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					CTMS: MATLAB Modeling Tutorial                                   http://www.engin.umich.edu/class/ctms/model/model.htm




                             MATLAB Modeling Tutorial
         Train system
         Free body diagram and Newton's law
         State-variable and output equations
         MATLAB representation
         MATLAB can be used to represent a physical system or a model. In this tutorial, you will
         learn how to enter a differential equation model into MATLAB. Let's start with a review of
         how to represent a physical system as a set of differential equations.

         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.




         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


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CTMS: MATLAB Modeling Tutorial                                  http://www.engin.umich.edu/class/ctms/model/model.htm


         applied by the engine. The forces acting on M2 are the spring and the friction. In the vertical
         direction, the gravitational force is canceled by the normal force applied by the ground, so
         that there will be no acceleration in the vertical direction. The equations of motion in the
         horizontal direction are the following:




         State-variable and output equations
         This set of system equations can now be manipulated into state-variable form. The state
         variab are the positions, X1 and X2, and the velocities, V1 and V2; the input is F. The state
         variable equations will look like the following:




         Let the output of the system be the velocity of the engine. Then the output equation will be:




         1. Transfer function
         To find the transfer function of the system, we first take the Laplace transforms of the
         differential equations.




         The output is Y(s) = V2(s) = s X2(s). The variable X1 should be algebraically eliminated to
         leave an expression for Y(s)/F(s). When finding the transfer function, zero initial
         conditions must be assumed. The transfer function should look like the one shown below.




         2. State-space

         Another method to solve the problem is to use the state-space form. Four matrices A, B, C,
         and D characterize the system behavior, and will be used to solve the problem. The


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CTMS: MATLAB Modeling Tutorial                                http://www.engin.umich.edu/class/ctms/model/model.htm


         state-space form which is found from the state-variable and the output equations is shown
         below.




         MATLAB representation
         Now we will show you how to enter the equations derived above into an m-file for MATLAB.
         Since MATLAB can not manipulate symbolic variab, let's assign numerical values to each of
         the variab. Let

               M1 = 1 kg
               M2 = 0.5 kg
               k = 1 N/m
               F= 1 N
               u = 0.002 sec/m
               g = 9.8 m/s^2



         Create an new m-file and enter the following commands.
               M1=1;
               M2=0.5;
               k=1;
               F=1;
               u=0.002;
               g=9.8;

         Now you have one of two choices: 1) Use the transfer function, or 2) Use the state-space
         form to solve the problem. If you choose to use the transfer function, add the following
         commands onto the end of the m-file which you have just created.
         num=[M2 M2*u*g 1];
         den=[M1*M2 2*M1*M2*u*g M1*k+M1*M2*u*u*g*g+M2*k M1*k*u*g+M2*k*u*g];
         train=tf(num,den)


         If you choose to use the state-space form, add the following commands at the end of the


3 of 4                                                                                          1/11/2011 5:39 PM
CTMS: MATLAB Modeling Tutorial                                 http://www.engin.umich.edu/class/ctms/model/model.htm


         m-file, instead of num and den matrices shown above.
         A=[    0    1     0     0;
          -k/M1 -u*g k/M1     0;
             0    0     0     1;
           k/M2   0   -k/M2 -u*g];
         B=[ 0;
         1/M1;
           0;
           0];
         C=[0 1 0 0];
         D=[0];
         train=ss(A,B,C,D)

         See the MATLAB basics tutorial to learn more about entering matrices into MATLAB.

         Continue solving the problem
         Once the differential equation representing the system model has been entered into MATLAB
         in either transfer-function or state-space form, the open loop and closed loop system
         behavior can be studied.

         Most operations can be done using either the transfer function or the state-space model.
         Furthermore, it is simple to transfer between the two if the other form of representation is
         required. If you need to learn how to convert from one representation to the other, see the
         Conversions page.

         This tutorial contains seven examples which allow you to learn more about modeling. You
         can link to them from below.


         Modeling Examples
               Cruise Control | Motor Speed | Motor Position | Bus Suspension | Inverted Pendulum |
               Pitch Controller | Ball and Beam

         Tutorials
               MATLAB Basics | MATLAB Modeling | PID Control | Root Locus | Frequency
               Response | State Space | Digital Control | Simulink Basics | Simulink Modeling |
               Examples




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