1

Simulink Basics Tutorial

Starting Simulink Model Files Basic Elements Running Simulations Building Systems

Simulink is a graphical extension to MATLAB for modeling and simulation of systems.

In Simulink, systems are drawn on screen as block diagrams. Many elements of block diagrams are available, such as transfer functions, summing junctions, etc., as well as virtual input and output devices such as function generators and oscilloscopes.

Simulink is integrated with MATLAB and data can be easily transfered between the programs.

Simulink is supported on Unix, Macintosh, and Windows environments; and is included in the student version of MATLAB for personal computers

2

STARTING SIMULINK Simulink is started from the MATLAB command prompt

>> Simulink

Alternatively, you can hit the New Simulink Model button at the top of the MATLAB command window

When it starts, Simulink brings up the main Simulink window

and a blank, untitled, model window. This is the window into which a new model can be drawn.

3

MODEL FILES In Simulink, a model is a collection of blocks which, in general, represents a system.

In addition, to drawing a model into a blank model window, previously saved model files can be loaded either from the File menu or from the MATLAB command prompt.

Example:

simple.mdl

The following model window should appear.

4

BASIC ELEMENTS

There are two major classes of items in Simulink: blocks and lines. Blocks are used to generate, modify, combine, output, and display signals. Lines are used to transfer signals from one block to another.

Blocks

There are several general classes of blocks:

• Sources: Used to generate various signals • Sinks: Used to output or display signals • Discrete: Linear, discrete-time system elements (transfer

functions, state-space models, etc.) • Linear: Linear, continuous-time system elements and

connections (summing junctions, gains, etc.) • Nonlinear: Nonlinear operators (arbitrary functions,

saturation, delay, etc.) • Connections: Multiplex, Demultiplex, System Macros, etc.

Lines

Lines transmit signals in the direction indicated by the arrow. Lines must always transmit signals from the output terminal of one block to the input terminal of another block. One exception to this is a line can tap off of another line, splitting the signal to each of two destination blocks, as shown below

5

Lines can never inject a signal into another line; lines must be combined through the use of a block such as a summing junction.

A signal can be either a scalar signal or a vector signal.

Simple Example

The simple model consists of three blocks: Step, Transfer Fcn, and Scope. The Step is a source block from which a step input signal originates. The Transfer Function linear block. The Transfer Function modifies its input signal and outputs a new signal. The Scope is a sink block used to display a signal much like an oscilloscope.

6

MODIFYING BLOCKS A block can be modified by double-clicking on it.

This dialog box contains fields for the numerator and the denominator of the block's transfer function. By entering a vector containing the coefficients of the desired numerator or denominator polynomial, the desired transfer function can be entered. For example, to change the denominator to s2+2s+1, enter the following into the denominator field: [1 2 1]

7

The "step" block can also be double-clicked, bringing up the following dialog box.

The default parameters in this dialog box generate a step function occurring at time =1 s, from an initial level of zero to a level of 1.

The most complicated of these three blocks is the "Scope" block. Double clicking on this brings up a blank oscilloscope screen.

8

When a simulation is performed, the signal which feeds into the scope will be displayed in this window. Use of the autoscale button, which appears as a pair of binoculars.

9

RUNNING SIMULATIONS To run a simulation, we will work with the following model file:

simple2.mdl

Before running a simulation of this system, first open the scope window by double-clicking on the scope block. Then, to start the simulation, either select Start from the Simulation menu

The simulation should run very quickly and the scope window will appear.

10

Note that the simulation output (shown in yellow) is at a very low level relative to the axes of the scope. To fix this, hit the autoscale button (binoculars)

We can change the parameters of the simulation also. In the model window, select Parameters from the Simulation menu. You will see the following dialog box.

11

There are many simulation parameter options; we will only be concerned with the start and stop times, which tell Simulink over what time period to perform the simulation. Change Start time from 0.0 to 0.. Change Stop time from 10.0 to 2.0, which should be only shortly after the system settles.

.

12

BUILDING SYSTEMS Learn how to build systems in Simulink using the building blocks in Simulink's Block Libraries.

1. Gather all the necessary blocks from the block libraries. 2. Modify the blocks so they correspond to the blocks in the desired model 3. Connect the blocks with lines to form the complete system. 4. Simulate the complete system to verify that it works.

13

Gathering Blocks

Create a new model (New from the File menu or Ctrl-N).

Double-click on the Sources icon in the main Simulink window.

14

Drag the the Sum, Gain, and two instances of the Transfer Fcn (drag it two times) and a Sink into your model window

15

Modifying Blocks

Follow these steps to properly modify the blocks in your model.

• Modify the Sum block. • Modify the Gain block. Change the gain to 2.5. • Modify the left-most Transfer Function block. Change the

numerator to [1 2] and the denominator to [1 0]. • Modify the rightmost Transfer Function block. Leave the

numerator [1], but change the denominator to [1 2 4].

• Change the name of the first Transfer Function to "PI Controller", and the name of the second Transfer Function block to "Plant".

16

Connecting Blocks with Lines

Now that the blocks are properly laid out, connect them together.

The line remaining to be drawn is the feedback signal connecting the output of the Plant to the negative input of the Sum block

Finally, labels can be placed in the model to identify the signals. To place a label anywhere in your model, double click at the point you want the label to be.

17

To save the model, select Save As in the File menu and type in any desired model name.

Simulation

Now that the model is complete, you can simulate the model. Select Start from the Simulation menu to run the simulation. Double-click on the Scope block to view its output.

18

Taking Variables from MATLAB

In some cases, parameters may be calculated in MATLAB to be used in a Simulink model. If this is the case, it is not necessary to enter the result of the MATLAB calculation directly into Simulink. For example, suppose we calculated the gain in MATLAB in the variable K >> K=2.5 This variable can now be used in the Simulink Gain block.

Re-run the simulation and view the output on the Scope. The result should be the same as before.

19

Now, if any calculations are done in MATLAB to change any of the variables used in the Simulink model, the simulation will use the new values the next time it is run. >> K=5 Start the Simulink simulation again, bring up the Scope window

20

Simulink Modeling Tutorial

In Simulink, it is very straightforward to represent a physical system or a model. In general, a dynamic system can be constructed from just basic physical laws. We will demonstrate through an example.

Train system

In this example, we will consider a toy train consisting of an engine and a car. Assuming that the train only travels in one direction, we want to apply control to the train so that it has a smooth start-up and stop, along with a constant-speed ride.

The mass of the engine and the car will be represented by M1 and M2, respectively. The two are held together by a spring, which has the stiffness coefficient of k. F represents the force applied by the engine, and the Greek letter, mu (which will also be represented by the letter u), represents the coefficient of rolling friction.

21

Free body diagram and Newton's law

The system can be represented by following Free Body Diagrams.

From Newton's law, you know that the sum of forces acting on a mass equals the mass times its acceleration.

In this case, the forces acting on M1 are the spring, the friction and the force applied by the engine. The forces acting on M2 are the spring and the friction.

We will begin to construct the model simply from

Sum(forces_on_M1)=M1*x1'' Sum(forces_on_M2)=M1*x2''

Constructing The Model

First, we will construct two copies (one for each mass) of the expressions

a=x’’=1/M*sum_F.

22

The outputs of these gain blocks are the accelerations of each of the masses. We are interested in both the velocities and the positions of the masses. Since velocity is the integral of acceleration, and position is the integral of velocity, we can generate these signals using integrator blocks.

23

Now, drag two Scopes from the Sinks library into your model and connect them to the outputs of these integrators. Label them "View_x1" and "View_x2".

Now we are ready to add in the forces acting on each mass. First, you need to adjust the inputs on each Sum block to represent the proper number.

24

The first force acting on M1 is just the input force, F.

The next force acting on M1 is the friction force. This force is equal to: F_friction_1=mu*g*M1*v1 and acts in the negative x1-direction.

25

The last force acting on M1 is the spring force between masses. This is equal to: F_spring = k*(x1-x2) First, we need to generate (x1-x2). Since this summation comes from right to left, we need to flip the block around. Select the bloc by single-clicking on it and select Flip from the Format menu

26

Now, we can apply forces to M2. For the first force, we will use the same spring force we just generated, except that it adds in with positive sign.

The last force to add in the the friction on M2. This is done in the exact same manner as the friction on M1, tapping off v2, multiplying by a gain of mu*g*M2

27

Now the model is complete. We simply need to supply the proper input and view the proper output. The input of the system will be the force, F, provided by the engine. We already have placed the function generator at the input. The output of the system will be the velocity of the engine.

Running the Model

Before running the model, we need to assign numerical values to each of the variables used in the model. Create an new m-file and enter the following commands.

M1=1; M2=0.5; k=1; F=1; mu=0.002; g=9.8;

Execute your m-file to define these values. Simulink will recognize MATLAB variables for use in the model.

Now, we need to give an appropriate input to the engine. Double-click on the function generator (F block). Select a square wave

28

with frequency .001Hz and amplitude -1 (positive amplitude steps negative before stepping positive).

The last step before running the simulation is to select an appropriate simulation time. To view one cycle of the .001Hz square wave, we should simulate for 1000 seconds. Select Parameters from the Simulation menu Now, run the simulation and open the View_v1 scope

29

Simulink Basics Tutorial - Block Libraries

Sources Sinks Discrete Linear Nonlinear Connections

Simulink contains a large number of blocks from which models can be built. These blocks are arranged in Block Libraries which are accessed in the main Simulink window.

Each icon in the main Simulink window can be double clicked to bring up the corresponding block library. Blocks in each library can then be dragged into a model window to build a model.

30

SOURCES Source Blocks are used to generate signals. Double-click on the Sources icon in the main Simulink window to bring up the Sources window.

Notice that all of the source blocks have a single output and no inputs. While parameters in each of these blocks in the library can be modified by double-clicking the block, it is best to not modify the blocks until they have been copied into a model window.

31

Constant

The Constant Source Block simply generates a constant signal. The constant output value is displayed in the middle of the block, with a default value of 1.

To change the constant output value, double-click on the block in your model window to bring up the following dialog box.

Change the constant value field from 1 to some other value, say, 5, and close the dialog box. Your model window will reflect the update by displaying a 5 in the middle of the constant block.

32

Signal Generator

The Signal Generator Source Block is a general-purpose source which encompasses some of the other blocks' functions. It generates periodic waveforms such as sine, square, and saw-tooth waves as well as a random signal.

By default, the Signal Generator generates a sine wave with an amplitude of 1 and a frequency of 1 Hz. To change this, double-click the Signal Generator in your model window to bring up the following dialog box.

33

The Amplitude and Frequency can be changed in this dialog box, as well as the type of waveform. To change the waveform, click on the Waveform field to bring up a list of possible waveforms.

Ramp

The Ramp Source Block generates a signal which is initially constant and begins to increase (or decrease) at a constant rate at a specified time. The slope, start time, and initial output can be specified.

Sine Wave

The Sine Wave Source Block generates a sinusoidal signal. The Amplitude and Frequency can be specified, as well as the Phase (unlike the Signal Generator). There is a fourth parameter, the Sample Time, which can be used to force the Sine Wave Source to operate in discrete-time mode (more about Discrete Time systems in Simulink later.)

Step

As described earlier, the Step Source Block generates a step function. The initial and final values can be specified, as well as the step time.

34

Chirp Signal

The Chirp Signal Source Block generates a sinusoidal signal which scans over a range of frequencies. The initial and final frequencies as well as the scan time can be specified. The amplitude is always 1, and the chirp signal repeats itself after each frequency scan.

Pulse Generator

The Pulse Generator Source Block generates a pulse train of varying duty cycle. The signal switches between 0 and the specified value starting at a particular time. The Period, Duty Cycle, Amplitude, and Start Time can be specified.

Repeating Sequence

An arbitrary set of points (t,y) can be specified. These points are entered as a vector specifying the time values, and a vector specifying the corresponding output values at those times. The output is linearly interpolated between the specified time values. At the last time value, the output immediately starts over, possibly with a discontinuous transition.

Clock

The Clock Source Block generates a signal equal to the current time in the simulation. This is useful when the output of a simulation is exported to MATLAB but occurs at uneven time steps. The clock's output reflects the times at which the other signals outputs occur.

Digital Clock

The Digital Clock Source Block generates a strictly periodic time signal at a specified sampling interval.

35

From File

The From File Source Block outputs a signal taken from a specified .mat file. A matrix saved in MATLAB as a .mat file will become a signal where the first row of the matrix specifies the time values. This is similar to the Repeating Sequence Source Block.

From Workspace

The From Workspace Source Block is identical to the From File Source Block except the values are taken from a variable (or expression) in the MATLAB Workspace.

Random Number

The Random Number Source Block generates a sequence of random numbers generated with the specified random number seed. Because of the seed, the same sequence can be applied to more than one simulation.

Band-Limited White Noise

The Band-Limited White Noise Source Block generates a random signal which changes at a specified sample period. The strength of the signal and a random number seed can also be specified.

36

SINKS Sink Blocks are used to display or output signals.

Notice that all of the sink blocks have inputs and no outputs. Most have a single input.

Scope

The Scope Sink Block was described earlier. It is used to display a signal as a function of time.

XY Graph

The XY Graph Sink Block plots one signal against another. It is useful for phase-plane plots, etc.

37

Display

The Display Sink Block is a digital readout of a signal at the current simulation time.

To File

The To File Sink Block saves a signal to a .mat file in the same way that the From File Source Block reads from a file. The sampling time can be specified, but is not necessary.

To Workspace

The To Workspace Sink Block stores a signal in a specified workspace variable. Unlike the To File Sink Block, the time is not saved in the variable, and must be stored separately.

Stop Simulation

This is a special control block which is triggered to stop the current simulation when its input is non-zero.

38

DISCRETE Discrete Blocks are elements of discrete time dynamic systems. Double-click on the Discrete icon in the main Simulink window to bring up the Discrete window.

In order for these block to interact with continuous time blocks (sources and sinks, for example) the sample time can be specified in all of the Discrete Blocks.

Unit Delay

The Unit Delay's output is equal to the input delayed by one sample time.

39

Discrete-Time Integrator

This is the discrete time approximation of a continuous-time integrator. The approximation method can be specified as well as the initial condition and saturation limits.

Zero-Order Hold

Outputs a stepwise-constant version of the input signal with a specified sampling period.

First-Order Hold

Outputs a piecewise-linear version of the input signal with a specified sampling period.

Discrete State-Space

This is a discrete-time dynamic system in state-space form. A, B, C, and D matrices can be specified, as well as initial conditions.

Discrete Filter

This is a discrete-time filter in rational function form. Vectors containing coefficients of the polynomials in z^-1 are specified.

Discrete Transfer Fcn

This is the standard form of a SISO LTI discrete time system. The transfer function polynomials are represented as coefficient vectors in terms of z.

Discrete Zero-Pole

A discrete-time transfer function can be represented as list of z-plane poles and zeros. The gain can also be specified.

40

LINEAR

Linear Blocks are elements of linear continuous-time dynamic systems

Gain

This is a scalar or vector gain. The specified gain multiplies the input. The output is either a scalar or vector signal following normal vector-scalar multiplication rules.

Sum

The Sum Block adds (or subtracts) two (or more) signals and outputs their sum (or difference). The two inputs must either all be scalars, or all be vectors of the same dimension. The output is the same dimension as the inputs.

41

By default, the Sum block adds two signals. The list of signs field represents both the number of inputs and whether to add or subtract them. To make the Sum block add two signals and subtract a third, change the list of signs to the following:

++- Each element in the list corresponds to one of the signals. Close the dialog box. The Sum block in your model window should now have three inputs, one of which has a minus sign as shown below.

Integrator

The output of the Integrator is the integral of the input. An initial condition can be specified, as well as saturation limits. This block is very useful for modeling systems.

Transfer Function

Numerator and denominator polynomials can be specified to create a standard SISO LTI system transfer function.

42

State Space

A, B, C, and D matrices can be specified to create a LTI state space system. Inputs and outputs may be vector signals depending on the sizes of the matrices.

Zero-Pole

Vector lists of zeros and poles can be specified to create a transfer function. DC gain is also specified.

Derivative

The output is equal to the derivative of the input.

Dot Product

The output is equal to the dot product of two vector signals.

Matrix Gain

The output is equal to the input times a specified constant matrix. The size of the input and output vector signals must match the size of the matrix.

Slider Gain

This multiplies the input by a scalar constant which is specified by moving a slider on the screen as shown below. The limits of the slider can be specified.

43

NONLINEAR

Nonlinear Blocks are elements of nonlinear continuous-time dynamic systems. Most of these have special-purpose applications and will not be used in the tutorials. Only the most relevant Nonlinear blocks will be discussed here. Double-click on the Nonlinear icon in the main Simulink window to bring up the Nonlinear window.

44

Elementary Math

The output of this block is a basic mathematical function applied to the input. Copy this block into your model window and double click on it. The resulting dialog box has one field, which when clicked on, give the following menu of functions.

The desired function can be selected from the list, and the choice will be displayed in the block in your model window.

Product

The output is equal to the product of the inputs. The number of inputs can be specified.

Fcn

Arbitrary MATLAB expressions can be represented by this block.

45

CONNECTIONS

Connection Blocks are used to organize and combine signals and systems. These also have special purpose applications, and only some of these will be described here. Double-click on the Connections icon in the main Simulink window to bring up the Connections window.

46

Mux, Demux

The Mux (Multiplexer) block is used to combine two or more scalar signals into a single vector signal. Similarly, a Demux (Demultiplexer) block breaks a vector signal into scalar signal components. The number of vector components must be specified in each case.

Subsystem, In, Out

The Subsystem block is used as a system "macro", where one Subsystem block can be used to represent an entire set of blocks. When double-clicked, the subsystem block brings up a blank model window. Note that the Subsystem block has no inputs or outputs. For each input a Subsystem has, an In block is used. For each output a Subsystem has, an Out block is used. For more information on Subsystems and In and Out blocks, see the Interaction with MATLAB tutorial page.

47

Simulink Basics Tutorial - Interaction With MATLAB

Defining Block Parameters Using MATLAB Variables Exchanging Signals With MATLAB Extracting Models From Simulink Into MATLAB

We will examine three of the ways in which Simulink can interact with MATLAB.

• Block parameters can be defined from MATLAB variab. • Signals can be exchanged between Simulink and MATLAB. • Entire systems can be extracted from Simulink into

MATLAB.

48

BLOCK PARAMETERS FROM MATLAB VARIABLES Often, a controller will be designed in MATLAB and verified in a Simulink model. Normally, numerical parameters such as gains and controller transfer functions are entered into simulink manually by entering the numbers in the block dialog boxes. Rather than enter numbers directly, it is possible to use MATLAB variables in Simulink block dialog boxes.

In this case, the complete controller transfer function is:

s+2 2.5 ----- s Suppose this transfer function were generated by some computation in MATLAB. In this case, there would most likely be three variables, the numerator polynomial, the denominator polynomial, and the gain. Enter the following commands in MATLAB to generate these variables.

K=2.5 num=[1 2] den=[1 0]

49

These variables can now be used in the blocks in Simulink.

Double-click on the PI Controller block. Enter num in the Numerator field and den into the Denominator field.

50

You can simulate the model with the MATLAB variable parameters. Select Start from the Simulation menu to run the simulation. Double-click on the Scope block to view its output. Hit the autoscale button (binoculars) and you should see the following.

51

52

EXCHANGING SIGNALS WITH MATLAB VARIABLES Sometimes, we would like to use the results of a Simulink simulation in the MATLAB command window for further calculations and plotting. Less often, we would like to generate signals in MATLAB which we then use as inputs in a Simulink model. These tasks are accomplished through the use of the To Workspace Sink Block and the From Workspace Source Block. We will only transfer signals from Simulink to MATLAB. Doing the reverse is a very similar process.

The To Workspace Sink Block saves a signal as a vector in the MATLAB Workspace.

Suppose we would like to use both the output signal and the control signal for calculations in MATLAB. We will save these two variab as well as a time signal from our Simulink model. First, you need to generate a time signal

53

Before connecting these blocks to the rest of your system, first you will name the variable to which they output.

54

The To Workspace block near the Plant block will output the control signal to the MATLAB variable u. Edit this block to output to the variable u. The last To Workspace block will output the output signal to the MATLAB variable y.

55

You can now examine the outputted variab in the MATLAB window with plot(t,u,t,y);

56

EXTRACTING MODELS FROM SIMULINK INTO MATLAB

Sometimes, we may build a complicated model in simulink and would like to derive either a transfer function or a state space model of the entire system. In order to do this, you first need to define the input and output signals of the model to be extracted. These virtual signals can be any signal in a model, for example, if we can generate an input-to-output transfer function or a disturbance-to-error transfer function. These signals are defined using the In and Out Connection Blocks.

Once the input/output model is defined, the Simulink model must be saved to a .mdl file.

You will be extracting a closed-loop reference-to-output model. Therefore, The virtual input will be put in place of the step input to the system.

First, delete the Step block and connect an In block from the Connections window to your model window in place of the Step block you just deleted.

57

The virtual output does not need to replace an existing block - the signal can be tapped off an existing line. Drag an Out block from the Connections window and place it just above the Scope block.

Now, save this model under a new name. Call it mymodel.mdl.

At the MATLAB prompt, LINMOD obtains linear models from systems of ord. diff. equations (ODEs).

>> [A,B,C,D]=linmod('mymodel') You should see the following output.

58

A = -2 -9 2 1 0 0 0 -5 0 B = 5 0 5 C = 0 1 0 D = 0

>>step(A,B,C,D)

You should see the following plot which is similar to the previous Simulink Scope output.