Simulink Simulation

MATLAB and Simulink:

Dynamic Modeling

ChE 446

Simulink blockso Sources

• Constant (output a constant)• Step (output a step change)• Clock ( Outputs the time )

o Sinks• Scope (simulation time plot of input)• Display (real-time display of input value during

a simulation)• To Workspace (saves the input to the matlab

workspace)o Linear

• Sum (add or subtract inputs)o Nonlinear

• S-Function (user defined function)

Simulink M-file S-Functions

o Primary purposes• Simulating non-linear dynamics with MATLAB

o How they work• Example M-file S-function script (Simulink/User-

Defined Functions/S-function/examples/M-files/Level-1 M-files) explains the basics

• Each iteration, the S-function performs calculations based on the value of a flag (initialize, find derivatives, update actual values, etc.); it returns the answer, then changes the flag for the next iteration.

• The code is reasonably well-documented as to what to enter where; we’ll help later.


o Switch statementso switch flag

case 0Statements

case 1statements

case 2statements

otherwisestatements

end


o Case 0: initializationo First, Simulink sends flag=0, which:

o Declares things (x, t, y, u)o Sets x to an initial value (x0)o Returns ‘sys’, it’s standard return variable, set

equal to:[#cont. states., #discrete states, #outputs, #inputs, direct feedthrough?, #sample times]

o See http://www.mathworks.com/access/helpdesk/help/toolbox/simulink/sfg/f7-60056.html for more information.


o Case 1: calculate derivativeso When Simulink sends flag=1, it expects the

function to return time derivativeso You enter these derivatives like so:

o Derivatives are returned (as ‘sys’) as a vectoro See

http://www.mathworks.com/access/helpdesk/help/toolbox/simulink/sfg/f7-60056.html for more information.

,2,1,2,1 21 xxfxxfdxdtuxfdt

xd


o Case 3: calculate outputso When Simulink sends flag=2, it expects the

function to return outputso You enter these like so:

o Outputs are returned (as ‘sys’) as a vector or as a scalar, depending on the number of inputs


,2,1,2,1 21 xxgxxgyuxgy


o Cases 2,4,9: not usedo Case 2 updates discrete states, sample timeso Case 4 calculates the time that the next discrete

variable update occurs.o Case 9 executes any statements you want to

perform at the END of the simulation. Plot the output, for example.

o Cases 5-8 are not used; accordingly, there is an “otherwise” statement to catch these cases (which signal an error)


Dynamic Simulation: Linear vs. Nonlinear

o Solution to HW 2 will be posted on website, including S-functions for open loop and closed loop simulation.

o In-class exercise: compare the nonlinear model from the homework to a linearized model

Nonlinear model

21

1

3158

12

3

52

2

11

3

32

3

31

3

32

3

31

10

xx

xy

xu

ux

ux

dt

xd

xxx

xxxxxxxxx

Linearized model

xxxy

ux

uxxx

ux

ux

dt

xd

0

71

00

00

7

325

163

2325

1163

23

25

811

43

6415

325

3811

2143

23

36415

25

3325

811

43

6415

325

1

00

00

A

723

25

B 0325

163 C 0D

85

33

23

2225

11

3

yyxx

xxxx

r

1

r_ss

0.625

To Workspace

D

State -Space

x' = Ax + Bu y = Cx + Du

Scope 1

Scope

S-Function

bioreact

Input

0.76

D_ss

0.75

If you build it, …

Bioreactor S-functionfunction [sys,x0] = bioreact(t,x,u,flag)

Kx=1.0; Ky=5.0; Yx=0.5; Yy=0.75;muxmax=1.0; muymax=2.0; Si=10.0;switch flag,

case 1,D=u; X=x(1); Y=x(2); S=x(3);mux=muxmax*S/(Kx+S);muy=muymax*S/(Ky+S);dxdt = [-D*X+mux*X, -D*Y+muy*Y, D*(Si-S)-mux*X/Yx-muy*Y/Yy];

sys = dxdt;

Bioreactor S-function cont.case 3,X=x(1); Y=x(2); r=X/(X+Y); y = r;sys = y;

case 0,NumContStates = 3; NumOutputs = 1; NumInputs = 1;sys = [NumContStates,0,NumOutputs,NumInputs,0,0];x0 = [2.5 1.5 3.0];

case { 2, 4, 9 },sys = [];

Otherwise% error([’Unhandled flag = ’,num2str(flag)]);end

Tasks

o Compare response of linear model to non-linear model for:o Constant inputo Step inputo Sine input

o Compare time-varying functions (step, sine, etc.) for both large and small deviations from steady-state

Documents

Simulink Simulation