1. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group TLTE.3120 Computer Simulation in Communication and Systems (5 ECTS) http://www.uva.fi/~timan/tlte3120/ Lecture 5 – 07.10.2015 Timo Mantere Professor, Communications & systems University of Vaasa http://www.uva.fi/~timan timan@uva.fi 1

2. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES 2 Model structures relates on mathematical modelling and system identification, see additional information e.g. from: Mathematical model: http://en.wikipedia.org/wiki/Mathematical_model Dynamical system: http://en.wikipedia.org/wiki/Dynamical_system System that changes over time, e.g. pendulum, quite often they might have unpredictable behavior due that the system parts interact with each others

3. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  Static models  Linear  Nonlinear  Dynamic models  Continuous differential equations  Continuous time-delay equations  Difference equations  Partial differential equations  Transfer functions  SIMULINK – starters  Stochastic processes  Parameterized models – Curve fitting 3

4. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  STATIC MODELS  Linear, scalar y = k u + b y, u real numbers y output; u input; k gain; b bias EXAMPLE: OHM’s Law U=RI U = voltage (output) I = current (input) R = resistor 4 Interferece etc. b Output y Input u System k

5. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  STATIC MODELS  Nonlinear, scalar y = f(u; a), a is a parameter EXAMPLE: OHM’s Law with nonlinear resistor R = R(I) = R 0 I 2 ; Resistor depends on current

6. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  STATIC MODELS  Linear, multi-input multi-output (MIMO) y = Au + b y is output vector (m elements) u input vector (n elements) b constant vector (m elements) A matrix (n*m)

7. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  EXAMPLE  Linear, multi-input multi-output (MIMO) y = Au 7

8. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  MATLAB - EXAMPLE >> A=[2 0;3 6] A = 2 0 3 6 >> u=[0.5; 0.4] u = 0.5000 0.4000 >> y=A*u y = 1.0000 3.9000 8

9. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Linear, scalar  If the system input changes as a function of time we need dynamic model a and b are constants 9

10. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS - EXAMPLE  1798: Model of population dynamics (Malthus, 1766-1834) : Assume that the growth is directly proportional to population size. Population at year 1960 was 3.09. 10 9 and its growth 2%. Compute how population evolved up to year 2007. 10

11. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group Year Population Average annual growth rate (%) Average annual population change 19603,039,669,3301.3340,792,172 19693,632,780,6142.0575,286,491 19703,708,067,1052.0777,587,001 19713,785,654,1062.0176,694,660 19723,862,348,7661.9576,183,283 19733,938,532,0491.9075,547,218 20006,085,478,7781.2174,220,528 20016,159,699,3061.1873,002,863 20026,232,702,1691.1672,442,511 20036,305,144,6801.1472,496,962 20046,377,641,6421.1372,578,164 20056,450,219,8061.1272,540,568 20066,522,760,3741.1072,466,183 20076,595,226,5571.0972,442,792 20086,667,669,3491.0872,368,570 20096,740,037,9191.0772,210,364

12. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES 12

13. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES 13

14. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  MATLAB – SIMULINK SOLUTION  Consider the left-hand side of the equation – integrate it:  In SIMULINK, the integral operator is – in goes out comes x(t). x(t)

15. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  MATLAB – SIMULINK SOLUTION  What about x(t 0 )? Click the integrator block open. In the Initial condition type 3.04*10^9.  Until now, we have handled the left-hand side of the equation.  In order to have balance in the equation, we need to complete the SIMULINK diagram with the term 0.02x(t).

16. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  MATLAB – SIMULINK SOLUTION x(t) 0.02x(t)

17. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  MATLAB – SIMULINK SOLUTION – SOLUTION DISPLAYED WITH A SCOPE x(t) 0.02x(t) Simulation result

18. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  Logistic population model  Population cannot grow without limit. One way to restrict the growth is to introduce a negative quadratic term (Verhulst). 18

19. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  Logistic population model  Verhulst used the model to predict quite accurately the population growth in USA with the following model. The initial population is from year 1790. 19

20. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  SIMULINK configuration of Verhulst logistics equation Simulation result 20

21. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group Model Structures, nonlinearity  Non-parametric models  Weight- and frequency functions can be measured directly from the process  A prior knowledge about the model structure is not required  Polynomial models  Based on preliminary prior knowledge about the model structure  Nonlinear systems m ay be simplified such that the parameter estimation problem becomes linear  Linearization  Simplification have its risks and it is not always possible  Computation programs  In-build optimization and estimation routines (Matlab etc.) 21

22. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Nonlinear, multi-input multi-output (MIMO)  Vector differential equation is in form:  Where x, u, f, and x(dot) are column vectors, X 0 is the start value at time point 0  In dynamic systems often part of the states are measured, so y is measurement data 22

23. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Nonlinear, multi-input multi-output (MIMO)

24. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Nonlinear, multi-input multi-output (MIMO) 24

25. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Nonlinear, multi-input multi-output (MIMO) 25

26. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Linear, multi-input multi-output (MIMO) A (nxn), B (nxm), C (pxn) and D (pxm) real, constant 26

27. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Delay equations, multi-input multi-output (MIMO) Initial datahistory 27

28. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Delay equations L QiQi Input mass flow QoQo Output mass flow vSpeed Length 28

29. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Delay equations

30. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group MODEL STRUCTURES  DYNAMIC MODELS  Discrete time or 30

31. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group ELLIPTICAL membrane PARTIAL DIFFERENTIAL EQUATIONS 31 PARABOLIC

32. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS 32 DISCRETIZATION

33. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group Discretization w.r.t. space variable PARTIAL DIFFERENTIAL EQUATIONS where 33

34. U NIVERSITY of V AASA Communications and Systems Engineering Group U NIVERSITY of V AASA Communications and Systems Engineering Group Discretization w.r.t. space variable leads to PARTIAL DIFFERENTIAL EQUATIONS 34