A Simulink Tutorial
Contents
1. selecting the arrow and dragging the mouse and releasing the mouse button and doing it again in another direction Perhaps you ll end up with something like this File Edit Options Simulation Style Code APR nh th a A Ranh ith Anh Atl ee Which I really don t want there at all because the mass spring model doesn t require a second output from the gain block To get rid of it just select it showing its handles and then select cut from the edit menu along the window border You can experiment with the other edit functions This works in general for any connection and any block What happened to the first gain block It is now about twice its original size That s just to illustrate that you can also change the size of the block icons too in the same way Just select the icon showing its four handles at the corners and drag one of the handles to size the icon how you want it You can also drag the icon whereever you want it by selecting not a single handle but dragging the body of the icon to where you want it The connections move along with it Now let s just finish off the diagram since we know pretty much how to use the simulink interface Now we said before that the output from the integrator block was going to be x2 Thus if we were to assign the gain 1 block to be the viscous damping a constant times the velocity or x2 then we want to have a connection from the output of the integrator to the input of g2. A Simulink Tutorial www cee uc edu juber e_reserves cee627 simulink_tut html With simulink building a dynamic model can actually be fun It is also powerful enough to do real work once you learn some of its advanced features i e it is not a waste of time to learn In fact we will only use some of the rather basic features of simulink for the models we build and you are free and encouraged to explore some other features that will help you with problems you want to solve Basically you can think of a stmulink model as like describing a wiring diagram of your system where the variables flow on wires and are changed or transformed by various devices The notes on the spring mass system provide perhaps a better brief introduction to this idea Let s build a simulink model for the linear spring mass system with viscous damping 0 Bot k pl m m First you need to start the matlab application which you do in Unix by typing matlab at the shell prompt or in Windows by double clicking rapidly clicking the left mouse button twice on the matlab icon which you will find buried somewhere with a bunch of other icons This will get you to the gt gt matlab prompt Type simulink to start simulink and bring up the main simulink window your windows may look somewhat different depending on the machine and operating system File Edit Options Simulation Style Code BESS E Sources Sinks Discrete Linear NonlinearConnectio
3. ain 1 Do this now and then position the line to where you d like it But now what is the input to the gain block the position of the mass or x1 Well this has to be constructed as the integration of dx 1 dt or the right hand side of the first o d e above which is just x2 Thus we need to drag another integrator block over and connect that as well to the output of the first integrator block Do this and then connect the output of the second integrator block to the gain block so the diagram looks something like this File Edit Options Simulation Style Code 1 Mass position Graph Sum Integrator Integrator Mux X 2 Mass Velocity XY Graph aD p nahh hah hahahah nah tah hath hah hl SSS Now stop and convince yourself that the above graphical depiction is actually and accurate representation of the dynamical system we are trying to solve You ll notice that I ve put in labels for the two main lines in the diagram a feature that is usually helpful And easy Just click the left mouse button about where you want the text and then start typing You can then move the selected text just like you can move a block actually it is a text block While you re at it open up the output and the connections icons and select the mux multiplexer block the graph block and the XY graph block as shown above The mux block just takes multiple inputs and produces a single vector output This is hand
4. are going to do first is graphically build the right hand side of the second ordinary differential equation 0 d e listed above This will be what goes into the left side of the integrator block and then what comes out the right side will be x2 Got it So let s proceed to build up the right hand side of this o d e which is just the linear summation of the two state variables x1 and x2 So what we ll need is two gain blocks and a sum block to sum the result The sum block will be first because it assembles the two parts of 2 dx2 dt So drag a sum block over from the linear menu and position it to the left of the integrator hkg F Sum Integrator How did the connecting line get there Easy just use the mouse and first click on the output side of the sum block the little out arrow and next on the input side of the integrator i e from tail to head of the vector Go ahead and do that Next notice that the sum block by default has two inputs and both are added together We want two inputs but we want them to be subtracted this is because we want the constants k m and k1 m to be positive as by convention we often wish that model parameters are positive So to open up the sum dialog box double click on it you ll get the following Block type Sum eee Revert Number of signs is number of inputs Example Help List of signs Notice that you can change the number of inputs by simply editing the n
5. ays to connect the wires and blocks in your diagram Now what we d like to do is open a new workspace to start a new model To do this press the mouse button to bring up the file menu and select new to create a new workspace You ll be prompted later to name the model when you want to save it What you ll get is a blank workspace with the same headings as the main simulink window Now go up to the main window and double click on the linear icon which will bring up another window containing all the possible linear transformations Arrange these three windows how you want them You ll get something like the following So rcgs Sika Dsurels Linea Nun newCunec ivis Exsas Co EEEE EE N EAE EE EAS i 1 nse ire y Th 3 j Sur lener Product i4 dufd Integrater Derivative Inegratar t It tk gt gt 1 9 7 a eadbosccsnccccconse cocosesoescoceooossoseosessessosssssssesssssessesossossessesessessossosessessossosesseoso so Actually you ll not have the integrator block in your model workspace yet To get it there is simple just press and hold the left mouse button over the integrator block and then drag a copy over the to workspace wherever you want it to be don t worry you can move it again in just the same way Go ahead and do this now But why did we start with the integrator block You ll recall that the integrator integrates its input to produce its output Thus if the input is dx dt then the output is x What we
6. ctions moving connections etc Notice that the graphical output will now give us both a time series plot of the system state and a plot of the state space the two traditionaly ways of graphically illustrating the orbits or trajectories of a dynamical system Now we re about ready to run the simulation and see the output But before we do this we need to set the parameter values and initial conditions how we want them Change the gain 1 parameter to be zero this corresponds to zero friction resistance correct File Edit Options Simulation Style Code 1 Mass position SE a I SR PO Block name Gaint 7 Block type Gain po e UNIES sccaeabaewieeeiondeida jae Gain X2 Ma EAE ee ee an Also set the initial conditions to be zero velocity and an initial stretch of the spring to position 10 corresponding to stretching the guitar string 10 units holding it there and then letting go Ee j File Edit Options Simulation Style Code 1 Mass position f Block name Integrator po i Gain Block type Integrator Bi i E s aaaea S 4 Integrates input starting with id _ ginitial value EB id Initial value Graph i 10 k K2 Mif 4 You can see that the initial conditions for the state variables are stored in the integrator blocks By the way the input to the integrator could be a vector in which case I d specifiy the initial condition in matlab mat
7. elect the 5th order Runge Kutta scheme which is accurate and an efficient method so long as different components of the solution the state space are not changing on very different time scales one much more rapidly than the others this is a very informal definition of stiff problems Note that most of the simulink numerical algorithms are variable step size algorithms meaning they take a variable length integration step along the trajectory to approximate the solution Bigger steps are taken when the solution is not changing very much and smaller steps otherwise Thus the minimum and maximum step size parameters are user controllable The algorithms will select the actual step size automatically within these bounds to satisfy a local error criterion The tolerance parameter sets a bound on the relative local integration error meaning basically the error that can occur at any one integration step Note that obviously this kind of error control which is very useful and practical can not say much by itself about the global error over the entire time range of integration 9 Now you can have some fun run the simulation by selecting start from the simulation pull down menu Adjust the graph windows as they pop up automatically and right on top of one another on my workstation For the current parameter values you ll get something like this e le Edi Silo Brule se roe 7H 2 5 41 bass pester eeler
8. ns Extras Let s go though the basic functions of this window first On top is a list of items that are associated with pull down menus These will be duplicated on your model workspace see below so for the most part they are not too important here Exept for the file menu which you will use to create a new model workspace or to bring up a file browser to access an existing saved model Each of the block icons correspond to a category of transformations that will be used to describe the dynamic system You will mostly use the transformations described in Sources Sinks Linear Nonlinear and Connections The Sources icon contains a suite of signal generators such as random number generators and wave sinusoidal square etc generators that can be used to provide time varying inputs that drive your dynamic system these are exogenous inputs or external inputs meaning that their dynamics are described externally to the model The Sinks category will mainly be used to access the Scope and XY graphs to which you will connect the system wires to get a readout The Linear category contains all the possible ways to transform a variable linearly such as different types of gains It also has the basic integrator block that we will use extensively to model our first order dynamic systems The nonlinear block contains 1 various specific and general nonlinear transformations and the connections block includes various w
9. oes ne oe ee eee eR te ete EEE a oo BSN pare LL eee File Edit Options Simulation Style Code i 1 Mass position xz Mlass Velocity xY Graph Clock To Workspace This will take the vector output of the state from the output of the mux block and put it into a matrix variable named x available from the matlab workspace as usual you open the to workspace block to set the name of the variable Similarly the output of the clock icon which you can think of as a timer that runs exactly at the simulation speed is put into a vector variable named t that is available from the matlab workspace Run the simulation again now and stop it after some time Go to the main matlab workspace window and type the command whos This should show the variables that are now available Then type the command plot t x which will plot the time versus each state variable which are each stored as a column in the matrix x See the plot O K now you can just type print and that plot will be sent to the default printer You can type help print in the matlab workspace for details about options and also help plot for details about the plot command 11
10. ow its handles by clicking on the desired line Anyway the significance of the handles is how they allow you to move the position of the connection to make it look better For example positioning the mouse pointer on one of the handles and dragging it repositions only that vertex of the connection with the other lines stretching to meet it Sum Integrator AAP ah tah ra earl nani enh anh ah tah th Ant A de But sometimes you d rather move an entire line without changing its orientation To do this select a connection segment that terminates in a handle at each end Then drag it to its new position like this I moved the vertical line between the gain and the sum blocks File Edit Options Simulation Style Code ARN Ana AAA Anh Ah Atl Atl ea Another thing that is useful to do is create a new line from an existing one i e tap into a line with a bit of solder This is easy to do just position the pointer where you want to create the new line and drag the right mouse button this creates a new connection oriented vertically horizontally or at a 45 degree angle depending on which direction you drag Experiment with this Once you release the mouse button you are left with a line segment with an arrow on the end presuming you didn t connect the line to another block yet You can then select this arrow just like the output of a block to draw another line segment perhaps in another direction Practice this
11. recties Ieee slay ises An SLX ver Perl Where the phase space is on the left and the time series on the right You can stop the simulation any time using the stop button on the simulation menu and then start again if you like using start Or you can restart from the beginning without stopping by selecting restart Or you can pause the solution and then resume from the paused point not the initial conditions by using pause and continue Another neat trick is to use a floating scope block one that is not connected from the Sinks library and then when you click any line while a simulation is running the scope will change to display the state on the line over time a disadvantage is that you don t get any scales just the signal It is possible to change simulation or block parameters on the fly while a simulation is running and see the effect Play with these things for a while until you get bored with this example How about printing a graph of the solution The best way is to run a simulation and place the values of the state variables along with the time of the simulation on the matlab workspace This gives you all the flexibility of matlab with which to process the graphics Let s learn by an example First open the Sinks icon again and drag over two to workspace blocks Then drag a clock block from the Sources library Connect them up like the following 10 Re ea re nr ee Ee Re aren
12. rix notation I did change the value in the correct integrator block right It s also probably a good idea to change the scale of the plots since we know that stretching the spring 10 units will produce a range of positions from 10 to 10 without friction Open the dialog box for the Graph and change it and verify that the scale for the XY graph phase space plot is o k too Block name Graph Block type Graph scope Mask Graph scope using MATLAB graph window Enter plotting ranges and line type Help Time range P i y min gymar a4 Graph 10 Line type tgbw Seperate each plot by y g c w m ro b In general you also need to set up the simulation parameters for the numerical algorithms used to integrate the dynamical system to determine the trajectory Open the simulation control dialog by selecting parameters from the simulation menu along the top of the window Euler Runge Kutta 3 Runge Kutta 5 Adams i Gear Adams Gear Linsim Start Time 0 0 Stop Time 999999 Min Step Size 9 0001 Max Step Size 10 fT Simulink offers you 7 different numerical algorithms We will learn more about some of these a bit later But you need to know that the particular numerical algorithms can make a big difference in terms of the efficiency with which a problem is solved The Simulink user s manual has a nice general introduction to this topic For now you can s
13. umber of signs that appear in the list Change the two signs to two signs and then select done to close the dialog input This is the way double clicking on an icon that you bring up the dialog box for any icon Each icon has its own dialog box for changing parameters and describing briefly its function The help button gives a bit more detail but to really know what each icon does in detail you ll have to consult the simulink reference guide For our purposes we ll mostly be able to go on the brief help that is available on the dialog Now we re going to get some more practice connecting blocks and just using the simulink interface First drag two gain blocks from the linear library to the workspace Alternatively you can drag one block over and then press and hold the right mouse button over the gain block to drag a copy of it This is sometimes useful since then the new block will have the same parameter values of its parent Connect both the gain blocks to the sum block in the usual way You should get something like this File Edit Options Simulation Style Code Sum Integrator Gain ANAND A hh Ah Aa Rah Aah Aa ah Aah Aah oh Notice that the lower connection between gain 1 and sum is highlighted with little square blocks These are significant and are called handles You can deselect that line by clicking the mouse button on the white background on nothing or you can select a different line sh
14. y because most simulink blocks can handle vector inputs as well as scalar inputs and sometimes it is useful or essential to have connections carry vector quantities For example what we will do is to multiplex the two variables position and velocity to produce a single output vector containing two elements the position and the velocity This single vector will be sent to the graph block which is set up to display any number of signals simultaneously Thus if we send it a two element vector it will display those two as output over time and if we sent it a 10 element vector it would display all of those This is just handier than having two separate graphs and also teaches us a very powerful aspect of simulink that you might use to advantage in the future that most of its blocks can process vector as well as scalar inputs Open up the mux dialog and change the number of inputs to 2 instead of 3 j j j Dee etn ES cae a eae act eet aaa htt he aaa aetna Block name Mux Block type Mux pene Revert i Multiplexer Help Number of inputs i S 2 4 Then connect the output of the mux to the graph and connect both the mux and the XY graph inputs to the position and velocity lines something like this File Edit Options Simulation Style Code 1 Mass position 2 Mass Velocity xY Graph You should be able to do this with the tools we ve discussed so far tapping off of existing conne
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