p> (t0)=0, y(t0)=y0, m(t0)=m0, V(t0)=V0, Vy(t0)=Vy0. </p><p> , U - . , , . </p><p>2.2. </p><p> . , . , , . D- dynamic (). </p></li><li><p> 30 </p><p> (, ) (t)={1(t),2(t),...,m(t)}, Y(t)={y1(t),y2(t),..., yr(t)}. , , . , </p><p> i</p><p>i 1 n 1 mdz = f (t, z (t), ..., z (t), x (t), ..., x (t)), i = 1,ndt</p><p>. (2.6) </p><p> i- fi, t, Z={z1(t),z2(t),,zn(t)} (t)={1(t),2(t),...,m(t)}. , </p><p>j j 1 n 1 my = g (t,z (t), ..., z (t),x (t), ...,x (t)), j = 1,r . (2.7) </p><p>(2.6), t(0)=t0 0 0 01 0 1 2 0 2 n 0 nz (t ) = z , z (t ) = z , ..., z (t ) = z , , (t) (t0,t]: </p><p>0 0 0 0</p><p>t t t tt 1 t 2 t m t(X(t)] = {(x (t)] ,(x (t)] , ...,(x (t)] } . </p><p> (2.6) , </p><p>o</p><p>0 0 0 ti i 0 1 2 n tz (t) = j (t, t , z , z , ..., z ,(X(t)] ), i = 1,n . (2.8) </p><p> (2.6), t0 </p></li><li><p> 31 </p><p> )z,...,z,z(Z 0n02</p><p>010 = , F. </p><p>) )]t(X(,Z,t,t(F)t(Z tot00</p><p>= . (2.9) ) )]t(X(,Z,t,t( t</p><p>ot00 </p><p> Z(t), t-t0 (t0,Z0) t</p><p>ot)]t(X( . </p><p>),)]t(X(,Z,t,t(G)t(y tot00</p><p>= (2.10) G ))]t(X(,Z,t,t( t</p><p>ot00 </p><p> yt=y(t). </p><p> , , , . </p><p> z(t) , q- , , : </p><p>=</p><p>z...dt</p><p>zddt</p><p>zddt</p><p>zdq2q</p><p>2q</p><p>21q</p><p>1q</p><p>1q</p><p>q</p><p>x...dt</p><p>xddt</p><p>xddt</p><p>xdr2r</p><p>2r</p><p>21r</p><p>1r</p><p>1r</p><p>r</p><p>0 ++++= </p><p> (2.11) </p><p> dtdp = , </p><p> v(t) (2.11) </p></li><li><p> 32 </p><p>z()=-1()()()+v(), -1()=q-1q-1-2q-2--q, ()=0r+1r-1 + + r. </p><p> . , .. . , , : </p><p>dZ =Z + GX +W, Y = HZ + Vdt</p><p>. (2.12) </p><p> (2.12): Z={z1(t),z2(t),,zn(t)} - ; (t)={1(t),2(t),...,m(t)} ; Y(t)={y1(t),y2(t),..., yr(t)} ; W={w1(t),w2(t),,wn(t)} - </p><p>; T1 2 ndZ dz dz dz= { , , ..., }dt dt dt dt</p><p> - </p><p> ; , G, H , Z, (t), Y(t), W. , G, H , . , (2.12). </p><p> (2.12) , , G, H . </p><p> V.),t,X,Z(Y ,W),t(),t,X,Z(</p><p>dtdZ</p><p>+=+= (2.13) </p></li><li><p> 33 </p><p> (), () (...) , . , . , </p><p> , . Simulink MatLab. , . , . </p><p>2.3. ( ) . </p><p>2.3.1. . n- </p><p>n n-1</p><p>n n-1</p><p>d z(t) dz(t) d z(t)= f[t, z(t), , ..., , z(t - ),dt dt dt </p><p>n-1</p><p>n-1</p><p>dz(t - ) d z(t - ), ..., ]dt dt</p><p>. (2.14) </p><p> (2.14) </p><p>. : z=z1; 2dz(t) = z</p><p>dt; </p><p>2</p><p>32</p><p>dz (t) = zdt</p></li><li><p> 34 </p><p>.. (2.14) : </p><p>n</p><p>1 2 n 1 2 n</p><p>d z(t)= f[t, z (t), z (t), ..., z (t), z (t - ), z (t - ), ..., z (t - )]</p><p>dt. </p><p>)]t(z),t(z,t[fdt</p><p>)t(dz= , (2.15) </p><p> &gt;0, =onst, , , z(t) t&gt;t0. (2.15) </p><p> +=t</p><p>0t0 .d)](z),(z,[f)t(z)t(z (2.16) </p><p> (2.16) . t0 z0=z(t0) </p><p> z(t) t0-tt0 . (2.16) . </p><p> z(t) t&gt;t0, , z(t)=W(t) t[t0-,t0). f W z, . </p></li><li><p> 35 </p><p> W(t) t0-t</p></li><li><p> 36 </p><p>=+ =</p><p>=)tk(v)titk(x)ti(h)tk(z</p><p>0i </p><p>).tk(v)ti(x)titk(hk</p><p>i+ =</p><p>= </p><p> t=1, </p><p>=+ =</p><p>=)k(v)ik(x)i(h)k(z</p><p>0i </p><p>).k(v)i(x)ik(hk</p><p>i+ =</p><p>= (2.17) </p><p> (2.17) , h(i) , , i . . , </p><p> (2.17) h(k-i). </p><p>),k(v)i(x)i,k(h)k(zk</p><p>i+=</p><p>= </p><p> h(k,i) - k i. , </p><p> , , .. , .. . , : </p></li><li><p> 37 </p><p>- </p><p>t</p><p>=0 =-</p><p>z(t) = h()x(t - )d + v(t) = h(t - )x()d + v(t) ; (2.18) </p><p>- </p><p> +==</p><p>t).t(vd)(x),t(h)t(z </p><p> v(t). , . ( ) ( ) - </p><p>,)t(),t(hp</p><p>1iii=</p><p>= (2.19) </p><p> i(t) - t, [t1,t2]. , i, . i . </p><p>: (2.19) ; i(t) ; i . </p><p>2</p><p>1</p><p>t</p><p>i jit</p><p>0 i j (t) (t)dt =</p><p>c i = j, (2.20) </p><p> i i(t). </p><p>, </p></li><li><p> 38 </p><p>i i i (t) = (t)/ c . (2.20) </p><p>2</p><p>1</p><p>t</p><p>i j ijit</p><p>1 i = j (t) (t)dt = = </p><p>0 i j , (2.21) </p><p> ij - . i </p><p> (2.19) i(t) </p><p> 2 2</p><p>1 1</p><p>t t</p><p>i k k ik = 0t t</p><p>h(t) (t)dt = (t) (t)dt . </p><p> k=i , </p><p>2</p><p>1</p><p>t</p><p>i it</p><p> = S(t) (t)dt . (2.22) </p><p> . </p><p>2.4. </p><p> , ( ) (2.17). Z- </p><p>v(n)=vn </p><p> -nnn=0</p><p>v(z) = v z . </p><p> Z- , </p></li><li><p> 39 </p><p> z(z) = h(z)x(z) + v(z) / (2.23) Z- </p><p> . z z=es, Z- . (2.23) </p><p> Z- )z(z z(k) Z- )z(x (k). </p><p>)z(h - ( ), Z- h(k). )z(v - Z- v(k). </p><p> (2.18) , z(s)=h(s)(s)+v(s). z(s), h(s), (s), v(s) - z(t), h(t), (t), v(t); h(s) - , . </p><p>+-js</p><p>-</p><p>x(s) = x(t)e dt . </p><p> (2.18) , z(jw)=h(jw)(jw)+v(jw), z(jw), (jw), v(jw) - , , h(jw) - , , . </p></li><li><p> 40 </p><p>+-jwt</p><p>-</p><p>x(jw) = x(t)e dt . </p><p> , , z, s, j. , , , . </p><p>2.5. , </p><p> , [7]. , . t, ti z(ti), Z. F- finite automata . ti (i=1,2,...) </p><p> - (ti) , y(t) - Y. </p><p>A=, ={1,2,...,m} - ; Z={z1,z2,...,zn} - ; Y={y1,y2,...,yr} - . , Z, Y , </p></li><li><p> 41 </p><p>. , : </p><p>z(t)=[z(t-1),(t)] z(t)=[z(t),(t)], y(t) , : y(t)=[z(t),(t)]; y(t)=[z(t-1),z(t) (t)]; y(t)=[z(t-1), (t)]. </p><p>- , . . ={1,2,3}, Z={z1,z2,z3,z4}, Y={y1,y2,y3,y4}. </p><p>. 2.1, y(t)=[z(t),(t)] . 2.2. 2.1 </p><p> Z z1 Z2 z3 z4 </p><p>1 z1 Z2 z3 z4 2 z2 Z3 z4 z1 3 z4 Z1 z2 z3 </p><p> 2.2 </p><p>Z z1 z2 z3 z4 1 y1 y4 y3 y2 2 y2 y1 y4 y3 3 y3 y2 y1 y4 </p><p> i j zk, i </p></li><li><p> 42 </p><p> t, , t-1 zj. . 2.4 </p><p> . </p><p>x1 x1 </p><p>x1 </p><p>x3 x3 </p><p>x3 </p><p>x2 </p><p>x2 </p><p>x2 </p><p>z1 </p><p>x1 </p><p>x2 </p><p>x3 z2 </p><p>z3 </p><p>z4 </p><p>. 2.4 </p><p> ( ) Y. , z0, (t0), (t1), (t2),, y(t0), y(t1), y(t2), </p><p>, . , </p></li><li><p> 43 </p><p> , z(t)=[z(t),(t)], - y(t)=[z(t),(t)]. </p><p>: z(t)=[z(t),(t)], - y(t)=[z(t-1),(t)]. , </p><p> - y(t)=[z(t)], . </p><p> . , . . - y(t)=[(t)]. Y , </p><p> . </p><p> . 1. </p><p> 1,2 5 5 . Z=(0,1,2,3), =(1,2,5) Y=(0,1), 0 , 1 . (t) </p><p>Z(t)=(z(t-1)+(t))mod5, (t) </p><p>( ) ( )( ) ( )</p><p>0, z t -1 + x t &lt; 4,Y =</p><p>1, z t -1 + x t &gt; 4. </p></li><li><p> 44 </p><p> 2. - , . i- i- </p><p>(i=1,2,,n). . . n- Z=(Z1,Z2,,Zn), Zi - i- . (n+1) =(1,2,, </p><p>n,), i - i- , . =1, = -1. n Y=(Y1,Y2,,Yn), Yi(t)=Zi(t) . </p><p>Zi(t)=Zi(t-1)+(t), </p><p>Yi(t)=Zi(t). . 2.5 </p><p> , z(t)=[z(t-1),(t)], - y(t)=[z(t),(t)]. WWOD F(i,j), k zk, i t, , t-1 zj. </p></li><li><p> 45 </p><p> k 5 </p><p>j=k 3 </p><p>1 </p><p> WWOD 1 </p><p> () X, i </p><p>2 </p><p>0 7 </p><p> k=FP(i,j) 4 </p><p> w=FW(i,k) </p><p>6 </p><p>. 2.5 </p><p> WWOD FW(i,k), w () yw, i t, , t zk. WWOD , .. z0=zk. 3 . </p></li><li><p> 46 </p><p> 3. </p><p>3.1. </p><p> , , . , , . . , . , </p><p> , . , , . , . , . </p><p>3.1.1. . ( ). , (). , </p></li><li><p> 47 </p><p> . , ( ), [8]. [0,1], </p><p> , . (), 0()1. . , </p><p> . , , x1, x2, , xn, xi P(xi). , . 3.1, . </p><p> P(x3) P(x2) P(x1) </p><p>. . xn x4 x3 </p><p>x2 x1 </p><p>P(xn) </p><p> . 3.1 </p><p> x ....</p></li></ul>

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