Solving Differential Equations Using Simulink
R. HERMANS O LV I N G D I F F E R E N T I A L E Q U AT I O N SUSING SIMULINKR . L . H E R M A N - V E R S I O N D AT E : D E C E M B E R 1 8 , 2 0 2 0
Copyright 2020 by R. Hermanpublished by r. l. hermanThis text has been reformatted from the original using a modification of the Tufte-book documentclass in LATEX.See tufte-latex.googlecode.com.solving differential equations using simulink by Russell Herman is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. These notes have resided athttp://people.uncw.edu/hermanr/mat361/simulink since Summer 2015.First printing, 2015
Contents12Introduction to Simulink1Solving an ODE . . . . . . . . . . . . . . . . . . . . . .2Handling Time in First Order Differential Equations3Working with Simulink Output . . . . . . . . . . . . .4Printing Simulink Scope Images . . . . . . . . . . . .5Scilab and Xcos . . . . . . . . . . . . . . . . . . . . . .6First Order ODEs in MATLAB . . . . . . . . . . . . .Symbolic Solutions . . . . . . . . . . . . . . . . . . . .ODE45 and Other Solvers. . . . . . . . . . . . . . . . .Direction Fields . . . . . . . . . . . . . . . . . . . . . .7Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .1161112161919212325First Order Differential Equations1Exponential Growth and Decay . .2Newton’s Law of Cooling . . . . .3Free Fall with Drag . . . . . . . . .4Pursuit Curves . . . . . . . . . . . .5The Logistic Equation . . . . . . .6The Logistic Equation with Delay .7Exercises . . . . . . . . . . . . . . .2727293335383941Second Order Differential Equations1Constant Coefficient Equations .Harmonic Oscillation . . . . . . .2Projectile Motion . . . . . . . . .3The Bouncing Ball . . . . . . . . .4Nonlinear Pendulum Animation5Second Order ODEs in MATLAB6Exercises . . . . . . . . . . . . . .43444554565864674Transfer Functions and State Space Blocks1State Space Formulation . . . . . . . . . . . . . . . . . . . . . .2Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . .6969705Systems of Differential Equations1Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .75753.
426Nonlinear Models . . . . . . . . . . .The Jerk Equation . . . . . . . . . . .Van der Pol Equation . . . . . . . . .Lorenz Equations . . . . . . . . . . .Lotka-Volterra Predator-Prey ModelSIR Model of Disease . . . . . . . . .Michaelis-Menten Kinetics . . . . . .The Chua Circuit . . . . . . . . . . .Index.767677788182848591
1Introduction to SimulinkThere are several computer packages for finding solutions of differential equations, such as Maple, Mathematica, Maxima, MATLAB, etc.These systems provide both symbolic and numeric approaches to findingsolutions. They often require a bit of coding. However, there are graphicalenvironments for solving problems, including differential equations. Onesuch environment is Simulink, which is closely connected to MATLAB. Inthese notes we will first lead the reader through Simulink examples of solutions of first and second order differential equations usually encounteredin a differential equations course. We will then look at examples of morecomplicated systems.Most of these models were createdusing Version 2015. Some changes inVersions 2017-2020 are noted.Examples of MATLAB solutions ofdifferential equations will also beprovided.1.1 Solving an ODESimulink is a graphical environment for designing simulationsof systems. As an example, we will use Simulink to solve the first orderdifferential equation (ODE)dx 2 sin 3t 4x.dt(1.1)We will also need an initial condition of the form x (t0 ) x0 at t t0 . Forthis problem we will let x (0) 0.dxto formally obtainWe can solve Equation (1.1) by integratingdtx (t) Z(2 sin 3t 4x (t)) dt.We will view this as a system in which the input, x 0 2 sin 3t 4x, is fedinto an integrator and the output will be x (t). Generally, we havex (t) Zx 0 (t) dt.This process is depicted in Figure 1.1.inputx0RxoutputIn order to carry this out, we separately insert the terms 2 sin 3t and 4x into the integration procedure. Since we do not know 4x, we takeFigure 1.1: Schematic for a generalsystem in which the block takes theinput and produces an output.
2solving differential equations using simulinkthe output from the integrator, multiply it by 4, and subtract that from2 sin 3t. This combined set of terms is then feed back into the integrator.This is shown schematically in Figure 1.2.2 sin 3t x0R xFigure 1.2: Schematic for solvingx 0 2 sin 3t 4x. The terms 2 sin 3t and4x are fed into the integrator and x isoutput.output 4When you have access to Simulink and MATLAB, you can start MATLAB by typing simulink on the command line to bring up Simulink. Alternatively, you can select Simulink on the MATLAB icon bar to launchSimulink. Starting in 2017 Simulink opens with a start screen in whichthere are several selections as shown in Figure 1.3.1 Pick the Blank Modelto begin a new model or select a recently opened model. Then, you will bein the Simulink workspace [see Figure 1.4].In earlier versions the Simulink Library Browser in Figure 1.5 wouldappear.1Figure 1.3: The Simulink Start screen.Pick the Blank Model to begin a newmodel or select a recently openedmodel.From the workspace you can open the Simulink Library Browser asshown in Figure 1.52 . Next, click the yellow plus to bring up a new model.We build models by dragging and connecting the needed components, orblocks, from groups such as the Continuous, Math Operations, Sinks, orSources.Now we can create the model for simulating Equation (1.1) in Simulinkas described in Figure schema2 using Simulink blocks and a differentialequation (ODE) solver. In the background Simulink uses one of MATLAB’s ODE solvers, numerical routines for solving first order differentialequations, such as ode45. This system uses the Integrator block31sIntegratorintegratedx, producing x (t).dtto2The notation on the Integrator block isrelated to the Laplace transform Z t 1Lf (τ ) dτ F (s),s03where F (s) is the Laplace transform off ( t ).
introduction to simulink3Figure 1.4: A blank model in Simulink.The input for the Integrator is the right side of the differential Equation(1.1), 2 sin 3t 4x. The sine function can be provided by using the SineWave block, whose parameters are set in the Sine Wave block. In order toget 4x, we grab the output of the Integrator (x) and boost it by changingthe Gain value to “4” Then, using the Sum component, these terms areadded, or subtracted, and fed into the integrator. The Scope is used toplot the output of the Integrator block, x (t). That is the main idea behindsolving this system using the model in Figure 1.6.For this example, we implement the following detailed steps in Simulink: Drag needed blocks into the model region [Figure 1.7.]:– Integrator block from the Continuous group;– Sum block from the Math Operations group,– Gain block from the Math Operations group,– Sine Wave block from the Math Operations group; and,– Scope block from the Sink group. Connect the output of the Sum block to the input of the Integratorblock. [Figure 1.8.] Connect the Integrator to the Scope by clicking on the Integrator output and dragging to the Scope until they are connected. In more recentversions it is easier to double-click the unattached arrow to get a connection. Right-click the Gain control and choose Flip Block under Rotate &Flip. Double-click the Gain block and change the Gain block valuefrom 1 to 4. It should change on the control.
4solving differential equations using simulinkFigure 1.5: The Simulink LibraryBrowser. This is where various blockscan be found for constructing models.[As seen in MATLAB 2015a.]1sSine WaveFunctionIntegratorScope4Gain Double-click the Sum control to bring up Block Parameters as shown inFigure 1.9 and change from to - in order to set addition/subtractionnodes. [Note that the symbol ‘ ’ is a blank node. Also, one can changethe block to rectangular form. This is often useful in displaying an overall flow direction to the model. In this case the spacer, , is not needed.] Double-click the Sine Wave block and change the frequency to 3 rad/sand the amplitude to 2. [See Figure 1.10] Set the time dropdown menuto Use Simulation Time. Connect the Gain output to the negative input of Sum and the SineWave output to the positive input on the Sum control. [Note: The Gaincan be set to a negative value and connected to a node in the Sumblock to obtain the same effect.] To add a node to route an x value to the Gain, hold the CTRL key andclick on the Output line of the Integrator and drag towards the inputFigure 1.6: System for solving firstorder ODE dxdt 2 sin 3t 4x as aSimulink simulation.
introduction to simulinkFigure 1.7: Add needed components tothe model window.1sSine WaveIntegratorScope1Gain1sSine WaveIntegratorScope1sSine WaveIntegrator5ScopeFigure 1.8: Example of connectingtwo components: Align the components, Click on output of one and dragto another. Then, release to finalizeconnection. Sometimes it is easier todouble-click the temporary arrow toconnect the blocks.of the Gain. You can also Right-Click the line where you want the nodeand drag from there to the Gain block. See Figure 1.11. The initial value, x (0), of x is inserted by double-clicking the Integratorand setting the value. For this example we set x (0) 0. One can annotate the diagram by clicking near where labels are neededand typing in the text box. This leads to the model in Figure 1.12. Inmore current versions the default is to hide block names. Save the file under a useable file name. This file can be called inMATLAB, or one can use the run button to run the simulation. Double-click the Scope to see the solution. Figure 1.13 shows the Scopeplot after using the autoscale () feature to rescale the scope view.A little effort is needed to change the plot attributes and to import theplots into working documents. This will be discussed in Section 1.4. Also, one can make further changes to the system by checking the Configuration Parameters under the Simulation menu item. See Figures1.14-1.15. In particular, changing the Refine Factor can lead to smoothersolutions. The solution shown in Figure 1.13 had a setting of 1 and thatin Figure 1.16 is the result of setting the Refine Factor to 10.As noted in setting the initial value, one can double-click the Integrator block and set the initial condition. However, sometimes it is useful toexternally feed the initial condition into the block. Double-click the Integrator block and change the initial condition source from internal to external. This adds another input to the block. Drag a Constant block from theSources group into the model, connect it to the new input, and change theconstant value to the desired initial value. This results in the simulationshown in Figure 1.17.In Release 2017b the names of blocksare hidden. One can change this bygoing to the block Properties andchanging the block parameter for’HideAutomaticName’ or change themodel parameter ’HideAutomaticNames’.
6solving differential equations using simulinkFigure 1.9: Block Parameters for theSum control. In many cases it is best toalso select the rectangular shape overthe default round shape.1.2 Handling Time in First Order Differential EquationsIn this section we review the solutions of first order differential equations, separable first order differential equations and linear first order differential equations involving explicit time dependence. The time dependent functions are obtained using the Clock block and a Math Functionblock. Double-clicking the Math Function block allows for the selection ofa number of common functions.Example 1.1. Solve the initial value problemdy2 y, where y(1) 1.dtt(1.2)This is a separable equation. Placing y-variables on the left andt-variables on the right side, we haveZdy yZ2dt.tIntegrating both sides,ln y 2 ln t C ln t2 C.Exponentiating, we obtain the general solution,y(t) At2 ,where A eC .Using the initial condition, we have the solution, y(t) t2 .We can set up the problem in Simulink as shown in Figure 1.18 for theinitial value problemdy2 y,dttThe independent variable is obtainedusing the Clock block.
introduction to simulinkFigure 1.10: Parameters for the SineWave block. Select the amplitude andfrequency desired.1sSine WaveIntegratorScope1Gainwhere y(1) 1. Running the simulation, we obtain the solution shown inFigure 1.19.The solution looks like y(t) t2 . We can verify this by plotting t2 alongwith the solution t see if they are the same. Another method would be tocompute the difference between the numerical and exact solution, y(t) t2 .In order to do this, we add a Math Function block, selecting the squarefunction and connect it to the time route and a Sum Block. The solution isalso fed into the latter block and the difference is fed into a second ScopeBlock. This is shown in Figure 1.20.The result of the simulation is shown in Figure 1.21. We note that this isthe numerical error, though the solution is only off by 1.4 10 5 over thegiven interval. Considering that the solution at t 10 is Y (10) 100, thisis a relative error of roughly 10 7 . That seems perfectly acceptable.It is simple to change the differential equation (1.2) in the previousexample to a linear first order differential equation.dy2 y t2 .dttFigure 1.11: Add a node by rightclicking one the line and dragging tothe input of a block.7
8solving differential equations using simulinkFigure 1.12: Connections for First OrderODE model for dxdt 2 sin 3t 4x.1sSine WaveFunctionIntegratorScope4GainFigure 1.13: Scope plot of the solutionof dxdt 2 sin 3t 4x, x (0) 0, withRefine Factor 1.Example 1.2. Solve the linear first order differential equation,2dy y t2 ,dtt(1.3)satisfying y(1) 1.We first rewrite Equation (1.3) in standard form,dy 2 y t2 .dtt(1.4)We can now determine the integrating factor, Z t 2µ(t) exp dττ exp [ 2 ln t] t 2 .Multiplying Equation (1.4) by the integrating factor, µ(t), we canfind the solution: dy 2t 2 y t 2 t 2dttd 2 t y 1dtt 2 y ( t ) t Cy(t) t3 Ct2 .(1.5)Using the initial condition, y(1) 1, we obtain C 0. Therefore,the solution is y(t) t3 .
introduction to simulinkFigure 1.14: System ConfigurationParameters.The model for this problem is shown in Figure 1.22. Running the simulation, we obtain the numerical solution, y(t) t3 , as shown in Figure1.23. Computing the difference between the numerical and exact solutionsin this case, we find the error is about 6 10 5 .Example 1.3. Consider the initial value problem,dx 2 sin 3t 4x,dtx (0) 0.(1.6)This is the example that we first solved using Simulink. It is another linear first order differential equation. In standard form is iswritten asdx 4x 2 sin 3t.dtThe integrating factor is found to beµ(t) exp Z 4 dt e4t .Figure 1.15: Configure Data Import/Export Parameters. ChangingtheRefine Factor can lead to smoothersolutions.9
10solving differential equations using simulinkFigure 1.16: Scope plot of the solutionof dxdt 2 sin 3t 4x, x (0) 0, withRefine Factor 10.Figure 1.17: Connections for the FirstOrder ODE model for dxdt 2 sin 3t 4xshowing how to provide an externalinitial value.1xosSine ng Equation (1.6) by the integrating factor, we can obtainthe general solution:d 4t e xdte4t x 2e4t sin 3t 2 x (t) Ze4t sin 3t dt C2 4te (4 sin 3t 3 cos 3t) C252(4 sin 3t 3 cos 3t) Ce 4t .25Using the initial condition, x (0) 0, we find C the particular solution isx (t) 26(4 sin 3t 3 cos 3t) e 4t .2525625 .(1.7)Therefore,(1.8)The solution can be found using Simulink. The model for thisexact solution is shown in Figure 1.24. The plot on the scope matchesthe solution we obtained earlier as seen in Figure 1.13.
introduction to t1sIntegrator11Figure 1.18: First order separabledifferential equation model.yScopey' 2/t y, y(1) 1Exact solution: y(t) t2Figure 1.19: Scope plot of the solutiondyof initial value problem (1.2), dt 2t y,where y(1) 1.1.3 Working with Simulink OutputOften we might want to access the solutions in MATLAB. Usingthe model in Figure 1.18 for a first oder separable equation, we can addthe To Workspace block. This is shown in Figure 1.25. Double-click andrename the variable as y and change the output type to array. When yourun the simulation, it will send the data to MATLAB for further analysis orplotting. This will put tout and y data into the MATLAB workspace.In MATLAB you can plot the data using plot(tout,y). You can addlabels with xlabel(‘t’), ylabel(‘y’), title(‘y vs t’). Adding the commandset(gcf,‘Color’,[1,1,1]) makes the plot background white. The result isshown in Figure le(‘y vs t’)set(gcf,‘Color’,[1,1,1])Once you have exported your data to the MATLAB workspace andcreated a plot, then you can use the menu items under Tools to annotatethe plot. Once you are satisfied with the figure, go to the Edit menu and
12solving differential equations using n2Gain2/tdy/dtProduct1sIntegratorFigure 1.20: First order separabledifferential equation model with extrablocks to plot the difference betweenthe numerical and exact solution,y(t) t2 , for Equation (1.2).yScopey' 2/t y, y(1) 1Exact solution: y(t) t2Figure 1.21: Scope plot of the differencebetween the numerical and exactsolution, y(t) t2 , for Equation (1.2)select Copy Figure. Go to your report document and Paste (CTRL-V) thefigure into your document. You can then resize the figure, center it, andadd a numbered Figure caption describing the figure. Other methods forrecording Simulink Scope images and the Simulink model are describednext.1.4 Printing Simulink Scope ImagesIn this section we discuss different methods for transferring the plotsgenerated in Simulink models to a document or report. For example, youmight want to copy images produced by the scope or your model into anMS Word document. There are several ways you can do this. You mightbe able to use the Print icon to print to a file or printer, or you can followone of the following methods. Note: In 2015 it was not easy to export plotsfrom Simulink. In the 2017 versions, it is easier to do so and perhaps the
introduction to 11sFigure 1.22: Linear first order differential equation model.yIntegrator113Scope12/t y1/t2Gain12/t2y' 2/t y t , y(1) 1Product1Exact solution: y(t) t3Figure 1.23: Scope plot of the differencebetween the numerical and exactsolution, y(t) t2 .preferred method unless you are using earlier methods.We compare the 2017a Scope figure windows to those shown laterfrom 2015a. In Figure 1.27 one might see slight differences in the Scopeicons. In Figure 1.28 the File menu shows a menu item for Print to Figure.Here one can put the scope figure in a MATLAB figure environment andSave As a figure file of different types such as the png image in Figure1.29. Other methods are provided below for producing output useful forreports.Method 1:Select the Scope figure window in Figure 1.30, then hit ALT PrintScrnto copy the figure to a clipboard and paste the figure into your application.You might want to change the colors before copying the scope image.Click the Scope Parameters icon (2nd icon) and go to the Style tab as seenin Figure 1.31. Change the Figure Color to black, Axes C
lutions of first and second order differential equations usually encountered in a differential equations course. We will then look at examples of more Examples of MATLAB solutions of differential equations will also be provided. complicated systems. 1.1 Solving an ODE Simulink is a graphical environment for designing simulations of systems.
9.1 Properties of Radicals 9.2 Solving Quadratic Equations by Graphing 9.3 Solving Quadratic Equations Using Square Roots 9.4 Solving Quadratic Equations by Completing the Square 9.5 Solving Quadratic Equations Using the Quadratic Formula 9.6 Solving Nonlinear Systems of Equations 9 Solving Quadratic Equations
(iii) introductory differential equations. Familiarity with the following topics is especially desirable: From basic differential equations: separable differential equations and separa-tion of variables; and solving linear, constant-coefficient differential equations using characteristic equations.
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