Solving Differential Equations with Simulink

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1 Solving Differential Equation with Simulink Dr. R. L. Herman UNC Wilmington, Wilmington, NC March, 206 Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 /9

2 Outline Simulink 2 Solution of ODE 3 Firt Order Differential Equation 4 Second Order Differential Equation 5 Linear Stem 6 Nonlinear Stem 7 Summar Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 2/9

3 Simulink What i Simulink Graphical environment for deigning imulation Product of Mathwork Select and connect block Gain Logitic Equation ' Ue in Differential Equation Project component of cla Modeling application ' = r (-) Product - Contant Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 3/9

4 Solving a Differential Equation Conider initial value problem: dx dt = f (x), x(0) = x 0. Solution t x(t) = x 0 + f (x(t)) dt. 0 Think of the olution a x(t) = x (t) dt. input x x output Figure : Schematic for a general tem. Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 4/9

5 Simulink Model of a Differential Equation Modeling x(t) = x (t) dt = f (x(t)) dt. Schematic input x x output Simulink Model f (x) x x Output Figure : Model for olving x = f (x). Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 5/9

6 Firt Order Differential Equation - Example Solve x = 4x, x(0) =. x x 4 Gain Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 6/9

7 Simulink Workpace Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 7/9

8 Scilab Xco Workpace Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 8/9

9 Firt Order Differential Equation - Example Solve x = 2 in 3t 4x, x(0) = 0. Sine Wave Function 4 Gain Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 9/9

10 Firt Order Differential Equation - Example 3 Solve = 2 t + t2, () =. u 2 t 2 d/dt Clock t Math Function2 /t 2 2/t u Math Gain Function Product 2/t 2 ' = 2/t +t, ()= Exact olution: (t) = t 3 Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 0/9

11 Second Order ODE Solve a + b + c = 0, (0) = 0, (0) = v 0. = dx, = dx, = b a c a. Second Order Contant Coefficient ODE '' ' b/a ' 5 b/a c/a 6 c/a Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 /9

12 Linear Stem of Differential Equation x = ax + b = cx + d. b 0 a x x(0) x x o 2 (0) x o XY Graph Linear Stem of Differential Equation 0 d - x x'=ax+b '=cx+d c Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 2/9

13 Coupled Ocillator x = k x b x + k 2 ( x) + b 2 ( x ) = k 2 ( x) b 2 ( x ). Damped Ocillator m x'' = - k x - b x' + k2 (-x) + b2 ('-x') /2 /m x'' x' x 0 b 5 k b2 ('-x') 0 b2 k2 (-x) '' -/2 -/m2 2 ' k2 5 3 m2 '' = - k2 (-x) - b2 ('-x') Damped Ocillator 2 Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 3/9

Coupled Ocillator - Solution x(t) = co (t) = 5 ( ) 5 2t co 4 5 in ( 5 4 ( ) 0t 3 ( ) ( ) 5 5 in 2t in 0t 4 5

14 Coupled Ocillator - Solution x(t) = co (t) = 5 ( ) 5 2t co 4 5 in ( 5 4 ( ) 0t 3 ( ) ( ) 5 5 in 2t in 0t ( ) ( ) ( ) 5 0t co 2t co 0t t ) in Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 4/9

15 Linear Stem - Matrix Form [ x ] = [ 0 0 ] [ x ] [x';'] [;2] IC xo [x;] x XY Graph 0-0 * u Linear Stem of Differential Equation Gain Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 5/9

16 Application - Lotka Volterra Predator-Pre Model x = x ax = + bx a 50 IC x' xo Rabbit x Predator-Pre Model x' = x - ax ' = - + bx x x Product 0.0 b ' 0 xo Foxe XY Graph IC Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 6/9

17 Application - SIR Epidemic Model S = β SI N, where N = S + I + R. I = β SI N γi, R = γi..5 - beta Gain Product S SIR Model Divide N I.33 gamma Product 2 R Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 7/9

18 Application - Logitic Equation with Dela Solve = r(t)( (t τ)) where τ > 0 provide a delaed repone. 2 Alpha >= ' Clock 0 Switch to enter =2 for t<= Solve '=0 (-(t-)) Logitic Equation with Dela ' = alpha (t)(-(t-)) Product -(t-) (t-) Tranport Dela Contant Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 8/9

19 Summar Demo of Simulink Model for Simulation of Differential Equation Graphical Interface Stem Deign Quick Entr to Application Sine Wave Function 4 Gain Preparation to Engineering Application Thank You! hermanr@uncw.edu Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 9/9

Correction for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002

Correction for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002 Correction for Simple Sytem Example and Note on Laplace Tranform / Deviation Variable ECHE 55 Fall 22 Conider a tank draining from an initial height of h o at time t =. With no flow into the tank (F in