微分代数方程式とINDEXの低減 - cybernet.co.jp

1 sice . INDEX . 2009/09/25.. 1.. 2.. 3. INDEX . 4. INDEX . 5.. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 2. 1.. y x 2 FEM . y y x x .. y t . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 3. 1. / .. y = f ( y ,t ). Euler y n +1 = y n + hf ( y n , tn ). Euler y n +1 = y n + hf ( y n +1 , tn +1 ). * .. [0,b] Euler .. * 3) . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 4.. max ( A ) max ( AT A ). MATLAB . min ( A ) min ( AT A ). * A . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 5.. y2. m2. k2 c2. y1 , f1. m1. k1 c1. 2 . x1 = c1 x 1 k1 x1 c2 ( x 1 x 2 ) k2 ( x1 x2 ). m1 .. x2 = c2 ( x 1 x 2 ) + k2 ( x1 x2 ). m2 . m1 = 2(kg ), c1 = 5 10, k1 = 5 103.. m2 = 1(kg ), c2 = 1 10 10 , k2 = 1 108. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 6. Simulink . Simulink . 9. 10 . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 7.

2 A) 5 10 5 b) 2 10 5. Dormand-Prince . Dormand-Prince Runge-Kutta 5 .. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 8.. a) ode45 (Dormand-Prince) b) ode23 (Bogacki-Shampine). c) ode15s (Stiff Solver) d) ode23s (Stiff / Rosenbrock .. ode45 . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 9. Newton-Raphson . y 0 x3 x x1 x0 x 2.. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 10. Simulink . y2 + y = x x y . +1 . Simulink . -1 . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 11. 2.. 3) . DAE: Differential-Algebraic Equation DAE . F (t, y, y ) = 0 (1). (2) .. x = f (t, x, z ) (2a). 0 = g (t, x, z ) (2b). x(t) (2a) z(t) . (2b) .. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 12. DAE . DAE Differential Algebraic Equation .. DAE . 1 Runge-Kutta, etc . Newton-Laphson, etc .. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 13.

3 X1,x2 . x1. Lagrange Newton o (3) .. x 1 = x 1. (3) l x 2 = x 2 g (4) .. x 12 + x 22 = 1 (4) x2. 1 (5) . (2) . x 1 = x 3. x 2 = x 4 mg (5). x 3 = x 1. x 4 = x 2 g * 1 . 1 . x 12 + x 22 = 1. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 14. 3. INDEX . INDEX DAE ODE . INDEX y y t . F (t, y, y ) = 0. dF. dt (t, y, y , y ) = 0. #. d pF. dt ( ). t, y, y , ", y ( p +1) = 0. INDEX p . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 15.. y1 = q (t ) . y2 = y1 . y2 = y1 = q (t ). y2 = y1 = q (t ). q (t) 2 . INDEX 2. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 16. 4. INDEX . Dymola INDEX Pantelides . 5) . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 17. 1/2. x1,x2 . x1. Lagrange Newton o (3) .. x 1 = x 1. (3) l x 2 = x 2 g (4) .. x 12 + x 22 = 1 (4) x2. 1 (5) . (2) . x 1 = x 3. x 2 = x 4 mg (5). x 3 = x 1. x 4 = x 2 g * 1 . 1 . x 12 + x 22 = 1.

4 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 18. 2/2.. x 1 = x3 x . 1 x3 .. x 2 = x4 x x4 . ODE d . 2 . x 3 = x 1 x 3 = x 1 . dt . x 4 = x 2 g x 4 . x 2. g .. x 12 + x 22 = 1 4 (x 1x 3 + x 2x 4 ) 3gx 4 .. 1 . 2x 1x 1 + 2x 2x 2 = 0. 2x 1x 3 + 2x 2x 4 = 0.. 2 . Index DAE . 2 (x 1x 3 + x 1 x 3 ) + 2 (x 2x 4 + x 2 x 4 ) = 0 .. (x 2. 3 ) ( ). + x 42 x 12 + x 22 gx 2 = 0.. 3 . ( ). 2x 3x 3 + 2x 4x 4 (2x 1x 1 + 2x 2x 2 ) x 12 + x 22 gx 2 = 0. = 4 (x 1x 3 + x 2x 4 ) 3gx 4. INDEX-3 . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 19.. x 1 = x3. x 2 = x4. x 3 = x 1. x 4 = x 2 g x 12 + x 22 = 1.. 1 . 2x 1x 1 + 2x 2x 2 = 0.. 2x 1x 3 + 2x 2x 4 = 0.. 2 . 2 (x 1x 3 + x 1 x 3 ) + 2 (x 2x 4 + x 2 x 4 ) = 0. (x 2. 3 ) ( ). + x 42 x 12 + x 22 gx 2 = 0.. 3 . ( ). 2x 3x 3 + 2x 4x 4 (2x 1x 1 + 2x 2x 2 ) x 12 + x 22 gx 2 = 0. = 4 (x 1x 3 + x 2x 4 ) 3gx 4. 2009 CYBERNET SYSTEMS CO.

5 ,LTD. All Rights Reserved. 20. Linear DAE INDEX . Linear DAE . Ez = Az + b Non-singular P Q . E . PEQ = 11 . 0 . y=Qx P . E . 11 z = A11 z + b1 . 0 A b . 21 2 . A21z + b2 . E . 11 z = A11 z + b1 .. A21 0 b2 . INDEX. New E . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 21. DAE . x1. o . o I = T . l ml 2 = mg sin . l x2. sin . sin .. mg . * 1 . 1 . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 22.. x 1 = x 1. x 12 + x 22 = 1? .. x 2 = x 2 g 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 23. 5.. Robertson DAE. INDEX . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 24. Robertson DAE. x1 = x1 + 1 104 x2 x3. x2 = x1 1 104 x2 x3 3 107 x22. 0 = x1 + x2 + x3 1. 1 ODE . INDEX-1 DAE. x1 = x1 + 1 104 x2 x3. x2 = x1 1 104 x2 x3 3 107 x22. x3 = 3 107 x22.. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 25. ODE vs DAE with Simulinik 1. Out1.

6 1. s Integrator Gain 2. Out2 1. 1 y_1. s 1e4. y_1 y_1. Integrator1 Gain1 1/s y_1'>y_1. u2 3e7. 1. Math Gain4 2. Function f(z)Solve z 3. y_2 f(z) = 0. 1e4 y_3. 1/s Algebraic Constraint y_2. y_2 y_2'>y_2. 2. u 3e7. Math 1. 3e7 s 3 Function Out3. Gain5 Integrator2. y_3. x1 = x1 + 1 104 x2 x3 x1 = x1 + 1 104 x2 x3. x2 = x1 1 104 x2 x3 3 107 x22 x2 = x1 1 104 x2 x3 3 107 x22. x3 = 3 107 x22 0 = x1 + x2 + x3 1. ODE IINEX-1 DAE . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 26. DAE vs ODE: Simulation Results 1. Robertson DAE problem converting to ODE 10-5 order 1 x1 x 10. -5 Constrain error Out1 2. 1. s x2. 1. error: 0=x 1 +x 2 +x 3 -1. Integrator Gain x3. Error divergence x 1 , 1e4*x 2 , x 3. 2 0. Out2 1. 0 = x1 + x2 + x3 1. s 1e4. Integrator1 Gain1 -1. u2 3e7. Math Gain4. -2. Function -3. 1. 3e7 3. Gain5. s Integrator2. Out3. -4 -3 -1 1 3 5 7. 10 10 10 10 10 10.

7 0 0 5 Time (s). ODE 10. Time (s). 10.. 1. Robertson DAE problem with a Conservation Law 10-16 order x1 x 10. -16 Constrain error 0. x2. error: 0=x 1 +x 2 +x 3 -1. x3. 0 = x1 + x2 + x3 1. 1. y_1. x 1 , 1e4*x 2 , x 3. y_1 y_1. 1/s y_1'>y_1. 1. 2. f(z)Solve z 3. y_2 f(z) = 0. 1e4 y_3. 1/s y_2 Algebraic Constraint y_2. u2 3e7. y_2'>y_2. Math Function -1. y_3. DAE 10. -3. 10. -1. 10. 1. 10. 3. 10. 5. 10. 7. 0. 10. 0. 10. 5 Time (s). Time (s). 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 27. INDEX . ODE ODE . 2. ml = mg sin 1 1. s s Integrator Integrator1 Scope sin Gain Trigonometric Function INDEX DAE INDEX . x1 = x1 INDEX-1 ODE . x 2 = x 2 g x12 + x 22 = 1.. INDEX . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 28. ODE vs DAE: Simulation Results x-position y-position 1. position x, y 0. -1. 0 1 2 3 4 5. Time (s). 0 5(s) 0 104(s). ODE . x-position y-position 1.

8 0. 1. 0 1 2 3 4 5. Time (s). 0 5(s) 0 104(s). DAE . 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 29.. DAE . DAE ODE .. 9 DAE . 9 INDEX DAE . 9 INDEX DAE INDEX . ODE . 9 . / .. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 30.. 1) 1998. 2) 2003. 3) . 2006. 4) 1991. 5) Michael M. Tilller Modelica . 2003. 2009 CYBERNET SYSTEMS CO.,LTD. All Rights Reserved. 31.